One Sunday morning, I figured I’d revisit my C# language implementation of linear regression with two-way interactions. I made several changes, including adding a method to compute the accuracy of the trained model grouped by target y values.

To cut to the chase, my demo model scored an overall accuracy (where a prediction is scored correct if it’s within 10% of the true target y value) of 143 correct of of 200 items, which is 71.50% accuracy. But when looking at the data when placed into five groups of target y values, the accuracy varied quite a bit, from poor to excellent:

From    To      Correct  Wrong   Accuracy
0.0779  0.2234     32      38     0.4571
0.2234  0.3689     35      17     0.6731
0.3689  0.5144     34       1     0.9714
0.5144  0.6599     23       1     0.9583
0.6599  0.8054     19       0     1.0000

Standard linear regression has one weight per predictor variable. for example, if you are predicting the bank account balance of a person based on age, height, years-education, income, there are four model weights. If you add two-way interactions, you get six additional weights: (age * height), (age * education), (age * income), (height * education), (height * income), (education * income). In general, if there are n predictors, two-way interactions adds n * (n-1) / 2 interaction weights.

Analyzing model accuracy by groups of target y values isn’t difficult conceptually, but the code is a bit tricky. Suppose the target y values range from 0.20 to 0.80 and you want to analyze into 4 groups. The width of each group is (0.80 – 0.20) / 4 = 0.15 and so the y intervals are (0.20 to 0.35), (0.35 to 0.50), (0.50 to 0.65), (0.65 to 0.80). You compute accuracy as usual except you look at each group of y values instead of all the data.

For my demo, I used synthetic data that looks like:

-0.1660,  0.4406, -0.9998, -0.3953, -0.7065,  0.4840
 0.0776, -0.1616,  0.3704, -0.5911,  0.7562,  0.1568
-0.9452,  0.3409, -0.1654,  0.1174, -0.7192,  0.8054
. . .

The first five values on each line are predictor values. The last value is the target y value to predict. The data was generated by a 5-10-1 neural network with random weights and bias values. There are 200 training items and 40 test items.

The output of my demo is:

Begin C# linear regression, with interactions, SGD training

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 Linear Regression with two-way interactions model

Setting lrnRate = 0.0010
Setting maxEpochs = 100

Starting training
epoch =     0  MSE =   0.1129
epoch =    20  MSE =   0.0032
epoch =    40  MSE =   0.0012
epoch =    60  MSE =   0.0009
epoch =    80  MSE =   0.0008
Done

Model base weights:
 -0.2626   0.0338  -0.0459   0.0327  -0.1153

Model interaction weights:
 -0.0014   0.0306   0.0167   0.0949   0.0104
 -0.0107   0.0322   0.0025  -0.0448   0.0007

Model bias/intercept: 0.3620

Evaluating model

Accuracy train (within 0.10) = 0.7150
Accuracy test (within 0.10) = 0.7000

Accuracy train (within 0.10) by target y interval

From    To      Correct  Wrong   Accuracy
0.0779  0.2234     32      38     0.4571
0.2234  0.3689     35      17     0.6731
0.3689  0.5144     34       1     0.9714
0.5144  0.6599     23       1     0.9583
0.6599  0.8054     19       0     1.0000

MSE train = 0.0007
MSE test = 0.0006

Predicting for x =
  -0.1660   0.4406  -0.9998  -0.3953  -0.7065

Predicted y = 0.5091

End demo

Good fun on a Sunday morning.

I like to work with machine learning prediction models. When I was a young man, I loved to construct scale models. Here are two that are especially memorable to me.

Right: The Nike Zeus had the ability to shoot down incoming Soviet ICBM nuclear missiles — at least in theory — luckily it was never used. There were 192 made but none were ever placed in service. To the best of my knowledge, no company ever made a scale model of the Zeus.

Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols.

using System;
using System.Collections.Generic;
using System.IO;

namespace LinearRegressionInteractions
{
  internal class LinearRegressionInteractionsProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin C# linear regression," +
        " with interactions, SGD training ");

      // 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 };
      double[][] trainX =
        MatLoad(trainFile, colsX, ',', "#");
      double[] trainY =
        MatToVec(MatLoad(trainFile,
        new int[] { 5 }, ',', "#"));

      string testFile =
        "..\\..\\..\\Data\\synthetic_test_40.txt";
      double[][] testX =
        MatLoad(testFile, colsX, ',', "#");
      double[] testY =
        MatToVec(MatLoad(testFile,
        new int[] { 5 }, ',', "#"));
      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 model
      Console.WriteLine("\nCreating Linear Regression" +
        " with two-way interactions model ");
      int seed = 0;
      LinearRegressor model = new LinearRegressor(seed);

      // 3. train model
      double lrnRate = 0.001;
      int maxEpochs = 100;
      Console.WriteLine("\nSetting lrnRate = " +
        lrnRate.ToString("F4"));
      Console.WriteLine("Setting maxEpochs = " +
        maxEpochs);
      Console.WriteLine("\nStarting training ");
      model.Train(trainX, trainY, lrnRate, maxEpochs);
      Console.WriteLine("Done ");

      // 4. show model weights
      Console.WriteLine("\nModel base weights: ");
      int dim = trainX[0].Length; int p = 0;
      for (; p "lt" dim; ++p)
        Console.Write(model.weights[p].
          ToString("F4").PadLeft(8) + " ");
      Console.WriteLine("");
      Console.WriteLine("\nModel interaction weights: ");
      for (; p "lt" model.weights.Length; ++p)
        Console.Write(model.weights[p].
          ToString("F4").PadLeft(8) + " ");
      Console.WriteLine("");
      Console.Write("\nModel bias/intercept: ");
      Console.WriteLine(model.bias.ToString("F4"));

      // 5. evaluate model
      Console.WriteLine("\nEvaluating model ");
      double accTrain = model.Accuracy(trainX, trainY, 0.10);
      Console.WriteLine("\nAccuracy 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"));

      Console.WriteLine("\nAccuracy train (within 0.10) by" +
        " target y interval ");
      double[] accsTrain = model.Accuracies(5, trainX,
        trainY, 0.10);

      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"));

      // 6. 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"));

      Console.WriteLine("\nEnd demo ");
      Console.ReadLine();
    } // Main

    // ------------------------------------------------------
    // helpers for Main()
    // ------------------------------------------------------

    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[][] 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;
    }

    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 LinearRegressor
  {
    public double[] weights;  // regular and interactions
    public double bias;
    private Random rnd;

    public LinearRegressor(int seed)
    {
      this.rnd = new Random(seed);
    }

    // ------------------------------------------------------

    public void Train(double[][] trainX, double[] trainY,
      double lrnRate, int maxEpochs)
    {
      int nRows = trainX.Length;
      int dim = trainX[0].Length;
      int nInteractions = (dim * (dim - 1)) / 2;
      this.weights = new double[dim + nInteractions];
 
      double low = -0.01; double hi = 0.01;
      for (int i = 0; i "lt" dim; ++i)
        this.weights[i] = (hi - low) *
          this.rnd.NextDouble() + low;

      this.bias = (hi - low) *
          this.rnd.NextDouble() + low;

      int[] indices = new int[nRows];
      for (int i = 0; i "lt" nRows; ++i)
        indices[i] = i;
      
      for (int epoch = 0; epoch "lt" maxEpochs; ++epoch)
      {
        Shuffle(indices, this.rnd);
        for (int i = 0; i "lt" nRows; ++i)
        {
          int ii = indices[i];
          double[] x = trainX[ii];
          double predY = this.Predict(x);
          double actualY = trainY[ii];

          // update regular weights
          for (int j = 0; j "lt" dim; ++j)
            this.weights[j] -= lrnRate *
              (predY - actualY) * x[j];

          // update interaction weights
          int p = dim;  // pts to 1st interaction wt
          for (int k = 0; k "lt" dim - 1; ++k)
          {
            for (int kk = k + 1; kk "lt" dim; ++kk)
            {
              double xx = x[k] * x[kk];
              this.weights[p] -= lrnRate *
                (predY - actualY) * xx;
              ++p;
            }
          } 
 
          // update the bias
          this.bias -= lrnRate * (predY - actualY) * 1.0;
        }
        if (epoch % (int)(maxEpochs / 5) == 0)
        {
          double mse = this.MSE(trainX, trainY);
          string s = "";
          s += "epoch = " + epoch.ToString().PadLeft(5);
          s += "  MSE = " + mse.ToString("F4").PadLeft(8);
          Console.WriteLine(s);
        }
      }

    } // Train()

    // ------------------------------------------------------

    //public double Predict(double[] x)
    //{
    //  int dim = x.Length;
    //  double result = 0.0;
    //  for (int j = 0; j "lt" dim; ++j)
    //    result += x[j] * this.weights[j];

    //  // add interactions
    //  for (int i = 0; i "lt" dim; ++i)
    //    for (int j = 0; j "lt" i; ++j)
    //      result += x[i] * x[j] * this.interactionWts[i][j];

    //  result += this.bias;
    //  return result;
    //}

    // ------------------------------------------------------

    public double Predict(double[] x)
    {
      if (this.weights.Length == 0)
        throw new Exception("\nCannot predict before train ");
      int dim = x.Length;
      double result = 0.0;

      for (int i = 0; i "lt" dim; ++i)  // 
        result += x[i] * this.weights[i]; // regular wts

      // add interactions
      int p = dim;  // pts to 1st interaction wt
      for (int k = 0; k "lt" dim - 1; ++k)
      {
        for (int kk = k + 1; kk "lt" dim; ++kk)
        {
          double xx = x[k] * x[kk];
          result += xx * this.weights[p];
          ++p;
        }
      }
 
      result += this.bias;
      return result;
    }

    // ------------------------------------------------------

    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) "lt"
          Math.Abs(pctClose * actualY))
          ++numCorrect;
        else
          ++numWrong;
      }
      return (numCorrect * 1.0) / (numWrong + numCorrect);
    }

    // ------------------------------------------------------

    public double[] Accuracies(int nGroups, 
      double[][] dataX, double[] dataY, double pctClose)
    {
      int n = dataY.Length;
      // find min and max dataY
      double minY = double.MaxValue;
      double maxY = double.MinValue;
      for (int i = 0; i "lt" n; ++i)
      {
        if (dataY[i] "lt" minY) minY = dataY[i];
        if (dataY[i] "gt" maxY) maxY = dataY[i];
      }
      
      // set up interval endppoints
      double width = (maxY - minY) / nGroups;
      int[] nCorrect = new int[nGroups];
      int[] nWrong = new int[nGroups];

      double[] ep = new double[nGroups+1];  // note
      ep[0] = minY;
      ep[nGroups] = maxY;
      for (int i = 1; i "lt" nGroups; ++i)
        ep[i] = ep[i-1] + width;

      // scan
      for (int i = 0; i "lt" n; ++i)
      {
        double[] x = dataX[i];
        double predY = this.Predict(x);
        double actY = dataY[i];
        bool correct = true;
        if (Math.Abs(predY-actY) "lt" 
          Math.Abs(pctClose * actY))
          correct = true;
        else
          correct = false;
        for (int j = 0; j "lt" nGroups; ++j) {
          if (actY "gte" ep[j] && actY "lte" ep[j+1]) {
            if (correct == true)
              ++nCorrect[j];
            else
              ++nWrong[j];
          } // if
        } // j
      } // i scan

      // accuracies
      double[] result = new double[nGroups];
      for (int j = 0; j "lt" nGroups; ++j)
      {
        int count = nCorrect[j] + nWrong[j];
        if (count == 0)
          result[j] = 0.0;
        else
          result[j] = (nCorrect[j] *1.0) / 
         (nCorrect[j] + nWrong[j]);
      } // j

      Console.WriteLine("\nFrom    To " +
        "Correct  Wrong   Accuracy");
      for (int j = 0; j "lt" nGroups; ++j)
      {
        string s = "";
        s += ep[j].ToString("F4") + "  ";
        s += ep[j+1].ToString("F4") + "  ";
        s += nCorrect[j].ToString().PadLeft(5) + " ";
        s += nWrong[j].ToString().PadLeft(7) + " ";
        s += result[j].ToString("F4").PadLeft(10);
        Console.WriteLine(s);
      }
 
      return result;
    } // Accuracies()

    // ------------------------------------------------------

    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;
    }

    // ------------------------------------------------------

    private static void Shuffle(int[] indices, Random rnd)
    {
      int n = indices.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        int ri = rnd.Next(i, n);
        int tmp = indices[i];
        indices[i] = indices[ri];
        indices[ri] = tmp;
      }
    }

  } // class LinearRegressor

  // ========================================================

} // 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:

 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