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, 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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, 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-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
.NET Test Automation Recipes
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