James D. McCaffrey

An Example When a Machine Learning Regression Model Has a Negative R2 Score

Posted on June 26, 2026 by jamesdmccaffrey

If you have a machine learning regression model that predicts a single numeric value, the three most common ways to evaluate the model are mean squared error (MSE), accuracy, and coefficient of determination (R2).

Note that root mean squared error (RMSE) is just the square root of MSE. This is useful when the target variable to predict has units, such as dollars. MSE has units “dollars-squared” but RMSE has units “dollars”.

MSE is reasonably interpretable, for example, if MSE = 0, the model predicts perfectly. But MSE has no upper limit, and MSE depends on how the data items are scaled.

Accuracy is very interpretable, for example, if accuracy = 75% and there are 200 data items, the model predicts 150 out of 200 correctly. But accuracy requires an arbitrary percentage closeness (perhaps 10%) that determines if a prediction is correct or not.

R2 is sort of a cross between MSE and accuracy. It’s often stated that “R2 is a value between 0 and 1 where higher values indicate a more accurate model.” So, an R2 score of 1 means a model predicts perfectly (which is never possible in practice). R2 is calculated as 1 – (SSres / SStot). The SSres is the sum of the squared differences between target y and predicted y values. The SStot is the sum of squared differences between average of target y and predicted y values. (Note R2 is not the same as classical statistics r2 for correlation).

The R2 score doesn’t isn’t affected by data scaling (good) but R2 can be negative (not good), and there’s no theoretical limit to how negative R2 can be.

As far as I know, all scikit-learn library regression modules (LinearRegression, Ridge, KernelRidge, GradientBoostingRegressor, etc.) have a built in score() function that gives the R2 value for the trained model. Implementing R2 from scratch is easy.

Suppose you have a set of training data. The simplest possible regression model is to just predict the average of the target y values in the training data, for any input x. But if you have a really bad regression model that predicts even worse than just returning the average of the target y values, the R2 score will be negative.

Here’s an example:

x0    x1    y        pred y   ss res   ss tot
0.2   0.3   0.34      0.19    0.0225   0.0841
0.6   0.5   0.86      0.61    0.0625   0.0529
0.3   0.7   0.79      0.44    0.1225   0.0256
0.6   0.2   0.56      0.46    0.0100   0.0049
0.1   0.6   0.61      0.31    0.0900   0.0004
0.3   0.4   0.49      0.29    0.0400   0.0196
0.5   0.5   0.75      0.50    0.0625   0.0144
0.2   0.6   0.64      0.34    0.0900   0.0001
                  
            5.04              0.5000   0.2020
            0.63            
                              R2 = -1.4752

The target y values are calculated by an unseen y = (x0 * x0) + x1. The predicted y values are computed by the terrible model y’ = (x0 * x0) – x1. R2 = 1 – (0.5000 / 0.2020) = 1 – 2.4752 = -1.4752.

The moral of this blog post is that there’s no single best way to evaluate a regression model.

Evaluating a regression model is mostly objective. Evaluating the name of a hotel is subjective. But here are a couple of names of foreign hotels that don’t resonate well in English.

Demo program.

# ridge_scikit_scratch_r2.py
# scratch R2 with scikit Ridge
# just for fun

import numpy as np
from sklearn.linear_model import Ridge

# Ridge(alpha=1.0, *, fit_intercept=True, copy_X=True,
# max_iter=None, tol=0.0001, solver='auto', positive=False,
# random_state=None)

# -----------------------------------------------------------

def r2(model, data_X, data_y):
  n = len(data_X)
  ss_res = 0.0
  ss_tot = 0.0
  mean_y = np.mean(data_y)
  for i in range(n):
    x = data_X[i]
    y = data_y[i]
    pred_y = model.predict([x])[0]
    ss_res += (y - pred_y) * (y -pred_y) # inefficient . .
    ss_tot += (y - mean_y) * (y -mean_y)

  r2 = 1.0 - (ss_res / ss_tot) # assume non-zero . .
  return r2

# -----------------------------------------------------------

print("\nBegin from-scratch R2 demo ")

np.set_printoptions(precision=4, suppress=True,
    floatmode='fixed')

print("\nLoading synthetic train (20) data ")
train_Xy = np.loadtxt(".\\Data\\synthetic_train_20.txt",
  usecols=[0,1,2,3,4,5], delimiter=",")
train_X = train_Xy[:,[0,1,2,3,4]]
train_y = train_Xy[:,5]

print("\nFirst three train X: ")
for i in range(3):
  print(train_X[i])
print("\nFirst three train y: ")
for i in range(3):
  print("%0.4f " % train_y[i])

print("\nCreating scikit Ridge model ")
alpha = 1.0
print("Using default L2 alpha = %0.4f " % alpha)
model = Ridge(alpha=alpha, solver='sag',
  fit_intercept=True, random_state=0)
print("Done ")

print("\nTraining scikit Ridge model ")
print("Using SAG with default training params ")
model.fit(train_X, train_y)
print("Done. Used " + str(model.n_iter_) + " iterations" )

print("\nModel weights: ")
print(model.coef_)
print("Model bias = %0.4f " % model.intercept_)

r2_scikit = model.score(train_X, train_y)
r2_scratch = r2(model, train_X, train_y)

print("\nR2 using built-in score() = %0.4f " % r2_scikit)
print("R2 using from-scratch = %0.4f " % r2_scratch)

print("\nEnd demo ")

Demo data:

# synthetic_train_200.txt
#
-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.0776, -0.1616,  0.3704, -0.5911,  0.7562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
 0.9365, -0.3732,  0.3846,  0.7528,  0.7892,  0.1345
-0.8299, -0.9219, -0.6603,  0.7563, -0.8033,  0.7955
 0.0663,  0.3838, -0.3690,  0.3730,  0.6693,  0.3206
-0.9634,  0.5003,  0.9777,  0.4963, -0.4391,  0.7377
-0.1042,  0.8172, -0.4128, -0.4244, -0.7399,  0.4801
-0.9613,  0.3577, -0.5767, -0.4689, -0.0169,  0.6861
-0.7065,  0.1786,  0.3995, -0.7953, -0.1719,  0.5569
 0.3888, -0.1716, -0.9001,  0.0718,  0.3276,  0.2500
 0.1731,  0.8068, -0.7251, -0.7214,  0.6148,  0.3297
-0.2046, -0.6693,  0.8550, -0.3045,  0.5016,  0.2129
 0.2473,  0.5019, -0.3022, -0.4601,  0.7918,  0.2613
-0.1438,  0.9297,  0.3269,  0.2434, -0.7705,  0.5171
 0.1568, -0.1837, -0.5259,  0.8068,  0.1474,  0.3307
-0.9943,  0.2343, -0.3467,  0.0541,  0.7719,  0.5581
 0.2467, -0.9684,  0.8589,  0.3818,  0.9946,  0.1092
-0.6553, -0.7257,  0.8652,  0.3936, -0.8680,  0.7018
 0.8460,  0.4230, -0.7515, -0.9602, -0.9476,  0.1996

Posted in Machine Learning | Leave a comment

Why You Should Never Use L1 Regularization or Lasso Regression

Posted on June 25, 2026 by jamesdmccaffrey

There is absolutely no reason to ever use L1 regularization. Well, maybe that’s a bit too extreme of a statement. But, “I never use L1 regularization or lasso regression unless I’m forced to by a legacy system” is accurate.

I am mildly baffled that the L1 regularization technique even exists. L2 regularization is superior to L1 in almost every scenario that I have ever encountered in several decades of machine learning experience.

Note that L1 regularization applied to linear regression is called lasso (the somewhat pretentious “Least Absolute Shrinkage and Selection Operator”) regression, and L2 regularization applied to linear regression is called ridge regression. The ugly elastic net regression uses L1 and L2 regularization together.

The only argument for using L1 regularization instead of L2 is, “L1 regularization can drive model weight(s) to exactly zero but L2 cannot.” This is true, but because L2 regularization can drive weight(s) to arbitrarily close to zero, you can use L2 and then, only if absolutely necessary, manually set the very small weight(s) to zero.

Setting a model weight to zero throws away information, and it’s very rare to encounter a scenario where this was a good idea. A predictor variable has “zero importance” only if there is exactly zero correlation to the target variable, which almost never happens with real-life data.

An example of L1 regularization for linear regression (lasso regression) using the scikit Lasso module. Not recommended.

The only L1 scenario I can even remotely imagine is one where you have thousands of predictor variables and you absolutely must get rid of some. I have never seen this scenario in practice, and even if it did occur, L2 regularization can do the job. Or you can preprocess the data using PCA or similar dimensionality reduction.

There are three reasons why L2 regularization is technically better than L1.

1.) L2 regularization works with any kind of training — iterative SGD training and closed form training (left pseudo-inverse via normal equations, and Moore-Penrose pseudo-inverse). L1 works only with SGD, and even then you must use a minor variation called sub-gradient stochastic gradient descent (SSGD). When closed form training works, it is usually much better than SGD because you don’t need to tune the learning rate and number of training iterations.

2.) L2 regularization is significantly better at dealing with multicollinearity in data. If two features are correlated, L1 will arbitrarily pick one feature and zero out the other, but L2 distributes weights evenly.

3.) Because L2 regularization doesn’t eliminate any predictor weights by setting them to zero, L2 regularized models often have better prediction accuracy than L1 models on new, non-training data. I have never seen L1 regularization out-perform L2 regularization (even though such scenarios must exist).

In short, L2 regularization is significantly better technically than L1, and L2 can identify unimportant predictors better than L1 can. In other words, there is really no reason to ever use L1 regularization, unless you are forced to by some sort of architecture design, or a legacy requirment.

By the way, a minor disadvantage of both L1 and L2 regularization is that they both require the training data to be scaled, typically to mean = 0 and unit variance. This is isn’t a huge deal but can be mildly annoying in practice.

The vast majority of machine learning techniques are good, but there are techniques like L1 regularization that are technical embarrassments when used naively by beginners.

I know very little about, and have very little interest in, politics in my country (US). But I do know that the vast majority of former First Ladies of the United States were examples of poise, grace, charm, and class. Melania Trump, Jacqueline Kennedy, and Hilary Clinton come to mind. But then there was an absolute disgrace and embarrassment to the entire cuntry, at least in terms of class and charm.

Demo program. Replace “lt” in accuracy() with less-than operator symbol. (My blog editor chokes om symbols).

# lasso_scikit.py
# L1 regularization. Not recommended!

import numpy as np
from sklearn.preprocessing import StandardScaler
from sklearn import linear_model

np.set_printoptions(precision=4, suppress=True,
  floatmode='fixed', linewidth=60)

# -----------------------------------------------------------

def accuracy(model, data_X, data_y, pct_close):
  n = len(data_X)
  n_correct = 0; n_wrong = 0
  for i in range(n):
    x = data_X[i].reshape(1,-1)
    y = data_y[i]
    y_pred = model.predict(x)[0]

    if np.abs(y - y_pred) "lt" np.abs(y * pct_close):
      n_correct += 1
    else: 
      n_wrong += 1
  return n_correct / (n_correct + n_wrong)

def mse(model, data_X, data_y):
  n = len(data_X)
  sum = 0.0
  for i in range(n):
    actual_y = data_y[i]
    pred_y = model.predict(data_X[i].reshape(1, -1))[0]
    diff = actual_y - pred_y
    sum += diff * diff
  return sum /n

# -----------------------------------------------------------

print("\nBegin lasso (L1) regression using scikit ")
print("Note: Not a recommended technique ")
print("L2 regularization is better in all scenarios ")

print("\nLoading train (200) and test (40) data ")
train_Xy = np.loadtxt(".\\Data\\synthetic_train_200.txt",
  usecols=[0,1,2,3,4,5], delimiter=",")
train_X = train_Xy[:,[0,1,2,3,4]]
train_y = train_Xy[:,5]

test_Xy = np.loadtxt(".\\Data\\synthetic_test_40.txt",
  usecols=[0,1,2,3,4,5], delimiter=",")
test_X = test_Xy[:,[0,1,2,3,4]]
test_y = test_Xy[:,5]

print("\nFirst three train X: ")
for i in range(3):
  print(train_X[i])
print("\nFirst three train y: ")
for i in range(3):
  print("%0.4f " % train_y[i])

# Lasso(alpha=1.0, *, fit_intercept=True, precompute=False,
# copy_X=True, max_iter=1000, tol=0.0001, warm_start=False,
# positive=False, random_state=None, selection='cyclic')

print("\nCreating and using data scaler on train X ")
scaler = StandardScaler()
scaler.fit(train_X)
scaled_train_X = scaler.transform(train_X)
print("\nFirst three scaled train X: ")
for i in range(3):
  print(scaled_train_X[i])
scaled_test_X = scaler.transform(test_X)

alpha = 0.01
print("\nCreating lasso model, alpha = %0.4f" % alpha)
model = linear_model.Lasso(alpha=alpha, random_state=0)
print("Done ")

print("\nTraining model ")
model.fit(scaled_train_X, train_y)
print("Done. ")

print("\nModel weights: ")
print(model.coef_)
print("Model bias = %0.4f " % model.intercept_)

print("\nEvaluating model ")
acc_train = accuracy(model, scaled_train_X, train_y, 0.10)
acc_test = accuracy(model, scaled_test_X, test_y, 0.10)
print("\nAccuracy (within 0.10) train = %0.4f " % \
  acc_train)
print("Accuracy (within 0.10) test = %0.4f " % \
  acc_test)

mse_train = mse(model, scaled_train_X, train_y)
mse_test = mse(model, scaled_test_X, test_y)
print("\nMSE train = %0.4f " % mse_train)
print("MSE test = %0.4f " % mse_test)

x = train_X[0]
print("\nPredicting for x = ")
print(x)
scaled_x = scaler.transform([x])
pred_y = model.predict(scaled_x)[0]
print("Predicted y = %0.4f " % pred_y)

print("\nEnd demo ")

Training data:

# synthetic_train_200.txt
#
-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.0776, -0.1616,  0.3704, -0.5911,  0.7562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
 0.9365, -0.3732,  0.3846,  0.7528,  0.7892,  0.1345
-0.8299, -0.9219, -0.6603,  0.7563, -0.8033,  0.7955
 0.0663,  0.3838, -0.3690,  0.3730,  0.6693,  0.3206
-0.9634,  0.5003,  0.9777,  0.4963, -0.4391,  0.7377
-0.1042,  0.8172, -0.4128, -0.4244, -0.7399,  0.4801
-0.9613,  0.3577, -0.5767, -0.4689, -0.0169,  0.6861
-0.7065,  0.1786,  0.3995, -0.7953, -0.1719,  0.5569
 0.3888, -0.1716, -0.9001,  0.0718,  0.3276,  0.2500
 0.1731,  0.8068, -0.7251, -0.7214,  0.6148,  0.3297
-0.2046, -0.6693,  0.8550, -0.3045,  0.5016,  0.2129
 0.2473,  0.5019, -0.3022, -0.4601,  0.7918,  0.2613
-0.1438,  0.9297,  0.3269,  0.2434, -0.7705,  0.5171
 0.1568, -0.1837, -0.5259,  0.8068,  0.1474,  0.3307
-0.9943,  0.2343, -0.3467,  0.0541,  0.7719,  0.5581
 0.2467, -0.9684,  0.8589,  0.3818,  0.9946,  0.1092
-0.6553, -0.7257,  0.8652,  0.3936, -0.8680,  0.7018
 0.8460,  0.4230, -0.7515, -0.9602, -0.9476,  0.1996
-0.9434, -0.5076,  0.7201,  0.0777,  0.1056,  0.5664
 0.9392,  0.1221, -0.9627,  0.6013, -0.5341,  0.1533
 0.6142, -0.2243,  0.7271,  0.4942,  0.1125,  0.1661
 0.4260,  0.1194, -0.9749, -0.8561,  0.9346,  0.2230
 0.1362, -0.5934, -0.4953,  0.4877, -0.6091,  0.3810
 0.6937, -0.5203, -0.0125,  0.2399,  0.6580,  0.1460
-0.6864, -0.9628, -0.8600, -0.0273,  0.2127,  0.5387
 0.9772,  0.1595, -0.2397,  0.1019,  0.4907,  0.1611
 0.3385, -0.4702, -0.8673, -0.2598,  0.2594,  0.2270
-0.8669, -0.4794,  0.6095, -0.6131,  0.2789,  0.4700
 0.0493,  0.8496, -0.4734, -0.8681,  0.4701,  0.3516
 0.8639, -0.9721, -0.5313,  0.2336,  0.8980,  0.1412
 0.9004,  0.1133,  0.8312,  0.2831, -0.2200,  0.1782
 0.0991,  0.8524,  0.8375, -0.2102,  0.9265,  0.2150
-0.6521, -0.7473, -0.7298,  0.0113, -0.9570,  0.7422
 0.6190, -0.3105,  0.8802,  0.1640,  0.7577,  0.1056
 0.6895,  0.8108, -0.0802,  0.0927,  0.5972,  0.2214
 0.1982, -0.9689,  0.1870, -0.1326,  0.6147,  0.1310
-0.3695,  0.7858,  0.1557, -0.6320,  0.5759,  0.3773
-0.1596,  0.3581,  0.8372, -0.9992,  0.9535,  0.2071
-0.2468,  0.9476,  0.2094,  0.6577,  0.1494,  0.4132
 0.1737,  0.5000,  0.7166,  0.5102,  0.3961,  0.2611
 0.7290, -0.3546,  0.3416, -0.0983, -0.2358,  0.1332
-0.3652,  0.2438, -0.1395,  0.9476,  0.3556,  0.4170
-0.6029, -0.1466, -0.3133,  0.5953,  0.7600,  0.4334
-0.4596, -0.4953,  0.7098,  0.0554,  0.6043,  0.2775
 0.1450,  0.4663,  0.0380,  0.5418,  0.1377,  0.2931
-0.8636, -0.2442, -0.8407,  0.9656, -0.6368,  0.7429
 0.6237,  0.7499,  0.3768,  0.1390, -0.6781,  0.2185
-0.5499,  0.1850, -0.3755,  0.8326,  0.8193,  0.4399
-0.4858, -0.7782, -0.6141, -0.0008,  0.4572,  0.4197
 0.7033, -0.1683,  0.2334, -0.5327, -0.7961,  0.1776
 0.0317, -0.0457, -0.6947,  0.2436,  0.0880,  0.3345
 0.5031, -0.5559,  0.0387,  0.5706, -0.9553,  0.3107
-0.3513,  0.7458,  0.6894,  0.0769,  0.7332,  0.3170
 0.2205,  0.5992, -0.9309,  0.5405,  0.4635,  0.3532
-0.4806, -0.4859,  0.2646, -0.3094,  0.5932,  0.3202
 0.9809, -0.3995, -0.7140,  0.8026,  0.0831,  0.1600
 0.9495,  0.2732,  0.9878,  0.0921,  0.0529,  0.1289
-0.9476, -0.6792,  0.4913, -0.9392, -0.2669,  0.5966
 0.7247,  0.3854,  0.3819, -0.6227, -0.1162,  0.1550
-0.5922, -0.5045, -0.4757,  0.5003, -0.0860,  0.5863
-0.8861,  0.0170, -0.5761,  0.5972, -0.4053,  0.7301
 0.6877, -0.2380,  0.4997,  0.0223,  0.0819,  0.1404
 0.9189,  0.6079, -0.9354,  0.4188, -0.0700,  0.1907
-0.1428, -0.7820,  0.2676,  0.6059,  0.3936,  0.2790
 0.5324, -0.3151,  0.6917, -0.1425,  0.6480,  0.1071
-0.8432, -0.9633, -0.8666, -0.0828, -0.7733,  0.7784
-0.9444,  0.5097, -0.2103,  0.4939, -0.0952,  0.6787
-0.0520,  0.6063, -0.1952,  0.8094, -0.9259,  0.4836
 0.5477, -0.7487,  0.2370, -0.9793,  0.0773,  0.1241
 0.2450,  0.8116,  0.9799,  0.4222,  0.4636,  0.2355
 0.8186, -0.1983, -0.5003, -0.6531, -0.7611,  0.1511
-0.4714,  0.6382, -0.3788,  0.9648, -0.4667,  0.5950
 0.0673, -0.3711,  0.8215, -0.2669, -0.1328,  0.2677
-0.9381,  0.4338,  0.7820, -0.9454,  0.0441,  0.5518
-0.3480,  0.7190,  0.1170,  0.3805, -0.0943,  0.4724
-0.9813,  0.1535, -0.3771,  0.0345,  0.8328,  0.5438
-0.1471, -0.5052, -0.2574,  0.8637,  0.8737,  0.3042
-0.5454, -0.3712, -0.6505,  0.2142, -0.1728,  0.5783
 0.6327, -0.6297,  0.4038, -0.5193,  0.1484,  0.1153
-0.5424,  0.3282, -0.0055,  0.0380, -0.6506,  0.6613
 0.1414,  0.9935,  0.6337,  0.1887,  0.9520,  0.2540
-0.9351, -0.8128, -0.8693, -0.0965, -0.2491,  0.7353
 0.9507, -0.6640,  0.9456,  0.5349,  0.6485,  0.1059
-0.0462, -0.9737, -0.2940, -0.0159,  0.4602,  0.2606
-0.0627, -0.0852, -0.7247, -0.9782,  0.5166,  0.2977
 0.0478,  0.5098, -0.0723, -0.7504, -0.3750,  0.3335
 0.0090,  0.3477,  0.5403, -0.7393, -0.9542,  0.4415
-0.9748,  0.3449,  0.3736, -0.1015,  0.8296,  0.4358
 0.2887, -0.9895, -0.0311,  0.7186,  0.6608,  0.2057
 0.1570, -0.4518,  0.1211,  0.3435, -0.2951,  0.3244
 0.7117, -0.6099,  0.4946, -0.4208,  0.5476,  0.1096
-0.2929, -0.5726,  0.5346, -0.3827,  0.4665,  0.2465
 0.4889, -0.5572, -0.5718, -0.6021, -0.7150,  0.2163
-0.7782,  0.3491,  0.5996, -0.8389, -0.5366,  0.6516
-0.5847,  0.8347,  0.4226,  0.1078, -0.3910,  0.6134
 0.8469,  0.4121, -0.0439, -0.7476,  0.9521,  0.1571
-0.6803, -0.5948, -0.1376, -0.1916, -0.7065,  0.7156
 0.2878,  0.5086, -0.5785,  0.2019,  0.4979,  0.2980
 0.2764,  0.1943, -0.4090,  0.4632,  0.8906,  0.2960
-0.8877,  0.6705, -0.6155, -0.2098, -0.3998,  0.7107
-0.8398,  0.8093, -0.2597,  0.0614, -0.0118,  0.6502
-0.8476,  0.0158, -0.4769, -0.2859, -0.7839,  0.7715
 0.5751, -0.7868,  0.9714, -0.6457,  0.1448,  0.1175
 0.4802, -0.7001,  0.1022, -0.5668,  0.5184,  0.1090
 0.4458, -0.6469,  0.7239, -0.9604,  0.7205,  0.0779
 0.5175,  0.4339,  0.9747, -0.4438, -0.9924,  0.2879
 0.8678,  0.7158,  0.4577,  0.0334,  0.4139,  0.1678
 0.5406,  0.5012,  0.2264, -0.1963,  0.3946,  0.2088
-0.9938,  0.5498,  0.7928, -0.5214, -0.7585,  0.7687
 0.7661,  0.0863, -0.4266, -0.7233, -0.4197,  0.1466
 0.2277, -0.3517, -0.0853, -0.1118,  0.6563,  0.1767
 0.3499, -0.5570, -0.0655, -0.3705,  0.2537,  0.1632
 0.7547, -0.1046,  0.5689, -0.0861,  0.3125,  0.1257
 0.8186,  0.2110,  0.5335,  0.0094, -0.0039,  0.1391
 0.6858, -0.8644,  0.1465,  0.8855,  0.0357,  0.1845
-0.4967,  0.4015,  0.0805,  0.8977,  0.2487,  0.4663
 0.6760, -0.9841,  0.9787, -0.8446, -0.3557,  0.1509
-0.1203, -0.4885,  0.6054, -0.0443, -0.7313,  0.4854
 0.8557,  0.7919, -0.0169,  0.7134, -0.1628,  0.2002
 0.0115, -0.6209,  0.9300, -0.4116, -0.7931,  0.4052
-0.7114, -0.9718,  0.4319,  0.1290,  0.5892,  0.3661
 0.3915,  0.5557, -0.1870,  0.2955, -0.6404,  0.2954
-0.3564, -0.6548, -0.1827, -0.5172, -0.1862,  0.4622
 0.2392, -0.4959,  0.5857, -0.1341, -0.2850,  0.2470
-0.3394,  0.3947, -0.4627,  0.6166, -0.4094,  0.5325
 0.7107,  0.7768, -0.6312,  0.1707,  0.7964,  0.2757
-0.1078,  0.8437, -0.4420,  0.2177,  0.3649,  0.4028
-0.3139,  0.5595, -0.6505, -0.3161, -0.7108,  0.5546
 0.4335,  0.3986,  0.3770, -0.4932,  0.3847,  0.1810
-0.2562, -0.2894, -0.8847,  0.2633,  0.4146,  0.4036
 0.2272,  0.2966, -0.6601, -0.7011,  0.0284,  0.2778
-0.0743, -0.1421, -0.0054, -0.6770, -0.3151,  0.3597
-0.4762,  0.6891,  0.6007, -0.1467,  0.2140,  0.4266
-0.4061,  0.7193,  0.3432,  0.2669, -0.7505,  0.6147
-0.0588,  0.9731,  0.8966,  0.2902, -0.6966,  0.4955
-0.0627, -0.1439,  0.1985,  0.6999,  0.5022,  0.3077
 0.1587,  0.8494, -0.8705,  0.9827, -0.8940,  0.4263
-0.7850,  0.2473, -0.9040, -0.4308, -0.8779,  0.7199
 0.4070,  0.3369, -0.2428, -0.6236,  0.4940,  0.2215
-0.0242,  0.0513, -0.9430,  0.2885, -0.2987,  0.3947
-0.5416, -0.1322, -0.2351, -0.0604,  0.9590,  0.3683
 0.1055,  0.7783, -0.2901, -0.5090,  0.8220,  0.2984
-0.9129,  0.9015,  0.1128, -0.2473,  0.9901,  0.4776
-0.9378,  0.1424, -0.6391,  0.2619,  0.9618,  0.5368
 0.7498, -0.0963,  0.4169,  0.5549, -0.0103,  0.1614
-0.2612, -0.7156,  0.4538, -0.0460, -0.1022,  0.3717
 0.7720,  0.0552, -0.1818, -0.4622, -0.8560,  0.1685
-0.4177,  0.0070,  0.9319, -0.7812,  0.3461,  0.3052
-0.0001,  0.5542, -0.7128, -0.8336, -0.2016,  0.3803
 0.5356, -0.4194, -0.5662, -0.9666, -0.2027,  0.1776
-0.2378,  0.3187, -0.8582, -0.6948, -0.9668,  0.5474
-0.1947, -0.3579,  0.1158,  0.9869,  0.6690,  0.2992
 0.3992,  0.8365, -0.9205, -0.8593, -0.0520,  0.3154
-0.0209,  0.0793,  0.7905, -0.1067,  0.7541,  0.1864
-0.4928, -0.4524, -0.3433,  0.0951, -0.5597,  0.6261
-0.8118,  0.7404, -0.5263, -0.2280,  0.1431,  0.6349
 0.0516, -0.8480,  0.7483,  0.9023,  0.6250,  0.1959
-0.3212,  0.1093,  0.9488, -0.3766,  0.3376,  0.2735
-0.3481,  0.5490, -0.3484,  0.7797,  0.5034,  0.4379
-0.5785, -0.9170, -0.3563, -0.9258,  0.3877,  0.4121
 0.3407, -0.1391,  0.5356,  0.0720, -0.9203,  0.3458
-0.3287, -0.8954,  0.2102,  0.0241,  0.2349,  0.3247
-0.1353,  0.6954, -0.0919, -0.9692,  0.7461,  0.3338
 0.9036, -0.8982, -0.5299, -0.8733, -0.1567,  0.1187
 0.7277, -0.8368, -0.0538, -0.7489,  0.5458,  0.0830
 0.9049,  0.8878,  0.2279,  0.9470, -0.3103,  0.2194
 0.7957, -0.1308, -0.5284,  0.8817,  0.3684,  0.2172
 0.4647, -0.4931,  0.2010,  0.6292, -0.8918,  0.3371
-0.7390,  0.6849,  0.2367,  0.0626, -0.5034,  0.7039
-0.1567, -0.8711,  0.7940, -0.5932,  0.6525,  0.1710
 0.7635, -0.0265,  0.1969,  0.0545,  0.2496,  0.1445
 0.7675,  0.1354, -0.7698, -0.5460,  0.1920,  0.1728
-0.5211, -0.7372, -0.6763,  0.6897,  0.2044,  0.5217
 0.1913,  0.1980,  0.2314, -0.8816,  0.5006,  0.1998
 0.8964,  0.0694, -0.6149,  0.5059, -0.9854,  0.1825
 0.1767,  0.7104,  0.2093,  0.6452,  0.7590,  0.2832
-0.3580, -0.7541,  0.4426, -0.1193, -0.7465,  0.5657
-0.5996,  0.5766, -0.9758, -0.3933, -0.9572,  0.6800
 0.9950,  0.1641, -0.4132,  0.8579,  0.0142,  0.2003
-0.4717, -0.3894, -0.2567, -0.5111,  0.1691,  0.4266
 0.3917, -0.8561,  0.9422,  0.5061,  0.6123,  0.1212
-0.0366, -0.1087,  0.3449, -0.1025,  0.4086,  0.2475
 0.3633,  0.3943,  0.2372, -0.6980,  0.5216,  0.1925
-0.5325, -0.6466, -0.2178, -0.3589,  0.6310,  0.3568
 0.2271,  0.5200, -0.1447, -0.8011, -0.7699,  0.3128
 0.6415,  0.1993,  0.3777, -0.0178, -0.8237,  0.2181
-0.5298, -0.0768, -0.6028, -0.9490,  0.4588,  0.4356
 0.6870, -0.1431,  0.7294,  0.3141,  0.1621,  0.1632
-0.5985,  0.0591,  0.7889, -0.3900,  0.7419,  0.2945
 0.3661,  0.7984, -0.8486,  0.7572, -0.6183,  0.3449
 0.6995,  0.3342, -0.3113, -0.6972,  0.2707,  0.1712
 0.2565,  0.9126,  0.1798, -0.6043, -0.1413,  0.2893
-0.3265,  0.9839, -0.2395,  0.9854,  0.0376,  0.4770
 0.2690, -0.1722,  0.9818,  0.8599, -0.7015,  0.3954
-0.2102, -0.0768,  0.1219,  0.5607, -0.0256,  0.3949
 0.8216, -0.9555,  0.6422, -0.6231,  0.3715,  0.0801
-0.2896,  0.9484, -0.7545, -0.6249,  0.7789,  0.4370
-0.9985, -0.5448, -0.7092, -0.5931,  0.7926,  0.5402

Test data:

# synthetic_test_40.txt
#
 0.7462,  0.4006, -0.0590,  0.6543, -0.0083,  0.1935
 0.8495, -0.2260, -0.0142, -0.4911,  0.7699,  0.1078
-0.2335, -0.4049,  0.4352, -0.6183, -0.7636,  0.5088
 0.1810, -0.5142,  0.2465,  0.2767, -0.3449,  0.3136
-0.8650,  0.7611, -0.0801,  0.5277, -0.4922,  0.7140
-0.2358, -0.7466, -0.5115, -0.8413, -0.3943,  0.4533
 0.4834,  0.2300,  0.3448, -0.9832,  0.3568,  0.1360
-0.6502, -0.6300,  0.6885,  0.9652,  0.8275,  0.3046
-0.3053,  0.5604,  0.0929,  0.6329, -0.0325,  0.4756
-0.7995,  0.0740, -0.2680,  0.2086,  0.9176,  0.4565
-0.2144, -0.2141,  0.5813,  0.2902, -0.2122,  0.4119
-0.7278, -0.0987, -0.3312, -0.5641,  0.8515,  0.4438
 0.3793,  0.1976,  0.4933,  0.0839,  0.4011,  0.1905
-0.8568,  0.9573, -0.5272,  0.3212, -0.8207,  0.7415
-0.5785,  0.0056, -0.7901, -0.2223,  0.0760,  0.5551
 0.0735, -0.2188,  0.3925,  0.3570,  0.3746,  0.2191
 0.1230, -0.2838,  0.2262,  0.8715,  0.1938,  0.2878
 0.4792, -0.9248,  0.5295,  0.0366, -0.9894,  0.3149
-0.4456,  0.0697,  0.5359, -0.8938,  0.0981,  0.3879
 0.8629, -0.8505, -0.4464,  0.8385,  0.5300,  0.1769
 0.1995,  0.6659,  0.7921,  0.9454,  0.9970,  0.2330
-0.0249, -0.3066, -0.2927, -0.4923,  0.8220,  0.2437
 0.4513, -0.9481, -0.0770, -0.4374, -0.9421,  0.2879
-0.3405,  0.5931, -0.3507, -0.3842,  0.8562,  0.3987
 0.9538,  0.0471,  0.9039,  0.7760,  0.0361,  0.1706
-0.0887,  0.2104,  0.9808,  0.5478, -0.3314,  0.4128
-0.8220, -0.6302,  0.0537, -0.1658,  0.6013,  0.4306
-0.4123, -0.2880,  0.9074, -0.0461, -0.4435,  0.5144
 0.0060,  0.2867, -0.7775,  0.5161,  0.7039,  0.3599
-0.7968, -0.5484,  0.9426, -0.4308,  0.8148,  0.2979
 0.7811,  0.8450, -0.6877,  0.7594,  0.2640,  0.2362
-0.6802, -0.1113, -0.8325, -0.6694, -0.6056,  0.6544
 0.3821,  0.1476,  0.7466, -0.5107,  0.2592,  0.1648
 0.7265,  0.9683, -0.9803, -0.4943, -0.5523,  0.2454
-0.9049, -0.9797, -0.0196, -0.9090, -0.4433,  0.6447
-0.4607,  0.1811, -0.2389,  0.4050, -0.0078,  0.5229
 0.2664, -0.2932, -0.4259, -0.7336,  0.8742,  0.1834
-0.4507,  0.1029, -0.6294, -0.1158, -0.6294,  0.6081
 0.8948, -0.0124,  0.9278,  0.2899, -0.0314,  0.1534
-0.1323, -0.8813, -0.0146, -0.0697,  0.6135,  0.2386

Posted in Machine Learning, Scikit | Leave a comment

Support Vector Regression Using From-Scratch-Python SVR with SGD Applied to the Diabetes Dataset

Posted on June 24, 2026 by jamesdmccaffrey

I write code almost every day. Like most skills, writing code is something that must be practiced. And besides that, I enjoy writing code. I had recently implemented a version of kernel support vector regression (SVR), from-scratch, using Python. So, one evening after work, I figured I’d run the scikit Diabetes Dataset through my from-scratch support vector regression model.

Based on previous experiments with linear regression, quadratic regression, neural network regression, kernel ridge regression, random forest regression, and AdaBoost regression, I was certain that the from-scratch support vector regression model would give poor prediction accuracy. My goal was not to create a good regression model — I wanted to validate my from-scratch SVR implementation by comparing results with the scikit library SVR module.

The raw Diabetes Dataset looks like:

59, 2, 32.1, 101.00, 157,  93.2, 38, 4.00, 4.8598, 87, 151
48, 1, 21.6,  87.00, 183, 103.2, 70, 3.00, 3.8918, 69,  75
72, 2, 30.5,  93.00, 156,  93.6, 41, 4.00, 4.6728, 85, 141
. . .

The dataset has 442 items. Each item represents a patient and has 10 predictor values followed by a target value to predict. The 10 predictor variables are age in column [0], sex [1], body mass index [2], blood pressure [3], serum cholesterol [4], low-density lipoproteins [5], high-density lipoproteins [6], total cholesterol [7], triglycerides [8], blood sugar [9]. The stated (in the documentation) target value to predict in the last column is a measure of diabetes [10].

The sex encoding isn’t explained anywhere but I suspect male = 1, female = 2 because there are 235 1 values and 206 2 values).

Note that this Diabetes Dataset, which is included as an example dataset in the Python language scikit-learn library, is not the same as the Pima Diabetes Dataset from the UCI dataset repository. See https://jamesmccaffreyblog.com/2026/02/03/the-origin-and-history-of-scikit-learn-diabetes-dataset/.

When using SVR, it’s a good idea to normalize predictor values so that a predictor variable with very large magnitude values doesn’t overwhelm the other predictor variables.

I converted the sex values from 1,2 into 0,1. Then I applied divide-by-constant normalization by dividing the 10 predictor columns by (100, 1, 100, 1000, 1000, 1000, 100, 10, 10, 1000) and the target y values by 1000. The resulting encoded and normalized data looks like:

0.5900, 1.0000, 0.3210, . . . 0.1510
0.4800, 0.0000, 0.2160, . . . 0.0750
0.7200, 1.0000, 0.3050, . . . 0.1410
. . .

I split the 442-items into a 342-item training set and a 100-item test set.

There are two versions of support vector regression. The kernel version is much more powerful than the linear version — so much so that linear SVR is essentially useless. I use kernel SVR.

First I ran the scikit SVR model. The key output is:

Begin scikit SVR on Diabetes Dataset demo

Loading diabetes train (342), test (100) data
Done

Creating scikit SVR model
Setting gamma = 0.3000
Setting C = 1.0000
Setting epsilon = 0.0100
Done

Training SVR model
Done.

Number support vectors =
299

Evaluating model

Accuracy (within 0.10) train = 0.1959
Accuracy (within 0.10) test = 0.1848

MSE train = 0.0029
MSE test = 0.0031

End demo

The number of support vectors generated by scikit SVR (299 items out of 342 training items) is determined implicitly by complex interactions between the gamma, C, and epsilon parameters. So I adjusted my epsilon value to 0.05 so that my from-scratch version ended up with roughly the same number of support vectors (292) as the scikit model (299).

The key output of my from-scratch SVR demo:

Begin scratch SVR using SGD on Diabetes Dataset

Loading diabetes train (342), test (100) data
Done

First three X predictors:
[0.5900 1.0000 0.3210 0.1010 0.1570 0.0932 0.3800 0.4000
 0.4860 0.0870]
[0.4800 0.0000 0.2160 0.0870 0.1830 0.1032 0.7000 0.3000
 0.3892 0.0690]
[0.7200 1.0000 0.3050 0.0930 0.1560 0.0936 0.4100 0.4000
 0.4673 0.0850]

First three y targets:
0.1510
0.0750
0.1410

Creating scratch Python SVR model
Setting gamma = 0.3000
Setting epsilon = 0.0500
Setting C = 1.0000
Setting lrn_rate = 0.0100
Setting max_epochs = 10000
Setting KKT tol = 0.000010

Training SVR model using SGD
epoch =    0  |  MSE = 0.0052
epoch = 2000  |  MSE = 0.0032
epoch = 4000  |  MSE = 0.0030
epoch = 6000  |  MSE = 0.0030
epoch = 8000  |  MSE = 0.0033
Done

Number support vectors = 292

Train accuracy (0.10) = 0.1696
Test accuracy (0.10) = 0.1522

MSE train = 0.0030
MSE test = 0.0031

End demo

My from-scratch SVR implementation gave similar results to the scikit SVR, and so, my experiment gave me some evidence that my from-scratch SVR implementation is probably correct

If you use the built-in load_diabetes() function with a return_X_y=True parameter, column [10] is automatically the target to predict. But I discovered that column [10] cannot be predicted with any meaningful accuracy.

But columns [4], [5], [6], [7], and [8] can be predicted meaningfully. You can load the data as a DataFrame and then specify a different column as the target. Alternatively, you can use preprocessed training data.

Good fun for me.

It’s common for me and my colleagues to refactor software systems many times. It’s rarely possible to a get a non-trivial system completely correct on the first effort. Therefore, a system is usually a collection of software sequels, so to speak, where each sequel is (hopefully) a bit better than its predecessor.

I’m a big fan of old science fiction movies. There have been dozens of sci-fi sequels. “Star Wars” (1977) and “Star Wars: The Empire Strikes Back” (1980). “Alien” (1979) and “Aliens” (1986). And so on. But, in my opinion, most sci-fi movie sequels are worse than their predecessor. However there are exceptions where the sequel is as-good-as, or even better than the original.

Left: In “The Magnetic Monster” (1953), agents from the fictitious Office of Scientific Investigation (OSI) investigate strange magnetic phenomena. A rouge scientist has created a new element that feeds on energy. It doubles its mass every 11 hours. If the menace can’t be stopped, it will eventually grow so large that it will throw Earth out of orbit. My grade = B-

Right: In “Riders to the Stars” (1954), the fictitious Office of Scientific Investigation (OSI) recruits men for a top secret project. The goal: Take a manned spacecraft to capture a meteor to learn why meteors do not disintegrate like steel does, when under constant bombardment from cosmic radiation. My grade = B.

Demo program. Replace “lt” (less than) in the accuracy() function with the Boolean less-than operator symbol. (My blog editor chokes on symbols).

# svr_sgd_scratch_diabetes.py
# SGD training

import numpy as np
from sklearn.svm import SVR  # for validation

# -----------------------------------------------------------

np.set_printoptions(precision=4, suppress=True,
  floatmode='fixed', linewidth=60)

# -----------------------------------------------------------

def accuracy(model, data_X, data_y, pct_close):
  if data_X.size == 0: return 0.0
  n = len(data_X)
  n_correct = 0; n_wrong = 0
  for i in range(n):
    x = data_X[i].reshape(1,-1)
    y = data_y[i]
    pred_y = model.predict(x)[0]
    if np.abs(y - pred_y) "lt np.abs(y * pct_close):
      n_correct += 1
    else: 
      n_wrong += 1
  return n_correct / (n_correct + n_wrong)

# -----------------------------------------------------------

def mse(model, data_X, data_y):
  if data_X.size == 0: return -1.0
  n = len(data_X)
  sum = 0.0
  for i in range(n):
    x = data_X[i].reshape(1,-1)
    y = data_y[i]
    pred_y = model.predict(x)[0]
    diff = pred_y - y
    sum += diff * diff
  return sum /n

# ===========================================================

class KernelSVR:
  def __init__(self, gamma=0.5, epsilon=0.001, C=1.0,
    lr=0.01, max_epochs=100, tol=1.0e-3, seed=0):
    self.gamma = gamma
    self.epsilon = epsilon
    self.C = C  
    self.lr = lr
    self.max_epochs = max_epochs
    self.tol = tol      # for in-epsilon tube
    self.alpha = None   # model weights
    self.b = 0.0
    self.supp_X = None  # pruned train X
    self.supp_y = None
    self.rnd = np.random.RandomState(seed)

  # ---------------------------------------------------------

  def kernel_matrix(self, X):
    n = len(X)
    result = np.zeros((n,n))
    for i in range(0,n):
      for j in range(i,n):
        z = self.rbf(X[i], X[j])
        result[i,j] = z
        result[j,i] = z
    return result

  # ---------------------------------------------------------

  def fit(self, X, y):
    n, dim = X.shape
    self.supp_X = X  # by ref
    self.supp_y = y
    
    self.alpha = np.zeros(n)
    lo = -0.01; hi = 0.01
    for i in range(n):
      self.alpha[i] = (hi - lo) * self.rnd.random() + lo
    self.b = 0.0
    
    # precompute kernel matrix and set regularization
    K = self.kernel_matrix(X)
    lamda = 1.0 / self.C         # do not allow 0
    freq = self.max_epochs // 5  # progress messages
    indices = np.arange(n)

    for epoch in range(self.max_epochs):
      self.rnd.shuffle(indices)
      
      for i in range(len(indices)):
        idx = indices[i]
        pred_y = np.dot(self.alpha, K[:, idx]) + self.b
        error = pred_y - y[idx]
        
        inside_tube = False
        if error "gt" self.epsilon:
          grad_loss = 1.0
        elif error "lt" -self.epsilon:
          grad_loss = -1.0
        else:
          grad_loss = 0.0
          inside_tube = True
          
        # local kernel regularization gradient
        grad_reg = self.alpha[idx] * K[idx, idx]
        
        # decoupled update to the active index
        self.alpha[idx] -= self.lr * \
          (lamda * grad_reg + grad_loss)
        self.b -= self.lr * grad_loss

        if inside_tube == True and \
          abs(self.alpha[idx]) "lt" self.tol:
          self.alpha[idx] = 0.0  # force small wt to zero
        
        # in-loop clip to bound updates mid-flight
        if self.alpha[idx] "lt" -self.C:
          self.alpha[idx] = -self.C
        elif self.alpha[idx] "gt" self.C:
          self.alpha[idx] = self.C

      if epoch % freq == 0:
        m = mse(self, X, y)
        print("epoch = %4d  |  MSE = %0.4f " % (epoch,m))
        pass

    # final global clip to all alphas
    self.alpha = np.clip(self.alpha, -self.C, self.C)

    # prune: store only explicit support vectors
    sv_mask = (np.abs(self.alpha) > 1.0e-5)
    self.supp_X = X[sv_mask]
    self.supp_y = y[sv_mask]
    self.alpha = self.alpha[sv_mask]

    return  # all done

  # ---------------------------------------------------------

  def rbf(self, v1, v2):
    sum = 0.0
    for i in range(len(v1)):
      sum += (v1[i] - v2[i]) * (v1[i] - v2[i])
    return np.exp(-1 * self.gamma * sum)

  def predict_one(self, x):
    # x is a vector
    n = len(self.supp_X)
    sum = 0.0
    for i in range(n):
      xx = self.supp_X[i]
      k = self.rbf(x, xx)
      sum += self.alpha[i] * k
    result = sum + self.b
    return result

  def predict(self, X):
    # X is a matrix
    n = len(X)
    result = np.zeros(n)
    for i in range(n):
      result[i] = self.predict_one(X[i])
    return result

  # ---------------------------------------------------------

  def r2_score(self, data_X, data_y):
    # coefficient of determination == scikit score()
    ss_res = 0.0; ss_tot = 0.0
    n = len(data_X)
    mean_y = np.mean(data_y)
    for i in range(n):
      x = data_X[i].reshape(1,-1)
      y = data_y[i]
      pred_y = self.predict(x)[0]
      ss_res += (y - pred_y) * (y - pred_y)
      ss_tot += (y - mean_y) * (y - mean_y)
    result = 1.0 - (ss_res / ss_tot)
    return result

# ===========================================================

def main():
  print("\nBegin scratch SVR using SGD on Diabetes Dataset ")

  print("\nLoading diabetes train (342), test (100) data ")
  train_file = ".\\Data\\diabetes_norm_train_342.txt"

  cols_X = [0,1,2,3,4,5,6,7,8,9]  
  col_y = 10  # cols # 4 5 6 7 8 are much better
  train_X = np.loadtxt(train_file, comments="#",
    usecols=cols_X, delimiter=",",  dtype=np.float64)
  train_y = np.loadtxt(train_file, comments="#", usecols=col_y,
    delimiter=",", dtype=np.float64)

  test_file = ".\\Data\\diabetes_norm_test_100.txt"
  test_X = np.loadtxt(test_file, comments="#",
    usecols=cols_X, delimiter=",",  dtype=np.float64)
  test_y = np.loadtxt(test_file, comments="#", usecols=col_y,
    delimiter=",",  dtype=np.float64)
  print("Done ")

  # alternative normalization and split
  # from sklearn.datasets import load_diabetes
  # from sklearn.model_selection import train_test_split
  # X, y = load_diabetes(return_X_y=True, scaled=True)
  # train_X, test_X, train_y, test_y = \
  #   train_test_split(X, y, random_state=0)  # 25% test

  print("\nFirst three X predictors: ")
  for i in range(3):
    print(train_X[i])
  print("\nFirst three y targets: ")
  for i in range(3):
    print("%0.4f" % train_y[i])

  # create and train model
  print("\nCreating scratch Python SVR model ")
  gamma = 0.30
  # epsilon = 0.001
  epsilon = 0.05 # 299 -- 
  C = 1.0
  lr = 0.01
  max_epochs = 10000
  tol = 0.00001

  print("Setting gamma = %0.4f " % gamma)
  print("Setting epsilon = %0.4f " % epsilon)
  print("Setting C = %0.4f " % C)
  print("Setting lrn_rate = %0.4f " % lr)
  print("Setting max_epochs = " + str(max_epochs))
  print("Setting KKT tol = %0.6f " % tol)

  print("\nTraining SVR model using SGD ")
  model = KernelSVR(gamma=gamma, epsilon=epsilon, C=C, lr=lr,
    max_epochs=max_epochs, tol=tol)
  model.fit(train_X, train_y)
  print("Done ")

  n_supp = len(model.alpha)
  print("\nNumber support vectors = " + str(n_supp))
    
  acc_train = accuracy(model, train_X, train_y, 0.10)
  print("\nTrain accuracy (0.10) = %0.4f" % acc_train)
  acc_test = accuracy(model, test_X, test_y, 0.10)
  print("Test accuracy (0.10) = %0.4f" % acc_test)

  mse_train = mse(model, train_X, train_y)
  mse_test = mse(model, test_X, test_y)
  print("\nMSE train = %0.4f " % mse_train)
  print("MSE test = %0.4f " % mse_test)

  print("\nEnd demo ")

# -----------------------------------------------------------

if __name__ == "__main__":
  main()

Training data:


# diabetes_norm_train_342.txt
# cols [0] to [9] predictors. col [10] target
# norm division constants:
# 100, -1, 100, 1000, 1000, 1000, 100, 10, 10, 1000, 1000
#
0.5900, 1.0000, 0.3210, 0.1010, 0.1570, 0.0932, 0.3800, 0.4000, 0.4860, 0.0870, 0.1510
0.4800, 0.0000, 0.2160, 0.0870, 0.1830, 0.1032, 0.7000, 0.3000, 0.3892, 0.0690, 0.0750
0.7200, 1.0000, 0.3050, 0.0930, 0.1560, 0.0936, 0.4100, 0.4000, 0.4673, 0.0850, 0.1410
0.2400, 0.0000, 0.2530, 0.0840, 0.1980, 0.1314, 0.4000, 0.5000, 0.4890, 0.0890, 0.2060
0.5000, 0.0000, 0.2300, 0.1010, 0.1920, 0.1254, 0.5200, 0.4000, 0.4291, 0.0800, 0.1350
0.2300, 0.0000, 0.2260, 0.0890, 0.1390, 0.0648, 0.6100, 0.2000, 0.4190, 0.0680, 0.0970
0.3600, 1.0000, 0.2200, 0.0900, 0.1600, 0.0996, 0.5000, 0.3000, 0.3951, 0.0820, 0.1380
0.6600, 1.0000, 0.2620, 0.1140, 0.2550, 0.1850, 0.5600, 0.4550, 0.4249, 0.0920, 0.0630
0.6000, 1.0000, 0.3210, 0.0830, 0.1790, 0.1194, 0.4200, 0.4000, 0.4477, 0.0940, 0.1100
0.2900, 0.0000, 0.3000, 0.0850, 0.1800, 0.0934, 0.4300, 0.4000, 0.5385, 0.0880, 0.3100
0.2200, 0.0000, 0.1860, 0.0970, 0.1140, 0.0576, 0.4600, 0.2000, 0.3951, 0.0830, 0.1010
0.5600, 1.0000, 0.2800, 0.0850, 0.1840, 0.1448, 0.3200, 0.6000, 0.3584, 0.0770, 0.0690
0.5300, 0.0000, 0.2370, 0.0920, 0.1860, 0.1092, 0.6200, 0.3000, 0.4304, 0.0810, 0.1790
0.5000, 1.0000, 0.2620, 0.0970, 0.1860, 0.1054, 0.4900, 0.4000, 0.5063, 0.0880, 0.1850
0.6100, 0.0000, 0.2400, 0.0910, 0.2020, 0.1154, 0.7200, 0.3000, 0.4291, 0.0730, 0.1180
0.3400, 1.0000, 0.2470, 0.1180, 0.2540, 0.1842, 0.3900, 0.7000, 0.5037, 0.0810, 0.1710
0.4700, 0.0000, 0.3030, 0.1090, 0.2070, 0.1002, 0.7000, 0.3000, 0.5215, 0.0980, 0.1660
0.6800, 1.0000, 0.2750, 0.1110, 0.2140, 0.1470, 0.3900, 0.5000, 0.4942, 0.0910, 0.1440
0.3800, 0.0000, 0.2540, 0.0840, 0.1620, 0.1030, 0.4200, 0.4000, 0.4443, 0.0870, 0.0970
0.4100, 0.0000, 0.2470, 0.0830, 0.1870, 0.1082, 0.6000, 0.3000, 0.4543, 0.0780, 0.1680
0.3500, 0.0000, 0.2110, 0.0820, 0.1560, 0.0878, 0.5000, 0.3000, 0.4511, 0.0950, 0.0680
0.2500, 1.0000, 0.2430, 0.0950, 0.1620, 0.0986, 0.5400, 0.3000, 0.3850, 0.0870, 0.0490
0.2500, 0.0000, 0.2600, 0.0920, 0.1870, 0.1204, 0.5600, 0.3000, 0.3970, 0.0880, 0.0680
0.6100, 1.0000, 0.3200, 0.1037, 0.2100, 0.0852, 0.3500, 0.6000, 0.6107, 0.1240, 0.2450
0.3100, 0.0000, 0.2970, 0.0880, 0.1670, 0.1034, 0.4800, 0.4000, 0.4357, 0.0780, 0.1840
0.3000, 1.0000, 0.2520, 0.0830, 0.1780, 0.1184, 0.3400, 0.5000, 0.4852, 0.0830, 0.2020
0.1900, 0.0000, 0.1920, 0.0870, 0.1240, 0.0540, 0.5700, 0.2000, 0.4174, 0.0900, 0.1370
0.4200, 0.0000, 0.3190, 0.0830, 0.1580, 0.0876, 0.5300, 0.3000, 0.4466, 0.1010, 0.0850
0.6300, 0.0000, 0.2440, 0.0730, 0.1600, 0.0914, 0.4800, 0.3000, 0.4635, 0.0780, 0.1310
0.6700, 1.0000, 0.2580, 0.1130, 0.1580, 0.0542, 0.6400, 0.2000, 0.5293, 0.1040, 0.2830
0.3200, 0.0000, 0.3050, 0.0890, 0.1820, 0.1106, 0.5600, 0.3000, 0.4344, 0.0890, 0.1290
0.4200, 0.0000, 0.2030, 0.0710, 0.1610, 0.0812, 0.6600, 0.2000, 0.4234, 0.0810, 0.0590
0.5800, 1.0000, 0.3800, 0.1030, 0.1500, 0.1072, 0.2200, 0.7000, 0.4644, 0.0980, 0.3410
0.5700, 0.0000, 0.2170, 0.0940, 0.1570, 0.0580, 0.8200, 0.2000, 0.4443, 0.0920, 0.0870
0.5300, 0.0000, 0.2050, 0.0780, 0.1470, 0.0842, 0.5200, 0.3000, 0.3989, 0.0750, 0.0650
0.6200, 1.0000, 0.2350, 0.0803, 0.2250, 0.1128, 0.8600, 0.2620, 0.4875, 0.0960, 0.1020
0.5200, 0.0000, 0.2850, 0.1100, 0.1950, 0.0972, 0.6000, 0.3000, 0.5242, 0.0850, 0.2650
0.4600, 0.0000, 0.2740, 0.0780, 0.1710, 0.0880, 0.5800, 0.3000, 0.4828, 0.0900, 0.2760
0.4800, 1.0000, 0.3300, 0.1230, 0.2530, 0.1636, 0.4400, 0.6000, 0.5425, 0.0970, 0.2520
0.4800, 1.0000, 0.2770, 0.0730, 0.1910, 0.1194, 0.4600, 0.4000, 0.4852, 0.0920, 0.0900
0.5000, 1.0000, 0.2560, 0.1010, 0.2290, 0.1622, 0.4300, 0.5000, 0.4779, 0.1140, 0.1000
0.2100, 0.0000, 0.2010, 0.0630, 0.1350, 0.0690, 0.5400, 0.3000, 0.4094, 0.0890, 0.0550
0.3200, 1.0000, 0.2540, 0.0903, 0.1530, 0.1004, 0.3400, 0.4500, 0.4533, 0.0830, 0.0610
0.5400, 0.0000, 0.2420, 0.0740, 0.2040, 0.1090, 0.8200, 0.2000, 0.4174, 0.1090, 0.0920
0.6100, 1.0000, 0.3270, 0.0970, 0.1770, 0.1184, 0.2900, 0.6000, 0.4997, 0.0870, 0.2590
0.5600, 1.0000, 0.2310, 0.1040, 0.1810, 0.1164, 0.4700, 0.4000, 0.4477, 0.0790, 0.0530
0.3300, 0.0000, 0.2530, 0.0850, 0.1550, 0.0850, 0.5100, 0.3000, 0.4554, 0.0700, 0.1900
0.2700, 0.0000, 0.1960, 0.0780, 0.1280, 0.0680, 0.4300, 0.3000, 0.4443, 0.0710, 0.1420
0.6700, 1.0000, 0.2250, 0.0980, 0.1910, 0.1192, 0.6100, 0.3000, 0.3989, 0.0860, 0.0750
0.3700, 1.0000, 0.2770, 0.0930, 0.1800, 0.1194, 0.3000, 0.6000, 0.5030, 0.0880, 0.1420
0.5800, 0.0000, 0.2570, 0.0990, 0.1570, 0.0916, 0.4900, 0.3000, 0.4407, 0.0930, 0.1550
0.6500, 1.0000, 0.2790, 0.1030, 0.1590, 0.0968, 0.4200, 0.4000, 0.4615, 0.0860, 0.2250
0.3400, 0.0000, 0.2550, 0.0930, 0.2180, 0.1440, 0.5700, 0.4000, 0.4443, 0.0880, 0.0590
0.4600, 0.0000, 0.2490, 0.1150, 0.1980, 0.1296, 0.5400, 0.4000, 0.4277, 0.1030, 0.1040
0.3500, 0.0000, 0.2870, 0.0970, 0.2040, 0.1268, 0.6400, 0.3000, 0.4190, 0.0930, 0.1820
0.3700, 0.0000, 0.2180, 0.0840, 0.1840, 0.1010, 0.7300, 0.3000, 0.3912, 0.0930, 0.1280
0.3700, 0.0000, 0.3020, 0.0870, 0.1660, 0.0960, 0.4000, 0.4150, 0.5011, 0.0870, 0.0520
0.4100, 0.0000, 0.2050, 0.0800, 0.1240, 0.0488, 0.6400, 0.2000, 0.4025, 0.0750, 0.0370
0.6000, 0.0000, 0.2040, 0.1050, 0.1980, 0.0784, 0.9900, 0.2000, 0.4635, 0.0790, 0.1700
0.6600, 1.0000, 0.2400, 0.0980, 0.2360, 0.1464, 0.5800, 0.4000, 0.5063, 0.0960, 0.1700
0.2900, 0.0000, 0.2600, 0.0830, 0.1410, 0.0652, 0.6400, 0.2000, 0.4078, 0.0830, 0.0610
0.3700, 1.0000, 0.2680, 0.0790, 0.1570, 0.0980, 0.2800, 0.6000, 0.5043, 0.0960, 0.1440
0.4100, 1.0000, 0.2570, 0.0830, 0.1810, 0.1066, 0.6600, 0.3000, 0.3738, 0.0850, 0.0520
0.3900, 0.0000, 0.2290, 0.0770, 0.2040, 0.1432, 0.4600, 0.4000, 0.4304, 0.0740, 0.1280
0.6700, 1.0000, 0.2400, 0.0830, 0.1430, 0.0772, 0.4900, 0.3000, 0.4431, 0.0940, 0.0710
0.3600, 1.0000, 0.2410, 0.1120, 0.1930, 0.1250, 0.3500, 0.6000, 0.5106, 0.0950, 0.1630
0.4600, 1.0000, 0.2470, 0.0850, 0.1740, 0.1232, 0.3000, 0.6000, 0.4644, 0.0960, 0.1500
0.6000, 1.0000, 0.2500, 0.0897, 0.1850, 0.1208, 0.4600, 0.4020, 0.4511, 0.0920, 0.0970
0.5900, 1.0000, 0.2360, 0.0830, 0.1650, 0.1000, 0.4700, 0.4000, 0.4500, 0.0920, 0.1600
0.5300, 0.0000, 0.2210, 0.0930, 0.1340, 0.0762, 0.4600, 0.3000, 0.4078, 0.0960, 0.1780
0.4800, 0.0000, 0.1990, 0.0910, 0.1890, 0.1096, 0.6900, 0.3000, 0.3951, 0.1010, 0.0480
0.4800, 0.0000, 0.2950, 0.1310, 0.2070, 0.1322, 0.4700, 0.4000, 0.4935, 0.1060, 0.2700
0.6600, 1.0000, 0.2600, 0.0910, 0.2640, 0.1466, 0.6500, 0.4000, 0.5568, 0.0870, 0.2020
0.5200, 1.0000, 0.2450, 0.0940, 0.2170, 0.1494, 0.4800, 0.5000, 0.4585, 0.0890, 0.1110
0.5200, 1.0000, 0.2660, 0.1110, 0.2090, 0.1264, 0.6100, 0.3000, 0.4682, 0.1090, 0.0850
0.4600, 1.0000, 0.2350, 0.0870, 0.1810, 0.1148, 0.4400, 0.4000, 0.4710, 0.0980, 0.0420
0.4000, 1.0000, 0.2900, 0.1150, 0.0970, 0.0472, 0.3500, 0.2770, 0.4304, 0.0950, 0.1700
0.2200, 0.0000, 0.2300, 0.0730, 0.1610, 0.0978, 0.5400, 0.3000, 0.3829, 0.0910, 0.2000
0.5000, 0.0000, 0.2100, 0.0880, 0.1400, 0.0718, 0.3500, 0.4000, 0.5112, 0.0710, 0.2520
0.2000, 0.0000, 0.2290, 0.0870, 0.1910, 0.1282, 0.5300, 0.4000, 0.3892, 0.0850, 0.1130
0.6800, 0.0000, 0.2750, 0.1070, 0.2410, 0.1496, 0.6400, 0.4000, 0.4920, 0.0900, 0.1430
0.5200, 1.0000, 0.2430, 0.0860, 0.1970, 0.1336, 0.4400, 0.5000, 0.4575, 0.0910, 0.0510
0.4400, 0.0000, 0.2310, 0.0870, 0.2130, 0.1264, 0.7700, 0.3000, 0.3871, 0.0720, 0.0520
0.3800, 0.0000, 0.2730, 0.0810, 0.1460, 0.0816, 0.4700, 0.3000, 0.4466, 0.0810, 0.2100
0.4900, 0.0000, 0.2270, 0.0653, 0.1680, 0.0962, 0.6200, 0.2710, 0.3892, 0.0600, 0.0650
0.6100, 0.0000, 0.3300, 0.0950, 0.1820, 0.1148, 0.5400, 0.3000, 0.4190, 0.0740, 0.1410
0.2900, 1.0000, 0.1940, 0.0830, 0.1520, 0.1058, 0.3900, 0.4000, 0.3584, 0.0830, 0.0550
0.6100, 0.0000, 0.2580, 0.0980, 0.2350, 0.1258, 0.7600, 0.3000, 0.5112, 0.0820, 0.1340
0.3400, 1.0000, 0.2260, 0.0750, 0.1660, 0.0918, 0.6000, 0.3000, 0.4263, 0.1080, 0.0420
0.3600, 0.0000, 0.2190, 0.0890, 0.1890, 0.1052, 0.6800, 0.3000, 0.4369, 0.0960, 0.1110
0.5200, 0.0000, 0.2400, 0.0830, 0.1670, 0.0866, 0.7100, 0.2000, 0.3850, 0.0940, 0.0980
0.6100, 0.0000, 0.3120, 0.0790, 0.2350, 0.1568, 0.4700, 0.5000, 0.5050, 0.0960, 0.1640
0.4300, 0.0000, 0.2680, 0.1230, 0.1930, 0.1022, 0.6700, 0.3000, 0.4779, 0.0940, 0.0480
0.3500, 0.0000, 0.2040, 0.0650, 0.1870, 0.1056, 0.6700, 0.2790, 0.4277, 0.0780, 0.0960
0.2700, 0.0000, 0.2480, 0.0910, 0.1890, 0.1068, 0.6900, 0.3000, 0.4190, 0.0690, 0.0900
0.2900, 0.0000, 0.2100, 0.0710, 0.1560, 0.0970, 0.3800, 0.4000, 0.4654, 0.0900, 0.1620
0.6400, 1.0000, 0.2730, 0.1090, 0.1860, 0.1076, 0.3800, 0.5000, 0.5308, 0.0990, 0.1500
0.4100, 0.0000, 0.3460, 0.0873, 0.2050, 0.1426, 0.4100, 0.5000, 0.4673, 0.1100, 0.2790
0.4900, 1.0000, 0.2590, 0.0910, 0.1780, 0.1066, 0.5200, 0.3000, 0.4575, 0.0750, 0.0920
0.4800, 0.0000, 0.2040, 0.0980, 0.2090, 0.1394, 0.4600, 0.5000, 0.4771, 0.0780, 0.0830
0.5300, 0.0000, 0.2800, 0.0880, 0.2330, 0.1438, 0.5800, 0.4000, 0.5050, 0.0910, 0.1280
0.5300, 1.0000, 0.2220, 0.1130, 0.1970, 0.1152, 0.6700, 0.3000, 0.4304, 0.1000, 0.1020
0.2300, 0.0000, 0.2900, 0.0900, 0.2160, 0.1314, 0.6500, 0.3000, 0.4585, 0.0910, 0.3020
0.6500, 1.0000, 0.3020, 0.0980, 0.2190, 0.1606, 0.4000, 0.5000, 0.4522, 0.0840, 0.1980
0.4100, 0.0000, 0.3240, 0.0940, 0.1710, 0.1044, 0.5600, 0.3000, 0.3970, 0.0760, 0.0950
0.5500, 1.0000, 0.2340, 0.0830, 0.1660, 0.1016, 0.4600, 0.4000, 0.4522, 0.0960, 0.0530
0.2200, 0.0000, 0.1930, 0.0820, 0.1560, 0.0932, 0.5200, 0.3000, 0.3989, 0.0710, 0.1340
0.5600, 0.0000, 0.3100, 0.0787, 0.1870, 0.1414, 0.3400, 0.5500, 0.4060, 0.0900, 0.1440
0.5400, 1.0000, 0.3060, 0.1033, 0.1440, 0.0798, 0.3000, 0.4800, 0.5142, 0.1010, 0.2320
0.5900, 1.0000, 0.2550, 0.0953, 0.1900, 0.1394, 0.3500, 0.5430, 0.4357, 0.1170, 0.0810
0.6000, 1.0000, 0.2340, 0.0880, 0.1530, 0.0898, 0.5800, 0.3000, 0.3258, 0.0950, 0.1040
0.5400, 0.0000, 0.2680, 0.0870, 0.2060, 0.1220, 0.6800, 0.3000, 0.4382, 0.0800, 0.0590
0.2500, 0.0000, 0.2830, 0.0870, 0.1930, 0.1280, 0.4900, 0.4000, 0.4382, 0.0920, 0.2460
0.5400, 1.0000, 0.2770, 0.1130, 0.2000, 0.1284, 0.3700, 0.5000, 0.5153, 0.1130, 0.2970
0.5500, 0.0000, 0.3660, 0.1130, 0.1990, 0.0944, 0.4300, 0.4630, 0.5730, 0.0970, 0.2580
0.4000, 1.0000, 0.2650, 0.0930, 0.2360, 0.1470, 0.3700, 0.7000, 0.5561, 0.0920, 0.2290
0.6200, 1.0000, 0.3180, 0.1150, 0.1990, 0.1286, 0.4400, 0.5000, 0.4883, 0.0980, 0.2750
0.6500, 0.0000, 0.2440, 0.1200, 0.2220, 0.1356, 0.3700, 0.6000, 0.5509, 0.1240, 0.2810
0.3300, 1.0000, 0.2540, 0.1020, 0.2060, 0.1410, 0.3900, 0.5000, 0.4868, 0.1050, 0.1790
0.5300, 0.0000, 0.2200, 0.0940, 0.1750, 0.0880, 0.5900, 0.3000, 0.4942, 0.0980, 0.2000
0.3500, 0.0000, 0.2680, 0.0980, 0.1620, 0.1036, 0.4500, 0.4000, 0.4205, 0.0860, 0.2000
0.6600, 0.0000, 0.2800, 0.1010, 0.1950, 0.1292, 0.4000, 0.5000, 0.4860, 0.0940, 0.1730
0.6200, 1.0000, 0.3390, 0.1010, 0.2210, 0.1564, 0.3500, 0.6000, 0.4997, 0.1030, 0.1800
0.5000, 1.0000, 0.2960, 0.0943, 0.3000, 0.2424, 0.3300, 0.9090, 0.4812, 0.1090, 0.0840
0.4700, 0.0000, 0.2860, 0.0970, 0.1640, 0.0906, 0.5600, 0.3000, 0.4466, 0.0880, 0.1210
0.4700, 1.0000, 0.2560, 0.0940, 0.1650, 0.0748, 0.4000, 0.4000, 0.5526, 0.0930, 0.1610
0.2400, 0.0000, 0.2070, 0.0870, 0.1490, 0.0806, 0.6100, 0.2000, 0.3611, 0.0780, 0.0990
0.5800, 1.0000, 0.2620, 0.0910, 0.2170, 0.1242, 0.7100, 0.3000, 0.4691, 0.0680, 0.1090
0.3400, 0.0000, 0.2060, 0.0870, 0.1850, 0.1122, 0.5800, 0.3000, 0.4304, 0.0740, 0.1150
0.5100, 0.0000, 0.2790, 0.0960, 0.1960, 0.1222, 0.4200, 0.5000, 0.5069, 0.1200, 0.2680
0.3100, 1.0000, 0.3530, 0.1250, 0.1870, 0.1124, 0.4800, 0.4000, 0.4890, 0.1090, 0.2740
0.2200, 0.0000, 0.1990, 0.0750, 0.1750, 0.1086, 0.5400, 0.3000, 0.4127, 0.0720, 0.1580
0.5300, 1.0000, 0.2440, 0.0920, 0.2140, 0.1460, 0.5000, 0.4000, 0.4500, 0.0970, 0.1070
0.3700, 1.0000, 0.2140, 0.0830, 0.1280, 0.0696, 0.4900, 0.3000, 0.3850, 0.0840, 0.0830
0.2800, 0.0000, 0.3040, 0.0850, 0.1980, 0.1156, 0.6700, 0.3000, 0.4344, 0.0800, 0.1030
0.4700, 0.0000, 0.3160, 0.0840, 0.1540, 0.0880, 0.3000, 0.5100, 0.5199, 0.1050, 0.2720
0.2300, 0.0000, 0.1880, 0.0780, 0.1450, 0.0720, 0.6300, 0.2000, 0.3912, 0.0860, 0.0850
0.5000, 0.0000, 0.3100, 0.1230, 0.1780, 0.1050, 0.4800, 0.4000, 0.4828, 0.0880, 0.2800
0.5800, 1.0000, 0.3670, 0.1170, 0.1660, 0.0938, 0.4400, 0.4000, 0.4949, 0.1090, 0.3360
0.5500, 0.0000, 0.3210, 0.1100, 0.1640, 0.0842, 0.4200, 0.4000, 0.5242, 0.0900, 0.2810
0.6000, 1.0000, 0.2770, 0.1070, 0.1670, 0.1146, 0.3800, 0.4000, 0.4277, 0.0950, 0.1180
0.4100, 0.0000, 0.3080, 0.0810, 0.2140, 0.1520, 0.2800, 0.7600, 0.5136, 0.1230, 0.3170
0.6000, 1.0000, 0.2750, 0.1060, 0.2290, 0.1438, 0.5100, 0.4000, 0.5142, 0.0910, 0.2350
0.4000, 0.0000, 0.2690, 0.0920, 0.2030, 0.1198, 0.7000, 0.3000, 0.4190, 0.0810, 0.0600
0.5700, 1.0000, 0.3070, 0.0900, 0.2040, 0.1478, 0.3400, 0.6000, 0.4710, 0.0930, 0.1740
0.3700, 0.0000, 0.3830, 0.1130, 0.1650, 0.0946, 0.5300, 0.3000, 0.4466, 0.0790, 0.2590
0.4000, 1.0000, 0.3190, 0.0950, 0.1980, 0.1356, 0.3800, 0.5000, 0.4804, 0.0930, 0.1780
0.3300, 0.0000, 0.3500, 0.0890, 0.2000, 0.1304, 0.4200, 0.4760, 0.4927, 0.1010, 0.1280
0.3200, 1.0000, 0.2780, 0.0890, 0.2160, 0.1462, 0.5500, 0.4000, 0.4304, 0.0910, 0.0960
0.3500, 1.0000, 0.2590, 0.0810, 0.1740, 0.1024, 0.3100, 0.6000, 0.5313, 0.0820, 0.1260
0.5500, 0.0000, 0.3290, 0.1020, 0.1640, 0.1062, 0.4100, 0.4000, 0.4431, 0.0890, 0.2880
0.4900, 0.0000, 0.2600, 0.0930, 0.1830, 0.1002, 0.6400, 0.3000, 0.4543, 0.0880, 0.0880
0.3900, 1.0000, 0.2630, 0.1150, 0.2180, 0.1582, 0.3200, 0.7000, 0.4935, 0.1090, 0.2920
0.6000, 1.0000, 0.2230, 0.1130, 0.1860, 0.1258, 0.4600, 0.4000, 0.4263, 0.0940, 0.0710
0.6700, 1.0000, 0.2830, 0.0930, 0.2040, 0.1322, 0.4900, 0.4000, 0.4736, 0.0920, 0.1970
0.4100, 1.0000, 0.3200, 0.1090, 0.2510, 0.1706, 0.4900, 0.5000, 0.5056, 0.1030, 0.1860
0.4400, 0.0000, 0.2540, 0.0950, 0.1620, 0.0926, 0.5300, 0.3000, 0.4407, 0.0830, 0.0250
0.4800, 1.0000, 0.2330, 0.0893, 0.2120, 0.1428, 0.4600, 0.4610, 0.4754, 0.0980, 0.0840
0.4500, 0.0000, 0.2030, 0.0743, 0.1900, 0.1262, 0.4900, 0.3880, 0.4304, 0.0790, 0.0960
0.4700, 0.0000, 0.3040, 0.1200, 0.1990, 0.1200, 0.4600, 0.4000, 0.5106, 0.0870, 0.1950
0.4600, 0.0000, 0.2060, 0.0730, 0.1720, 0.1070, 0.5100, 0.3000, 0.4249, 0.0800, 0.0530
0.3600, 1.0000, 0.3230, 0.1150, 0.2860, 0.1994, 0.3900, 0.7000, 0.5472, 0.1120, 0.2170
0.3400, 0.0000, 0.2920, 0.0730, 0.1720, 0.1082, 0.4900, 0.4000, 0.4304, 0.0910, 0.1720
0.5300, 1.0000, 0.3310, 0.1170, 0.1830, 0.1190, 0.4800, 0.4000, 0.4382, 0.1060, 0.1310
0.6100, 0.0000, 0.2460, 0.1010, 0.2090, 0.1068, 0.7700, 0.3000, 0.4836, 0.0880, 0.2140
0.3700, 0.0000, 0.2020, 0.0810, 0.1620, 0.0878, 0.6300, 0.3000, 0.4025, 0.0880, 0.0590
0.3300, 1.0000, 0.2080, 0.0840, 0.1250, 0.0702, 0.4600, 0.3000, 0.3784, 0.0660, 0.0700
0.6800, 0.0000, 0.3280, 0.1057, 0.2050, 0.1164, 0.4000, 0.5130, 0.5493, 0.1170, 0.2200
0.4900, 1.0000, 0.3190, 0.0940, 0.2340, 0.1558, 0.3400, 0.7000, 0.5398, 0.1220, 0.2680
0.4800, 0.0000, 0.2390, 0.1090, 0.2320, 0.1052, 0.3700, 0.6000, 0.6107, 0.0960, 0.1520
0.5500, 1.0000, 0.2450, 0.0840, 0.1790, 0.1058, 0.6600, 0.3000, 0.3584, 0.0870, 0.0470
0.4300, 0.0000, 0.2210, 0.0660, 0.1340, 0.0772, 0.4500, 0.3000, 0.4078, 0.0800, 0.0740
0.6000, 1.0000, 0.3300, 0.0970, 0.2170, 0.1256, 0.4500, 0.5000, 0.5447, 0.1120, 0.2950
0.3100, 1.0000, 0.1900, 0.0930, 0.1370, 0.0730, 0.4700, 0.3000, 0.4443, 0.0780, 0.1010
0.5300, 1.0000, 0.2730, 0.0820, 0.1190, 0.0550, 0.3900, 0.3000, 0.4828, 0.0930, 0.1510
0.6700, 0.0000, 0.2280, 0.0870, 0.1660, 0.0986, 0.5200, 0.3000, 0.4344, 0.0920, 0.1270
0.6100, 1.0000, 0.2820, 0.1060, 0.2040, 0.1320, 0.5200, 0.4000, 0.4605, 0.0960, 0.2370
0.6200, 0.0000, 0.2890, 0.0873, 0.2060, 0.1272, 0.3300, 0.6240, 0.5434, 0.0990, 0.2250
0.6000, 0.0000, 0.2560, 0.0870, 0.2070, 0.1258, 0.6900, 0.3000, 0.4111, 0.0840, 0.0810
0.4200, 0.0000, 0.2490, 0.0910, 0.2040, 0.1418, 0.3800, 0.5000, 0.4796, 0.0890, 0.1510
0.3800, 1.0000, 0.2680, 0.1050, 0.1810, 0.1192, 0.3700, 0.5000, 0.4820, 0.0910, 0.1070
0.6200, 0.0000, 0.2240, 0.0790, 0.2220, 0.1474, 0.5900, 0.4000, 0.4357, 0.0760, 0.0640
0.6100, 1.0000, 0.2690, 0.1110, 0.2360, 0.1724, 0.3900, 0.6000, 0.4812, 0.0890, 0.1380
0.6100, 1.0000, 0.2310, 0.1130, 0.1860, 0.1144, 0.4700, 0.4000, 0.4812, 0.1050, 0.1850
0.5300, 0.0000, 0.2860, 0.0880, 0.1710, 0.0988, 0.4100, 0.4000, 0.5050, 0.0990, 0.2650
0.2800, 1.0000, 0.2470, 0.0970, 0.1750, 0.0996, 0.3200, 0.5000, 0.5380, 0.0870, 0.1010
0.2600, 1.0000, 0.3030, 0.0890, 0.2180, 0.1522, 0.3100, 0.7000, 0.5159, 0.0820, 0.1370
0.3000, 0.0000, 0.2130, 0.0870, 0.1340, 0.0630, 0.6300, 0.2000, 0.3689, 0.0660, 0.1430
0.5000, 0.0000, 0.2610, 0.1090, 0.2430, 0.1606, 0.6200, 0.4000, 0.4625, 0.0890, 0.1410
0.4800, 0.0000, 0.2020, 0.0950, 0.1870, 0.1174, 0.5300, 0.4000, 0.4419, 0.0850, 0.0790
0.5100, 0.0000, 0.2520, 0.1030, 0.1760, 0.1122, 0.3700, 0.5000, 0.4898, 0.0900, 0.2920
0.4700, 1.0000, 0.2250, 0.0820, 0.1310, 0.0668, 0.4100, 0.3000, 0.4754, 0.0890, 0.1780
0.6400, 1.0000, 0.2350, 0.0970, 0.2030, 0.1290, 0.5900, 0.3000, 0.4318, 0.0770, 0.0910
0.5100, 1.0000, 0.2590, 0.0760, 0.2400, 0.1690, 0.3900, 0.6000, 0.5075, 0.0960, 0.1160
0.3000, 0.0000, 0.2090, 0.1040, 0.1520, 0.0838, 0.4700, 0.3000, 0.4663, 0.0970, 0.0860
0.5600, 1.0000, 0.2870, 0.0990, 0.2080, 0.1464, 0.3900, 0.5000, 0.4727, 0.0970, 0.1220
0.4200, 0.0000, 0.2210, 0.0850, 0.2130, 0.1386, 0.6000, 0.4000, 0.4277, 0.0940, 0.0720
0.6200, 1.0000, 0.2670, 0.1150, 0.1830, 0.1240, 0.3500, 0.5000, 0.4788, 0.1000, 0.1290
0.3400, 0.0000, 0.3140, 0.0870, 0.1490, 0.0938, 0.4600, 0.3000, 0.3829, 0.0770, 0.1420
0.6000, 0.0000, 0.2220, 0.1047, 0.2210, 0.1054, 0.6000, 0.3680, 0.5628, 0.0930, 0.0900
0.6400, 0.0000, 0.2100, 0.0923, 0.2270, 0.1468, 0.6500, 0.3490, 0.4331, 0.1020, 0.1580
0.3900, 1.0000, 0.2120, 0.0900, 0.1820, 0.1104, 0.6000, 0.3000, 0.4060, 0.0980, 0.0390
0.7100, 1.0000, 0.2650, 0.1050, 0.2810, 0.1736, 0.5500, 0.5000, 0.5568, 0.0840, 0.1960
0.4800, 1.0000, 0.2920, 0.1100, 0.2180, 0.1516, 0.3900, 0.6000, 0.4920, 0.0980, 0.2220
0.7900, 1.0000, 0.2700, 0.1030, 0.1690, 0.1108, 0.3700, 0.5000, 0.4663, 0.1100, 0.2770
0.4000, 0.0000, 0.3070, 0.0990, 0.1770, 0.0854, 0.5000, 0.4000, 0.5338, 0.0850, 0.0990
0.4900, 1.0000, 0.2880, 0.0920, 0.2070, 0.1400, 0.4400, 0.5000, 0.4745, 0.0920, 0.1960
0.5100, 0.0000, 0.3060, 0.1030, 0.1980, 0.1066, 0.5700, 0.3000, 0.5148, 0.1000, 0.2020
0.5700, 0.0000, 0.3010, 0.1170, 0.2020, 0.1396, 0.4200, 0.5000, 0.4625, 0.1200, 0.1550
0.5900, 1.0000, 0.2470, 0.1140, 0.1520, 0.1048, 0.2900, 0.5000, 0.4511, 0.0880, 0.0770
0.5100, 0.0000, 0.2770, 0.0990, 0.2290, 0.1456, 0.6900, 0.3000, 0.4277, 0.0770, 0.1910
0.7400, 0.0000, 0.2980, 0.1010, 0.1710, 0.1048, 0.5000, 0.3000, 0.4394, 0.0860, 0.0700
0.6700, 0.0000, 0.2670, 0.1050, 0.2250, 0.1354, 0.6900, 0.3000, 0.4635, 0.0960, 0.0730
0.4900, 0.0000, 0.1980, 0.0880, 0.1880, 0.1148, 0.5700, 0.3000, 0.4394, 0.0930, 0.0490
0.5700, 0.0000, 0.2330, 0.0880, 0.1550, 0.0636, 0.7800, 0.2000, 0.4205, 0.0780, 0.0650
0.5600, 1.0000, 0.3510, 0.1230, 0.1640, 0.0950, 0.3800, 0.4000, 0.5043, 0.1170, 0.2630
0.5200, 1.0000, 0.2970, 0.1090, 0.2280, 0.1628, 0.3100, 0.8000, 0.5142, 0.1030, 0.2480
0.6900, 0.0000, 0.2930, 0.1240, 0.2230, 0.1390, 0.5400, 0.4000, 0.5011, 0.1020, 0.2960
0.3700, 0.0000, 0.2030, 0.0830, 0.1850, 0.1246, 0.3800, 0.5000, 0.4719, 0.0880, 0.2140
0.2400, 0.0000, 0.2250, 0.0890, 0.1410, 0.0680, 0.5200, 0.3000, 0.4654, 0.0840, 0.1850
0.5500, 1.0000, 0.2270, 0.0930, 0.1540, 0.0942, 0.5300, 0.3000, 0.3526, 0.0750, 0.0780
0.3600, 0.0000, 0.2280, 0.0870, 0.1780, 0.1160, 0.4100, 0.4000, 0.4654, 0.0820, 0.0930
0.4200, 1.0000, 0.2400, 0.1070, 0.1500, 0.0850, 0.4400, 0.3000, 0.4654, 0.0960, 0.2520
0.2100, 0.0000, 0.2420, 0.0760, 0.1470, 0.0770, 0.5300, 0.3000, 0.4443, 0.0790, 0.1500
0.4100, 0.0000, 0.2020, 0.0620, 0.1530, 0.0890, 0.5000, 0.3000, 0.4249, 0.0890, 0.0770
0.5700, 1.0000, 0.2940, 0.1090, 0.1600, 0.0876, 0.3100, 0.5000, 0.5333, 0.0920, 0.2080
0.2000, 1.0000, 0.2210, 0.0870, 0.1710, 0.0996, 0.5800, 0.3000, 0.4205, 0.0780, 0.0770
0.6700, 1.0000, 0.2360, 0.1113, 0.1890, 0.1054, 0.7000, 0.2700, 0.4220, 0.0930, 0.1080
0.3400, 0.0000, 0.2520, 0.0770, 0.1890, 0.1206, 0.5300, 0.4000, 0.4344, 0.0790, 0.1600
0.4100, 1.0000, 0.2490, 0.0860, 0.1920, 0.1150, 0.6100, 0.3000, 0.4382, 0.0940, 0.0530
0.3800, 1.0000, 0.3300, 0.0780, 0.3010, 0.2150, 0.5000, 0.6020, 0.5193, 0.1080, 0.2200
0.5100, 0.0000, 0.2350, 0.1010, 0.1950, 0.1210, 0.5100, 0.4000, 0.4745, 0.0940, 0.1540
0.5200, 1.0000, 0.2640, 0.0913, 0.2180, 0.1520, 0.3900, 0.5590, 0.4905, 0.0990, 0.2590
0.6700, 0.0000, 0.2980, 0.0800, 0.1720, 0.0934, 0.6300, 0.3000, 0.4357, 0.0820, 0.0900
0.6100, 0.0000, 0.3000, 0.1080, 0.1940, 0.1000, 0.5200, 0.3730, 0.5347, 0.1050, 0.2460
0.6700, 1.0000, 0.2500, 0.1117, 0.1460, 0.0934, 0.3300, 0.4420, 0.4585, 0.1030, 0.1240
0.5600, 0.0000, 0.2700, 0.1050, 0.2470, 0.1606, 0.5400, 0.5000, 0.5088, 0.0940, 0.0670
0.6400, 0.0000, 0.2000, 0.0747, 0.1890, 0.1148, 0.6200, 0.3050, 0.4111, 0.0910, 0.0720
0.5800, 1.0000, 0.2550, 0.1120, 0.1630, 0.1106, 0.2900, 0.6000, 0.4762, 0.0860, 0.2570
0.5500, 0.0000, 0.2820, 0.0910, 0.2500, 0.1402, 0.6700, 0.4000, 0.5366, 0.1030, 0.2620
0.6200, 1.0000, 0.3330, 0.1140, 0.1820, 0.1140, 0.3800, 0.5000, 0.5011, 0.0960, 0.2750
0.5700, 1.0000, 0.2560, 0.0960, 0.2000, 0.1330, 0.5200, 0.3850, 0.4318, 0.1050, 0.1770
0.2000, 1.0000, 0.2420, 0.0880, 0.1260, 0.0722, 0.4500, 0.3000, 0.3784, 0.0740, 0.0710
0.5300, 1.0000, 0.2210, 0.0980, 0.1650, 0.1052, 0.4700, 0.4000, 0.4159, 0.0810, 0.0470
0.3200, 1.0000, 0.3140, 0.0890, 0.1530, 0.0842, 0.5600, 0.3000, 0.4159, 0.0900, 0.1870
0.4100, 0.0000, 0.2310, 0.0860, 0.1480, 0.0780, 0.5800, 0.3000, 0.4094, 0.0600, 0.1250
0.6000, 0.0000, 0.2340, 0.0767, 0.2470, 0.1480, 0.6500, 0.3800, 0.5136, 0.0770, 0.0780
0.2600, 0.0000, 0.1880, 0.0830, 0.1910, 0.1036, 0.6900, 0.3000, 0.4522, 0.0690, 0.0510
0.3700, 0.0000, 0.3080, 0.1120, 0.2820, 0.1972, 0.4300, 0.7000, 0.5342, 0.1010, 0.2580
0.4500, 0.0000, 0.3200, 0.1100, 0.2240, 0.1342, 0.4500, 0.5000, 0.5412, 0.0930, 0.2150
0.6700, 0.0000, 0.3160, 0.1160, 0.1790, 0.0904, 0.4100, 0.4000, 0.5472, 0.1000, 0.3030
0.3400, 1.0000, 0.3550, 0.1200, 0.2330, 0.1466, 0.3400, 0.7000, 0.5568, 0.1010, 0.2430
0.5000, 0.0000, 0.3190, 0.0783, 0.2070, 0.1492, 0.3800, 0.5450, 0.4595, 0.0840, 0.0910
0.7100, 0.0000, 0.2950, 0.0970, 0.2270, 0.1516, 0.4500, 0.5000, 0.5024, 0.1080, 0.1500
0.5700, 1.0000, 0.3160, 0.1170, 0.2250, 0.1076, 0.4000, 0.6000, 0.5958, 0.1130, 0.3100
0.4900, 0.0000, 0.2030, 0.0930, 0.1840, 0.1030, 0.6100, 0.3000, 0.4605, 0.0930, 0.1530
0.3500, 0.0000, 0.4130, 0.0810, 0.1680, 0.1028, 0.3700, 0.5000, 0.4949, 0.0940, 0.3460
0.4100, 1.0000, 0.2120, 0.1020, 0.1840, 0.1004, 0.6400, 0.3000, 0.4585, 0.0790, 0.0630
0.7000, 1.0000, 0.2410, 0.0823, 0.1940, 0.1492, 0.3100, 0.6260, 0.4234, 0.1050, 0.0890
0.5200, 0.0000, 0.2300, 0.1070, 0.1790, 0.1237, 0.4250, 0.4210, 0.4159, 0.0930, 0.0500
0.6000, 0.0000, 0.2560, 0.0780, 0.1950, 0.0954, 0.9100, 0.2000, 0.3761, 0.0870, 0.0390
0.6200, 0.0000, 0.2250, 0.1250, 0.2150, 0.0990, 0.9800, 0.2000, 0.4500, 0.0950, 0.1030
0.4400, 1.0000, 0.3820, 0.1230, 0.2010, 0.1266, 0.4400, 0.5000, 0.5024, 0.0920, 0.3080
0.2800, 1.0000, 0.1920, 0.0810, 0.1550, 0.0946, 0.5100, 0.3000, 0.3850, 0.0870, 0.1160
0.5800, 1.0000, 0.2900, 0.0850, 0.1560, 0.1092, 0.3600, 0.4000, 0.3989, 0.0860, 0.1450
0.3900, 1.0000, 0.2400, 0.0897, 0.1900, 0.1136, 0.5200, 0.3650, 0.4804, 0.1010, 0.0740
0.3400, 1.0000, 0.2060, 0.0980, 0.1830, 0.0920, 0.8300, 0.2000, 0.3689, 0.0920, 0.0450
0.6500, 0.0000, 0.2630, 0.0700, 0.2440, 0.1662, 0.5100, 0.5000, 0.4898, 0.0980, 0.1150
0.6600, 1.0000, 0.3460, 0.1150, 0.2040, 0.1394, 0.3600, 0.6000, 0.4963, 0.1090, 0.2640
0.5100, 0.0000, 0.2340, 0.0870, 0.2200, 0.1088, 0.9300, 0.2000, 0.4511, 0.0820, 0.0870
0.5000, 1.0000, 0.2920, 0.1190, 0.1620, 0.0852, 0.5400, 0.3000, 0.4736, 0.0950, 0.2020
0.5900, 1.0000, 0.2720, 0.1070, 0.1580, 0.1020, 0.3900, 0.4000, 0.4443, 0.0930, 0.1270
0.5200, 0.0000, 0.2700, 0.0783, 0.1340, 0.0730, 0.4400, 0.3050, 0.4443, 0.0690, 0.1820
0.6900, 1.0000, 0.2450, 0.1080, 0.2430, 0.1364, 0.4000, 0.6000, 0.5808, 0.1000, 0.2410
0.5300, 0.0000, 0.2410, 0.1050, 0.1840, 0.1134, 0.4600, 0.4000, 0.4812, 0.0950, 0.0660
0.4700, 1.0000, 0.2530, 0.0980, 0.1730, 0.1056, 0.4400, 0.4000, 0.4762, 0.1080, 0.0940
0.5200, 0.0000, 0.2880, 0.1130, 0.2800, 0.1740, 0.6700, 0.4000, 0.5273, 0.0860, 0.2830
0.3900, 0.0000, 0.2090, 0.0950, 0.1500, 0.0656, 0.6800, 0.2000, 0.4407, 0.0950, 0.0640
0.6700, 1.0000, 0.2300, 0.0700, 0.1840, 0.1280, 0.3500, 0.5000, 0.4654, 0.0990, 0.1020
0.5900, 1.0000, 0.2410, 0.0960, 0.1700, 0.0986, 0.5400, 0.3000, 0.4466, 0.0850, 0.2000
0.5100, 1.0000, 0.2810, 0.1060, 0.2020, 0.1222, 0.5500, 0.4000, 0.4820, 0.0870, 0.2650
0.2300, 1.0000, 0.1800, 0.0780, 0.1710, 0.0960, 0.4800, 0.4000, 0.4905, 0.0920, 0.0940
0.6800, 0.0000, 0.2590, 0.0930, 0.2530, 0.1812, 0.5300, 0.5000, 0.4543, 0.0980, 0.2300
0.4400, 0.0000, 0.2150, 0.0850, 0.1570, 0.0922, 0.5500, 0.3000, 0.3892, 0.0840, 0.1810
0.6000, 1.0000, 0.2430, 0.1030, 0.1410, 0.0866, 0.3300, 0.4000, 0.4673, 0.0780, 0.1560
0.5200, 0.0000, 0.2450, 0.0900, 0.1980, 0.1290, 0.2900, 0.7000, 0.5298, 0.0860, 0.2330
0.3800, 0.0000, 0.2130, 0.0720, 0.1650, 0.0602, 0.8800, 0.2000, 0.4431, 0.0900, 0.0600
0.6100, 0.0000, 0.2580, 0.0900, 0.2800, 0.1954, 0.5500, 0.5000, 0.4997, 0.0900, 0.2190
0.6800, 1.0000, 0.2480, 0.1010, 0.2210, 0.1514, 0.6000, 0.4000, 0.3871, 0.0870, 0.0800
0.2800, 1.0000, 0.3150, 0.0830, 0.2280, 0.1494, 0.3800, 0.6000, 0.5313, 0.0830, 0.0680
0.6500, 1.0000, 0.3350, 0.1020, 0.1900, 0.1262, 0.3500, 0.5000, 0.4970, 0.1020, 0.3320
0.6900, 0.0000, 0.2810, 0.1130, 0.2340, 0.1428, 0.5200, 0.4000, 0.5278, 0.0770, 0.2480
0.5100, 0.0000, 0.2430, 0.0853, 0.1530, 0.0716, 0.7100, 0.2150, 0.3951, 0.0820, 0.0840
0.2900, 0.0000, 0.3500, 0.0983, 0.2040, 0.1426, 0.5000, 0.4080, 0.4043, 0.0910, 0.2000
0.5500, 1.0000, 0.2350, 0.0930, 0.1770, 0.1268, 0.4100, 0.4000, 0.3829, 0.0830, 0.0550
0.3400, 1.0000, 0.3000, 0.0830, 0.1850, 0.1072, 0.5300, 0.3000, 0.4820, 0.0920, 0.0850
0.6700, 0.0000, 0.2070, 0.0830, 0.1700, 0.0998, 0.5900, 0.3000, 0.4025, 0.0770, 0.0890
0.4900, 0.0000, 0.2560, 0.0760, 0.1610, 0.0998, 0.5100, 0.3000, 0.3932, 0.0780, 0.0310
0.5500, 1.0000, 0.2290, 0.0810, 0.1230, 0.0672, 0.4100, 0.3000, 0.4304, 0.0880, 0.1290
0.5900, 1.0000, 0.2510, 0.0900, 0.1630, 0.1014, 0.4600, 0.4000, 0.4357, 0.0910, 0.0830
0.5300, 0.0000, 0.3320, 0.0827, 0.1860, 0.1068, 0.4600, 0.4040, 0.5112, 0.1020, 0.2750
0.4800, 1.0000, 0.2410, 0.1100, 0.2090, 0.1346, 0.5800, 0.4000, 0.4407, 0.1000, 0.0650
0.5200, 0.0000, 0.2950, 0.1043, 0.2110, 0.1328, 0.4900, 0.4310, 0.4984, 0.0980, 0.1980
0.6900, 0.0000, 0.2960, 0.1220, 0.2310, 0.1284, 0.5600, 0.4000, 0.5451, 0.0860, 0.2360
0.6000, 1.0000, 0.2280, 0.1100, 0.2450, 0.1898, 0.3900, 0.6000, 0.4394, 0.0880, 0.2530
0.4600, 1.0000, 0.2270, 0.0830, 0.1830, 0.1258, 0.3200, 0.6000, 0.4836, 0.0750, 0.1240
0.5100, 1.0000, 0.2620, 0.1010, 0.1610, 0.0996, 0.4800, 0.3000, 0.4205, 0.0880, 0.0440
0.6700, 1.0000, 0.2350, 0.0960, 0.2070, 0.1382, 0.4200, 0.5000, 0.4898, 0.1110, 0.1720
0.4900, 0.0000, 0.2210, 0.0850, 0.1360, 0.0634, 0.6200, 0.2190, 0.3970, 0.0720, 0.1140
0.4600, 1.0000, 0.2650, 0.0940, 0.2470, 0.1602, 0.5900, 0.4000, 0.4935, 0.1110, 0.1420
0.4700, 0.0000, 0.3240, 0.1050, 0.1880, 0.1250, 0.4600, 0.4090, 0.4443, 0.0990, 0.1090
0.7500, 0.0000, 0.3010, 0.0780, 0.2220, 0.1542, 0.4400, 0.5050, 0.4779, 0.0970, 0.1800
0.2800, 0.0000, 0.2420, 0.0930, 0.1740, 0.1064, 0.5400, 0.3000, 0.4220, 0.0840, 0.1440
0.6500, 1.0000, 0.3130, 0.1100, 0.2130, 0.1280, 0.4700, 0.5000, 0.5247, 0.0910, 0.1630
0.4200, 0.0000, 0.3010, 0.0910, 0.1820, 0.1148, 0.4900, 0.4000, 0.4511, 0.0820, 0.1470
0.5100, 0.0000, 0.2450, 0.0790, 0.2120, 0.1286, 0.6500, 0.3000, 0.4522, 0.0910, 0.0970
0.5300, 1.0000, 0.2770, 0.0950, 0.1900, 0.1018, 0.4100, 0.5000, 0.5464, 0.1010, 0.2200
0.5400, 0.0000, 0.2320, 0.1107, 0.2380, 0.1628, 0.4800, 0.4960, 0.4913, 0.1080, 0.1900
0.7300, 0.0000, 0.2700, 0.1020, 0.2110, 0.1210, 0.6700, 0.3000, 0.4745, 0.0990, 0.1090
0.5400, 0.0000, 0.2680, 0.1080, 0.1760, 0.0806, 0.6700, 0.3000, 0.4956, 0.1060, 0.1910
0.4200, 0.0000, 0.2920, 0.0930, 0.2490, 0.1742, 0.4500, 0.6000, 0.5004, 0.0920, 0.1220
0.7500, 0.0000, 0.3120, 0.1177, 0.2290, 0.1388, 0.2900, 0.7900, 0.5724, 0.1060, 0.2300
0.5500, 1.0000, 0.3210, 0.1127, 0.2070, 0.0924, 0.2500, 0.8280, 0.6105, 0.1110, 0.2420
0.6800, 1.0000, 0.2570, 0.1090, 0.2330, 0.1126, 0.3500, 0.7000, 0.6057, 0.1050, 0.2480
0.5700, 0.0000, 0.2690, 0.0980, 0.2460, 0.1652, 0.3800, 0.7000, 0.5366, 0.0960, 0.2490
0.4800, 0.0000, 0.3140, 0.0753, 0.2420, 0.1516, 0.3800, 0.6370, 0.5568, 0.1030, 0.1920
0.6100, 1.0000, 0.2560, 0.0850, 0.1840, 0.1162, 0.3900, 0.5000, 0.4970, 0.0980, 0.1310
0.6900, 0.0000, 0.3700, 0.1030, 0.2070, 0.1314, 0.5500, 0.4000, 0.4635, 0.0900, 0.2370
0.3800, 0.0000, 0.3260, 0.0770, 0.1680, 0.1006, 0.4700, 0.4000, 0.4625, 0.0960, 0.0780
0.4500, 1.0000, 0.2120, 0.0940, 0.1690, 0.0968, 0.5500, 0.3000, 0.4454, 0.1020, 0.1350
0.5100, 1.0000, 0.2920, 0.1070, 0.1870, 0.1390, 0.3200, 0.6000, 0.4382, 0.0950, 0.2440
0.7100, 1.0000, 0.2400, 0.0840, 0.1380, 0.0858, 0.3900, 0.4000, 0.4190, 0.0900, 0.1990
0.5700, 0.0000, 0.3610, 0.1170, 0.1810, 0.1082, 0.3400, 0.5000, 0.5268, 0.1000, 0.2700
0.5600, 1.0000, 0.2580, 0.1030, 0.1770, 0.1144, 0.3400, 0.5000, 0.4963, 0.0990, 0.1640
0.3200, 1.0000, 0.2200, 0.0880, 0.1370, 0.0786, 0.4800, 0.3000, 0.3951, 0.0780, 0.0720
0.5000, 0.0000, 0.2190, 0.0910, 0.1900, 0.1112, 0.6700, 0.3000, 0.4078, 0.0770, 0.0960
0.4300, 0.0000, 0.3430, 0.0840, 0.2560, 0.1726, 0.3300, 0.8000, 0.5529, 0.1040, 0.3060
0.5400, 1.0000, 0.2520, 0.1150, 0.1810, 0.1200, 0.3900, 0.5000, 0.4701, 0.0920, 0.0910
0.3100, 0.0000, 0.2330, 0.0850, 0.1900, 0.1308, 0.4300, 0.4000, 0.4394, 0.0770, 0.2140
0.5600, 0.0000, 0.2570, 0.0800, 0.2440, 0.1516, 0.5900, 0.4000, 0.5118, 0.0950, 0.0950
0.4400, 0.0000, 0.2510, 0.1330, 0.1820, 0.1130, 0.5500, 0.3000, 0.4249, 0.0840, 0.2160
0.5700, 1.0000, 0.3190, 0.1110, 0.1730, 0.1162, 0.4100, 0.4000, 0.4369, 0.0870, 0.2630

Test data:


# diabetes_norm_test_100.txt
#
0.6400, 1.0000, 0.2840, 0.1110, 0.1840, 0.1270, 0.4100, 0.4000, 0.4382, 0.0970, 0.1780
0.4300, 0.0000, 0.2810, 0.1210, 0.1920, 0.1210, 0.6000, 0.3000, 0.4007, 0.0930, 0.1130
0.1900, 0.0000, 0.2530, 0.0830, 0.2250, 0.1566, 0.4600, 0.5000, 0.4719, 0.0840, 0.2000
0.7100, 1.0000, 0.2610, 0.0850, 0.2200, 0.1524, 0.4700, 0.5000, 0.4635, 0.0910, 0.1390
0.5000, 1.0000, 0.2800, 0.1040, 0.2820, 0.1968, 0.4400, 0.6000, 0.5328, 0.0950, 0.1390
0.5900, 1.0000, 0.2360, 0.0730, 0.1800, 0.1074, 0.5100, 0.4000, 0.4682, 0.0840, 0.0880
0.5700, 0.0000, 0.2450, 0.0930, 0.1860, 0.0966, 0.7100, 0.3000, 0.4522, 0.0910, 0.1480
0.4900, 1.0000, 0.2100, 0.0820, 0.1190, 0.0854, 0.2300, 0.5000, 0.3970, 0.0740, 0.0880
0.4100, 1.0000, 0.3200, 0.1260, 0.1980, 0.1042, 0.4900, 0.4000, 0.5412, 0.1240, 0.2430
0.2500, 1.0000, 0.2260, 0.0850, 0.1300, 0.0710, 0.4800, 0.3000, 0.4007, 0.0810, 0.0710
0.5200, 1.0000, 0.1970, 0.0810, 0.1520, 0.0534, 0.8200, 0.2000, 0.4419, 0.0820, 0.0770
0.3400, 0.0000, 0.2120, 0.0840, 0.2540, 0.1134, 0.5200, 0.5000, 0.6094, 0.0920, 0.1090
0.4200, 1.0000, 0.3060, 0.1010, 0.2690, 0.1722, 0.5000, 0.5000, 0.5455, 0.1060, 0.2720
0.2800, 1.0000, 0.2550, 0.0990, 0.1620, 0.1016, 0.4600, 0.4000, 0.4277, 0.0940, 0.0600
0.4700, 1.0000, 0.2330, 0.0900, 0.1950, 0.1258, 0.5400, 0.4000, 0.4331, 0.0730, 0.0540
0.3200, 1.0000, 0.3100, 0.1000, 0.1770, 0.0962, 0.4500, 0.4000, 0.5187, 0.0770, 0.2210
0.4300, 0.0000, 0.1850, 0.0870, 0.1630, 0.0936, 0.6100, 0.2670, 0.3738, 0.0800, 0.0900
0.5900, 1.0000, 0.2690, 0.1040, 0.1940, 0.1266, 0.4300, 0.5000, 0.4804, 0.1060, 0.3110
0.5300, 0.0000, 0.2830, 0.1010, 0.1790, 0.1070, 0.4800, 0.4000, 0.4788, 0.1010, 0.2810
0.6000, 0.0000, 0.2570, 0.1030, 0.1580, 0.0846, 0.6400, 0.2000, 0.3850, 0.0970, 0.1820
0.5400, 1.0000, 0.3610, 0.1150, 0.1630, 0.0984, 0.4300, 0.4000, 0.4682, 0.1010, 0.3210
0.3500, 1.0000, 0.2410, 0.0947, 0.1550, 0.0974, 0.3200, 0.4840, 0.4852, 0.0940, 0.0580
0.4900, 1.0000, 0.2580, 0.0890, 0.1820, 0.1186, 0.3900, 0.5000, 0.4804, 0.1150, 0.2620
0.5800, 0.0000, 0.2280, 0.0910, 0.1960, 0.1188, 0.4800, 0.4000, 0.4984, 0.1150, 0.2060
0.3600, 1.0000, 0.3910, 0.0900, 0.2190, 0.1358, 0.3800, 0.6000, 0.5421, 0.1030, 0.2330
0.4600, 1.0000, 0.4220, 0.0990, 0.2110, 0.1370, 0.4400, 0.5000, 0.5011, 0.0990, 0.2420
0.4400, 1.0000, 0.2660, 0.0990, 0.2050, 0.1090, 0.4300, 0.5000, 0.5580, 0.1110, 0.1230
0.4600, 0.0000, 0.2990, 0.0830, 0.1710, 0.1130, 0.3800, 0.4500, 0.4585, 0.0980, 0.1670
0.5400, 0.0000, 0.2100, 0.0780, 0.1880, 0.1074, 0.7000, 0.3000, 0.3970, 0.0730, 0.0630
0.6300, 1.0000, 0.2550, 0.1090, 0.2260, 0.1032, 0.4600, 0.5000, 0.5951, 0.0870, 0.1970
0.4100, 1.0000, 0.2420, 0.0900, 0.1990, 0.1236, 0.5700, 0.4000, 0.4522, 0.0860, 0.0710
0.2800, 0.0000, 0.2540, 0.0930, 0.1410, 0.0790, 0.4900, 0.3000, 0.4174, 0.0910, 0.1680
0.1900, 0.0000, 0.2320, 0.0750, 0.1430, 0.0704, 0.5200, 0.3000, 0.4635, 0.0720, 0.1400
0.6100, 1.0000, 0.2610, 0.1260, 0.2150, 0.1298, 0.5700, 0.4000, 0.4949, 0.0960, 0.2170
0.4800, 0.0000, 0.3270, 0.0930, 0.2760, 0.1986, 0.4300, 0.6420, 0.5148, 0.0910, 0.1210
0.5400, 1.0000, 0.2730, 0.1000, 0.2000, 0.1440, 0.3300, 0.6000, 0.4745, 0.0760, 0.2350
0.5300, 1.0000, 0.2660, 0.0930, 0.1850, 0.1224, 0.3600, 0.5000, 0.4890, 0.0820, 0.2450
0.4800, 0.0000, 0.2280, 0.1010, 0.1100, 0.0416, 0.5600, 0.2000, 0.4127, 0.0970, 0.0400
0.5300, 0.0000, 0.2880, 0.1117, 0.1450, 0.0872, 0.4600, 0.3150, 0.4078, 0.0850, 0.0520
0.2900, 1.0000, 0.1810, 0.0730, 0.1580, 0.0990, 0.4100, 0.4000, 0.4500, 0.0780, 0.1040
0.6200, 0.0000, 0.3200, 0.0880, 0.1720, 0.0690, 0.3800, 0.4000, 0.5784, 0.1000, 0.1320
0.5000, 1.0000, 0.2370, 0.0920, 0.1660, 0.0970, 0.5200, 0.3000, 0.4443, 0.0930, 0.0880
0.5800, 1.0000, 0.2360, 0.0960, 0.2570, 0.1710, 0.5900, 0.4000, 0.4905, 0.0820, 0.0690
0.5500, 1.0000, 0.2460, 0.1090, 0.1430, 0.0764, 0.5100, 0.3000, 0.4357, 0.0880, 0.2190
0.5400, 0.0000, 0.2260, 0.0900, 0.1830, 0.1042, 0.6400, 0.3000, 0.4304, 0.0920, 0.0720
0.3600, 0.0000, 0.2780, 0.0730, 0.1530, 0.1044, 0.4200, 0.4000, 0.3497, 0.0730, 0.2010
0.6300, 1.0000, 0.2410, 0.1110, 0.1840, 0.1122, 0.4400, 0.4000, 0.4935, 0.0820, 0.1100
0.4700, 1.0000, 0.2650, 0.0700, 0.1810, 0.1048, 0.6300, 0.3000, 0.4190, 0.0700, 0.0510
0.5100, 1.0000, 0.3280, 0.1120, 0.2020, 0.1006, 0.3700, 0.5000, 0.5775, 0.1090, 0.2770
0.4200, 0.0000, 0.1990, 0.0760, 0.1460, 0.0832, 0.5500, 0.3000, 0.3664, 0.0790, 0.0630
0.3700, 1.0000, 0.2360, 0.0940, 0.2050, 0.1388, 0.5300, 0.4000, 0.4190, 0.1070, 0.1180
0.2800, 0.0000, 0.2210, 0.0820, 0.1680, 0.1006, 0.5400, 0.3000, 0.4205, 0.0860, 0.0690
0.5800, 0.0000, 0.2810, 0.1110, 0.1980, 0.0806, 0.3100, 0.6000, 0.6068, 0.0930, 0.2730
0.3200, 0.0000, 0.2650, 0.0860, 0.1840, 0.1016, 0.5300, 0.4000, 0.4990, 0.0780, 0.2580
0.2500, 1.0000, 0.2350, 0.0880, 0.1430, 0.0808, 0.5500, 0.3000, 0.3584, 0.0830, 0.0430
0.6300, 0.0000, 0.2600, 0.0857, 0.1550, 0.0782, 0.4600, 0.3370, 0.5037, 0.0970, 0.1980
0.5200, 0.0000, 0.2780, 0.0850, 0.2190, 0.1360, 0.4900, 0.4000, 0.5136, 0.0750, 0.2420
0.6500, 1.0000, 0.2850, 0.1090, 0.2010, 0.1230, 0.4600, 0.4000, 0.5075, 0.0960, 0.2320
0.4200, 0.0000, 0.3060, 0.1210, 0.1760, 0.0928, 0.6900, 0.3000, 0.4263, 0.0890, 0.1750
0.5300, 0.0000, 0.2220, 0.0780, 0.1640, 0.0810, 0.7000, 0.2000, 0.4174, 0.1010, 0.0930
0.7900, 1.0000, 0.2330, 0.0880, 0.1860, 0.1284, 0.3300, 0.6000, 0.4812, 0.1020, 0.1680
0.4300, 0.0000, 0.3540, 0.0930, 0.1850, 0.1002, 0.4400, 0.4000, 0.5318, 0.1010, 0.2750
0.4400, 0.0000, 0.3140, 0.1150, 0.1650, 0.0976, 0.5200, 0.3000, 0.4344, 0.0890, 0.2930
0.6200, 1.0000, 0.3780, 0.1190, 0.1130, 0.0510, 0.3100, 0.4000, 0.5043, 0.0840, 0.2810
0.3300, 0.0000, 0.1890, 0.0700, 0.1620, 0.0918, 0.5900, 0.3000, 0.4025, 0.0580, 0.0720
0.5600, 0.0000, 0.3500, 0.0793, 0.1950, 0.1408, 0.4200, 0.4640, 0.4111, 0.0960, 0.1400
0.6600, 0.0000, 0.2170, 0.1260, 0.2120, 0.1278, 0.4500, 0.4710, 0.5278, 0.1010, 0.1890
0.3400, 1.0000, 0.2530, 0.1110, 0.2300, 0.1620, 0.3900, 0.6000, 0.4977, 0.0900, 0.1810
0.4600, 1.0000, 0.2380, 0.0970, 0.2240, 0.1392, 0.4200, 0.5000, 0.5366, 0.0810, 0.2090
0.5000, 0.0000, 0.3180, 0.0820, 0.1360, 0.0692, 0.5500, 0.2000, 0.4078, 0.0850, 0.1360
0.6900, 0.0000, 0.3430, 0.1130, 0.2000, 0.1238, 0.5400, 0.4000, 0.4710, 0.1120, 0.2610
0.3400, 0.0000, 0.2630, 0.0870, 0.1970, 0.1200, 0.6300, 0.3000, 0.4249, 0.0960, 0.1130
0.7100, 1.0000, 0.2700, 0.0933, 0.2690, 0.1902, 0.4100, 0.6560, 0.5242, 0.0930, 0.1310
0.4700, 0.0000, 0.2720, 0.0800, 0.2080, 0.1456, 0.3800, 0.6000, 0.4804, 0.0920, 0.1740
0.4100, 0.0000, 0.3380, 0.1233, 0.1870, 0.1270, 0.4500, 0.4160, 0.4318, 0.1000, 0.2570
0.3400, 0.0000, 0.3300, 0.0730, 0.1780, 0.1146, 0.5100, 0.3490, 0.4127, 0.0920, 0.0550
0.5100, 0.0000, 0.2410, 0.0870, 0.2610, 0.1756, 0.6900, 0.4000, 0.4407, 0.0930, 0.0840
0.4300, 0.0000, 0.2130, 0.0790, 0.1410, 0.0788, 0.5300, 0.3000, 0.3829, 0.0900, 0.0420
0.5500, 0.0000, 0.2300, 0.0947, 0.1900, 0.1376, 0.3800, 0.5000, 0.4277, 0.1060, 0.1460
0.5900, 1.0000, 0.2790, 0.1010, 0.2180, 0.1442, 0.3800, 0.6000, 0.5187, 0.0950, 0.2120
0.2700, 1.0000, 0.3360, 0.1100, 0.2460, 0.1566, 0.5700, 0.4000, 0.5088, 0.0890, 0.2330
0.5100, 1.0000, 0.2270, 0.1030, 0.2170, 0.1624, 0.3000, 0.7000, 0.4812, 0.0800, 0.0910
0.4900, 1.0000, 0.2740, 0.0890, 0.1770, 0.1130, 0.3700, 0.5000, 0.4905, 0.0970, 0.1110
0.2700, 0.0000, 0.2260, 0.0710, 0.1160, 0.0434, 0.5600, 0.2000, 0.4419, 0.0790, 0.1520
0.5700, 1.0000, 0.2320, 0.1073, 0.2310, 0.1594, 0.4100, 0.5630, 0.5030, 0.1120, 0.1200
0.3900, 1.0000, 0.2690, 0.0930, 0.1360, 0.0754, 0.4800, 0.3000, 0.4143, 0.0990, 0.0670
0.6200, 1.0000, 0.3460, 0.1200, 0.2150, 0.1292, 0.4300, 0.5000, 0.5366, 0.1230, 0.3100
0.3700, 0.0000, 0.2330, 0.0880, 0.2230, 0.1420, 0.6500, 0.3400, 0.4357, 0.0820, 0.0940
0.4600, 0.0000, 0.2110, 0.0800, 0.2050, 0.1444, 0.4200, 0.5000, 0.4533, 0.0870, 0.1830
0.6800, 1.0000, 0.2350, 0.1010, 0.1620, 0.0854, 0.5900, 0.3000, 0.4477, 0.0910, 0.0660
0.5100, 0.0000, 0.3150, 0.0930, 0.2310, 0.1440, 0.4900, 0.4700, 0.5252, 0.1170, 0.1730
0.4100, 0.0000, 0.2080, 0.0860, 0.2230, 0.1282, 0.8300, 0.3000, 0.4078, 0.0890, 0.0720
0.5300, 0.0000, 0.2650, 0.0970, 0.1930, 0.1224, 0.5800, 0.3000, 0.4143, 0.0990, 0.0490
0.4500, 0.0000, 0.2420, 0.0830, 0.1770, 0.1184, 0.4500, 0.4000, 0.4220, 0.0820, 0.0640
0.3300, 0.0000, 0.1950, 0.0800, 0.1710, 0.0854, 0.7500, 0.2000, 0.3970, 0.0800, 0.0480
0.6000, 1.0000, 0.2820, 0.1120, 0.1850, 0.1138, 0.4200, 0.4000, 0.4984, 0.0930, 0.1780
0.4700, 1.0000, 0.2490, 0.0750, 0.2250, 0.1660, 0.4200, 0.5000, 0.4443, 0.1020, 0.1040
0.6000, 1.0000, 0.2490, 0.0997, 0.1620, 0.1066, 0.4300, 0.3770, 0.4127, 0.0950, 0.1320
0.3600, 0.0000, 0.3000, 0.0950, 0.2010, 0.1252, 0.4200, 0.4790, 0.5130, 0.0850, 0.2200
0.3600, 0.0000, 0.1960, 0.0710, 0.2500, 0.1332, 0.9700, 0.3000, 0.4595, 0.0920, 0.0570

Posted in Machine Learning, Scikit | Leave a comment

Using Trimmed Kernel Ridge Regression to Approximate Support Vector Regression With C#

Posted on June 23, 2026 by jamesdmccaffrey

Bottom line: I refactored a C# implementation of trimmed kernel ridge regression to approximate support vector regression. The old version had ugly repeated code in two different SGD training phases, so I created a helper method to remove the code repeated-ness.

As is often the case with interesting topics, explaining what the problem is, takes longer than explaining the solution. So bear with me. The goal of a machine learning regression problem is to predict a single numeric value. For example, a bank might want to predict a maximum loan amount based on applicant age, sex, annual income, debt, and so on.

Common regression techniques include linear regression, quadratic regression, nearest neighbors regression, kernel ridge regression, Gaussian process regression, kernel support vector regression, neural network regression, random forest regression, and gradient boost regression. Each technique has dozens of variations, and each technique has pros and cons.

Two closely related machine learning regression techniques are kernel ridge regression (KRR) and support vector regression (SVR). Both techniques use a kernel function (usually the radial basis function, RBF) to compare two data items for similarity. Both techniques must store training data in order to make predictions. KRR must store all training items, while SVR eliminates some of the items during training, leaving just the “support vectors” that need to be stored. So, SVR uses less memory than KRR.

On the other hand, the loss function used by SVR is not differentiable, so SVR cannot be trained using stochastic gradient descent, and therefore, SVR does not scale well to very large datasets. So, KRR is easier to train than SVR.

Note: In addition to the kernel support vector regression technique discussed in this blog post, there is a linear support vector regression technique which is absolutely useless in practice.

One morning before work, I got the idea of combining KRR and SVR. Briefly, I train a KRR model as normal using all data. Then I identify training data items that are predicted “too well” and remove them, leaving just pseudo support vectors. The somewhat counterintuitive SVR idea is that items that are predicted too well don’t help model accuracy very much. Then I retrain a new KRR model using only the reduced training data. This gives a trimmed/sparse KRR model that approximates an SVR model.

In my mind, this technique gives the advantages of KRR (ability to handle very large datasets via SGD training) and the advantages of SVR (fewer stored model items and weights than KRR). I first implemented the idea using the scikit KernelRidge module and the idea worked well. Then I implemented the idea using C#, just to see if I could — using C# I had to implement a lot of code from scratch that is built-in to Python and scikit. The C# code worked but it was clunky so I refactored the clunky C# code to make it more rational. But the refactored C# code was awkward because the SGD part was repeated in the preliminary training phase and the second/final training phase. I refactored the code to avoid the duplicate code.

Let me point out that the idea of trimming a KRR model to approximate an SVR model is fairly obvious, and so I’m reasonably sure other people have used the idea, but I could find no examples or discussion on the Internet.

The output of the new, re-re-factored C# demo is:

Begin kernel ridge regression trim approximation
 to SVR using C#

Loading train (200) and test (40) data
Done

First three X predictors:
  -0.1660   0.4406  -0.9998  -0.3953  -0.7065
   0.0776  -0.1616   0.3704  -0.5911   0.7562
  -0.9452   0.3409  -0.1654   0.1174  -0.7192

First three target y:
  0.4840
  0.1568
  0.8054

Creating KRR model
Setting RBF gamma = 0.30
Done

Setting lrnRate = 0.0500
Setting maxEpochs = 5000
Setting alpha decay =  0.00001
Setting auto-exit tol = 0.0010
Setting trim epsilon = 0.0035

Training model
epoch =      0 MSE = 0.0181 acc = 0.1700
Auto-exit at epoch 723
epoch =      0 MSE = 0.0377 acc = 0.1009
Auto-exit at epoch 707
Done

Final model weights:
 -0.7406  -0.4260  -0.3825 . . . 
. . .      0.3299   0.8800

Evaluating trimmed KRR model
Number model item/weights = 109

Train acc = 0.9817
Test acc = 0.9500

Train MSE = 0.0001
Test MSE = 0.0003

End KRR trim/prune demo

The demo data is synthetic. There are 200 training items and 40 test items. The demo data was generated by a neural network with random weights and biases. The data is normalized, which is strongly recommended for KRR and SVR because items are compared mathematically and you don’t want a column with huge magnitudes to dominate. Both KRR and SVR can handle non-numeric data, but that’s another story.

The key calling statements are:

double gamma = 0.30;
KRR krr = new KRR(gamma);

double lrnRate = 0.05;
int maxEpochs = 5000;
double alpha = 1.0e-5;  // wt decay regularization
double exitTol = 0.001;  // auto-exit
double epsilon = 0.0035; // trimming
// larger epsilon: fewer wts retained
// smaller epsilon: more wts retained
krr.Train(trainX, trainY, lrnRate, maxEpochs, alpha,
  exitTol, epsilon);

The large number of parameters is a significant weakness of KRR (and SVR too).

After training on 200 data items, those items that were predicted with a small error of less than 0.0035 were removed. There were 91 such well-predicted items, leaving a reduced training dataset of 109 items. These would be called the support vectors in SVR. After training a new KRR model on the reduced dataset, the trimmed/sparse KRR model scored 98.17% accuracy on the reduced training data (107 out of 109 correct) and 95.00% accuracy on the test data (38 out of 40 correct). Very nice.

To validate my C# trimmed KRR approximation to SVR, I ran the data through the scikit SVR module. A grid-search optimized scikit SVR model created 90 support vectors (better than the trimmed KRR 109 items), with 96.50% accuracy on the training data (not quite as good as the C# trimmed KRR model), and 95.00% accuracy on the test data (identical to the trimmed KRR model). In short, the C# trimmed KRR model and the SVR model are very similar. The large number of hyperparameters involved with KRR and SVR makes an exact comparison impossible in practice.

An interesting experiment.

It’s common for me to refactor the software systems I create many times. It’s rarely possible to a get a non-trivial system completely correct on the first effort, and so a system is usually a collection of software sequels, so to speak, where each sequel is a bit better than its predecessor.

I’m a big fan of science fiction movies. There have been many sci fi sequels. Star Wars (1977) and Star Wars: The Empire Strikes Back (1980). Alien (1979) and Aliens (1986). And so on. But most movie sequels are worse than their predecessor, however there are exceptions where the sequel is better than the original.

Left: In “Pitch Black” (2000), a group of people crash land on a planet that is infested by dangerous creatures. The survivors are picked off one by one. A classic movie theme. My grade = C+.

Right: In the sequel, “The Chronicles of Riddick” (2004), an alien race, called the Necromongers, are religious fanatics. The story is wildly creative, with very good special effects, great set design, and OK acting. The movie is not well-liked by critics, typically about 1 star out of 4. But I like it a lot and give it my personal B+ grade.

The (re-re-factored) C# trimmed KRR demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols. (My blog editor chokes on symbols).

using System;
using System.IO;
using System.Collections.Generic;

namespace KernelRidgeRegressionTrim
{
  internal class KernelRidgeRegressionProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin kernel ridge " +
        "regression trim approximation to SVR using C# ");

      Console.WriteLine("\nLoading train (200) and" +
        " test (40) data ");
      string trainFile =
        "..\\..\\..\\Data\\synthetic_train_200.txt";
      double[][] trainX =
        Utils.MatLoad(trainFile,
        new int[] { 0, 1, 2, 3, 4 }, ',', "#");
      double[] trainY =
        Utils.MatToVec(Utils.MatLoad(trainFile,
        new int[] { 5 }, ',', "#"));

      string testFile =
        "..\\..\\..\\Data\\synthetic_test_40.txt";
      double[][] testX =
        Utils.MatLoad(testFile,
        new int[] { 0, 1, 2, 3, 4 }, ',', "#");
      double[] testY =
        Utils.MatToVec(Utils.MatLoad(testFile,
        new int[] { 5 }, ',', "#"));
      Console.WriteLine("Done ");

      Console.WriteLine("\nFirst three X predictors: ");
      for (int i = 0; i "lt" 3; ++i)
        Utils.VecShow(trainX[i], 4, 9);
      Console.WriteLine("\nFirst three target y: ");
      for (int i = 0; i "lt" 3; ++i)
        Console.WriteLine(trainY[i].ToString("F4").
          PadLeft(8));

      Console.WriteLine("\nCreating KRR model ");
      double gamma = 0.3;    // RBF param
      Console.WriteLine("Setting RBF gamma = " +
        gamma.ToString("F2"));
      KRR krr = new KRR(gamma);
      Console.WriteLine("Done ");

      double lrnRate = 0.05;
      int maxEpochs = 5000;
      double alpha = 1.0e-5;  // wt decay regularization
      double exitTol = 0.001;  // auto-exit
      double epsilon = 0.0035; // trimming
      // larger epsilon: fewer wts retained
      // smaller epsilon: more wts retained
      Console.WriteLine("\nSetting lrnRate = " +
        lrnRate.ToString("F4"));
      Console.WriteLine("Setting maxEpochs = " + maxEpochs);
      Console.WriteLine("Setting alpha decay =  " +
        alpha.ToString("F5"));
      Console.WriteLine("Setting auto-exit tol = " +
        exitTol.ToString("F4"));
      Console.WriteLine("Setting trim epsilon = " +
        epsilon.ToString("F4"));
      Console.WriteLine("\nTraining model ");
      krr.Train(trainX, trainY, lrnRate, maxEpochs, alpha,
        exitTol, epsilon);
      Console.WriteLine("Done ");
      Console.WriteLine("\nFinal model weights: ");
      Utils.VecShow(krr.wts, 4, 9);

      Console.WriteLine("\nEvaluating trimmed KRR model ");
      Console.WriteLine("Number model item/weights = " +
        krr.wts.Length);
      double trainAcc = 
        krr.Accuracy(krr.trainX, krr.trainY, 0.10);
      double testAcc = krr.Accuracy(testX, testY, 0.10);

      Console.WriteLine("\nTrain acc = " +
        trainAcc.ToString("F4"));
      Console.WriteLine("Test acc = " +
        testAcc.ToString("F4"));

      double trainMSE = krr.MSE(krr.trainX, krr.trainY);
      double testMSE = krr.MSE(testX, testY);

      Console.WriteLine("\nTrain MSE = " +
        trainMSE.ToString("F4"));
      Console.WriteLine("Test MSE = " +
        testMSE.ToString("F4"));

      Console.WriteLine("\nEnd KRR trim/prune demo ");
      Console.ReadLine();
    } // Main()

  } // class Program

  /*
  Creating scikit SVR model
  Setting gamma = 0.1000
  Setting C = 10.0000
  Setting epsilon = 0.0100

  Number support vectors = 90
  Accuracy (within 0.10) train = 0.9650
  Accuracy (within 0.10) test = 0.9500
  MSE train = 0.0001
  MSE test = 0.0002 
  */

  // ========================================================

  public class KRR
  {
    public double gamma;  // for RBF kernel
    public double[][] trainX;  // need for prediction
    public double[] trainY; // not used this version
    public double[] wts;  // one per trainX item
    public Random rnd; // for SGD

    // ------------------------------------------------------

    public KRR(double gamma, int seed = 1)
    {
      this.gamma = gamma;
      this.trainX = new double[0][];
      this.trainY = new double[0];
      this.wts = new double[0];
      this.rnd = new Random(seed);  // shuffle train order
    } // ctor

    // ------------------------------------------------------

    private void Shuffle(int[] indices)
    {
      // Fisher-Yates for SGD
      for (int i = 0; i "lt" indices.Length; ++i)
      {
        int ri = this.rnd.Next(i, indices.Length);
        int tmp = indices[i];
        indices[i] = indices[ri];
        indices[ri] = tmp;
      }
    } // Shuffle

    // ------------------------------------------------------

    private void TrainSGD(double[][] trainX, double[] trainY,
     double lrnRate, int maxEpochs, double alpha,
     double exitTol, double epsilon)
    {
      // helper for Train()
      // assign to this.trainX, this.trainY, this.wts

      // trainx, trainY by ref:
      this.trainX = trainX;
      this.trainY = trainY;

      int freq = maxEpochs / 5;  // when to show progress

      // 1. init weights
      this.wts = new double[trainX.Length];
      double lo = -0.10; double hi = 0.10;
      for (int i = 0; i "lt" this.wts.Length; ++i)
        this.wts[i] = (hi - lo) *
          this.rnd.NextDouble() + lo;

      // 2. set up indices for shuffling
      int[] indices = new int[trainX.Length];
      for (int i = 0; i "lt" indices.Length; ++i)
        indices[i] = i;

      // 3. set up prev weights for auto-exit
      double[] prevWeights = new double[trainX.Length];
      for (int j = 0; j "lt" trainX.Length; ++j)
        prevWeights[j] = this.wts[j];

      // 4. SGD prelim train
      for (int epoch = 0; epoch "lt" maxEpochs; ++epoch)
      {
        Shuffle(indices);
        for (int i = 0; i "lt" trainX.Length; ++i)
        {
          int idx = indices[i];
          double[] x = trainX[idx];
          double predY = this.Predict(x);
          double actualY = trainY[idx];

          // update wt assoc with x
          this.wts[idx] -= lrnRate * (predY - actualY);
        } // each item

        if (epoch % freq == 0)
        {
          double rmse = this.MSE(trainX, trainY);
          double acc = this.Accuracy(trainX, trainY, 0.10);
          string s1 = "epoch = " +
            epoch.ToString().PadLeft(6);
          string s2 = " MSE = " +
            rmse.ToString("F4");
          string s3 = " acc = " + acc.ToString("F4");
          Console.WriteLine(s1 + s2 + s3);
        }
                
        // if max_change_in_wts / max_weights "lt" tol
        int numWts = this.wts.Length;
        double[] weightDeltas = new double[numWts];
        for (int j = 0; j "lt" numWts; ++j)
          weightDeltas[j] = 
            Math.Abs(prevWeights[j] - this.wts[j]);
        double maxChange = weightDeltas[0];
        for (int j = 0; j "lt" numWts; ++j)
          if (weightDeltas[j] "gt" maxChange)
            maxChange = weightDeltas[j];

        double maxWeightMag = Math.Abs(this.wts[0]);
        for (int j = 0; j "lt" numWts; ++j)
          if (Math.Abs(this.wts[j]) "gt" maxWeightMag)
            maxWeightMag = Math.Abs(this.wts[j]);

        if (maxWeightMag != 0.0 &&
          (maxChange / maxWeightMag) "lt" exitTol)
        {
          Console.WriteLine("Auto-exit at epoch " + epoch);
          break;
        }

        // an auto-exit didn't happen
        for (int j = 0; j "lt" numWts; ++j)
          prevWeights[j] = this.wts[j];
      
      } // each epoch

      // 5. apply one final wt decay regularization
      // as opposed to small decay after every epoch
      for (int j = 0; j "lt" this.wts.Length; ++j)
        this.wts[j] *= (1.0 - alpha);
    }

    // ------------------------------------------------------

    public void Train(double[][] trainX, double[] trainY,
     double lrnRate, int maxEpochs, double alpha,
     double exitTol, double epsilon)
    {
      // 1. preliminary train
      this.TrainSGD(trainX, trainY, lrnRate,
        maxEpochs, alpha, exitTol, epsilon);

      // 2. construct trimmed data
      int[] isSupportVec = new int[trainX.Length]; 
      for (int i = 0; i "lt" trainX.Length; ++i)
      {
        if (Math.Abs(this.Predict(trainX[i]) - 
          trainY[i]) "gt" epsilon)
          isSupportVec[i] = 1; // just OK prediction
      }
      List"lt"double[]"gt" tmpTrainX = 
        new List"lt"double[]"gt"();
      List"lt"double"gt" tmpTrainY = 
        new List"lt"double"gt"();
      for (int i = 0; i "lt" trainX.Length; ++i)
      {
        if (isSupportVec[i] == 1)
        {
          tmpTrainX.Add(trainX[i]);
          tmpTrainY.Add(trainY[i]);
        }
      }
      double[][] newTrainX = tmpTrainX.ToArray();
      double[] newTrainY = tmpTrainY.ToArray();
      
      // 3. re-train using new small dataset
      this.TrainSGD(newTrainX, newTrainY, lrnRate,
        maxEpochs, alpha, exitTol, epsilon);
    }

    // ------------------------------------------------------

    private double Rbf(double[] v1, double[] v2)
    {
      // the gamma version aot len_scale version
      int dim = v1.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" dim; ++i)
      {
        sum += (v1[i] - v2[i]) * (v1[i] - v2[i]);
      }
      return Math.Exp(-1 * this.gamma * sum);
    }

    // ------------------------------------------------------

    public double Predict(double[] x)
    {
      int N = this.trainX.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" N; ++i)
      {
        double[] xx = this.trainX[i];
        double k = this.Rbf(x, xx);
        sum += this.wts[i] * k;
      }
      return sum;
    }

    // ------------------------------------------------------

    public double Accuracy(double[][] dataX,
      double[] dataY, double pctClose)
    {
      int numCorrect = 0; int numWrong = 0;
      int n = dataX.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        double[] x = dataX[i];
        double actualY = dataY[i];
        double predY = this.Predict(x);
        if (Math.Abs(actualY - predY) "lt"
          Math.Abs(actualY * pctClose))
          ++numCorrect;
        else
          ++numWrong;
      }
      return (numCorrect * 1.0) / n;
    }

    // ------------------------------------------------------

    public double MSE(double[][] dataX, double[] dataY)
    {
      double sum = 0.0;
      int n = dataX.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        double[] x = dataX[i];
        double actualY = dataY[i];
        double predY = this.Predict(x);
        sum += (actualY - predY) * (actualY - predY);
      }
      return sum / n;
    }
  } // class KRR

  // ========================================================

  public class Utils
  {
    // ------------------------------------------------------

    public static double[][] MatLoad(string fn,
      int[] usecols, char sep, string comment)
    {
      List"lt"double[]"gt" result = 
        new List"lt"double[]"gt"();
      string line = "";
      FileStream ifs = new FileStream(fn, FileMode.Open);
      StreamReader sr = new StreamReader(ifs);
      while ((line = sr.ReadLine()) != null)
      {
        if (line.StartsWith(comment) == true)
          continue;
        string[] tokens = line.Split(sep);
        List"lt"double"gt" lst = new List"lt"double"gt"();
        for (int j = 0; j "lt" usecols.Length; ++j)
          lst.Add(double.Parse(tokens[usecols[j]]));
        double[] row = lst.ToArray();
        result.Add(row);
      }
      sr.Close(); ifs.Close();
      return result.ToArray();
    }

    // ------------------------------------------------------

    public static double[] MatToVec(double[][] mat)
    {
      int nRows = mat.Length;
      int nCols = mat[0].Length;
      double[] result = new double[nRows * nCols];
      int k = 0;
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          result[k++] = mat[i][j];
      return result;
    }

    // ------------------------------------------------------

    public static void MatShow(double[][] m, int dec,
      int wid)
    {
      int nRows = m.Length; int nCols = m[0].Length;
      double small = 1.0 / Math.Pow(10, dec);
      for (int i = 0; i "lt" nRows; ++i)
      {
        for (int j = 0; j "lt" nCols; ++j)
        {
          double v = m[i][j];
          if (Math.Abs(v) "lt" small) v = 0.0;
          Console.Write(v.ToString("F" + dec).
            PadLeft(wid));
        }
        Console.WriteLine("");
      }
    }

    // ------------------------------------------------------

    public static void VecShow(double[] vec, int dec,
      int wid)
    {
      for (int i = 0; i "lt" vec.Length; ++i)
        Console.Write(vec[i].ToString("F" + dec).
        PadLeft(wid));
      Console.WriteLine("");
    }

    // ------------------------------------------------------

  } // class Utils

  // ========================================================

} // ns

Training data:

# synthetic_train_200.txt
#
-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.0776, -0.1616,  0.3704, -0.5911,  0.7562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
 0.9365, -0.3732,  0.3846,  0.7528,  0.7892,  0.1345
-0.8299, -0.9219, -0.6603,  0.7563, -0.8033,  0.7955
 0.0663,  0.3838, -0.3690,  0.3730,  0.6693,  0.3206
-0.9634,  0.5003,  0.9777,  0.4963, -0.4391,  0.7377
-0.1042,  0.8172, -0.4128, -0.4244, -0.7399,  0.4801
-0.9613,  0.3577, -0.5767, -0.4689, -0.0169,  0.6861
-0.7065,  0.1786,  0.3995, -0.7953, -0.1719,  0.5569
 0.3888, -0.1716, -0.9001,  0.0718,  0.3276,  0.2500
 0.1731,  0.8068, -0.7251, -0.7214,  0.6148,  0.3297
-0.2046, -0.6693,  0.8550, -0.3045,  0.5016,  0.2129
 0.2473,  0.5019, -0.3022, -0.4601,  0.7918,  0.2613
-0.1438,  0.9297,  0.3269,  0.2434, -0.7705,  0.5171
 0.1568, -0.1837, -0.5259,  0.8068,  0.1474,  0.3307
-0.9943,  0.2343, -0.3467,  0.0541,  0.7719,  0.5581
 0.2467, -0.9684,  0.8589,  0.3818,  0.9946,  0.1092
-0.6553, -0.7257,  0.8652,  0.3936, -0.8680,  0.7018
 0.8460,  0.4230, -0.7515, -0.9602, -0.9476,  0.1996
-0.9434, -0.5076,  0.7201,  0.0777,  0.1056,  0.5664
 0.9392,  0.1221, -0.9627,  0.6013, -0.5341,  0.1533
 0.6142, -0.2243,  0.7271,  0.4942,  0.1125,  0.1661
 0.4260,  0.1194, -0.9749, -0.8561,  0.9346,  0.2230
 0.1362, -0.5934, -0.4953,  0.4877, -0.6091,  0.3810
 0.6937, -0.5203, -0.0125,  0.2399,  0.6580,  0.1460
-0.6864, -0.9628, -0.8600, -0.0273,  0.2127,  0.5387
 0.9772,  0.1595, -0.2397,  0.1019,  0.4907,  0.1611
 0.3385, -0.4702, -0.8673, -0.2598,  0.2594,  0.2270
-0.8669, -0.4794,  0.6095, -0.6131,  0.2789,  0.4700
 0.0493,  0.8496, -0.4734, -0.8681,  0.4701,  0.3516
 0.8639, -0.9721, -0.5313,  0.2336,  0.8980,  0.1412
 0.9004,  0.1133,  0.8312,  0.2831, -0.2200,  0.1782
 0.0991,  0.8524,  0.8375, -0.2102,  0.9265,  0.2150
-0.6521, -0.7473, -0.7298,  0.0113, -0.9570,  0.7422
 0.6190, -0.3105,  0.8802,  0.1640,  0.7577,  0.1056
 0.6895,  0.8108, -0.0802,  0.0927,  0.5972,  0.2214
 0.1982, -0.9689,  0.1870, -0.1326,  0.6147,  0.1310
-0.3695,  0.7858,  0.1557, -0.6320,  0.5759,  0.3773
-0.1596,  0.3581,  0.8372, -0.9992,  0.9535,  0.2071
-0.2468,  0.9476,  0.2094,  0.6577,  0.1494,  0.4132
 0.1737,  0.5000,  0.7166,  0.5102,  0.3961,  0.2611
 0.7290, -0.3546,  0.3416, -0.0983, -0.2358,  0.1332
-0.3652,  0.2438, -0.1395,  0.9476,  0.3556,  0.4170
-0.6029, -0.1466, -0.3133,  0.5953,  0.7600,  0.4334
-0.4596, -0.4953,  0.7098,  0.0554,  0.6043,  0.2775
 0.1450,  0.4663,  0.0380,  0.5418,  0.1377,  0.2931
-0.8636, -0.2442, -0.8407,  0.9656, -0.6368,  0.7429
 0.6237,  0.7499,  0.3768,  0.1390, -0.6781,  0.2185
-0.5499,  0.1850, -0.3755,  0.8326,  0.8193,  0.4399
-0.4858, -0.7782, -0.6141, -0.0008,  0.4572,  0.4197
 0.7033, -0.1683,  0.2334, -0.5327, -0.7961,  0.1776
 0.0317, -0.0457, -0.6947,  0.2436,  0.0880,  0.3345
 0.5031, -0.5559,  0.0387,  0.5706, -0.9553,  0.3107
-0.3513,  0.7458,  0.6894,  0.0769,  0.7332,  0.3170
 0.2205,  0.5992, -0.9309,  0.5405,  0.4635,  0.3532
-0.4806, -0.4859,  0.2646, -0.3094,  0.5932,  0.3202
 0.9809, -0.3995, -0.7140,  0.8026,  0.0831,  0.1600
 0.9495,  0.2732,  0.9878,  0.0921,  0.0529,  0.1289
-0.9476, -0.6792,  0.4913, -0.9392, -0.2669,  0.5966
 0.7247,  0.3854,  0.3819, -0.6227, -0.1162,  0.1550
-0.5922, -0.5045, -0.4757,  0.5003, -0.0860,  0.5863
-0.8861,  0.0170, -0.5761,  0.5972, -0.4053,  0.7301
 0.6877, -0.2380,  0.4997,  0.0223,  0.0819,  0.1404
 0.9189,  0.6079, -0.9354,  0.4188, -0.0700,  0.1907
-0.1428, -0.7820,  0.2676,  0.6059,  0.3936,  0.2790
 0.5324, -0.3151,  0.6917, -0.1425,  0.6480,  0.1071
-0.8432, -0.9633, -0.8666, -0.0828, -0.7733,  0.7784
-0.9444,  0.5097, -0.2103,  0.4939, -0.0952,  0.6787
-0.0520,  0.6063, -0.1952,  0.8094, -0.9259,  0.4836
 0.5477, -0.7487,  0.2370, -0.9793,  0.0773,  0.1241
 0.2450,  0.8116,  0.9799,  0.4222,  0.4636,  0.2355
 0.8186, -0.1983, -0.5003, -0.6531, -0.7611,  0.1511
-0.4714,  0.6382, -0.3788,  0.9648, -0.4667,  0.5950
 0.0673, -0.3711,  0.8215, -0.2669, -0.1328,  0.2677
-0.9381,  0.4338,  0.7820, -0.9454,  0.0441,  0.5518
-0.3480,  0.7190,  0.1170,  0.3805, -0.0943,  0.4724
-0.9813,  0.1535, -0.3771,  0.0345,  0.8328,  0.5438
-0.1471, -0.5052, -0.2574,  0.8637,  0.8737,  0.3042
-0.5454, -0.3712, -0.6505,  0.2142, -0.1728,  0.5783
 0.6327, -0.6297,  0.4038, -0.5193,  0.1484,  0.1153
-0.5424,  0.3282, -0.0055,  0.0380, -0.6506,  0.6613
 0.1414,  0.9935,  0.6337,  0.1887,  0.9520,  0.2540
-0.9351, -0.8128, -0.8693, -0.0965, -0.2491,  0.7353
 0.9507, -0.6640,  0.9456,  0.5349,  0.6485,  0.1059
-0.0462, -0.9737, -0.2940, -0.0159,  0.4602,  0.2606
-0.0627, -0.0852, -0.7247, -0.9782,  0.5166,  0.2977
 0.0478,  0.5098, -0.0723, -0.7504, -0.3750,  0.3335
 0.0090,  0.3477,  0.5403, -0.7393, -0.9542,  0.4415
-0.9748,  0.3449,  0.3736, -0.1015,  0.8296,  0.4358
 0.2887, -0.9895, -0.0311,  0.7186,  0.6608,  0.2057
 0.1570, -0.4518,  0.1211,  0.3435, -0.2951,  0.3244
 0.7117, -0.6099,  0.4946, -0.4208,  0.5476,  0.1096
-0.2929, -0.5726,  0.5346, -0.3827,  0.4665,  0.2465
 0.4889, -0.5572, -0.5718, -0.6021, -0.7150,  0.2163
-0.7782,  0.3491,  0.5996, -0.8389, -0.5366,  0.6516
-0.5847,  0.8347,  0.4226,  0.1078, -0.3910,  0.6134
 0.8469,  0.4121, -0.0439, -0.7476,  0.9521,  0.1571
-0.6803, -0.5948, -0.1376, -0.1916, -0.7065,  0.7156
 0.2878,  0.5086, -0.5785,  0.2019,  0.4979,  0.2980
 0.2764,  0.1943, -0.4090,  0.4632,  0.8906,  0.2960
-0.8877,  0.6705, -0.6155, -0.2098, -0.3998,  0.7107
-0.8398,  0.8093, -0.2597,  0.0614, -0.0118,  0.6502
-0.8476,  0.0158, -0.4769, -0.2859, -0.7839,  0.7715
 0.5751, -0.7868,  0.9714, -0.6457,  0.1448,  0.1175
 0.4802, -0.7001,  0.1022, -0.5668,  0.5184,  0.1090
 0.4458, -0.6469,  0.7239, -0.9604,  0.7205,  0.0779
 0.5175,  0.4339,  0.9747, -0.4438, -0.9924,  0.2879
 0.8678,  0.7158,  0.4577,  0.0334,  0.4139,  0.1678
 0.5406,  0.5012,  0.2264, -0.1963,  0.3946,  0.2088
-0.9938,  0.5498,  0.7928, -0.5214, -0.7585,  0.7687
 0.7661,  0.0863, -0.4266, -0.7233, -0.4197,  0.1466
 0.2277, -0.3517, -0.0853, -0.1118,  0.6563,  0.1767
 0.3499, -0.5570, -0.0655, -0.3705,  0.2537,  0.1632
 0.7547, -0.1046,  0.5689, -0.0861,  0.3125,  0.1257
 0.8186,  0.2110,  0.5335,  0.0094, -0.0039,  0.1391
 0.6858, -0.8644,  0.1465,  0.8855,  0.0357,  0.1845
-0.4967,  0.4015,  0.0805,  0.8977,  0.2487,  0.4663
 0.6760, -0.9841,  0.9787, -0.8446, -0.3557,  0.1509
-0.1203, -0.4885,  0.6054, -0.0443, -0.7313,  0.4854
 0.8557,  0.7919, -0.0169,  0.7134, -0.1628,  0.2002
 0.0115, -0.6209,  0.9300, -0.4116, -0.7931,  0.4052
-0.7114, -0.9718,  0.4319,  0.1290,  0.5892,  0.3661
 0.3915,  0.5557, -0.1870,  0.2955, -0.6404,  0.2954
-0.3564, -0.6548, -0.1827, -0.5172, -0.1862,  0.4622
 0.2392, -0.4959,  0.5857, -0.1341, -0.2850,  0.2470
-0.3394,  0.3947, -0.4627,  0.6166, -0.4094,  0.5325
 0.7107,  0.7768, -0.6312,  0.1707,  0.7964,  0.2757
-0.1078,  0.8437, -0.4420,  0.2177,  0.3649,  0.4028
-0.3139,  0.5595, -0.6505, -0.3161, -0.7108,  0.5546
 0.4335,  0.3986,  0.3770, -0.4932,  0.3847,  0.1810
-0.2562, -0.2894, -0.8847,  0.2633,  0.4146,  0.4036
 0.2272,  0.2966, -0.6601, -0.7011,  0.0284,  0.2778
-0.0743, -0.1421, -0.0054, -0.6770, -0.3151,  0.3597
-0.4762,  0.6891,  0.6007, -0.1467,  0.2140,  0.4266
-0.4061,  0.7193,  0.3432,  0.2669, -0.7505,  0.6147
-0.0588,  0.9731,  0.8966,  0.2902, -0.6966,  0.4955
-0.0627, -0.1439,  0.1985,  0.6999,  0.5022,  0.3077
 0.1587,  0.8494, -0.8705,  0.9827, -0.8940,  0.4263
-0.7850,  0.2473, -0.9040, -0.4308, -0.8779,  0.7199
 0.4070,  0.3369, -0.2428, -0.6236,  0.4940,  0.2215
-0.0242,  0.0513, -0.9430,  0.2885, -0.2987,  0.3947
-0.5416, -0.1322, -0.2351, -0.0604,  0.9590,  0.3683
 0.1055,  0.7783, -0.2901, -0.5090,  0.8220,  0.2984
-0.9129,  0.9015,  0.1128, -0.2473,  0.9901,  0.4776
-0.9378,  0.1424, -0.6391,  0.2619,  0.9618,  0.5368
 0.7498, -0.0963,  0.4169,  0.5549, -0.0103,  0.1614
-0.2612, -0.7156,  0.4538, -0.0460, -0.1022,  0.3717
 0.7720,  0.0552, -0.1818, -0.4622, -0.8560,  0.1685
-0.4177,  0.0070,  0.9319, -0.7812,  0.3461,  0.3052
-0.0001,  0.5542, -0.7128, -0.8336, -0.2016,  0.3803
 0.5356, -0.4194, -0.5662, -0.9666, -0.2027,  0.1776
-0.2378,  0.3187, -0.8582, -0.6948, -0.9668,  0.5474
-0.1947, -0.3579,  0.1158,  0.9869,  0.6690,  0.2992
 0.3992,  0.8365, -0.9205, -0.8593, -0.0520,  0.3154
-0.0209,  0.0793,  0.7905, -0.1067,  0.7541,  0.1864
-0.4928, -0.4524, -0.3433,  0.0951, -0.5597,  0.6261
-0.8118,  0.7404, -0.5263, -0.2280,  0.1431,  0.6349
 0.0516, -0.8480,  0.7483,  0.9023,  0.6250,  0.1959
-0.3212,  0.1093,  0.9488, -0.3766,  0.3376,  0.2735
-0.3481,  0.5490, -0.3484,  0.7797,  0.5034,  0.4379
-0.5785, -0.9170, -0.3563, -0.9258,  0.3877,  0.4121
 0.3407, -0.1391,  0.5356,  0.0720, -0.9203,  0.3458
-0.3287, -0.8954,  0.2102,  0.0241,  0.2349,  0.3247
-0.1353,  0.6954, -0.0919, -0.9692,  0.7461,  0.3338
 0.9036, -0.8982, -0.5299, -0.8733, -0.1567,  0.1187
 0.7277, -0.8368, -0.0538, -0.7489,  0.5458,  0.0830
 0.9049,  0.8878,  0.2279,  0.9470, -0.3103,  0.2194
 0.7957, -0.1308, -0.5284,  0.8817,  0.3684,  0.2172
 0.4647, -0.4931,  0.2010,  0.6292, -0.8918,  0.3371
-0.7390,  0.6849,  0.2367,  0.0626, -0.5034,  0.7039
-0.1567, -0.8711,  0.7940, -0.5932,  0.6525,  0.1710
 0.7635, -0.0265,  0.1969,  0.0545,  0.2496,  0.1445
 0.7675,  0.1354, -0.7698, -0.5460,  0.1920,  0.1728
-0.5211, -0.7372, -0.6763,  0.6897,  0.2044,  0.5217
 0.1913,  0.1980,  0.2314, -0.8816,  0.5006,  0.1998
 0.8964,  0.0694, -0.6149,  0.5059, -0.9854,  0.1825
 0.1767,  0.7104,  0.2093,  0.6452,  0.7590,  0.2832
-0.3580, -0.7541,  0.4426, -0.1193, -0.7465,  0.5657
-0.5996,  0.5766, -0.9758, -0.3933, -0.9572,  0.6800
 0.9950,  0.1641, -0.4132,  0.8579,  0.0142,  0.2003
-0.4717, -0.3894, -0.2567, -0.5111,  0.1691,  0.4266
 0.3917, -0.8561,  0.9422,  0.5061,  0.6123,  0.1212
-0.0366, -0.1087,  0.3449, -0.1025,  0.4086,  0.2475
 0.3633,  0.3943,  0.2372, -0.6980,  0.5216,  0.1925
-0.5325, -0.6466, -0.2178, -0.3589,  0.6310,  0.3568
 0.2271,  0.5200, -0.1447, -0.8011, -0.7699,  0.3128
 0.6415,  0.1993,  0.3777, -0.0178, -0.8237,  0.2181
-0.5298, -0.0768, -0.6028, -0.9490,  0.4588,  0.4356
 0.6870, -0.1431,  0.7294,  0.3141,  0.1621,  0.1632
-0.5985,  0.0591,  0.7889, -0.3900,  0.7419,  0.2945
 0.3661,  0.7984, -0.8486,  0.7572, -0.6183,  0.3449
 0.6995,  0.3342, -0.3113, -0.6972,  0.2707,  0.1712
 0.2565,  0.9126,  0.1798, -0.6043, -0.1413,  0.2893
-0.3265,  0.9839, -0.2395,  0.9854,  0.0376,  0.4770
 0.2690, -0.1722,  0.9818,  0.8599, -0.7015,  0.3954
-0.2102, -0.0768,  0.1219,  0.5607, -0.0256,  0.3949
 0.8216, -0.9555,  0.6422, -0.6231,  0.3715,  0.0801
-0.2896,  0.9484, -0.7545, -0.6249,  0.7789,  0.4370
-0.9985, -0.5448, -0.7092, -0.5931,  0.7926,  0.5402

Test data:

# synthetic_test_40.txt
#
 0.7462,  0.4006, -0.0590,  0.6543, -0.0083,  0.1935
 0.8495, -0.2260, -0.0142, -0.4911,  0.7699,  0.1078
-0.2335, -0.4049,  0.4352, -0.6183, -0.7636,  0.5088
 0.1810, -0.5142,  0.2465,  0.2767, -0.3449,  0.3136
-0.8650,  0.7611, -0.0801,  0.5277, -0.4922,  0.7140
-0.2358, -0.7466, -0.5115, -0.8413, -0.3943,  0.4533
 0.4834,  0.2300,  0.3448, -0.9832,  0.3568,  0.1360
-0.6502, -0.6300,  0.6885,  0.9652,  0.8275,  0.3046
-0.3053,  0.5604,  0.0929,  0.6329, -0.0325,  0.4756
-0.7995,  0.0740, -0.2680,  0.2086,  0.9176,  0.4565
-0.2144, -0.2141,  0.5813,  0.2902, -0.2122,  0.4119
-0.7278, -0.0987, -0.3312, -0.5641,  0.8515,  0.4438
 0.3793,  0.1976,  0.4933,  0.0839,  0.4011,  0.1905
-0.8568,  0.9573, -0.5272,  0.3212, -0.8207,  0.7415
-0.5785,  0.0056, -0.7901, -0.2223,  0.0760,  0.5551
 0.0735, -0.2188,  0.3925,  0.3570,  0.3746,  0.2191
 0.1230, -0.2838,  0.2262,  0.8715,  0.1938,  0.2878
 0.4792, -0.9248,  0.5295,  0.0366, -0.9894,  0.3149
-0.4456,  0.0697,  0.5359, -0.8938,  0.0981,  0.3879
 0.8629, -0.8505, -0.4464,  0.8385,  0.5300,  0.1769
 0.1995,  0.6659,  0.7921,  0.9454,  0.9970,  0.2330
-0.0249, -0.3066, -0.2927, -0.4923,  0.8220,  0.2437
 0.4513, -0.9481, -0.0770, -0.4374, -0.9421,  0.2879
-0.3405,  0.5931, -0.3507, -0.3842,  0.8562,  0.3987
 0.9538,  0.0471,  0.9039,  0.7760,  0.0361,  0.1706
-0.0887,  0.2104,  0.9808,  0.5478, -0.3314,  0.4128
-0.8220, -0.6302,  0.0537, -0.1658,  0.6013,  0.4306
-0.4123, -0.2880,  0.9074, -0.0461, -0.4435,  0.5144
 0.0060,  0.2867, -0.7775,  0.5161,  0.7039,  0.3599
-0.7968, -0.5484,  0.9426, -0.4308,  0.8148,  0.2979
 0.7811,  0.8450, -0.6877,  0.7594,  0.2640,  0.2362
-0.6802, -0.1113, -0.8325, -0.6694, -0.6056,  0.6544
 0.3821,  0.1476,  0.7466, -0.5107,  0.2592,  0.1648
 0.7265,  0.9683, -0.9803, -0.4943, -0.5523,  0.2454
-0.9049, -0.9797, -0.0196, -0.9090, -0.4433,  0.6447
-0.4607,  0.1811, -0.2389,  0.4050, -0.0078,  0.5229
 0.2664, -0.2932, -0.4259, -0.7336,  0.8742,  0.1834
-0.4507,  0.1029, -0.6294, -0.1158, -0.6294,  0.6081
 0.8948, -0.0124,  0.9278,  0.2899, -0.0314,  0.1534
-0.1323, -0.8813, -0.0146, -0.0697,  0.6135,  0.2386

Posted in Machine Learning | Leave a comment

Support Vector Regression with SGD Training From Scratch Using C#

Posted on June 22, 2026 by jamesdmccaffrey

The goal of a machine learning regression problem is to predict a single numeric value. For example, a bank might want to predict the maximum safe load amount for a customer, based on age, account balance, annual income, and so on.

One of about a dozen common regression techniques is (kernel) support vector regression (SVR). One day before work, I realized that I had not implemented SVR from scratch, using C#, for a long time. So I figured I’d take a stab at it.

There are three main ways to train a support regression model: quadratic programming (QP) optimization, the sequential minimal optimization (SMO) algorithm, and stochastic sub-gradient descent (SGD). I used SGD, which is far by the simplest SVR training technique.

The output of my demo is:

Begin C# kernel support vector regression with SGD

Loading train (200) and test (40) data
Done

First three X predictors:
  -0.1660   0.4406  -0.9998  -0.3953  -0.7065
   0.0776  -0.1616   0.3704  -0.5911   0.7562
  -0.9452   0.3409  -0.1654   0.1174  -0.7192

First three target y:
  0.4840
  0.1568
  0.8054

Creating SVR object
Setting RBF gamma = 0.3000
Setting epsilon = 0.007500
Setting C = 1.00
Setting lrnRate = 0.0010
Setting maxEpochs = 5000
Setting tol = 0.000100
Done

Training SVR model using SGD
epoch =      0 MSE = 0.0430 acc = 0.1300
epoch =   1000 MSE = 0.0001 acc = 0.9850
epoch =   2000 MSE = 0.0001 acc = 0.9850
epoch =   3000 MSE = 0.0001 acc = 0.9800
epoch =   4000 MSE = 0.0001 acc = 0.9800
Done


Model alpha (weights):
 -0.9256  -0.0443  -0.0041  -0.6581  . . .   0.0049
  0.2983   0.4573  -0.0488   0.0956  . . .   0.1471
. . .
  0.0017   0.0023   0.3193   0.0063  . . .  -0.0078
 -0.0690   0.9895   . . .    0.3843
Model bias = 0.4030

Number supp vectors = 194

Evaluating model

Train acc (within 0.10) = 0.9900
Test acc (within 0.10) = 0.9250

Train MSE = 0.0001
Test MSE = 0.0001

Train R2 = 0.9985
Test R2 = 0.9949

Predicting for trainX[0]
Predicted y = 0.4922

End SVR with SGD training demo

The demo data is synthetic. It was generated by a 5-10-1 neural network with random weights and bias values. The idea here is that the synthetic data does have an underlying, but complex, non-linear structure which can be predicted.

During training, the SVR model assigns one weight value, into a vector called alpha, to each training item, plus a special weight called the bias. After training, the SVR model determined that 6 of the 200 alpha weights were very close to zero, and so those 6 alpha values and their 6 associated training items were removed. This left 194 weight values and training items, called the support vectors.

All of the parameter values must be determined by trial and error. The gamma parameter defines the RBF function that is used to measure the similarity between data item vectors. Larger values of gamma shrinks the radius of influence of individual training points. This tends to increase model accuracy at the expense of increased risk of model overfitting.

The epsilon value defines how close to correct a prediction must be to be considered a non-support vector. Larger values of epsilon create fewer support vectors.

The C value is used for model regularization, which prevents model alpha weights from becoming very large, which often leads to model overfitting. Larger values of C have a smaller regularization affect.

The lrnRate value controls how much alpha weight values change at each update during training. Larger values of lrnRate increase the speed of training, at the risk of jumping over good weight values.

The maxEpochs value controls how many iterations are performed during training. The effect of larger values of SVR maxEpochs can vary greatly.

The tol (“tolerance”) value controls pruning away training vectors to support vectors, by defining how to close to 0 an alpha weight value must be in order to be pruned away. Larger values of tol allow more alpha weights to be defined as zero, which reduces the number of support vectors.

The biggest weakness of support vector regression is the difficulty of tuning the hyperparameters. Small changes in parameter values can have extremely large changes in the model, and the hyperparameters interact in complex ways.

Support vector regression had a brief surge of popularity in the late 1990s and early 2000s. However, data scientists realized that the closely related kernel ridge regression (KRR) has several significant advantages over SVR, and so the use of SVR declined to the point where it is not used very much today.

SVR is more difficult to implement than KRR, SVR is much more difficult to tune than KRR (KRR can use true SGD, which is easier to tune than SVR sub-gradient descent), and SVR often gives slightly worse prediction accuracy than KRR (due mostly to the difficulty in parameter tuning). That said, there are some problem scenarios where kernel SVR is highly effective.

One of the advantages of implementing a machine learning regression system from scratch is that it gives you a full explanation and understanding of exactly how the system works.

I’m a big fan of science fiction movies from the 1950s and 1960s. I especially like scenes where a scientist explains what the threat is.

Left: In “Crack in the World” (1965), a scientist explains how his project that will drill a hole into the Earth’s magma core (using a nuclear missile), to gain access to unlimited power. Well, based on the title of this movie, you can correctly guess that this idea did not work out very well. My grade for this movie = B.

Right: In “Gorath” (1962), a scientist explains that a runaway star named Gorath is on a collision course with Earth. All governments unite to build gigantic rocket thrusters at the South Pole to move Earth out of the way, and then back again. The plan succeeds. My grade for this movie = B.

Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols (my lame blog editor often chokes on symbols).

using System;
using System.IO;
using System.Collections.Generic;

// kernel SVR with SGD training
// hard-wired RBF kernel function

namespace SupportVectorRegressionSGD
{
  internal class SupportVectorRegressionSGDProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin C# kernel support " +
        "vector regression with SGD ");

      Console.WriteLine("\nLoading train (200) and" +
        " test (40) data ");
      string trainFile =
        "..\\..\\..\\Data\\synthetic_train_200.txt";
      double[][] trainX =
        Utils.MatLoad(trainFile,
        new int[] { 0, 1, 2, 3, 4 }, ',', "#");
      double[] trainY =
        Utils.MatToVec(Utils.MatLoad(trainFile,
        new int[] { 5 }, ',', "#"));

      string testFile =
        "..\\..\\..\\Data\\synthetic_test_40.txt";
      double[][] testX =
        Utils.MatLoad(testFile,
        new int[] { 0, 1, 2, 3, 4 }, ',', "#");
      double[] testY =
        Utils.MatToVec(Utils.MatLoad(testFile,
        new int[] { 5 }, ',', "#"));
      Console.WriteLine("Done ");

      Console.WriteLine("\nFirst three X predictors: ");
      for (int i = 0; i "lt" 3; ++i)
        Utils.VecShow(trainX[i], 4, 9);
      Console.WriteLine("\nFirst three target y: ");
      for (int i = 0; i "lt" 3; ++i)
        Console.WriteLine(trainY[i].ToString("F4").
          PadLeft(8));

      Console.WriteLine("\nCreating SVR object");
      double gamma = 0.30;    // RBF param
      double epsilon = 0.0075;
      double C = 1.0;
      double lrnRate = 0.001;
      int maxEpochs = 5000;
      double tol = 1.0e-4;

      Console.WriteLine("Setting RBF gamma = " +
        gamma.ToString("F4"));
      Console.WriteLine("Setting epsilon = " +
        epsilon.ToString("F6"));
      Console.WriteLine("Setting C = " +
        C.ToString("F2"));
      Console.WriteLine("Setting lrnRate = " +
        lrnRate.ToString("F4"));
      Console.WriteLine("Setting maxEpochs = " +
        maxEpochs);
      Console.WriteLine("Setting tol = " +
        tol.ToString("F6"));

      SVR model = new SVR(gamma, epsilon, C,
        lrnRate, maxEpochs, tol, seed: 0);
      Console.WriteLine("Done ");

      Console.WriteLine("\nTraining SVR model using" +
        " SGD ");
      model.Train(trainX, trainY);
      Console.WriteLine("Done ");
      Console.WriteLine("\nModel alpha (weights): ");
      Utils.VecShow(model.alpha, 4, 9);
      Console.WriteLine("\nModel bias = " + 
        model.b.ToString("F4"));

      Console.WriteLine("\nNumber supp vectors = " +
        model.suppX.Length);

      Console.WriteLine("\nEvaluating model ");
      double trainAcc =
        model.Accuracy(trainX, trainY, 0.10);
      double testAcc =
        model.Accuracy(testX, testY, 0.10);

      Console.WriteLine("\nTrain acc (within 0.10) = " +
        trainAcc.ToString("F4"));
      Console.WriteLine("Test acc (within 0.10) = " +
        testAcc.ToString("F4"));

      double trainMSE = model.MSE(trainX, trainY);
      double testMSE = model.MSE(testX, testY);

      Console.WriteLine("\nTrain MSE = " +
        trainMSE.ToString("F4"));
      Console.WriteLine("Test MSE = " +
        testMSE.ToString("F4"));

      double trainR2 = model.R2(trainX, trainY);
      double testR2 = model.R2(testX, testY);

      Console.WriteLine("\nTrain R2 = " +
        trainR2.ToString("F4"));
      Console.WriteLine("Test R2 = " +
        testR2.ToString("F4"));

      double[] x = trainX[0];
      Console.WriteLine("\nPredicting for trainX[0] ");
      double y = model.Predict(x);
      Console.WriteLine("Predicted y = " +
        y.ToString("F4"));

      Console.WriteLine("\nEnd SVR with SGD training demo ");
      Console.ReadLine();
    } // Main()

  } // class Program

  // ========================================================

  public class SVR
  {
    public double gamma;  // for RBF kernel
    public double epsilon;
    public double C; // weight regularization
    public double[][] suppX;  // needed for pred
    public double[] suppY;
    public double[] alpha;  // one per trainX item
    public double b;   // bias
    public double lrnRate;  // for SGD training
    public int maxEpochs;
    public double tol;
    public Random rnd;

    // ------------------------------------------------------

    public SVR(double gamma, double epsilon, double C,
      double lrnRate, int maxEpochs, double tol,
      int seed = 0)
    {
      this.gamma = gamma;
      this.epsilon = epsilon;
      this.C = C;
      this.suppX = new double[0][]; // compiler happy
      this.suppY = new double[0];
      this.lrnRate = lrnRate;
      this.maxEpochs = maxEpochs;
      this.tol = tol;
      this.alpha = new double[0];
      this.b = 0.0;
      this.rnd = new Random(seed);  // shuffle train order
    } // ctor

    // ------------------------------------------------------

    public void Train(double[][] trainX, double[] trainY)
    {
      this.suppX = trainX;
      this.suppY = trainY;
      int n = trainX.Length;

      // init weights
      this.alpha = new double[n];
      double lo = -0.01; double hi = 0.01; // not needed
      for (int i = 0; i "lt" n; ++i)
        this.alpha[i] =
          (hi - lo) * this.rnd.NextDouble() + lo;
      this.b = 0.0;

      // precompute all rbf values to K for fast train
      // not feasible for huge datasets
      double[][] K = this.MakeK(trainX);

      // set up indices for random order SGD training
      int[] indices = Utils.VecRange(n); // 0, 1, 2, ..
      double lamda = 1.0 / this.C;
      int progressFreq = (int)(this.maxEpochs / 5);

      // main sub-gradient processing loop
      for (int epoch = 0; epoch "lt" this.maxEpochs; ++epoch)
      {
        this.Shuffle(indices);
        for (int i = 0; i "lt" indices.Length; ++i)
        {
          int idx = indices[i];
          double predY = 0.0;
          for (int j = 0; j "lt" this.alpha.Length; ++j)
            predY += this.alpha[j] * K[idx][j]; // fast
          predY += this.b;
          double error = predY - trainY[idx];

          double gradLoss;
          bool insideTube = false;
          if (error "gt" this.epsilon)
            gradLoss = 1.0;
          else if (error "lt" -this.epsilon)
            gradLoss = -1.0;
          else
          {
            gradLoss = 0.0;
            insideTube = true;
          }

          // local kernel regularization gradient
          double gradReg = this.alpha[idx] * K[idx][idx];

          //  decoupled updates to the active index
          this.alpha[idx] -= this.lrnRate *
            (lamda * gradReg + gradLoss);
          this.b -= this.lrnRate * gradLoss;

          // force tiny weights to 0
          if (insideTube == true &&
            Math.Abs(this.alpha[idx]) "lt" this.tol)
            this.alpha[idx] = 0.0;

          // in-loop clip to bound updates mid-flight
          if (this.alpha[idx] "lt" -this.C)
            this.alpha[idx] = -this.C;
          else if (this.alpha[idx] "gt" this.C)
            this.alpha[idx] = this.C;

        } // each item

        // show training progress every few epochs
        if (epoch % progressFreq == 0)
        {
          double mse =
            this.MSE(trainX, trainY);
          double acc =
            this.Accuracy(trainX, trainY, 0.10);
          string s1 = "epoch = " +
            epoch.ToString().PadLeft(6);
          string s2 = " MSE = " +
            mse.ToString("F4");
          string s3 = " acc = " + acc.ToString("F4");
          Console.WriteLine(s1 + s2 + s3);
        }

      } // each epoch

      // final global clip
      for (int i = 0; i "lt" n; ++i)
      {
        if (this.alpha[i] "lt" -this.C)
          this.alpha[i] = -this.C;
        else if (this.alpha[i] "gt" this.C)
          this.alpha[i] = this.C;
      }

      // prune: store only explicit support vectors
      List"lt"int"gt" svLst = new List"lt"int"gt"();
      for (int i = 0; i "lt" this.alpha.Length; ++i)
      {
        if (Math.Abs(this.alpha[i]) "gt" 1.0e-5)
          svLst.Add(i);
      }
      int[] svMask = svLst.ToArray();

      this.suppX = Utils.MatSelectRows(trainX, svMask);
      this.suppY = Utils.VecSelectItems(trainY, svMask);
      this.alpha = Utils.VecSelectItems(this.alpha, svMask);

      return;  // all done
    } // Train

    // ------------------------------------------------------

    private void Shuffle(int[] indices)
    {
      // Fisher-Yates helper for Train()
      for (int i = 0; i "lt" indices.Length; ++i)
      {
        int ri = this.rnd.Next(i, indices.Length);
        int tmp = indices[i];
        indices[i] = indices[ri];
        indices[ri] = tmp;
      }
    } // Shuffle

    // ------------------------------------------------------

    private double RBF(double[] v1, double[] v2)
    {
      int n = v1.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" n; ++i)
      {
        double d = v1[i] - v2[i];
        sum += d * d;
      }
      double result = Math.Exp(-1 * this.gamma * sum);
      return result;
    }

    // ------------------------------------------------------

    private double[][] MakeK(double[][] X)
    {
      // Kernel-Gram matrix helper for Train()
      // pre-compute all similarities, to avoid re-computes
      int n = X.Length;
      double[][] result = Utils.MatMake(n, n);
      for (int i = 0; i "lt" n; ++i)
        for (int j = 0; j "lt" n; ++j)
          result[i][j] = this.RBF(X[i], X[j]);
      return result;
    }

    // ------------------------------------------------------

    public double Predict(double[] x)
    {
      int n = this.suppX.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" n; ++i)
      {
        double[] xx = this.suppX[i];
        double k = this.RBF(x, xx);
        sum += this.alpha[i] * k;
      }
      return sum + this.b;
    }

    // ------------------------------------------------------

    public double Accuracy(double[][] dataX,
      double[] dataY, double pctClose)
    {
      int numCorrect = 0; int numWrong = 0;
      int n = dataX.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        double[] x = dataX[i];
        double actualY = dataY[i];
        double predY = this.Predict(x);
        if (Math.Abs(actualY - predY) "lt"
          Math.Abs(actualY * pctClose))
          ++numCorrect;
        else
          ++numWrong;
      }
      return (numCorrect * 1.0) / n;
    }

    // ------------------------------------------------------

    public double MSE(double[][] dataX, double[] dataY)
    {
      double sum = 0.0;
      int n = dataX.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        double[] x = dataX[i];
        double actualY = dataY[i];
        double predY = this.Predict(x);
        sum += (actualY - predY) * (actualY - predY);
      }
      return sum / n;
    }

    // ------------------------------------------------------

    public double R2(double[][] dataX, double[] dataY)
    {
      // coefficient of determination
      int n = dataX.Length;
      double ssRes = 0.0; double ssTot = 0.0;
      double meanY = Utils.VecMean(dataY);

      for (int i = 0; i "lt" n; ++i)
      {
        double[] x = dataX[i];
        double actualY = dataY[i];
        double predY = this.Predict(x);
        ssRes += (actualY - predY) * (actualY - predY);
        ssTot += (actualY - meanY) * (actualY - meanY);
      }
      double result = 1.0 - (ssRes / ssTot);
      return result;
    }

  } // class KRR

  // ========================================================

  public class Utils
  {
    // ------------------------------------------------------

    public static double[][] MatLoad(string fn,
      int[] usecols, char sep, string comment)
    {
      List"lt"double[]"gt" result =
        new List"lt"double[]"gt"();
      string line = "";
      FileStream ifs = new FileStream(fn, FileMode.Open);
      StreamReader sr = new StreamReader(ifs);
      while ((line = sr.ReadLine()) != null)
      {
        if (line.StartsWith(comment) == true)
          continue;
        string[] tokens = line.Split(sep);
        List"lt"double"gt" lst = new List"lt"double"gt"();
        for (int j = 0; j "lt" usecols.Length; ++j)
          lst.Add(double.Parse(tokens[usecols[j]]));
        double[] row = lst.ToArray();
        result.Add(row);
      }
      sr.Close(); ifs.Close();
      return result.ToArray();
    }

    // ------------------------------------------------------

    public static double[] MatToVec(double[][] X)
    {
      int nRows = X.Length;
      int nCols = X[0].Length;
      double[] result = new double[nRows * nCols];
      int k = 0;
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          result[k++] = X[i][j];
      return result;
    }

    // ------------------------------------------------------

    //public static double[] MatGetColumn(double[][] X,
    //  int col)
    //{
    //  int n = X.Length;
    //  double[] result = new double[n];
    //  for (int i = 0; i "lt" n; ++i)
    //    result[i] = X[i][col];
    //  return result;
    //}

    // ------------------------------------------------------

    public static double[][] MatSelectRows(double[][] X,
      int[] rows)
    {
      int nRowsSrc = X.Length;
      int nColsSrc = X[0].Length;
      int n = rows.Length;
      double[][] result = MatMake(n, nColsSrc);

      for (int i = 0; i "lt" n; ++i) // i pts into result
      {
        int srcRow = rows[i];
        for (int j = 0; j "lt" nColsSrc; ++j)
        {
          result[i][j] = X[srcRow][j];
        }
      }
      return result;
    }

    // ------------------------------------------------------

    public static double[][] MatMake(int nRows, int nCols)
    {
      double[][] result = new double[nRows][];
      for (int i = 0; i "lt" nRows; ++i)
        result[i] = new double[nCols];
      return result;
    }

    // ------------------------------------------------------

    public static double VecMean(double[] vec)
    {
      int n = vec.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" n; ++i)
        sum += vec[i];
      double result = sum / n;
      return result;
    }

    // ------------------------------------------------------

    public static int[] VecRange(int n)
    {
      int[] result = new int[n];
      for (int i = 0; i "lt" n; ++i)
        result[i] = i;
      return result;
    }

    // ------------------------------------------------------

    //public static double VecDot(double[] v1, double[] v2)
    //{
    //  int n = v1.Length;
    //  double sum = 0.0;
    //  for (int i = 0; i "lt" n; ++i)
    //    sum += v1[i] * v2[i];
    //  return sum;
    //}

    // ------------------------------------------------------

    public static double[] VecSelectItems(double[] vec,
      int[] idxs)
    {
      int n = idxs.Length;
      double[] result = new double[n];
      for (int i = 0; i "lt" n; ++i)
      {
        result[i] = vec[idxs[i]];
      }
      return result;
    }

    // ------------------------------------------------------

    public static void VecShow(int[] vec, int wid)
    {
      for (int i = 0; i "lt" vec.Length; ++i)
        Console.Write(vec[i].ToString().PadLeft(wid));
      Console.WriteLine("");
    }

    // ------------------------------------------------------

    public static void MatShow(double[][] m, int dec,
      int wid)
    {
      int nRows = m.Length; int nCols = m[0].Length;
      double small = 1.0 / Math.Pow(10, dec);
      for (int i = 0; i "lt" nRows; ++i)
      {
        for (int j = 0; j "lt" nCols; ++j)
        {
          double v = m[i][j];
          if (Math.Abs(v) "lt" small) v = 0.0;
          Console.Write(v.ToString("F" + dec).
            PadLeft(wid));
        }
        Console.WriteLine("");
      }
    }

    // ------------------------------------------------------

    public static void VecShow(double[] vec, int dec,
      int wid)
    {
      for (int i = 0; i "lt" vec.Length; ++i)
        Console.Write(vec[i].ToString("F" + dec).PadLeft(wid));
      Console.WriteLine("");
    }

    // ------------------------------------------------------

  } // class Utils

  // ========================================================

} // ns

Training data:

# synthetic_train_200.txt
#
-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.0776, -0.1616,  0.3704, -0.5911,  0.7562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
 0.9365, -0.3732,  0.3846,  0.7528,  0.7892,  0.1345
-0.8299, -0.9219, -0.6603,  0.7563, -0.8033,  0.7955
 0.0663,  0.3838, -0.3690,  0.3730,  0.6693,  0.3206
-0.9634,  0.5003,  0.9777,  0.4963, -0.4391,  0.7377
-0.1042,  0.8172, -0.4128, -0.4244, -0.7399,  0.4801
-0.9613,  0.3577, -0.5767, -0.4689, -0.0169,  0.6861
-0.7065,  0.1786,  0.3995, -0.7953, -0.1719,  0.5569
 0.3888, -0.1716, -0.9001,  0.0718,  0.3276,  0.2500
 0.1731,  0.8068, -0.7251, -0.7214,  0.6148,  0.3297
-0.2046, -0.6693,  0.8550, -0.3045,  0.5016,  0.2129
 0.2473,  0.5019, -0.3022, -0.4601,  0.7918,  0.2613
-0.1438,  0.9297,  0.3269,  0.2434, -0.7705,  0.5171
 0.1568, -0.1837, -0.5259,  0.8068,  0.1474,  0.3307
-0.9943,  0.2343, -0.3467,  0.0541,  0.7719,  0.5581
 0.2467, -0.9684,  0.8589,  0.3818,  0.9946,  0.1092
-0.6553, -0.7257,  0.8652,  0.3936, -0.8680,  0.7018
 0.8460,  0.4230, -0.7515, -0.9602, -0.9476,  0.1996
-0.9434, -0.5076,  0.7201,  0.0777,  0.1056,  0.5664
 0.9392,  0.1221, -0.9627,  0.6013, -0.5341,  0.1533
 0.6142, -0.2243,  0.7271,  0.4942,  0.1125,  0.1661
 0.4260,  0.1194, -0.9749, -0.8561,  0.9346,  0.2230
 0.1362, -0.5934, -0.4953,  0.4877, -0.6091,  0.3810
 0.6937, -0.5203, -0.0125,  0.2399,  0.6580,  0.1460
-0.6864, -0.9628, -0.8600, -0.0273,  0.2127,  0.5387
 0.9772,  0.1595, -0.2397,  0.1019,  0.4907,  0.1611
 0.3385, -0.4702, -0.8673, -0.2598,  0.2594,  0.2270
-0.8669, -0.4794,  0.6095, -0.6131,  0.2789,  0.4700
 0.0493,  0.8496, -0.4734, -0.8681,  0.4701,  0.3516
 0.8639, -0.9721, -0.5313,  0.2336,  0.8980,  0.1412
 0.9004,  0.1133,  0.8312,  0.2831, -0.2200,  0.1782
 0.0991,  0.8524,  0.8375, -0.2102,  0.9265,  0.2150
-0.6521, -0.7473, -0.7298,  0.0113, -0.9570,  0.7422
 0.6190, -0.3105,  0.8802,  0.1640,  0.7577,  0.1056
 0.6895,  0.8108, -0.0802,  0.0927,  0.5972,  0.2214
 0.1982, -0.9689,  0.1870, -0.1326,  0.6147,  0.1310
-0.3695,  0.7858,  0.1557, -0.6320,  0.5759,  0.3773
-0.1596,  0.3581,  0.8372, -0.9992,  0.9535,  0.2071
-0.2468,  0.9476,  0.2094,  0.6577,  0.1494,  0.4132
 0.1737,  0.5000,  0.7166,  0.5102,  0.3961,  0.2611
 0.7290, -0.3546,  0.3416, -0.0983, -0.2358,  0.1332
-0.3652,  0.2438, -0.1395,  0.9476,  0.3556,  0.4170
-0.6029, -0.1466, -0.3133,  0.5953,  0.7600,  0.4334
-0.4596, -0.4953,  0.7098,  0.0554,  0.6043,  0.2775
 0.1450,  0.4663,  0.0380,  0.5418,  0.1377,  0.2931
-0.8636, -0.2442, -0.8407,  0.9656, -0.6368,  0.7429
 0.6237,  0.7499,  0.3768,  0.1390, -0.6781,  0.2185
-0.5499,  0.1850, -0.3755,  0.8326,  0.8193,  0.4399
-0.4858, -0.7782, -0.6141, -0.0008,  0.4572,  0.4197
 0.7033, -0.1683,  0.2334, -0.5327, -0.7961,  0.1776
 0.0317, -0.0457, -0.6947,  0.2436,  0.0880,  0.3345
 0.5031, -0.5559,  0.0387,  0.5706, -0.9553,  0.3107
-0.3513,  0.7458,  0.6894,  0.0769,  0.7332,  0.3170
 0.2205,  0.5992, -0.9309,  0.5405,  0.4635,  0.3532
-0.4806, -0.4859,  0.2646, -0.3094,  0.5932,  0.3202
 0.9809, -0.3995, -0.7140,  0.8026,  0.0831,  0.1600
 0.9495,  0.2732,  0.9878,  0.0921,  0.0529,  0.1289
-0.9476, -0.6792,  0.4913, -0.9392, -0.2669,  0.5966
 0.7247,  0.3854,  0.3819, -0.6227, -0.1162,  0.1550
-0.5922, -0.5045, -0.4757,  0.5003, -0.0860,  0.5863
-0.8861,  0.0170, -0.5761,  0.5972, -0.4053,  0.7301
 0.6877, -0.2380,  0.4997,  0.0223,  0.0819,  0.1404
 0.9189,  0.6079, -0.9354,  0.4188, -0.0700,  0.1907
-0.1428, -0.7820,  0.2676,  0.6059,  0.3936,  0.2790
 0.5324, -0.3151,  0.6917, -0.1425,  0.6480,  0.1071
-0.8432, -0.9633, -0.8666, -0.0828, -0.7733,  0.7784
-0.9444,  0.5097, -0.2103,  0.4939, -0.0952,  0.6787
-0.0520,  0.6063, -0.1952,  0.8094, -0.9259,  0.4836
 0.5477, -0.7487,  0.2370, -0.9793,  0.0773,  0.1241
 0.2450,  0.8116,  0.9799,  0.4222,  0.4636,  0.2355
 0.8186, -0.1983, -0.5003, -0.6531, -0.7611,  0.1511
-0.4714,  0.6382, -0.3788,  0.9648, -0.4667,  0.5950
 0.0673, -0.3711,  0.8215, -0.2669, -0.1328,  0.2677
-0.9381,  0.4338,  0.7820, -0.9454,  0.0441,  0.5518
-0.3480,  0.7190,  0.1170,  0.3805, -0.0943,  0.4724
-0.9813,  0.1535, -0.3771,  0.0345,  0.8328,  0.5438
-0.1471, -0.5052, -0.2574,  0.8637,  0.8737,  0.3042
-0.5454, -0.3712, -0.6505,  0.2142, -0.1728,  0.5783
 0.6327, -0.6297,  0.4038, -0.5193,  0.1484,  0.1153
-0.5424,  0.3282, -0.0055,  0.0380, -0.6506,  0.6613
 0.1414,  0.9935,  0.6337,  0.1887,  0.9520,  0.2540
-0.9351, -0.8128, -0.8693, -0.0965, -0.2491,  0.7353
 0.9507, -0.6640,  0.9456,  0.5349,  0.6485,  0.1059
-0.0462, -0.9737, -0.2940, -0.0159,  0.4602,  0.2606
-0.0627, -0.0852, -0.7247, -0.9782,  0.5166,  0.2977
 0.0478,  0.5098, -0.0723, -0.7504, -0.3750,  0.3335
 0.0090,  0.3477,  0.5403, -0.7393, -0.9542,  0.4415
-0.9748,  0.3449,  0.3736, -0.1015,  0.8296,  0.4358
 0.2887, -0.9895, -0.0311,  0.7186,  0.6608,  0.2057
 0.1570, -0.4518,  0.1211,  0.3435, -0.2951,  0.3244
 0.7117, -0.6099,  0.4946, -0.4208,  0.5476,  0.1096
-0.2929, -0.5726,  0.5346, -0.3827,  0.4665,  0.2465
 0.4889, -0.5572, -0.5718, -0.6021, -0.7150,  0.2163
-0.7782,  0.3491,  0.5996, -0.8389, -0.5366,  0.6516
-0.5847,  0.8347,  0.4226,  0.1078, -0.3910,  0.6134
 0.8469,  0.4121, -0.0439, -0.7476,  0.9521,  0.1571
-0.6803, -0.5948, -0.1376, -0.1916, -0.7065,  0.7156
 0.2878,  0.5086, -0.5785,  0.2019,  0.4979,  0.2980
 0.2764,  0.1943, -0.4090,  0.4632,  0.8906,  0.2960
-0.8877,  0.6705, -0.6155, -0.2098, -0.3998,  0.7107
-0.8398,  0.8093, -0.2597,  0.0614, -0.0118,  0.6502
-0.8476,  0.0158, -0.4769, -0.2859, -0.7839,  0.7715
 0.5751, -0.7868,  0.9714, -0.6457,  0.1448,  0.1175
 0.4802, -0.7001,  0.1022, -0.5668,  0.5184,  0.1090
 0.4458, -0.6469,  0.7239, -0.9604,  0.7205,  0.0779
 0.5175,  0.4339,  0.9747, -0.4438, -0.9924,  0.2879
 0.8678,  0.7158,  0.4577,  0.0334,  0.4139,  0.1678
 0.5406,  0.5012,  0.2264, -0.1963,  0.3946,  0.2088
-0.9938,  0.5498,  0.7928, -0.5214, -0.7585,  0.7687
 0.7661,  0.0863, -0.4266, -0.7233, -0.4197,  0.1466
 0.2277, -0.3517, -0.0853, -0.1118,  0.6563,  0.1767
 0.3499, -0.5570, -0.0655, -0.3705,  0.2537,  0.1632
 0.7547, -0.1046,  0.5689, -0.0861,  0.3125,  0.1257
 0.8186,  0.2110,  0.5335,  0.0094, -0.0039,  0.1391
 0.6858, -0.8644,  0.1465,  0.8855,  0.0357,  0.1845
-0.4967,  0.4015,  0.0805,  0.8977,  0.2487,  0.4663
 0.6760, -0.9841,  0.9787, -0.8446, -0.3557,  0.1509
-0.1203, -0.4885,  0.6054, -0.0443, -0.7313,  0.4854
 0.8557,  0.7919, -0.0169,  0.7134, -0.1628,  0.2002
 0.0115, -0.6209,  0.9300, -0.4116, -0.7931,  0.4052
-0.7114, -0.9718,  0.4319,  0.1290,  0.5892,  0.3661
 0.3915,  0.5557, -0.1870,  0.2955, -0.6404,  0.2954
-0.3564, -0.6548, -0.1827, -0.5172, -0.1862,  0.4622
 0.2392, -0.4959,  0.5857, -0.1341, -0.2850,  0.2470
-0.3394,  0.3947, -0.4627,  0.6166, -0.4094,  0.5325
 0.7107,  0.7768, -0.6312,  0.1707,  0.7964,  0.2757
-0.1078,  0.8437, -0.4420,  0.2177,  0.3649,  0.4028
-0.3139,  0.5595, -0.6505, -0.3161, -0.7108,  0.5546
 0.4335,  0.3986,  0.3770, -0.4932,  0.3847,  0.1810
-0.2562, -0.2894, -0.8847,  0.2633,  0.4146,  0.4036
 0.2272,  0.2966, -0.6601, -0.7011,  0.0284,  0.2778
-0.0743, -0.1421, -0.0054, -0.6770, -0.3151,  0.3597
-0.4762,  0.6891,  0.6007, -0.1467,  0.2140,  0.4266
-0.4061,  0.7193,  0.3432,  0.2669, -0.7505,  0.6147
-0.0588,  0.9731,  0.8966,  0.2902, -0.6966,  0.4955
-0.0627, -0.1439,  0.1985,  0.6999,  0.5022,  0.3077
 0.1587,  0.8494, -0.8705,  0.9827, -0.8940,  0.4263
-0.7850,  0.2473, -0.9040, -0.4308, -0.8779,  0.7199
 0.4070,  0.3369, -0.2428, -0.6236,  0.4940,  0.2215
-0.0242,  0.0513, -0.9430,  0.2885, -0.2987,  0.3947
-0.5416, -0.1322, -0.2351, -0.0604,  0.9590,  0.3683
 0.1055,  0.7783, -0.2901, -0.5090,  0.8220,  0.2984
-0.9129,  0.9015,  0.1128, -0.2473,  0.9901,  0.4776
-0.9378,  0.1424, -0.6391,  0.2619,  0.9618,  0.5368
 0.7498, -0.0963,  0.4169,  0.5549, -0.0103,  0.1614
-0.2612, -0.7156,  0.4538, -0.0460, -0.1022,  0.3717
 0.7720,  0.0552, -0.1818, -0.4622, -0.8560,  0.1685
-0.4177,  0.0070,  0.9319, -0.7812,  0.3461,  0.3052
-0.0001,  0.5542, -0.7128, -0.8336, -0.2016,  0.3803
 0.5356, -0.4194, -0.5662, -0.9666, -0.2027,  0.1776
-0.2378,  0.3187, -0.8582, -0.6948, -0.9668,  0.5474
-0.1947, -0.3579,  0.1158,  0.9869,  0.6690,  0.2992
 0.3992,  0.8365, -0.9205, -0.8593, -0.0520,  0.3154
-0.0209,  0.0793,  0.7905, -0.1067,  0.7541,  0.1864
-0.4928, -0.4524, -0.3433,  0.0951, -0.5597,  0.6261
-0.8118,  0.7404, -0.5263, -0.2280,  0.1431,  0.6349
 0.0516, -0.8480,  0.7483,  0.9023,  0.6250,  0.1959
-0.3212,  0.1093,  0.9488, -0.3766,  0.3376,  0.2735
-0.3481,  0.5490, -0.3484,  0.7797,  0.5034,  0.4379
-0.5785, -0.9170, -0.3563, -0.9258,  0.3877,  0.4121
 0.3407, -0.1391,  0.5356,  0.0720, -0.9203,  0.3458
-0.3287, -0.8954,  0.2102,  0.0241,  0.2349,  0.3247
-0.1353,  0.6954, -0.0919, -0.9692,  0.7461,  0.3338
 0.9036, -0.8982, -0.5299, -0.8733, -0.1567,  0.1187
 0.7277, -0.8368, -0.0538, -0.7489,  0.5458,  0.0830
 0.9049,  0.8878,  0.2279,  0.9470, -0.3103,  0.2194
 0.7957, -0.1308, -0.5284,  0.8817,  0.3684,  0.2172
 0.4647, -0.4931,  0.2010,  0.6292, -0.8918,  0.3371
-0.7390,  0.6849,  0.2367,  0.0626, -0.5034,  0.7039
-0.1567, -0.8711,  0.7940, -0.5932,  0.6525,  0.1710
 0.7635, -0.0265,  0.1969,  0.0545,  0.2496,  0.1445
 0.7675,  0.1354, -0.7698, -0.5460,  0.1920,  0.1728
-0.5211, -0.7372, -0.6763,  0.6897,  0.2044,  0.5217
 0.1913,  0.1980,  0.2314, -0.8816,  0.5006,  0.1998
 0.8964,  0.0694, -0.6149,  0.5059, -0.9854,  0.1825
 0.1767,  0.7104,  0.2093,  0.6452,  0.7590,  0.2832
-0.3580, -0.7541,  0.4426, -0.1193, -0.7465,  0.5657
-0.5996,  0.5766, -0.9758, -0.3933, -0.9572,  0.6800
 0.9950,  0.1641, -0.4132,  0.8579,  0.0142,  0.2003
-0.4717, -0.3894, -0.2567, -0.5111,  0.1691,  0.4266
 0.3917, -0.8561,  0.9422,  0.5061,  0.6123,  0.1212
-0.0366, -0.1087,  0.3449, -0.1025,  0.4086,  0.2475
 0.3633,  0.3943,  0.2372, -0.6980,  0.5216,  0.1925
-0.5325, -0.6466, -0.2178, -0.3589,  0.6310,  0.3568
 0.2271,  0.5200, -0.1447, -0.8011, -0.7699,  0.3128
 0.6415,  0.1993,  0.3777, -0.0178, -0.8237,  0.2181
-0.5298, -0.0768, -0.6028, -0.9490,  0.4588,  0.4356
 0.6870, -0.1431,  0.7294,  0.3141,  0.1621,  0.1632
-0.5985,  0.0591,  0.7889, -0.3900,  0.7419,  0.2945
 0.3661,  0.7984, -0.8486,  0.7572, -0.6183,  0.3449
 0.6995,  0.3342, -0.3113, -0.6972,  0.2707,  0.1712
 0.2565,  0.9126,  0.1798, -0.6043, -0.1413,  0.2893
-0.3265,  0.9839, -0.2395,  0.9854,  0.0376,  0.4770
 0.2690, -0.1722,  0.9818,  0.8599, -0.7015,  0.3954
-0.2102, -0.0768,  0.1219,  0.5607, -0.0256,  0.3949
 0.8216, -0.9555,  0.6422, -0.6231,  0.3715,  0.0801
-0.2896,  0.9484, -0.7545, -0.6249,  0.7789,  0.4370
-0.9985, -0.5448, -0.7092, -0.5931,  0.7926,  0.5402

Test data:

# synthetic_test_40.txt
#
 0.7462,  0.4006, -0.0590,  0.6543, -0.0083,  0.1935
 0.8495, -0.2260, -0.0142, -0.4911,  0.7699,  0.1078
-0.2335, -0.4049,  0.4352, -0.6183, -0.7636,  0.5088
 0.1810, -0.5142,  0.2465,  0.2767, -0.3449,  0.3136
-0.8650,  0.7611, -0.0801,  0.5277, -0.4922,  0.7140
-0.2358, -0.7466, -0.5115, -0.8413, -0.3943,  0.4533
 0.4834,  0.2300,  0.3448, -0.9832,  0.3568,  0.1360
-0.6502, -0.6300,  0.6885,  0.9652,  0.8275,  0.3046
-0.3053,  0.5604,  0.0929,  0.6329, -0.0325,  0.4756
-0.7995,  0.0740, -0.2680,  0.2086,  0.9176,  0.4565
-0.2144, -0.2141,  0.5813,  0.2902, -0.2122,  0.4119
-0.7278, -0.0987, -0.3312, -0.5641,  0.8515,  0.4438
 0.3793,  0.1976,  0.4933,  0.0839,  0.4011,  0.1905
-0.8568,  0.9573, -0.5272,  0.3212, -0.8207,  0.7415
-0.5785,  0.0056, -0.7901, -0.2223,  0.0760,  0.5551
 0.0735, -0.2188,  0.3925,  0.3570,  0.3746,  0.2191
 0.1230, -0.2838,  0.2262,  0.8715,  0.1938,  0.2878
 0.4792, -0.9248,  0.5295,  0.0366, -0.9894,  0.3149
-0.4456,  0.0697,  0.5359, -0.8938,  0.0981,  0.3879
 0.8629, -0.8505, -0.4464,  0.8385,  0.5300,  0.1769
 0.1995,  0.6659,  0.7921,  0.9454,  0.9970,  0.2330
-0.0249, -0.3066, -0.2927, -0.4923,  0.8220,  0.2437
 0.4513, -0.9481, -0.0770, -0.4374, -0.9421,  0.2879
-0.3405,  0.5931, -0.3507, -0.3842,  0.8562,  0.3987
 0.9538,  0.0471,  0.9039,  0.7760,  0.0361,  0.1706
-0.0887,  0.2104,  0.9808,  0.5478, -0.3314,  0.4128
-0.8220, -0.6302,  0.0537, -0.1658,  0.6013,  0.4306
-0.4123, -0.2880,  0.9074, -0.0461, -0.4435,  0.5144
 0.0060,  0.2867, -0.7775,  0.5161,  0.7039,  0.3599
-0.7968, -0.5484,  0.9426, -0.4308,  0.8148,  0.2979
 0.7811,  0.8450, -0.6877,  0.7594,  0.2640,  0.2362
-0.6802, -0.1113, -0.8325, -0.6694, -0.6056,  0.6544
 0.3821,  0.1476,  0.7466, -0.5107,  0.2592,  0.1648
 0.7265,  0.9683, -0.9803, -0.4943, -0.5523,  0.2454
-0.9049, -0.9797, -0.0196, -0.9090, -0.4433,  0.6447
-0.4607,  0.1811, -0.2389,  0.4050, -0.0078,  0.5229
 0.2664, -0.2932, -0.4259, -0.7336,  0.8742,  0.1834
-0.4507,  0.1029, -0.6294, -0.1158, -0.6294,  0.6081
 0.8948, -0.0124,  0.9278,  0.2899, -0.0314,  0.1534
-0.1323, -0.8813, -0.0146, -0.0697,  0.6135,  0.2386

Posted in Machine Learning | Leave a comment

Example of Anomaly Detection Using scikit IsolationForest

Posted on June 19, 2026 by jamesdmccaffrey

I came across a relatively obscure module in the scikit-learn library called IsolationForest. The module is used for anomaly detection. In a nutshell, if you apply a decision tree to a dataset, because of the way the branches are created, after the tree is created, tree nodes that are near the root, and which hold a single value, are anomalous.

I decided to put a demo together.

For my demo I used a set of synthetic data that looks like:

-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.9776, -0.9616,  0.9704, -0.9911,  0.9562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
 0.9365, -0.3732,  0.3846,  0.7528,  0.7892,  0.1345
-0.8299, -0.9219, -0.6603,  0.7563, -0.8033,  0.7955
. . .

The last value in each column is a dependent variable, used for regression explorations, so I used only the first five values on each line. There are 200 data items.

I deliberately modified data item [1] to make the magnitudes of each value very large — so data item [1] is anomalous. Notice all the values in item [1] are very close to -1 or +1.

The output of my IsolationForest demo is:

Begin scikit IsolationForest demo

Loading synthetic data (200)

First three data items:
[-0.1660  0.4406 -0.9998 -0.3953 -0.7065]
[ 0.9776 -0.9616  0.9704 -0.9911  0.9562]
[-0.9452  0.3409 -0.1654  0.1174 -0.7192]

Creating IsolationForest
Done

Analyzing dataset
Done

First three anomaly scores
  0.0044
 -0.1226
  0.0079

Most anomalous item = [1]
Anomaly score = -0.1226

End demo

Smaller values (more negative) are more anomalous. The demo correctly identified item [1] as the most anomalous.

When I first discovered the scikit IsolationForest, I didn’t like some of the technical details and so I used the key ideas to implement a C# system that I called an Anomaly Forest. When I ran the data through the C# program, it also picked data item [1] as most anomalous. (See the screenshot above).

Different anomaly detection algorithms identify different types of anomalies. The scikit IsolationForest module is just one of many techniques.

In some sense, machine learning is all about finding hidden patterns in data. Every cover of Playboy Magazine (except for the very first issue in December 1953) has a logo of a bunny. On most covers, the bunny logo is fairly obvious, but sometimes the logo is very cleverly hidden.

Left: On the July 1976 (the 200th anniversary of the United States), the bunny logo is disguised as one of the stars on the U.S. flag.

Right: This is the most famous cover for the hidden logo. Many thousands of readers wrote to the magazine saying that they had examined every square inch of the cover and there is no logo. But there is — the trickiest hidden logo ever, up until that time. The logo is hidden in the text “The Playmate of the Year”. The bunny head is the upper-left part of the “Y” of “Year” and the ears are the bottom part of the “P” and the “y” in Playmate”. Because of this tricky cover, starting a few months later, the magazine explained where the hidden logo is on the Contents page of each issue.

Demo program.

# isolation_forest_scikit.py
# anomaly detection

import numpy as np
from sklearn.ensemble import IsolationForest

np.set_printoptions(precision=4, suppress=True,
  floatmode='fixed', linewidth=120)

print("\nBegin scikit IsolationForest demo ")

print("\nLoading synthetic data (200) ")
data_file = ".\\Data\\synthetic_200.txt"

cols_X = [0,1,2,3,4]  
data_X = np.loadtxt(data_file, comments="#",
  usecols=cols_X, delimiter=",",  dtype=np.float64)

print("\nFirst three data items: ")
for i in range(3):
  print(data_X[i])

print("\nCreating IsolationForest ")
detector = IsolationForest(n_estimators=50, random_state=0)
print("Done ")

print("\nAnalyzing dataset ")
detector.fit(data_X)
print("Done ")

scores = detector.decision_function(data_X)
print("\nFirst three anomaly scores ")

for i in range(3):
  print("%8.4f " % scores[i])

min_indx = 0
min_score = scores[0]
for i in range(len(scores)):
  if scores[i] < min_score:
    min_score = scores[i]
    min_indx = i

print("\nMost anomalous item = [" + str(min_indx) + "]")
print("Anomaly score = %0.4f " % min_score)

# print("")
# for i in range(len(scores)):
#   print("%4d  %0.4f " % (i, scores[i]))

print("\nEnd demo ")

Data:

# synthetic_200.txt
#
-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.9776, -0.9616,  0.9704, -0.9911,  0.9562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
 0.9365, -0.3732,  0.3846,  0.7528,  0.7892,  0.1345
-0.8299, -0.9219, -0.6603,  0.7563, -0.8033,  0.7955
 0.0663,  0.3838, -0.3690,  0.3730,  0.6693,  0.3206
-0.9634,  0.5003,  0.9777,  0.4963, -0.4391,  0.7377
-0.1042,  0.8172, -0.4128, -0.4244, -0.7399,  0.4801
-0.9613,  0.3577, -0.5767, -0.4689, -0.0169,  0.6861
-0.7065,  0.1786,  0.3995, -0.7953, -0.1719,  0.5569
 0.3888, -0.1716, -0.9001,  0.0718,  0.3276,  0.2500
 0.1731,  0.8068, -0.7251, -0.7214,  0.6148,  0.3297
-0.2046, -0.6693,  0.8550, -0.3045,  0.5016,  0.2129
 0.2473,  0.5019, -0.3022, -0.4601,  0.7918,  0.2613
-0.1438,  0.9297,  0.3269,  0.2434, -0.7705,  0.5171
 0.1568, -0.1837, -0.5259,  0.8068,  0.1474,  0.3307
-0.9943,  0.2343, -0.3467,  0.0541,  0.7719,  0.5581
 0.2467, -0.9684,  0.8589,  0.3818,  0.9946,  0.1092
-0.6553, -0.7257,  0.8652,  0.3936, -0.8680,  0.7018
 0.8460,  0.4230, -0.7515, -0.9602, -0.9476,  0.1996
-0.9434, -0.5076,  0.7201,  0.0777,  0.1056,  0.5664
 0.9392,  0.1221, -0.9627,  0.6013, -0.5341,  0.1533
 0.6142, -0.2243,  0.7271,  0.4942,  0.1125,  0.1661
 0.4260,  0.1194, -0.9749, -0.8561,  0.9346,  0.2230
 0.1362, -0.5934, -0.4953,  0.4877, -0.6091,  0.3810
 0.6937, -0.5203, -0.0125,  0.2399,  0.6580,  0.1460
-0.6864, -0.9628, -0.8600, -0.0273,  0.2127,  0.5387
 0.9772,  0.1595, -0.2397,  0.1019,  0.4907,  0.1611
 0.3385, -0.4702, -0.8673, -0.2598,  0.2594,  0.2270
-0.8669, -0.4794,  0.6095, -0.6131,  0.2789,  0.4700
 0.0493,  0.8496, -0.4734, -0.8681,  0.4701,  0.3516
 0.8639, -0.9721, -0.5313,  0.2336,  0.8980,  0.1412
 0.9004,  0.1133,  0.8312,  0.2831, -0.2200,  0.1782
 0.0991,  0.8524,  0.8375, -0.2102,  0.9265,  0.2150
-0.6521, -0.7473, -0.7298,  0.0113, -0.9570,  0.7422
 0.6190, -0.3105,  0.8802,  0.1640,  0.7577,  0.1056
 0.6895,  0.8108, -0.0802,  0.0927,  0.5972,  0.2214
 0.1982, -0.9689,  0.1870, -0.1326,  0.6147,  0.1310
-0.3695,  0.7858,  0.1557, -0.6320,  0.5759,  0.3773
-0.1596,  0.3581,  0.8372, -0.9992,  0.9535,  0.2071
-0.2468,  0.9476,  0.2094,  0.6577,  0.1494,  0.4132
 0.1737,  0.5000,  0.7166,  0.5102,  0.3961,  0.2611
 0.7290, -0.3546,  0.3416, -0.0983, -0.2358,  0.1332
-0.3652,  0.2438, -0.1395,  0.9476,  0.3556,  0.4170
-0.6029, -0.1466, -0.3133,  0.5953,  0.7600,  0.4334
-0.4596, -0.4953,  0.7098,  0.0554,  0.6043,  0.2775
 0.1450,  0.4663,  0.0380,  0.5418,  0.1377,  0.2931
-0.8636, -0.2442, -0.8407,  0.9656, -0.6368,  0.7429
 0.6237,  0.7499,  0.3768,  0.1390, -0.6781,  0.2185
-0.5499,  0.1850, -0.3755,  0.8326,  0.8193,  0.4399
-0.4858, -0.7782, -0.6141, -0.0008,  0.4572,  0.4197
 0.7033, -0.1683,  0.2334, -0.5327, -0.7961,  0.1776
 0.0317, -0.0457, -0.6947,  0.2436,  0.0880,  0.3345
 0.5031, -0.5559,  0.0387,  0.5706, -0.9553,  0.3107
-0.3513,  0.7458,  0.6894,  0.0769,  0.7332,  0.3170
 0.2205,  0.5992, -0.9309,  0.5405,  0.4635,  0.3532
-0.4806, -0.4859,  0.2646, -0.3094,  0.5932,  0.3202
 0.9809, -0.3995, -0.7140,  0.8026,  0.0831,  0.1600
 0.9495,  0.2732,  0.9878,  0.0921,  0.0529,  0.1289
-0.9476, -0.6792,  0.4913, -0.9392, -0.2669,  0.5966
 0.7247,  0.3854,  0.3819, -0.6227, -0.1162,  0.1550
-0.5922, -0.5045, -0.4757,  0.5003, -0.0860,  0.5863
-0.8861,  0.0170, -0.5761,  0.5972, -0.4053,  0.7301
 0.6877, -0.2380,  0.4997,  0.0223,  0.0819,  0.1404
 0.9189,  0.6079, -0.9354,  0.4188, -0.0700,  0.1907
-0.1428, -0.7820,  0.2676,  0.6059,  0.3936,  0.2790
 0.5324, -0.3151,  0.6917, -0.1425,  0.6480,  0.1071
-0.8432, -0.9633, -0.8666, -0.0828, -0.7733,  0.7784
-0.9444,  0.5097, -0.2103,  0.4939, -0.0952,  0.6787
-0.0520,  0.6063, -0.1952,  0.8094, -0.9259,  0.4836
 0.5477, -0.7487,  0.2370, -0.9793,  0.0773,  0.1241
 0.2450,  0.8116,  0.9799,  0.4222,  0.4636,  0.2355
 0.8186, -0.1983, -0.5003, -0.6531, -0.7611,  0.1511
-0.4714,  0.6382, -0.3788,  0.9648, -0.4667,  0.5950
 0.0673, -0.3711,  0.8215, -0.2669, -0.1328,  0.2677
-0.9381,  0.4338,  0.7820, -0.9454,  0.0441,  0.5518
-0.3480,  0.7190,  0.1170,  0.3805, -0.0943,  0.4724
-0.9813,  0.1535, -0.3771,  0.0345,  0.8328,  0.5438
-0.1471, -0.5052, -0.2574,  0.8637,  0.8737,  0.3042
-0.5454, -0.3712, -0.6505,  0.2142, -0.1728,  0.5783
 0.6327, -0.6297,  0.4038, -0.5193,  0.1484,  0.1153
-0.5424,  0.3282, -0.0055,  0.0380, -0.6506,  0.6613
 0.1414,  0.9935,  0.6337,  0.1887,  0.9520,  0.2540
-0.9351, -0.8128, -0.8693, -0.0965, -0.2491,  0.7353
 0.9507, -0.6640,  0.9456,  0.5349,  0.6485,  0.1059
-0.0462, -0.9737, -0.2940, -0.0159,  0.4602,  0.2606
-0.0627, -0.0852, -0.7247, -0.9782,  0.5166,  0.2977
 0.0478,  0.5098, -0.0723, -0.7504, -0.3750,  0.3335
 0.0090,  0.3477,  0.5403, -0.7393, -0.9542,  0.4415
-0.9748,  0.3449,  0.3736, -0.1015,  0.8296,  0.4358
 0.2887, -0.9895, -0.0311,  0.7186,  0.6608,  0.2057
 0.1570, -0.4518,  0.1211,  0.3435, -0.2951,  0.3244
 0.7117, -0.6099,  0.4946, -0.4208,  0.5476,  0.1096
-0.2929, -0.5726,  0.5346, -0.3827,  0.4665,  0.2465
 0.4889, -0.5572, -0.5718, -0.6021, -0.7150,  0.2163
-0.7782,  0.3491,  0.5996, -0.8389, -0.5366,  0.6516
-0.5847,  0.8347,  0.4226,  0.1078, -0.3910,  0.6134
 0.8469,  0.4121, -0.0439, -0.7476,  0.9521,  0.1571
-0.6803, -0.5948, -0.1376, -0.1916, -0.7065,  0.7156
 0.2878,  0.5086, -0.5785,  0.2019,  0.4979,  0.2980
 0.2764,  0.1943, -0.4090,  0.4632,  0.8906,  0.2960
-0.8877,  0.6705, -0.6155, -0.2098, -0.3998,  0.7107
-0.8398,  0.8093, -0.2597,  0.0614, -0.0118,  0.6502
-0.8476,  0.0158, -0.4769, -0.2859, -0.7839,  0.7715
 0.5751, -0.7868,  0.9714, -0.6457,  0.1448,  0.1175
 0.4802, -0.7001,  0.1022, -0.5668,  0.5184,  0.1090
 0.4458, -0.6469,  0.7239, -0.9604,  0.7205,  0.0779
 0.5175,  0.4339,  0.9747, -0.4438, -0.9924,  0.2879
 0.8678,  0.7158,  0.4577,  0.0334,  0.4139,  0.1678
 0.5406,  0.5012,  0.2264, -0.1963,  0.3946,  0.2088
-0.9938,  0.5498,  0.7928, -0.5214, -0.7585,  0.7687
 0.7661,  0.0863, -0.4266, -0.7233, -0.4197,  0.1466
 0.2277, -0.3517, -0.0853, -0.1118,  0.6563,  0.1767
 0.3499, -0.5570, -0.0655, -0.3705,  0.2537,  0.1632
 0.7547, -0.1046,  0.5689, -0.0861,  0.3125,  0.1257
 0.8186,  0.2110,  0.5335,  0.0094, -0.0039,  0.1391
 0.6858, -0.8644,  0.1465,  0.8855,  0.0357,  0.1845
-0.4967,  0.4015,  0.0805,  0.8977,  0.2487,  0.4663
 0.6760, -0.9841,  0.9787, -0.8446, -0.3557,  0.1509
-0.1203, -0.4885,  0.6054, -0.0443, -0.7313,  0.4854
 0.8557,  0.7919, -0.0169,  0.7134, -0.1628,  0.2002
 0.0115, -0.6209,  0.9300, -0.4116, -0.7931,  0.4052
-0.7114, -0.9718,  0.4319,  0.1290,  0.5892,  0.3661
 0.3915,  0.5557, -0.1870,  0.2955, -0.6404,  0.2954
-0.3564, -0.6548, -0.1827, -0.5172, -0.1862,  0.4622
 0.2392, -0.4959,  0.5857, -0.1341, -0.2850,  0.2470
-0.3394,  0.3947, -0.4627,  0.6166, -0.4094,  0.5325
 0.7107,  0.7768, -0.6312,  0.1707,  0.7964,  0.2757
-0.1078,  0.8437, -0.4420,  0.2177,  0.3649,  0.4028
-0.3139,  0.5595, -0.6505, -0.3161, -0.7108,  0.5546
 0.4335,  0.3986,  0.3770, -0.4932,  0.3847,  0.1810
-0.2562, -0.2894, -0.8847,  0.2633,  0.4146,  0.4036
 0.2272,  0.2966, -0.6601, -0.7011,  0.0284,  0.2778
-0.0743, -0.1421, -0.0054, -0.6770, -0.3151,  0.3597
-0.4762,  0.6891,  0.6007, -0.1467,  0.2140,  0.4266
-0.4061,  0.7193,  0.3432,  0.2669, -0.7505,  0.6147
-0.0588,  0.9731,  0.8966,  0.2902, -0.6966,  0.4955
-0.0627, -0.1439,  0.1985,  0.6999,  0.5022,  0.3077
 0.1587,  0.8494, -0.8705,  0.9827, -0.8940,  0.4263
-0.7850,  0.2473, -0.9040, -0.4308, -0.8779,  0.7199
 0.4070,  0.3369, -0.2428, -0.6236,  0.4940,  0.2215
-0.0242,  0.0513, -0.9430,  0.2885, -0.2987,  0.3947
-0.5416, -0.1322, -0.2351, -0.0604,  0.9590,  0.3683
 0.1055,  0.7783, -0.2901, -0.5090,  0.8220,  0.2984
-0.9129,  0.9015,  0.1128, -0.2473,  0.9901,  0.4776
-0.9378,  0.1424, -0.6391,  0.2619,  0.9618,  0.5368
 0.7498, -0.0963,  0.4169,  0.5549, -0.0103,  0.1614
-0.2612, -0.7156,  0.4538, -0.0460, -0.1022,  0.3717
 0.7720,  0.0552, -0.1818, -0.4622, -0.8560,  0.1685
-0.4177,  0.0070,  0.9319, -0.7812,  0.3461,  0.3052
-0.0001,  0.5542, -0.7128, -0.8336, -0.2016,  0.3803
 0.5356, -0.4194, -0.5662, -0.9666, -0.2027,  0.1776
-0.2378,  0.3187, -0.8582, -0.6948, -0.9668,  0.5474
-0.1947, -0.3579,  0.1158,  0.9869,  0.6690,  0.2992
 0.3992,  0.8365, -0.9205, -0.8593, -0.0520,  0.3154
-0.0209,  0.0793,  0.7905, -0.1067,  0.7541,  0.1864
-0.4928, -0.4524, -0.3433,  0.0951, -0.5597,  0.6261
-0.8118,  0.7404, -0.5263, -0.2280,  0.1431,  0.6349
 0.0516, -0.8480,  0.7483,  0.9023,  0.6250,  0.1959
-0.3212,  0.1093,  0.9488, -0.3766,  0.3376,  0.2735
-0.3481,  0.5490, -0.3484,  0.7797,  0.5034,  0.4379
-0.5785, -0.9170, -0.3563, -0.9258,  0.3877,  0.4121
 0.3407, -0.1391,  0.5356,  0.0720, -0.9203,  0.3458
-0.3287, -0.8954,  0.2102,  0.0241,  0.2349,  0.3247
-0.1353,  0.6954, -0.0919, -0.9692,  0.7461,  0.3338
 0.9036, -0.8982, -0.5299, -0.8733, -0.1567,  0.1187
 0.7277, -0.8368, -0.0538, -0.7489,  0.5458,  0.0830
 0.9049,  0.8878,  0.2279,  0.9470, -0.3103,  0.2194
 0.7957, -0.1308, -0.5284,  0.8817,  0.3684,  0.2172
 0.4647, -0.4931,  0.2010,  0.6292, -0.8918,  0.3371
-0.7390,  0.6849,  0.2367,  0.0626, -0.5034,  0.7039
-0.1567, -0.8711,  0.7940, -0.5932,  0.6525,  0.1710
 0.7635, -0.0265,  0.1969,  0.0545,  0.2496,  0.1445
 0.7675,  0.1354, -0.7698, -0.5460,  0.1920,  0.1728
-0.5211, -0.7372, -0.6763,  0.6897,  0.2044,  0.5217
 0.1913,  0.1980,  0.2314, -0.8816,  0.5006,  0.1998
 0.8964,  0.0694, -0.6149,  0.5059, -0.9854,  0.1825
 0.1767,  0.7104,  0.2093,  0.6452,  0.7590,  0.2832
-0.3580, -0.7541,  0.4426, -0.1193, -0.7465,  0.5657
-0.5996,  0.5766, -0.9758, -0.3933, -0.9572,  0.6800
 0.9950,  0.1641, -0.4132,  0.8579,  0.0142,  0.2003
-0.4717, -0.3894, -0.2567, -0.5111,  0.1691,  0.4266
 0.3917, -0.8561,  0.9422,  0.5061,  0.6123,  0.1212
-0.0366, -0.1087,  0.3449, -0.1025,  0.4086,  0.2475
 0.3633,  0.3943,  0.2372, -0.6980,  0.5216,  0.1925
-0.5325, -0.6466, -0.2178, -0.3589,  0.6310,  0.3568
 0.2271,  0.5200, -0.1447, -0.8011, -0.7699,  0.3128
 0.6415,  0.1993,  0.3777, -0.0178, -0.8237,  0.2181
-0.5298, -0.0768, -0.6028, -0.9490,  0.4588,  0.4356
 0.6870, -0.1431,  0.7294,  0.3141,  0.1621,  0.1632
-0.5985,  0.0591,  0.7889, -0.3900,  0.7419,  0.2945
 0.3661,  0.7984, -0.8486,  0.7572, -0.6183,  0.3449
 0.6995,  0.3342, -0.3113, -0.6972,  0.2707,  0.1712
 0.2565,  0.9126,  0.1798, -0.6043, -0.1413,  0.2893
-0.3265,  0.9839, -0.2395,  0.9854,  0.0376,  0.4770
 0.2690, -0.1722,  0.9818,  0.8599, -0.7015,  0.3954
-0.2102, -0.0768,  0.1219,  0.5607, -0.0256,  0.3949
 0.8216, -0.9555,  0.6422, -0.6231,  0.3715,  0.0801
-0.2896,  0.9484, -0.7545, -0.6249,  0.7789,  0.4370
-0.9985, -0.5448, -0.7092, -0.5931,  0.7926,  0.5402

Posted in Machine Learning, Scikit | Leave a comment

“The AI Coding Revolution: Productivity Boom and Employment Crisis” on the Pure AI Web Site

Posted on June 18, 2026 by jamesdmccaffrey

I contributed some technical content and opinions to an article titled “The AI Coding Revolution: Productivity Boom and Employment Crisis” on the Pure AI web site. See https://pureai.com/articles/2026/06/01/the-ai-coding-revolution-productivity-boom-and-employment-crisis.aspx.

AI-assisted coding is easily the biggest technology change that I’ve seen in my 50+ years of writing code. I helped write an article that discusses what some of the impacts are. Briefly:

1.) AI coding assistants are rapidly evolving from code-completion tools into autonomous software development agents.

2.) The technology is increasing software development productivity while reducing demand for routine programming work.

3.) The long-term benefits of AI-generated software may be significant, but the short-term impact on technology employment will be severe.

McCaffrey commented, “AI-assisted coding is particularly effective in SQL development and web development, where large amounts of structured training data and established coding patterns exist. AI coding is weakest at user interface design, which is highly subjective and a moving target.”

The AI coding market is currently dominated by six powerful competitors: Microsoft, Google, Anthropic, Amazon, OpenAI, and Cursor.

McCaffrey said, “Microsoft’s GitHub Copilot is the dominant enterprise solution. Google’s Gemini Code Assist is known for its ability to analyze extremely large codebases. Anthropic’s Claude Code has emerged as a leading agentic coding platform capable of operating autonomously.”

“Amazon Q Developer specializes in cloud-native development within AWS environments. OpenAI has evolved from primarily supplying the underlying models for other products to offering its own autonomous coding agents through Codex.”

“Cursor has become one of the fastest-growing software products in history, transforming from an AI-powered editor into a platform where developers manage teams of autonomous software agents, using large language models from different companies.”

Industry layoff trackers estimate that well over 400,000 technology workers were laid off in the U.S. during the 18 months between January 2025 and June 2026.

McCaffrey commented, “AI coding assistants are fueling a technology employment Armageddon. Some research firms suggest that AI will eventually create more tech jobs than it eliminates. I am extremely pessimistic about that claim, at least for the short term of the next five years or so.”

“The hundreds of thousands of jobs that have already been eliminated will not reappear in another form for the foreseeable future, and many tens or even hundreds of thousands more tech jobs will be wiped out in the near future.”

AI coding is having a profound effect on society.

The early James Bond films had a profound effect on culture. The early movies were released in the 1960s, just a few years after the end of World War II, at a time when Japan was almost completely unknown to audiences. These early films have not held up too well over time, but their impact at the time was huge.

Left: In “Thunderball” (1965), Bond briefly cooperates with Madame LaPorte (actress Maryse Guy Mitsouko), a French secret service agent. Bond must recover two stolen nuclear bombs that have been hidden in the Caribbean. He does.

Right: In “You Only Live Twice” (1967), Bond cooperates with Aki (actress Akiko Wakabayashi), a Japanese secret service agent. Bond must stop a plan by the evil SPECTRE organization to start a nuclear war between the U.S. and the Soviet Union. He does.

Posted in Uncategorized | Leave a comment

Why I Never Use a Voting Regressor Model for Machine Learning Regression

Posted on June 17, 2026 by jamesdmccaffrey

Bottom line: I never use a voting regressor model because the number of meta parameters (which regression models to use) and model parameters (such as a model learning rate) is astronomically large and therefore unmanageable in practice.

A voting regressor model is conceptually simple. Instead of using a particular model (linear regression, quadratic regression, kernel ridge, nearest neighbors, neural network, random forest, adaptive boosting, gradient boosting, etc., etc., etc.), you use several models and then use the average of their predictions.

The problem is that there are just too many parameters. First you have to choose which models to use, and then each model has on average about three to five parameters to specify. This just isn’t feasible.

I put together a demo using the VotingRegressor model from the scikit-learn library. For my demo I used the Diabetes Dataset, and I used three models: nearest neighbors, kernel ridge, gradient boosting. I knew from previous explorations that the default target in the Diabetes Dataset cannot be predicted with meaningful accuracy. As expected, the VotingRegressor model failed to give good accuracy.

The raw Diabetes Dataset looks like:

59, 2, 32.1, 101.00, 157,  93.2, 38, 4.00, 4.8598, 87, 151
48, 1, 21.6,  87.00, 183, 103.2, 70, 3.00, 3.8918, 69,  75
72, 2, 30.5,  93.00, 156,  93.6, 41, 4.00, 4.6728, 85, 141
. . .

Note: The sex encoding isn’t explained but I suspect male = 1, female = 2 because there are 235 1 values and 206 2 values). Also, the scikit Diabetes Dataset is not the same as the Pima Diabetes dataset.

I re-encoded the sex predictor from 1,2 to 0,1. I normalized the other predictor values using divide-by-constant normalization (100, 1, 100, 1000, 1000, 1000, 100, 10, 10, 1000, 1000). I normalized the target y values by dividing by 1000. The resulting data is:

0.5900, 1.0000, 0.3210, . . . 0.1510
0.4800, 0.0000, 0.2160, . . . 0.0750
0.7200, 1.0000, 0.3050, . . . 0.1410
. . .

I used the first 342 data items for training and the remaining 100 items for testing. I experimented with other splitting ideas and normalization techniques, but none had any significant effect.

The output of my demo is:

Begin Diabetes Dataset using VotingRegressor

Loading diabetes train (342), test (100) data
Done

First three X predictors:
[0.5900 1.0000 0.3210 0.1010 0.1570 . . . 0.0870]
[0.4800 0.0000 0.2160 0.0870 0.1830 . . . 0.0690]
[0.7200 1.0000 0.3050 0.0930 0.1560 . . . 0.0850]

First three y targets:
0.1510
0.0750
0.1410

Creating nearest neighbors, kernel ridge, 
  gradient boost regressors

Creating voting regressor model
Done

acc train (0.10) = 0.2895
acc test (0.10) = 0.1630

End demo

The voting regressor model had poor accuracy as expected. A prediction is scored as correct if it’s within 10% of the true target value.

If you use the built-in load_diabetes() function with a return_X_y=True parameter, column [10] is automatically the target to predict. But I discovered that columns [4], [5], [6], [7], and [8] can be predicted meaningfully. You can load the data as a DataFrame and then specify a different column as the target. Alternatively, you can use preprocessed training data.

The key lines in the demo program are:

. . .
r1 = KNeighborsRegressor(n_neighbors=3)
r2 = KernelRidge(kernel='rbf', gamma=0.005, alpha=0.01)
r3 = GradientBoostingRegressor(n_estimators=100,
  max_depth=5, learning_rate=0.10, random_state=0)

model = VotingRegressor([('knn', r1), ('krr', r2), ('gbr', r3)])
model.fit(train_X, train_y)
. . .

An interesting exploration.

Different regression techniques have different characteristics, and unfortunately, there’s no principled way to select which techniques to use in voting ression.

My two favorite movie genres are science fiction, and comedy. Here are three films that, in my opinion, have a good balance of science fiction and comedy. I’ve noticed that opinion among my friends about these three movies is quite polarized.

Left: In “Galaxy Quest” (1999), a group of past-their-prime actors from a Star Trek like TV series are recruited by aliens to save them from a really evil other alien race. I give this movie my personal A grade.

Center: In “The Fifth Element” (1997), a retired soldier is called back to service to literally save the Earth. He is aided by a female alien. I give this movie my personal A grade. I actually have a tattoo of the four elemental symbols used in the movie on my back. Seriously, I really do.

Right: In “Dude, Where’s My Car?” (2000), two losers have all kinds of outlandish adventures and meet different alien races, while saving the Universe from extinction. I give this movie my personal B+ grade.

Demo program. Replace the “lt” in the accuracy() function with the Boolean less-than symbol (my blog editor chokes on symbols).

# diabetes_voting_scikit.py
# various techniques for the Diabetes Dataset

import numpy as np
from sklearn.neighbors import KNeighborsRegressor
from sklearn.kernel_ridge import KernelRidge
# from sklearn.tree import DecisionTreeRegressor
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.ensemble import VotingRegressor

np.set_printoptions(precision=4, suppress=True,
  floatmode='fixed', linewidth=120)

# -----------------------------------------------------------

def accuracy(model, data_X, data_y, pct_close):
  n = len(data_X)
  n_correct = 0; n_wrong = 0
  for i in range(n):
    x = data_X[i].reshape(1,-1)
    y = data_y[i]
    y_pred = model.predict(x)

    if np.abs(y - y_pred) "lt" np.abs(y * pct_close):
      n_correct += 1
    else: 
      n_wrong += 1
  # print("Correct = " + str(n_correct))
  # print("Wrong   = " + str(n_wrong))
  return n_correct / (n_correct + n_wrong)

# -----------------------------------------------------------

def MSE(model, data_X, data_y):
  n = len(data_X)
  sum = 0.0
  for i in range(n):
    x = data_X[i].reshape(1,-1)
    y = data_y[i]
    y_pred = model.predict(x)[0]
    # print(y_pred); input()
    sum += (y - y_pred) * (y - y_pred)

  return sum / n

# -----------------------------------------------------------

print("\nBegin Diabetes Dataset using VotingRegressor ")

print("\nLoading diabetes train (342), test (100) data ")
train_file = ".\\Data\\diabetes_norm_train_342.txt"

cols_X = [0,1,2,3,4,5,6,7,8,9]  
col_y = 10  # cols # 4 5 6 7 8 are much better
train_X = np.loadtxt(train_file, comments="#",
  usecols=cols_X, delimiter=",",  dtype=np.float64)
train_y = np.loadtxt(train_file, comments="#",
  usecols=col_y, delimiter=",",  dtype=np.float64)

test_file = ".\\Data\\diabetes_norm_test_100.txt"
test_X = np.loadtxt(test_file, comments="#",
  usecols=cols_X, delimiter=",",  dtype=np.float64)
test_y = np.loadtxt(test_file, comments="#",
  usecols=col_y, delimiter=",",  dtype=np.float64)
print("Done ")

# alternative normalization and split
# from sklearn.datasets import load_diabetes
# from sklearn.model_selection import train_test_split
# X, y = load_diabetes(return_X_y=True, scaled=True)
# train_X, test_X, train_y, test_y = \
#   train_test_split(X, y, random_state=0)  # 25% test

print("\nFirst three X predictors: ")
for i in range(3):
  print(train_X[i])
print("\nFirst three y targets: ")
for i in range(3):
  print("%0.4f" % train_y[i])

print("\nCreating nearest neighbors, kernel ridge, " + \
  "gradient boost regressors ")
r1 = KNeighborsRegressor(n_neighbors=3)
r2 = KernelRidge(kernel='rbf', gamma=0.005, alpha=0.01)
r3 = GradientBoostingRegressor(n_estimators=100,
  max_depth=5, learning_rate=0.10, random_state=0)

print("\nCreating voting regressor model ")
model = VotingRegressor([('knn', r1), ('krr', r2), ('gbr', r3)])
model.fit(train_X, train_y)
print("Done ")

acc_train = accuracy(model, train_X, train_y, 0.10)
print("\nacc train (0.10) = %0.4f " % acc_train)
acc_test = accuracy(model, test_X, test_y, 0.10)
print("acc test (0.10) = %0.4f " % acc_test)

print("\nEnd demo ")

Training data:


# diabetes_norm_train_342.txt
# cols [0] to [9] predictors. col [10] target
# norm division constants:
# 100, -1, 100, 1000, 1000, 1000, 100, 10, 10, 1000, 1000
#
0.5900, 1.0000, 0.3210, 0.1010, 0.1570, 0.0932, 0.3800, 0.4000, 0.4860, 0.0870, 0.1510
0.4800, 0.0000, 0.2160, 0.0870, 0.1830, 0.1032, 0.7000, 0.3000, 0.3892, 0.0690, 0.0750
0.7200, 1.0000, 0.3050, 0.0930, 0.1560, 0.0936, 0.4100, 0.4000, 0.4673, 0.0850, 0.1410
0.2400, 0.0000, 0.2530, 0.0840, 0.1980, 0.1314, 0.4000, 0.5000, 0.4890, 0.0890, 0.2060
0.5000, 0.0000, 0.2300, 0.1010, 0.1920, 0.1254, 0.5200, 0.4000, 0.4291, 0.0800, 0.1350
0.2300, 0.0000, 0.2260, 0.0890, 0.1390, 0.0648, 0.6100, 0.2000, 0.4190, 0.0680, 0.0970
0.3600, 1.0000, 0.2200, 0.0900, 0.1600, 0.0996, 0.5000, 0.3000, 0.3951, 0.0820, 0.1380
0.6600, 1.0000, 0.2620, 0.1140, 0.2550, 0.1850, 0.5600, 0.4550, 0.4249, 0.0920, 0.0630
0.6000, 1.0000, 0.3210, 0.0830, 0.1790, 0.1194, 0.4200, 0.4000, 0.4477, 0.0940, 0.1100
0.2900, 0.0000, 0.3000, 0.0850, 0.1800, 0.0934, 0.4300, 0.4000, 0.5385, 0.0880, 0.3100
0.2200, 0.0000, 0.1860, 0.0970, 0.1140, 0.0576, 0.4600, 0.2000, 0.3951, 0.0830, 0.1010
0.5600, 1.0000, 0.2800, 0.0850, 0.1840, 0.1448, 0.3200, 0.6000, 0.3584, 0.0770, 0.0690
0.5300, 0.0000, 0.2370, 0.0920, 0.1860, 0.1092, 0.6200, 0.3000, 0.4304, 0.0810, 0.1790
0.5000, 1.0000, 0.2620, 0.0970, 0.1860, 0.1054, 0.4900, 0.4000, 0.5063, 0.0880, 0.1850
0.6100, 0.0000, 0.2400, 0.0910, 0.2020, 0.1154, 0.7200, 0.3000, 0.4291, 0.0730, 0.1180
0.3400, 1.0000, 0.2470, 0.1180, 0.2540, 0.1842, 0.3900, 0.7000, 0.5037, 0.0810, 0.1710
0.4700, 0.0000, 0.3030, 0.1090, 0.2070, 0.1002, 0.7000, 0.3000, 0.5215, 0.0980, 0.1660
0.6800, 1.0000, 0.2750, 0.1110, 0.2140, 0.1470, 0.3900, 0.5000, 0.4942, 0.0910, 0.1440
0.3800, 0.0000, 0.2540, 0.0840, 0.1620, 0.1030, 0.4200, 0.4000, 0.4443, 0.0870, 0.0970
0.4100, 0.0000, 0.2470, 0.0830, 0.1870, 0.1082, 0.6000, 0.3000, 0.4543, 0.0780, 0.1680
0.3500, 0.0000, 0.2110, 0.0820, 0.1560, 0.0878, 0.5000, 0.3000, 0.4511, 0.0950, 0.0680
0.2500, 1.0000, 0.2430, 0.0950, 0.1620, 0.0986, 0.5400, 0.3000, 0.3850, 0.0870, 0.0490
0.2500, 0.0000, 0.2600, 0.0920, 0.1870, 0.1204, 0.5600, 0.3000, 0.3970, 0.0880, 0.0680
0.6100, 1.0000, 0.3200, 0.1037, 0.2100, 0.0852, 0.3500, 0.6000, 0.6107, 0.1240, 0.2450
0.3100, 0.0000, 0.2970, 0.0880, 0.1670, 0.1034, 0.4800, 0.4000, 0.4357, 0.0780, 0.1840
0.3000, 1.0000, 0.2520, 0.0830, 0.1780, 0.1184, 0.3400, 0.5000, 0.4852, 0.0830, 0.2020
0.1900, 0.0000, 0.1920, 0.0870, 0.1240, 0.0540, 0.5700, 0.2000, 0.4174, 0.0900, 0.1370
0.4200, 0.0000, 0.3190, 0.0830, 0.1580, 0.0876, 0.5300, 0.3000, 0.4466, 0.1010, 0.0850
0.6300, 0.0000, 0.2440, 0.0730, 0.1600, 0.0914, 0.4800, 0.3000, 0.4635, 0.0780, 0.1310
0.6700, 1.0000, 0.2580, 0.1130, 0.1580, 0.0542, 0.6400, 0.2000, 0.5293, 0.1040, 0.2830
0.3200, 0.0000, 0.3050, 0.0890, 0.1820, 0.1106, 0.5600, 0.3000, 0.4344, 0.0890, 0.1290
0.4200, 0.0000, 0.2030, 0.0710, 0.1610, 0.0812, 0.6600, 0.2000, 0.4234, 0.0810, 0.0590
0.5800, 1.0000, 0.3800, 0.1030, 0.1500, 0.1072, 0.2200, 0.7000, 0.4644, 0.0980, 0.3410
0.5700, 0.0000, 0.2170, 0.0940, 0.1570, 0.0580, 0.8200, 0.2000, 0.4443, 0.0920, 0.0870
0.5300, 0.0000, 0.2050, 0.0780, 0.1470, 0.0842, 0.5200, 0.3000, 0.3989, 0.0750, 0.0650
0.6200, 1.0000, 0.2350, 0.0803, 0.2250, 0.1128, 0.8600, 0.2620, 0.4875, 0.0960, 0.1020
0.5200, 0.0000, 0.2850, 0.1100, 0.1950, 0.0972, 0.6000, 0.3000, 0.5242, 0.0850, 0.2650
0.4600, 0.0000, 0.2740, 0.0780, 0.1710, 0.0880, 0.5800, 0.3000, 0.4828, 0.0900, 0.2760
0.4800, 1.0000, 0.3300, 0.1230, 0.2530, 0.1636, 0.4400, 0.6000, 0.5425, 0.0970, 0.2520
0.4800, 1.0000, 0.2770, 0.0730, 0.1910, 0.1194, 0.4600, 0.4000, 0.4852, 0.0920, 0.0900
0.5000, 1.0000, 0.2560, 0.1010, 0.2290, 0.1622, 0.4300, 0.5000, 0.4779, 0.1140, 0.1000
0.2100, 0.0000, 0.2010, 0.0630, 0.1350, 0.0690, 0.5400, 0.3000, 0.4094, 0.0890, 0.0550
0.3200, 1.0000, 0.2540, 0.0903, 0.1530, 0.1004, 0.3400, 0.4500, 0.4533, 0.0830, 0.0610
0.5400, 0.0000, 0.2420, 0.0740, 0.2040, 0.1090, 0.8200, 0.2000, 0.4174, 0.1090, 0.0920
0.6100, 1.0000, 0.3270, 0.0970, 0.1770, 0.1184, 0.2900, 0.6000, 0.4997, 0.0870, 0.2590
0.5600, 1.0000, 0.2310, 0.1040, 0.1810, 0.1164, 0.4700, 0.4000, 0.4477, 0.0790, 0.0530
0.3300, 0.0000, 0.2530, 0.0850, 0.1550, 0.0850, 0.5100, 0.3000, 0.4554, 0.0700, 0.1900
0.2700, 0.0000, 0.1960, 0.0780, 0.1280, 0.0680, 0.4300, 0.3000, 0.4443, 0.0710, 0.1420
0.6700, 1.0000, 0.2250, 0.0980, 0.1910, 0.1192, 0.6100, 0.3000, 0.3989, 0.0860, 0.0750
0.3700, 1.0000, 0.2770, 0.0930, 0.1800, 0.1194, 0.3000, 0.6000, 0.5030, 0.0880, 0.1420
0.5800, 0.0000, 0.2570, 0.0990, 0.1570, 0.0916, 0.4900, 0.3000, 0.4407, 0.0930, 0.1550
0.6500, 1.0000, 0.2790, 0.1030, 0.1590, 0.0968, 0.4200, 0.4000, 0.4615, 0.0860, 0.2250
0.3400, 0.0000, 0.2550, 0.0930, 0.2180, 0.1440, 0.5700, 0.4000, 0.4443, 0.0880, 0.0590
0.4600, 0.0000, 0.2490, 0.1150, 0.1980, 0.1296, 0.5400, 0.4000, 0.4277, 0.1030, 0.1040
0.3500, 0.0000, 0.2870, 0.0970, 0.2040, 0.1268, 0.6400, 0.3000, 0.4190, 0.0930, 0.1820
0.3700, 0.0000, 0.2180, 0.0840, 0.1840, 0.1010, 0.7300, 0.3000, 0.3912, 0.0930, 0.1280
0.3700, 0.0000, 0.3020, 0.0870, 0.1660, 0.0960, 0.4000, 0.4150, 0.5011, 0.0870, 0.0520
0.4100, 0.0000, 0.2050, 0.0800, 0.1240, 0.0488, 0.6400, 0.2000, 0.4025, 0.0750, 0.0370
0.6000, 0.0000, 0.2040, 0.1050, 0.1980, 0.0784, 0.9900, 0.2000, 0.4635, 0.0790, 0.1700
0.6600, 1.0000, 0.2400, 0.0980, 0.2360, 0.1464, 0.5800, 0.4000, 0.5063, 0.0960, 0.1700
0.2900, 0.0000, 0.2600, 0.0830, 0.1410, 0.0652, 0.6400, 0.2000, 0.4078, 0.0830, 0.0610
0.3700, 1.0000, 0.2680, 0.0790, 0.1570, 0.0980, 0.2800, 0.6000, 0.5043, 0.0960, 0.1440
0.4100, 1.0000, 0.2570, 0.0830, 0.1810, 0.1066, 0.6600, 0.3000, 0.3738, 0.0850, 0.0520
0.3900, 0.0000, 0.2290, 0.0770, 0.2040, 0.1432, 0.4600, 0.4000, 0.4304, 0.0740, 0.1280
0.6700, 1.0000, 0.2400, 0.0830, 0.1430, 0.0772, 0.4900, 0.3000, 0.4431, 0.0940, 0.0710
0.3600, 1.0000, 0.2410, 0.1120, 0.1930, 0.1250, 0.3500, 0.6000, 0.5106, 0.0950, 0.1630
0.4600, 1.0000, 0.2470, 0.0850, 0.1740, 0.1232, 0.3000, 0.6000, 0.4644, 0.0960, 0.1500
0.6000, 1.0000, 0.2500, 0.0897, 0.1850, 0.1208, 0.4600, 0.4020, 0.4511, 0.0920, 0.0970
0.5900, 1.0000, 0.2360, 0.0830, 0.1650, 0.1000, 0.4700, 0.4000, 0.4500, 0.0920, 0.1600
0.5300, 0.0000, 0.2210, 0.0930, 0.1340, 0.0762, 0.4600, 0.3000, 0.4078, 0.0960, 0.1780
0.4800, 0.0000, 0.1990, 0.0910, 0.1890, 0.1096, 0.6900, 0.3000, 0.3951, 0.1010, 0.0480
0.4800, 0.0000, 0.2950, 0.1310, 0.2070, 0.1322, 0.4700, 0.4000, 0.4935, 0.1060, 0.2700
0.6600, 1.0000, 0.2600, 0.0910, 0.2640, 0.1466, 0.6500, 0.4000, 0.5568, 0.0870, 0.2020
0.5200, 1.0000, 0.2450, 0.0940, 0.2170, 0.1494, 0.4800, 0.5000, 0.4585, 0.0890, 0.1110
0.5200, 1.0000, 0.2660, 0.1110, 0.2090, 0.1264, 0.6100, 0.3000, 0.4682, 0.1090, 0.0850
0.4600, 1.0000, 0.2350, 0.0870, 0.1810, 0.1148, 0.4400, 0.4000, 0.4710, 0.0980, 0.0420
0.4000, 1.0000, 0.2900, 0.1150, 0.0970, 0.0472, 0.3500, 0.2770, 0.4304, 0.0950, 0.1700
0.2200, 0.0000, 0.2300, 0.0730, 0.1610, 0.0978, 0.5400, 0.3000, 0.3829, 0.0910, 0.2000
0.5000, 0.0000, 0.2100, 0.0880, 0.1400, 0.0718, 0.3500, 0.4000, 0.5112, 0.0710, 0.2520
0.2000, 0.0000, 0.2290, 0.0870, 0.1910, 0.1282, 0.5300, 0.4000, 0.3892, 0.0850, 0.1130
0.6800, 0.0000, 0.2750, 0.1070, 0.2410, 0.1496, 0.6400, 0.4000, 0.4920, 0.0900, 0.1430
0.5200, 1.0000, 0.2430, 0.0860, 0.1970, 0.1336, 0.4400, 0.5000, 0.4575, 0.0910, 0.0510
0.4400, 0.0000, 0.2310, 0.0870, 0.2130, 0.1264, 0.7700, 0.3000, 0.3871, 0.0720, 0.0520
0.3800, 0.0000, 0.2730, 0.0810, 0.1460, 0.0816, 0.4700, 0.3000, 0.4466, 0.0810, 0.2100
0.4900, 0.0000, 0.2270, 0.0653, 0.1680, 0.0962, 0.6200, 0.2710, 0.3892, 0.0600, 0.0650
0.6100, 0.0000, 0.3300, 0.0950, 0.1820, 0.1148, 0.5400, 0.3000, 0.4190, 0.0740, 0.1410
0.2900, 1.0000, 0.1940, 0.0830, 0.1520, 0.1058, 0.3900, 0.4000, 0.3584, 0.0830, 0.0550
0.6100, 0.0000, 0.2580, 0.0980, 0.2350, 0.1258, 0.7600, 0.3000, 0.5112, 0.0820, 0.1340
0.3400, 1.0000, 0.2260, 0.0750, 0.1660, 0.0918, 0.6000, 0.3000, 0.4263, 0.1080, 0.0420
0.3600, 0.0000, 0.2190, 0.0890, 0.1890, 0.1052, 0.6800, 0.3000, 0.4369, 0.0960, 0.1110
0.5200, 0.0000, 0.2400, 0.0830, 0.1670, 0.0866, 0.7100, 0.2000, 0.3850, 0.0940, 0.0980
0.6100, 0.0000, 0.3120, 0.0790, 0.2350, 0.1568, 0.4700, 0.5000, 0.5050, 0.0960, 0.1640
0.4300, 0.0000, 0.2680, 0.1230, 0.1930, 0.1022, 0.6700, 0.3000, 0.4779, 0.0940, 0.0480
0.3500, 0.0000, 0.2040, 0.0650, 0.1870, 0.1056, 0.6700, 0.2790, 0.4277, 0.0780, 0.0960
0.2700, 0.0000, 0.2480, 0.0910, 0.1890, 0.1068, 0.6900, 0.3000, 0.4190, 0.0690, 0.0900
0.2900, 0.0000, 0.2100, 0.0710, 0.1560, 0.0970, 0.3800, 0.4000, 0.4654, 0.0900, 0.1620
0.6400, 1.0000, 0.2730, 0.1090, 0.1860, 0.1076, 0.3800, 0.5000, 0.5308, 0.0990, 0.1500
0.4100, 0.0000, 0.3460, 0.0873, 0.2050, 0.1426, 0.4100, 0.5000, 0.4673, 0.1100, 0.2790
0.4900, 1.0000, 0.2590, 0.0910, 0.1780, 0.1066, 0.5200, 0.3000, 0.4575, 0.0750, 0.0920
0.4800, 0.0000, 0.2040, 0.0980, 0.2090, 0.1394, 0.4600, 0.5000, 0.4771, 0.0780, 0.0830
0.5300, 0.0000, 0.2800, 0.0880, 0.2330, 0.1438, 0.5800, 0.4000, 0.5050, 0.0910, 0.1280
0.5300, 1.0000, 0.2220, 0.1130, 0.1970, 0.1152, 0.6700, 0.3000, 0.4304, 0.1000, 0.1020
0.2300, 0.0000, 0.2900, 0.0900, 0.2160, 0.1314, 0.6500, 0.3000, 0.4585, 0.0910, 0.3020
0.6500, 1.0000, 0.3020, 0.0980, 0.2190, 0.1606, 0.4000, 0.5000, 0.4522, 0.0840, 0.1980
0.4100, 0.0000, 0.3240, 0.0940, 0.1710, 0.1044, 0.5600, 0.3000, 0.3970, 0.0760, 0.0950
0.5500, 1.0000, 0.2340, 0.0830, 0.1660, 0.1016, 0.4600, 0.4000, 0.4522, 0.0960, 0.0530
0.2200, 0.0000, 0.1930, 0.0820, 0.1560, 0.0932, 0.5200, 0.3000, 0.3989, 0.0710, 0.1340
0.5600, 0.0000, 0.3100, 0.0787, 0.1870, 0.1414, 0.3400, 0.5500, 0.4060, 0.0900, 0.1440
0.5400, 1.0000, 0.3060, 0.1033, 0.1440, 0.0798, 0.3000, 0.4800, 0.5142, 0.1010, 0.2320
0.5900, 1.0000, 0.2550, 0.0953, 0.1900, 0.1394, 0.3500, 0.5430, 0.4357, 0.1170, 0.0810
0.6000, 1.0000, 0.2340, 0.0880, 0.1530, 0.0898, 0.5800, 0.3000, 0.3258, 0.0950, 0.1040
0.5400, 0.0000, 0.2680, 0.0870, 0.2060, 0.1220, 0.6800, 0.3000, 0.4382, 0.0800, 0.0590
0.2500, 0.0000, 0.2830, 0.0870, 0.1930, 0.1280, 0.4900, 0.4000, 0.4382, 0.0920, 0.2460
0.5400, 1.0000, 0.2770, 0.1130, 0.2000, 0.1284, 0.3700, 0.5000, 0.5153, 0.1130, 0.2970
0.5500, 0.0000, 0.3660, 0.1130, 0.1990, 0.0944, 0.4300, 0.4630, 0.5730, 0.0970, 0.2580
0.4000, 1.0000, 0.2650, 0.0930, 0.2360, 0.1470, 0.3700, 0.7000, 0.5561, 0.0920, 0.2290
0.6200, 1.0000, 0.3180, 0.1150, 0.1990, 0.1286, 0.4400, 0.5000, 0.4883, 0.0980, 0.2750
0.6500, 0.0000, 0.2440, 0.1200, 0.2220, 0.1356, 0.3700, 0.6000, 0.5509, 0.1240, 0.2810
0.3300, 1.0000, 0.2540, 0.1020, 0.2060, 0.1410, 0.3900, 0.5000, 0.4868, 0.1050, 0.1790
0.5300, 0.0000, 0.2200, 0.0940, 0.1750, 0.0880, 0.5900, 0.3000, 0.4942, 0.0980, 0.2000
0.3500, 0.0000, 0.2680, 0.0980, 0.1620, 0.1036, 0.4500, 0.4000, 0.4205, 0.0860, 0.2000
0.6600, 0.0000, 0.2800, 0.1010, 0.1950, 0.1292, 0.4000, 0.5000, 0.4860, 0.0940, 0.1730
0.6200, 1.0000, 0.3390, 0.1010, 0.2210, 0.1564, 0.3500, 0.6000, 0.4997, 0.1030, 0.1800
0.5000, 1.0000, 0.2960, 0.0943, 0.3000, 0.2424, 0.3300, 0.9090, 0.4812, 0.1090, 0.0840
0.4700, 0.0000, 0.2860, 0.0970, 0.1640, 0.0906, 0.5600, 0.3000, 0.4466, 0.0880, 0.1210
0.4700, 1.0000, 0.2560, 0.0940, 0.1650, 0.0748, 0.4000, 0.4000, 0.5526, 0.0930, 0.1610
0.2400, 0.0000, 0.2070, 0.0870, 0.1490, 0.0806, 0.6100, 0.2000, 0.3611, 0.0780, 0.0990
0.5800, 1.0000, 0.2620, 0.0910, 0.2170, 0.1242, 0.7100, 0.3000, 0.4691, 0.0680, 0.1090
0.3400, 0.0000, 0.2060, 0.0870, 0.1850, 0.1122, 0.5800, 0.3000, 0.4304, 0.0740, 0.1150
0.5100, 0.0000, 0.2790, 0.0960, 0.1960, 0.1222, 0.4200, 0.5000, 0.5069, 0.1200, 0.2680
0.3100, 1.0000, 0.3530, 0.1250, 0.1870, 0.1124, 0.4800, 0.4000, 0.4890, 0.1090, 0.2740
0.2200, 0.0000, 0.1990, 0.0750, 0.1750, 0.1086, 0.5400, 0.3000, 0.4127, 0.0720, 0.1580
0.5300, 1.0000, 0.2440, 0.0920, 0.2140, 0.1460, 0.5000, 0.4000, 0.4500, 0.0970, 0.1070
0.3700, 1.0000, 0.2140, 0.0830, 0.1280, 0.0696, 0.4900, 0.3000, 0.3850, 0.0840, 0.0830
0.2800, 0.0000, 0.3040, 0.0850, 0.1980, 0.1156, 0.6700, 0.3000, 0.4344, 0.0800, 0.1030
0.4700, 0.0000, 0.3160, 0.0840, 0.1540, 0.0880, 0.3000, 0.5100, 0.5199, 0.1050, 0.2720
0.2300, 0.0000, 0.1880, 0.0780, 0.1450, 0.0720, 0.6300, 0.2000, 0.3912, 0.0860, 0.0850
0.5000, 0.0000, 0.3100, 0.1230, 0.1780, 0.1050, 0.4800, 0.4000, 0.4828, 0.0880, 0.2800
0.5800, 1.0000, 0.3670, 0.1170, 0.1660, 0.0938, 0.4400, 0.4000, 0.4949, 0.1090, 0.3360
0.5500, 0.0000, 0.3210, 0.1100, 0.1640, 0.0842, 0.4200, 0.4000, 0.5242, 0.0900, 0.2810
0.6000, 1.0000, 0.2770, 0.1070, 0.1670, 0.1146, 0.3800, 0.4000, 0.4277, 0.0950, 0.1180
0.4100, 0.0000, 0.3080, 0.0810, 0.2140, 0.1520, 0.2800, 0.7600, 0.5136, 0.1230, 0.3170
0.6000, 1.0000, 0.2750, 0.1060, 0.2290, 0.1438, 0.5100, 0.4000, 0.5142, 0.0910, 0.2350
0.4000, 0.0000, 0.2690, 0.0920, 0.2030, 0.1198, 0.7000, 0.3000, 0.4190, 0.0810, 0.0600
0.5700, 1.0000, 0.3070, 0.0900, 0.2040, 0.1478, 0.3400, 0.6000, 0.4710, 0.0930, 0.1740
0.3700, 0.0000, 0.3830, 0.1130, 0.1650, 0.0946, 0.5300, 0.3000, 0.4466, 0.0790, 0.2590
0.4000, 1.0000, 0.3190, 0.0950, 0.1980, 0.1356, 0.3800, 0.5000, 0.4804, 0.0930, 0.1780
0.3300, 0.0000, 0.3500, 0.0890, 0.2000, 0.1304, 0.4200, 0.4760, 0.4927, 0.1010, 0.1280
0.3200, 1.0000, 0.2780, 0.0890, 0.2160, 0.1462, 0.5500, 0.4000, 0.4304, 0.0910, 0.0960
0.3500, 1.0000, 0.2590, 0.0810, 0.1740, 0.1024, 0.3100, 0.6000, 0.5313, 0.0820, 0.1260
0.5500, 0.0000, 0.3290, 0.1020, 0.1640, 0.1062, 0.4100, 0.4000, 0.4431, 0.0890, 0.2880
0.4900, 0.0000, 0.2600, 0.0930, 0.1830, 0.1002, 0.6400, 0.3000, 0.4543, 0.0880, 0.0880
0.3900, 1.0000, 0.2630, 0.1150, 0.2180, 0.1582, 0.3200, 0.7000, 0.4935, 0.1090, 0.2920
0.6000, 1.0000, 0.2230, 0.1130, 0.1860, 0.1258, 0.4600, 0.4000, 0.4263, 0.0940, 0.0710
0.6700, 1.0000, 0.2830, 0.0930, 0.2040, 0.1322, 0.4900, 0.4000, 0.4736, 0.0920, 0.1970
0.4100, 1.0000, 0.3200, 0.1090, 0.2510, 0.1706, 0.4900, 0.5000, 0.5056, 0.1030, 0.1860
0.4400, 0.0000, 0.2540, 0.0950, 0.1620, 0.0926, 0.5300, 0.3000, 0.4407, 0.0830, 0.0250
0.4800, 1.0000, 0.2330, 0.0893, 0.2120, 0.1428, 0.4600, 0.4610, 0.4754, 0.0980, 0.0840
0.4500, 0.0000, 0.2030, 0.0743, 0.1900, 0.1262, 0.4900, 0.3880, 0.4304, 0.0790, 0.0960
0.4700, 0.0000, 0.3040, 0.1200, 0.1990, 0.1200, 0.4600, 0.4000, 0.5106, 0.0870, 0.1950
0.4600, 0.0000, 0.2060, 0.0730, 0.1720, 0.1070, 0.5100, 0.3000, 0.4249, 0.0800, 0.0530
0.3600, 1.0000, 0.3230, 0.1150, 0.2860, 0.1994, 0.3900, 0.7000, 0.5472, 0.1120, 0.2170
0.3400, 0.0000, 0.2920, 0.0730, 0.1720, 0.1082, 0.4900, 0.4000, 0.4304, 0.0910, 0.1720
0.5300, 1.0000, 0.3310, 0.1170, 0.1830, 0.1190, 0.4800, 0.4000, 0.4382, 0.1060, 0.1310
0.6100, 0.0000, 0.2460, 0.1010, 0.2090, 0.1068, 0.7700, 0.3000, 0.4836, 0.0880, 0.2140
0.3700, 0.0000, 0.2020, 0.0810, 0.1620, 0.0878, 0.6300, 0.3000, 0.4025, 0.0880, 0.0590
0.3300, 1.0000, 0.2080, 0.0840, 0.1250, 0.0702, 0.4600, 0.3000, 0.3784, 0.0660, 0.0700
0.6800, 0.0000, 0.3280, 0.1057, 0.2050, 0.1164, 0.4000, 0.5130, 0.5493, 0.1170, 0.2200
0.4900, 1.0000, 0.3190, 0.0940, 0.2340, 0.1558, 0.3400, 0.7000, 0.5398, 0.1220, 0.2680
0.4800, 0.0000, 0.2390, 0.1090, 0.2320, 0.1052, 0.3700, 0.6000, 0.6107, 0.0960, 0.1520
0.5500, 1.0000, 0.2450, 0.0840, 0.1790, 0.1058, 0.6600, 0.3000, 0.3584, 0.0870, 0.0470
0.4300, 0.0000, 0.2210, 0.0660, 0.1340, 0.0772, 0.4500, 0.3000, 0.4078, 0.0800, 0.0740
0.6000, 1.0000, 0.3300, 0.0970, 0.2170, 0.1256, 0.4500, 0.5000, 0.5447, 0.1120, 0.2950
0.3100, 1.0000, 0.1900, 0.0930, 0.1370, 0.0730, 0.4700, 0.3000, 0.4443, 0.0780, 0.1010
0.5300, 1.0000, 0.2730, 0.0820, 0.1190, 0.0550, 0.3900, 0.3000, 0.4828, 0.0930, 0.1510
0.6700, 0.0000, 0.2280, 0.0870, 0.1660, 0.0986, 0.5200, 0.3000, 0.4344, 0.0920, 0.1270
0.6100, 1.0000, 0.2820, 0.1060, 0.2040, 0.1320, 0.5200, 0.4000, 0.4605, 0.0960, 0.2370
0.6200, 0.0000, 0.2890, 0.0873, 0.2060, 0.1272, 0.3300, 0.6240, 0.5434, 0.0990, 0.2250
0.6000, 0.0000, 0.2560, 0.0870, 0.2070, 0.1258, 0.6900, 0.3000, 0.4111, 0.0840, 0.0810
0.4200, 0.0000, 0.2490, 0.0910, 0.2040, 0.1418, 0.3800, 0.5000, 0.4796, 0.0890, 0.1510
0.3800, 1.0000, 0.2680, 0.1050, 0.1810, 0.1192, 0.3700, 0.5000, 0.4820, 0.0910, 0.1070
0.6200, 0.0000, 0.2240, 0.0790, 0.2220, 0.1474, 0.5900, 0.4000, 0.4357, 0.0760, 0.0640
0.6100, 1.0000, 0.2690, 0.1110, 0.2360, 0.1724, 0.3900, 0.6000, 0.4812, 0.0890, 0.1380
0.6100, 1.0000, 0.2310, 0.1130, 0.1860, 0.1144, 0.4700, 0.4000, 0.4812, 0.1050, 0.1850
0.5300, 0.0000, 0.2860, 0.0880, 0.1710, 0.0988, 0.4100, 0.4000, 0.5050, 0.0990, 0.2650
0.2800, 1.0000, 0.2470, 0.0970, 0.1750, 0.0996, 0.3200, 0.5000, 0.5380, 0.0870, 0.1010
0.2600, 1.0000, 0.3030, 0.0890, 0.2180, 0.1522, 0.3100, 0.7000, 0.5159, 0.0820, 0.1370
0.3000, 0.0000, 0.2130, 0.0870, 0.1340, 0.0630, 0.6300, 0.2000, 0.3689, 0.0660, 0.1430
0.5000, 0.0000, 0.2610, 0.1090, 0.2430, 0.1606, 0.6200, 0.4000, 0.4625, 0.0890, 0.1410
0.4800, 0.0000, 0.2020, 0.0950, 0.1870, 0.1174, 0.5300, 0.4000, 0.4419, 0.0850, 0.0790
0.5100, 0.0000, 0.2520, 0.1030, 0.1760, 0.1122, 0.3700, 0.5000, 0.4898, 0.0900, 0.2920
0.4700, 1.0000, 0.2250, 0.0820, 0.1310, 0.0668, 0.4100, 0.3000, 0.4754, 0.0890, 0.1780
0.6400, 1.0000, 0.2350, 0.0970, 0.2030, 0.1290, 0.5900, 0.3000, 0.4318, 0.0770, 0.0910
0.5100, 1.0000, 0.2590, 0.0760, 0.2400, 0.1690, 0.3900, 0.6000, 0.5075, 0.0960, 0.1160
0.3000, 0.0000, 0.2090, 0.1040, 0.1520, 0.0838, 0.4700, 0.3000, 0.4663, 0.0970, 0.0860
0.5600, 1.0000, 0.2870, 0.0990, 0.2080, 0.1464, 0.3900, 0.5000, 0.4727, 0.0970, 0.1220
0.4200, 0.0000, 0.2210, 0.0850, 0.2130, 0.1386, 0.6000, 0.4000, 0.4277, 0.0940, 0.0720
0.6200, 1.0000, 0.2670, 0.1150, 0.1830, 0.1240, 0.3500, 0.5000, 0.4788, 0.1000, 0.1290
0.3400, 0.0000, 0.3140, 0.0870, 0.1490, 0.0938, 0.4600, 0.3000, 0.3829, 0.0770, 0.1420
0.6000, 0.0000, 0.2220, 0.1047, 0.2210, 0.1054, 0.6000, 0.3680, 0.5628, 0.0930, 0.0900
0.6400, 0.0000, 0.2100, 0.0923, 0.2270, 0.1468, 0.6500, 0.3490, 0.4331, 0.1020, 0.1580
0.3900, 1.0000, 0.2120, 0.0900, 0.1820, 0.1104, 0.6000, 0.3000, 0.4060, 0.0980, 0.0390
0.7100, 1.0000, 0.2650, 0.1050, 0.2810, 0.1736, 0.5500, 0.5000, 0.5568, 0.0840, 0.1960
0.4800, 1.0000, 0.2920, 0.1100, 0.2180, 0.1516, 0.3900, 0.6000, 0.4920, 0.0980, 0.2220
0.7900, 1.0000, 0.2700, 0.1030, 0.1690, 0.1108, 0.3700, 0.5000, 0.4663, 0.1100, 0.2770
0.4000, 0.0000, 0.3070, 0.0990, 0.1770, 0.0854, 0.5000, 0.4000, 0.5338, 0.0850, 0.0990
0.4900, 1.0000, 0.2880, 0.0920, 0.2070, 0.1400, 0.4400, 0.5000, 0.4745, 0.0920, 0.1960
0.5100, 0.0000, 0.3060, 0.1030, 0.1980, 0.1066, 0.5700, 0.3000, 0.5148, 0.1000, 0.2020
0.5700, 0.0000, 0.3010, 0.1170, 0.2020, 0.1396, 0.4200, 0.5000, 0.4625, 0.1200, 0.1550
0.5900, 1.0000, 0.2470, 0.1140, 0.1520, 0.1048, 0.2900, 0.5000, 0.4511, 0.0880, 0.0770
0.5100, 0.0000, 0.2770, 0.0990, 0.2290, 0.1456, 0.6900, 0.3000, 0.4277, 0.0770, 0.1910
0.7400, 0.0000, 0.2980, 0.1010, 0.1710, 0.1048, 0.5000, 0.3000, 0.4394, 0.0860, 0.0700
0.6700, 0.0000, 0.2670, 0.1050, 0.2250, 0.1354, 0.6900, 0.3000, 0.4635, 0.0960, 0.0730
0.4900, 0.0000, 0.1980, 0.0880, 0.1880, 0.1148, 0.5700, 0.3000, 0.4394, 0.0930, 0.0490
0.5700, 0.0000, 0.2330, 0.0880, 0.1550, 0.0636, 0.7800, 0.2000, 0.4205, 0.0780, 0.0650
0.5600, 1.0000, 0.3510, 0.1230, 0.1640, 0.0950, 0.3800, 0.4000, 0.5043, 0.1170, 0.2630
0.5200, 1.0000, 0.2970, 0.1090, 0.2280, 0.1628, 0.3100, 0.8000, 0.5142, 0.1030, 0.2480
0.6900, 0.0000, 0.2930, 0.1240, 0.2230, 0.1390, 0.5400, 0.4000, 0.5011, 0.1020, 0.2960
0.3700, 0.0000, 0.2030, 0.0830, 0.1850, 0.1246, 0.3800, 0.5000, 0.4719, 0.0880, 0.2140
0.2400, 0.0000, 0.2250, 0.0890, 0.1410, 0.0680, 0.5200, 0.3000, 0.4654, 0.0840, 0.1850
0.5500, 1.0000, 0.2270, 0.0930, 0.1540, 0.0942, 0.5300, 0.3000, 0.3526, 0.0750, 0.0780
0.3600, 0.0000, 0.2280, 0.0870, 0.1780, 0.1160, 0.4100, 0.4000, 0.4654, 0.0820, 0.0930
0.4200, 1.0000, 0.2400, 0.1070, 0.1500, 0.0850, 0.4400, 0.3000, 0.4654, 0.0960, 0.2520
0.2100, 0.0000, 0.2420, 0.0760, 0.1470, 0.0770, 0.5300, 0.3000, 0.4443, 0.0790, 0.1500
0.4100, 0.0000, 0.2020, 0.0620, 0.1530, 0.0890, 0.5000, 0.3000, 0.4249, 0.0890, 0.0770
0.5700, 1.0000, 0.2940, 0.1090, 0.1600, 0.0876, 0.3100, 0.5000, 0.5333, 0.0920, 0.2080
0.2000, 1.0000, 0.2210, 0.0870, 0.1710, 0.0996, 0.5800, 0.3000, 0.4205, 0.0780, 0.0770
0.6700, 1.0000, 0.2360, 0.1113, 0.1890, 0.1054, 0.7000, 0.2700, 0.4220, 0.0930, 0.1080
0.3400, 0.0000, 0.2520, 0.0770, 0.1890, 0.1206, 0.5300, 0.4000, 0.4344, 0.0790, 0.1600
0.4100, 1.0000, 0.2490, 0.0860, 0.1920, 0.1150, 0.6100, 0.3000, 0.4382, 0.0940, 0.0530
0.3800, 1.0000, 0.3300, 0.0780, 0.3010, 0.2150, 0.5000, 0.6020, 0.5193, 0.1080, 0.2200
0.5100, 0.0000, 0.2350, 0.1010, 0.1950, 0.1210, 0.5100, 0.4000, 0.4745, 0.0940, 0.1540
0.5200, 1.0000, 0.2640, 0.0913, 0.2180, 0.1520, 0.3900, 0.5590, 0.4905, 0.0990, 0.2590
0.6700, 0.0000, 0.2980, 0.0800, 0.1720, 0.0934, 0.6300, 0.3000, 0.4357, 0.0820, 0.0900
0.6100, 0.0000, 0.3000, 0.1080, 0.1940, 0.1000, 0.5200, 0.3730, 0.5347, 0.1050, 0.2460
0.6700, 1.0000, 0.2500, 0.1117, 0.1460, 0.0934, 0.3300, 0.4420, 0.4585, 0.1030, 0.1240
0.5600, 0.0000, 0.2700, 0.1050, 0.2470, 0.1606, 0.5400, 0.5000, 0.5088, 0.0940, 0.0670
0.6400, 0.0000, 0.2000, 0.0747, 0.1890, 0.1148, 0.6200, 0.3050, 0.4111, 0.0910, 0.0720
0.5800, 1.0000, 0.2550, 0.1120, 0.1630, 0.1106, 0.2900, 0.6000, 0.4762, 0.0860, 0.2570
0.5500, 0.0000, 0.2820, 0.0910, 0.2500, 0.1402, 0.6700, 0.4000, 0.5366, 0.1030, 0.2620
0.6200, 1.0000, 0.3330, 0.1140, 0.1820, 0.1140, 0.3800, 0.5000, 0.5011, 0.0960, 0.2750
0.5700, 1.0000, 0.2560, 0.0960, 0.2000, 0.1330, 0.5200, 0.3850, 0.4318, 0.1050, 0.1770
0.2000, 1.0000, 0.2420, 0.0880, 0.1260, 0.0722, 0.4500, 0.3000, 0.3784, 0.0740, 0.0710
0.5300, 1.0000, 0.2210, 0.0980, 0.1650, 0.1052, 0.4700, 0.4000, 0.4159, 0.0810, 0.0470
0.3200, 1.0000, 0.3140, 0.0890, 0.1530, 0.0842, 0.5600, 0.3000, 0.4159, 0.0900, 0.1870
0.4100, 0.0000, 0.2310, 0.0860, 0.1480, 0.0780, 0.5800, 0.3000, 0.4094, 0.0600, 0.1250
0.6000, 0.0000, 0.2340, 0.0767, 0.2470, 0.1480, 0.6500, 0.3800, 0.5136, 0.0770, 0.0780
0.2600, 0.0000, 0.1880, 0.0830, 0.1910, 0.1036, 0.6900, 0.3000, 0.4522, 0.0690, 0.0510
0.3700, 0.0000, 0.3080, 0.1120, 0.2820, 0.1972, 0.4300, 0.7000, 0.5342, 0.1010, 0.2580
0.4500, 0.0000, 0.3200, 0.1100, 0.2240, 0.1342, 0.4500, 0.5000, 0.5412, 0.0930, 0.2150
0.6700, 0.0000, 0.3160, 0.1160, 0.1790, 0.0904, 0.4100, 0.4000, 0.5472, 0.1000, 0.3030
0.3400, 1.0000, 0.3550, 0.1200, 0.2330, 0.1466, 0.3400, 0.7000, 0.5568, 0.1010, 0.2430
0.5000, 0.0000, 0.3190, 0.0783, 0.2070, 0.1492, 0.3800, 0.5450, 0.4595, 0.0840, 0.0910
0.7100, 0.0000, 0.2950, 0.0970, 0.2270, 0.1516, 0.4500, 0.5000, 0.5024, 0.1080, 0.1500
0.5700, 1.0000, 0.3160, 0.1170, 0.2250, 0.1076, 0.4000, 0.6000, 0.5958, 0.1130, 0.3100
0.4900, 0.0000, 0.2030, 0.0930, 0.1840, 0.1030, 0.6100, 0.3000, 0.4605, 0.0930, 0.1530
0.3500, 0.0000, 0.4130, 0.0810, 0.1680, 0.1028, 0.3700, 0.5000, 0.4949, 0.0940, 0.3460
0.4100, 1.0000, 0.2120, 0.1020, 0.1840, 0.1004, 0.6400, 0.3000, 0.4585, 0.0790, 0.0630
0.7000, 1.0000, 0.2410, 0.0823, 0.1940, 0.1492, 0.3100, 0.6260, 0.4234, 0.1050, 0.0890
0.5200, 0.0000, 0.2300, 0.1070, 0.1790, 0.1237, 0.4250, 0.4210, 0.4159, 0.0930, 0.0500
0.6000, 0.0000, 0.2560, 0.0780, 0.1950, 0.0954, 0.9100, 0.2000, 0.3761, 0.0870, 0.0390
0.6200, 0.0000, 0.2250, 0.1250, 0.2150, 0.0990, 0.9800, 0.2000, 0.4500, 0.0950, 0.1030
0.4400, 1.0000, 0.3820, 0.1230, 0.2010, 0.1266, 0.4400, 0.5000, 0.5024, 0.0920, 0.3080
0.2800, 1.0000, 0.1920, 0.0810, 0.1550, 0.0946, 0.5100, 0.3000, 0.3850, 0.0870, 0.1160
0.5800, 1.0000, 0.2900, 0.0850, 0.1560, 0.1092, 0.3600, 0.4000, 0.3989, 0.0860, 0.1450
0.3900, 1.0000, 0.2400, 0.0897, 0.1900, 0.1136, 0.5200, 0.3650, 0.4804, 0.1010, 0.0740
0.3400, 1.0000, 0.2060, 0.0980, 0.1830, 0.0920, 0.8300, 0.2000, 0.3689, 0.0920, 0.0450
0.6500, 0.0000, 0.2630, 0.0700, 0.2440, 0.1662, 0.5100, 0.5000, 0.4898, 0.0980, 0.1150
0.6600, 1.0000, 0.3460, 0.1150, 0.2040, 0.1394, 0.3600, 0.6000, 0.4963, 0.1090, 0.2640
0.5100, 0.0000, 0.2340, 0.0870, 0.2200, 0.1088, 0.9300, 0.2000, 0.4511, 0.0820, 0.0870
0.5000, 1.0000, 0.2920, 0.1190, 0.1620, 0.0852, 0.5400, 0.3000, 0.4736, 0.0950, 0.2020
0.5900, 1.0000, 0.2720, 0.1070, 0.1580, 0.1020, 0.3900, 0.4000, 0.4443, 0.0930, 0.1270
0.5200, 0.0000, 0.2700, 0.0783, 0.1340, 0.0730, 0.4400, 0.3050, 0.4443, 0.0690, 0.1820
0.6900, 1.0000, 0.2450, 0.1080, 0.2430, 0.1364, 0.4000, 0.6000, 0.5808, 0.1000, 0.2410
0.5300, 0.0000, 0.2410, 0.1050, 0.1840, 0.1134, 0.4600, 0.4000, 0.4812, 0.0950, 0.0660
0.4700, 1.0000, 0.2530, 0.0980, 0.1730, 0.1056, 0.4400, 0.4000, 0.4762, 0.1080, 0.0940
0.5200, 0.0000, 0.2880, 0.1130, 0.2800, 0.1740, 0.6700, 0.4000, 0.5273, 0.0860, 0.2830
0.3900, 0.0000, 0.2090, 0.0950, 0.1500, 0.0656, 0.6800, 0.2000, 0.4407, 0.0950, 0.0640
0.6700, 1.0000, 0.2300, 0.0700, 0.1840, 0.1280, 0.3500, 0.5000, 0.4654, 0.0990, 0.1020
0.5900, 1.0000, 0.2410, 0.0960, 0.1700, 0.0986, 0.5400, 0.3000, 0.4466, 0.0850, 0.2000
0.5100, 1.0000, 0.2810, 0.1060, 0.2020, 0.1222, 0.5500, 0.4000, 0.4820, 0.0870, 0.2650
0.2300, 1.0000, 0.1800, 0.0780, 0.1710, 0.0960, 0.4800, 0.4000, 0.4905, 0.0920, 0.0940
0.6800, 0.0000, 0.2590, 0.0930, 0.2530, 0.1812, 0.5300, 0.5000, 0.4543, 0.0980, 0.2300
0.4400, 0.0000, 0.2150, 0.0850, 0.1570, 0.0922, 0.5500, 0.3000, 0.3892, 0.0840, 0.1810
0.6000, 1.0000, 0.2430, 0.1030, 0.1410, 0.0866, 0.3300, 0.4000, 0.4673, 0.0780, 0.1560
0.5200, 0.0000, 0.2450, 0.0900, 0.1980, 0.1290, 0.2900, 0.7000, 0.5298, 0.0860, 0.2330
0.3800, 0.0000, 0.2130, 0.0720, 0.1650, 0.0602, 0.8800, 0.2000, 0.4431, 0.0900, 0.0600
0.6100, 0.0000, 0.2580, 0.0900, 0.2800, 0.1954, 0.5500, 0.5000, 0.4997, 0.0900, 0.2190
0.6800, 1.0000, 0.2480, 0.1010, 0.2210, 0.1514, 0.6000, 0.4000, 0.3871, 0.0870, 0.0800
0.2800, 1.0000, 0.3150, 0.0830, 0.2280, 0.1494, 0.3800, 0.6000, 0.5313, 0.0830, 0.0680
0.6500, 1.0000, 0.3350, 0.1020, 0.1900, 0.1262, 0.3500, 0.5000, 0.4970, 0.1020, 0.3320
0.6900, 0.0000, 0.2810, 0.1130, 0.2340, 0.1428, 0.5200, 0.4000, 0.5278, 0.0770, 0.2480
0.5100, 0.0000, 0.2430, 0.0853, 0.1530, 0.0716, 0.7100, 0.2150, 0.3951, 0.0820, 0.0840
0.2900, 0.0000, 0.3500, 0.0983, 0.2040, 0.1426, 0.5000, 0.4080, 0.4043, 0.0910, 0.2000
0.5500, 1.0000, 0.2350, 0.0930, 0.1770, 0.1268, 0.4100, 0.4000, 0.3829, 0.0830, 0.0550
0.3400, 1.0000, 0.3000, 0.0830, 0.1850, 0.1072, 0.5300, 0.3000, 0.4820, 0.0920, 0.0850
0.6700, 0.0000, 0.2070, 0.0830, 0.1700, 0.0998, 0.5900, 0.3000, 0.4025, 0.0770, 0.0890
0.4900, 0.0000, 0.2560, 0.0760, 0.1610, 0.0998, 0.5100, 0.3000, 0.3932, 0.0780, 0.0310
0.5500, 1.0000, 0.2290, 0.0810, 0.1230, 0.0672, 0.4100, 0.3000, 0.4304, 0.0880, 0.1290
0.5900, 1.0000, 0.2510, 0.0900, 0.1630, 0.1014, 0.4600, 0.4000, 0.4357, 0.0910, 0.0830
0.5300, 0.0000, 0.3320, 0.0827, 0.1860, 0.1068, 0.4600, 0.4040, 0.5112, 0.1020, 0.2750
0.4800, 1.0000, 0.2410, 0.1100, 0.2090, 0.1346, 0.5800, 0.4000, 0.4407, 0.1000, 0.0650
0.5200, 0.0000, 0.2950, 0.1043, 0.2110, 0.1328, 0.4900, 0.4310, 0.4984, 0.0980, 0.1980
0.6900, 0.0000, 0.2960, 0.1220, 0.2310, 0.1284, 0.5600, 0.4000, 0.5451, 0.0860, 0.2360
0.6000, 1.0000, 0.2280, 0.1100, 0.2450, 0.1898, 0.3900, 0.6000, 0.4394, 0.0880, 0.2530
0.4600, 1.0000, 0.2270, 0.0830, 0.1830, 0.1258, 0.3200, 0.6000, 0.4836, 0.0750, 0.1240
0.5100, 1.0000, 0.2620, 0.1010, 0.1610, 0.0996, 0.4800, 0.3000, 0.4205, 0.0880, 0.0440
0.6700, 1.0000, 0.2350, 0.0960, 0.2070, 0.1382, 0.4200, 0.5000, 0.4898, 0.1110, 0.1720
0.4900, 0.0000, 0.2210, 0.0850, 0.1360, 0.0634, 0.6200, 0.2190, 0.3970, 0.0720, 0.1140
0.4600, 1.0000, 0.2650, 0.0940, 0.2470, 0.1602, 0.5900, 0.4000, 0.4935, 0.1110, 0.1420
0.4700, 0.0000, 0.3240, 0.1050, 0.1880, 0.1250, 0.4600, 0.4090, 0.4443, 0.0990, 0.1090
0.7500, 0.0000, 0.3010, 0.0780, 0.2220, 0.1542, 0.4400, 0.5050, 0.4779, 0.0970, 0.1800
0.2800, 0.0000, 0.2420, 0.0930, 0.1740, 0.1064, 0.5400, 0.3000, 0.4220, 0.0840, 0.1440
0.6500, 1.0000, 0.3130, 0.1100, 0.2130, 0.1280, 0.4700, 0.5000, 0.5247, 0.0910, 0.1630
0.4200, 0.0000, 0.3010, 0.0910, 0.1820, 0.1148, 0.4900, 0.4000, 0.4511, 0.0820, 0.1470
0.5100, 0.0000, 0.2450, 0.0790, 0.2120, 0.1286, 0.6500, 0.3000, 0.4522, 0.0910, 0.0970
0.5300, 1.0000, 0.2770, 0.0950, 0.1900, 0.1018, 0.4100, 0.5000, 0.5464, 0.1010, 0.2200
0.5400, 0.0000, 0.2320, 0.1107, 0.2380, 0.1628, 0.4800, 0.4960, 0.4913, 0.1080, 0.1900
0.7300, 0.0000, 0.2700, 0.1020, 0.2110, 0.1210, 0.6700, 0.3000, 0.4745, 0.0990, 0.1090
0.5400, 0.0000, 0.2680, 0.1080, 0.1760, 0.0806, 0.6700, 0.3000, 0.4956, 0.1060, 0.1910
0.4200, 0.0000, 0.2920, 0.0930, 0.2490, 0.1742, 0.4500, 0.6000, 0.5004, 0.0920, 0.1220
0.7500, 0.0000, 0.3120, 0.1177, 0.2290, 0.1388, 0.2900, 0.7900, 0.5724, 0.1060, 0.2300
0.5500, 1.0000, 0.3210, 0.1127, 0.2070, 0.0924, 0.2500, 0.8280, 0.6105, 0.1110, 0.2420
0.6800, 1.0000, 0.2570, 0.1090, 0.2330, 0.1126, 0.3500, 0.7000, 0.6057, 0.1050, 0.2480
0.5700, 0.0000, 0.2690, 0.0980, 0.2460, 0.1652, 0.3800, 0.7000, 0.5366, 0.0960, 0.2490
0.4800, 0.0000, 0.3140, 0.0753, 0.2420, 0.1516, 0.3800, 0.6370, 0.5568, 0.1030, 0.1920
0.6100, 1.0000, 0.2560, 0.0850, 0.1840, 0.1162, 0.3900, 0.5000, 0.4970, 0.0980, 0.1310
0.6900, 0.0000, 0.3700, 0.1030, 0.2070, 0.1314, 0.5500, 0.4000, 0.4635, 0.0900, 0.2370
0.3800, 0.0000, 0.3260, 0.0770, 0.1680, 0.1006, 0.4700, 0.4000, 0.4625, 0.0960, 0.0780
0.4500, 1.0000, 0.2120, 0.0940, 0.1690, 0.0968, 0.5500, 0.3000, 0.4454, 0.1020, 0.1350
0.5100, 1.0000, 0.2920, 0.1070, 0.1870, 0.1390, 0.3200, 0.6000, 0.4382, 0.0950, 0.2440
0.7100, 1.0000, 0.2400, 0.0840, 0.1380, 0.0858, 0.3900, 0.4000, 0.4190, 0.0900, 0.1990
0.5700, 0.0000, 0.3610, 0.1170, 0.1810, 0.1082, 0.3400, 0.5000, 0.5268, 0.1000, 0.2700
0.5600, 1.0000, 0.2580, 0.1030, 0.1770, 0.1144, 0.3400, 0.5000, 0.4963, 0.0990, 0.1640
0.3200, 1.0000, 0.2200, 0.0880, 0.1370, 0.0786, 0.4800, 0.3000, 0.3951, 0.0780, 0.0720
0.5000, 0.0000, 0.2190, 0.0910, 0.1900, 0.1112, 0.6700, 0.3000, 0.4078, 0.0770, 0.0960
0.4300, 0.0000, 0.3430, 0.0840, 0.2560, 0.1726, 0.3300, 0.8000, 0.5529, 0.1040, 0.3060
0.5400, 1.0000, 0.2520, 0.1150, 0.1810, 0.1200, 0.3900, 0.5000, 0.4701, 0.0920, 0.0910
0.3100, 0.0000, 0.2330, 0.0850, 0.1900, 0.1308, 0.4300, 0.4000, 0.4394, 0.0770, 0.2140
0.5600, 0.0000, 0.2570, 0.0800, 0.2440, 0.1516, 0.5900, 0.4000, 0.5118, 0.0950, 0.0950
0.4400, 0.0000, 0.2510, 0.1330, 0.1820, 0.1130, 0.5500, 0.3000, 0.4249, 0.0840, 0.2160
0.5700, 1.0000, 0.3190, 0.1110, 0.1730, 0.1162, 0.4100, 0.4000, 0.4369, 0.0870, 0.2630

Test data:


# diabetes_norm_test_100.txt
#
0.6400, 1.0000, 0.2840, 0.1110, 0.1840, 0.1270, 0.4100, 0.4000, 0.4382, 0.0970, 0.1780
0.4300, 0.0000, 0.2810, 0.1210, 0.1920, 0.1210, 0.6000, 0.3000, 0.4007, 0.0930, 0.1130
0.1900, 0.0000, 0.2530, 0.0830, 0.2250, 0.1566, 0.4600, 0.5000, 0.4719, 0.0840, 0.2000
0.7100, 1.0000, 0.2610, 0.0850, 0.2200, 0.1524, 0.4700, 0.5000, 0.4635, 0.0910, 0.1390
0.5000, 1.0000, 0.2800, 0.1040, 0.2820, 0.1968, 0.4400, 0.6000, 0.5328, 0.0950, 0.1390
0.5900, 1.0000, 0.2360, 0.0730, 0.1800, 0.1074, 0.5100, 0.4000, 0.4682, 0.0840, 0.0880
0.5700, 0.0000, 0.2450, 0.0930, 0.1860, 0.0966, 0.7100, 0.3000, 0.4522, 0.0910, 0.1480
0.4900, 1.0000, 0.2100, 0.0820, 0.1190, 0.0854, 0.2300, 0.5000, 0.3970, 0.0740, 0.0880
0.4100, 1.0000, 0.3200, 0.1260, 0.1980, 0.1042, 0.4900, 0.4000, 0.5412, 0.1240, 0.2430
0.2500, 1.0000, 0.2260, 0.0850, 0.1300, 0.0710, 0.4800, 0.3000, 0.4007, 0.0810, 0.0710
0.5200, 1.0000, 0.1970, 0.0810, 0.1520, 0.0534, 0.8200, 0.2000, 0.4419, 0.0820, 0.0770
0.3400, 0.0000, 0.2120, 0.0840, 0.2540, 0.1134, 0.5200, 0.5000, 0.6094, 0.0920, 0.1090
0.4200, 1.0000, 0.3060, 0.1010, 0.2690, 0.1722, 0.5000, 0.5000, 0.5455, 0.1060, 0.2720
0.2800, 1.0000, 0.2550, 0.0990, 0.1620, 0.1016, 0.4600, 0.4000, 0.4277, 0.0940, 0.0600
0.4700, 1.0000, 0.2330, 0.0900, 0.1950, 0.1258, 0.5400, 0.4000, 0.4331, 0.0730, 0.0540
0.3200, 1.0000, 0.3100, 0.1000, 0.1770, 0.0962, 0.4500, 0.4000, 0.5187, 0.0770, 0.2210
0.4300, 0.0000, 0.1850, 0.0870, 0.1630, 0.0936, 0.6100, 0.2670, 0.3738, 0.0800, 0.0900
0.5900, 1.0000, 0.2690, 0.1040, 0.1940, 0.1266, 0.4300, 0.5000, 0.4804, 0.1060, 0.3110
0.5300, 0.0000, 0.2830, 0.1010, 0.1790, 0.1070, 0.4800, 0.4000, 0.4788, 0.1010, 0.2810
0.6000, 0.0000, 0.2570, 0.1030, 0.1580, 0.0846, 0.6400, 0.2000, 0.3850, 0.0970, 0.1820
0.5400, 1.0000, 0.3610, 0.1150, 0.1630, 0.0984, 0.4300, 0.4000, 0.4682, 0.1010, 0.3210
0.3500, 1.0000, 0.2410, 0.0947, 0.1550, 0.0974, 0.3200, 0.4840, 0.4852, 0.0940, 0.0580
0.4900, 1.0000, 0.2580, 0.0890, 0.1820, 0.1186, 0.3900, 0.5000, 0.4804, 0.1150, 0.2620
0.5800, 0.0000, 0.2280, 0.0910, 0.1960, 0.1188, 0.4800, 0.4000, 0.4984, 0.1150, 0.2060
0.3600, 1.0000, 0.3910, 0.0900, 0.2190, 0.1358, 0.3800, 0.6000, 0.5421, 0.1030, 0.2330
0.4600, 1.0000, 0.4220, 0.0990, 0.2110, 0.1370, 0.4400, 0.5000, 0.5011, 0.0990, 0.2420
0.4400, 1.0000, 0.2660, 0.0990, 0.2050, 0.1090, 0.4300, 0.5000, 0.5580, 0.1110, 0.1230
0.4600, 0.0000, 0.2990, 0.0830, 0.1710, 0.1130, 0.3800, 0.4500, 0.4585, 0.0980, 0.1670
0.5400, 0.0000, 0.2100, 0.0780, 0.1880, 0.1074, 0.7000, 0.3000, 0.3970, 0.0730, 0.0630
0.6300, 1.0000, 0.2550, 0.1090, 0.2260, 0.1032, 0.4600, 0.5000, 0.5951, 0.0870, 0.1970
0.4100, 1.0000, 0.2420, 0.0900, 0.1990, 0.1236, 0.5700, 0.4000, 0.4522, 0.0860, 0.0710
0.2800, 0.0000, 0.2540, 0.0930, 0.1410, 0.0790, 0.4900, 0.3000, 0.4174, 0.0910, 0.1680
0.1900, 0.0000, 0.2320, 0.0750, 0.1430, 0.0704, 0.5200, 0.3000, 0.4635, 0.0720, 0.1400
0.6100, 1.0000, 0.2610, 0.1260, 0.2150, 0.1298, 0.5700, 0.4000, 0.4949, 0.0960, 0.2170
0.4800, 0.0000, 0.3270, 0.0930, 0.2760, 0.1986, 0.4300, 0.6420, 0.5148, 0.0910, 0.1210
0.5400, 1.0000, 0.2730, 0.1000, 0.2000, 0.1440, 0.3300, 0.6000, 0.4745, 0.0760, 0.2350
0.5300, 1.0000, 0.2660, 0.0930, 0.1850, 0.1224, 0.3600, 0.5000, 0.4890, 0.0820, 0.2450
0.4800, 0.0000, 0.2280, 0.1010, 0.1100, 0.0416, 0.5600, 0.2000, 0.4127, 0.0970, 0.0400
0.5300, 0.0000, 0.2880, 0.1117, 0.1450, 0.0872, 0.4600, 0.3150, 0.4078, 0.0850, 0.0520
0.2900, 1.0000, 0.1810, 0.0730, 0.1580, 0.0990, 0.4100, 0.4000, 0.4500, 0.0780, 0.1040
0.6200, 0.0000, 0.3200, 0.0880, 0.1720, 0.0690, 0.3800, 0.4000, 0.5784, 0.1000, 0.1320
0.5000, 1.0000, 0.2370, 0.0920, 0.1660, 0.0970, 0.5200, 0.3000, 0.4443, 0.0930, 0.0880
0.5800, 1.0000, 0.2360, 0.0960, 0.2570, 0.1710, 0.5900, 0.4000, 0.4905, 0.0820, 0.0690
0.5500, 1.0000, 0.2460, 0.1090, 0.1430, 0.0764, 0.5100, 0.3000, 0.4357, 0.0880, 0.2190
0.5400, 0.0000, 0.2260, 0.0900, 0.1830, 0.1042, 0.6400, 0.3000, 0.4304, 0.0920, 0.0720
0.3600, 0.0000, 0.2780, 0.0730, 0.1530, 0.1044, 0.4200, 0.4000, 0.3497, 0.0730, 0.2010
0.6300, 1.0000, 0.2410, 0.1110, 0.1840, 0.1122, 0.4400, 0.4000, 0.4935, 0.0820, 0.1100
0.4700, 1.0000, 0.2650, 0.0700, 0.1810, 0.1048, 0.6300, 0.3000, 0.4190, 0.0700, 0.0510
0.5100, 1.0000, 0.3280, 0.1120, 0.2020, 0.1006, 0.3700, 0.5000, 0.5775, 0.1090, 0.2770
0.4200, 0.0000, 0.1990, 0.0760, 0.1460, 0.0832, 0.5500, 0.3000, 0.3664, 0.0790, 0.0630
0.3700, 1.0000, 0.2360, 0.0940, 0.2050, 0.1388, 0.5300, 0.4000, 0.4190, 0.1070, 0.1180
0.2800, 0.0000, 0.2210, 0.0820, 0.1680, 0.1006, 0.5400, 0.3000, 0.4205, 0.0860, 0.0690
0.5800, 0.0000, 0.2810, 0.1110, 0.1980, 0.0806, 0.3100, 0.6000, 0.6068, 0.0930, 0.2730
0.3200, 0.0000, 0.2650, 0.0860, 0.1840, 0.1016, 0.5300, 0.4000, 0.4990, 0.0780, 0.2580
0.2500, 1.0000, 0.2350, 0.0880, 0.1430, 0.0808, 0.5500, 0.3000, 0.3584, 0.0830, 0.0430
0.6300, 0.0000, 0.2600, 0.0857, 0.1550, 0.0782, 0.4600, 0.3370, 0.5037, 0.0970, 0.1980
0.5200, 0.0000, 0.2780, 0.0850, 0.2190, 0.1360, 0.4900, 0.4000, 0.5136, 0.0750, 0.2420
0.6500, 1.0000, 0.2850, 0.1090, 0.2010, 0.1230, 0.4600, 0.4000, 0.5075, 0.0960, 0.2320
0.4200, 0.0000, 0.3060, 0.1210, 0.1760, 0.0928, 0.6900, 0.3000, 0.4263, 0.0890, 0.1750
0.5300, 0.0000, 0.2220, 0.0780, 0.1640, 0.0810, 0.7000, 0.2000, 0.4174, 0.1010, 0.0930
0.7900, 1.0000, 0.2330, 0.0880, 0.1860, 0.1284, 0.3300, 0.6000, 0.4812, 0.1020, 0.1680
0.4300, 0.0000, 0.3540, 0.0930, 0.1850, 0.1002, 0.4400, 0.4000, 0.5318, 0.1010, 0.2750
0.4400, 0.0000, 0.3140, 0.1150, 0.1650, 0.0976, 0.5200, 0.3000, 0.4344, 0.0890, 0.2930
0.6200, 1.0000, 0.3780, 0.1190, 0.1130, 0.0510, 0.3100, 0.4000, 0.5043, 0.0840, 0.2810
0.3300, 0.0000, 0.1890, 0.0700, 0.1620, 0.0918, 0.5900, 0.3000, 0.4025, 0.0580, 0.0720
0.5600, 0.0000, 0.3500, 0.0793, 0.1950, 0.1408, 0.4200, 0.4640, 0.4111, 0.0960, 0.1400
0.6600, 0.0000, 0.2170, 0.1260, 0.2120, 0.1278, 0.4500, 0.4710, 0.5278, 0.1010, 0.1890
0.3400, 1.0000, 0.2530, 0.1110, 0.2300, 0.1620, 0.3900, 0.6000, 0.4977, 0.0900, 0.1810
0.4600, 1.0000, 0.2380, 0.0970, 0.2240, 0.1392, 0.4200, 0.5000, 0.5366, 0.0810, 0.2090
0.5000, 0.0000, 0.3180, 0.0820, 0.1360, 0.0692, 0.5500, 0.2000, 0.4078, 0.0850, 0.1360
0.6900, 0.0000, 0.3430, 0.1130, 0.2000, 0.1238, 0.5400, 0.4000, 0.4710, 0.1120, 0.2610
0.3400, 0.0000, 0.2630, 0.0870, 0.1970, 0.1200, 0.6300, 0.3000, 0.4249, 0.0960, 0.1130
0.7100, 1.0000, 0.2700, 0.0933, 0.2690, 0.1902, 0.4100, 0.6560, 0.5242, 0.0930, 0.1310
0.4700, 0.0000, 0.2720, 0.0800, 0.2080, 0.1456, 0.3800, 0.6000, 0.4804, 0.0920, 0.1740
0.4100, 0.0000, 0.3380, 0.1233, 0.1870, 0.1270, 0.4500, 0.4160, 0.4318, 0.1000, 0.2570
0.3400, 0.0000, 0.3300, 0.0730, 0.1780, 0.1146, 0.5100, 0.3490, 0.4127, 0.0920, 0.0550
0.5100, 0.0000, 0.2410, 0.0870, 0.2610, 0.1756, 0.6900, 0.4000, 0.4407, 0.0930, 0.0840
0.4300, 0.0000, 0.2130, 0.0790, 0.1410, 0.0788, 0.5300, 0.3000, 0.3829, 0.0900, 0.0420
0.5500, 0.0000, 0.2300, 0.0947, 0.1900, 0.1376, 0.3800, 0.5000, 0.4277, 0.1060, 0.1460
0.5900, 1.0000, 0.2790, 0.1010, 0.2180, 0.1442, 0.3800, 0.6000, 0.5187, 0.0950, 0.2120
0.2700, 1.0000, 0.3360, 0.1100, 0.2460, 0.1566, 0.5700, 0.4000, 0.5088, 0.0890, 0.2330
0.5100, 1.0000, 0.2270, 0.1030, 0.2170, 0.1624, 0.3000, 0.7000, 0.4812, 0.0800, 0.0910
0.4900, 1.0000, 0.2740, 0.0890, 0.1770, 0.1130, 0.3700, 0.5000, 0.4905, 0.0970, 0.1110
0.2700, 0.0000, 0.2260, 0.0710, 0.1160, 0.0434, 0.5600, 0.2000, 0.4419, 0.0790, 0.1520
0.5700, 1.0000, 0.2320, 0.1073, 0.2310, 0.1594, 0.4100, 0.5630, 0.5030, 0.1120, 0.1200
0.3900, 1.0000, 0.2690, 0.0930, 0.1360, 0.0754, 0.4800, 0.3000, 0.4143, 0.0990, 0.0670
0.6200, 1.0000, 0.3460, 0.1200, 0.2150, 0.1292, 0.4300, 0.5000, 0.5366, 0.1230, 0.3100
0.3700, 0.0000, 0.2330, 0.0880, 0.2230, 0.1420, 0.6500, 0.3400, 0.4357, 0.0820, 0.0940
0.4600, 0.0000, 0.2110, 0.0800, 0.2050, 0.1444, 0.4200, 0.5000, 0.4533, 0.0870, 0.1830
0.6800, 1.0000, 0.2350, 0.1010, 0.1620, 0.0854, 0.5900, 0.3000, 0.4477, 0.0910, 0.0660
0.5100, 0.0000, 0.3150, 0.0930, 0.2310, 0.1440, 0.4900, 0.4700, 0.5252, 0.1170, 0.1730
0.4100, 0.0000, 0.2080, 0.0860, 0.2230, 0.1282, 0.8300, 0.3000, 0.4078, 0.0890, 0.0720
0.5300, 0.0000, 0.2650, 0.0970, 0.1930, 0.1224, 0.5800, 0.3000, 0.4143, 0.0990, 0.0490
0.4500, 0.0000, 0.2420, 0.0830, 0.1770, 0.1184, 0.4500, 0.4000, 0.4220, 0.0820, 0.0640
0.3300, 0.0000, 0.1950, 0.0800, 0.1710, 0.0854, 0.7500, 0.2000, 0.3970, 0.0800, 0.0480
0.6000, 1.0000, 0.2820, 0.1120, 0.1850, 0.1138, 0.4200, 0.4000, 0.4984, 0.0930, 0.1780
0.4700, 1.0000, 0.2490, 0.0750, 0.2250, 0.1660, 0.4200, 0.5000, 0.4443, 0.1020, 0.1040
0.6000, 1.0000, 0.2490, 0.0997, 0.1620, 0.1066, 0.4300, 0.3770, 0.4127, 0.0950, 0.1320
0.3600, 0.0000, 0.3000, 0.0950, 0.2010, 0.1252, 0.4200, 0.4790, 0.5130, 0.0850, 0.2200
0.3600, 0.0000, 0.1960, 0.0710, 0.2500, 0.1332, 0.9700, 0.3000, 0.4595, 0.0920, 0.0570

Posted in Machine Learning, Scikit | Leave a comment

Implementing Ridge Regression from Scratch Python in the Scikit Style Using SAG Training

Posted on June 16, 2026 by jamesdmccaffrey

Ridge regression, aka linear ridge regression, is just linear regression with L2 regularization integrated. The idea is to discourage large model weights, which is a cause of model overfitting. Note: linear ridge regression is sometimes confused with kernel ridge regression, but the two techniques are very, very different.

Basic ridge regression is quite simple. It can be implemented from scratch, with SGD training, in only a few dozen lines of code.

I took a look at the Python scikit library Ridge module. I was extremely surprised at how complicated the module is. First, it uses a very complex variation of SGD training called SAG (stochastic average gradient). And it uses a very tricky algorithm to automatically determine a learning rate. And it uses a non-obvious technique when implementing the L2 alpha constant. And it uses a non-trivial algorithm to auto-stop the training iteration. When I looked at the scikit source code, it was spread over many files and hundreds (more likely thousands — I didn’t try to count) of lines of code.

For reasons that aren’t clear to me, I decided to implement an approximation to the scikit Ridge module, from scratch using Python. I was mostly interested in gaining an understanding of the SAG algorithm. Bottom line: After far too many hours spent on the investigation, I didn’t see any significant advantage of SAG over the much simpler SGD.

SAG is kind of a cross between online (one line of training data at a time for a single gradient) SGD and batch (process all lines to compute an overall gradient) SGD. The SAG algorithm processes one line of data at a time, but stores computed gradients in memory, and updates model weights using a running average gradient.

I used the scikit approach to use the L2 alpha constant. Briefly, the key trick is to scale alpha by dividing by the number of training items.

I used the scikit approach for early stopping for training. Briefly, the training exits if max_change_in_wts / max_weights less-than tolerance (typically 0.0001 or 0.00001). Simple idea, but it was tricky to implement.

I explored the automatic setting of the learning rate, and after a lot of digging through the scikit source code, I figured it out. But I am moderately skeptical that the technique is useful — I think parameterizing the learning rate is a better approach. However, that is a very long and complicated topic which is beyond this blog post.

The output of a demo program that compares my from-scratch ridge regression using SAG training with the scikit Ridge module shows nearly identical (to 4 decimals) results for the model weights and bias:

Begin SAG ridge regression scratch Python

Loading train (200) and test (40) data

First three train X:
[-0.1660  0.4406 -0.9998 -0.3953 -0.7065]
[ 0.0776 -0.1616  0.3704 -0.5911  0.7562]
[-0.9452  0.3409 -0.1654  0.1174 -0.7192]

First three train y:
0.4840
0.1568
0.8054

==========================

Creating scratch Python ridge model
Done

Training model with SAG
Setting lrn_rate = 0.0010
Setting max_epochs = 200
Setting L2 alpha = 0.5000
Done. Used 127 epochs

Model weights:
[-0.2637  0.0332 -0.0454  0.0355 -0.1139]
Model bias = 0.3620

Evaluating model

Accuracy (within 0.10) train = 0.4650
Accuracy (within 0.10) test = 0.6250

MSE train = 0.00258
MSE test = 0.00195

Predicting for x =
[-0.1660  0.4406 -0.9998 -0.3953 -0.7065]
Predicted y = 0.5322

==========================

Creating scikit Ridge model
Setting L2 alpha = 0.5000
Done

Training scikit Ridge model (using SAG)
Using default training params
Done. Used [36] iterations

Model weights:
[-0.2637  0.0332 -0.0453  0.0355 -0.1139]
Model bias = 0.3619

End demo

Note that for simplicity, I did not scale the training data to mean = 0 and variance = 1, which is more or less required so that all model weights are penalized equally.

I’m not sure what the moral of this blog post is, but it points out the trade-offs between code simplicity and system performance and ease-of-use.

It seems like most people have a passion for something in life. For me, I can’t resist exploring interesting algorithms. For many of my friends, they love games of all types. Here’s an image of the “James Bond Thunderball” board game (1965) from the Milton Bradley company. According to the Internet, this game is quite rare and is highly prized by collectors. I’m pretty sure I would enjoy playing this game, but only once — I don’t think the game is rich enough in strategy to make me want to play it multiple times.

Demo program. Replace

# ridge_regression_sag_scratch.py
# Stochastic Average Gradient training

import numpy as np

# -----------------------------------------------------------

def accuracy(model, data_X, data_y, pct_close):
  n = len(data_X)
  n_correct = 0; n_wrong = 0
  for i in range(n):
    # x = data_X[i].reshape(1,-1)
    x = data_X[i]
    y = data_y[i]
    y_pred = model.predict(x)

    if np.abs(y - y_pred) "lt" np.abs(y * pct_close):
      n_correct += 1
    else: 
      n_wrong += 1
  return n_correct / (n_correct + n_wrong)

def mse(model, data_X, data_y):
  n = len(data_X)
  sum = 0.0
  for i in range(n):
    y = data_y[i]
    pred_y = model.predict(data_X[i])
    diff = y - pred_y
    sum += diff * diff
  return sum /n

# -----------------------------------------------------------

class MyRidge:
  # ridge regression using stochastic average gradient (SAG)
  def __init__(self, seed=0):
    self.weights = None
    self.bias = 0.0
    self.rnd = np.random.RandomState(seed)

  def predict(self, x):
    sum = 0.0
    for j in range(len(x)):
      sum += self.weights[j] * x[j]
    sum += self.bias
    return sum

  def train(self, train_X, train_y, lrn_rate=0.001,
    max_epochs=1000, tol=0.0001, alpha=1.0):
    # using SAG algorithm
    n = len(train_X)
    dim = len(train_X[0])
    self.weights = np.zeros(dim)
    lo = -0.01; hi = 0.01
    for j in range(dim):
      self.weights[j] = (hi - lo) * self.rnd.random() + lo

    # center X and y
    # col_means = np.mean(train_X, axis=0)
    # centered_X = train_X - col_means
    # y_mean = np.mean(train_y)
    # centered_y = train_y - y_mean
    # train_X = centered_X
    # train_y = centered_y

    grad_mem = np.zeros((n, dim))
    tot_grad = np.zeros(dim)
    new_grad = np.zeros(dim)
    indices = np.arange(n)
    alpha_scaled = alpha / n

    prev_weights = np.zeros(dim)

    for epoch in range(max_epochs): # each epoch
      self.rnd.shuffle(indices)
      prev_weights = np.copy(self.weights)
      for i in range(n): # each item
        idx = indices[i]
        x = train_X[idx]
        y = train_y[idx]
        pred_y = self.predict(x)
        err = pred_y - y
        new_grad = err * x
        
        for j in range(dim): # each weight
          tot_grad[j] = tot_grad[j] - grad_mem[idx][j] + \
            new_grad[j]
          grad_mem[idx][j] = new_grad[j]
          self.weights[j] -= lrn_rate * (tot_grad[j] / n)
          self.weights[j] -= lrn_rate * (alpha_scaled * \
            self.weights[j])
        self.bias -= lrn_rate * err

      # check for early stop
      # print("new weights"); print(self.weights)
      # print("prev weights "); print(prev_weights); input()
      max_change_in_wts = np.abs(self.weights - \
        prev_weights).max()
      max_weights = np.abs(self.weights).max()
      # print("max_change_in_wts"); print(max_change_in_wts);
      # print("max_weights"); print(max_weights); input()
      if max_change_in_wts / max_weights "lt" tol:
        return epoch
        # print("Early exit at epoch " + str(epoch))
        # break
      
    return epoch

# -----------------------------------------------------------

print("\nBegin SAG ridge regression scratch Python ")

np.set_printoptions(precision=4, suppress=True,
    floatmode='fixed')

print("\nLoading train (200) and test (40) data ")
train_Xy = np.loadtxt(".\\Data\\synthetic_train_200.txt",
  usecols=[0,1,2,3,4,5], delimiter=",")
train_X = train_Xy[:,[0,1,2,3,4]]
train_y = train_Xy[:,5]

test_Xy = np.loadtxt(".\\Data\\synthetic_test_40.txt",
  usecols=[0,1,2,3,4,5], delimiter=",")
test_X = test_Xy[:,[0,1,2,3,4]]
test_y = test_Xy[:,5]

print("\nFirst three train X: ")
for i in range(3):
  print(train_X[i])
print("\nFirst three train y: ")
for i in range(3):
  print("%0.4f " % train_y[i])

print("\n========================== ")

print("\nCreating scratch Python ridge model ")
model = MyRidge()
print("Done ")

print("\nTraining model with SAG ")
lrn_rate = 0.001
max_epochs = 200
alpha = 0.5
print("Setting lrn_rate = %0.4f " % lrn_rate)
print("Setting max_epochs = " + str(max_epochs))
print("Setting L2 alpha = %0.4f " % alpha)
eps = model.train(train_X, train_y, lrn_rate,
  max_epochs, 0.00001, alpha)
print("Done. Used " + str(eps) + " epochs " )

print("\nModel weights: ")
print(model.weights)
print("Model bias = %0.4f " % model.bias)

print("\nEvaluating model ")
acc_train = accuracy(model, train_X, train_y, 0.10)
acc_test = accuracy(model, test_X, test_y, 0.10)
print("\nAccuracy (within 0.10) train = %0.4f " % \
  acc_train)
print("Accuracy (within 0.10) test = %0.4f " % \
  acc_test)

mse_train = mse(model, train_X, train_y)
mse_test = mse(model, test_X, test_y)
print("\nMSE train = %0.5f " % mse_train)
print("MSE test = %0.5f " % mse_test)

x = train_X[0]
print("\nPredicting for x = ")
print(x)
pred_y = model.predict(x)
print("Predicted y = %0.4f " % pred_y)

print("\n========================== ")

from sklearn.linear_model import Ridge

# Ridge(alpha=1.0, *, fit_intercept=True, copy_X=True,
# max_iter=None, tol=0.0001, solver='auto', positive=False,
# random_state=None)

print("\nCreating scikit Ridge model ")
alpha = 0.5
print("Setting L2 alpha = %0.4f " % alpha)
model = Ridge(alpha=alpha, solver='sag', fit_intercept=True)
print("Done ")

print("\nTraining scikit Ridge model (using SAG) ")
print("Using default training params ")
model.fit(train_X, train_y)
print("Done. Used " + str(model.n_iter_) + " iterations" )

print("\nModel weights: ")
print(model.coef_)
print("Model bias = %0.4f " % model.intercept_)

print("\nEnd demo ")

Training data:

# synthetic_train_200.txt
#
-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.0776, -0.1616,  0.3704, -0.5911,  0.7562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
 0.9365, -0.3732,  0.3846,  0.7528,  0.7892,  0.1345
-0.8299, -0.9219, -0.6603,  0.7563, -0.8033,  0.7955
 0.0663,  0.3838, -0.3690,  0.3730,  0.6693,  0.3206
-0.9634,  0.5003,  0.9777,  0.4963, -0.4391,  0.7377
-0.1042,  0.8172, -0.4128, -0.4244, -0.7399,  0.4801
-0.9613,  0.3577, -0.5767, -0.4689, -0.0169,  0.6861
-0.7065,  0.1786,  0.3995, -0.7953, -0.1719,  0.5569
 0.3888, -0.1716, -0.9001,  0.0718,  0.3276,  0.2500
 0.1731,  0.8068, -0.7251, -0.7214,  0.6148,  0.3297
-0.2046, -0.6693,  0.8550, -0.3045,  0.5016,  0.2129
 0.2473,  0.5019, -0.3022, -0.4601,  0.7918,  0.2613
-0.1438,  0.9297,  0.3269,  0.2434, -0.7705,  0.5171
 0.1568, -0.1837, -0.5259,  0.8068,  0.1474,  0.3307
-0.9943,  0.2343, -0.3467,  0.0541,  0.7719,  0.5581
 0.2467, -0.9684,  0.8589,  0.3818,  0.9946,  0.1092
-0.6553, -0.7257,  0.8652,  0.3936, -0.8680,  0.7018
 0.8460,  0.4230, -0.7515, -0.9602, -0.9476,  0.1996
-0.9434, -0.5076,  0.7201,  0.0777,  0.1056,  0.5664
 0.9392,  0.1221, -0.9627,  0.6013, -0.5341,  0.1533
 0.6142, -0.2243,  0.7271,  0.4942,  0.1125,  0.1661
 0.4260,  0.1194, -0.9749, -0.8561,  0.9346,  0.2230
 0.1362, -0.5934, -0.4953,  0.4877, -0.6091,  0.3810
 0.6937, -0.5203, -0.0125,  0.2399,  0.6580,  0.1460
-0.6864, -0.9628, -0.8600, -0.0273,  0.2127,  0.5387
 0.9772,  0.1595, -0.2397,  0.1019,  0.4907,  0.1611
 0.3385, -0.4702, -0.8673, -0.2598,  0.2594,  0.2270
-0.8669, -0.4794,  0.6095, -0.6131,  0.2789,  0.4700
 0.0493,  0.8496, -0.4734, -0.8681,  0.4701,  0.3516
 0.8639, -0.9721, -0.5313,  0.2336,  0.8980,  0.1412
 0.9004,  0.1133,  0.8312,  0.2831, -0.2200,  0.1782
 0.0991,  0.8524,  0.8375, -0.2102,  0.9265,  0.2150
-0.6521, -0.7473, -0.7298,  0.0113, -0.9570,  0.7422
 0.6190, -0.3105,  0.8802,  0.1640,  0.7577,  0.1056
 0.6895,  0.8108, -0.0802,  0.0927,  0.5972,  0.2214
 0.1982, -0.9689,  0.1870, -0.1326,  0.6147,  0.1310
-0.3695,  0.7858,  0.1557, -0.6320,  0.5759,  0.3773
-0.1596,  0.3581,  0.8372, -0.9992,  0.9535,  0.2071
-0.2468,  0.9476,  0.2094,  0.6577,  0.1494,  0.4132
 0.1737,  0.5000,  0.7166,  0.5102,  0.3961,  0.2611
 0.7290, -0.3546,  0.3416, -0.0983, -0.2358,  0.1332
-0.3652,  0.2438, -0.1395,  0.9476,  0.3556,  0.4170
-0.6029, -0.1466, -0.3133,  0.5953,  0.7600,  0.4334
-0.4596, -0.4953,  0.7098,  0.0554,  0.6043,  0.2775
 0.1450,  0.4663,  0.0380,  0.5418,  0.1377,  0.2931
-0.8636, -0.2442, -0.8407,  0.9656, -0.6368,  0.7429
 0.6237,  0.7499,  0.3768,  0.1390, -0.6781,  0.2185
-0.5499,  0.1850, -0.3755,  0.8326,  0.8193,  0.4399
-0.4858, -0.7782, -0.6141, -0.0008,  0.4572,  0.4197
 0.7033, -0.1683,  0.2334, -0.5327, -0.7961,  0.1776
 0.0317, -0.0457, -0.6947,  0.2436,  0.0880,  0.3345
 0.5031, -0.5559,  0.0387,  0.5706, -0.9553,  0.3107
-0.3513,  0.7458,  0.6894,  0.0769,  0.7332,  0.3170
 0.2205,  0.5992, -0.9309,  0.5405,  0.4635,  0.3532
-0.4806, -0.4859,  0.2646, -0.3094,  0.5932,  0.3202
 0.9809, -0.3995, -0.7140,  0.8026,  0.0831,  0.1600
 0.9495,  0.2732,  0.9878,  0.0921,  0.0529,  0.1289
-0.9476, -0.6792,  0.4913, -0.9392, -0.2669,  0.5966
 0.7247,  0.3854,  0.3819, -0.6227, -0.1162,  0.1550
-0.5922, -0.5045, -0.4757,  0.5003, -0.0860,  0.5863
-0.8861,  0.0170, -0.5761,  0.5972, -0.4053,  0.7301
 0.6877, -0.2380,  0.4997,  0.0223,  0.0819,  0.1404
 0.9189,  0.6079, -0.9354,  0.4188, -0.0700,  0.1907
-0.1428, -0.7820,  0.2676,  0.6059,  0.3936,  0.2790
 0.5324, -0.3151,  0.6917, -0.1425,  0.6480,  0.1071
-0.8432, -0.9633, -0.8666, -0.0828, -0.7733,  0.7784
-0.9444,  0.5097, -0.2103,  0.4939, -0.0952,  0.6787
-0.0520,  0.6063, -0.1952,  0.8094, -0.9259,  0.4836
 0.5477, -0.7487,  0.2370, -0.9793,  0.0773,  0.1241
 0.2450,  0.8116,  0.9799,  0.4222,  0.4636,  0.2355
 0.8186, -0.1983, -0.5003, -0.6531, -0.7611,  0.1511
-0.4714,  0.6382, -0.3788,  0.9648, -0.4667,  0.5950
 0.0673, -0.3711,  0.8215, -0.2669, -0.1328,  0.2677
-0.9381,  0.4338,  0.7820, -0.9454,  0.0441,  0.5518
-0.3480,  0.7190,  0.1170,  0.3805, -0.0943,  0.4724
-0.9813,  0.1535, -0.3771,  0.0345,  0.8328,  0.5438
-0.1471, -0.5052, -0.2574,  0.8637,  0.8737,  0.3042
-0.5454, -0.3712, -0.6505,  0.2142, -0.1728,  0.5783
 0.6327, -0.6297,  0.4038, -0.5193,  0.1484,  0.1153
-0.5424,  0.3282, -0.0055,  0.0380, -0.6506,  0.6613
 0.1414,  0.9935,  0.6337,  0.1887,  0.9520,  0.2540
-0.9351, -0.8128, -0.8693, -0.0965, -0.2491,  0.7353
 0.9507, -0.6640,  0.9456,  0.5349,  0.6485,  0.1059
-0.0462, -0.9737, -0.2940, -0.0159,  0.4602,  0.2606
-0.0627, -0.0852, -0.7247, -0.9782,  0.5166,  0.2977
 0.0478,  0.5098, -0.0723, -0.7504, -0.3750,  0.3335
 0.0090,  0.3477,  0.5403, -0.7393, -0.9542,  0.4415
-0.9748,  0.3449,  0.3736, -0.1015,  0.8296,  0.4358
 0.2887, -0.9895, -0.0311,  0.7186,  0.6608,  0.2057
 0.1570, -0.4518,  0.1211,  0.3435, -0.2951,  0.3244
 0.7117, -0.6099,  0.4946, -0.4208,  0.5476,  0.1096
-0.2929, -0.5726,  0.5346, -0.3827,  0.4665,  0.2465
 0.4889, -0.5572, -0.5718, -0.6021, -0.7150,  0.2163
-0.7782,  0.3491,  0.5996, -0.8389, -0.5366,  0.6516
-0.5847,  0.8347,  0.4226,  0.1078, -0.3910,  0.6134
 0.8469,  0.4121, -0.0439, -0.7476,  0.9521,  0.1571
-0.6803, -0.5948, -0.1376, -0.1916, -0.7065,  0.7156
 0.2878,  0.5086, -0.5785,  0.2019,  0.4979,  0.2980
 0.2764,  0.1943, -0.4090,  0.4632,  0.8906,  0.2960
-0.8877,  0.6705, -0.6155, -0.2098, -0.3998,  0.7107
-0.8398,  0.8093, -0.2597,  0.0614, -0.0118,  0.6502
-0.8476,  0.0158, -0.4769, -0.2859, -0.7839,  0.7715
 0.5751, -0.7868,  0.9714, -0.6457,  0.1448,  0.1175
 0.4802, -0.7001,  0.1022, -0.5668,  0.5184,  0.1090
 0.4458, -0.6469,  0.7239, -0.9604,  0.7205,  0.0779
 0.5175,  0.4339,  0.9747, -0.4438, -0.9924,  0.2879
 0.8678,  0.7158,  0.4577,  0.0334,  0.4139,  0.1678
 0.5406,  0.5012,  0.2264, -0.1963,  0.3946,  0.2088
-0.9938,  0.5498,  0.7928, -0.5214, -0.7585,  0.7687
 0.7661,  0.0863, -0.4266, -0.7233, -0.4197,  0.1466
 0.2277, -0.3517, -0.0853, -0.1118,  0.6563,  0.1767
 0.3499, -0.5570, -0.0655, -0.3705,  0.2537,  0.1632
 0.7547, -0.1046,  0.5689, -0.0861,  0.3125,  0.1257
 0.8186,  0.2110,  0.5335,  0.0094, -0.0039,  0.1391
 0.6858, -0.8644,  0.1465,  0.8855,  0.0357,  0.1845
-0.4967,  0.4015,  0.0805,  0.8977,  0.2487,  0.4663
 0.6760, -0.9841,  0.9787, -0.8446, -0.3557,  0.1509
-0.1203, -0.4885,  0.6054, -0.0443, -0.7313,  0.4854
 0.8557,  0.7919, -0.0169,  0.7134, -0.1628,  0.2002
 0.0115, -0.6209,  0.9300, -0.4116, -0.7931,  0.4052
-0.7114, -0.9718,  0.4319,  0.1290,  0.5892,  0.3661
 0.3915,  0.5557, -0.1870,  0.2955, -0.6404,  0.2954
-0.3564, -0.6548, -0.1827, -0.5172, -0.1862,  0.4622
 0.2392, -0.4959,  0.5857, -0.1341, -0.2850,  0.2470
-0.3394,  0.3947, -0.4627,  0.6166, -0.4094,  0.5325
 0.7107,  0.7768, -0.6312,  0.1707,  0.7964,  0.2757
-0.1078,  0.8437, -0.4420,  0.2177,  0.3649,  0.4028
-0.3139,  0.5595, -0.6505, -0.3161, -0.7108,  0.5546
 0.4335,  0.3986,  0.3770, -0.4932,  0.3847,  0.1810
-0.2562, -0.2894, -0.8847,  0.2633,  0.4146,  0.4036
 0.2272,  0.2966, -0.6601, -0.7011,  0.0284,  0.2778
-0.0743, -0.1421, -0.0054, -0.6770, -0.3151,  0.3597
-0.4762,  0.6891,  0.6007, -0.1467,  0.2140,  0.4266
-0.4061,  0.7193,  0.3432,  0.2669, -0.7505,  0.6147
-0.0588,  0.9731,  0.8966,  0.2902, -0.6966,  0.4955
-0.0627, -0.1439,  0.1985,  0.6999,  0.5022,  0.3077
 0.1587,  0.8494, -0.8705,  0.9827, -0.8940,  0.4263
-0.7850,  0.2473, -0.9040, -0.4308, -0.8779,  0.7199
 0.4070,  0.3369, -0.2428, -0.6236,  0.4940,  0.2215
-0.0242,  0.0513, -0.9430,  0.2885, -0.2987,  0.3947
-0.5416, -0.1322, -0.2351, -0.0604,  0.9590,  0.3683
 0.1055,  0.7783, -0.2901, -0.5090,  0.8220,  0.2984
-0.9129,  0.9015,  0.1128, -0.2473,  0.9901,  0.4776
-0.9378,  0.1424, -0.6391,  0.2619,  0.9618,  0.5368
 0.7498, -0.0963,  0.4169,  0.5549, -0.0103,  0.1614
-0.2612, -0.7156,  0.4538, -0.0460, -0.1022,  0.3717
 0.7720,  0.0552, -0.1818, -0.4622, -0.8560,  0.1685
-0.4177,  0.0070,  0.9319, -0.7812,  0.3461,  0.3052
-0.0001,  0.5542, -0.7128, -0.8336, -0.2016,  0.3803
 0.5356, -0.4194, -0.5662, -0.9666, -0.2027,  0.1776
-0.2378,  0.3187, -0.8582, -0.6948, -0.9668,  0.5474
-0.1947, -0.3579,  0.1158,  0.9869,  0.6690,  0.2992
 0.3992,  0.8365, -0.9205, -0.8593, -0.0520,  0.3154
-0.0209,  0.0793,  0.7905, -0.1067,  0.7541,  0.1864
-0.4928, -0.4524, -0.3433,  0.0951, -0.5597,  0.6261
-0.8118,  0.7404, -0.5263, -0.2280,  0.1431,  0.6349
 0.0516, -0.8480,  0.7483,  0.9023,  0.6250,  0.1959
-0.3212,  0.1093,  0.9488, -0.3766,  0.3376,  0.2735
-0.3481,  0.5490, -0.3484,  0.7797,  0.5034,  0.4379
-0.5785, -0.9170, -0.3563, -0.9258,  0.3877,  0.4121
 0.3407, -0.1391,  0.5356,  0.0720, -0.9203,  0.3458
-0.3287, -0.8954,  0.2102,  0.0241,  0.2349,  0.3247
-0.1353,  0.6954, -0.0919, -0.9692,  0.7461,  0.3338
 0.9036, -0.8982, -0.5299, -0.8733, -0.1567,  0.1187
 0.7277, -0.8368, -0.0538, -0.7489,  0.5458,  0.0830
 0.9049,  0.8878,  0.2279,  0.9470, -0.3103,  0.2194
 0.7957, -0.1308, -0.5284,  0.8817,  0.3684,  0.2172
 0.4647, -0.4931,  0.2010,  0.6292, -0.8918,  0.3371
-0.7390,  0.6849,  0.2367,  0.0626, -0.5034,  0.7039
-0.1567, -0.8711,  0.7940, -0.5932,  0.6525,  0.1710
 0.7635, -0.0265,  0.1969,  0.0545,  0.2496,  0.1445
 0.7675,  0.1354, -0.7698, -0.5460,  0.1920,  0.1728
-0.5211, -0.7372, -0.6763,  0.6897,  0.2044,  0.5217
 0.1913,  0.1980,  0.2314, -0.8816,  0.5006,  0.1998
 0.8964,  0.0694, -0.6149,  0.5059, -0.9854,  0.1825
 0.1767,  0.7104,  0.2093,  0.6452,  0.7590,  0.2832
-0.3580, -0.7541,  0.4426, -0.1193, -0.7465,  0.5657
-0.5996,  0.5766, -0.9758, -0.3933, -0.9572,  0.6800
 0.9950,  0.1641, -0.4132,  0.8579,  0.0142,  0.2003
-0.4717, -0.3894, -0.2567, -0.5111,  0.1691,  0.4266
 0.3917, -0.8561,  0.9422,  0.5061,  0.6123,  0.1212
-0.0366, -0.1087,  0.3449, -0.1025,  0.4086,  0.2475
 0.3633,  0.3943,  0.2372, -0.6980,  0.5216,  0.1925
-0.5325, -0.6466, -0.2178, -0.3589,  0.6310,  0.3568
 0.2271,  0.5200, -0.1447, -0.8011, -0.7699,  0.3128
 0.6415,  0.1993,  0.3777, -0.0178, -0.8237,  0.2181
-0.5298, -0.0768, -0.6028, -0.9490,  0.4588,  0.4356
 0.6870, -0.1431,  0.7294,  0.3141,  0.1621,  0.1632
-0.5985,  0.0591,  0.7889, -0.3900,  0.7419,  0.2945
 0.3661,  0.7984, -0.8486,  0.7572, -0.6183,  0.3449
 0.6995,  0.3342, -0.3113, -0.6972,  0.2707,  0.1712
 0.2565,  0.9126,  0.1798, -0.6043, -0.1413,  0.2893
-0.3265,  0.9839, -0.2395,  0.9854,  0.0376,  0.4770
 0.2690, -0.1722,  0.9818,  0.8599, -0.7015,  0.3954
-0.2102, -0.0768,  0.1219,  0.5607, -0.0256,  0.3949
 0.8216, -0.9555,  0.6422, -0.6231,  0.3715,  0.0801
-0.2896,  0.9484, -0.7545, -0.6249,  0.7789,  0.4370
-0.9985, -0.5448, -0.7092, -0.5931,  0.7926,  0.5402

Test data:

# synthetic_test_40.txt
#
 0.7462,  0.4006, -0.0590,  0.6543, -0.0083,  0.1935
 0.8495, -0.2260, -0.0142, -0.4911,  0.7699,  0.1078
-0.2335, -0.4049,  0.4352, -0.6183, -0.7636,  0.5088
 0.1810, -0.5142,  0.2465,  0.2767, -0.3449,  0.3136
-0.8650,  0.7611, -0.0801,  0.5277, -0.4922,  0.7140
-0.2358, -0.7466, -0.5115, -0.8413, -0.3943,  0.4533
 0.4834,  0.2300,  0.3448, -0.9832,  0.3568,  0.1360
-0.6502, -0.6300,  0.6885,  0.9652,  0.8275,  0.3046
-0.3053,  0.5604,  0.0929,  0.6329, -0.0325,  0.4756
-0.7995,  0.0740, -0.2680,  0.2086,  0.9176,  0.4565
-0.2144, -0.2141,  0.5813,  0.2902, -0.2122,  0.4119
-0.7278, -0.0987, -0.3312, -0.5641,  0.8515,  0.4438
 0.3793,  0.1976,  0.4933,  0.0839,  0.4011,  0.1905
-0.8568,  0.9573, -0.5272,  0.3212, -0.8207,  0.7415
-0.5785,  0.0056, -0.7901, -0.2223,  0.0760,  0.5551
 0.0735, -0.2188,  0.3925,  0.3570,  0.3746,  0.2191
 0.1230, -0.2838,  0.2262,  0.8715,  0.1938,  0.2878
 0.4792, -0.9248,  0.5295,  0.0366, -0.9894,  0.3149
-0.4456,  0.0697,  0.5359, -0.8938,  0.0981,  0.3879
 0.8629, -0.8505, -0.4464,  0.8385,  0.5300,  0.1769
 0.1995,  0.6659,  0.7921,  0.9454,  0.9970,  0.2330
-0.0249, -0.3066, -0.2927, -0.4923,  0.8220,  0.2437
 0.4513, -0.9481, -0.0770, -0.4374, -0.9421,  0.2879
-0.3405,  0.5931, -0.3507, -0.3842,  0.8562,  0.3987
 0.9538,  0.0471,  0.9039,  0.7760,  0.0361,  0.1706
-0.0887,  0.2104,  0.9808,  0.5478, -0.3314,  0.4128
-0.8220, -0.6302,  0.0537, -0.1658,  0.6013,  0.4306
-0.4123, -0.2880,  0.9074, -0.0461, -0.4435,  0.5144
 0.0060,  0.2867, -0.7775,  0.5161,  0.7039,  0.3599
-0.7968, -0.5484,  0.9426, -0.4308,  0.8148,  0.2979
 0.7811,  0.8450, -0.6877,  0.7594,  0.2640,  0.2362
-0.6802, -0.1113, -0.8325, -0.6694, -0.6056,  0.6544
 0.3821,  0.1476,  0.7466, -0.5107,  0.2592,  0.1648
 0.7265,  0.9683, -0.9803, -0.4943, -0.5523,  0.2454
-0.9049, -0.9797, -0.0196, -0.9090, -0.4433,  0.6447
-0.4607,  0.1811, -0.2389,  0.4050, -0.0078,  0.5229
 0.2664, -0.2932, -0.4259, -0.7336,  0.8742,  0.1834
-0.4507,  0.1029, -0.6294, -0.1158, -0.6294,  0.6081
 0.8948, -0.0124,  0.9278,  0.2899, -0.0314,  0.1534
-0.1323, -0.8813, -0.0146, -0.0697,  0.6135,  0.2386

Posted in Machine Learning, Scikit | Leave a comment

Refactoring My Trimmed Kernel Ridge Regression to Approximate Support Vector Regression Using C#

Posted on June 15, 2026 by jamesdmccaffrey

Bottom line: I refactored my C# implementation of trimmed kernel ridge regression to approximate support vector regression. The refactoring works fine, but the code is so long, there are multiple ways to organize it, and so I might have to refactor at some point in the future.

As is often the case with interesting topics, explaining what the problem is, takes longer than explaining the solution. So bear with me.

The goal of a machine learning regression problem is to predict a single numeric value. For example, a bank might want to predict a maximum loan amount based on applicant age, sex, annual income, debt, and so on.

Two closely related machine learning regression techniques are kernel ridge regression (KRR) and support vector regression (SVR). Both techniques use a kernel function (usually the radial basis function, RBF) to compare two data items for similarity. Both techniques must store training data in order to make predictions, but KRR must store all training items, while SVR eliminates some of the items during training, leaving just the “support vectors” that need to be stored.

On the other hand, the loss function used by SVR is not differentiable, so SVR cannot be trained using stochastic gradient descent, and therefore, SVR does not scale well to very large datasets.

Note: In addition to the kernel support vector regression technique discussed in this blog post, there is a linear support vector regression technique which is absolutely useless in practice.

KRR often gives slightly better prediction accuracy than SVR, is easier to train than SVR, can handle very large datasets via SGD training, and is much, much easier to implement than SVR.

In my mind, this technique gives the advantages of KRR (ability to handle very large datasets) and the advantages of SVR (fewer stored model items and weights than KRR). I first implemented the idea using the scikit KernelRidge module and the idea worked well. Then I implemented the idea using C#, just to see if I could — using C# I had to implent a lot of code from scratch that is built-in to Python and scikit. The C# code worked but it was clunky so — the point of this blog post — I refactored the clunky C# code to make it more rational.

The output of the new, refactored C# demo is:

Begin kernel ridge regression trim 
approximation to SVR using C#

Loading train (200) and test (40) data
Done

First three X predictors:
  -0.1660   0.4406  -0.9998  -0.3953  -0.7065
   0.0776  -0.1616   0.3704  -0.5911   0.7562
  -0.9452   0.3409  -0.1654   0.1174  -0.7192

First three target y:
  0.4840
  0.1568
  0.8054

Creating KRR model
Setting RBF gamma = 0.30
Done

Setting lrnRate = 0.0500
Setting maxEpochs = 5000
Setting alpha decay =  0.00001
Setting auto-exit tol = 0.0010
Setting trim epsilon = 0.0035

Training model
epoch =      0 MSE = 0.0181 acc = 0.1700
Auto-exit at epoch 723
Auto-exit at epoch 707
Done

Final model weights:
 -0.7406  -0.4260  -0.3825 . . .
. . .      0.3299   0.8800

Evaluating trimmed KRR model
Number model item/weights = 109

Train acc = 0.9817
Test acc = 0.9500

Train MSE = 0.0001
Test MSE = 0.0003

End KRR trim/prune demo

The demo data is synthetic demo. There are 200 training items and 40 test items. The demo data was generated by a neural network with random weights and biases. The data is normalized, which is strogly recommended for KRR and SVR because items are compared mathematically and you don’t want a column with huge magnitudes to dominate.

Both KRR and SVR can handle non-numeric data, but that’s another story.

The key calling statements are:

KRR krr = new KRR(gamma);

double lrnRate = 0.05;
int maxEpochs = 5000;
double alpha = 1.0e-5;  // wt decay regularization
double exitTol = 0.001;  // auto-exit
double epsilon = 0.0035; // trimming
// larger epsilon: fewer wts retained
// smaller epsilon: more wts retained
krr.Train(trainX, trainY, lrnRate, maxEpochs, alpha,
  exitTol, epsilon);

The large number of parameters is a significant weakness of KRR (and SVR too).

After training on 200 data items, those items that were predicted with a small error of less than 0.0035 were removed. There were 91 such well-predicted items, leaving a reduced training dataset of 109 items. These would be called the support vectors in SVR. After training a new model on the reduced dataset, the sparse model scored 98.17% accuracy on the reduced training data (107 out of 109 correct) and 95.00% accuracy on the test data (38 out of 40 correct). Nice!

To validate my C# trimmed KRR approximation to SVR, I ran the data through the scikit SVR module. A grid-search optimized scikit SVR model created 90 support vectors (better than the trimmed KRR 109 items), with 96.50% accuracy on the training data (not quite as good as the C# trimmed KRR model), and 95.00% accuracy on the test data (identical to the sparse KRR model). In short, the C# trimmed KRR model and the SVR model are very similar. The large number of hyperparameters involved with KRR and SVR makes exact comparison impossible in practice.

An interesting experiment.

Like most people I know, I get obsessed with all kinds of strange things. I have been obsessed with machine learning for many years.

For reason(s) unknown to me, I am mildly obsessed with science fiction movies that feature a creature that makes noises that are translated to “chittering” in the closed caption text. The are an amazing number of chittering instances in science fiction movies, that you’ll only notice if you’re looking for them.

Left: In “Tarantua” (1955), a scientist experiments with making animals grow and shrink. What could possibly go wrong? The giant tarantula is eventually disposed of via bombs and missiles from an F-80 jet that is piloted by actor Clint Eastwood.

Right: I avoided watching “Predator: Badlands” (2025) for a long time because all of the previous Predator movies were awful (except for the first in 1987). My good friend Ken L convinced me to watch “Badlands” in his fancy home theater. It was really good, and on top of that, it was a bonanza of “chittering”. The very first opening scene had a small chittering creature and then on the Death Plant, basically every plant and animal was a chitterer.

The (refacored) C# trimmed KRR demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols. (My blog editor chokes on symbols).

using System;
using System.IO;
using System.Collections.Generic;

namespace KernelRidgeRegressionTrim
{
  internal class KernelRidgeRegressionProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin kernel ridge " +
        "regression trim approximation to SVR using C# ");

      Console.WriteLine("\nLoading train (200) and" +
        " test (40) data ");
      string trainFile =
        "..\\..\\..\\Data\\synthetic_train_200.txt";
      double[][] trainX =
        Utils.MatLoad(trainFile,
        new int[] { 0, 1, 2, 3, 4 }, ',', "#");
      double[] trainY =
        Utils.MatToVec(Utils.MatLoad(trainFile,
        new int[] { 5 }, ',', "#"));

      string testFile =
        "..\\..\\..\\Data\\synthetic_test_40.txt";
      double[][] testX =
        Utils.MatLoad(testFile,
        new int[] { 0, 1, 2, 3, 4 }, ',', "#");
      double[] testY =
        Utils.MatToVec(Utils.MatLoad(testFile,
        new int[] { 5 }, ',', "#"));
      Console.WriteLine("Done ");

      Console.WriteLine("\nFirst three X predictors: ");
      for (int i = 0; i "lt" 3; ++i)
        Utils.VecShow(trainX[i], 4, 9);
      Console.WriteLine("\nFirst three target y: ");
      for (int i = 0; i "lt" 3; ++i)
        Console.WriteLine(trainY[i].ToString("F4").
          PadLeft(8));

      Console.WriteLine("\nCreating KRR model ");
      double gamma = 0.3;    // RBF param
      Console.WriteLine("Setting RBF gamma = " +
        gamma.ToString("F2"));
      KRR krr = new KRR(gamma);
      Console.WriteLine("Done ");

      double lrnRate = 0.05;
      int maxEpochs = 5000;
      double alpha = 1.0e-5;  // wt decay regularization
      double exitTol = 0.001;  // auto-exit
      double epsilon = 0.0035; // trimming
      // larger epsilon: more wts retained
      // smaller epsilon: fewer wts retained
      Console.WriteLine("\nSetting lrnRate = " +
        lrnRate.ToString("F4"));
      Console.WriteLine("Setting maxEpochs = " + maxEpochs);
      Console.WriteLine("Setting alpha decay =  " +
        alpha.ToString("F5"));
      Console.WriteLine("Setting auto-exit tol = " +
        exitTol.ToString("F4"));
      Console.WriteLine("Setting trim epsilon = " +
        epsilon.ToString("F4"));
      Console.WriteLine("\nTraining model ");
      krr.Train(trainX, trainY, lrnRate, maxEpochs, alpha,
        exitTol, epsilon);
      Console.WriteLine("Done ");
      Console.WriteLine("\nFinal model weights: ");
      Utils.VecShow(krr.wts, 4, 9);

      Console.WriteLine("\nEvaluating trimmed KRR model ");
      Console.WriteLine("Number model item/weights = " +
        krr.wts.Length);
      double trainAcc = 
        krr.Accuracy(krr.trainX, krr.trainY, 0.10);
      double testAcc = krr.Accuracy(testX, testY, 0.10);

      Console.WriteLine("\nTrain acc = " +
        trainAcc.ToString("F4"));
      Console.WriteLine("Test acc = " +
        testAcc.ToString("F4"));

      double trainMSE = krr.MSE(krr.trainX, krr.trainY);
      double testMSE = krr.MSE(testX, testY);

      Console.WriteLine("\nTrain MSE = " +
        trainMSE.ToString("F4"));
      Console.WriteLine("Test MSE = " +
        testMSE.ToString("F4"));

      Console.WriteLine("\nEnd KRR trim/prune demo ");
      Console.ReadLine();
    } // Main()

  } // class Program

  /*
  Creating scikit SVR model
  Setting gamma = 0.1000
  Setting C = 10.0000
  Setting epsilon = 0.0100

  Number support vectors = 90
  Accuracy (within 0.10) train = 0.9650
  Accuracy (within 0.10) test = 0.9500
  MSE train = 0.0001
  MSE test = 0.0002 
  */

  // ========================================================

  public class KRR
  {
    public double gamma;  // for RBF kernel
    public double[][] trainX;  // need for prediction
    public double[] trainY; // not used this version
    public double[] wts;  // one per trainX item
    public Random rnd; // for SGD

    // ------------------------------------------------------

    public KRR(double gamma, int seed = 1)
    {
      this.gamma = gamma;
      this.trainX = new double[0][];
      this.trainY = new double[0];
      this.wts = new double[0];
      this.rnd = new Random(seed);  // shuffle train order
    } // ctor

    // ------------------------------------------------------

    private void Shuffle(int[] indices)
    {
      // Fisher-Yates for SGD
      for (int i = 0; i "lt" indices.Length; ++i)
      {
        int ri = this.rnd.Next(i, indices.Length);
        int tmp = indices[i];
        indices[i] = indices[ri];
        indices[ri] = tmp;
      }
    } // Shuffle

    // ------------------------------------------------------

    public void Train(double[][] trainX, double[] trainY,
     double lrnRate, int maxEpochs, double alpha,
     double exitTol, double epsilon)
    {
      int N = trainX.Length; // not used this version
      int freq = maxEpochs / 5;  // when to show progress

      // 0. store trainX -- needed by Predict()
      // copy by value for safety. change to copy
      // by ref if short of memory.
      this.trainX = new double[trainX.Length][];
      for (int i = 0; i "lt" trainX.Length; ++i)
        this.trainX[i] = new double[trainX[0].Length];
      for (int i = 0; i "lt" trainX.Length; ++i)
        for (int j = 0; j "lt" trainX[0].Length; ++j)
          this.trainX[i][j] = trainX[i][j];

      this.trainY = new double[trainY.Length];
      for (int i = 0; i "lt" this.trainY.Length; ++i)
        this.trainY[i] = trainY[i];

      // 1. init weights
      this.wts = new double[trainX.Length];
      double lo = -0.10; double hi = 0.10;
      for (int i = 0; i "lt" this.wts.Length; ++i)
        this.wts[i] = (hi - lo) *
          this.rnd.NextDouble() + lo;

      // 2. set up indices for shuffling
      int[] indices = new int[trainX.Length];
      for (int i = 0; i "lt" indices.Length; ++i)
        indices[i] = i;

      // 3. set up prev weights for auto-exit
      double[] prevWeights = new double[trainX.Length];
      for (int j = 0; j "lt" trainX.Length; ++j)
        prevWeights[j] = this.wts[j];

      // 4. SGD prelim train
      for (int epoch = 0; epoch "lt" maxEpochs; ++epoch)
      {
        Shuffle(indices);
        for (int i = 0; i "lt" trainX.Length; ++i)
        {
          int idx = indices[i];
          double[] x = trainX[idx];
          double predY = this.Predict(x);
          double actualY = trainY[idx];

          // update wt assoc with x
          this.wts[idx] -= lrnRate * (predY - actualY);
        } // each item

        if (epoch % freq == 0)
        {
          double rmse = this.MSE(trainX, trainY);
          double acc = this.Accuracy(trainX, trainY, 0.10);
          string s1 = "epoch = " +
            epoch.ToString().PadLeft(6);
          string s2 = " MSE = " +
            rmse.ToString("F4");
          string s3 = " acc = " + acc.ToString("F4");
          Console.WriteLine(s1 + s2 + s3);
        }
                
        // if max_change_in_wts / max_weights "lt" tol
        int numWts = this.wts.Length;
        double[] weightDeltas = new double[numWts];
        for (int j = 0; j "lt" numWts; ++j)
          weightDeltas[j] = 
            Math.Abs(prevWeights[j] - this.wts[j]);
        double maxChange = weightDeltas[0];
        for (int j = 0; j "lt" numWts; ++j)
          if (weightDeltas[j] "gt" maxChange)
            maxChange = weightDeltas[j];

        double maxWeightMag = Math.Abs(this.wts[0]);
        for (int j = 0; j "lt" numWts; ++j)
          if (Math.Abs(this.wts[j]) "gt" maxWeightMag)
            maxWeightMag = Math.Abs(this.wts[j]);

        if (maxWeightMag != 0.0 &&
          (maxChange / maxWeightMag) "lt" exitTol)
        {
          Console.WriteLine("Auto-exit at epoch " + epoch);
          break;
        }

        // an auto-exit didn't happen
        for (int j = 0; j "lt" numWts; ++j)
          prevWeights[j] = this.wts[j];
      
      } // each epoch

      // 5. apply one final wt decay regularization
      // as opposed to small decay after every epoch
      for (int j = 0; j "lt" this.wts.Length; ++j)
        this.wts[j] *= (1.0 - alpha);
      
      // trimming phase

      // 6. find idxs of items that are predicted too well
      // 0 = not supp vec == predicted too well.
      // 1 = supp vec == predicts only OK.
      int[] isSupportVec = new int[trainX.Length]; 
      for (int i = 0; i "lt" trainX.Length; ++i)
      {
        if (Math.Abs(this.Predict(trainX[i]) - 
          trainY[i]) "gt" epsilon)
          isSupportVec[i] = 1; // just OK prediction
      }
      
      // 7. create a new, smaller training dataset
      List"lt"double[]"gt" tmpTrainX = 
        new List"lt"double[]"gt"();
      List"lt"double"gt" tmpTrainY = 
        new List"lt"double"gt"();
      for (int i = 0; i "lt" trainX.Length; ++i)
      {
        if (isSupportVec[i] == 1)
        {
          tmpTrainX.Add(trainX[i]);
          tmpTrainY.Add(trainY[i]);
        }
      }
      double[][] newTrainX = tmpTrainX.ToArray();
      double[] newTrainY = tmpTrainY.ToArray();

      // 8. re-train using new small dataset
      // store trainX -- needed by Predict()
      this.trainX = newTrainX;  // by ref -- could copy
      this.trainY = newTrainY;  // not used this version

      // re-init new, smaller set of weights
      this.wts = new double[newTrainX.Length];
      for (int i = 0; i "lt" this.wts.Length; ++i)
        this.wts[i] = (hi - lo) *
          this.rnd.NextDouble() + lo;

      // set up new, small indices for shuffling
      int[] newIndices = new int[newTrainX.Length];
      for (int i = 0; i "lt" newIndices.Length; ++i)
        newIndices[i] = i;

      // set up new prev weights for auto-exit
      double[] newPrevWeights = new double[newTrainX.Length];
      for (int j = 0; j "lt" newTrainX.Length; ++j)
        newPrevWeights[j] = this.wts[j];

      // SGD final train
      for (int epoch = 0; epoch "lt" maxEpochs; ++epoch)
      {
        Shuffle(newIndices);
        for (int i = 0; i "lt" newTrainX.Length; ++i)
        {
          int idx = newIndices[i];
          double[] x = newTrainX[idx];
          double predY = this.Predict(x);
          double actualY = newTrainY[idx];

          // update wt assoc with x
          this.wts[idx] -= lrnRate * (predY - actualY);
        } // each item

        // show progress on second SGD
        //if (epoch % freq == 0)
        //{
        //  double rmse = this.MSE(trainX, trainY);
        //  double acc = this.Accuracy(trainX, trainY, 0.10);
        //  string s1 = "epoch = " +
        //    epoch.ToString().PadLeft(6);
        //  string s2 = " MSE = " +
        //    rmse.ToString("F4");
        //  string s3 = " acc = " + acc.ToString("F4");
        //  Console.WriteLine(s1 + s2 + s3);
        //}
                
        // if max_change_in_wts / max_weights "lt" tol
        int numWts = this.wts.Length;
        double[] weightDeltas = new double[numWts];
        for (int j = 0; j "lt" numWts; ++j)
          weightDeltas[j] = 
            Math.Abs(newPrevWeights[j] - this.wts[j]);
        double maxChange = weightDeltas[0];
        for (int j = 0; j "lt" numWts; ++j)
          if (weightDeltas[j] "gt" maxChange)
            maxChange = weightDeltas[j];

        double maxWeightMag = Math.Abs(this.wts[0]);
        for (int j = 0; j "lt" numWts; ++j)
          if (Math.Abs(this.wts[j]) "gt" maxWeightMag)
            maxWeightMag = Math.Abs(this.wts[j]);

        if (maxWeightMag != 0.0 &&
          (maxChange / maxWeightMag) "lt" exitTol)
        {
          Console.WriteLine("Auto-exit at epoch " + epoch);
          break;
        }

        // an auto-exit didn't happen
        for (int j = 0; j "lt" numWts; ++j)
          newPrevWeights[j] = this.wts[j];
      
      } // each epoch

      // apply one final wt decay regularization
      // as opposed to small decay after every epoch
      for (int j = 0; j "lt" this.wts.Length; ++j)
        this.wts[j] *= (1.0 - alpha);

    } // Train()

    // ------------------------------------------------------

    private double Rbf(double[] v1, double[] v2)
    {
      // the gamma version aot len_scale version
      int dim = v1.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" dim; ++i)
      {
        sum += (v1[i] - v2[i]) * (v1[i] - v2[i]);
      }
      return Math.Exp(-1 * this.gamma * sum);
    }

    // ------------------------------------------------------

    public double Predict(double[] x)
    {
      int N = this.trainX.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" N; ++i)
      {
        double[] xx = this.trainX[i];
        double k = this.Rbf(x, xx);
        sum += this.wts[i] * k;
      }
      return sum;
    }

    // ------------------------------------------------------

    public double Accuracy(double[][] dataX,
      double[] dataY, double pctClose)
    {
      int numCorrect = 0; int numWrong = 0;
      int n = dataX.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        double[] x = dataX[i];
        double actualY = dataY[i];
        double predY = this.Predict(x);
        if (Math.Abs(actualY - predY) "lt"
          Math.Abs(actualY * pctClose))
          ++numCorrect;
        else
          ++numWrong;
      }
      return (numCorrect * 1.0) / n;
    }

    // ------------------------------------------------------

    public double MSE(double[][] dataX, double[] dataY)
    {
      double sum = 0.0;
      int n = dataX.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        double[] x = dataX[i];
        double actualY = dataY[i];
        double predY = this.Predict(x);
        sum += (actualY - predY) * (actualY - predY);
      }
      return sum / n;
    }
  } // class KRR

  // ========================================================

  public class Utils
  {
    // ------------------------------------------------------

    public static double[][] MatLoad(string fn,
      int[] usecols, char sep, string comment)
    {
      List"lt"double[]"gt" result = 
        new List"lt"double[]"gt"();
      string line = "";
      FileStream ifs = new FileStream(fn, FileMode.Open);
      StreamReader sr = new StreamReader(ifs);
      while ((line = sr.ReadLine()) != null)
      {
        if (line.StartsWith(comment) == true)
          continue;
        string[] tokens = line.Split(sep);
        List"lt"double"gt" lst = new List"lt"double"gt"();
        for (int j = 0; j "lt" usecols.Length; ++j)
          lst.Add(double.Parse(tokens[usecols[j]]));
        double[] row = lst.ToArray();
        result.Add(row);
      }
      sr.Close(); ifs.Close();
      return result.ToArray();
    }

    // ------------------------------------------------------

    public static double[] MatToVec(double[][] mat)
    {
      int nRows = mat.Length;
      int nCols = mat[0].Length;
      double[] result = new double[nRows * nCols];
      int k = 0;
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          result[k++] = mat[i][j];
      return result;
    }

    // ------------------------------------------------------

    public static void MatShow(double[][] m, int dec,
      int wid)
    {
      int nRows = m.Length; int nCols = m[0].Length;
      double small = 1.0 / Math.Pow(10, dec);
      for (int i = 0; i "lt" nRows; ++i)
      {
        for (int j = 0; j "lt" nCols; ++j)
        {
          double v = m[i][j];
          if (Math.Abs(v) "lt" small) v = 0.0;
          Console.Write(v.ToString("F" + dec).
            PadLeft(wid));
        }
        Console.WriteLine("");
      }
    }

    // ------------------------------------------------------

    public static void VecShow(double[] vec, int dec,
      int wid)
    {
      for (int i = 0; i "lt" vec.Length; ++i)
        Console.Write(vec[i].ToString("F" + dec).
          PadLeft(wid));
      Console.WriteLine("");
    }

    // ------------------------------------------------------

  } // class Utils

  // ========================================================

} // ns

Training data:

# synthetic_train_200.txt
#
-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.0776, -0.1616,  0.3704, -0.5911,  0.7562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
 0.9365, -0.3732,  0.3846,  0.7528,  0.7892,  0.1345
-0.8299, -0.9219, -0.6603,  0.7563, -0.8033,  0.7955
 0.0663,  0.3838, -0.3690,  0.3730,  0.6693,  0.3206
-0.9634,  0.5003,  0.9777,  0.4963, -0.4391,  0.7377
-0.1042,  0.8172, -0.4128, -0.4244, -0.7399,  0.4801
-0.9613,  0.3577, -0.5767, -0.4689, -0.0169,  0.6861
-0.7065,  0.1786,  0.3995, -0.7953, -0.1719,  0.5569
 0.3888, -0.1716, -0.9001,  0.0718,  0.3276,  0.2500
 0.1731,  0.8068, -0.7251, -0.7214,  0.6148,  0.3297
-0.2046, -0.6693,  0.8550, -0.3045,  0.5016,  0.2129
 0.2473,  0.5019, -0.3022, -0.4601,  0.7918,  0.2613
-0.1438,  0.9297,  0.3269,  0.2434, -0.7705,  0.5171
 0.1568, -0.1837, -0.5259,  0.8068,  0.1474,  0.3307
-0.9943,  0.2343, -0.3467,  0.0541,  0.7719,  0.5581
 0.2467, -0.9684,  0.8589,  0.3818,  0.9946,  0.1092
-0.6553, -0.7257,  0.8652,  0.3936, -0.8680,  0.7018
 0.8460,  0.4230, -0.7515, -0.9602, -0.9476,  0.1996
-0.9434, -0.5076,  0.7201,  0.0777,  0.1056,  0.5664
 0.9392,  0.1221, -0.9627,  0.6013, -0.5341,  0.1533
 0.6142, -0.2243,  0.7271,  0.4942,  0.1125,  0.1661
 0.4260,  0.1194, -0.9749, -0.8561,  0.9346,  0.2230
 0.1362, -0.5934, -0.4953,  0.4877, -0.6091,  0.3810
 0.6937, -0.5203, -0.0125,  0.2399,  0.6580,  0.1460
-0.6864, -0.9628, -0.8600, -0.0273,  0.2127,  0.5387
 0.9772,  0.1595, -0.2397,  0.1019,  0.4907,  0.1611
 0.3385, -0.4702, -0.8673, -0.2598,  0.2594,  0.2270
-0.8669, -0.4794,  0.6095, -0.6131,  0.2789,  0.4700
 0.0493,  0.8496, -0.4734, -0.8681,  0.4701,  0.3516
 0.8639, -0.9721, -0.5313,  0.2336,  0.8980,  0.1412
 0.9004,  0.1133,  0.8312,  0.2831, -0.2200,  0.1782
 0.0991,  0.8524,  0.8375, -0.2102,  0.9265,  0.2150
-0.6521, -0.7473, -0.7298,  0.0113, -0.9570,  0.7422
 0.6190, -0.3105,  0.8802,  0.1640,  0.7577,  0.1056
 0.6895,  0.8108, -0.0802,  0.0927,  0.5972,  0.2214
 0.1982, -0.9689,  0.1870, -0.1326,  0.6147,  0.1310
-0.3695,  0.7858,  0.1557, -0.6320,  0.5759,  0.3773
-0.1596,  0.3581,  0.8372, -0.9992,  0.9535,  0.2071
-0.2468,  0.9476,  0.2094,  0.6577,  0.1494,  0.4132
 0.1737,  0.5000,  0.7166,  0.5102,  0.3961,  0.2611
 0.7290, -0.3546,  0.3416, -0.0983, -0.2358,  0.1332
-0.3652,  0.2438, -0.1395,  0.9476,  0.3556,  0.4170
-0.6029, -0.1466, -0.3133,  0.5953,  0.7600,  0.4334
-0.4596, -0.4953,  0.7098,  0.0554,  0.6043,  0.2775
 0.1450,  0.4663,  0.0380,  0.5418,  0.1377,  0.2931
-0.8636, -0.2442, -0.8407,  0.9656, -0.6368,  0.7429
 0.6237,  0.7499,  0.3768,  0.1390, -0.6781,  0.2185
-0.5499,  0.1850, -0.3755,  0.8326,  0.8193,  0.4399
-0.4858, -0.7782, -0.6141, -0.0008,  0.4572,  0.4197
 0.7033, -0.1683,  0.2334, -0.5327, -0.7961,  0.1776
 0.0317, -0.0457, -0.6947,  0.2436,  0.0880,  0.3345
 0.5031, -0.5559,  0.0387,  0.5706, -0.9553,  0.3107
-0.3513,  0.7458,  0.6894,  0.0769,  0.7332,  0.3170
 0.2205,  0.5992, -0.9309,  0.5405,  0.4635,  0.3532
-0.4806, -0.4859,  0.2646, -0.3094,  0.5932,  0.3202
 0.9809, -0.3995, -0.7140,  0.8026,  0.0831,  0.1600
 0.9495,  0.2732,  0.9878,  0.0921,  0.0529,  0.1289
-0.9476, -0.6792,  0.4913, -0.9392, -0.2669,  0.5966
 0.7247,  0.3854,  0.3819, -0.6227, -0.1162,  0.1550
-0.5922, -0.5045, -0.4757,  0.5003, -0.0860,  0.5863
-0.8861,  0.0170, -0.5761,  0.5972, -0.4053,  0.7301
 0.6877, -0.2380,  0.4997,  0.0223,  0.0819,  0.1404
 0.9189,  0.6079, -0.9354,  0.4188, -0.0700,  0.1907
-0.1428, -0.7820,  0.2676,  0.6059,  0.3936,  0.2790
 0.5324, -0.3151,  0.6917, -0.1425,  0.6480,  0.1071
-0.8432, -0.9633, -0.8666, -0.0828, -0.7733,  0.7784
-0.9444,  0.5097, -0.2103,  0.4939, -0.0952,  0.6787
-0.0520,  0.6063, -0.1952,  0.8094, -0.9259,  0.4836
 0.5477, -0.7487,  0.2370, -0.9793,  0.0773,  0.1241
 0.2450,  0.8116,  0.9799,  0.4222,  0.4636,  0.2355
 0.8186, -0.1983, -0.5003, -0.6531, -0.7611,  0.1511
-0.4714,  0.6382, -0.3788,  0.9648, -0.4667,  0.5950
 0.0673, -0.3711,  0.8215, -0.2669, -0.1328,  0.2677
-0.9381,  0.4338,  0.7820, -0.9454,  0.0441,  0.5518
-0.3480,  0.7190,  0.1170,  0.3805, -0.0943,  0.4724
-0.9813,  0.1535, -0.3771,  0.0345,  0.8328,  0.5438
-0.1471, -0.5052, -0.2574,  0.8637,  0.8737,  0.3042
-0.5454, -0.3712, -0.6505,  0.2142, -0.1728,  0.5783
 0.6327, -0.6297,  0.4038, -0.5193,  0.1484,  0.1153
-0.5424,  0.3282, -0.0055,  0.0380, -0.6506,  0.6613
 0.1414,  0.9935,  0.6337,  0.1887,  0.9520,  0.2540
-0.9351, -0.8128, -0.8693, -0.0965, -0.2491,  0.7353
 0.9507, -0.6640,  0.9456,  0.5349,  0.6485,  0.1059
-0.0462, -0.9737, -0.2940, -0.0159,  0.4602,  0.2606
-0.0627, -0.0852, -0.7247, -0.9782,  0.5166,  0.2977
 0.0478,  0.5098, -0.0723, -0.7504, -0.3750,  0.3335
 0.0090,  0.3477,  0.5403, -0.7393, -0.9542,  0.4415
-0.9748,  0.3449,  0.3736, -0.1015,  0.8296,  0.4358
 0.2887, -0.9895, -0.0311,  0.7186,  0.6608,  0.2057
 0.1570, -0.4518,  0.1211,  0.3435, -0.2951,  0.3244
 0.7117, -0.6099,  0.4946, -0.4208,  0.5476,  0.1096
-0.2929, -0.5726,  0.5346, -0.3827,  0.4665,  0.2465
 0.4889, -0.5572, -0.5718, -0.6021, -0.7150,  0.2163
-0.7782,  0.3491,  0.5996, -0.8389, -0.5366,  0.6516
-0.5847,  0.8347,  0.4226,  0.1078, -0.3910,  0.6134
 0.8469,  0.4121, -0.0439, -0.7476,  0.9521,  0.1571
-0.6803, -0.5948, -0.1376, -0.1916, -0.7065,  0.7156
 0.2878,  0.5086, -0.5785,  0.2019,  0.4979,  0.2980
 0.2764,  0.1943, -0.4090,  0.4632,  0.8906,  0.2960
-0.8877,  0.6705, -0.6155, -0.2098, -0.3998,  0.7107
-0.8398,  0.8093, -0.2597,  0.0614, -0.0118,  0.6502
-0.8476,  0.0158, -0.4769, -0.2859, -0.7839,  0.7715
 0.5751, -0.7868,  0.9714, -0.6457,  0.1448,  0.1175
 0.4802, -0.7001,  0.1022, -0.5668,  0.5184,  0.1090
 0.4458, -0.6469,  0.7239, -0.9604,  0.7205,  0.0779
 0.5175,  0.4339,  0.9747, -0.4438, -0.9924,  0.2879
 0.8678,  0.7158,  0.4577,  0.0334,  0.4139,  0.1678
 0.5406,  0.5012,  0.2264, -0.1963,  0.3946,  0.2088
-0.9938,  0.5498,  0.7928, -0.5214, -0.7585,  0.7687
 0.7661,  0.0863, -0.4266, -0.7233, -0.4197,  0.1466
 0.2277, -0.3517, -0.0853, -0.1118,  0.6563,  0.1767
 0.3499, -0.5570, -0.0655, -0.3705,  0.2537,  0.1632
 0.7547, -0.1046,  0.5689, -0.0861,  0.3125,  0.1257
 0.8186,  0.2110,  0.5335,  0.0094, -0.0039,  0.1391
 0.6858, -0.8644,  0.1465,  0.8855,  0.0357,  0.1845
-0.4967,  0.4015,  0.0805,  0.8977,  0.2487,  0.4663
 0.6760, -0.9841,  0.9787, -0.8446, -0.3557,  0.1509
-0.1203, -0.4885,  0.6054, -0.0443, -0.7313,  0.4854
 0.8557,  0.7919, -0.0169,  0.7134, -0.1628,  0.2002
 0.0115, -0.6209,  0.9300, -0.4116, -0.7931,  0.4052
-0.7114, -0.9718,  0.4319,  0.1290,  0.5892,  0.3661
 0.3915,  0.5557, -0.1870,  0.2955, -0.6404,  0.2954
-0.3564, -0.6548, -0.1827, -0.5172, -0.1862,  0.4622
 0.2392, -0.4959,  0.5857, -0.1341, -0.2850,  0.2470
-0.3394,  0.3947, -0.4627,  0.6166, -0.4094,  0.5325
 0.7107,  0.7768, -0.6312,  0.1707,  0.7964,  0.2757
-0.1078,  0.8437, -0.4420,  0.2177,  0.3649,  0.4028
-0.3139,  0.5595, -0.6505, -0.3161, -0.7108,  0.5546
 0.4335,  0.3986,  0.3770, -0.4932,  0.3847,  0.1810
-0.2562, -0.2894, -0.8847,  0.2633,  0.4146,  0.4036
 0.2272,  0.2966, -0.6601, -0.7011,  0.0284,  0.2778
-0.0743, -0.1421, -0.0054, -0.6770, -0.3151,  0.3597
-0.4762,  0.6891,  0.6007, -0.1467,  0.2140,  0.4266
-0.4061,  0.7193,  0.3432,  0.2669, -0.7505,  0.6147
-0.0588,  0.9731,  0.8966,  0.2902, -0.6966,  0.4955
-0.0627, -0.1439,  0.1985,  0.6999,  0.5022,  0.3077
 0.1587,  0.8494, -0.8705,  0.9827, -0.8940,  0.4263
-0.7850,  0.2473, -0.9040, -0.4308, -0.8779,  0.7199
 0.4070,  0.3369, -0.2428, -0.6236,  0.4940,  0.2215
-0.0242,  0.0513, -0.9430,  0.2885, -0.2987,  0.3947
-0.5416, -0.1322, -0.2351, -0.0604,  0.9590,  0.3683
 0.1055,  0.7783, -0.2901, -0.5090,  0.8220,  0.2984
-0.9129,  0.9015,  0.1128, -0.2473,  0.9901,  0.4776
-0.9378,  0.1424, -0.6391,  0.2619,  0.9618,  0.5368
 0.7498, -0.0963,  0.4169,  0.5549, -0.0103,  0.1614
-0.2612, -0.7156,  0.4538, -0.0460, -0.1022,  0.3717
 0.7720,  0.0552, -0.1818, -0.4622, -0.8560,  0.1685
-0.4177,  0.0070,  0.9319, -0.7812,  0.3461,  0.3052
-0.0001,  0.5542, -0.7128, -0.8336, -0.2016,  0.3803
 0.5356, -0.4194, -0.5662, -0.9666, -0.2027,  0.1776
-0.2378,  0.3187, -0.8582, -0.6948, -0.9668,  0.5474
-0.1947, -0.3579,  0.1158,  0.9869,  0.6690,  0.2992
 0.3992,  0.8365, -0.9205, -0.8593, -0.0520,  0.3154
-0.0209,  0.0793,  0.7905, -0.1067,  0.7541,  0.1864
-0.4928, -0.4524, -0.3433,  0.0951, -0.5597,  0.6261
-0.8118,  0.7404, -0.5263, -0.2280,  0.1431,  0.6349
 0.0516, -0.8480,  0.7483,  0.9023,  0.6250,  0.1959
-0.3212,  0.1093,  0.9488, -0.3766,  0.3376,  0.2735
-0.3481,  0.5490, -0.3484,  0.7797,  0.5034,  0.4379
-0.5785, -0.9170, -0.3563, -0.9258,  0.3877,  0.4121
 0.3407, -0.1391,  0.5356,  0.0720, -0.9203,  0.3458
-0.3287, -0.8954,  0.2102,  0.0241,  0.2349,  0.3247
-0.1353,  0.6954, -0.0919, -0.9692,  0.7461,  0.3338
 0.9036, -0.8982, -0.5299, -0.8733, -0.1567,  0.1187
 0.7277, -0.8368, -0.0538, -0.7489,  0.5458,  0.0830
 0.9049,  0.8878,  0.2279,  0.9470, -0.3103,  0.2194
 0.7957, -0.1308, -0.5284,  0.8817,  0.3684,  0.2172
 0.4647, -0.4931,  0.2010,  0.6292, -0.8918,  0.3371
-0.7390,  0.6849,  0.2367,  0.0626, -0.5034,  0.7039
-0.1567, -0.8711,  0.7940, -0.5932,  0.6525,  0.1710
 0.7635, -0.0265,  0.1969,  0.0545,  0.2496,  0.1445
 0.7675,  0.1354, -0.7698, -0.5460,  0.1920,  0.1728
-0.5211, -0.7372, -0.6763,  0.6897,  0.2044,  0.5217
 0.1913,  0.1980,  0.2314, -0.8816,  0.5006,  0.1998
 0.8964,  0.0694, -0.6149,  0.5059, -0.9854,  0.1825
 0.1767,  0.7104,  0.2093,  0.6452,  0.7590,  0.2832
-0.3580, -0.7541,  0.4426, -0.1193, -0.7465,  0.5657
-0.5996,  0.5766, -0.9758, -0.3933, -0.9572,  0.6800
 0.9950,  0.1641, -0.4132,  0.8579,  0.0142,  0.2003
-0.4717, -0.3894, -0.2567, -0.5111,  0.1691,  0.4266
 0.3917, -0.8561,  0.9422,  0.5061,  0.6123,  0.1212
-0.0366, -0.1087,  0.3449, -0.1025,  0.4086,  0.2475
 0.3633,  0.3943,  0.2372, -0.6980,  0.5216,  0.1925
-0.5325, -0.6466, -0.2178, -0.3589,  0.6310,  0.3568
 0.2271,  0.5200, -0.1447, -0.8011, -0.7699,  0.3128
 0.6415,  0.1993,  0.3777, -0.0178, -0.8237,  0.2181
-0.5298, -0.0768, -0.6028, -0.9490,  0.4588,  0.4356
 0.6870, -0.1431,  0.7294,  0.3141,  0.1621,  0.1632
-0.5985,  0.0591,  0.7889, -0.3900,  0.7419,  0.2945
 0.3661,  0.7984, -0.8486,  0.7572, -0.6183,  0.3449
 0.6995,  0.3342, -0.3113, -0.6972,  0.2707,  0.1712
 0.2565,  0.9126,  0.1798, -0.6043, -0.1413,  0.2893
-0.3265,  0.9839, -0.2395,  0.9854,  0.0376,  0.4770
 0.2690, -0.1722,  0.9818,  0.8599, -0.7015,  0.3954
-0.2102, -0.0768,  0.1219,  0.5607, -0.0256,  0.3949
 0.8216, -0.9555,  0.6422, -0.6231,  0.3715,  0.0801
-0.2896,  0.9484, -0.7545, -0.6249,  0.7789,  0.4370
-0.9985, -0.5448, -0.7092, -0.5931,  0.7926,  0.5402

Test data:

# synthetic_test_40.txt
#
 0.7462,  0.4006, -0.0590,  0.6543, -0.0083,  0.1935
 0.8495, -0.2260, -0.0142, -0.4911,  0.7699,  0.1078
-0.2335, -0.4049,  0.4352, -0.6183, -0.7636,  0.5088
 0.1810, -0.5142,  0.2465,  0.2767, -0.3449,  0.3136
-0.8650,  0.7611, -0.0801,  0.5277, -0.4922,  0.7140
-0.2358, -0.7466, -0.5115, -0.8413, -0.3943,  0.4533
 0.4834,  0.2300,  0.3448, -0.9832,  0.3568,  0.1360
-0.6502, -0.6300,  0.6885,  0.9652,  0.8275,  0.3046
-0.3053,  0.5604,  0.0929,  0.6329, -0.0325,  0.4756
-0.7995,  0.0740, -0.2680,  0.2086,  0.9176,  0.4565
-0.2144, -0.2141,  0.5813,  0.2902, -0.2122,  0.4119
-0.7278, -0.0987, -0.3312, -0.5641,  0.8515,  0.4438
 0.3793,  0.1976,  0.4933,  0.0839,  0.4011,  0.1905
-0.8568,  0.9573, -0.5272,  0.3212, -0.8207,  0.7415
-0.5785,  0.0056, -0.7901, -0.2223,  0.0760,  0.5551
 0.0735, -0.2188,  0.3925,  0.3570,  0.3746,  0.2191
 0.1230, -0.2838,  0.2262,  0.8715,  0.1938,  0.2878
 0.4792, -0.9248,  0.5295,  0.0366, -0.9894,  0.3149
-0.4456,  0.0697,  0.5359, -0.8938,  0.0981,  0.3879
 0.8629, -0.8505, -0.4464,  0.8385,  0.5300,  0.1769
 0.1995,  0.6659,  0.7921,  0.9454,  0.9970,  0.2330
-0.0249, -0.3066, -0.2927, -0.4923,  0.8220,  0.2437
 0.4513, -0.9481, -0.0770, -0.4374, -0.9421,  0.2879
-0.3405,  0.5931, -0.3507, -0.3842,  0.8562,  0.3987
 0.9538,  0.0471,  0.9039,  0.7760,  0.0361,  0.1706
-0.0887,  0.2104,  0.9808,  0.5478, -0.3314,  0.4128
-0.8220, -0.6302,  0.0537, -0.1658,  0.6013,  0.4306
-0.4123, -0.2880,  0.9074, -0.0461, -0.4435,  0.5144
 0.0060,  0.2867, -0.7775,  0.5161,  0.7039,  0.3599
-0.7968, -0.5484,  0.9426, -0.4308,  0.8148,  0.2979
 0.7811,  0.8450, -0.6877,  0.7594,  0.2640,  0.2362
-0.6802, -0.1113, -0.8325, -0.6694, -0.6056,  0.6544
 0.3821,  0.1476,  0.7466, -0.5107,  0.2592,  0.1648
 0.7265,  0.9683, -0.9803, -0.4943, -0.5523,  0.2454
-0.9049, -0.9797, -0.0196, -0.9090, -0.4433,  0.6447
-0.4607,  0.1811, -0.2389,  0.4050, -0.0078,  0.5229
 0.2664, -0.2932, -0.4259, -0.7336,  0.8742,  0.1834
-0.4507,  0.1029, -0.6294, -0.1158, -0.6294,  0.6081
 0.8948, -0.0124,  0.9278,  0.2899, -0.0314,  0.1534
-0.1323, -0.8813, -0.0146, -0.0697,  0.6135,  0.2386

Posted in Machine Learning | Leave a comment