The goal of a machine learning regression problem is to predict a single numeric value. For example, you might want to predict the bank account balance of a person based on his annual income, age, years of education, and so on.
Linear regression is the simplest form of machine learning regression. The prediction equation has the form y’ = (w0 * x0) + (w1 * x1) + . . + b, where the wi are weight values, the xi are predictor values, and b is the bias. For example, y’ = (0.56 * income) + (-0.37 * age) + (0.04 * education) + 1.66.
Training is the process of finding the values of the weights and the bias so that predicted y values closely match the known correct y values in a set of training data.
There are four main linear regression training techniques (and dozens of less-common techniques): stochastic gradient descent, relaxed Moore-Penrose pseudo-inverse, closed-form pseudo-inverse via normal equations, L-BFGS. Each of these four techniques has dozens of significant variations. For example, MP pseudo-inverse training can use singular value decomposition (SVD — many algorithms such as Jacobi, GKR, etc.) or QR decomposition (many algorithms such as Householder, Gram-Schmidt, Givens, etc.) In short, there are a bewildering number of different ways to train a linear regression prediction model.
Each training technique has pros and cons related to ease/difficulty of implementation and customization, ability to deal with large or small datasets, accuracy, performance, sensitivity to data normalization, ability to handle categorical data, and more. If there was one best training technique, it would be the only technique used.
One evening, I was sitting in my living room, waiting for the rain to stop (it never did) so I could walk my two dogs. While I was waiting, I figured I’d implement linear regression, using relaxed MP pseudo-inverse (via SVD-Jacobi) training, using JavaScript.
For my demo, 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 0.9365, -0.3732, 0.3846, 0.7528, 0.7892, 0.1345 . . .
The first five values on each line are the predictors (“features”) and the last value is the target y to predict. The data was generated by a 5-10-1 neural network with random weights and biases. There are 200 training items and 40 test items.
The equation for training is:
w = pinv(DX) * y
The w is a vector that holds the weights and the bias you want, pinv() is the relaxed Moore-Penrose pseudo-inverse function, DX is the design matrix derived from the X training data, and y is a vector of the known target values to predict.
The design matrix is the X predictor values with a leading column of 1.0s added. The leading column of 1.0s deals with the bias term.
The output of my demo:
Linear regression with pseudo-inverse (SVD-Jacobi) training using JavaScript Loading synthetic train (200) and test (40) data from file 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 model Done Model weights/coefficients: -0.2656 0.0333 -0.0454 0.0358 -0.1146 Model bias/intercept: 0.3619 Evaluating model Train acc (within 0.10) = 0.4600 Test acc (within 0.10) = 0.6500 Train MSE = 0.0026 Test MSE = 0.0020 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 Predicted y = 0.5329 End demo
The output matches the output from several other training techniques, validating the results are correct. Good fun on a rainy evening in the Pacific Northwest.
I’ve always been fascinated by pinball machines. The “Saratoga” (1948) by Williams featured the first electrically-powered “pop” bumpers. Before this innovation, bumpers were spring-powered and barely moved the ball when struck. Notice that the two flippers are reversed from the modern geometry (which appeared two years later in a pair of machines by Gottlieb). Also, and no reel scoring, and no in-lanes, and no drop targets — features to appear in years to come.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols (my blog editor chokes on symbols).
// linear_regression_pinverse_svd.js
// linear regression, pseudo-inverse SVD-Jacobi training
// node.js
let FS = require("fs") // for loadTxt()
// ----------------------------------------------------------
class LinearRegressor
{
constructor(seed)
{
this.seed = seed + 0.5; // virtual RNG, not used
this.weights; // allocated in train()
this.bias = 0.0; // supplied in train()
}
// --------------------------------------------------------
// methods: predict(), train(), accuracy(), mse()
// --------------------------------------------------------
predict(x)
{
let sum = 0.0;
for (let i = 0; i "lt" x.length; ++i) {
sum += x[i] * this.weights[i];
}
sum += this.bias;
return sum;
}
// --------------------------------------------------------
train(trainX, trainY)
{
// w = pinv(DX) * y
let dim = trainX[0].length; // number predictors
this.weights = this.vecMake(dim, 0.0); // allocate wts
let DX = this.matToDesign(trainX); // design matrix
let Xpinv = this.matPinv(DX); // pinv of design X
let biasAndWts = this.matVecProd(Xpinv, trainY);
this.bias = biasAndWts[0];
for (let i = 1; i "lt" biasAndWts.length; ++i)
this.weights[i-1] = biasAndWts[i];
return;
}
// --------------------------------------------------------
accuracy(dataX, dataY, pctClose)
{
let nCorrect = 0; let nWrong = 0;
let N = dataX.length;
for (let i = 0; i "lt" N; ++i) {
let x = dataX[i];
let actualY = dataY[i];
let predY = this.predict(x);
if (Math.abs(predY - actualY) "lt"
Math.abs(pctClose * actualY)) {
++nCorrect;
}
else {
++nWrong;
}
}
return (nCorrect * 1.0) / (nCorrect + nWrong);
}
// --------------------------------------------------------
mse(dataX, dataY)
{
let N = dataX.length;
let sum = 0.0;
for (let i = 0; i "lt" N; ++i) {
let x = dataX[i];
let actualY = dataY[i];
let predY = this.predict(x);
sum += (actualY - predY) * (actualY - predY);
}
return sum / N;
}
// --------------------------------------------------------
// primary helpers for train()
// matVecProd, matPinv, matToDesign, svdJacobi
// --------------------------------------------------------
matVecProd(M, v)
{
// return a regular 1D vector
let nRows = M.length;
let nCols = M[0].length;
let n = v.length; // assume nCols = n
let result = this.vecMake(nRows, 0.0);
for (let i = 0; i "lt" nRows; ++i)
for (let k = 0; k "lt" nCols; ++k)
result[i] += M[i][k] * v[k];
return result;
}
// --------------------------------------------------------
matToDesign(M)
{
// add a leading column of 1.0s to M
let nRows = M.length;
let nCols = M[0].length;
let result = this.matMake(nRows, nCols+1, 1.0);
for (let i = 0; i "lt" nRows; ++i) {
for (let j = 1; j "lt" nCols+1; ++j) { // note 1s
result[i][j] = M[i][j-1];
}
}
return result;
}
// --------------------------------------------------------
matPinv(M) // pseudo-inverse of matrix M, using SVD-Jacobi
{
let decomp = this.svdJacobi(M); // SVD, Jacobi algorithm
let U = decomp[0];
let s = decomp[1]; // singluar values
let Vh = decomp[2];
let Sinv = this.matMake(s.length, s.length, 0.0);
for (let i = 0; i "lt" Sinv.length; ++i)
Sinv[i][i] = 1.0 / s[i]; // SVD fails if any s is 0.0
let V = this.matTranspose(Vh);
let Uh = this.matTranspose(U);
let tmp = this.matMult(Sinv, Uh);
let result = this.matMult(V, tmp);
return result;
}
// --------------------------------------------------------
svdJacobi(M)
{
// A = U * S * Vh
// returns an array where result[0] = U matrix,
// result[1] = s vector, result[2] = Vh matrix
// see github.com/ampl/gsl/blob/master/linalg/svd.c
// numpy svd() full_matrices = False version
let DBL_EPSILON = 1.0e-15;
let A = this.matCopy(M); // working U
let m = A.length;
let n = A[0].length;
let Q = this.matIdentity(n); // working V
let t = this.vecMake(n, 0.0); // working s
// initialize the rotation and sweep counters
let count = 1;
let sweep = 0;
let tolerance = 10 * m * DBL_EPSILON; // heuristic
// always do at least 12 sweeps
let sweepmax = Math.max(5 * n, 12); // heuristic
// store the column error estimates in t, for use
// during orthogonalization
for (let j = 0; j "lt" n; ++j) {
let cj = this.matGetColumn(A, j);
let sj = this.vecNorm(cj);
t[j] = DBL_EPSILON * sj;
}
// orthogonalize A by plane rotations
while (count "gt" 0 && sweep "lte" sweepmax) {
// initialize rotation counter
count = n * (n - 1) / 2;
for (let j = 0; j "lt" n - 1; ++j) {
for (let k = j + 1; k "lt" n; ++k) {
let cosine = 0.0; let sine = 0.0;
let cj = this.matGetColumn(A, j);
let ck = this.matGetColumn(A, k);
let p = 2.0 * this.vecDot(cj, ck);
let a = this.vecNorm(cj);
let b = this.vecNorm(ck);
let q = a * a - b * b;
let v = this.hypot(p, q);
// test for columns j,k orthogonal,
// or dominant errors
let abserr_a = t[j];
let abserr_b = t[k];
let sorted = (a "gte" b);
let orthog = (Math.abs(p) "lte"
tolerance * (a * b));
let noisya = (a "lt" abserr_a);
let 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 (let i = 0; i "lt" m; ++i) {
let Aik = A[i][k];
let 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 (Vh)
for (let i = 0; i "lt" n; ++i) {
let Qij = Q[i][j];
let 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
let prevNorm = -1.0;
for (let j = 0; j "lt" n; ++j) {
let column = this.matGetColumn(A, j);
let norm = this.vecNorm(column);
// determine if singular value is zero
if (norm == 0.0 || prevNorm == 0.0
|| (j "gt" 0 && norm "lte" tolerance * prevNorm)) {
t[j] = 0.0;
for (let i = 0; i "lt" m; ++i)
A[i][j] = 0.0;
prevNorm = 0.0;
}
else {
t[j] = norm;
for (let i = 0; i "lt" m; ++i)
A[i][j] = A[i][j] * 1.0 / norm;
prevNorm = norm;
}
}
if (count "gt" 0) {
console.log("Jacobi iterations did not" +
" converge");
}
let U = A;
let s = t;
let Vh = this.matTranspose(Q);
if (m "lt" n) {
U = this.matExtractFirstColumns(U, m);
s = this.vecExtractFirst(s, m);
Vh = this.matExtractFirstRows(Vh, m);
}
let result = [];
result[0] = U;
result[1] = s;
result[2] = Vh;
return result;
} // svdJacobi()
// === minor helper functions =============================
//
// matMake, matCopy, matIdentity, matGetColumn,
// matExtractFirstColumns, matExtractFirstRows,
// matTranspose, matMult, vecMake, vecNorm, vecDot,
// hypot, matShow, vecShow, matAreEqual, matLoad
//
// all of these could be declared 'static'
// ========================================================
matMake(nRows, nCols, val)
{
let result = [];
for (let i = 0; i "lt" nRows; ++i) {
result[i] = [];
for (let j = 0; j "lt" nCols; ++j) {
result[i][j] = val;
}
}
return result;
}
// --------------------------------------------------------
matCopy(M)
{
let nRows = M.length;
let nCols = M[0].length;
let result = this.matMake(nRows, nCols, 0.0);
for (let i = 0; i "lt" nRows; ++i)
for (let j = 0; j "lt" nCols; ++j)
result[i][j] = M[i][j];
return result;
}
// --------------------------------------------------------
matIdentity(n)
{
let result = this.matMake(n, n, 0.0);
for (let i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
return result;
}
// --------------------------------------------------------
matGetColumn(M, j)
{
let nRows = M.length;
let result = this.vecMake(nRows, 0.0);
for (let i = 0; i "lt" nRows; ++i)
result[i] = M[i][j];
return result;
}
// --------------------------------------------------------
matExtractFirstColumns(M, nc)
{
let nRows = M.length;
let result = this.matMake(nRows, nc);
for (let i = 0; i "lt" nRows; ++i)
for (let j = 0; j "lt" nc; ++j)
result[i][j] = M[i][j];
return result;
}
// --------------------------------------------------------
matExtractFirstRows(M, nr)
{
let nCols = M[0].length;
let result = this.matMake(nr, nCols);
for (let i = 0; i "lt" nr; ++i)
for (let j = 0; j "lt" nCols; ++j)
result[i][j] = M[i][j];
return result;
}
// --------------------------------------------------------
matTranspose(M)
{
let nRows = M.length;
let nCols = M[0].length;
let result = this.matMake(nCols, nRows, 0.0);
for (let i = 0; i "lt" nRows; ++i)
for (let j = 0; j "lt" nCols; ++j)
result[j][i] = M[i][j];
return result;
}
// --------------------------------------------------------
matMult(matA, matB)
{
let aRows = matA.length;
let aCols = matA[0].length;
let bRows = matB.length;
let bCols = matB[0].length;
let result = this.matMake(aRows, bCols, 0.0);
for (let i = 0; i "lt" aRows; ++i)
for (let j = 0; j "lt" bCols; ++j)
for (let k = 0; k "lt" aCols; ++k)
result[i][j] += matA[i][k] * matB[k][j];
return result;
}
// --------------------------------------------------------
vecMake(n, val)
{
let result = [];
for (let i = 0; i "lt" n; ++i) {
result[i] = val;
}
return result;
}
// --------------------------------------------------------
vecNorm(vec)
{
let sum = 0.0;
let n = vec.length;
for (let i = 0; i "lt" n; ++i)
sum += vec[i] * vec[i];
return Math.sqrt(sum);
}
// --------------------------------------------------------
vecDot(v1, v2)
{
let n = v1.length; // assume len(v1) == len(v2)
let sum = 0.0;
for (let i = 0; i "lt" n; ++i)
sum += v1[i] * v2[i];
return sum;
}
// --------------------------------------------------------
hypot(a, b)
{
// wacky sqrt(a^2 + b^2) used by numerical implementations
let xabs = Math.abs(a); let yabs = Math.abs(b);
let min, max;
if (xabs "lt" yabs) {
min = xabs; max = yabs;
}
else {
min = yabs; max = xabs;
}
if (min == 0)
return max;
else {
let u = min / max;
return max * Math.sqrt(1 + u * u);
}
}
// --------------------------------------------------------
// helpers for debugging: matShow, vecShow
// --------------------------------------------------------
matShow(M, dec, wid) // for debugging
{
let small = 1.0 / Math.pow(10, dec);
let nr = M.length;
let nc = M[0].length;
for (let i = 0; i "lt" nr; ++i) {
for (let j = 0; j "lt" nc; ++j) {
let x = M[i][j];
if (Math.abs(x) "lt" small) x = 0.0;
let xx = x.toFixed(dec);
let s = xx.toString().padStart(wid, ' ');
process.stdout.write(s);
process.stdout.write(" ");
}
process.stdout.write("\n");
}
}
// --------------------------------------------------------
vecShow(vec, dec, wid) // for debugging
{
let small = 1.0 / Math.pow(10, dec);
for (let i = 0; i "lt" vec.length; ++i) {
let x = vec[i];
if (Math.abs(x) "lt" small) x = 0.0 // avoid -0.00
let xx = x.toFixed(dec);
let s = xx.toString().padStart(wid, ' ');
process.stdout.write(s);
process.stdout.write(" ");
}
process.stdout.write("\n");
}
} // end class LinearRegressor
// ----------------------------------------------------------
// helper functions for main()
// matShow, vecShow, matLoad, matToVec
// ----------------------------------------------------------
function vecShow(vec, dec, wid, nl)
{
let small = 1.0 / Math.pow(10, dec);
for (let i = 0; i "lt" vec.length; ++i) {
let x = vec[i];
if (Math.abs(x) "lt" small) x = 0.0 // avoid -0.00
let xx = x.toFixed(dec);
let s = xx.toString().padStart(wid, ' ');
process.stdout.write(s);
process.stdout.write(" ");
}
if (nl == true)
process.stdout.write("\n");
}
// ----------------------------------------------------------
function matShow(A, dec, wid)
{
let small = 1.0 / Math.pow(10, dec);
let nr = A.length;
let nc = A[0].length;
for (let i = 0; i "lt" nr; ++i) {
for (let j = 0; j "lt" nc; ++j) {
let x = A[i][j];
if (Math.abs(x) "lt" small) x = 0.0;
let xx = x.toFixed(dec);
let s = xx.toString().padStart(wid, ' ');
process.stdout.write(s);
process.stdout.write(" ");
}
process.stdout.write("\n");
}
}
// ----------------------------------------------------------
function matToVec(M)
{
let nr = M.length;
let nc = M[0].length;
let result = [];
for (let i = 0; i "lt" nr*nc; ++i) { // vecMake(r*c, 0.0);
result[i] = 0.0;
}
let k = 0;
for (let i = 0; i "lt" nr; ++i) {
for (let j = 0; j "lt" nc; ++j) {
result[k++] = M[i][j];
}
}
return result;
}
// ----------------------------------------------------------
function matLoad(fn, delimit, usecols, comment)
{
// efficient but mildly complicated
let all = FS.readFileSync(fn, "utf8"); // giant string
all = all.trim(); // strip final crlf in file
let lines = all.split("\n"); // array of lines
// count number non-comment lines
let nRows = 0;
for (let i = 0; i "lt" lines.length; ++i) {
if (!lines[i].startsWith(comment))
++nRows;
}
let nCols = usecols.length;
// let result = matMake(nRows, nCols, 0.0);
let result = [];
for (let i = 0; i "lt" nRows; ++i) {
result[i] = [];
for (let j = 0; j "lt" nCols; ++j) {
result[i][j] = 0.0;
}
}
let r = 0; // into lines
let i = 0; // into result[][]
while (r "lt" lines.length) {
if (lines[r].startsWith(comment)) {
++r; // next row
}
else {
let tokens = lines[r].split(delimit);
for (let j = 0; j "lt" nCols; ++j) {
result[i][j] = parseFloat(tokens[usecols[j]]);
}
++r;
++i;
}
}
return result;
}
// ----------------------------------------------------------
function main()
{
console.log("\nLinear regression with " +
"pseudo-inverse (SVD-Jacobi) using JavaScript ");
// 1. load data
console.log("\nLoading synthetic train (200) and" +
" test (40) data from file ");
let trainFile = ".\\Data\\synthetic_train_200.txt";
let trainX = matLoad(trainFile, ",", [0,1,2,3,4], "#");
let trainY = matLoad(trainFile, ",", [5], "#");
trainY = matToVec(trainY);
let testFile = ".\\Data\\synthetic_test_40.txt";
let testX = matLoad(testFile, ",", [0,1,2,3,4], "#");
let testY = matLoad(testFile, ",", [5], "#");
testY = matToVec(testY);
console.log("\nFirst three train X: ");
for (let i = 0; i "lt" 3; ++i)
vecShow(trainX[i], 4, 8, true);
console.log("\nFirst three train y: ");
for (let i = 0; i "lt" 3; ++i)
console.log(trainY[i].toFixed(4).toString().
padStart(9, ' '));
// 2. create and train linear regression model
let seed = 0; // not used this version
console.log("\nCreating and training model ");
let model = new LinearRegressor(seed);
model.train(trainX, trainY);
console.log("Done ");
console.log("\nModel weights/coefficients: ");
vecShow(model.weights, 4, 8, true);
console.log("Model bias/intercept: " +
model.bias.toFixed(4).toString());
// 3. evaluate
console.log("\nEvaluating model ");
let trainAcc = model.accuracy(trainX, trainY, 0.10);
let testAcc = model.accuracy(testX, testY, 0.10);
console.log("\nTrain acc (within 0.10) = " +
trainAcc.toFixed(4).toString());
console.log("Test acc (within 0.10) = " +
testAcc.toFixed(4).toString());
let trainMSE = model.mse(trainX, trainY);
let testMSE = model.mse(testX, testY);
console.log("\nTrain MSE = " +
trainMSE.toFixed(4).toString());
console.log("Test MSE = " +
testMSE.toFixed(4).toString());
// 4. use model
let x = trainX[0];
console.log("\nPredicting for x = ");
vecShow(x, 4, 9, true); // add newline
let predY = model.predict(x);
console.log("Predicted y = " +
predY.toFixed(4).toString());
console.log("\nEnd demo");
}
main();
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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