The form of a linear regression prediction equation is y’ = (w0 * x0) + (w1 * x1) + . . + (wn * xn) + b where y’ is the predicted value, the xi are predictor values, the wi are constants called model weights, and b is a constant called the bias. For example, y’ = predicted balance = (-0.54 * age) + (0.38 * height) + (0.11 * experience) + 0.72.
Training the model is the process of finding the values of the weights and bias so that predicted y values are close to known correct target y values in a set of training data.
There are several ways to train a linear regression model. Two of the most common techniques are stochastic gradient descent (SGD) training, and closed form training using matrix pseudo-inverse. This blog post demonstrates how to implement the closed form training technique, using the singular value decomposition (SVD) algorithm, implemented using the Jacobi technique.
Briefly, the math equation is:
w = pinv(DX) * y
Here w is a vector of the weights and the bias that you want. The pinv() function is the Moore-Penrose pseudo-inverse of a matrix. DX is a design matrix of the X predictors of the training data. The * is matrix-to-vector multiplication, and y is a vector of target y values of the training data.
A design matrix takes the bias into account by adding a leading column of 1.0 values. For example, if a set of training predictors stored in a matrix X is:
0.50 0.77 0.32 0.92 0.41 0.84 0.64 0.73 0.43
The associated design matrix is:
1.0 0.50 0.77 0.32 1.0 0.92 0.41 0.84 1.0 0.64 0.73 0.43
For a demo, I used one of my standard sets of synthetic data that 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 five values on each line are the predictors (sometimes called features). The last value on each line is the target y value to predict. There are 200 training items and 40 test items.
The output of my demo is:
Begin C# linear regression using pseudo-inverse closed-form training 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 and training Linear Regression model using p-inverse Done Coefficients/weights: -0.2656 0.0333 -0.0454 0.0358 -0.1146 Bias/constant: 0.3619 Evaluating model Accuracy train (within 0.10) = 0.4600 Accuracy test (within 0.10) = 0.6500 MSE train = 0.0026 MSE test = 0.0020 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 Predicted y = 0.5329 End demo
The accuracy of the trained model is poor because the synthetic data has complex, non-linear relationships. Linear regression doesn’t always work well, but it’s usually a good place to start because at the very least it gives you a baseline result for comparison with more sophisticated techniques such as neural network regression.
Implementing a pseudo-inverse function is very difficult. Many mathematicians worked for many years on the problem, and came up with many solutions. The one used by the demo program is to compute the pseudo-inverse of a matrix using singular value decomposition (SVD) via the Jacobi algorithm. Other pseudo-inverse techniques include SVD with QR, direct QR via Householder, direct QR via modified Gram-Schmidt, direct QR via Givens.
Note: instead of using explicit pseudo-inverse training, it’s possible to mimic a pseudo-inverse function using regular matrix inverse. That form is w = inv(Xt * X) * Xt * y, where X is the design matrix, Xt is the transpose of X, inv() is regular matrix inverse, * is matrix multiplication, and y is a vector of target values from the training data. This technique is sometimes called pseudo-inverse via normal equations. Because Xt * X is square symmetric positive definite, it’s inverse can be computed using the special Cholesky decomposition.
Machine learning is a combination of math techniques that are hundreds of years old, and new ideas that have been developed recently. Here’s a somewhat-random image of a past+future steampunk automobile — very nice.
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 LinearRegressionPseudoInverse
{
internal class LinearRegressionProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin C# linear regression" +
" using pseudo-inverse (SVD-Jacobi) closed-form" +
" training ");
// 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 };
double[][] trainX =
MatLoad(trainFile, colsX, ',', "#");
double[] trainY =
MatToVec(MatLoad(trainFile,
new int[] { 5 }, ',', "#"));
string testFile =
"..\\..\\..\\Data\\synthetic_test_40.txt";
double[][] testX =
MatLoad(testFile, colsX, ',', "#");
double[] testY =
MatToVec(MatLoad(testFile,
new int[] { 5 }, ',', "#"));
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 and train model using pseudo-inverse
Console.WriteLine("\nCreating and training" +
" Linear Regression model using p-inverse ");
LinearRegressor model = new LinearRegressor();
model.Train(trainX, trainY);
Console.WriteLine("Done ");
// 2b. show model parameters
Console.WriteLine("\nCoefficients/weights: ");
for (int i = 0; i "lt" model.weights.Length; ++i)
Console.Write(model.weights[i].ToString("F4") + " ");
Console.WriteLine("\nBias/constant: " +
model.bias.ToString("F4"));
// 3. evaluate model
Console.WriteLine("\nEvaluating model ");
double accTrain = model.Accuracy(trainX, trainY, 0.10);
Console.WriteLine("\nAccuracy train (within 0.10) = " +
accTrain.ToString("F4"));
double accTest = model.Accuracy(testX, testY, 0.10);
Console.WriteLine("Accuracy test (within 0.10) = " +
accTest.ToString("F4"));
double mseTrain = model.MSE(trainX, trainY);
Console.WriteLine("\nMSE train = " +
mseTrain.ToString("F4"));
double mseTest = model.MSE(testX, testY);
Console.WriteLine("MSE test = " +
mseTest.ToString("F4"));
// 4. use model to predict first training item
double[] x = trainX[0];
Console.WriteLine("\nPredicting for x = ");
VecShow(x, 4, 9);
double predY = model.Predict(x);
Console.WriteLine("\nPredicted 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[][] 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;
}
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 LinearRegressor
{
public double[] weights;
public double bias;
private Random rnd;
// ------------------------------------------------------
public LinearRegressor(int seed = 0) // ctor
{
this.weights = new double[0];
this.bias = 0;
this.rnd = new Random(seed); // no need this version
}
// ------------------------------------------------------
public double Predict(double[] x)
{
double result = 0.0;
for (int j = 0; j "lt" x.Length; ++j)
result += x[j] * this.weights[j];
result += this.bias;
return result;
}
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
// closed-form w = pinv(designX) * y
int dim = trainX[0].Length;
this.weights = new double[dim];
double[][] X = MatToDesign(trainX); // design X
double[][] Xpinv = MatPinv(X);
double[] biasAndWts = MatVecProd(Xpinv, trainY);
this.bias = biasAndWts[0];
for (int i = 1; i "lt" biasAndWts.Length; ++i)
this.weights[i - 1] = biasAndWts[i];
return;
}
// ------------------------------------------------------
public double Accuracy(double[][] dataX, double[] dataY,
double pctClose)
{
int numCorrect = 0; int numWrong = 0;
for (int i = 0; i "lt" dataX.Length; ++i)
{
double actualY = dataY[i];
double predY = this.Predict(dataX[i]);
if (Math.Abs(predY - actualY) "lt"
Math.Abs(pctClose * actualY))
++numCorrect;
else
++numWrong;
}
return (numCorrect * 1.0) / (numWrong + numCorrect);
}
// ------------------------------------------------------
public double MSE(double[][] dataX, double[] dataY)
{
int n = dataX.Length;
double sum = 0.0;
for (int i = 0; i "lt" n; ++i)
{
double actualY = dataY[i];
double predY = this.Predict(dataX[i]);
sum += (actualY - predY) * (actualY - predY);
}
return sum / n;
}
// ------------------------------------------------------
//
// helpers: MatPinv, MatMake, MatTranspose, MatProduct
// MatDecompSVD, MatVecProd, MatToDesign
//
// ------------------------------------------------------
private static double[][] MatPinv(double[][] M)
{
// Moore-Penrose pseudo-inverse from SVD (Jacobi)
double[][] U; double[] s; double[][] Vh;
MatDecompSVD(M, out U, out s, out Vh);
double[][] Sinv = MatMake(s.Length, s.Length);
for (int i = 0; i "lt" Sinv.Length; ++i)
Sinv[i][i] = 1.0 / (s[i] + 1.0e-12); // avoid 0
// pinv = V * Sinv * Uh
double[][] V = MatTranspose(Vh);
double[][] Uh = MatTranspose(U);
double[][] tmp = MatProduct(Sinv, Uh);
double[][] result = MatProduct(V, tmp);
return result;
} // MatPinv()
// ------------------------------------------------------
private static void MatDecompSVD(double[][] M,
out double[][] U, out double[] s, out double[][] Vh)
{
// Jacobi algorithm
// see github.com/ampl/gsl/blob/master/linalg/svd.c
// "full_matrices=False" version
double DBL_EPSILON = 1.0e-15;
int m = M.Length; // nRows
int n = M[0].Length; // nCols
// A = MatCopy(M);
double[][] A = new double[m][]; // working U
for (int i = 0; i "lt" m; ++i)
A[i] = new double[n];
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
A[i][j] = M[i][j];
// Q = MatIdentity(n);
double[][] Q = new double[n][]; // working V
for (int i = 0; i "lt" n; ++i)
Q[i] = new double[n];
for (int i = 0; i "lt" n; ++i)
Q[i][i] = 1.0;
//
double[] t = new double[n]; // working s
// initialize counters
int count = 1;
int sweep = 0;
double tol = 10 * m * DBL_EPSILON; // heuristic
// Always do at least 12 sweeps
int sweepmax = Math.Max(5 * n, 12); // heuristic
// store the column error estimates in St for use
// during orthogonalization
for (int j = 0; j "lt" n; ++j)
{
// double[] cj = MatGetColumn(A, j);
double[] cj = new double[m];
for (int i = 0; i "lt" m; ++i)
cj[i] = A[i][j];
// double sj = VecNorm(cj);
double sj = 0.0;
for (int i = 0; i "lt" cj.Length; ++i)
sj += cj[i] * cj[i];
sj = Math.Sqrt(sj);
t[j] = DBL_EPSILON * sj;
} // j
// orthogonalize A by plane rotations
while (count "gt" 0 && sweep "lte" sweepmax)
{
// initialize rotation counter
count = n * (n - 1) / 2;
for (int j = 0; j "lt" n - 1; ++j)
{
for (int k = j + 1; k "lt" n; ++k)
{
double cosine, sine;
// double[] cj = MatGetColumn(A, j);
double[] cj = new double[m];
for (int i = 0; i "lt" m; ++i)
cj[i] = A[i][j];
// double[] ck = MatGetColumn(A, k);
double[] ck = new double[m];
for (int i = 0; i "lt" m; ++i)
ck[i] = A[i][k];
// double p = 2.0 * VecDot(cj, ck);
double p = 0.0;
for (int i = 0; i "lt" cj.Length; ++i)
p += cj[i] * ck[i];
p *= 2.0;
// double a = VecNorm(cj);
double a = 0.0;
for (int i = 0; i "lt" cj.Length; ++i)
a += cj[i] * cj[i];
a = Math.Sqrt(a);
//double b = VecNorm(ck);
double b = 0.0;
for (int i = 0; i "lt" ck.Length; ++i)
b += ck[i] * ck[i];
b = Math.Sqrt(b);
double q = a * a - b * b;
// double v = Hypot(p, q); // sqrt(x^2 + y^2)
double v = Math.Sqrt((p * p) + (q * q));
// test for columns j,k orthogonal,
// or dominant errors
double abserr_a = t[j];
double abserr_b = t[k];
bool sorted = (a "gte" b);
bool orthog = (Math.Abs(p) "lte"
tol * (a * b));
bool noisya = (a "lt" abserr_a);
bool noisyb = (b "lt" abserr_b);
if (sorted && (orthog ||
noisya || noisyb))
{
--count; continue;
}
// calculate rotation angles
if (v == 0 || !sorted)
{
cosine = 0.0; sine = 1.0;
}
else
{
cosine = Math.Sqrt((v + q) / (2.0 * v));
sine = p / (2.0 * v * cosine);
}
// apply rotation to A (U)
for (int i = 0; i "lt" m; ++i)
{
double Aik = A[i][k];
double Aij = A[i][j];
A[i][j] = Aij * cosine + Aik * sine;
A[i][k] = -Aij * sine + Aik * cosine;
}
// update singular values
t[j] = Math.Abs(cosine) * abserr_a +
Math.Abs(sine) * abserr_b;
t[k] = Math.Abs(sine) * abserr_a +
Math.Abs(cosine) * abserr_b;
// apply rotation to Q (V)
for (int i = 0; i "lt" n; ++i)
{
double Qij = Q[i][j];
double Qik = Q[i][k];
Q[i][j] = Qij * cosine + Qik * sine;
Q[i][k] = -Qij * sine + Qik * cosine;
} // i
} // k
} // j
++sweep;
} // while
// compute singular values
double prevNorm = -1.0;
for (int j = 0; j "lt" n; ++j)
{
// double[] column = MatGetColumn(A, j);
double[] column = new double[m];
for (int i = 0; i "lt" m; ++i)
column[i] = A[i][j];
// double norm = VecNorm(column);
double norm = 0.0;
for (int i = 0; i "lt" column.Length; ++i)
norm += column[i] * column[i];
norm = Math.Sqrt(norm);
// determine if singular value is zero
if (norm == 0.0 || prevNorm == 0.0
|| (j "gt" 0 &&
norm "lte" tol * prevNorm))
{
t[j] = 0.0;
for (int i = 0; i "lt" m; ++i)
A[i][j] = 0.0;
prevNorm = 0.0;
}
else
{
t[j] = norm;
for (int i = 0; i "lt" m; ++i)
A[i][j] = A[i][j] * 1.0 / norm;
prevNorm = norm;
}
}
if (count "gt" 0)
{
Console.WriteLine("Jacobi iterations did not" +
" converge");
}
// U = A;
U = new double[m][];
for (int i = 0; i "lt" m; ++i)
U[i] = new double[n];
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
U[i][j] = A[i][j];
// s = t;
s = new double[n];
for (int i = 0; i "lt" n; ++i)
s[i] = t[i];
// Vh = MatTranspose(Q); // Q is nxn
Vh = new double[n][];
for (int i = 0; i "lt" n; ++i)
Vh[i] = new double[n];
for (int i = 0; i "lt" n; ++i)
for (int j = 0; j "lt" n; ++j)
Vh[i][j] = Q[j][i];
// to sync with np.linalg.svd() shapes
// for full_matrices=False (not the default)
// if m "lt" n,
// extract 1st m columns of U
// extract 1st m values of s
// extract 1st m rows of Vh
if (m "lt" n)
{
// U = MatExtractFirstColumns(U, m);
double[][] newU = new double[U.Length][]; // all rows
for (int i = 0; i "lt" U.Length; ++i)
newU[i] = new double[m]; // just m cols
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" m; ++j) // m cols
newU[i][j] = U[i][j];
U = newU;
// s = VecExtractFirst(s, m);
double[] newS = new double[m]; // just m vals
for (int i = 0; i "lt" m; ++i)
newS[i] = s[i];
s = newS;
// Vh = MatExtractFirstRows(Vh, m);
double[][] newVh = new double[m][]; // just m rows
for (int i = 0; i "lt" m; ++i)
newVh[i] = new double[Vh[0].Length]; // all cols
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" Vh[0].Length; ++j)
newVh[i][j] = Vh[i][j];
Vh = newVh;
} // m "lt" n
} // MatDecompSVD()
private static double[] MatVecProd(double[][] M,
double[] v)
{
// return a regular vector
int nRows = M.Length;
int nCols = M[0].Length;
int n = v.Length;
if (nCols != n)
throw new Exception("non-comform in MatVecProd");
double[] result = new double[nRows];
for (int i = 0; i "lt" nRows; ++i)
for (int k = 0; k "lt" nCols; ++k)
result[i] += M[i][k] * v[k];
return result;
}
private static double[][] MatToDesign(double[][] M)
{
// add a column of 1s
int nRows = M.Length;
int nCols = M[0].Length;
double[][] result = new double[M.Length][];
for (int i = 0; i "lt" nRows; ++i)
result[i] = new double[nCols + 1];
for (int i = 0; i "lt" nRows; ++i)
{
result[i][0] = 1.0;
for (int j = 1; j "lt" nCols + 1; ++j)
result[i][j] = M[i][j - 1];
}
return result;
}
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;
}
private static double[][] MatTranspose(double[][] M)
{
int nRows = M.Length;
int nCols = M[0].Length;
double[][] result = MatMake(nCols, nRows);
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[j][i] = M[i][j];
return result;
}
private static double[][] MatProduct(double[][] A,
double[][] B)
{
int aRows = A.Length;
int aCols = A[0].Length;
int bRows = B.Length;
int bCols = B[0].Length;
if (aCols != bRows)
throw new Exception("Non-conformable matrices");
double[][] result = MatMake(aRows, bCols);
for (int i = 0; i "lt" aRows; ++i)
for (int j = 0; j "lt" bCols; ++j)
for (int k = 0; k "lt" aCols; ++k)
result[i][j] += A[i][k] * B[k][j];
return result;
}
} // class LinearRegressor
} // 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
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