When I was a college student, I remember studying linear regression with two-way interactions. It’s a valuable technique, but one which is rarely used nowadays.
The goal of a machine learning regression problem is to predict a single numeric value. For example, you might want to predict an employee’s bank account balance based on age, annual income, and so on. The most basic technique is called linear regression, or sometimes multiple linear regression, where the “multiple” term indicates two or more predictor variables.
The form of a basic linear regression prediction model is y’ = (w0 * x0) + (w1 * x1) + . . . + (wn * xn) + b, where y’ is the predicted value, the xi are predictor values, the wi are weights (also called coefficients), and b is the bias (also called the intercept).
The form of a linear regression with two-way interactions model is y’ = (w0 * x0) + . . . + (wn * xn) + (w01 * x0 * x1) + (w02 * x0 * x2) + . . . + b. The interaction terms are the multiplication products of all combinations of pairs of the predictor variables.
Compared to basic linear regression, linear regression with two-way interactions can handle more complex data. Compared to more sophisticated regression techniques, linear regression with interactions often has slightly worse prediction accuracy, but has better model interpretability. This can be important in scenarios where interpretability is useful, or even required by law.
For my demo, I used synthetic data that was generated by a neural network. It 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. There are 200 training items and 40 test items.
My demo was designed to run in a node.js environment. The output of the demo is:
Begin linear regression with two-way interactions using node.js JavaScript Loading train (200) and test (40) 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 Setting SGD lrnRate = 0.001 Setting SGD maxEpochs = 100 epoch = 0 MSE = 0.1116 acc = 0.0000 epoch = 20 MSE = 0.0033 acc = 0.4400 epoch = 40 MSE = 0.0012 acc = 0.5950 epoch = 60 MSE = 0.0009 acc = 0.6750 epoch = 80 MSE = 0.0008 acc = 0.7000 Done Model base weights: -0.2625 0.0341 -0.0460 0.0324 -0.1151 Model bias: 0.3619 Model interaction weights: 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0008 0.0000 0.0000 0.0000 0.0000 0.0313 0.0107 0.0000 0.0000 0.0000 0.0169 -0.0104 0.0013 0.0000 0.0000 0.0952 0.0315 -0.0445 0.0003 0.0000 Computing model accuracy Train acc (within 0.15) = 0.8350 Test acc (within 0.15) = 0.8000 Train MSE = 0.0008 Test MSE = 0.0006 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 Predicted y = 0.5090 End demo
Because the demo data has five predictor variables, there are five basic weights. With two-way interactions, there are an additional 10 interaction weights: w01, w02, w03, w04, w12, w13, w14, w23, w24, w34. In general, if there are n predictor variables, there are (n * (n-1)) / 2 interaction weights.
The demo stores the interaction weights in the lower left part of an n-by-n matrix where the row index is the first predictor and the column index is the second predictor. For instance, the -0.0104 value at [3][1] is the weight for the x3 and x1 interaction.
Linear regression with two-way interactions has a nice balance of prediction power and interpretability. The model weights/coefficients are easy to interpret. If the predictor values have been normalized to the same scale, larger magnitudes mean larger effect, and the sign of the weights indicate the direction of the effect. Also, linear regression (regular and with two-way interactions) is less susceptible to model overfitting than other techniques.
There are several different types of linear regression. There are several different types of software engineers. Here are three puzzling examples I found doing an Internet image search for “female software engineer stock photo”. Thank you Internet for 40+ years of entertainment.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols (my blog editor usually chokes on symbols).
// linear_regression_interactions.js
// linear regression with two-way interactions (using SGD)
// node.js
let FS = require("fs") // for loadTxt()
// ----------------------------------------------------------
class LinearRegressor
{
constructor(seed)
{
this.seed = seed + 0.5; // avoid 0
this.weights = null; // allocated in train()
this.interactionWts = null;
this.bias = 0.0;
}
// --------------------------------------------------------
predict(x)
{
let dim = x.length; // number of predictors
let sum = 0.0;
for (let i = 0; i "lt" x.length; ++i) {
sum += x[i] * this.weights[i];
}
// add interactions
for (let i = 0; i "lt" dim; ++i) // each pair predictors
for (let j = 0; j "lt" i; ++j)
sum += x[i] * x[j] * this.interactionWts[i][j];
sum += this.bias; // don't forget the bias
return sum;
}
// --------------------------------------------------------
train(trainX, trainY, lrnRate, maxEpochs)
{
let dim = trainX[0].length; // number predictors
this.weights = vecMake(dim, 0.0); // allocate model wts
let freq = maxEpochs / 5; // when to show progress
let lo = -0.01; let hi = 0.01;
for (let i = 0; i "lt" dim; ++i) {
this.weights[i] = (hi - lo) * this.next() + lo;
}
this.interactionWts = matMake(dim, dim, 0.0);
for (let i = 0; i "lt" dim; ++i) // all rows
for (let j = 0; j "lt" i; ++j) // small = lower left
this.interactionWts[i][j] = (hi - lo) *
this.next() + lo;
this.bias = (hi - lo) * this.next() + lo;
// set up indices for shuffling
let N = trainX.length;
let indices = vecMake(N, 0.0);
for (let i = 0; i "lt" N; ++i)
indices[i] = i;
for (let epoch = 0; epoch "lt" maxEpochs; ++epoch) {
this.shuffle(indices);
for (let i = 0; i "lt" N; ++i) {
let idx = indices[i];
let x = trainX[idx];
let predY = this.predict(x);
let actualY = trainY[idx];
// update regular wts
for (let j = 0; j "lt" dim; ++j) {
this.weights[j] -=
lrnRate * (predY - actualY) * x[j];
}
// update interaction weights
for (let r = 0; r "lt" dim; ++r)
for (let c = 0; c "lt" r; ++c)
this.interactionWts[r][c] -= lrnRate *
(predY - actualY) * x[r] * x[c];
// don't forget to update bias
this.bias -= lrnRate * (predY - actualY);
} // each item
if (epoch % freq == 0) // show progress
{
let mse = this.meanSqError(trainX, trainY);
let acc = this.accuracy(trainX, trainY, 0.10);
let s1 = "epoch = " +
epoch.toString().padStart(6, ' ');
let s2 = " MSE = " +
mse.toFixed(4).toString();
let s3 = " acc = " + acc.toFixed(4).toString();
console.log(s1 + s2 + s3);
}
} // each epoch
// apply wt decay aka L2 regularization
// for (let j = 0; j "lt" dim; ++j)
// this.weights[j] *= (1.0 - this.alpha);
// for (let i = 0; i "lt" dim; ++i) // all rows
// for (let j = 0; j "lt" i; ++j) // small = lower left
// this.interactionWts[i][j] *= (1.0 - this.alpha);
} // train()
// --------------------------------------------------------
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);
}
// --------------------------------------------------------
meanSqError(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;
}
// --------------------------------------------------------
next()
{
// return sort-of-random in [0.0, 1.0)
let x = Math.sin(this.seed) * 1000;
let result = x - Math.floor(x); // [0.0,1.0)
this.seed = result; // for next call
return result;
}
// --------------------------------------------------------
nextInt(lo, hi)
{
let x = this.next();
return Math.trunc((hi - lo) * x + lo);
}
// --------------------------------------------------------
shuffle(indices)
{
// Fisher-Yates
for (let i = 0; i "lt" indices.length; ++i) {
let ri = this.nextInt(i, indices.length);
let tmp = indices[ri];
indices[ri] = indices[i];
indices[i] = tmp;
//indices[i] = i; // for testing
}
}
} // end class LinearRegressor
// ----------------------------------------------------------
// 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 r = m.length;
let c = m[0].length;
let result = vecMake(r*c, 0.0);
let k = 0;
for (let i = 0; i "lt" r; ++i) {
for (let j = 0; j "lt" c; ++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 linear regression with two-way " +
"interactions using node.js JavaScript ");
// 1. load data
console.log("\nLoading 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 linear regression model
let seed = 0;
console.log("\nCreating and training model ");
let model = new LinearRegressor(seed);
let lrnRate = 0.001;
let maxEpochs = 100;
console.log("\nSetting SGD lrnRate = " +
lrnRate.toFixed(3).toString());
console.log("Setting SGD maxEpochs = " +
maxEpochs.toString());
model.train(trainX, trainY, lrnRate, maxEpochs);
console.log("Done ");
console.log("\nModel base weights: ");
vecShow(model.weights, 4, 8, true);
console.log("Model bias: " +
model.bias.toFixed(4).toString());
console.log("\nModel interaction weights: ");
matShow(model.interactionWts, 4, 8);
// 3. evaluate
console.log("\nComputing model accuracy ");
let trainAcc = model.accuracy(trainX, trainY, 0.15);
let testAcc = model.accuracy(testX, testY, 0.15);
console.log("\nTrain acc (within 0.15) = " +
trainAcc.toFixed(4).toString());
console.log("Test acc (within 0.15) = " +
testAcc.toFixed(4).toString());
let trainMSE = model.meanSqError(trainX, trainY);
let testMSE = model.meanSqError(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
2026 Visual Studio Live
2025 Summer MLADS Conference
2025 DevIntersection Conference
2025 Machine Learning Week
2025 Ai4 Conference
2025 G2E Conference
2025 iSC West Conference
You must be logged in to post a comment.