My personal C# machine learning library has code to perform Gaussian process regression (GPR). That code computes the inverse of a covariance matrix using basic LUP decomposition (Crout’s algorithm). Just to explore, I figured I’d look at using matrix inverse using Cholesky decomposition.
Cholesky decomposition/inverse is possible because a covariance is (usually) square, symmetric, positive definite — a requirement to use Cholesky. Cholesky is simpler than LUP but could be more brittle because if a non-PD covariance matrix arises, the matrix inverse will fail.
For my demo, I generated synthetic data using a 6-10-1 PyTorch neural network with random weights. Each data item has six predictor variables with values between -1.0 and +1.0. The target y values are between 0 and 1. There are 200 training items and 40 test items. The data looks like:
-0.2065, 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.4374 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, -0.6038, 0.8437 0.7528, 0.7892, -0.8299, -0.9219, -0.6603, 0.7563, 0.8765 -0.8033, -0.1578, 0.9158, 0.0663, 0.3838, -0.3690, 0.5514 . . .
My GPR demo worked as expected, meaning the new Cholesky inverse version gave identical results as the LUP inverse version. I didn’t do any formal performance measurements, but the Cholesky inverse version of my demo seemed to run a little bit faster than the LUP inverse version. I also dropped in a QR inverse function, but that version was significantly slower than the Cholesky and LUP inverse versions.
Good fun.
I went to a Catholic boys high school (Servite HS in Anaheim, CA) and was required to take four years of Latin. It was very difficult for me but very interesting. By the time we all became Seniors, we were reading things like “De Bello Gallico” (the Gallic Wars), Julius Caesar’s description of his campaigns against the Gauls in the 50s BC. Reading ancient books in Latin really gave me a feel for what people might have been like a couple of thousand years ago.
Many regression problems predict a money value of some sort — the price of a house, the income of a employee, and so on. Here’s a nice image from Wikipedia.
Demo code. Very long! Replace “lt”, “gt”, “lte”, “gte” with Boolean operator symbols.
using System;
using System.IO;
namespace GaussianProcessRegression
{
internal class GPRProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nGaussian Process Regression" +
" using Cholesky inverse ");
// 1. Load data
Console.WriteLine("\nLoading data from file ");
string trainFile =
"..\\..\\..\\Data\\synthetic_train_200.txt";
double[][] trainX = Utils.MatLoad(trainFile,
new int[] { 0, 1, 2, 3, 4, 5 }, ',', "#");
double[] trainY = Utils.MatToVec(Utils.MatLoad(trainFile,
new int[] { 6 }, ',', "#"));
string testFile =
"..\\..\\..\\Data\\synthetic_test_40.txt";
double[][] testX = Utils.MatLoad(testFile,
new int[] { 0, 1, 2, 3, 4, 5 }, ',', "#");
double[] testY = Utils.MatToVec(Utils.MatLoad(testFile,
new int[] { 6 }, ',', "#"));
Console.WriteLine("Done ");
Console.WriteLine("\nFirst four X predictors: ");
for (int i = 0; i "lt" 4; ++i)
Utils.VecShow(trainX[i], 4, 8);
Console.WriteLine("\nFirst four y targets: ");
for (int i = 0; i "lt" 4; ++i)
Console.WriteLine(trainY[i].ToString("F4").PadLeft(8));
// 2. explore alpha, theta, lenScale hyperparameters
Console.WriteLine("\nExploring hyperparameters ");
double[] alphas = new double[] { 0.01, 0.05 };
double[] thetas = new double[] { 0.50, 1.00 };
double[] lenScales = new double[] { 0.60, 1.00, 2.00 };
double trainRMSE = 0.0;
double testRMSE = 0.0;
double trainAcc = 0.0;
double testAcc = 0.0;
foreach (double t in thetas)
{
foreach (double s in lenScales)
{
foreach (double a in alphas)
{
GPR gprModel = new GPR(t, s, a, trainX, trainY);
trainRMSE = gprModel.RootMSE(trainX, trainY);
testRMSE = gprModel.RootMSE(testX, testY);
trainAcc = gprModel.Accuracy(trainX,
trainY, 0.10);
testAcc = gprModel.Accuracy(testX,
testY, 0.10);
Console.Write(" theta = " +
t.ToString("F2"));
Console.Write(" lenScale = " +
s.ToString("F2"));
Console.Write(" alpha = " +
a.ToString("F2"));
Console.Write(" | train RMSE = " +
trainRMSE.ToString("F4"));
Console.Write(" test RMSE = " +
testRMSE.ToString("F4"));
Console.Write(" train acc = " +
trainAcc.ToString("F4"));
Console.Write(" test acc = " +
testAcc.ToString("F4"));
Console.WriteLine("");
}
}
}
// 3. create and train model
Console.WriteLine("\nCreating and training GPR model ");
double theta = 0.50; // "constant kernel"
double lenScale = 2.0; // RBF parameter
double alpha = 0.01; // noise
Console.WriteLine("Setting theta = " +
theta.ToString("F2") +
", lenScale = " +
lenScale.ToString("F2") +
", alpha = " + alpha.ToString("F2"));
GPR model = new GPR(theta, lenScale, alpha,
trainX, trainY); // create and train
// 4. evaluate model
Console.WriteLine("\nEvaluate model acc" +
" (within 0.10 of true)");
trainAcc = model.Accuracy(trainX, trainY, 0.10);
testAcc = model.Accuracy(testX, testY, 0.10);
Console.WriteLine("Train acc = " +
trainAcc.ToString("F4"));
Console.WriteLine("Test acc = " +
testAcc.ToString("F4"));
// 5. use model for previously unseen data
Console.WriteLine("\nPredicting for (0.1, 0.2, 0.3," +
" 0.4, 0.5, 0.6) ");
double[][] X = Utils.MatCreate(1, 6);
X[0] = new double[] { 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 };
double[][] results = model.Predict(X);
Console.WriteLine("Predicted y value: ");
double[] means = results[0];
Utils.VecShow(means, 4, 8);
Console.WriteLine("Predicted std: ");
double[] stds = results[1];
Utils.VecShow(stds, 4, 8);
Console.WriteLine("\nEnd GPR demo ");
Console.ReadLine();
} // Main()
} // Program
public class GPR
{
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 GPR(double theta, double lenScale,
double noise, double[][] trainX, double[] trainY)
{
this.theta = theta;
this.lenScale = lenScale;
this.noise = noise;
this.trainX = trainX;
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 =
// Utils.MatInverseLUP(covarMat); // basic
// this.invCovarMat =
// Utils.MatInverseQR(covarMat); // slow, robust
this.invCovarMat =
Utils.MatInverseCholesky(covarMat); // fast, brittle
} // ctor()
public double[][] ComputeCovarMat(double[][] X1,
double[][] X2)
{
int n1 = X1.Length; int n2 = X2.Length;
double[][] result = Utils.MatCreate(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;
}
public 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 / (2 * (this.lenScale * this.lenScale));
return this.theta * Math.Exp(sum * term);
}
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);
double[][] b =
Utils.MatProduct(a, this.invCovarMat);
double[][] c = Utils.VecToMat(this.trainY,
trainY.Length, 1); // (200,1)
double[][] d = Utils.MatProduct(b, c);
double[] means = Utils.MatToVec(d);
// sigmas matrix = K(X,X) - [ a * invCoverMat * (a)T ]
double[][] e = this.ComputeCovarMat(X, X);
double[][] f = Utils.MatProduct(a, this.invCovarMat);
double[][] g =
Utils.MatProduct(f, Utils.MatTranspose(a));
double[][] h = Utils.MatDifference(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;
} // Predict()
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);
}
public double RootMSE(double[][] dataX, double[] dataY)
{
// get all predictions
double[][] results = this.Predict(dataX);
double[] y_preds = results[0]; // no need stds
double sumSquaredErr = 0.0;
for (int i = 0; i "lt" y_preds.Length; ++i)
sumSquaredErr += (y_preds[i] - dataY[i]) *
(y_preds[i] - dataY[i]);
return Math.Sqrt(sumSquaredErr / dataY.Length);
}
} // class GPR
public class Utils
{
public static double[][] VecToMat(double[] vec,
int nRows, int nCols)
{
double[][] result = MatCreate(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[][] MatCreate(int rows,
int cols)
{
double[][] result = new double[rows][];
for (int i = 0; i "lt" rows; ++i)
result[i] = new double[cols];
return result;
}
public static double[] MatToVec(double[][] m)
{
int rows = m.Length;
int cols = m[0].Length;
double[] result = new double[rows * cols];
int k = 0;
for (int i = 0; i "lt" rows; ++i)
for (int j = 0; j "lt" cols; ++j)
{
result[k++] = m[i][j];
}
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 = MatCreate(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[][] MatTranspose(double[][] m)
{
int nr = m.Length;
int nc = m[0].Length;
double[][] result = MatCreate(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[][] MatDifference(double[][] matA,
double[][] matB)
{
int nr = matA.Length;
int nc = matA[0].Length;
double[][] result = MatCreate(nc, nr); // note
for (int i = 0; i "lt" nr; ++i)
for (int j = 0; j "lt" nc; ++j)
result[j][i] = matA[i][j] - matB[i][j];
return result;
}
// -------
//
// Cholesky decomp inverse
//
// -------
public static double[][] MatInverseCholesky(double[][] A)
{
double[][] L = MatCholeskyDecomp(A);
double[][] result = MatInverseFromCholesky(L);
return result;
}
public static double[][]
MatInverseFromCholesky(double[][] L)
{
// L is a lower triangular result of Cholesky decomp
// direct version
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;
}
public static double[][] MatIdentity(int n)
{
double[][] result = MatCreate(n, n);
for (int i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
return result;
}
static double[][] MatCholeskyDecomp(double[][] m)
{
// Cholesky decomposition
// m is square, symmetric, positive definite
int n = m.Length;
double[][] result = MatCreate(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;
} // MatCholesky
// -------
static int NumNonCommentLines(string fn,
string comment)
{
int ct = 0;
string line = "";
FileStream ifs = new FileStream(fn,
FileMode.Open);
StreamReader sr = new StreamReader(ifs);
while ((line = sr.ReadLine()) != null)
if (line.StartsWith(comment) == false)
++ct;
sr.Close(); ifs.Close();
return ct;
}
public static double[][] MatLoad(string fn,
int[] usecols, char sep, string comment)
{
// count number of non-comment lines
int nRows = NumNonCommentLines(fn, comment);
int nCols = usecols.Length;
double[][] result = MatCreate(nRows, nCols);
string line = "";
string[] tokens = null;
FileStream ifs = new FileStream(fn, FileMode.Open);
StreamReader sr = new StreamReader(ifs);
int i = 0;
while ((line = sr.ReadLine()) != null)
{
if (line.StartsWith(comment) == true)
continue;
tokens = line.Split(sep);
for (int j = 0; j "lt" nCols; ++j)
{
int k = usecols[j]; // into tokens
result[i][j] = double.Parse(tokens[k]);
}
++i;
}
sr.Close(); ifs.Close();
return result;
}
public static void MatShow(double[][] m,
int dec, int wid)
{
for (int i = 0; i "lt" m.Length; ++i)
{
for (int j = 0; j "lt" m[0].Length; ++j)
{
double v = m[i][j];
if (Math.Abs(v) "lt" 1.0e-8) v = 0.0; // hack
Console.Write(v.ToString("F" +
dec).PadLeft(wid));
}
Console.WriteLine("");
}
}
public static void VecShow(int[] vec, int wid)
{
for (int i = 0; i "lt" vec.Length; ++i)
Console.Write(vec[i].ToString().PadLeft(wid));
Console.WriteLine("");
}
public static void VecShow(double[] vec,
int dec, int wid)
{
for (int i = 0; i "lt" vec.Length; ++i)
{
double x = vec[i];
if (Math.Abs(x) "lt" 1.0e-8) x = 0.0; // hack
Console.Write(x.ToString("F" +
dec).PadLeft(wid));
}
Console.WriteLine("");
}
} // class Utils
} // ns
Training data:
# synthetic_train_200.txt # -0.1660, 0.4406, -0.9998, -0.3953, -0.7065, -0.8153, 0.9386 -0.2065, 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.4374 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, -0.6038, 0.8437 0.7528, 0.7892, -0.8299, -0.9219, -0.6603, 0.7563, 0.8765 -0.8033, -0.1578, 0.9158, 0.0663, 0.3838, -0.3690, 0.5514 -0.9634, 0.5003, 0.9777, 0.4963, -0.4391, 0.5786, 0.0463 -0.7935, -0.1042, 0.8172, -0.4128, -0.4244, -0.7399, 0.6770 -0.0169, -0.8933, 0.1482, -0.7065, 0.1786, 0.3995, 0.8482 -0.7953, -0.1719, 0.3888, -0.1716, -0.9001, 0.0718, 0.5987 0.8892, 0.1731, 0.8068, -0.7251, -0.7214, 0.6148, 0.5553 -0.2046, -0.6693, 0.8550, -0.3045, 0.5016, 0.4520, 0.5287 0.5019, -0.3022, -0.4601, 0.7918, -0.1438, 0.9297, 0.5783 0.3269, 0.2434, -0.7705, 0.8990, -0.1002, 0.1568, 0.6709 0.8068, 0.1474, -0.9943, 0.2343, -0.3467, 0.0541, 0.8099 0.7719, -0.2855, 0.8171, 0.2467, -0.9684, 0.8589, 0.1221 0.8652, 0.3936, -0.8680, 0.5109, 0.5078, 0.8460, 0.8698 0.4230, -0.7515, -0.9602, -0.9476, -0.9434, -0.5076, 0.9699 0.1056, 0.6841, -0.7517, -0.4416, 0.1715, 0.9392, 0.9652 0.1221, -0.9627, 0.6013, -0.5341, 0.6142, -0.2243, 0.7474 0.1125, -0.7271, -0.8802, -0.7573, -0.9109, -0.7850, 0.9604 -0.5486, 0.4260, 0.1194, -0.9749, -0.8561, 0.9346, 0.7454 -0.4953, 0.4877, -0.6091, 0.1627, 0.9400, 0.6937, 0.9583 -0.5203, -0.0125, 0.2399, 0.6580, -0.6864, -0.9628, 0.5269 0.2127, 0.1377, -0.3653, 0.9772, 0.1595, -0.2397, 0.6514 0.1019, 0.4907, 0.3385, -0.4702, -0.8673, -0.2598, 0.6933 0.5055, -0.8669, -0.4794, 0.6095, -0.6131, 0.2789, 0.5812 0.0493, 0.8496, -0.4734, -0.8681, 0.4701, 0.5444, 0.9899 0.9004, 0.1133, 0.8312, 0.2831, -0.2200, -0.0280, 0.2632 0.2086, 0.0991, 0.8524, 0.8375, -0.2102, 0.9265, 0.0389 -0.7298, 0.0113, -0.9570, 0.8959, 0.6542, -0.9700, 0.9881 -0.6476, -0.3359, -0.7380, 0.6190, -0.3105, 0.8802, 0.7334 0.6895, 0.8108, -0.0802, 0.0927, 0.5972, -0.4286, 0.8753 -0.0195, 0.1982, -0.9689, 0.1870, -0.1326, 0.6147, 0.9221 0.1557, -0.6320, 0.5759, 0.2241, -0.8922, -0.1596, 0.3223 0.3581, 0.8372, -0.9992, 0.9535, -0.2468, 0.9476, 0.6502 0.1494, 0.2562, -0.4288, 0.1737, 0.5000, 0.7166, 0.9090 0.5102, 0.3961, 0.7290, -0.3546, 0.3416, -0.0983, 0.8050 -0.1970, -0.3652, 0.2438, -0.1395, 0.9476, 0.3556, 0.8670 -0.6029, -0.1466, -0.3133, 0.5953, 0.7600, 0.8077, 0.8145 -0.4953, 0.7098, 0.0554, 0.6043, 0.1450, 0.4663, 0.4205 0.0380, 0.5418, 0.1377, -0.0686, -0.3146, -0.8636, 0.8325 0.9656, -0.6368, 0.6237, 0.7499, 0.3768, 0.1390, 0.2422 -0.6781, -0.0662, -0.3097, -0.5499, 0.1850, -0.3755, 0.9766 -0.6141, -0.0008, 0.4572, -0.5836, -0.5039, 0.7033, 0.6395 -0.1683, 0.2334, -0.5327, -0.7961, 0.0317, -0.0457, 0.9878 0.0880, 0.3083, -0.7109, 0.5031, -0.5559, 0.0387, 0.6264 0.5706, -0.9553, -0.3513, 0.7458, 0.6894, 0.0769, 0.8789 -0.8025, 0.3026, 0.4070, 0.2205, 0.5992, -0.9309, 0.8814 0.5405, 0.4635, -0.4806, -0.4859, 0.2646, -0.3094, 0.9626 0.5655, 0.9809, -0.3995, -0.7140, 0.8026, 0.0831, 0.9948 0.9495, 0.2732, 0.9878, 0.0921, 0.0529, -0.7291, 0.4988 -0.6792, 0.4913, -0.9392, -0.2669, 0.7247, 0.3854, 0.9909 0.3819, -0.6227, -0.1162, 0.1632, 0.9795, -0.5922, 0.9609 0.5003, -0.0860, -0.8861, 0.0170, -0.5761, 0.5972, 0.8313 -0.4053, -0.9448, 0.1869, 0.6877, -0.2380, 0.4997, 0.4024 0.9189, 0.6079, -0.9354, 0.4188, -0.0700, 0.8951, 0.7782 -0.5571, -0.4659, -0.8371, -0.1428, -0.7820, 0.2676, 0.8987 0.5324, -0.3151, 0.6917, -0.1425, 0.6480, 0.2530, 0.6724 -0.7132, -0.8432, -0.9633, -0.8666, -0.0828, -0.7733, 1.0051 -0.0952, -0.0998, -0.0439, -0.0520, 0.6063, -0.1952, 0.9472 0.8094, -0.9259, 0.5477, -0.7487, 0.2370, -0.9793, 0.8873 0.9024, 0.8108, 0.5919, 0.8305, -0.7089, -0.6845, 0.1543 -0.6247, 0.2450, 0.8116, 0.9799, 0.4222, 0.4636, 0.1227 -0.5003, -0.6531, -0.7611, 0.6252, -0.7064, -0.4714, 0.7072 0.6382, -0.3788, 0.9648, -0.4667, 0.0673, -0.3711, 0.6024 -0.1328, 0.0246, 0.8778, -0.9381, 0.4338, 0.7820, 0.7793 -0.9454, 0.0441, -0.3480, 0.7190, 0.1170, 0.3805, 0.7989 -0.4198, -0.9813, 0.1535, -0.3771, 0.0345, 0.8328, 0.7482 -0.1471, -0.5052, -0.2574, 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-0.1439, 0.1985, 0.6999, 0.5022, 0.1587, 0.8494, 0.1350 0.2473, -0.9040, -0.4308, -0.8779, 0.4070, 0.3369, 0.9716 -0.2428, -0.6236, 0.4940, -0.3192, 0.5906, -0.0242, 0.8032 0.2885, -0.2987, -0.5416, -0.1322, -0.2351, -0.0604, 0.9201 0.9590, -0.2712, 0.5488, 0.1055, 0.7783, -0.2901, 0.7501 -0.9129, 0.9015, 0.1128, -0.2473, 0.9901, -0.8833, 0.9796 0.0334, -0.9378, 0.1424, -0.6391, 0.2619, 0.9618, 0.7955 0.4169, 0.5549, -0.0103, 0.0571, -0.6984, -0.2612, 0.5340 -0.7156, 0.4538, -0.0460, -0.1022, 0.7720, 0.0552, 0.9432 -0.8560, -0.1637, -0.9485, -0.4177, 0.0070, 0.9319, 0.9602 -0.7812, 0.3461, -0.0001, 0.5542, -0.7128, -0.8336, 0.6613 -0.6166, 0.5356, -0.4194, -0.5662, -0.9666, -0.2027, 0.9244 -0.2378, 0.3187, -0.8582, -0.6948, -0.9668, -0.7724, 0.9198 -0.3579, 0.1158, 0.9869, 0.6690, 0.3992, 0.8365, 0.1103 -0.9205, -0.8593, -0.0520, -0.3017, 0.8745, -0.0209, 0.9639 -0.1067, 0.7541, -0.4928, -0.4524, -0.3433, 0.0951, 0.9502 -0.5597, 0.3429, -0.7144, -0.8118, 0.7404, -0.5263, 1.0098 0.0516, 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0.7590, -0.3580, -0.7541, 0.2429 -0.7465, 0.1796, -0.9279, -0.5996, 0.5766, -0.9758, 1.0155 -0.3933, -0.9572, 0.9950, 0.1641, -0.4132, 0.8579, 0.1326 0.1757, -0.4717, -0.3894, -0.2567, -0.5111, 0.1691, 0.8829 0.3917, -0.8561, 0.9422, 0.5061, 0.6123, 0.5033, 0.2117 -0.1087, 0.3449, -0.1025, 0.4086, 0.3633, 0.3943, 0.7479 0.2372, -0.6980, 0.5216, 0.5621, 0.8082, -0.5325, 0.6879 -0.3589, 0.6310, 0.2271, 0.5200, -0.1447, -0.8011, 0.6644 -0.7699, -0.2532, -0.6123, 0.6415, 0.1993, 0.3777, 0.8791 -0.5298, -0.0768, -0.6028, -0.9490, 0.4588, 0.4498, 0.9867 -0.3392, 0.6870, -0.1431, 0.7294, 0.3141, 0.1621, 0.5972 0.7889, -0.3900, 0.7419, 0.8175, -0.3403, 0.3661, 0.0714 0.7984, -0.8486, 0.7572, -0.6183, 0.6995, 0.3342, 0.5188 0.2707, 0.6956, 0.6437, 0.2565, 0.9126, 0.1798, 0.6748 -0.6043, -0.1413, -0.3265, 0.9839, -0.2395, 0.9854, 0.3455 -0.8509, -0.2594, -0.7532, 0.2690, -0.1722, 0.9818, 0.8388 0.8599, -0.7015, -0.2102, -0.0768, 0.1219, 0.5607, 0.8489 -0.4760, 0.8216, -0.9555, 0.6422, -0.6231, 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0.2992, 0.8629, -0.8505, -0.4464, 0.8385, 0.5300, 0.9865 0.1995, 0.6659, 0.7921, 0.9454, 0.9970, -0.7207, 0.3848 -0.3066, -0.2927, -0.4923, 0.8220, 0.4513, -0.9481, 0.9805 -0.0770, -0.4374, -0.9421, 0.7694, 0.5420, -0.3405, 0.9735 -0.3842, 0.8562, 0.9538, 0.0471, 0.9039, 0.7760, 0.4332 0.0361, -0.2545, 0.4207, -0.0887, 0.2104, 0.9808, 0.5813 -0.8220, -0.6302, 0.0537, -0.1658, 0.6013, 0.8664, 0.8076 -0.6443, 0.7201, 0.9148, 0.9189, -0.9243, -0.8848, 0.1623 -0.2880, 0.9074, -0.0461, -0.4435, 0.0060, 0.2867, 0.9211 -0.7775, 0.5161, 0.7039, 0.6885, 0.7810, -0.2363, 0.3703 -0.5484, 0.9426, -0.4308, 0.8148, 0.7811, 0.8450, 0.6650
Test data:
# synthetic_test_40.txt # -0.6877, 0.7594, 0.2640, -0.5787, -0.3098, -0.6802, 0.9262 -0.6694, -0.6056, 0.3821, 0.1476, 0.7466, -0.5107, 0.8790 0.2592, -0.9311, 0.0324, 0.7265, 0.9683, -0.9803, 0.9250 -0.9049, -0.9797, -0.0196, -0.9090, -0.4433, 0.2799, 0.8999 -0.4106, -0.4607, 0.1811, -0.2389, 0.4050, -0.0078, 0.8794 -0.4259, -0.7336, 0.8742, 0.6097, 0.8761, -0.6292, 0.6253 0.8663, 0.8715, -0.4329, -0.4507, 0.1029, -0.6294, 0.8922 0.8948, -0.0124, 0.9278, 0.2899, -0.0314, 0.9354, 0.2092 -0.7136, 0.2647, 0.3238, -0.1323, -0.8813, -0.0146, 0.6075 -0.4867, -0.2171, -0.5197, 0.3729, 0.9798, -0.6451, 0.9907 0.6429, -0.5380, -0.8840, -0.7224, 0.8703, 0.7771, 0.9871 0.6999, -0.1307, -0.0639, 0.2597, -0.6839, -0.9704, 0.5031 -0.4690, -0.9691, 0.3490, 0.1029, -0.3567, 0.5604, 0.5090 -0.4154, -0.6081, -0.8241, 0.7400, -0.8236, 0.3674, 0.5618 -0.7592, -0.9786, 0.1145, 0.8142, 0.7209, -0.3231, 0.8586 0.3393, 0.6156, 0.7950, -0.0923, 0.1157, 0.0123, 0.6244 0.3840, 0.3658, 0.0406, 0.6569, 0.0116, 0.6497, 0.3585 0.9397, 0.4839, -0.4804, 0.1625, 0.9105, -0.8385, 0.9460 -0.8329, 0.2383, -0.5510, 0.5304, 0.1363, 0.3324, 0.8796 -0.8255, -0.2579, 0.3443, -0.6208, 0.7915, 0.8997, 0.8454 0.9231, 0.4602, -0.1874, 0.4875, -0.4240, -0.3712, 0.3832 0.7573, -0.4908, 0.5324, 0.8820, -0.9979, -0.0478, 0.1087 0.3141, 0.6866, -0.6325, 0.7123, -0.2713, 0.7845, 0.5467 -0.1647, -0.6616, 0.2998, -0.9260, -0.3768, -0.3530, 0.8954 0.2149, 0.3017, 0.6921, 0.8552, 0.3209, 0.1563, 0.1767 -0.6918, 0.7902, -0.3780, 0.0970, 0.3641, -0.5271, 0.9735 -0.6645, 0.0170, 0.5837, 0.3848, -0.7621, 0.8015, 0.1184 0.1069, -0.8304, -0.5951, 0.7085, 0.4119, 0.7899, 0.8076 -0.3417, 0.0560, 0.3008, 0.1886, -0.5371, -0.1464, 0.5610 0.9734, -0.8669, 0.4279, -0.3398, 0.2509, -0.4837, 0.7845 0.3020, -0.2577, -0.4104, 0.8235, 0.8850, 0.2271, 0.9088 -0.5766, 0.6603, -0.5198, 0.2632, 0.4215, 0.4848, 0.9255 -0.2195, 0.5197, 0.8059, 0.1748, -0.8192, -0.7420, 0.4317 -0.9212, -0.5169, 0.7581, 0.9470, 0.2108, 0.9525, 0.1578 -0.9131, 0.8971, -0.3774, 0.5979, 0.6213, 0.7200, 0.7684 -0.4842, 0.8689, 0.2382, 0.9709, -0.9347, 0.4503, 0.0743 0.1311, -0.0152, -0.4816, -0.3463, -0.5011, -0.5615, 0.9131 -0.8336, 0.5540, 0.0673, 0.4788, 0.0308, -0.2001, 0.7141 0.9725, -0.9435, 0.8655, 0.8617, -0.2182, -0.5711, 0.1298 0.6064, -0.4921, -0.4184, 0.8318, 0.8058, 0.0708, 0.8992
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