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 * income) + (-0.38 * ago) + (0.17 * experience) + 0.62.
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 training using matrix pseudo-inverse. This blog post demonstrates how to implement one of several possible versions of the pseudo-inverse training technique, specifically relaxed Moore-Penrose pseudo-inverse training.
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 relaxed 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 adds a leading column of 1s, which takes the bias into account. For example, if a set of training predictors stored in a matrix X is:
A design matrix takes the bias into account. For example, if a set of training predictors stored in a matrix X is:
0.60 0.87 0.42 0.82 0.51 0.94 0.74 0.83 0.53
The associated design matrix DX is:
1.0 0.60 0.87 0.42 1.0 0.82 0.51 0.94 1.0 0.74 0.83 0.53
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:
Linear regression with pseudo-inverse (QR-Householder) 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 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 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 QR decomposition via the Householder algorithm.
Most library implementations of MP pseudo-inverse use an algorithm involving singular value decomposition (SVD) — one of the most complicated techniques in all of numerical programming — typically 10,000 lines of code. Using QR decomposition is dramatically simpler than SVD but less robust than SVD.
Note: Instead of using explicit relaxed Moore-Penrose pseudo-inverse training (via QR decomposition or SVD), it’s possible to use a related technique called pseudo-inverse via normal equations closed form training. The math equation is w = inv(Xt * X) * Xt * y, where X is the design matrix, Xt is the transpose of X, inv() is regular matrix inverse (but usually Cholesky decomposition), * is matrix multiplication, and y is a vector of target values from the training data. The disadvantage of this approach is that Xt * X can involve many thousands of multiplications, and so the operation can fail due to arithmetic underflow or overflow. I noticed that quite a few Internet resources call both relaxed MP pseudo-inverse and pseudo-inverse via normal equations, just “pseudo-inverse”. Therefore, “pseudo-inverse” in a machine learning scenario is an ambiguous term.
I know quite a bit about machine learning models. I know essentially nothing about fashion models. It’s pretty easy to evaluate machine learning models but not so easy to evaluate fashion models. Here are three examples from the Internet of “bad model fashion”. But I think all three (both the models and their fashion) are quite attractive.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols — my blog editor chokes on symbols.
// linear_regression_pinverse.js
// Moore-Penrose pseudo-inverse (QR-Householder) training
// node.js
let FS = require("fs") // for loadTxt()
// ----------------------------------------------------------
class LinearRegressor
{
constructor(seed)
{
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
// --------------------------------------------------------
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) // Moore-Penrose using QR-Householder decomp
{
// A = Q*R, pinv(A) = inv(R) * trans(Q)
let m = M.length; let n = M[0].length;
if (m "lt" n)
cosole.log("ERROR. works only m "gte" m");
let QR = this.matDecomposeQR(M, true); // reduced
let Rinv = this.matInvUpperTri(QR[1]);
let Qinv = this.matTranspose(QR[0]);
let result = this.matProduct(Rinv, Qinv);
return result;
}
// --------------------------------------------------------
// helpers for matPinv
// matDecomposeQR, matInvUpperTri, matTranspose, matProduct
// --------------------------------------------------------
matDecomposeQR(M, reduced)
{
// QR decomposition, Householder algorithm.
// result[0] = Q, result[1] = R
let m = M.length;
let n = M[0].length;
if (m "lt" n)
console.log("No rows less than cols");
let Q = this.matIdentity(m); // working Q
let R = this.matCopy(M); // working R
let end = 0;
if (m == n) end = n - 1;
else end = n;
for (let i = 0; i "lt" end; ++i) {
let H = this.matIdentity(m);
let a = this.vecMake(m-i, 0.0);
let k = 0;
for (let ii = i; ii "lt" m; ++ii) // corr
a[k++] = R[ii][i];
let normA = this.vecNorm(a);
if (a[0] "lt" 0.0 && normA "gt" 0.0) // corr
normA = -normA;
else if (a[0] "gt" 0.0 && normA "lt" 0.0)
normA = -normA;
let v = this.vecMake(a.length, 0.0);
for (let j = 0; j "lt" v.length; ++j)
v[j] = a[j] / (a[0] + normA);
v[0] = 1.0;
// Householder algorithm
let h = this.matIdentity(a.length);
let vvDot = this.vecDot(v, v);
let A = this.vecToMat(v, v.length, 1);
let B = this.vecToMat(v, 1, v.length);
let AB = this.matProduct(A, B);
for (let ii = 0; ii "lt" h.length; ++ii)
for (let jj = 0; jj "lt" h[0].length; ++jj)
h[ii][jj] -= (2.0 / vvDot) * AB[ii][jj];
// copy h[][] into lower right corner of H[][]
let d = m - h.length; // corr
for (let ii = 0; ii "lt" h.length; ++ii)
for (let jj = 0; jj "lt" h[0].length; ++jj)
H[ii + d][jj + d] = h[ii][jj];
Q = this.matProduct(Q, H);
R = this.matProduct(H, R);
} // i
let result = [];
if (reduced == false) {
result[0] = Q;
result[1] = R;
return result;
}
else if (reduced == true) {
let qRows = Q.length; let qCols = Q[0].length;
let rRows = R.length; let rCols = R[0].length;
// assumes m "gte" n !!
// square-up R
let dim = Math.min(rRows, rCols);
let Rsquared = this.matMake(dim, dim, 0.0);
for (let i = 0; i "lt" dim; ++i)
for (let j = 0; j "lt" dim; ++j)
Rsquared[i][j] = R[i][j];
// Q needs same number columns as R
// so that inv(R) * trans(Q) works
let Qtrimmed = this.matMake(qRows, dim, 0.0);
for (let i = 0; i "lt" qRows; ++i)
for (let j = 0; j "lt" dim; ++j)
Qtrimmed[i][j] = Q[i][j];
result[0] = Qtrimmed;
result[1] = Rsquared;
return result;
}
} // matDecomposeQR()
// --------------------------------------------------------
matInvUpperTri(U)
{
let n = U.length; // U must be square matrix
let result = this.matIdentity(n);
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] + 1.0e-8); // avoid 0
}
}
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;
}
// --------------------------------------------------------
matProduct(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;
}
// --------------------------------------------------------
// misc helper methods/functions:
// matMake, matCopy, matIdentity, vecMake,
// vecToMat, vecNorm, vecDot
//
// 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;
}
// --------------------------------------------------------
vecMake(n, val)
{
let result = [];
for (let i = 0; i "lt" n; ++i) {
result[i] = val;
}
return result;
}
// --------------------------------------------------------
vecToMat(vec, nRows, nCols)
{
let result = this.matMake(nRows, nCols, 0.0);
let k = 0;
for (let i = 0; i "lt" nRows; ++i)
for (let j = 0; j "lt" nCols; ++j)
result[i][j] = vec[k++];
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;
}
// --------------------------------------------------------
// --------------------------------------------------------
// member methods 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) {
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" +
" (QR-Householder) training 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
console.log("\nCreating and training model ");
let model = new LinearRegressor();
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
SciPy Programming Succinctly
Keras Succinctly
R Programming
Visual Studio Live
Microsoft MLADS Conference
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Machine Learning Week
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