One evening, for no reason I can really think of, I decided to implement a quadratic regression system, using relaxed Moore-Penrose pseudo-inverse via QR-Householder inverse, from scratch, using JavaScript.
So I did.
The goal of a machine learning regression problem is to predict a single numeric value. Quadratic regression is an enhanced form of basic linear regression. One of several ways to train a quadratic regression model is to use relaxed Moore-Penrose pseudo-inverse via the QR-Householder algorithm. I recently updated my QR-Householder implementation, so I decided to use it for quadratic regression.
Suppose there are five predictors, (x0, x1, x2, x3, x4). The prediction equation for basic linear regression is:
y’ = (w0 * x0) + (w1 * x1) + (w2 * x2) + (w3 * x3) + (w4 * x4) + b
The wi are model weights (aka coefficients), and b is the model bias (aka intercept). The values of the weights and the bias must be determined by training, so that predicted y’ values are close to the known, correct y values in a set of training data.
The prediction equation for quadratic regression with five predictors is:
y’ = (w0 * x0) + (w1 * x1) + (w2 * x2) + (w3 * x3) + (w4 * x4) +
(w5 * x0*x0) + (w6 * x1*x1) + (w7 * x2*x2) +
(w8 * x3*x3) + (w9 * x4*x4) +
(w10 * x0*x1) + (w11 * x0*x2) + (w12 * x0*x3) + (w13 * x0*x4) +
(w14 * x1*x2) + (w15 * x1*x3) + (w16 * x1*x4) +
(w17 * x2*x3) + (w18 * x2*x4) +
(w19 * x3*x4)
+ b
The squared (“quadratic”) xi^2 terms handle non-linear structure. If there are n predictors, there are also n squared terms. The xi * xj terms between all possible pairs of original predictors handle interactions between predictors. If there are n predictors, there (n * (n-1)) / 2 interaction terms.
Training is the process of finding values for the weights and the bias so that the model predicts well. There are several different training techniques, including the relaxed Moore-Penrose pseudo-inverse via QR-Householder technique in this blog post. Two significant alternatives are stochastic gradient descent training, and left pseudo-inverse via normal equations with Cholesky inverse.
The technique described in this blog post — relaxed MP pseudo-inverse — is technically correct, but is really more of an investigation because there is a closely related training technique — OLS (ordinary least squares) Solve — that is more efficient.
The math equation for relaxed MP pseudo-inverse training is w = pinv(X) * y where w is a vector that holds the weights and bias you are looking for, X is an augmented training data matrix that has quadratic columns and interaction columns, and a leading column of 1.0s which handles the bias term. The y is a vector of target y values from the training data, and * is matrix-to-vector multiplication.
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 x predictors. The last value on each line is the target y variable to predict. The data is synthetic, and was generated by a 5-10-1 neural network with randome weights and biases. There are 200 training items and 40 test items.
The output of my demo is:
Begin quadratic regression with MP pinv QR-Householder training using node.js JavaScript Loading synthetic train (200) and test (40) from file 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 quadratic regression model Done Training model Done Model base weights: -0.2630 0.0354 -0.0421 0.0341 -0.1124 Model quadratic weights: 0.0655 0.0194 0.0051 0.0047 0.0243 Model interaction weights: 0.0043 0.0249 0.0071 0.1081 -0.0012 -0.0093 0.0362 0.0085 -0.0568 0.0016 Model bias: 0.3220 Computing model accuracy Train acc (within 0.10) = 0.8850 Test acc (within 0.10) = 0.9250 Train MSE = 0.0003 Test MSE = 0.0005 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 Predicted y = 0.4843 End demo
The model accuracy of 88.50% on the training data (177 out of 200 correct) and 92.50% on the test data (37 out of 40 correct) is quite good compared to other regression techniques, and much better than simple linear regression. A prediction is scored as correct if it’s within 10% of the true target y value. Quadratic regression has a good balance between predictive accuracy and interpretability.
Good fun.
Quadratic regression is just one of many regression techniques to predict a single value. Another technique is decision tree regression.
It probably wasn’t too difficult to predict the irony involved in these two photos.
Left: “If you want to live forever, plant a tree.” Well, anywhere except in a city.
Right: “Tree of Life”. Until it’s not.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols. (My blog editor chokes on symbols).
// quadratic_regression_pinv_qr.js
// quadratic regression using MP pseudo-inv QR-Householder
// node.js
let FS = require("fs") // for loadTxt()
// ----------------------------------------------------------
class QuadraticRegressor
{
constructor(seed)
{
this.weights; // allocated in train()
this.bias = 0;
this.seed = seed + 0.5; // avoid 0
}
// --------------------------------------------------------
predict(x)
{
let dim = x.length;
let result = 0.0;
let p = 0; // ptr into wts
for (let i = 0; i "lt" dim; ++i) // regular
result += x[i] * this.weights[p++];
for (let i = 0; i "lt" dim; ++i) // quadratic
result += x[i] * x[i] * this.weights[p++];
for (let i = 0; i "lt" dim-1; ++i) // interactions
for (let j = i+1; j "lt" dim; ++j)
result += x[i] * x[j] * this.weights[p++];
result += this.bias;
return result;
}
// --------------------------------------------------------
train(trainX, trainY)
{
let nRows = trainX.length;
let dim = trainX[0].length; // number predictors
let nInteractions = (dim * (dim - 1)) / 2;
this.weights = vecMake(dim + dim + nInteractions, 0.0);
let Xa = this.matAugment(trainX); // add columns
let Xd = this.matToDesign(Xa); // add leading 1s
let Xpinv = QRHouseholder.matPinv(Xd);
let biasAndWts = this.matVecProd(Xpinv, trainY);
this.bias = biasAndWts[0]; // bias is at [0]
for (let i = 1; i "lt" biasAndWts.length; ++i)
this.weights[i - 1] = biasAndWts[i];
}
// --------------------------------------------------------
matAugment(trainX)
{
// add quadratic and interaction columns
let nRows = trainX.length; // src and dest
let dim = trainX[0].length; // src
let nInteractions = dim * (dim - 1) / 2;
let nColsDest = dim + dim + nInteractions;
let result = matMake(nRows, nColsDest, 0.0);
for (let i = 0; i "lt" nRows; ++i)
{
let p = 0; // points to column of result
for (let j = 0; j "lt" dim; ++j) // base
result[i][p++] = trainX[i][j];
for (let j = 0; j "lt" dim; ++j) // quadratic
result[i][p++] = trainX[i][j] * trainX[i][j];
for (let j = 0; j "lt" nInteractions-1; ++j)
for (let k = j + 1; k "lt" dim; ++k)
result[i][p++] = trainX[i][j] * trainX[i][k];
}
return result;
}
// --------------------------------------------------------
matToDesign(X)
{
// add leading column of 1.0s to handle bias term
let nRows = X.length; // src and dest
let dim = X[0].length;
let result = matMake(nRows, dim+1, 0.0); // extra col
for (let i = 0; i "lt" nRows; ++i) {
result[i][0] = 1.0;
for (let j = 1; j "lt" result[0].length; ++j)
result[i][j] = X[i][j - 1];
}
return result;
}
// --------------------------------------------------------
matVecProd(M, v)
{
// helper for train()
let nRows = M.length;
let nCols = M[0].length;
let n = v.length;
if (nCols != n)
console.log("FATAL: non-comform in matVecProd");
let result = 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;
}
// --------------------------------------------------------
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;
}
// --------------------------------------------------------
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;
}
// --------------------------------------------------------
} // class LinearRegressor
// ==========================================================
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)
static matPinv(M)
{
// awkward to call instance methods from static in JS
let self = new QRHouseholder();
let QR = self.matDecompQR(M); // Householder
let Q = QR[0];
let R = QR[1];
let Ri = self.matInvUpperTri(R);
let Qi = self.matTranspose(Q);
let result = self.matProduct(Ri, Qi);
return result;
}
// --------------------------------------------------------
matDecompQR(A)
{
// Householder algorithm
let m = A.length; let n = A[0].length;
if (m "lt" n)
console.log("FATAL: nRows must be gte nCols");
let QQ = matMake(m, m, 0.0); // working full Q
for (let i = 0; i "lt" m; ++i)
QQ[i][i] = 1.0; // identity matrix
let RR = matMake(m, n, 0.0);
for (let i = 0; i "lt" m; ++i)
for (let j = 0; j "lt" n; ++j)
RR[i][j] = A[i][j]; // copy of A is working R
let k = Math.min(m, n); // or just use n
for (let j = 0; j "lt" k; ++j) { // main processing loop
let xn = m - j;
let x = vecMake(xn, 0.0);
for (let i = 0; i "lt" xn; ++i)
x[i] = RR[j + i][j];
let ss = 0.0;
for (let i = 0; i "lt" xn; ++i)
ss += x[i] * x[i];
let normX = Math.sqrt(ss);
// if (normX == 0.0) continue; // risky
if (Math.abs(normX) "lt" 1.0e-12) continue;
let sign = 0.0;
if (x[0] "gte" 0.0)
sign = -1.0;
else
sign = 1.0; // counter-intuitive
let u = vecMake(xn, 0.0);
for (let 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)
let tau = -sign * (x[0] - sign * normX) / normX;
// dimensions for sub-matrices
let nRowsSubR = m - j; let nColsSubR = n - j;
let nRowsSubQ = m; let nColsSubQ = m - j;
let vr = vecMake(nColsSubR, 0.0);
for (let c = 0; c "lt" nColsSubR; ++c) {
let acc = 0.0; // accumulate
for (let r = 0; r "lt" nRowsSubR; ++r)
acc += u[r] * RR[j + r][j + c];
vr[c] = acc;
}
let vq = vecMake(nRowsSubQ, 0.0);
for (let r = 0; r "lt" nRowsSubQ; ++r) {
let acc = 0.0;
for (let c = 0; c "lt" nColsSubQ; ++c)
acc += u[c] * QQ[r][j + c];
vq[r] = acc;
}
// update sub-R
for (let r = 0; r "lt" nRowsSubR; ++r)
for (let c = 0; c "lt" nColsSubR; ++c)
RR[j + r][j + c] -= tau * u[r] * vr[c];
// update sub-Q
for (let r = 0; r "lt" nRowsSubQ; ++r)
for (let c = 0; c "lt" nColsSubQ; ++c)
QQ[r][j + c] -= tau * vq[r] * u[c];
} // j main loop
// extract QQ RR into [Q, R]
let Q = matMake(m, n, 0.0);
for (let i = 0; i "lt" m; ++i)
for (let j = 0; j "lt" n; ++j)
Q[i][j] = QQ[i][j];
let R = matMake(n, n, 0.0);
for (let i = 0; i "lt" n; ++i)
for (let j = 0; j "lt" n; ++j)
R[i][j] = RR[i][j];
return [Q, R];
} // matDecompQR()
// --------------------------------------------------------
matInvUpperTri(U)
{
let n = U.length; // must be square matrix
let result = matMake(n, n, 0.0);
for (let i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
for (let k = 0; k "lt" n; ++k) {
for (let j = 0; j "lt" n; ++j) {
for (let i = 0; i "lt" k; ++i) {
result[j][k] -= result[j][i] * U[i][k];
}
result[j][k] /= U[k][k];
}
}
return result;
}
// --------------------------------------------------------
matTranspose(M)
{
let nRows = M.length;
let nCols = M[0].length;
let result = matMake(nCols, nRows, 0.0); // note
for (let i = 0; i "lt" nRows; ++i)
for (let j = 0; j "lt" nCols; ++j)
result[j][i] = M[i][j]; // note
return result;
}
// --------------------------------------------------------
matProduct(A, B)
{
let aRows = A.length; let aCols = A[0].length;
let bRows = B.length; let bCols = B[0].length;
if (aCols != bRows)
console.log("FATAL: Non-conformable matrices");
let result = matMake(aRows, bCols, 0.0);
for (let i = 0; i "lt" aRows; ++i) // each row of A
for (let j = 0; j "lt" bCols; ++j) // each col of B
for (let k = 0; k "lt" aCols; ++k)
result[i][j] += A[i][k] * B[k][j];
return result;
}
} // class
// ==========================================================
// ----------------------------------------------------------
// vector and matrix functions
// ----------------------------------------------------------
function vecMake(n, val)
{
let result = [];
for (let i = 0; i "lt" n; ++i) {
result[i] = val;
}
return result;
}
// ----------------------------------------------------------
function 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;
}
// ----------------------------------------------------------
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 = vecMake(nr*nc, 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 loadTxt(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 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("\nBegin quadratic regression with MP pinv " +
"QR-Householder training using node.js JavaScript ");
// 1. load data
console.log("\nLoading synthetic train (200) and" +
" test (40) from file ");
let trainFile = ".\\Data\\synthetic_train_200.txt";
let trainX = loadTxt(trainFile, ",", [0,1,2,3,4], "#");
let trainY = loadTxt(trainFile, ",", [5], "#");
trainY = matToVec(trainY);
let testFile = ".\\Data\\synthetic_test_40.txt";
let testX = loadTxt(testFile, ",", [0,1,2,3,4], "#");
let testY = loadTxt(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 quadratic regression model
console.log("\nCreating quadratic regression model ");
let model = new QuadraticRegressor(0); // seed not used
console.log("Done ");
console.log("\nTraining model ");
model.train(trainX, trainY);
console.log("Done ");
// 3. show model weights
console.log("\nModel base weights: ");
let dim = trainX[0].length;
for (let i = 0; i "lt" dim; ++i)
process.stdout.write(model.weights[i].toFixed(4).
toString().padStart(8, ' '));
console.log("");
console.log("\nModel quadratic weights: ");
for (let i = dim; i "lt" dim + dim; ++i)
process.stdout.write(model.weights[i].toFixed(4).
toString().padStart(8, ' '));
console.log("");
console.log("\nModel interaction weights: ");
for (let i = dim + dim; i "lt" model.weights.length; ++i) {
process.stdout.write(model.weights[i].toFixed(4).
toString().padStart(8, ' '));
if (i "gt" dim+dim && i % dim == 0)
console.log("");
}
console.log("");
console.log("\nModel bias: " +
model.bias.toFixed(4).toString());
// 4. evaluate
console.log("\nComputing model accuracy ");
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());
// 5. 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
SciPy Programming Succinctly
Keras Succinctly
R Programming
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Microsoft MLADS Conference
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