I like to write code every day. Writing code is a skill that must be practiced, and besides, I enjoy writing code. Over the course of my career, I saw countless colleagues move from an engineering role to an engineering-management role, and they all lost their coding ability with surprising speed.
I decided to combine quadratic regression (QR) with gradient boosting regression (GBR). The idea is simple: create a container class that holds a QR model and a GBR model. After the two component models are trained independently, the hybrid model prediction is just a weighted average of the QR prediction and the GBR prediction.
For my demo, I used one of my standard synthetic datasets. The data looks like:
-0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.4840 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.1568 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, 0.8054 . . .
The first five values on each line are predictors. The last value on each line is the target to predict. There are 200 training items and 40 test items. The data was generated by a neural network with random weights and biases. Therefore, in theory, QR should do better than GBR.
The output of a demo, with the QR model prediction weight = 0.70 and the GBR prediction weight = 0.30:
Begin C# hybrid quadratic regression + gradient boost regression Loading synthetic train (200) and test (40) data Done First three train X: -0.1660 0.4406 -0.9998 -0.3953 -0.7065 0.0776 -0.1616 0.3704 -0.5911 0.7562 -0.9452 0.3409 -0.1654 0.1174 -0.7192 First three train y: 0.4840 0.1568 0.8054 Creating hybrid quadratic regression + gradient boost regression model Setting nTrees for GBR model = 200 Setting maxDepth GBR model = 3 Setting lrnRate GBR model = 0.1000 Setting QR predict weight = 0.70 Setting GBR predict weight = 0.30 Done Training hybrid model Done QR model base weights: -0.2630 0.0354 -0.0421 0.0341 -0.1124 Evaluating hybrid model Accuracy train (within 0.10) = 0.9300 Accuracy test (within 0.10) = 0.9250 MSE train = 0.0001 MSE test = 0.0004 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 Predicted y = 0.4839 End demo
The choice of combining QR and GBR is arbitrary. Any regression techniques can be used (for example, linear regression, kernel ridge regression, support vector regression, neural network regression, random forest regression, gradient boost regression). And you can use more than two regression techniques.
In practice, I like to use regression techniques that are much different from each other (because they can handle different types of data). As a practical matter, the fewer hyperparameters to tune the better.
With QR, I used a version that needs no parameters (no learning rate, no L2 regression, etc.) For gradient boosting regression, I used absolute minimal hyperparameters: nTrees, maxDepth, and lrnRate. I used sensible defaults for minSample (2), minLeaf (1), numSplitCols (-1 = all).
The Predict() and Train() methods are simple, because all of the work is done by the component models:
public class Hybid_QR_GBR
{
. . .
public double Predict(double[] x)
{
double predQR = this.qrModel.Predict(x);
double predGBR = this.gbrModel.Predict(x);
double result =
(predQR * this.weights[0]) +
(predGBR * this.weights[1]);
return result;
}
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
this.qrModel.Train(trainX, trainY);
this.gbrModel.Train(trainX, trainY);
}
. . .
public class QuadraticRegressor { . . . }
public class QRHouseholder { . . . } // helper for QR
public class GradientBoostRegressor { . . . }
public class DecisionTreeRegressor { . . . } // help GBR
Instead of wrapping the hybrid regressor in a container class, Hybid_QR_NN_Regressor, it’s perfectly possible to just use the QuadraticRegressor class and the GradientBoostRegressor class separately, and then compute a weighted average of their Predict() methods. But putting everything into a container class keeps things more organized and easier to work with.
The scikit-learn library provides a VotingRegressor module that combines two or more scikit regression models.
Most of my friends, and me, get obsessed with all kinds of strange things. For reasons that are unknown to me, I am mildly obsessed with findling movies that feature an animal or creature that makes noises that are translated to “chittering” in the closed caption text. The are an amazing number of chittering instances in science fiction movies, but you’ll only notice these chitterings if you’re actively looking for them.
In the “Legend of Ochi” (2025), a farm girl on a fictional rural island discovers a wounded baby creature (that chitters a lot). She learns its language and works to return it to its family, all while avoiding a dangerous hunting party. Everything about this movie should make me hate it — cute creature, tough adolescent girl, evil hunters, and so on. But I really liked this movie a lot. I give it a solid B+ rating.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols (my blog editor chokes on symbols).
using System;
using System.IO;
using System.Collections.Generic;
namespace Hybrid_QR_GBR
{
internal class HybridProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin C# hybrid quadratic " +
"regression + gradient boost regression ");
// 1. load data
Console.WriteLine("\nLoading synthetic train" +
" (200) and test (40) data");
string trainFile =
"..\\..\\..\\Data\\synthetic_train_200.txt";
int[] colsX = new int[] { 0, 1, 2, 3, 4 };
int colY = 5;
double[][] trainX =
MatLoad(trainFile, colsX, ',', "#");
double[] trainY =
MatToVec(MatLoad(trainFile,
new int[] { colY }, ',', "#"));
string testFile =
"..\\..\\..\\Data\\synthetic_test_40.txt";
double[][] testX =
MatLoad(testFile, colsX, ',', "#");
double[] testY =
MatToVec(MatLoad(testFile,
new int[] { colY }, ',', "#"));
Console.WriteLine("Done ");
Console.WriteLine("\nFirst three train X: ");
for (int i = 0; i "lt" 3; ++i)
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 model
Console.WriteLine("\nCreating hybrid quadratic " +
"regression + gradient boost regression model ");
int nTrees = 200;
int maxDepth = 3;
double lrnRate = 0.10;
int seed = 0;
double[] weights = new double[] { 0.70, 0.30 };
Console.WriteLine("Setting nTrees for GBR model = " +
nTrees);
Console.WriteLine("Setting maxDepth GBR model = " +
maxDepth);
Console.WriteLine("Setting lrnRate GBR model = " +
lrnRate.ToString("F4"));
Console.WriteLine("Setting QR predict weight = " +
weights[0].ToString("F2"));
Console.WriteLine("Setting GBR predict weight = " +
weights[1].ToString("F2"));
Hybid_QR_GBR_Regressor model =
new Hybid_QR_GBR_Regressor(nTrees, maxDepth,
lrnRate, seed, weights);
Console.WriteLine("Done ");
Console.WriteLine("\nTraining hybrid model ");
model.Train(trainX, trainY); // train components Console.WriteLine("Done ");
// 3. show model weights
Console.WriteLine("\nQR model base weights: ");
int dim = trainX[0].Length;
for (int i = 0; i "lt" dim; ++i)
Console.Write(model.qrModel.weights[i].
ToString("F4").PadLeft(8));
Console.WriteLine("");
// Console.WriteLine("\nModel quadratic weights: ");
// Console.WriteLine("\nModel interaction weights: ");
// 4. evaluate model
Console.WriteLine("\nEvaluating hybrid model ");
double accTrain = model.Accuracy(trainX, trainY, 0.10);
Console.WriteLine("Accuracy train (within 0.10) = " +
accTrain.ToString("F4"));
double accTest = model.Accuracy(testX, testY, 0.10);
Console.WriteLine("Accuracy test (within 0.10) = " +
accTest.ToString("F4"));
double mseTrain = model.MSE(trainX, trainY);
Console.WriteLine("\nMSE train = " +
mseTrain.ToString("F4"));
double mseTest = model.MSE(testX, testY);
Console.WriteLine("MSE test = " +
mseTest.ToString("F4"));
// 5. use model
double[] x = trainX[0];
Console.WriteLine("\nPredicting for x = ");
VecShow(x, 4, 9);
double predY = model.Predict(x);
Console.WriteLine("\nPredicted y = " +
predY.ToString("F4"));
// 6. TODO: implement model Save() and Load()
Console.WriteLine("\nEnd demo ");
Console.ReadLine();
} // Main()
// ------------------------------------------------------
// helpers for Main(): MatLoad(), MatToVec(), VecShow()
// ------------------------------------------------------
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();
}
static double[] MatToVec(double[][] M)
{
int nRows = M.Length;
int nCols = M[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++] = M[i][j];
return result;
}
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 Program
// ========================================================
public class Hybid_QR_GBR_Regressor
{
// training parameters for QR and GBR
// QR: no parameters needed
// GBR: nTrees, maxDepth, lrnRate.
// uses minSample = 2, minLeaf = 1
public int nNeighbors;
private Random rnd; // not used w/ Pinv training
public double[] weights = new double[0]; // each model
public QuadraticRegressor qrModel;
public GradientBoostRegressor gbrModel;
// ------------------------------------------------------
public Hybid_QR_GBR_Regressor(int nTrees, int maxDepth,
double lrnRate, int seed, double[] weights)
{
this.nNeighbors = nNeighbors;
this.rnd = new Random(seed);
this.weights = weights;
this.qrModel =
new QuadraticRegressor(seed);
this.gbrModel =
new GradientBoostRegressor(nTrees: nTrees,
maxDepth: maxDepth, minSamples: 2,
minLeaf: 1, numSplitCols: -1, lrnRate: lrnRate,
seed);
}
// ------------------------------------------------------
public double Predict(double[] x)
{
double predQR = this.qrModel.Predict(x);
double predGBR = this.gbrModel.Predict(x);
double result = (predQR * this.weights[0]) +
(predGBR * this.weights[1]);
return result;
}
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
this.qrModel.Train(trainX, trainY);
this.gbrModel.Train(trainX, trainY);
}
// ------------------------------------------------------
public double MSE(double[][] dataX,
double[] dataY)
{
int n = dataX.Length;
double sum = 0.0;
for (int i = 0; i "lt" n; ++i)
{
double actualY = dataY[i];
double predY = this.Predict(dataX[i]);
sum += (actualY - predY) * (actualY - predY);
}
return sum / n;
}
// ------------------------------------------------------
public double Accuracy(double[][] dataX, double[] dataY,
double pctClose)
{
int numCorrect = 0; int numWrong = 0;
for (int i = 0; i "lt" dataX.Length; ++i)
{
double actualY = dataY[i];
double predY = this.Predict(dataX[i]);
if (Math.Abs(predY - actualY) "lte"
Math.Abs(pctClose * actualY))
++numCorrect;
else
++numWrong;
}
return (numCorrect * 1.0) / (numWrong + numCorrect);
}
// ------------------------------------------------------
} // class Hybid_QR_NN
// ========================================================
public class QuadraticRegressor
{
public double[] weights; // regular, quad, interactions
public double bias;
private Random rnd; // not used w/ Pinv training
public QuadraticRegressor(int seed = 0)
{
this.weights = new double[0]; // empty, but not null
this.bias = 0; // dummy value
this.rnd = new Random(seed);
}
// ------------------------------------------------------
public double Predict(double[] x)
{
int dim = x.Length;
double result = 0.0;
int p = 0; // points into this.weights
for (int i = 0; i "lt" dim; ++i) // base terms
result += x[i] * this.weights[p++];
for (int i = 0; i "lt" dim; ++i) // quadratic terms
result += (x[i] * x[i]) * this.weights[p++];
for (int i = 0; i "lt" dim - 1; ++i) // interactions
for (int j = i + 1; j "lt" dim; ++j)
result += (x[i] * x[j]) * this.weights[p++];
result += this.bias;
return result;
}
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
// train using MP pseudo-inverse QR-Householder
// no regulaization
// w = pinv(designX) * y
int nRows = trainX.Length; // not used
int dim = trainX[0].Length;
int nInteractions = (dim * (dim - 1)) / 2;
this.weights = new double[dim + dim + nInteractions];
double[][] Xa = MatAugment(trainX); // add quad cols
double[][] X = MatToDesign(Xa); // add 1.0s col
double[][] Xpinv = QRHouseholder.MatPinv(X);
double[] biasAndWts = MatVecProd(Xpinv, trainY);
this.bias = biasAndWts[0]; // bias is at [0]
for (int i = 1; i "lt" biasAndWts.Length; ++i)
this.weights[i - 1] = biasAndWts[i];
return;
}
// ------------------------------------------------------
private static double[][] MatAugment(double[][] trainX)
{
// add quadratic and interaction columns
int nRows = trainX.Length; // src and dest
int dim = trainX[0].Length; // src
int nInteractions = dim * (dim - 1) / 2;
int nColsDest = dim + dim + nInteractions;
double[][] result = new double[nRows][];
for (int i = 0; i "lt" nRows; i++)
result[i] = new double[nColsDest];
for (int i = 0; i "lt" nRows; ++i)
{
int p = 0; // points to column of result
for (int j = 0; j "lt" dim; ++j) // base
result[i][p++] = trainX[i][j];
for (int j = 0; j "lt" dim; ++j) // quadratic
result[i][p++] = trainX[i][j] * trainX[i][j];
for (int j = 0; j "lt" nInteractions - 1; ++j)
for (int k = j + 1; k "lt" dim; ++k)
result[i][p++] = trainX[i][j] * trainX[i][k];
}
return result;
}
// ------------------------------------------------------
private static double[][] MatToDesign(double[][] X)
{
// add leading column of 1.0s to handle bias term
int nRows = X.Length; // src and dest
int dim = X[0].Length;
double[][] result = new double[nRows][];
for (int i = 0; i "lt" nRows; ++i)
result[i] = new double[dim + 1]; // extra col
for (int i = 0; i "lt" nRows; ++i)
{
result[i][0] = 1.0;
for (int j = 1; j "lt" result[0].Length; ++j)
{
result[i][j] = X[i][j - 1];
}
}
return result;
}
// ------------------------------------------------------
private static double[] MatVecProd(double[][] M,
double[] v)
{
// helper for Train()
int nRows = M.Length;
int nCols = M[0].Length;
int n = v.Length;
if (nCols != n)
throw new Exception("non-comform in MatVecProd");
double[] result = new double[nRows];
for (int i = 0; i "lt" nRows; ++i)
for (int k = 0; k "lt" nCols; ++k)
result[i] += M[i][k] * v[k];
return result;
}
// ------------------------------------------------------
} // class QuadraticRegressor
// ========================================================
public class QRHouseholder
{
// for class QuadraticRegressor Train()
// container for MP pseudo-inverse via QR-Householder
// A = Q * R
// pinv(A) = inv(R) * inv(Q) note order matters
// = inv upper tri (easy) * transpose (easy)
public static double[][] MatPinv(double[][] M)
{
double[][] Q; double[][] R;
MatDecompQR(M, out Q, out R); // Householder
double[][] Ri = MatInvUpperTri(R);
double[][] Qi = MatTranspose(Q);
double[][] result = MatProduct(Ri, Qi);
return result;
}
// ------------------------------------------------------
public static double[][] MatInvUpperTri(double[][] U)
{
int n = U.Length; // must be square matrix
double[][] result = MatMake(n, n);
for (int i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
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[j][k] -= result[j][i] * U[i][k];
}
result[j][k] /= U[k][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[][] MatTranspose(double[][] M)
{
int nRows = M.Length;
int nCols = M[0].Length;
double[][] result = MatMake(nCols, nRows);
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[j][i] = M[i][j];
return result;
}
// ------------------------------------------------------
public static double[][] MatProduct(double[][] A,
double[][] B)
{
int aRows = A.Length; int aCols = A[0].Length;
int bRows = B.Length; int bCols = B[0].Length;
if (aCols != bRows)
throw new Exception("Non-conformable matrices");
double[][] result = new double[aRows][];
for (int i = 0; i "lt" aRows; ++i)
result[i] = new double[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] += A[i][k] * B[k][j];
return result;
}
// ------------------------------------------------------
public static void MatDecompQR(double[][] A,
out double[][] Q, out double[][] R)
{
// Householder algorithm
int m = A.Length; int n = A[0].Length;
if (m "lt" n)
Console.WriteLine("FATAL: nRows must be gte nCols");
double[][] QQ = MatMake(m, m); // working full Q
for (int i = 0; i "lt" m; ++i)
QQ[i][i] = 1.0; // identity matrix
double[][] RR = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
RR[i][j] = A[i][j]; // copy of A is working R
int k = Math.Min(m, n); // or just use n
for (int j = 0; j "lt" k; ++j) // main processing loop
{
int xn = m - j;
double[] x = new double[xn];
for (int i = 0; i "lt" xn; ++i)
x[i] = RR[j + i][j];
double ss = 0.0;
for (int i = 0; i "lt" xn; ++i)
ss += x[i] * x[i];
double normX = Math.Sqrt(ss);
// if (normX == 0.0) continue; // risky
if (Math.Abs(normX) "lt" 1.0e-12) continue;
double sign;
if (x[0] "gte" 0.0) sign = -1.0;
else sign = 1.0; // counter-intuitive
double[] u = new double[xn];
for (int i = 0; i "lt" xn; ++i)
u[i] = x[i] / (x[0] - sign * normX); // check div 0
u[0] = 1.0;
// compute scaling factor tau = 2 / (u^T * u)
double tau = -sign * (x[0] - sign * normX) / normX;
// dimensions for sub-matrices
int nRowsSubR = m - j; int nColsSubR = n - j;
int nRowsSubQ = m; int nColsSubQ = m - j;
double[] vr = new double[nColsSubR];
for (int c = 0; c "lt" nColsSubR; ++c)
{
double acc = 0.0;
for (int r = 0; r "lt" nRowsSubR; ++r)
acc += u[r] * RR[j + r][j + c];
vr[c] = acc;
}
double[] vq = new double[nRowsSubQ];
for (int r = 0; r "lt" nRowsSubQ; ++r)
{
double acc = 0.0;
for (int c = 0; c "lt" nColsSubQ; ++c)
acc += u[c] * QQ[r][j + c];
vq[r] = acc;
}
// update sub-R
for (int r = 0; r "lt" nRowsSubR; ++r)
for (int c = 0; c "lt" nColsSubR; ++c)
RR[j + r][j + c] -= tau * u[r] * vr[c];
// update sub-Q
for (int r = 0; r "lt" nRowsSubQ; ++r)
for (int c = 0; c "lt" nColsSubQ; ++c)
QQ[r][j + c] -= tau * vq[r] * u[c];
} // j
// extract QQ RR into out params
Q = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
Q[i][j] = QQ[i][j];
R = MatMake(n, n);
for (int i = 0; i "lt" n; ++i)
for (int j = 0; j "lt" n; ++j)
R[i][j] = RR[i][j];
return;
} // MatDecompQR
} // class QRHouseholder
// ========================================================
public class GradientBoostRegressor
{
public double lrnRate;
public int nTrees;
public int maxDepth;
public int minSamples;
public int minLeaf;
public int numSplitCols;
public List"lt"DecisionTreeRegressor"gt" trees;
public double pred0; // initial prediction
private Random rnd;
public GradientBoostRegressor(int nTrees, int maxDepth,
int minSamples, int minLeaf, int numSplitCols,
double lrnRate, int seed = 0)
{
this.nTrees = nTrees;
this.maxDepth = maxDepth;
this.minSamples = minSamples;
this.minLeaf = minLeaf;
this.numSplitCols = numSplitCols;
this.trees = new List"lt"DecisionTreeRegressor"gt"();
this.lrnRate = lrnRate;
this.rnd = new Random(seed);
}
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
int n = trainX.Length;
this.pred0 = Mean(trainY);
double[] preds = new double[n]; //each data item
for (int i = 0; i "lt" n; ++i)
preds[i] = this.pred0;
double[] residuals = new double[n]; // for curr tree
for (int t = 0; t "lt" this.nTrees; ++t) // each tree
{
// compute residuals from prev tree predictions
for (int i = 0; i "lt" n; ++i)
residuals[i] = trainY[i] - preds[i];
int treeSeed = this.rnd.Next();
DecisionTreeRegressor dtr =
new DecisionTreeRegressor(
this.maxDepth,
this.minSamples,
this.minLeaf,
this.numSplitCols,
false, // don't save rows info
treeSeed);
dtr.Train(trainX, residuals); // predict residuals
// predict the residuals
for (int i = 0; i "lt" n; ++i)
{
double predResidual = dtr.Predict(trainX[i]);
preds[i] += this.lrnRate * predResidual;
}
this.trees.Add(dtr);
}
} // Train
// ------------------------------------------------------
public double Predict(double[] x)
{
double result = this.pred0;
for (int t = 0; t "lt" this.nTrees; ++t)
{
double predResidual = this.trees[t].Predict(x);
double delta = this.lrnRate * predResidual;
result += delta;
}
return result;
}
// ------------------------------------------------------
private static double Mean(double[] data)
{
int n = data.Length;
double sum = 0.0;
for (int i = 0; i "lt" n; ++i)
sum += data[i];
return sum / n;
}
} // class GradientBoostRegression
// ========================================================
public class DecisionTreeRegressor
{
// ChatGPT: "I don't see any fundamental tree-building or
// indexing bugs. If the inputs are guaranteed to be
// finite doubles and the parameters are validated
// elsewhere, the core regression algorithm is correct."
public int maxDepth;
public int minSamples;
public int minLeaf;
public int numSplitCols;
public List"lt"Node"gt" tree;
public Random rnd;
public bool saveRows;
public double[][] trainX;
public double[] trainY;
// ............................................
public class Node
{
public int id;
public int colIdx = -1;
public double thresh = 0.0;
public int left = -1;
public int right = -1;
public double value = 0.0;
public bool isLeaf = false;
public List"lt"int"gt" rows;
}
// ............................................
public DecisionTreeRegressor(int maxDepth = 3,
int minSamples = 2, int minLeaf = 1,
int numSplitCols = -1, bool saveRows = false,
int seed = 0)
{
this.maxDepth = maxDepth;
this.minSamples = minSamples;
this.minLeaf = minLeaf;
this.numSplitCols = numSplitCols;
this.saveRows = saveRows;
this.tree = new List"lt"Node"gt"();
int numNodes = (int)Math.Pow(2, (maxDepth + 1)) - 1;
for (int i = 0; i "lt" numNodes; ++i)
this.tree.Add(null);
this.rnd = new Random(seed);
}
// ------------------------------------------------------
// ------------------------------------------------------
// public: ctor(), Train(), Predict()
// private: BestSplit(), TreeTargetMean()
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
this.trainX = trainX;
this.trainY = trainY;
int maxParentID = (int)Math.Pow(2, this.maxDepth) - 2;
List"lt"int"gt" allRows =
new List"lt"int"gt"(this.trainX.Length);
for (int i = 0; i "lt" this.trainX.Length; ++i)
allRows.Add(i);
// construct root node
Node root = new Node()
{
id = 0,
value = this.TreeTargetMean(allRows),
rows = allRows
};
this.tree[0] = root;
for (int i = 0; i "lt" this.tree.Count; ++i)
{
Node currNode = this.tree[i];
if (currNode == null) continue;
int leftID = currNode.id * 2 + 1;
int rightID = currNode.id * 2 + 2;
// 1. check before expensive split calcs
if (currNode.id "gt" maxParentID ||
currNode.rows.Count "lt" this.minSamples ||
rightID "gte" this.tree.Count)
{
currNode.isLeaf = true;
continue;
}
// 2. find best split
double[] splitInfo = this.BestSplit(currNode.rows);
int colIdx = (int)splitInfo[0];
double splitVal = splitInfo[1];
if (colIdx == -1) // unable to split
{
currNode.isLeaf = true;
continue;
}
// 3. apply split properties
currNode.colIdx = colIdx; // aka split col
currNode.thresh = splitVal;
List"lt"int"gt" leftIdxs = new List"lt"int"gt"();
List"lt"int"gt" rightIdxs = new List"lt"int"gt"();
for (int k = 0; k "lt" currNode.rows.Count; ++k)
{
int r = currNode.rows[k];
if (this.trainX[r][colIdx] "lte" splitVal)
leftIdxs.Add(r);
else
rightIdxs.Add(r);
}
// 4. create and attach children
currNode.left = leftID;
this.tree[leftID] = new Node()
{
id = leftID,
rows = leftIdxs,
value = this.TreeTargetMean(leftIdxs)
};
currNode.right = rightID;
this.tree[rightID] = new Node()
{
id = rightID,
rows = rightIdxs,
value = this.TreeTargetMean(rightIdxs)
};
} // main loop
// delete row info (for ensembles)
if (this.saveRows == false)
{
for (int i = 0; i "lt" this.tree.Count; ++i)
if (this.tree[i] != null)
this.tree[i].rows = null;
}
} // Train()
// ------------------------------------------------------
public double Predict(double[] x)
{
int p = 0;
double lastValidValue = 0.0;
while (p != -1 && p "lt" this.tree.Count)
{
Node currNode = this.tree[p];
if (currNode == null) break;
lastValidValue = currNode.value;
if (currNode.isLeaf ||
currNode.colIdx == -1) break;
if (x[currNode.colIdx] "lte" currNode.thresh)
p = currNode.left;
else
p = currNode.right;
}
return lastValidValue;
}
// ------------------------------------------------------
private double[] BestSplit(List"lt"int"gt" rows)
{
int bestColIdx = -1;
double bestThresh = 0.0;
double bestGain = -1.0;
int nRows = rows.Count;
int nCols = this.trainX[0].Length;
int[] colIndices = new int[nCols];
for (int k = 0; k "lt" nCols; ++k)
colIndices[k] = k;
// feature sampling
for (int i = 0; i "lt" nCols; ++i)
{
int ri = rnd.Next(i, nCols);
int tmp = colIndices[i];
colIndices[i] = colIndices[ri];
colIndices[ri] = tmp;
}
int numColsToUse;
if (this.numSplitCols == -1)
numColsToUse = nCols;
else
numColsToUse = Math.Min(this.numSplitCols, nCols);
int[] sortedRows = new int[nRows];
double[] featureKeys = new double[nRows];
double totalSum = 0.0;
for (int i = 0; i "lt" nRows; ++i)
totalSum += this.trainY[rows[i]];
for (int j = 0; j "lt" numColsToUse; ++j)
{
int colIdx = colIndices[j];
for (int i = 0; i "lt" nRows; ++i)
{
int r = rows[i];
sortedRows[i] = r;
featureKeys[i] = this.trainX[r][colIdx];
}
Array.Sort(featureKeys, sortedRows);
double leftSum = 0.0;
for (int i = 0; i "lt" nRows - 1; ++i)
{
leftSum += this.trainY[sortedRows[i]];
int leftCount = i + 1;
int rightCount = nRows - leftCount;
if (featureKeys[i] == featureKeys[i + 1])
continue; // duplicate values
if (leftCount "lt" this.minLeaf ||
rightCount "lt" this.minLeaf)
continue;
double rightSum = totalSum - leftSum;
// maximizing Gain = (G_L^2 / N_L) + (G_R^2 / N_R)
// equivalent to maximizing explicit weighted
// variance reduction
double gain = (leftSum * leftSum / leftCount) +
(rightSum * rightSum / rightCount);
if (gain "gt" bestGain)
{
bestGain = gain;
bestColIdx = colIdx;
// bestThresh =
// (featureKeys[i] + featureKeys[i + 1]) / 2.0;
bestThresh = featureKeys[i] +
(featureKeys[i + 1] - featureKeys[i]) / 2.0;
}
}
}
// using a tuple would create version dependency
double[] result = new double[2];
result[0] = 1.0 * bestColIdx;
result[1] = bestThresh;
return result;
}
private double TreeTargetMean(List"lt"int"gt" rows)
{
if (rows == null || rows.Count == 0)
return 0.0;
double sum = 0.0;
for (int i = 0; i "lt" rows.Count; ++i)
sum += this.trainY[rows[i]];
return sum / rows.Count;
}
} // DecisionTreeRegressor
// ========================================================
} // 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, 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
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