Gaussian process regression (GPR) is a powerful machine learning technique to predict a single numeric value. When GPR works, it often works very well. The major disadvantages of GPR are: 1.) it uses matrix inversion, which means GPR is not well-suited for large datasets (say, over 2,000 training data items), and 2.) you have to specify a kernel function (radial basis function, polynomial function, etc.) and each kernel function will have at least one hyperparameter to tune, so tuning is time-consuming.
Some time ago, I implemented GPR from scratch, using C#, with the Cholesky matrix inverse algorithm. I was reviewing that implementation and saw a few design choices that I wasn’t completely happy with, and so I decided to refactor the implementation.
The new implementation worked quite well. The output of the demo is:
Begin Gaussian Process Regression with Cholesky matrix inverse Loading 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 Setting constant theta = 0.50 Setting RBF lenScale = 2.00 Setting alpha noise = 0.01 Create and train GPR model Done Evaluating model accuracy Train acc (within 0.10) = 0.9450 Test acc (within 0.10) = 0.9500 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 predicted y = 0.4897 prediction std dev = 0.0435 End demo
The demo data is synthetic. There are five predictor variables with values between -1 and +1. The target values to predict are between 0.0 and 1.0. The synthetic data was generated by a 5-10-1 neural network with random weights and biases. Therefore, the data has an underlying, but complex, structure which can, in theory, be predicted. There are 200 training items and 40 test items.
The key equations for Gaussian process regression. They’re simpler than they might first appear. The -1 exponent means matrix invese, which is where the Cholesky algorithm is used.
The demo program hard-codes a constant kernel * radial basis function (RBF) kernel function. RBF is a good general purpose kernel function. The values of theta (0.50) for the constant part of the kernel function and lenScale (2.00) for the RBF part of kernel function were determined by trial and error.
The demo program uses the trained GPR model to predict for x = (-0.1660, 0.4406, -0.9998, -0.3953, -0.7065) which is the first training item. The predicted y value is 0.4897 which is quite close to the actual y value of 0.4840.
One unusual characteristic of GPR is that a prediction gives you a point estimate but also gives you an estimated standard deviation of the prediction, 0.0435 for the example prediction. You can use the standard deviation to construct a confidence interval prediction if you wish. For example, a 95% confidence interval would be 0.4897 +/- (3 * 0.0435).
This was an interesting and fun exploration.
When I refactor a computer program, I have every expectation that the remake version will be better then the original. I’m a big fan of science fiction movies, but unfortunately, the vast majority of sci fi movies remakes are much worse than the originals. But some remakes are better, and even fewer are subjectively pretty much the same. Here are two remakes that I rate as equal to the original.
Left: The movie “It! The Terror from Beyond Space” (1958) tells the story of a spaceship that lands onn Mars but picks up an unwanted visitor. The movie “Alien” (1979) has a different title but has essentially the same story. I grade both movies out as A-.
Right: The movie “Dune” (1984) was not well-received when released, but I like it a lot. The remake, “Dune” (2012) benefits from a much larger budget and vastly improved special effects technology. I grade both movies out as A-.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte”, “and” with Boolean operator symbols. My blog editor chokes on symbols.
using System;
using System.IO;
using System.Collections.Generic;
namespace GaussianProcessRegressionCholesky
{
internal class GPRCholeskyProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin Gaussian Process " +
"Regression with Cholesky matrix inverse ");
// 1. Load data
Console.WriteLine("\nLoading train (200) and" +
" test (40) from file ");
string trainFile =
"..\\..\\..\\Data\\synthetic_train_200.txt";
double[][] trainX = MatUtils.MatLoad(trainFile,
new int[] { 0, 1, 2, 3, 4 }, ',', "#");
double[] trainY =
MatUtils.MatToVec(MatUtils.MatLoad(trainFile,
new int[] { 5 }, ',', "#"));
string testFile =
"..\\..\\..\\Data\\synthetic_test_40.txt";
double[][] testX = MatUtils.MatLoad(testFile,
new int[] { 0, 1, 2, 3, 4 }, ',', "#");
double[] testY =
MatUtils.MatToVec(MatUtils.MatLoad(testFile,
new int[] { 5 }, ',', "#"));
Console.WriteLine("Done ");
Console.WriteLine("\nFirst three train X: ");
for (int i = 0; i "lt" 3; ++i)
MatUtils.VecShow(trainX[i], 4, 8);
Console.WriteLine("\nFirst three train y: ");
for (int i = 0; i "lt" 3; ++i)
Console.WriteLine(trainY[i].ToString("F4").
PadLeft(8));
// 2. create and train GPR model
double theta = 0.50; // "constant kernel"
double lenScale = 2.0; // RBF parameter
double alpha = 0.01; // noise
Console.WriteLine("\nSetting constant theta = " +
theta.ToString("F2"));
Console.WriteLine("Setting RBF lenScale = " +
lenScale.ToString("F2"));
Console.WriteLine("Setting alpha noise = " +
alpha.ToString("F2"));
Console.WriteLine("\nCreate and train GPR model ");
GPRCholesky model =
new GPRCholesky(theta, lenScale, alpha);
model.Train(trainX, trainY);
Console.WriteLine("Done ");
// 3. evaluate model
Console.WriteLine("\nEvaluating model accuracy ");
double trainAcc = model.Accuracy(trainX, trainY, 0.10);
double testAcc = model.Accuracy(testX, testY, 0.10);
Console.WriteLine("Train acc (within 0.10) = " +
trainAcc.ToString("F4"));
Console.WriteLine("Test acc (within 0.10) = " +
testAcc.ToString("F4"));
// 4. use model
double[] x = trainX[0];
Console.WriteLine("\nPredicting for x = ");
MatUtils.VecShow(x, 4, 9);
double[] meanStd = model.Predict(x);
double predY = meanStd[0];
double std = meanStd[1];
Console.WriteLine("\npredicted y = " +
predY.ToString("F4"));
Console.WriteLine("prediction std dev = " +
std.ToString("F4"));
Console.WriteLine("\nEnd demo ");
Console.ReadLine();
} // Main()
} // Program class
// ========================================================
public class GPRCholesky
{
public double theta = 0.0;
public double lenScale = 0.0;
public double noise = 0.0;
public double[][] trainX;
public double[] trainY;
public double[][] invCovarMat;
public GPRCholesky(double theta, double lenScale,
double noise)
{
this.theta = theta; // constant kernel
this.lenScale = lenScale; // RBF
this.noise = noise;
} // ctor()
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
this.trainX = trainX; // by ref
this.trainY = trainY;
double[][] covarMat =
this.ComputeCovarMat(this.trainX, this.trainX);
int n = covarMat.Length;
for (int i = 0; i "lt" n; ++i)
covarMat[i][i] += this.noise; // aka alpha, lambda
this.invCovarMat =
MatUtils.MatInverseCholesky(covarMat);
}
// ------------------------------------------------------
public double[][] Predict(double[][] X)
{
double[][] result = new double[2][]; // means, stds
// X = to predict, X* = train X, Y* = trainY as matrix
// means = K(X,X*) * inv(K(X*,X*)) * Y*
double[][] a =
this.ComputeCovarMat(X, this.trainX); // (n,200)
double[][] b =
MatUtils.MatProduct(a, this.invCovarMat); // (n,200)
double[][] c = MatUtils.VecToMat(this.trainY,
trainY.Length, 1); // (200,1)
double[][] d = MatUtils.MatProduct(b, c); // (n,1)
double[] means = MatUtils.MatToVec(d); // (n)
// sigmas matrix = K(X,X) - [ a * invCoverMat * (a)T ]
double[][] e = this.ComputeCovarMat(X, X);
double[][] f = MatUtils.MatProduct(a, this.invCovarMat);
double[][] g =
MatUtils.MatProduct(f, MatUtils.MatTranspose(a));
double[][] h = MatUtils.MatSubtract(e, g);
int n = h.Length;
double[] stds = new double[n]; // sqrt of diag elements
for (int i = 0; i "lt" n; ++i)
stds[i] = Math.Sqrt(h[i][i]);
result[0] = means;
result[1] = stds;
return result;
}
// ------------------------------------------------------
public double[] Predict(double[] x)
{
// convert x to matrix and call matrix Predict(X)
double[] result = new double[2]; // mean at [0]
double[][] X = MatUtils.VecToMat(x, 1, x.Length);
double[][] results = this.Predict(X);
result[0] = results[0][0];
result[1] = results[1][0];
return result;
}
// ------------------------------------------------------
public double Accuracy(double[][] dataX, double[] dataY,
double pctClose)
{
int numCorrect = 0; int numWrong = 0;
// get all predictions
double[][] results = this.Predict(dataX);
double[] y_preds = results[0];
for (int i = 0; i "lt" y_preds.Length; ++i)
{
if (Math.Abs(y_preds[i] - dataY[i])
"lt" Math.Abs(pctClose * dataY[i]))
numCorrect += 1;
else
numWrong += 1;
}
return (numCorrect * 1.0) / (numCorrect + numWrong);
}
// ------------------------------------------------------
private double[][] ComputeCovarMat(double[][] X1,
double[][] X2)
{
int n1 = X1.Length; int n2 = X2.Length;
double[][] result = MatUtils.MatMake(n1, n2);
for (int i = 0; i "lt" n1; ++i)
for (int j = 0; j "lt" n2; ++j)
result[i][j] = this.KernelFunc(X1[i], X2[j]);
return result;
}
// ------------------------------------------------------
private double KernelFunc(double[] x1, double[] x2)
{
// constant * RBF
int dim = x1.Length;
double sum = 0.0; // Euclidean distance squared
for (int i = 0; i "lt" dim; ++i)
sum += (x1[i] - x2[i]) * (x1[i] - x2[i]);
double term =
-1.0 / (2 * (this.lenScale * this.lenScale));
return this.theta * Math.Exp(sum * term);
}
} // class GPRCholesky
// ========================================================
public class MatUtils
{
// ------------------------------------------------------
public static double[][] MatLoad(string fn,
int[] usecols, char sep, string comment)
{
List"lt"double[]"gt" result = new List"lt"double[]"gt"();
string line = "";
FileStream ifs = new FileStream(fn, FileMode.Open);
StreamReader sr = new StreamReader(ifs);
while ((line = sr.ReadLine()) != null)
{
if (line.StartsWith(comment) == true)
continue;
string[] tokens = line.Split(sep);
List"lt"double"gt" lst = new List"lt"double"gt"();
for (int j = 0; j "lt" usecols.Length; ++j)
lst.Add(double.Parse(tokens[usecols[j]]));
double[] row = lst.ToArray();
result.Add(row);
}
sr.Close(); ifs.Close();
return result.ToArray();
}
// ------------------------------------------------------
public static double[] MatToVec(double[][] mat)
{
int nRows = mat.Length;
int nCols = mat[0].Length;
double[] result = new double[nRows * nCols];
int k = 0;
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[k++] = mat[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] VecToMat(double[] vec,
int nRows, int nCols)
{
double[][] result = MatMake(nRows, nCols);
int k = 0;
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[i][j] = vec[k++];
return result;
}
// ------------------------------------------------------
public static double[][] MatMake(int nRows, int nCols)
{
double[][] result = new double[nRows][];
for (int i = 0; i "lt" nRows; ++i)
result[i] = new double[nCols];
return result;
}
// ------------------------------------------------------
public static double[][] MatProduct(double[][] matA,
double[][] matB)
{
int aRows = matA.Length;
int aCols = matA[0].Length;
int bRows = matB.Length;
int bCols = matB[0].Length;
if (aCols != bRows)
throw new Exception("Non-conformable matrices");
double[][] result = MatMake(aRows, bCols);
for (int i = 0; i "lt" aRows; ++i) // each row of A
for (int j = 0; j "lt" bCols; ++j) // each col of B
for (int k = 0; k "lt" aCols; ++k)
result[i][j] += matA[i][k] * matB[k][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatSubtract(double[][] matA,
double[][] matB)
{
// matA - matB
int nRows = matA.Length; int nCols = matA[0].Length;
double[][] result = MatMake(nRows, nCols);
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[i][j] = matA[i][j] - matB[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatTranspose(double[][] m)
{
int nr = m.Length; int nc = m[0].Length;
double[][] result = MatMake(nc, nr); // note
for (int i = 0; i "lt" nr; ++i)
for (int j = 0; j "lt" nc; ++j)
result[j][i] = m[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatCopyOf(double[][] m)
{
int nRows = m.Length; int nCols = m[0].Length;
double[][] result = MatMake(nRows, nCols);
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[i][j] = m[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatIdentity(int n)
{
double[][] result = MatMake(n, n);
for (int i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
return result;
}
// ------------------------------------------------------
public static double[][] MatScalarMult(double[][] m,
double u)
{
int nRows = m.Length; int nCols = m[0].Length;
double[][] result = MatMake(nRows, nCols);
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[i][j] = u * m[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatInverseCholesky(double[][] A)
{
double[][] L = MatCholeskyDecomp(A);
double[][] result = MatInverseFromCholeskyLower(L);
return result;
// ****************************************************
// nested helpers: MatCholeskyDecomp(),
// MatInverseFromCholeskyLower()
// ****************************************************
static double[][] MatCholeskyDecomp(double[][] m)
{
int n = m.Length;
double[][] result = MatMake(n, n); // all 0.0
for (int i = 0; i "lt" n; ++i)
{
for (int j = 0; j "lte" i; ++j)
{
double sum = 0.0;
for (int k = 0; k "lt" j; ++k)
sum += result[i][k] * result[j][k];
if (i == j)
{
double tmp = m[i][i] - sum;
if (tmp "lt" 0.0)
throw new
Exception("MatCholesky fatal error ");
result[i][j] = Math.Sqrt(tmp);
}
else
{
if (result[j][j] == 0.0)
throw new
Exception("MatCholesky fatal error ");
result[i][j] =
(1.0 / result[j][j] * (m[i][j] - sum));
}
} // j
} // i
return result;
} // MatCholeskyDecomp()
// ****************************************************
static double[][]
MatInverseFromCholeskyLower(double[][] L)
{
// L is a lower triangular result of Cholesky decomp
int n = L.Length;
double[][] result = MatIdentity(n);
for (int k = 0; k "lt" n; ++k)
{
for (int j = 0; j "lt" n; j++)
{
for (int i = 0; i "lt" k; i++)
{
result[k][j] -= result[i][j] * L[k][i];
}
result[k][j] /= L[k][k];
}
}
for (int k = n - 1; k "gte" 0; --k)
{
for (int j = 0; j "lt" n; j++)
{
for (int i = k + 1; i "lt" n; i++)
{
result[k][j] -= result[i][j] * L[i][k];
}
result[k][j] /= L[k][k];
}
}
return result;
} // MatInverseFromCholeskyLower()
// ****************************************************
} // MatInverseCholesky()
// ------------------------------------------------------
public static void MatShow(double[][] m, int dec, int wid)
{
int nRows = m.Length; int nCols = m[0].Length;
double small = 1.0 / Math.Pow(10, dec);
for (int i = 0; i "lt" nRows; ++i)
{
for (int j = 0; j "lt" nCols; ++j)
{
double v = m[i][j];
if (Math.Abs(v) "lt" small) v = 0.0;
Console.Write(v.ToString("F" + dec).
PadLeft(wid));
}
Console.WriteLine("");
}
}
// ------------------------------------------------------
public static void VecShow(double[] vec, int dec, int wid)
{
for (int i = 0; i "lt" vec.Length; ++i)
Console.Write(vec[i].ToString("F" + dec).PadLeft(wid));
Console.WriteLine("");
}
// ------------------------------------------------------
} // class MatUtils
// ========================================================
} // ns
Training data:
# synthetic_train_200.txt # -0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.4840 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.1568 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, 0.8054 0.9365, -0.3732, 0.3846, 0.7528, 0.7892, 0.1345 -0.8299, -0.9219, -0.6603, 0.7563, -0.8033, 0.7955 0.0663, 0.3838, -0.3690, 0.3730, 0.6693, 0.3206 -0.9634, 0.5003, 0.9777, 0.4963, -0.4391, 0.7377 -0.1042, 0.8172, -0.4128, -0.4244, -0.7399, 0.4801 -0.9613, 0.3577, -0.5767, -0.4689, -0.0169, 0.6861 -0.7065, 0.1786, 0.3995, -0.7953, -0.1719, 0.5569 0.3888, -0.1716, -0.9001, 0.0718, 0.3276, 0.2500 0.1731, 0.8068, -0.7251, -0.7214, 0.6148, 0.3297 -0.2046, -0.6693, 0.8550, -0.3045, 0.5016, 0.2129 0.2473, 0.5019, -0.3022, -0.4601, 0.7918, 0.2613 -0.1438, 0.9297, 0.3269, 0.2434, -0.7705, 0.5171 0.1568, -0.1837, -0.5259, 0.8068, 0.1474, 0.3307 -0.9943, 0.2343, -0.3467, 0.0541, 0.7719, 0.5581 0.2467, -0.9684, 0.8589, 0.3818, 0.9946, 0.1092 -0.6553, -0.7257, 0.8652, 0.3936, -0.8680, 0.7018 0.8460, 0.4230, -0.7515, -0.9602, -0.9476, 0.1996 -0.9434, -0.5076, 0.7201, 0.0777, 0.1056, 0.5664 0.9392, 0.1221, -0.9627, 0.6013, -0.5341, 0.1533 0.6142, -0.2243, 0.7271, 0.4942, 0.1125, 0.1661 0.4260, 0.1194, -0.9749, -0.8561, 0.9346, 0.2230 0.1362, -0.5934, -0.4953, 0.4877, -0.6091, 0.3810 0.6937, -0.5203, -0.0125, 0.2399, 0.6580, 0.1460 -0.6864, -0.9628, -0.8600, -0.0273, 0.2127, 0.5387 0.9772, 0.1595, -0.2397, 0.1019, 0.4907, 0.1611 0.3385, -0.4702, -0.8673, -0.2598, 0.2594, 0.2270 -0.8669, -0.4794, 0.6095, -0.6131, 0.2789, 0.4700 0.0493, 0.8496, -0.4734, -0.8681, 0.4701, 0.3516 0.8639, -0.9721, -0.5313, 0.2336, 0.8980, 0.1412 0.9004, 0.1133, 0.8312, 0.2831, -0.2200, 0.1782 0.0991, 0.8524, 0.8375, -0.2102, 0.9265, 0.2150 -0.6521, -0.7473, -0.7298, 0.0113, -0.9570, 0.7422 0.6190, -0.3105, 0.8802, 0.1640, 0.7577, 0.1056 0.6895, 0.8108, -0.0802, 0.0927, 0.5972, 0.2214 0.1982, -0.9689, 0.1870, -0.1326, 0.6147, 0.1310 -0.3695, 0.7858, 0.1557, -0.6320, 0.5759, 0.3773 -0.1596, 0.3581, 0.8372, -0.9992, 0.9535, 0.2071 -0.2468, 0.9476, 0.2094, 0.6577, 0.1494, 0.4132 0.1737, 0.5000, 0.7166, 0.5102, 0.3961, 0.2611 0.7290, -0.3546, 0.3416, -0.0983, -0.2358, 0.1332 -0.3652, 0.2438, -0.1395, 0.9476, 0.3556, 0.4170 -0.6029, -0.1466, -0.3133, 0.5953, 0.7600, 0.4334 -0.4596, -0.4953, 0.7098, 0.0554, 0.6043, 0.2775 0.1450, 0.4663, 0.0380, 0.5418, 0.1377, 0.2931 -0.8636, -0.2442, -0.8407, 0.9656, -0.6368, 0.7429 0.6237, 0.7499, 0.3768, 0.1390, -0.6781, 0.2185 -0.5499, 0.1850, -0.3755, 0.8326, 0.8193, 0.4399 -0.4858, -0.7782, -0.6141, -0.0008, 0.4572, 0.4197 0.7033, -0.1683, 0.2334, -0.5327, -0.7961, 0.1776 0.0317, -0.0457, -0.6947, 0.2436, 0.0880, 0.3345 0.5031, -0.5559, 0.0387, 0.5706, -0.9553, 0.3107 -0.3513, 0.7458, 0.6894, 0.0769, 0.7332, 0.3170 0.2205, 0.5992, -0.9309, 0.5405, 0.4635, 0.3532 -0.4806, -0.4859, 0.2646, -0.3094, 0.5932, 0.3202 0.9809, -0.3995, -0.7140, 0.8026, 0.0831, 0.1600 0.9495, 0.2732, 0.9878, 0.0921, 0.0529, 0.1289 -0.9476, -0.6792, 0.4913, -0.9392, -0.2669, 0.5966 0.7247, 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0.2215 -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, 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-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
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