The goal of a machine learning regression problem is to predict a single numeric value. For example, a bank might to predict the maximum safe loan amount to a customer based on customer age, current savings balance, and so on.
There are roughly a dozen major regression techniques, and each technique has several variations. Common techniques are linear regression, linear ridge regression, gradient boosting decision tree regression, and neural network regression. Each technique has pros and cons.
I use the non-standard term bistratum (“two layers”, borrowed from biology) to indicate a neural network with exactly two hidden layers.
I put together a demo using C#. I used one of my standard synthetic datasets. The data looks like:
-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 . . .
The first four fields on each line are predictors (often called features). The last value is the target to predict. There are 200 training items and 40 test items.
The output of the demo program is:
Neural network bistratum regression using C# Loading synthetic train (200) and test (40) data Done 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 5-(10-10)-1 tanh() identity() neural network regressor Setting lrnRate = 0.0600 Setting maxEpochs = 30000 Setting decay = 0.00000100 Starting training epoch: 0 MSE = 0.0354 acc = 0.0600 epoch: 6000 MSE = 0.0001 acc = 0.8100 epoch: 12000 MSE = 0.0001 acc = 0.7900 epoch: 18000 MSE = 0.0001 acc = 0.8100 epoch: 24000 MSE = 0.0001 acc = 0.8000 Done Evaluating model Accuracy (5%) on train data = 0.8100 Accuracy (5%) on test data = 0.8000 MSE on train data = 0.0001 MSE on test data = 0.0002 Predicting y for train[0] Predicted y = 0.4870 End demo
The demo program creates a 5-(10-10)-1 tanh() identity() neural network regression model. This means there are 5 input nodes, two hidden layers with 10 nodes each, and 1 output node. The hidden nodes uses tanh() activation, and the output node uses identity() activation.
The demo program trains the bistratum network using plain vanilla online (aka batch-1) stochastic gradient descent.
The lrnRate (learning rate) influences how quicky the model weights and biases are updated. The maxEpochs controls how many passes through the training data occur. The decay is a weight decay that discourages model weights from becoming extrememly large. When used with SGD, weight decay is mathematically equivalent to L2 regularization.
The model prediction accuracy on the training data is 81% (162 out of 200 correct), and 80% on the test data (32 out of 40 correct). A prediction is scored correct if it is withing 5% of the true target value.
I’m a big fan of 1950s science fiction movies. One of my favorite scenes in many of these old movies is “the monster in the window”.
Left: In “Beginning of the End” (1957), the U.S. Dept. of Agriculture experiments with radiation to grow supr-large vegetables. What could possibly go wrong? How about tens of thousands of 20-foot long hungry locusts? In “the monster in the window” scenes, there is always a woman who is oblivious to the danger.
Right: In “Tarantula” (1955), a biologist in a remote desert lab experiments with ways to grow animals to giant size. What could possibly . . . OK, how about a 50-foot hungry tarantula spider?
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols. (My blog editor often chokes on symbols).
using System;
using System.IO;
using System.Collections.Generic;
namespace NeuralNetworkBistratumRegression
{
internal class NeuralNetworkBistratumRegressionProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nNeural network bistratum " +
"regression using C# ");
// 1. load data
Console.WriteLine("\nLoading synthetic train (200)" +
" and test (40) data ");
string trainFile =
"..\\..\\..\\Data\\synthetic_train_200.txt";
int[] colsX = new int[] { 0, 1, 2, 3, 4 };
int colY = 5;
double[][] trainX = MatLoad(trainFile, colsX, ',', "#");
double[] trainY = MatToVec(MatLoad(trainFile,
new int[] { colY }, ',', "#"));
string testFile =
"..\\..\\..\\Data\\synthetic_test_40.txt";
double[][] testX = MatLoad(testFile, colsX, ',', "#");
double[] testY = MatToVec(MatLoad(testFile,
new int[] { colY }, ',', "#"));
Console.WriteLine("Done ");
Console.WriteLine("\nFirst three train X: ");
for (int i = 0; i "lt" 3; ++i)
VecShow(trainX[i], 4, 8);
Console.WriteLine("\nFirst three train y: ");
for (int i = 0; i "lt" 3; ++i)
Console.WriteLine(trainY[i].ToString("F4").
PadLeft(8));
// 2. create NN
Console.WriteLine("\nCreating 5-(10-10)-1 tanh()" +
" identity() neural network regressor ");
NeuralNetworkBistratumRegressor nn =
new NeuralNetworkBistratumRegressor(5, 10, 10, 1);
// 3. train NN
double lrnRate = 0.06;
int maxEpochs = 30_000;
double decay = 0.000001; // .81 .80 .0001 .0002
Console.WriteLine("\nSetting lrnRate = " +
lrnRate.ToString("F4"));
Console.WriteLine("Setting maxEpochs = " +
maxEpochs);
Console.WriteLine("Setting decay = " +
decay.ToString("F8"));
Console.WriteLine("\nStarting training ");
nn.Train(trainX, trainY, lrnRate, maxEpochs, decay);
Console.WriteLine("Done ");
// TODO: programmatically analyze weights and
// biases for extreme values
// 4. evaluate trained model
Console.WriteLine("\nEvaluating model ");
double trainAcc = nn.Accuracy(trainX, trainY, 0.05);
double testAcc = nn.Accuracy(testX, testY, 0.05);
Console.WriteLine("\nAccuracy (5%) on train data = " +
trainAcc.ToString("F4"));
Console.WriteLine("Accuracy (5%) on test data = " +
testAcc.ToString("F4"));
double trainMSE = nn.MSE(trainX, trainY);
double testMSE = nn.MSE(testX, testY);
Console.WriteLine("\nMSE on train data = " +
trainMSE.ToString("F4"));
Console.WriteLine("MSE on test data = " +
testMSE.ToString("F4"));
// 5. use model
Console.WriteLine("\nPredicting y for train[0] ");
double[] x = trainX[0];
double predY = nn.Predict(x);
Console.WriteLine("Predicted y = " +
predY.ToString("F4"));
Console.WriteLine("\nEnd demo ");
Console.ReadLine();
} // Main
// ------------------------------------------------------
// helpers for Main(): MatLoad(), MatToVec(), VecShow()
// ------------------------------------------------------
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();
}
static double[] MatToVec(double[][] M)
{
int nRows = M.Length;
int nCols = M[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++] = M[i][j];
return result;
}
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 Program
// ========================================================
public class NeuralNetworkBistratumRegressor
{
// two hidden layers
public int numInput;
public int numHiddenA;
public int numHiddenB;
public int numOutput;
public double[] iNodes; // input nodes
public double[] aNodes;
public double[] bNodes;
public double[] oNodes; // output nodes
public double[][] iaWeights; // input to hidden A
public double[][] abWeights; // hidden A to hidden B
public double[][] boWeights; // hidden B to output
public double[] aBiases;
public double[] bBiases;
public double[] oBiases;
private Random rnd; // wt init and train shuffle
// ------------------------------------------------------
public NeuralNetworkBistratumRegressor(int numInput,
int numHiddenA, int numHiddenB, int numOutput,
int seed = 0)
{
this.numInput = numInput;
this.numHiddenA = numHiddenA;
this.numHiddenB = numHiddenB;
this.numOutput = numOutput; // 1 for regression
this.iNodes = new double[numInput];
this.iaWeights = MatMake(numInput, numHiddenA);
this.aBiases = new double[numHiddenA];
this.aNodes = new double[numHiddenA];
this.abWeights = MatMake(numHiddenA, numHiddenB);
this.bBiases = new double[numHiddenB];
this.bNodes = new double[numHiddenB];
this.boWeights = MatMake(numHiddenB, numOutput);
this.oBiases = new double[numOutput];
this.oNodes = new double[numOutput];
this.rnd = new Random(seed);
}
// ------------------------------------------------------
private 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 double Predict(double[] x)
{
// copy input into iNodes
for (int i = 0; i "lt" this.numInput; ++i)
this.iNodes[i] = x[i];
// compute hidden layer A node values
for (int j = 0; j "lt" numHiddenA; ++j)
{
double sum = 0.0;
for (int i = 0; i "lt" this.numInput; ++i)
sum += this.iNodes[i] * this.iaWeights[i][j];
sum += this.aBiases[j];
this.aNodes[j] = HyperTan(sum);
}
// compute hidden layer B node values
for (int j = 0; j "lt" this.numHiddenB; ++j)
{
double sum = 0.0;
for (int i = 0; i "lt" this.numHiddenA; ++i)
sum += this.aNodes[i] * this.abWeights[i][j];
sum += this.bBiases[j];
this.bNodes[j] = HyperTan(sum);
}
// compute output layer node
for (int j = 0; j "lt" this.numOutput; ++j)
{
double sum = 0.0;
for (int i = 0; i "lt" this.numHiddenB; ++i)
sum += this.bNodes[i] * this.boWeights[i][j];
sum += this.oBiases[j];
this.oNodes[j] = Identity(sum);
}
return this.oNodes[0]; // a single value
}
// ------------------------------------------------------
private static double HyperTan(double x)
{
if (x "lt" -6.0) return -1.0;
else if (x "gt" 6.0) return 1.0;
else return Math.Tanh(x);
}
// ------------------------------------------------------
private static double Identity(double x)
{
return x;
}
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY,
double lrnRate, int maxEpochs, double decay)
{
// 'online' aka batch-1 SGD training (not batch)
// init weights, leave biases at 0
double lo = -0.01; double hi = 0.01;
for (int i = 0; i "lt" this.numInput; ++i)
for (int j = 0; j "lt" this.numHiddenA; ++j)
this.iaWeights[i][j] =
(hi - lo) * this.rnd.NextDouble() + lo;
for (int i = 0; i "lt" this.numHiddenA; ++i)
for (int j = 0; j "lt" this.numHiddenB; ++j)
this.abWeights[i][j] =
(hi - lo) * this.rnd.NextDouble() + lo;
for (int i = 0; i "lt" this.numHiddenB; ++i)
for (int j = 0; j "lt" this.numOutput; ++j)
this.boWeights[i][j] =
(hi - lo) * this.rnd.NextDouble() + lo;
// each weight and bias has a gradient
double[][] boGrads = MatMake(numHiddenB, numOutput);
double[][] abGrads = MatMake(numHiddenA, numHiddenB);
double[][] iaGrads = MatMake(numInput, numHiddenA);
double[] oBiasGrads = new double[numOutput];
double[] bBiasGrads = new double[numHiddenB];
double[] aBiasGrads = new double[numHiddenA];
// each output and hidden node has a 'signal'
// which is gradient without associated input
// (lower case delta in Wikipedia)
double[] oSignals = new double[numOutput];
double[] bSignals = new double[numHiddenB];
double[] aSignals = new double[numHiddenA];
// to shuffle order of items when training
int[] indices = new int[trainX.Length];
for (int i = 0; i "lt" indices.Length; ++i)
indices[i] = i;
int freq = (int)(maxEpochs / 5); // progress 5 times
for (int epoch = 0; epoch "lt" maxEpochs; ++epoch)
{
this.Shuffle(indices); // process in scrambled order
for (int ii = 0; ii "lt" trainX.Length; ++ii)
{
int idx = indices[ii];
double[] x = trainX[idx];
double actualY = trainY[idx];
double predY = this.Predict(x);
// output node signals depends on target values
for (int k = 0; k "lt" this.numOutput; ++k)
{
double error = predY - actualY; // standard
double derivative = 1.0; // identity activation
oSignals[k] = error * derivative;
}
// signal for B nodes depends on output signals
for (int j = 0; j "lt" numHiddenB; ++j)
{
// derivative for tanh() activation
double derivative =
(1 + this.bNodes[j]) * (1 - this.bNodes[j]);
double sum = 0.0;
for (int k = 0; k "lt" numOutput; ++k)
sum += oSignals[k] * this.boWeights[j][k];
bSignals[j] = derivative * sum;
}
// signal for A nodes, depends on B signals
for (int j = 0; j "lt" numHiddenA; ++j)
{
// derivative for tanh() activation
double derivative =
(1 + this.aNodes[j]) * (1 - this.aNodes[j]);
double sum = 0.0;
for (int k = 0; k "lt" numHiddenB; ++k)
sum += bSignals[k] * this.abWeights[j][k];
aSignals[j] = derivative * sum;
}
// at this point, all signals have been computed
// use signals to calculate gradients left-to-right
for (int i = 0; i "lt" numInput; ++i)
for (int j = 0; j "lt" numHiddenA; ++j)
iaGrads[i][j] = this.iNodes[i] * aSignals[j];
for (int i = 0; i "lt" numHiddenA; ++i)
for (int j = 0; j "lt" numHiddenB; ++j)
abGrads[i][j] = this.aNodes[i] * bSignals[j];
for (int i = 0; i "lt" numHiddenB; ++i)
for (int j = 0; j "lt" numOutput; ++j)
boGrads[i][j] = this.bNodes[i] * oSignals[j];
// compute bias gradients
for (int j = 0; j "lt" numHiddenA; ++j)
aBiasGrads[j] = 1.0 * aSignals[j];
for (int j = 0; j "lt" numHiddenB; ++j)
bBiasGrads[j] = 1.0 * bSignals[j];
for (int j = 0; j "lt" numOutput; ++j)
oBiasGrads[j] = 1.0 * oSignals[j];
// before update, decay all weights
// wt decay is (almost) L2 regularization
// when used with plain vanilla SGD
for (int i = 0; i "lt" numInput; ++i)
for (int j = 0; j "lt" numHiddenA; ++j)
this.iaWeights[i][j] *= (1 - decay);
for (int i = 0; i "lt" numHiddenA; ++i)
for (int j = 0; j "lt" numHiddenB; ++j)
this.abWeights[i][j] *= (1 - decay);
for (int i = 0; i "lt" numHiddenB; ++i)
for (int j = 0; j "lt" numOutput; ++j)
this.boWeights[i][j] *= (1 - decay);
// use gradients to update all weights
for (int i = 0; i "lt" numInput; ++i)
for (int j = 0; j "lt" numHiddenA; ++j)
this.iaWeights[i][j] -= iaGrads[i][j] * lrnRate;
for (int i = 0; i "lt" numHiddenA; ++i)
for (int j = 0; j "lt" numHiddenB; ++j)
this.abWeights[i][j] -= abGrads[i][j] * lrnRate;
for (int i = 0; i "lt" numHiddenB; ++i)
for (int j = 0; j "lt" numOutput; ++j)
this.boWeights[i][j] -= boGrads[i][j] * lrnRate;
// update all biases
for (int j = 0; j "lt" numHiddenA; ++j)
this.aBiases[j] -= aBiasGrads[j] * lrnRate;
for (int j = 0; j "lt" numHiddenB; ++j)
this.bBiases[j] -= bBiasGrads[j] * lrnRate;
for (int j = 0; j "lt" numOutput; ++j)
this.oBiases[j] -= oBiasGrads[j] * lrnRate;
} // ii each train item
// display progress
if (epoch % freq == 0 && epoch "lt" maxEpochs)
{
double mse = this.MSE(trainX, trainY);
double acc = this.Accuracy(trainX, trainY, 0.05);
string s1 = "epoch: " +
epoch.ToString().PadLeft(6);
string s2 = " MSE = " +
mse.ToString("F4");
string s3 = " acc = " +
acc.ToString("F4");
Console.WriteLine(s1 + s2 + s3);
}
} // epoch
return; // no explicit return values
}
// ------------------------------------------------------
private void Shuffle(int[] indices)
{
// Fisher-Yates algorithm
for (int i = 0; i "lt" indices.Length; ++i)
{
int r = this.rnd.Next(i, indices.Length); // note!
int tmp = indices[r];
indices[r] = indices[i];
indices[i] = tmp;
}
}
// ------------------------------------------------------
public double Accuracy(double[][] dataX, double[] dataY,
double pctClose)
{
// in this version, if target y is 0, only pred equal
// exactly 0 is scored as a correct prediction
int n = dataX.Length;
int nCorrect = 0; int nWrong = 0;
for (int i = 0; i "lt" n; ++i)
{
double predY = this.Predict(dataX[i]);
double actualY = dataY[i];
if (Math.Abs(predY - actualY) "lte"
pctClose * Math.Abs(actualY))
++nCorrect;
else
++nWrong;
}
return (nCorrect * 1.0) / (nCorrect + nWrong);
}
// ------------------------------------------------------
public double MSE(double[][] dataX, double[] dataY)
{
// mean squared error
int n = dataX.Length;
double sum = 0.0;
for (int i = 0; i "lt" n; ++i)
{
double predY = this.Predict(dataX[i]);
double actualY = dataY[i];
sum += (predY - actualY) * (predY - actualY);
}
return sum / n;
}
// ------------------------------------------------------
// ------------------------------------------------------
//public void SetWeights(double[] weights)
//{
// // used for a Load function, not-yet-implemented
// int ptr = 0;
// for (int i = 0; i "lt" this.numInput; ++i)
// for (int j = 0; j "lt" this.numHiddenA; ++j)
// this.iaWeights[i][j] = weights[ptr++];
// for (int i = 0; i "lt" numHiddenA; ++i)
// this.aBiases[i] = weights[ptr++];
// for (int i = 0; i "lt" this.numHiddenA; ++i)
// for (int j = 0; j "lt" this.numHiddenB; ++j)
// this.abWeights[i][j] = weights[ptr++];
// for (int i = 0; i "lt" this.numHiddenB; ++i)
// this.bBiases[i] = weights[ptr++];
// for (int i = 0; i "lt" this.numHiddenB; ++i)
// for (int j = 0; j "lt" this.numOutput; ++j)
// this.boWeights[i][j] = weights[ptr++];
// for (int i = 0; i "lt" this.numOutput; ++i)
// this.oBiases[i] = weights[ptr++];
//}
// ------------------------------------------------------
} // class NeuralNetworkBistratumRegressor
// ========================================================
} // 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
.NET Test Automation Recipes
Software Testing
SciPy Programming Succinctly
Keras Succinctly
R Programming
Visual Studio Live
Microsoft MLADS Conference
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Machine Learning Week
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