Arguably the most basic form of machine learning regression is linear regression. The liner regression prediction model is y’ = (x0 * w0) + (x1 * w1) + . . (xn * wn) + b. The y’ is the predicted value. The xi are the predictor values, the wi are the model weights (aka coefficients), and b is the bias (aka intercept).
If you have a set of training data with known correct target values, training the model is the process of finding good values for the weights and bias so that the mean squared error between predicted y values and target y values is minimized.
There are many training techniques. One category of techniques is iterative, with a training loop that gets closer and closer to optimal weight and bias values (SGD and its variations). Another category of techniques is called closed form solutions. The weights and bias are computed directly, using clever math. The three closed-form techniques I use most often are: 1.) The QR decomp + Solve OLS technique, 2.) The Moore-Penrose pseudo-inverse via QR technique, 3.) The left pseudo-inverse via Cholesky inverse technique.
1. The QR decomp + Solve OLS technique:
1. X * w = y
2. (Q * R) * w = y
3. Qt * (Q * R) * w = Qt * y
4. R * w = Qt * y
5. w = Solve(R, Qt * y) // back-substitution
The X is a matrix of predictor values, with a leading column of 1.0 values added to stand in for the model bias. The w is a vector that holds the bias in cell [0] and the weights in the remaining cells. Step 1 is the starting point. In step 2, X hs been decomposed using QR decomposition. In step 3, both sides are multiplied by Qt, which is the transpose of Q. Step 4 uses the fact that Qt is the matrix inverse of Q. Step 5 solves for the weight vector using an abbreviated form os sort-of matrix inverse called ordinary least squares solve, which uses a technique called back-substitution.
Example C# code:
public void Train(double[][] trainX, double[] trainY)
{
// w = Solve(R, Qt * y)
int dim = trainX[0].Length;
this.weights = new double[dim];
double[][] X = MatToDesign(trainX);
double[][] Q; double[][] R;
QRHouseholder.MatDecompQR(X, out Q, out R);
double[][] Qt = QRHouseholder.MatTranspose(Q);
double[] biasAndWts = MatSolveQR(R, Qt, trainY);
this.bias = biasAndWts[0];
for (int i = 1; i < biasAndWts.Length; ++i)
this.weights[i - 1] = biasAndWts[i];
}
public static double[] MatSolveQR(double[][] R,
double[][] Qt, double[] y)
{
double[] z = MatVecProd(Qt, y); // Qt * y
int n = R[0].Length;
double[] w = new double[n];
// back-substitution
for (int i = n - 1; i >= 0; --i)
{
double sum = z[i];
for (int j = i + 1; j < n; ++j)
sum -= R[i][j] * w[j];
w[i] = sum / R[i][i];
}
return w;
}
This technique is used by most library implementations. The solve() function is similar to a matrix inverse() function but uses far fewer operations and so is more stable.
2. The Moore-Penrose pseudo-inverse via QR technique:
1. X * w = y
2. inv(X) * X * w = inv(X) * y // inv() is matrix inverse
3. w = inv(X) * y
4. w = inv(Q * R) * y
5. w = inv(R) * inv(Q) * y
6. w = inv(R) * Qt * y // inv(R) easy, Qt is transpose Q
Step 1 is the starting point. In step 2, both sides are multiplied by inv(X), which is the pseudo-inverse. Step 3 uses the fact that inx(X) * X = I the identity matrix. Step 4 applies QR decomposition to X. Step 5 is a consequence of inv(A*B) = inv(B)*inv(A). Step 6 uses the fact that the transpose of Q, Qt, is the inverse of Q. The inverse of R is a special, easy case (because R is an upper triangular matrix).
Example C# code:
public void Train(double[][] trainX, double[] trainY)
{
// w = inv(R) * Qt * y
int dim = trainX[0].Length;
this.weights = new double[dim];
double[][] X = MatToDesign(trainX);
double[][] Q; double[][] R;
QRHouseholder.MatDecompQR(X, out Q, out R);
double[][] Ri = QRHouseholder.MatInvUpperTri(R);
double[][] Qt = QRHouseholder.MatTranspose(Q);
double[][] RiQt = QRHouseholder.MatProduct(Ri, Qt);
double[] biasAndWts = MatVecProd(RiQt, trainY);
this.bias = biasAndWts[0];
for (int i = 1; i < biasAndWts.Length; ++i)
this.weights[i - 1] = biasAndWts[i]; }
This technique is not bad but it explicitly computes inv(R). The solve() technique does not compute an explicit inverse and is faster and more stable.
3. The left pseudo-inverse via Cholesky inverse technique:
1. X * w = y
2. (Xt * X) * w = Xt * y // problematic
3. inv(Xt * X) * (Xt * X) * w = inv(Xt * X) * Xt * y
4. w = inv(Xt * X) * Xt * y
Step 1 is the starting point. In step 2, both sides are pre-multiplied by Xt, the transpose of X. In step 3, both sides are pre-multiplied by inv(Xt * X). Step 4 uses the fact that inv(A) * A = I, the identity matrix. The inv(Xt * X) inverse calculation uses Cholesky inverse and is relatively easy because (Xt * X) is square symmetic positive definite.
Example C# code:
public void Train(double[][] trainX, double[] trainY)
{
// w = inv(Xt * X) * Xt * y
int dim = trainX[0].Length;
this.weights = new double[dim];
double[][] X = MatToDesign(trainX);
double[][] Xt = Cholesky.MatTranspose(X);
double[][] XtX = Cholesky.MatProduct(Xt, X);
double[][] XtXinv = Cholesky.MatInvCholesky(XtX);
double[][] tmp = Cholesky.MatProduct(XtXinv, Xt);
double[] biasAndWts = MatVecProd(tmp, trainY);
// extract bias and weights
this.bias = biasAndWts[0];
for (int i = 1; i < biasAndWts.Length; ++i)
this.weights[i - 1] = biasAndWts[i];
}
The left pseudo-inverse training technique is rarely used except when X is small. If X has m rows and n columns, the Xt * X calculation requires m * n^2 multiplications and n^2 * (m-1) additions. For example, if m is 10,000 rows and n is 10 columns, there are 10,000 * 100 = 1,000,000 multiplication and approximately 100 * 10,000 = 1,000,000 additions. Arithmetic overflow, underflow, or a inaccurate result could easily happen.
I miss the days of newspapers
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 LinearRegressionThreeClosedFormTrainingTechniques
{
internal class LinearRegressionProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin C# linear regression" +
" with three closed-form training techniques ");
// 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));
// 1. QR OLS Solve training
Console.WriteLine("\n1. QR OLS Solve training ");
LinearRegressor model1 = new LinearRegressor();
model1.Train_QR_Solve(trainX, trainY);
Console.WriteLine("Done ");
Console.WriteLine("\nCoefficients/weights: ");
for (int i = 0; i "lt" model1.weights.Length; ++i)
Console.Write(model1.weights[i].
ToString("F4") + " ");
Console.WriteLine("\nBias/constant: " +
model1.bias.ToString("F4"));
// 2. MP pseudo-inverse training
Console.WriteLine("\n2. Moore-Penrose pinv via" +
" QR training ");
LinearRegressor model2 = new LinearRegressor();
model2.Train_MP_Pinv(trainX, trainY);
Console.WriteLine("Done ");
Console.WriteLine("\nCoefficients/weights: ");
for (int i = 0; i "lt" model2.weights.Length; ++i)
Console.Write(model2.weights[i].
ToString("F4") + " ");
Console.WriteLine("\nBias/constant: " +
model2.bias.ToString("F4"));
// 3. left pinv via normal equations (Cholesky)
Console.WriteLine("\n3. left pseudo-inv via Cholesky");
LinearRegressor model3 = new LinearRegressor();
model3.Train_Left_Pinv(trainX, trainY);
Console.WriteLine("Done ");
Console.WriteLine("\nCoefficients/weights: ");
for (int i = 0; i "lt" model3.weights.Length; ++i)
Console.Write(model3.weights[i].
ToString("F4") + " ");
Console.WriteLine("\nBias/constant: " +
model3.bias.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; // not used in most closed-form
// ------------------------------------------------------
public LinearRegressor(int seed = 0) // ctor
{
this.weights = new double[0];
this.bias = 0;
this.rnd = new Random(seed); // not used 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_QR_Solve(double[][] trainX,
double[] trainY)
{
// w = Solve(R, Qt * y)
int dim = trainX[0].Length;
this.weights = new double[dim];
double[][] X = MatToDesign(trainX);
double[][] Q; double[][] R;
QRHouseholder.MatDecompQR(X, out Q, out R);
double[][] Qt = QRHouseholder.MatTranspose(Q);
double[] biasAndWts = MatSolveQR(R, Qt, trainY);
this.bias = biasAndWts[0];
for (int i = 1; i "lt" biasAndWts.Length; ++i)
this.weights[i - 1] = biasAndWts[i];
}
private static double[] MatSolveQR(double[][] R,
double[][] Qt, double[] y)
{
double[] z = MatVecProd(Qt, y); // Qt * y
int n = R[0].Length;
double[] w = new double[n];
// back-substitution
for (int i = n - 1; i "gte" 0; --i)
{
double sum = z[i];
for (int j = i + 1; j "lt" n; ++j)
{
sum -= R[i][j] * w[j];
}
w[i] = sum / R[i][i];
}
return w;
}
// ------------------------------------------------------
public void Train_MP_Pinv(double[][] trainX,
double[] trainY)
{
// w = inv(R) * Qt * y
int dim = trainX[0].Length;
this.weights = new double[dim];
double[][] X = MatToDesign(trainX);
double[][] Q; double[][] R;
QRHouseholder.MatDecompQR(X, out Q, out R);
double[][] Ri = QRHouseholder.MatInvUpperTri(R);
double[][] Qt = QRHouseholder.MatTranspose(Q);
double[][] RiQt = QRHouseholder.MatProduct(Ri, Qt);
double[] biasAndWts = MatVecProd(RiQt, trainY);
this.bias = biasAndWts[0];
for (int i = 1; i "lt" biasAndWts.Length; ++i)
this.weights[i - 1] = biasAndWts[i];
}
// ------------------------------------------------------
public void Train_Left_Pinv(double[][] trainX,
double[] trainY)
{
// w = inv(Xt * X) * Xt * y
int dim = trainX[0].Length;
this.weights = new double[dim];
double[][] X = MatToDesign(trainX);
double[][] Xt = Cholesky.MatTranspose(X);
double[][] XtX = Cholesky.MatProduct(Xt, X);
double[][] XtXinv = Cholesky.MatInvCholesky(XtX);
double[][] tmp = Cholesky.MatProduct(XtXinv, Xt);
double[] biasAndWts = MatVecProd(tmp, trainY);
// extract bias and weights
this.bias = biasAndWts[0];
for (int i = 1; i "lt" biasAndWts.Length; ++i)
this.weights[i - 1] = biasAndWts[i];
}
// ------------------------------------------------------
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-conform 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[nRows][];
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;
}
} // class LinearRegressor
// ========================================================
public class QRHouseholder
{
// container for MP pseudo-inverse via QR-Householder
// A = Q * R
// pinv(A) = inv(R) * inv(Q) note order matters
// = inv upper tri (easy) * transpose (easy)
public static double[][] MatPinv(double[][] M)
{
double[][] Q; double[][] R;
MatDecompQR(M, out Q, out R); // Householder
double[][] Ri = MatInvUpperTri(R);
double[][] Qi = MatTranspose(Q);
double[][] result = MatProduct(Ri, Qi);
return result;
}
// ------------------------------------------------------
public static double[][] MatInvUpperTri(double[][] U)
{
int n = U.Length; // must be square matrix
double[][] result = MatMake(n, n);
for (int i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
for (int k = 0; k "lt" n; ++k)
{
for (int j = 0; j "lt" n; ++j)
{
for (int i = 0; i "lt" k; ++i)
{
result[j][k] -= result[j][i] * U[i][k];
}
result[j][k] /= U[k][k];
}
}
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[][] 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;
}
// ------------------------------------------------------
public 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 = new double[aRows][];
for (int i = 0; i "lt" aRows; ++i)
result[i] = new double[bCols];
for (int i = 0; i "lt" aRows; ++i) // each row of A
for (int j = 0; j "lt" bCols; ++j) // each col of B
for (int k = 0; k "lt" aCols; ++k)
result[i][j] += A[i][k] * B[k][j];
return result;
}
// ------------------------------------------------------
public static void MatDecompQR(double[][] A,
out double[][] Q, out double[][] R)
{
int m = A.Length; int n = A[0].Length;
if (m "lt" n)
Console.WriteLine("FATAL: nRows must be gte nCols");
double[][] QQ = MatMake(m, m); // working full Q
for (int i = 0; i "lt" m; ++i)
QQ[i][i] = 1.0; // identity matrix
double[][] RR = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
RR[i][j] = A[i][j]; // copy of A is working R
int k = Math.Min(m, n); // or just use n
for (int j = 0; j "lt" k; ++j) // main processing loop
{
int xn = m - j;
double[] x = new double[xn];
for (int i = 0; i "lt" xn; ++i)
x[i] = RR[j + i][j];
double ss = 0.0;
for (int i = 0; i "lt" xn; ++i)
ss += x[i] * x[i];
double normX = Math.Sqrt(ss);
// if (normX == 0.0) continue;
if (Math.Abs(normX) "lt" 1.0e-12) continue;
double sign;
if (x[0] "gte" 0.0) sign = -1.0;
else sign = 1.0; // counter-intuitive
double[] u = new double[xn];
for (int i = 0; i "lt" xn; ++i)
u[i] = x[i] / (x[0] - sign * normX); // check div 0
u[0] = 1.0;
// compute scaling factor tau = 2 / (u^T * u)
double tau = -sign * (x[0] - sign * normX) / normX;
// dimensions for sub-matrices
int nRowsSubR = m - j; int nColsSubR = n - j;
int nRowsSubQ = m; int nColsSubQ = m - j;
double[] vr = new double[nColsSubR];
for (int c = 0; c "lt" nColsSubR; ++c)
{
double acc = 0.0;
for (int r = 0; r "lt" nRowsSubR; ++r)
acc += u[r] * RR[j + r][j + c];
vr[c] = acc;
}
double[] vq = new double[nRowsSubQ];
for (int r = 0; r "lt" nRowsSubQ; ++r)
{
double acc = 0.0;
for (int c = 0; c "lt" nColsSubQ; ++c)
acc += u[c] * QQ[r][j + c];
vq[r] = acc;
}
// update sub-R
for (int r = 0; r "lt" nRowsSubR; ++r)
for (int c = 0; c "lt" nColsSubR; ++c)
RR[j + r][j + c] -= tau * u[r] * vr[c];
// update sub-Q
for (int r = 0; r "lt" nRowsSubQ; ++r)
for (int c = 0; c "lt" nColsSubQ; ++c)
QQ[r][j + c] -= tau * vq[r] * u[c];
} // j
// extract QQ RR into out params
Q = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
Q[i][j] = QQ[i][j];
R = MatMake(n, n);
for (int i = 0; i "lt" n; ++i)
for (int j = 0; j "lt" n; ++j)
R[i][j] = RR[i][j];
return;
} // MatDecompQR
} // class QRHouseholder
// ========================================================
public class Cholesky
{
// MatPseudoInv(), MatInvCholesky(),
// MatTranspose(), MatProduct()
public static double[][] MatPseudoInv(double[][] A)
{
// left pseudo-inverse via normal equations
// nRows must be gte nCols
// inv(At * A) * A
double[][] At = MatTranspose(A);
double[][] AtA = MatProduct(At, A);
for (int i = 0; i "lt" AtA.Length; ++i)
AtA[i][i] += 1.0e-8; /// condition before inv
double[][] AtAinv = MatInvCholesky(AtA);
double[][] pinv = MatProduct(AtAinv, At);
return pinv;
} // MatPseudoInv()
// ------------------------------------------------------
public static double[][] MatInvCholesky(double[][] A)
{
// A must be square, symmetric, positive definite
int m = A.Length; int n = A[0].Length; // m == n
// 1. decompose A to L
double[][] L = new double[n][];
for (int i = 0; i "lt" n; ++i)
L[i] = new double[n];
for (int i = 0; i "lt" n; ++i)
{
for (int j = 0; j "lte" i; ++j)
{
double sum = 0.0;
for (int k = 0; k "lt" j; ++k)
sum += L[i][k] * L[j][k];
if (i == j)
{
double tmp = A[i][i] - sum;
if (tmp "lt" 0.0)
throw new
Exception("decomp Cholesky fatal");
L[i][j] = Math.Sqrt(tmp);
}
else
{
if (L[j][j] == 0.0)
throw new
Exception("decomp Cholesky fatal ");
L[i][j] = (A[i][j] - sum) / L[j][j];
}
} // j
} // i
// 2. compute inverse from L
double[][] result = new double[n][]; // make Identity
for (int i = 0; i "lt" n; ++i)
result[i] = new double[n];
for (int i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
for (int k = 0; k "lt" n; ++k)
{
for (int j = 0; j "lt" n; j++)
{
for (int i = 0; i "lt" k; i++)
{
result[k][j] -= result[i][j] * L[k][i];
}
result[k][j] /= L[k][k];
}
}
for (int k = n - 1; k "gte" 0; --k)
{
for (int j = 0; j "lt" n; j++)
{
for (int i = k + 1; i "lt" n; i++)
{
result[k][j] -= result[i][j] * L[i][k];
}
result[k][j] /= L[k][k];
}
}
return result;
} // MatInvCholesky()
// ------------------------------------------------------
public static double[][] MatTranspose(double[][] M)
{
int nr = M.Length; int nc = M[0].Length;
double[][] result = new double[nc][]; // note
for (int i = 0; i "lt" nc; ++i)
result[i] = new double[nr];
for (int i = 0; i "lt" nr; ++i)
for (int j = 0; j "lt" nc; ++j)
result[j][i] = M[i][j]; // note
return result;
}
// ````````````````````````````````````````````````````
public 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 = new double[aRows][];
for (int i = 0; i "lt" aRows; ++i)
result[i] = new double[bCols];
for (int i = 0; i "lt" aRows; ++i) // each row of A
for (int j = 0; j "lt" bCols; ++j) // each col of B
for (int k = 0; k "lt" aCols; ++k)
result[i][j] += A[i][k] * B[k][j];
return result;
}
// ------------------------------------------------------
} // class Cholesky
} // 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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0.4038, -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, 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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 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.NET Test Automation Recipes
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