James D. McCaffrey

I write code almost every day. Like many skills, writing code is something that must be practiced, and anyway, I just enjoy writing code. One morning before work, I figured I’d run the well-known Diabetes Dataset through an AdaBoost regression model. Based on previous experiments with linear regression, quadratic regression, neural network regression, kernel ridge regression, and random forest regression, I was pretty sure the C# Adaboost regression model would give poor prediction accuracy. And that’s what happened.

The raw Diabetes Dataset looks like:

59, 2, 32.1, 101.00, 157,  93.2, 38, 4.00, 4.8598, 87, 151
48, 1, 21.6,  87.00, 183, 103.2, 70, 3.00, 3.8918, 69,  75
72, 2, 30.5,  93.00, 156,  93.6, 41, 4.00, 4.6728, 85, 141
. . .

Each line represents a patient. The first 10 values on each line are predictors. The last value on each line is the target value (a diabetes metric) to predict. The predictors are: age, sex, body mass index, blood pressure, serum cholesterol, low-density lipoproteins, high-density lipoproteins, total cholesterol, triglycerides, blood sugar. There are 442 data items.

The sex encoding isn’t explained anywhere but I suspect male = 1, female = 2 because there are 235 1 values and 206 2 values).

Note that this Diabetes Dataset, which is included as an example dataset in the Python language scikit-learn library, is not the same as the Pima Diabetes Dataset from the UCI dataset repository.

I converted the sex values from 1,2 into 0,1. Then I applied divide-by-constant normalization by dividing the 10 predictor columns by (100, 1, 100, 1000, 1000, 1000, 100, 10, 10, 1000) and the target y values by 1000. The resulting encoded and normalized data looks like:

0.5900, 1.0000, 0.3210, . . . 0.1510
0.4800, 0.0000, 0.2160, . . . 0.0750
0.7200, 1.0000, 0.3050, . . . 0.1410
. . .

Normalization isn’t necessary for ensemble models like AdaBoost that use decision trees, but normalization doesn’t hurt. I split the 442-items into a 342-item training set and a 100-item test set.

AdaBoost uses a collection of decision trees. Each tree is trained using data that was poorly predicted by previous trees, so in theory, each tree gets better. The final predicted value is a weighted median of all the tree predictions.

I implemented an AdaBoost regression model using C#. The algorithm is known as AdaBoost.R2 (“adaptive boosting for regression, version 2”). I used 100 decision trees. I set the maximum depth of each tree to 5, the minimum samples parameter to 2 (standard default), and the minimum leaf parameter to 1 (standard default).

The output of the demo is:

Begin AdaBoost.R2 regression on Diabetes Dataset

Loading diabetes train (342) and test (100) normalized data
Done

First three train X:
  0.5900  1.0000  0.3210  . . .  0.4860  0.0870
  0.4800  0.0000  0.2160  . . .  0.3892  0.0690
  0.7200  1.0000  0.3050  . . .  0.4673  0.0850

First three train y:
  0.1510
  0.0750
  0.1410

Setting maxLearners = 100
Setting tree maxDepth = 5
Setting tree minSamples = 2
Setting tree minLeaf = 1

Training AdaBoost.R2 model
Done
Created 100 learners

Accuracy train (within 0.10): 0.2807
Accuracy test (within 0.10): 0.2065

MSE train: 0.0015
MSE test: 0.0035

Predicting for x =
  0.5900  1.0000  0.3210  . . .  0.4860  0.0870
Predicted y = 0.1779

End demo

These poor results were essentially the same as the results I got using the scikit AdaBoostRegressor module (there was slight differences because AdaBoost has a random component when selecting training items for each tree).

I have done many experiments with the Diabetes Dataset and I’ve concluded the the default target value in the last column (a patient diabetes score) simply cannot be predicted well. But the variables in columns [4], [5], [6], [7], and [8] can be meaningfully predicted from the other columns.

I have a fascination with machine learning. And I have a strange fascination with “chittering”. Several years ago, I noticed that when I watch movies with closed captioning enabled, there are an amazing number of scenes where an animal is making noises and the closed captioning reads “chittering”. I have seen this dozens of times. Here’s an example from “Superhero Movie” (2008), a very funny (to me) spoof of all the annoying superhero movies. On the left, the Dragonfly hero sees a crowd of people gathered. He rushes over and sees the crowd being fascinated by a monkey — while a super villain is robbing a bank and is being completely ignored.

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 AdaBoostRegression
{
  internal class AdaBoostRegressionProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin AdaBoost.R2" +
        " regression on Diabetes Dataset ");

      // 1. load data
      Console.WriteLine("\nLoading diabetes train" +
        " (342) and test (100) normalized data");
      string trainFile = "..\\..\\..\\Data\\" +
        "diabetes_norm_train_342.txt";
      int[] colsX = new int[] { 0, 1, 2, 3, 4,
        5, 6, 7, 8, 9};
      int colY = 10;
      double[][] trainX =
        MatLoad(trainFile, colsX, ',', "#");
      double[] trainY =
        MatToVec(MatLoad(trainFile,
        new int[] { colY }, ',', "#"));

      string testFile = "..\\..\\..\\Data\\" +
        "diabetes_norm_test_100.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
      int maxLearners = 100;
      int maxDepth = 5;
      int minSamples = 2;
      int minLeaf = 1;

      Console.WriteLine("\nSetting maxLearners = " +
        maxLearners);
      Console.WriteLine("Setting tree maxDepth = " +
        maxDepth);
      Console.WriteLine("Setting tree minSamples = " +
        minSamples);
      Console.WriteLine("Setting tree minLeaf = " +
        minLeaf);

      Console.WriteLine("\nTraining AdaBoost.R2 model ");
      AdaBoostRegressor model =
        new AdaBoostRegressor(maxLearners, maxDepth,
        minSamples, minLeaf, seed: 0); // master seed
      model.Train(trainX, trainY);
      Console.WriteLine("Done ");
      Console.WriteLine("Created " +
        model.learners.Count() + " learners ");

      // 3. evaluate 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"));

      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"));

      // 4. use model to make a prediction
      double[] x = trainX[0];
      Console.WriteLine("\nPredicting for x = ");
      VecShow(x, 4, 8);
      double yPred = model.Predict(x);
      Console.WriteLine("Predicted y = " +
        yPred.ToString("F4"));
      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[][] 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

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

  class AdaBoostRegressor
  {
    public int maxLearners;
    public int maxDepth;
    public int minSamples;
    public int minLeaf;
    private Random rnd;
    private int seed; // master also passed to each tree
    public List"lt"DecisionTreeRegressor"gt" learners;
    public List"lt"double"gt" betas;

    public AdaBoostRegressor(int maxLearners, int maxDepth,
      int minSamples, int minLeaf, int seed)
    {
      this.maxLearners = maxLearners;
      this.maxDepth = maxDepth;
      this.minSamples = minSamples;
      this.minLeaf = minLeaf;
      this.rnd = new Random(seed);
      this.seed = seed;
      this.learners = new List"lt"DecisionTreeRegressor"gt"();
      this.betas = new List"lt"double"gt"();
      // each learner has a "model weight" derived
      // from its beta-confidence weight
    } // ctor

    public void Train(double[][] trainX, double[] trainY)
    {
      int N = trainX.Length;
      double[] weights = new double[N];
      for (int i = 0; i "lt" N; ++i)
        weights[i] = 1.0 / N;

      for (int t = 0; t "lt" this.maxLearners; ++t)
      {
        // 1. use weights to select a sample
        int[] sampleIdxs = this.SelectIdxs(weights, N);

        double[][] sampleX = ExtractRows(trainX,
          sampleIdxs);
        double[] sampleY = ExtractVals(trainY,
          sampleIdxs);

        // 2. construct regression machine t
        DecisionTreeRegressor lrnr =
          new DecisionTreeRegressor(this.maxDepth,
          this.minSamples, this.minLeaf, seed:this.seed);
        lrnr.Train(sampleX, sampleY);

        // 3. compute predicteds for all training items
        double[] predY = new double[N];
        for (int i = 0; i "lt" N; ++i)
          predY[i] = lrnr.Predict(trainX[i]);

        // 4. calculate loss each sample
        double[] absDiffs = new double[N];
        for (int i = 0; i "lt" N; ++i)
        {
          absDiffs[i] = Math.Abs(predY[i] - trainY[i]);
        }

        double D = 0.0;  // largest abs diff
        for (int i = 0; i "lt" N; ++i)
        {
          if (absDiffs[i] "gt" D)
            D = absDiffs[i];
        }

        double[] L = new double[N];
        for (int i = 0; i "lt" N; ++i)
          L[i] = absDiffs[i] / D;  // normalized

        // 5. calculate average loss
        double Lbar = 0.0;
        for (int i = 0; i "lt" N; ++i)
          Lbar += L[i] * weights[i];

        if (Lbar "gte" 0.5)  // bad learner
          continue;

        // 6. compute beta (low beta = high confidence)
        this.learners.Add(lrnr);  // lrnr is good
        double beta = Lbar / (1.0 - Lbar);
        this.betas.Add(beta); // each learner has a beta

        // 7. update training weights
        double sum = 0.0;
        for (int i = 0; i "lt" N; ++i)
        {
          weights[i] *= Math.Pow(beta, (1.0 - Lbar));
          sum += weights[i];
        }
        for (int i = 0; i "lt" N; ++i)
          weights[i] /= sum;

      } // t

    } // Train()

    public double Predict(double[] x)
    {
      // each learner makes a prediction
      double[] preds = new double[this.learners.Count];
      for (int i = 0; i "lt" this.learners.Count; ++i)
      {
        // preds[i] = lrnRate * this.learners[i].Predict(x);
        // scikit uses a lrnRate param, default val = 1.0
        preds[i] = this.learners[i].Predict(x);
      }

      // each learner has a "model weight" derived
      // from its beta-confidence weight
      double[] modelWts = new double[this.learners.Count];
      for (int i = 0; i "lt" this.learners.Count; ++i)
        modelWts[i] = Math.Log(1.0 / this.betas[i]);

      // final prediction is weighted median of 
      // the predictions
      double result = WeightedMedian(preds, modelWts);
      //double result = WeightedMean(preds, modelWts);

      return result;
    } // Predict()

    public double Accuracy(double[][] dataX,
        double[] dataY, double pctClose)
    {
      int nCorrect = 0; int nWrong = 0;
      for (int i = 0; i "lt" dataX.Length; ++i)
      {
        double predY = this.Predict(dataX[i]);
        double actuaY = dataY[i];
        if (Math.Abs(predY - actuaY) "lt"
          Math.Abs(pctClose * actuaY))
          ++nCorrect;
        else
          ++nWrong;
      }
      return (nCorrect * 1.0) / (nCorrect + nWrong);
    }

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

    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;
    }

    // ------------------------------------------------------
    // helper functions for AdaBoostRegressor
    // ------------------------------------------------------

    //private static double WeightedMean(double[] values,
    //  double[] weights)
    //{
    //  int n = values.Length;
    //  double sumWts = 0.0;
    //  double sum = 0.0;
    //  for (int i = 0; i "lt" n; ++i)
    //  {
    //    sumWts += weights[i];
    //    sum += values[i] * weights[i];
    //  }
    //  return sum / sumWts;
    //}

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

    private static double WeightedMedian(double[] values,
      double[] weights)
    {
      // no interpolation for even n
      // don't assume weights sum to 1.0
      int n = values.Length;
      double sumWts = 0.0;
      for (int i = 0; i "lt" n; ++i)
        sumWts += weights[i];
      double thresh = sumWts / 2;
      int[] sortedIdxs = ArgSort(values);

      double accum = 0.0;
      for (int j = 0; j "lt" n; ++j)
      {
        accum += weights[sortedIdxs[j]];
        if (accum "gte" thresh)
          return values[sortedIdxs[j]];
      }
      return values[sortedIdxs[n - 1]];
    }

    // helper for WeightedMedian()
    private static int[] ArgSort(double[] values)
    {
      int n = values.Length;
      double[] copy = new double[n];
      int[] indices = new int[n];
      for (int i = 0; i "lt" n; ++i)
      {
        copy[i] = values[i];
        indices[i] = i;
      }
      Array.Sort(copy, indices);  // in parallel
      return indices;
    }

    private 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;
    }

    private int SelectIdx(double[] probs)
    {
      // roulette wheel selection
      double p = this.rnd.NextDouble();
      double accum = 0.0;
      for (int i = 0; i "lt" probs.Length; ++i)
      {
        accum += probs[i];
        if (p "lt" accum)
          return i;
      }
      return probs.Length - 1;
    }

    private int[] SelectIdxs(double[] probs, int n)
    {
      // with replacement
      int[] result = new int[n];
      for (int i = 0; i "lt" n; ++i)
        result[i] = this.SelectIdx(probs);
      return result;
    }

    private static double[][] ExtractRows(double[][] mat,
      int[] idxs)
    {
      int nRows = idxs.Length;  // of the result
      int nCols = mat[0].Length;  // source and result
      double[][] result = MatMake(nRows, nCols);

      for (int i = 0; i "lt" nRows; ++i)
      {
        for (int j = 0; j "lt" nCols; ++j)
        {
          int srcRow = idxs[i];
          result[i][j] = mat[srcRow][j];
        }
      }
      return result;
    }

    private static double[] ExtractVals(double[] vec,
      int[] idxs)
    {
      int nVals = idxs.Length;
      double[] result = new double[nVals];
      for (int i = 0; i "lt" nVals; ++i)
      {
        int srcIdx = idxs[i];
        result[i] = vec[srcIdx];
      }
      return result;
    }

  } // class AdaBoostRegressor

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

  public class DecisionTreeRegressor
  {
    public int maxDepth;
    public int minSamples;  // aka min_samples_split
    public int minLeaf;  // min number of values in a leaf
    public int numSplitCols; // mostly for random forest
    public List"lt"Node"gt" tree = new List"lt"Node"gt"();
    public Random rnd;  // order in which cols are searched

    public double[][] trainX;  // store data by ref
    public double[] trainY;

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

    public class Node
    {
      public int id;
      public int colIdx;  // aka featureIdx
      public double thresh;
      public int left;  // index into List
      public int right;
      public double value;
      public bool isLeaf;
      public List"lt"int"gt" rows;  // assoc rows in train

      public Node()
      {
        this.id = -1;
        this.colIdx = -1;
        this.thresh = 0.0;  // aka split value
        this.left = -1;
        this.right = -1;
        this.value = 0.0;  // aka pred y
        this.isLeaf = false;
        this.rows = null;
      }
    } // class Node

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

    public DecisionTreeRegressor(int maxDepth = 2,
      int minSamples = 2, int minLeaf = 1,
      int numSplitCols = -1, int seed = 0)
    {
      // if maxDepth = 0, tree has just a root node
      // if maxDepth = 1, at most 3 nodes (root, l, r)
      // if maxDepth = n, at most 2^(n+1) - 1 nodes
      this.maxDepth = maxDepth;
      this.minSamples = minSamples;
      this.minLeaf = minLeaf;
      this.numSplitCols = numSplitCols;  // for ran. forest

      // create full tree List with null nodes
      int numNodes = (int)Math.Pow(2, (maxDepth + 1)) - 1;
      for (int i = 0; i "lt" numNodes; ++i)
      {
        this.tree.Add(null);  // empty nodes
      }
      this.rnd = new Random(seed);
    }

    // ------------------------------------------------------
    // public: Train(), Predict().
    // helpers: MakeTree(), BestSplit(), TreeTargetMean(),
    //   TreeTargetVariance().
    // ------------------------------------------------------

    public void Train(double[][] trainX, double[] trainY)
    {
      this.trainX = trainX; // 
      this.trainY = trainY;
      this.MakeTree();

      // optionally delete rows in each Node to save space
      // when tree is part of an ensemble
      for (int i = 0; i "lt" this.tree.Count; ++i)
        if (this.tree[i] != null)
          this.tree[i].rows = null;
    }

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

    public double Predict(double[] x)
    {
      int p = 0;
      Node currNode = this.tree[p];
      while (currNode != null &&
        currNode.isLeaf == false &&
        p "lt" this.tree.Count)
      {
        if (x[currNode.colIdx] "lte" currNode.thresh)
          p = currNode.left;
        else
          p = currNode.right;
        currNode = this.tree[p];
      }
      return this.tree[p].value;
    }

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

    // helpers: MakeTree(), BestSplit(),
    // TreeTargetMean(), TreeTargetVariance()

    private void MakeTree()
    {
      // no recursion, no pointers, List storage, no stack
      if (this.numSplitCols == -1) // use all cols
        this.numSplitCols = this.trainX[0].Length;

      // prepare root node
      List"lt"int"gt" allRows = new List"lt"int"gt"();
      for (int i = 0; i "lt" this.trainX.Length; ++i)
        allRows.Add(i);
      double grandMean = this.TreeTargetMean(allRows);

      // wait to supply colIdx and thresh in loop
      Node root = new Node();
      root.id = 0;
      root.left = 1;
      root.right = 2;
      root.value = grandMean;
      root.isLeaf = false; // already set
      root.rows = allRows;
      this.tree[0] = root;

      for (int i = 0; i "lt" this.tree.Count; ++i)
      {
        Node currNode = this.tree[i];
        // curr node has values except colIdx and thresh

        // curr node too deep to have children OR
        // curr node not enough rows to split then
        // leave both children as null
        if (currNode == null ||
          currNode.rows.Count == 0) { continue; }

        // if parent cannot be split, make parent a leaf
        if (currNode.id "gte" (int)Math.Pow(2,
          (this.maxDepth)) - 1 ||
          currNode.rows.Count "lt" this.minSamples)
        {
          currNode.isLeaf = true;
          continue;
        }

        // parent has enough rows to try to split
        double[] splitInfo = this.BestSplit(currNode.rows);
        int colIdx = (int)splitInfo[0];
        double splitVal = splitInfo[1]; //split value

        if (colIdx == -1)  // unable to split, parent leaf
        {
          currNode.isLeaf = true;
          continue;
        }

        // complete the fields for curr node
        currNode.colIdx = colIdx;
        currNode.thresh = splitVal;

        // construct the children, except colIdx and thresh
        Node leftNode = new Node();
        Node rightNode = new Node();

        // construct the children rows using split info
        // all info except colIdx and thresh
        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);
        }

        // assign -1 to children if out of range
        leftNode.id = currNode.id * 2 + 1;
        if (leftNode.id "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) leftNode.id = -1;
        leftNode.left = leftNode.id * 2 + 1;
        if (leftNode.left "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) leftNode.left = -1;
        leftNode.right = leftNode.id * 2 + 2;
        if (leftNode.right "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) leftNode.right = -1;
        leftNode.rows = leftIdxs;
        leftNode.value =
          this.TreeTargetMean(leftNode.rows);
        this.tree[leftNode.id] = leftNode;

        rightNode.id = currNode.id * 2 + 2;
        if (rightNode.id "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) rightNode.id = -1;
        rightNode.left = rightNode.id * 2 + 1;
        if (rightNode.left "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) rightNode.left = -1;
        rightNode.right = rightNode.id * 2 + 2;
        if (rightNode.right "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) rightNode.right = -1;
        rightNode.rows = rightIdxs;
        rightNode.value =
          this.TreeTargetMean(rightNode.rows);
        this.tree[rightNode.id] = rightNode;

      } // i
      return;
    }

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

    private double[] BestSplit(List"lt"int"gt" rows)
    {
      // implicit params numSplitCols, minLeaf, numsplitCols
      // result[0] = best col idx (as double)
      // result[1] = best split value
      rows.Sort();

      int bestColIdx = -1;  // indicates bad split
      double bestThresh = 0.0;
      double bestVar = double.MaxValue; // smaller better

      int nRows = rows.Count;  // or dataY.Length
      int nCols = this.trainX[0].Length;

      if (nRows == 0)
      {
        throw new Exception("empty data in BestSplit()");
      }

      // process cols in scrambled order
      int[] colIndices = new int[nCols];
      for (int k = 0; k "lt" nCols; ++k)
        colIndices[k] = k;
      // shuffle, inline Fisher-Yates
      int n = colIndices.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        int ri = rnd.Next(i, n);  // be careful
        int tmp = colIndices[i];
        colIndices[i] = colIndices[ri];
        colIndices[ri] = tmp;
      }

      // numSplitCols is usually all columns (-1)
      for (int j = 0; j "lt" this.numSplitCols; ++j)
      {
        int colIdx = colIndices[j];
        HashSet"lt"double"gt" examineds = 
          new HashSet"lt"double"gt"();

        for (int i = 0; i "lt" nRows; ++i) // each row
        {
          // if curr thresh been seen, skip it
          double thresh = this.trainX[rows[i]][colIdx];
          if (examineds.Contains(thresh)) continue;
          examineds.Add(thresh);

          // get row idxs where x is lte, gt thresh
          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" nRows; ++k)
          {
            if (this.trainX[rows[k]][colIdx] "lte" thresh)
              leftIdxs.Add(rows[k]);
            else
              rightIdxs.Add(rows[k]);
          }

          // Check if proposed split has too few values
          if (leftIdxs.Count "lt" this.minLeaf ||
            rightIdxs.Count "lt" this.minLeaf)
            continue;  // to next row

          double leftVar =
            this.TreeTargetVariance(leftIdxs);
          double rightVar =
            this.TreeTargetVariance(rightIdxs);
          double weightedVar = (leftIdxs.Count * leftVar +
            rightIdxs.Count * rightVar) / nRows;

          if (weightedVar "lt" bestVar)
          {
            // if this never happens, bestColIdx remains -1
            // which means a bad split. used in MakeTree()
            bestColIdx = colIdx;
            bestThresh = thresh;
            bestVar = weightedVar;
          }

        } // each row
      } // j each col

      double[] result = new double[2];  // out params ugly
      result[0] = 1.0 * bestColIdx;
      result[1] = bestThresh;
      return result;

    } // BestSplit()

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

    private double TreeTargetMean(List"lt"int"gt" rows)
    {
      // mean of rows items in trainY: for node prediction
      double sum = 0.0;
      for (int i = 0; i "lt" rows.Count; ++i)
      {
        int r = rows[i];
        sum += this.trainY[r];
      }
      return sum / rows.Count;
    }

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

    private double TreeTargetVariance(List"lt"int"gt" rows)
    {
      double mean = this.TreeTargetMean(rows);
      double sum = 0.0;
      for (int i = 0; i "lt" rows.Count; ++i)
      {
        int r = rows[i];
        sum += (this.trainY[r] - mean) *
          (this.trainY[r] - mean);
      }
      return sum / rows.Count;
    }

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

  } // class DecisionTreeRegressor

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

} // ns

Training data:

# diabetes_norm_train_342.txt
# cols [0] to [9] predictors. col [10] target
# norm division constants:
# 100, -1, 100, 1000, 1000, 1000, 100, 10, 10, 1000, 1000
#
0.5900, 1.0000, 0.3210, 0.1010, 0.1570, 0.0932, 0.3800, 0.4000, 0.4860, 0.0870, 0.1510
0.4800, 0.0000, 0.2160, 0.0870, 0.1830, 0.1032, 0.7000, 0.3000, 0.3892, 0.0690, 0.0750
0.7200, 1.0000, 0.3050, 0.0930, 0.1560, 0.0936, 0.4100, 0.4000, 0.4673, 0.0850, 0.1410
0.2400, 0.0000, 0.2530, 0.0840, 0.1980, 0.1314, 0.4000, 0.5000, 0.4890, 0.0890, 0.2060
0.5000, 0.0000, 0.2300, 0.1010, 0.1920, 0.1254, 0.5200, 0.4000, 0.4291, 0.0800, 0.1350
0.2300, 0.0000, 0.2260, 0.0890, 0.1390, 0.0648, 0.6100, 0.2000, 0.4190, 0.0680, 0.0970
0.3600, 1.0000, 0.2200, 0.0900, 0.1600, 0.0996, 0.5000, 0.3000, 0.3951, 0.0820, 0.1380
0.6600, 1.0000, 0.2620, 0.1140, 0.2550, 0.1850, 0.5600, 0.4550, 0.4249, 0.0920, 0.0630
0.6000, 1.0000, 0.3210, 0.0830, 0.1790, 0.1194, 0.4200, 0.4000, 0.4477, 0.0940, 0.1100
0.2900, 0.0000, 0.3000, 0.0850, 0.1800, 0.0934, 0.4300, 0.4000, 0.5385, 0.0880, 0.3100
0.2200, 0.0000, 0.1860, 0.0970, 0.1140, 0.0576, 0.4600, 0.2000, 0.3951, 0.0830, 0.1010
0.5600, 1.0000, 0.2800, 0.0850, 0.1840, 0.1448, 0.3200, 0.6000, 0.3584, 0.0770, 0.0690
0.5300, 0.0000, 0.2370, 0.0920, 0.1860, 0.1092, 0.6200, 0.3000, 0.4304, 0.0810, 0.1790
0.5000, 1.0000, 0.2620, 0.0970, 0.1860, 0.1054, 0.4900, 0.4000, 0.5063, 0.0880, 0.1850
0.6100, 0.0000, 0.2400, 0.0910, 0.2020, 0.1154, 0.7200, 0.3000, 0.4291, 0.0730, 0.1180
0.3400, 1.0000, 0.2470, 0.1180, 0.2540, 0.1842, 0.3900, 0.7000, 0.5037, 0.0810, 0.1710
0.4700, 0.0000, 0.3030, 0.1090, 0.2070, 0.1002, 0.7000, 0.3000, 0.5215, 0.0980, 0.1660
0.6800, 1.0000, 0.2750, 0.1110, 0.2140, 0.1470, 0.3900, 0.5000, 0.4942, 0.0910, 0.1440
0.3800, 0.0000, 0.2540, 0.0840, 0.1620, 0.1030, 0.4200, 0.4000, 0.4443, 0.0870, 0.0970
0.4100, 0.0000, 0.2470, 0.0830, 0.1870, 0.1082, 0.6000, 0.3000, 0.4543, 0.0780, 0.1680
0.3500, 0.0000, 0.2110, 0.0820, 0.1560, 0.0878, 0.5000, 0.3000, 0.4511, 0.0950, 0.0680
0.2500, 1.0000, 0.2430, 0.0950, 0.1620, 0.0986, 0.5400, 0.3000, 0.3850, 0.0870, 0.0490
0.2500, 0.0000, 0.2600, 0.0920, 0.1870, 0.1204, 0.5600, 0.3000, 0.3970, 0.0880, 0.0680
0.6100, 1.0000, 0.3200, 0.1037, 0.2100, 0.0852, 0.3500, 0.6000, 0.6107, 0.1240, 0.2450
0.3100, 0.0000, 0.2970, 0.0880, 0.1670, 0.1034, 0.4800, 0.4000, 0.4357, 0.0780, 0.1840
0.3000, 1.0000, 0.2520, 0.0830, 0.1780, 0.1184, 0.3400, 0.5000, 0.4852, 0.0830, 0.2020
0.1900, 0.0000, 0.1920, 0.0870, 0.1240, 0.0540, 0.5700, 0.2000, 0.4174, 0.0900, 0.1370
0.4200, 0.0000, 0.3190, 0.0830, 0.1580, 0.0876, 0.5300, 0.3000, 0.4466, 0.1010, 0.0850
0.6300, 0.0000, 0.2440, 0.0730, 0.1600, 0.0914, 0.4800, 0.3000, 0.4635, 0.0780, 0.1310
0.6700, 1.0000, 0.2580, 0.1130, 0.1580, 0.0542, 0.6400, 0.2000, 0.5293, 0.1040, 0.2830
0.3200, 0.0000, 0.3050, 0.0890, 0.1820, 0.1106, 0.5600, 0.3000, 0.4344, 0.0890, 0.1290
0.4200, 0.0000, 0.2030, 0.0710, 0.1610, 0.0812, 0.6600, 0.2000, 0.4234, 0.0810, 0.0590
0.5800, 1.0000, 0.3800, 0.1030, 0.1500, 0.1072, 0.2200, 0.7000, 0.4644, 0.0980, 0.3410
0.5700, 0.0000, 0.2170, 0.0940, 0.1570, 0.0580, 0.8200, 0.2000, 0.4443, 0.0920, 0.0870
0.5300, 0.0000, 0.2050, 0.0780, 0.1470, 0.0842, 0.5200, 0.3000, 0.3989, 0.0750, 0.0650
0.6200, 1.0000, 0.2350, 0.0803, 0.2250, 0.1128, 0.8600, 0.2620, 0.4875, 0.0960, 0.1020
0.5200, 0.0000, 0.2850, 0.1100, 0.1950, 0.0972, 0.6000, 0.3000, 0.5242, 0.0850, 0.2650
0.4600, 0.0000, 0.2740, 0.0780, 0.1710, 0.0880, 0.5800, 0.3000, 0.4828, 0.0900, 0.2760
0.4800, 1.0000, 0.3300, 0.1230, 0.2530, 0.1636, 0.4400, 0.6000, 0.5425, 0.0970, 0.2520
0.4800, 1.0000, 0.2770, 0.0730, 0.1910, 0.1194, 0.4600, 0.4000, 0.4852, 0.0920, 0.0900
0.5000, 1.0000, 0.2560, 0.1010, 0.2290, 0.1622, 0.4300, 0.5000, 0.4779, 0.1140, 0.1000
0.2100, 0.0000, 0.2010, 0.0630, 0.1350, 0.0690, 0.5400, 0.3000, 0.4094, 0.0890, 0.0550
0.3200, 1.0000, 0.2540, 0.0903, 0.1530, 0.1004, 0.3400, 0.4500, 0.4533, 0.0830, 0.0610
0.5400, 0.0000, 0.2420, 0.0740, 0.2040, 0.1090, 0.8200, 0.2000, 0.4174, 0.1090, 0.0920
0.6100, 1.0000, 0.3270, 0.0970, 0.1770, 0.1184, 0.2900, 0.6000, 0.4997, 0.0870, 0.2590
0.5600, 1.0000, 0.2310, 0.1040, 0.1810, 0.1164, 0.4700, 0.4000, 0.4477, 0.0790, 0.0530
0.3300, 0.0000, 0.2530, 0.0850, 0.1550, 0.0850, 0.5100, 0.3000, 0.4554, 0.0700, 0.1900
0.2700, 0.0000, 0.1960, 0.0780, 0.1280, 0.0680, 0.4300, 0.3000, 0.4443, 0.0710, 0.1420
0.6700, 1.0000, 0.2250, 0.0980, 0.1910, 0.1192, 0.6100, 0.3000, 0.3989, 0.0860, 0.0750
0.3700, 1.0000, 0.2770, 0.0930, 0.1800, 0.1194, 0.3000, 0.6000, 0.5030, 0.0880, 0.1420
0.5800, 0.0000, 0.2570, 0.0990, 0.1570, 0.0916, 0.4900, 0.3000, 0.4407, 0.0930, 0.1550
0.6500, 1.0000, 0.2790, 0.1030, 0.1590, 0.0968, 0.4200, 0.4000, 0.4615, 0.0860, 0.2250
0.3400, 0.0000, 0.2550, 0.0930, 0.2180, 0.1440, 0.5700, 0.4000, 0.4443, 0.0880, 0.0590
0.4600, 0.0000, 0.2490, 0.1150, 0.1980, 0.1296, 0.5400, 0.4000, 0.4277, 0.1030, 0.1040
0.3500, 0.0000, 0.2870, 0.0970, 0.2040, 0.1268, 0.6400, 0.3000, 0.4190, 0.0930, 0.1820
0.3700, 0.0000, 0.2180, 0.0840, 0.1840, 0.1010, 0.7300, 0.3000, 0.3912, 0.0930, 0.1280
0.3700, 0.0000, 0.3020, 0.0870, 0.1660, 0.0960, 0.4000, 0.4150, 0.5011, 0.0870, 0.0520
0.4100, 0.0000, 0.2050, 0.0800, 0.1240, 0.0488, 0.6400, 0.2000, 0.4025, 0.0750, 0.0370
0.6000, 0.0000, 0.2040, 0.1050, 0.1980, 0.0784, 0.9900, 0.2000, 0.4635, 0.0790, 0.1700
0.6600, 1.0000, 0.2400, 0.0980, 0.2360, 0.1464, 0.5800, 0.4000, 0.5063, 0.0960, 0.1700
0.2900, 0.0000, 0.2600, 0.0830, 0.1410, 0.0652, 0.6400, 0.2000, 0.4078, 0.0830, 0.0610
0.3700, 1.0000, 0.2680, 0.0790, 0.1570, 0.0980, 0.2800, 0.6000, 0.5043, 0.0960, 0.1440
0.4100, 1.0000, 0.2570, 0.0830, 0.1810, 0.1066, 0.6600, 0.3000, 0.3738, 0.0850, 0.0520
0.3900, 0.0000, 0.2290, 0.0770, 0.2040, 0.1432, 0.4600, 0.4000, 0.4304, 0.0740, 0.1280
0.6700, 1.0000, 0.2400, 0.0830, 0.1430, 0.0772, 0.4900, 0.3000, 0.4431, 0.0940, 0.0710
0.3600, 1.0000, 0.2410, 0.1120, 0.1930, 0.1250, 0.3500, 0.6000, 0.5106, 0.0950, 0.1630
0.4600, 1.0000, 0.2470, 0.0850, 0.1740, 0.1232, 0.3000, 0.6000, 0.4644, 0.0960, 0.1500
0.6000, 1.0000, 0.2500, 0.0897, 0.1850, 0.1208, 0.4600, 0.4020, 0.4511, 0.0920, 0.0970
0.5900, 1.0000, 0.2360, 0.0830, 0.1650, 0.1000, 0.4700, 0.4000, 0.4500, 0.0920, 0.1600
0.5300, 0.0000, 0.2210, 0.0930, 0.1340, 0.0762, 0.4600, 0.3000, 0.4078, 0.0960, 0.1780
0.4800, 0.0000, 0.1990, 0.0910, 0.1890, 0.1096, 0.6900, 0.3000, 0.3951, 0.1010, 0.0480
0.4800, 0.0000, 0.2950, 0.1310, 0.2070, 0.1322, 0.4700, 0.4000, 0.4935, 0.1060, 0.2700
0.6600, 1.0000, 0.2600, 0.0910, 0.2640, 0.1466, 0.6500, 0.4000, 0.5568, 0.0870, 0.2020
0.5200, 1.0000, 0.2450, 0.0940, 0.2170, 0.1494, 0.4800, 0.5000, 0.4585, 0.0890, 0.1110
0.5200, 1.0000, 0.2660, 0.1110, 0.2090, 0.1264, 0.6100, 0.3000, 0.4682, 0.1090, 0.0850
0.4600, 1.0000, 0.2350, 0.0870, 0.1810, 0.1148, 0.4400, 0.4000, 0.4710, 0.0980, 0.0420
0.4000, 1.0000, 0.2900, 0.1150, 0.0970, 0.0472, 0.3500, 0.2770, 0.4304, 0.0950, 0.1700
0.2200, 0.0000, 0.2300, 0.0730, 0.1610, 0.0978, 0.5400, 0.3000, 0.3829, 0.0910, 0.2000
0.5000, 0.0000, 0.2100, 0.0880, 0.1400, 0.0718, 0.3500, 0.4000, 0.5112, 0.0710, 0.2520
0.2000, 0.0000, 0.2290, 0.0870, 0.1910, 0.1282, 0.5300, 0.4000, 0.3892, 0.0850, 0.1130
0.6800, 0.0000, 0.2750, 0.1070, 0.2410, 0.1496, 0.6400, 0.4000, 0.4920, 0.0900, 0.1430
0.5200, 1.0000, 0.2430, 0.0860, 0.1970, 0.1336, 0.4400, 0.5000, 0.4575, 0.0910, 0.0510
0.4400, 0.0000, 0.2310, 0.0870, 0.2130, 0.1264, 0.7700, 0.3000, 0.3871, 0.0720, 0.0520
0.3800, 0.0000, 0.2730, 0.0810, 0.1460, 0.0816, 0.4700, 0.3000, 0.4466, 0.0810, 0.2100
0.4900, 0.0000, 0.2270, 0.0653, 0.1680, 0.0962, 0.6200, 0.2710, 0.3892, 0.0600, 0.0650
0.6100, 0.0000, 0.3300, 0.0950, 0.1820, 0.1148, 0.5400, 0.3000, 0.4190, 0.0740, 0.1410
0.2900, 1.0000, 0.1940, 0.0830, 0.1520, 0.1058, 0.3900, 0.4000, 0.3584, 0.0830, 0.0550
0.6100, 0.0000, 0.2580, 0.0980, 0.2350, 0.1258, 0.7600, 0.3000, 0.5112, 0.0820, 0.1340
0.3400, 1.0000, 0.2260, 0.0750, 0.1660, 0.0918, 0.6000, 0.3000, 0.4263, 0.1080, 0.0420
0.3600, 0.0000, 0.2190, 0.0890, 0.1890, 0.1052, 0.6800, 0.3000, 0.4369, 0.0960, 0.1110
0.5200, 0.0000, 0.2400, 0.0830, 0.1670, 0.0866, 0.7100, 0.2000, 0.3850, 0.0940, 0.0980
0.6100, 0.0000, 0.3120, 0.0790, 0.2350, 0.1568, 0.4700, 0.5000, 0.5050, 0.0960, 0.1640
0.4300, 0.0000, 0.2680, 0.1230, 0.1930, 0.1022, 0.6700, 0.3000, 0.4779, 0.0940, 0.0480
0.3500, 0.0000, 0.2040, 0.0650, 0.1870, 0.1056, 0.6700, 0.2790, 0.4277, 0.0780, 0.0960
0.2700, 0.0000, 0.2480, 0.0910, 0.1890, 0.1068, 0.6900, 0.3000, 0.4190, 0.0690, 0.0900
0.2900, 0.0000, 0.2100, 0.0710, 0.1560, 0.0970, 0.3800, 0.4000, 0.4654, 0.0900, 0.1620
0.6400, 1.0000, 0.2730, 0.1090, 0.1860, 0.1076, 0.3800, 0.5000, 0.5308, 0.0990, 0.1500
0.4100, 0.0000, 0.3460, 0.0873, 0.2050, 0.1426, 0.4100, 0.5000, 0.4673, 0.1100, 0.2790
0.4900, 1.0000, 0.2590, 0.0910, 0.1780, 0.1066, 0.5200, 0.3000, 0.4575, 0.0750, 0.0920
0.4800, 0.0000, 0.2040, 0.0980, 0.2090, 0.1394, 0.4600, 0.5000, 0.4771, 0.0780, 0.0830
0.5300, 0.0000, 0.2800, 0.0880, 0.2330, 0.1438, 0.5800, 0.4000, 0.5050, 0.0910, 0.1280
0.5300, 1.0000, 0.2220, 0.1130, 0.1970, 0.1152, 0.6700, 0.3000, 0.4304, 0.1000, 0.1020
0.2300, 0.0000, 0.2900, 0.0900, 0.2160, 0.1314, 0.6500, 0.3000, 0.4585, 0.0910, 0.3020
0.6500, 1.0000, 0.3020, 0.0980, 0.2190, 0.1606, 0.4000, 0.5000, 0.4522, 0.0840, 0.1980
0.4100, 0.0000, 0.3240, 0.0940, 0.1710, 0.1044, 0.5600, 0.3000, 0.3970, 0.0760, 0.0950
0.5500, 1.0000, 0.2340, 0.0830, 0.1660, 0.1016, 0.4600, 0.4000, 0.4522, 0.0960, 0.0530
0.2200, 0.0000, 0.1930, 0.0820, 0.1560, 0.0932, 0.5200, 0.3000, 0.3989, 0.0710, 0.1340
0.5600, 0.0000, 0.3100, 0.0787, 0.1870, 0.1414, 0.3400, 0.5500, 0.4060, 0.0900, 0.1440
0.5400, 1.0000, 0.3060, 0.1033, 0.1440, 0.0798, 0.3000, 0.4800, 0.5142, 0.1010, 0.2320
0.5900, 1.0000, 0.2550, 0.0953, 0.1900, 0.1394, 0.3500, 0.5430, 0.4357, 0.1170, 0.0810
0.6000, 1.0000, 0.2340, 0.0880, 0.1530, 0.0898, 0.5800, 0.3000, 0.3258, 0.0950, 0.1040
0.5400, 0.0000, 0.2680, 0.0870, 0.2060, 0.1220, 0.6800, 0.3000, 0.4382, 0.0800, 0.0590
0.2500, 0.0000, 0.2830, 0.0870, 0.1930, 0.1280, 0.4900, 0.4000, 0.4382, 0.0920, 0.2460
0.5400, 1.0000, 0.2770, 0.1130, 0.2000, 0.1284, 0.3700, 0.5000, 0.5153, 0.1130, 0.2970
0.5500, 0.0000, 0.3660, 0.1130, 0.1990, 0.0944, 0.4300, 0.4630, 0.5730, 0.0970, 0.2580
0.4000, 1.0000, 0.2650, 0.0930, 0.2360, 0.1470, 0.3700, 0.7000, 0.5561, 0.0920, 0.2290
0.6200, 1.0000, 0.3180, 0.1150, 0.1990, 0.1286, 0.4400, 0.5000, 0.4883, 0.0980, 0.2750
0.6500, 0.0000, 0.2440, 0.1200, 0.2220, 0.1356, 0.3700, 0.6000, 0.5509, 0.1240, 0.2810
0.3300, 1.0000, 0.2540, 0.1020, 0.2060, 0.1410, 0.3900, 0.5000, 0.4868, 0.1050, 0.1790
0.5300, 0.0000, 0.2200, 0.0940, 0.1750, 0.0880, 0.5900, 0.3000, 0.4942, 0.0980, 0.2000
0.3500, 0.0000, 0.2680, 0.0980, 0.1620, 0.1036, 0.4500, 0.4000, 0.4205, 0.0860, 0.2000
0.6600, 0.0000, 0.2800, 0.1010, 0.1950, 0.1292, 0.4000, 0.5000, 0.4860, 0.0940, 0.1730
0.6200, 1.0000, 0.3390, 0.1010, 0.2210, 0.1564, 0.3500, 0.6000, 0.4997, 0.1030, 0.1800
0.5000, 1.0000, 0.2960, 0.0943, 0.3000, 0.2424, 0.3300, 0.9090, 0.4812, 0.1090, 0.0840
0.4700, 0.0000, 0.2860, 0.0970, 0.1640, 0.0906, 0.5600, 0.3000, 0.4466, 0.0880, 0.1210
0.4700, 1.0000, 0.2560, 0.0940, 0.1650, 0.0748, 0.4000, 0.4000, 0.5526, 0.0930, 0.1610
0.2400, 0.0000, 0.2070, 0.0870, 0.1490, 0.0806, 0.6100, 0.2000, 0.3611, 0.0780, 0.0990
0.5800, 1.0000, 0.2620, 0.0910, 0.2170, 0.1242, 0.7100, 0.3000, 0.4691, 0.0680, 0.1090
0.3400, 0.0000, 0.2060, 0.0870, 0.1850, 0.1122, 0.5800, 0.3000, 0.4304, 0.0740, 0.1150
0.5100, 0.0000, 0.2790, 0.0960, 0.1960, 0.1222, 0.4200, 0.5000, 0.5069, 0.1200, 0.2680
0.3100, 1.0000, 0.3530, 0.1250, 0.1870, 0.1124, 0.4800, 0.4000, 0.4890, 0.1090, 0.2740
0.2200, 0.0000, 0.1990, 0.0750, 0.1750, 0.1086, 0.5400, 0.3000, 0.4127, 0.0720, 0.1580
0.5300, 1.0000, 0.2440, 0.0920, 0.2140, 0.1460, 0.5000, 0.4000, 0.4500, 0.0970, 0.1070
0.3700, 1.0000, 0.2140, 0.0830, 0.1280, 0.0696, 0.4900, 0.3000, 0.3850, 0.0840, 0.0830
0.2800, 0.0000, 0.3040, 0.0850, 0.1980, 0.1156, 0.6700, 0.3000, 0.4344, 0.0800, 0.1030
0.4700, 0.0000, 0.3160, 0.0840, 0.1540, 0.0880, 0.3000, 0.5100, 0.5199, 0.1050, 0.2720
0.2300, 0.0000, 0.1880, 0.0780, 0.1450, 0.0720, 0.6300, 0.2000, 0.3912, 0.0860, 0.0850
0.5000, 0.0000, 0.3100, 0.1230, 0.1780, 0.1050, 0.4800, 0.4000, 0.4828, 0.0880, 0.2800
0.5800, 1.0000, 0.3670, 0.1170, 0.1660, 0.0938, 0.4400, 0.4000, 0.4949, 0.1090, 0.3360
0.5500, 0.0000, 0.3210, 0.1100, 0.1640, 0.0842, 0.4200, 0.4000, 0.5242, 0.0900, 0.2810
0.6000, 1.0000, 0.2770, 0.1070, 0.1670, 0.1146, 0.3800, 0.4000, 0.4277, 0.0950, 0.1180
0.4100, 0.0000, 0.3080, 0.0810, 0.2140, 0.1520, 0.2800, 0.7600, 0.5136, 0.1230, 0.3170
0.6000, 1.0000, 0.2750, 0.1060, 0.2290, 0.1438, 0.5100, 0.4000, 0.5142, 0.0910, 0.2350
0.4000, 0.0000, 0.2690, 0.0920, 0.2030, 0.1198, 0.7000, 0.3000, 0.4190, 0.0810, 0.0600
0.5700, 1.0000, 0.3070, 0.0900, 0.2040, 0.1478, 0.3400, 0.6000, 0.4710, 0.0930, 0.1740
0.3700, 0.0000, 0.3830, 0.1130, 0.1650, 0.0946, 0.5300, 0.3000, 0.4466, 0.0790, 0.2590
0.4000, 1.0000, 0.3190, 0.0950, 0.1980, 0.1356, 0.3800, 0.5000, 0.4804, 0.0930, 0.1780
0.3300, 0.0000, 0.3500, 0.0890, 0.2000, 0.1304, 0.4200, 0.4760, 0.4927, 0.1010, 0.1280
0.3200, 1.0000, 0.2780, 0.0890, 0.2160, 0.1462, 0.5500, 0.4000, 0.4304, 0.0910, 0.0960
0.3500, 1.0000, 0.2590, 0.0810, 0.1740, 0.1024, 0.3100, 0.6000, 0.5313, 0.0820, 0.1260
0.5500, 0.0000, 0.3290, 0.1020, 0.1640, 0.1062, 0.4100, 0.4000, 0.4431, 0.0890, 0.2880
0.4900, 0.0000, 0.2600, 0.0930, 0.1830, 0.1002, 0.6400, 0.3000, 0.4543, 0.0880, 0.0880
0.3900, 1.0000, 0.2630, 0.1150, 0.2180, 0.1582, 0.3200, 0.7000, 0.4935, 0.1090, 0.2920
0.6000, 1.0000, 0.2230, 0.1130, 0.1860, 0.1258, 0.4600, 0.4000, 0.4263, 0.0940, 0.0710
0.6700, 1.0000, 0.2830, 0.0930, 0.2040, 0.1322, 0.4900, 0.4000, 0.4736, 0.0920, 0.1970
0.4100, 1.0000, 0.3200, 0.1090, 0.2510, 0.1706, 0.4900, 0.5000, 0.5056, 0.1030, 0.1860
0.4400, 0.0000, 0.2540, 0.0950, 0.1620, 0.0926, 0.5300, 0.3000, 0.4407, 0.0830, 0.0250
0.4800, 1.0000, 0.2330, 0.0893, 0.2120, 0.1428, 0.4600, 0.4610, 0.4754, 0.0980, 0.0840
0.4500, 0.0000, 0.2030, 0.0743, 0.1900, 0.1262, 0.4900, 0.3880, 0.4304, 0.0790, 0.0960
0.4700, 0.0000, 0.3040, 0.1200, 0.1990, 0.1200, 0.4600, 0.4000, 0.5106, 0.0870, 0.1950
0.4600, 0.0000, 0.2060, 0.0730, 0.1720, 0.1070, 0.5100, 0.3000, 0.4249, 0.0800, 0.0530
0.3600, 1.0000, 0.3230, 0.1150, 0.2860, 0.1994, 0.3900, 0.7000, 0.5472, 0.1120, 0.2170
0.3400, 0.0000, 0.2920, 0.0730, 0.1720, 0.1082, 0.4900, 0.4000, 0.4304, 0.0910, 0.1720
0.5300, 1.0000, 0.3310, 0.1170, 0.1830, 0.1190, 0.4800, 0.4000, 0.4382, 0.1060, 0.1310
0.6100, 0.0000, 0.2460, 0.1010, 0.2090, 0.1068, 0.7700, 0.3000, 0.4836, 0.0880, 0.2140
0.3700, 0.0000, 0.2020, 0.0810, 0.1620, 0.0878, 0.6300, 0.3000, 0.4025, 0.0880, 0.0590
0.3300, 1.0000, 0.2080, 0.0840, 0.1250, 0.0702, 0.4600, 0.3000, 0.3784, 0.0660, 0.0700
0.6800, 0.0000, 0.3280, 0.1057, 0.2050, 0.1164, 0.4000, 0.5130, 0.5493, 0.1170, 0.2200
0.4900, 1.0000, 0.3190, 0.0940, 0.2340, 0.1558, 0.3400, 0.7000, 0.5398, 0.1220, 0.2680
0.4800, 0.0000, 0.2390, 0.1090, 0.2320, 0.1052, 0.3700, 0.6000, 0.6107, 0.0960, 0.1520
0.5500, 1.0000, 0.2450, 0.0840, 0.1790, 0.1058, 0.6600, 0.3000, 0.3584, 0.0870, 0.0470
0.4300, 0.0000, 0.2210, 0.0660, 0.1340, 0.0772, 0.4500, 0.3000, 0.4078, 0.0800, 0.0740
0.6000, 1.0000, 0.3300, 0.0970, 0.2170, 0.1256, 0.4500, 0.5000, 0.5447, 0.1120, 0.2950
0.3100, 1.0000, 0.1900, 0.0930, 0.1370, 0.0730, 0.4700, 0.3000, 0.4443, 0.0780, 0.1010
0.5300, 1.0000, 0.2730, 0.0820, 0.1190, 0.0550, 0.3900, 0.3000, 0.4828, 0.0930, 0.1510
0.6700, 0.0000, 0.2280, 0.0870, 0.1660, 0.0986, 0.5200, 0.3000, 0.4344, 0.0920, 0.1270
0.6100, 1.0000, 0.2820, 0.1060, 0.2040, 0.1320, 0.5200, 0.4000, 0.4605, 0.0960, 0.2370
0.6200, 0.0000, 0.2890, 0.0873, 0.2060, 0.1272, 0.3300, 0.6240, 0.5434, 0.0990, 0.2250
0.6000, 0.0000, 0.2560, 0.0870, 0.2070, 0.1258, 0.6900, 0.3000, 0.4111, 0.0840, 0.0810
0.4200, 0.0000, 0.2490, 0.0910, 0.2040, 0.1418, 0.3800, 0.5000, 0.4796, 0.0890, 0.1510
0.3800, 1.0000, 0.2680, 0.1050, 0.1810, 0.1192, 0.3700, 0.5000, 0.4820, 0.0910, 0.1070
0.6200, 0.0000, 0.2240, 0.0790, 0.2220, 0.1474, 0.5900, 0.4000, 0.4357, 0.0760, 0.0640
0.6100, 1.0000, 0.2690, 0.1110, 0.2360, 0.1724, 0.3900, 0.6000, 0.4812, 0.0890, 0.1380
0.6100, 1.0000, 0.2310, 0.1130, 0.1860, 0.1144, 0.4700, 0.4000, 0.4812, 0.1050, 0.1850
0.5300, 0.0000, 0.2860, 0.0880, 0.1710, 0.0988, 0.4100, 0.4000, 0.5050, 0.0990, 0.2650
0.2800, 1.0000, 0.2470, 0.0970, 0.1750, 0.0996, 0.3200, 0.5000, 0.5380, 0.0870, 0.1010
0.2600, 1.0000, 0.3030, 0.0890, 0.2180, 0.1522, 0.3100, 0.7000, 0.5159, 0.0820, 0.1370
0.3000, 0.0000, 0.2130, 0.0870, 0.1340, 0.0630, 0.6300, 0.2000, 0.3689, 0.0660, 0.1430
0.5000, 0.0000, 0.2610, 0.1090, 0.2430, 0.1606, 0.6200, 0.4000, 0.4625, 0.0890, 0.1410
0.4800, 0.0000, 0.2020, 0.0950, 0.1870, 0.1174, 0.5300, 0.4000, 0.4419, 0.0850, 0.0790
0.5100, 0.0000, 0.2520, 0.1030, 0.1760, 0.1122, 0.3700, 0.5000, 0.4898, 0.0900, 0.2920
0.4700, 1.0000, 0.2250, 0.0820, 0.1310, 0.0668, 0.4100, 0.3000, 0.4754, 0.0890, 0.1780
0.6400, 1.0000, 0.2350, 0.0970, 0.2030, 0.1290, 0.5900, 0.3000, 0.4318, 0.0770, 0.0910
0.5100, 1.0000, 0.2590, 0.0760, 0.2400, 0.1690, 0.3900, 0.6000, 0.5075, 0.0960, 0.1160
0.3000, 0.0000, 0.2090, 0.1040, 0.1520, 0.0838, 0.4700, 0.3000, 0.4663, 0.0970, 0.0860
0.5600, 1.0000, 0.2870, 0.0990, 0.2080, 0.1464, 0.3900, 0.5000, 0.4727, 0.0970, 0.1220
0.4200, 0.0000, 0.2210, 0.0850, 0.2130, 0.1386, 0.6000, 0.4000, 0.4277, 0.0940, 0.0720
0.6200, 1.0000, 0.2670, 0.1150, 0.1830, 0.1240, 0.3500, 0.5000, 0.4788, 0.1000, 0.1290
0.3400, 0.0000, 0.3140, 0.0870, 0.1490, 0.0938, 0.4600, 0.3000, 0.3829, 0.0770, 0.1420
0.6000, 0.0000, 0.2220, 0.1047, 0.2210, 0.1054, 0.6000, 0.3680, 0.5628, 0.0930, 0.0900
0.6400, 0.0000, 0.2100, 0.0923, 0.2270, 0.1468, 0.6500, 0.3490, 0.4331, 0.1020, 0.1580
0.3900, 1.0000, 0.2120, 0.0900, 0.1820, 0.1104, 0.6000, 0.3000, 0.4060, 0.0980, 0.0390
0.7100, 1.0000, 0.2650, 0.1050, 0.2810, 0.1736, 0.5500, 0.5000, 0.5568, 0.0840, 0.1960
0.4800, 1.0000, 0.2920, 0.1100, 0.2180, 0.1516, 0.3900, 0.6000, 0.4920, 0.0980, 0.2220
0.7900, 1.0000, 0.2700, 0.1030, 0.1690, 0.1108, 0.3700, 0.5000, 0.4663, 0.1100, 0.2770
0.4000, 0.0000, 0.3070, 0.0990, 0.1770, 0.0854, 0.5000, 0.4000, 0.5338, 0.0850, 0.0990
0.4900, 1.0000, 0.2880, 0.0920, 0.2070, 0.1400, 0.4400, 0.5000, 0.4745, 0.0920, 0.1960
0.5100, 0.0000, 0.3060, 0.1030, 0.1980, 0.1066, 0.5700, 0.3000, 0.5148, 0.1000, 0.2020
0.5700, 0.0000, 0.3010, 0.1170, 0.2020, 0.1396, 0.4200, 0.5000, 0.4625, 0.1200, 0.1550
0.5900, 1.0000, 0.2470, 0.1140, 0.1520, 0.1048, 0.2900, 0.5000, 0.4511, 0.0880, 0.0770
0.5100, 0.0000, 0.2770, 0.0990, 0.2290, 0.1456, 0.6900, 0.3000, 0.4277, 0.0770, 0.1910
0.7400, 0.0000, 0.2980, 0.1010, 0.1710, 0.1048, 0.5000, 0.3000, 0.4394, 0.0860, 0.0700
0.6700, 0.0000, 0.2670, 0.1050, 0.2250, 0.1354, 0.6900, 0.3000, 0.4635, 0.0960, 0.0730
0.4900, 0.0000, 0.1980, 0.0880, 0.1880, 0.1148, 0.5700, 0.3000, 0.4394, 0.0930, 0.0490
0.5700, 0.0000, 0.2330, 0.0880, 0.1550, 0.0636, 0.7800, 0.2000, 0.4205, 0.0780, 0.0650
0.5600, 1.0000, 0.3510, 0.1230, 0.1640, 0.0950, 0.3800, 0.4000, 0.5043, 0.1170, 0.2630
0.5200, 1.0000, 0.2970, 0.1090, 0.2280, 0.1628, 0.3100, 0.8000, 0.5142, 0.1030, 0.2480
0.6900, 0.0000, 0.2930, 0.1240, 0.2230, 0.1390, 0.5400, 0.4000, 0.5011, 0.1020, 0.2960
0.3700, 0.0000, 0.2030, 0.0830, 0.1850, 0.1246, 0.3800, 0.5000, 0.4719, 0.0880, 0.2140
0.2400, 0.0000, 0.2250, 0.0890, 0.1410, 0.0680, 0.5200, 0.3000, 0.4654, 0.0840, 0.1850
0.5500, 1.0000, 0.2270, 0.0930, 0.1540, 0.0942, 0.5300, 0.3000, 0.3526, 0.0750, 0.0780
0.3600, 0.0000, 0.2280, 0.0870, 0.1780, 0.1160, 0.4100, 0.4000, 0.4654, 0.0820, 0.0930
0.4200, 1.0000, 0.2400, 0.1070, 0.1500, 0.0850, 0.4400, 0.3000, 0.4654, 0.0960, 0.2520
0.2100, 0.0000, 0.2420, 0.0760, 0.1470, 0.0770, 0.5300, 0.3000, 0.4443, 0.0790, 0.1500
0.4100, 0.0000, 0.2020, 0.0620, 0.1530, 0.0890, 0.5000, 0.3000, 0.4249, 0.0890, 0.0770
0.5700, 1.0000, 0.2940, 0.1090, 0.1600, 0.0876, 0.3100, 0.5000, 0.5333, 0.0920, 0.2080
0.2000, 1.0000, 0.2210, 0.0870, 0.1710, 0.0996, 0.5800, 0.3000, 0.4205, 0.0780, 0.0770
0.6700, 1.0000, 0.2360, 0.1113, 0.1890, 0.1054, 0.7000, 0.2700, 0.4220, 0.0930, 0.1080
0.3400, 0.0000, 0.2520, 0.0770, 0.1890, 0.1206, 0.5300, 0.4000, 0.4344, 0.0790, 0.1600
0.4100, 1.0000, 0.2490, 0.0860, 0.1920, 0.1150, 0.6100, 0.3000, 0.4382, 0.0940, 0.0530
0.3800, 1.0000, 0.3300, 0.0780, 0.3010, 0.2150, 0.5000, 0.6020, 0.5193, 0.1080, 0.2200
0.5100, 0.0000, 0.2350, 0.1010, 0.1950, 0.1210, 0.5100, 0.4000, 0.4745, 0.0940, 0.1540
0.5200, 1.0000, 0.2640, 0.0913, 0.2180, 0.1520, 0.3900, 0.5590, 0.4905, 0.0990, 0.2590
0.6700, 0.0000, 0.2980, 0.0800, 0.1720, 0.0934, 0.6300, 0.3000, 0.4357, 0.0820, 0.0900
0.6100, 0.0000, 0.3000, 0.1080, 0.1940, 0.1000, 0.5200, 0.3730, 0.5347, 0.1050, 0.2460
0.6700, 1.0000, 0.2500, 0.1117, 0.1460, 0.0934, 0.3300, 0.4420, 0.4585, 0.1030, 0.1240
0.5600, 0.0000, 0.2700, 0.1050, 0.2470, 0.1606, 0.5400, 0.5000, 0.5088, 0.0940, 0.0670
0.6400, 0.0000, 0.2000, 0.0747, 0.1890, 0.1148, 0.6200, 0.3050, 0.4111, 0.0910, 0.0720
0.5800, 1.0000, 0.2550, 0.1120, 0.1630, 0.1106, 0.2900, 0.6000, 0.4762, 0.0860, 0.2570
0.5500, 0.0000, 0.2820, 0.0910, 0.2500, 0.1402, 0.6700, 0.4000, 0.5366, 0.1030, 0.2620
0.6200, 1.0000, 0.3330, 0.1140, 0.1820, 0.1140, 0.3800, 0.5000, 0.5011, 0.0960, 0.2750
0.5700, 1.0000, 0.2560, 0.0960, 0.2000, 0.1330, 0.5200, 0.3850, 0.4318, 0.1050, 0.1770
0.2000, 1.0000, 0.2420, 0.0880, 0.1260, 0.0722, 0.4500, 0.3000, 0.3784, 0.0740, 0.0710
0.5300, 1.0000, 0.2210, 0.0980, 0.1650, 0.1052, 0.4700, 0.4000, 0.4159, 0.0810, 0.0470
0.3200, 1.0000, 0.3140, 0.0890, 0.1530, 0.0842, 0.5600, 0.3000, 0.4159, 0.0900, 0.1870
0.4100, 0.0000, 0.2310, 0.0860, 0.1480, 0.0780, 0.5800, 0.3000, 0.4094, 0.0600, 0.1250
0.6000, 0.0000, 0.2340, 0.0767, 0.2470, 0.1480, 0.6500, 0.3800, 0.5136, 0.0770, 0.0780
0.2600, 0.0000, 0.1880, 0.0830, 0.1910, 0.1036, 0.6900, 0.3000, 0.4522, 0.0690, 0.0510
0.3700, 0.0000, 0.3080, 0.1120, 0.2820, 0.1972, 0.4300, 0.7000, 0.5342, 0.1010, 0.2580
0.4500, 0.0000, 0.3200, 0.1100, 0.2240, 0.1342, 0.4500, 0.5000, 0.5412, 0.0930, 0.2150
0.6700, 0.0000, 0.3160, 0.1160, 0.1790, 0.0904, 0.4100, 0.4000, 0.5472, 0.1000, 0.3030
0.3400, 1.0000, 0.3550, 0.1200, 0.2330, 0.1466, 0.3400, 0.7000, 0.5568, 0.1010, 0.2430
0.5000, 0.0000, 0.3190, 0.0783, 0.2070, 0.1492, 0.3800, 0.5450, 0.4595, 0.0840, 0.0910
0.7100, 0.0000, 0.2950, 0.0970, 0.2270, 0.1516, 0.4500, 0.5000, 0.5024, 0.1080, 0.1500
0.5700, 1.0000, 0.3160, 0.1170, 0.2250, 0.1076, 0.4000, 0.6000, 0.5958, 0.1130, 0.3100
0.4900, 0.0000, 0.2030, 0.0930, 0.1840, 0.1030, 0.6100, 0.3000, 0.4605, 0.0930, 0.1530
0.3500, 0.0000, 0.4130, 0.0810, 0.1680, 0.1028, 0.3700, 0.5000, 0.4949, 0.0940, 0.3460
0.4100, 1.0000, 0.2120, 0.1020, 0.1840, 0.1004, 0.6400, 0.3000, 0.4585, 0.0790, 0.0630
0.7000, 1.0000, 0.2410, 0.0823, 0.1940, 0.1492, 0.3100, 0.6260, 0.4234, 0.1050, 0.0890
0.5200, 0.0000, 0.2300, 0.1070, 0.1790, 0.1237, 0.4250, 0.4210, 0.4159, 0.0930, 0.0500
0.6000, 0.0000, 0.2560, 0.0780, 0.1950, 0.0954, 0.9100, 0.2000, 0.3761, 0.0870, 0.0390
0.6200, 0.0000, 0.2250, 0.1250, 0.2150, 0.0990, 0.9800, 0.2000, 0.4500, 0.0950, 0.1030
0.4400, 1.0000, 0.3820, 0.1230, 0.2010, 0.1266, 0.4400, 0.5000, 0.5024, 0.0920, 0.3080
0.2800, 1.0000, 0.1920, 0.0810, 0.1550, 0.0946, 0.5100, 0.3000, 0.3850, 0.0870, 0.1160
0.5800, 1.0000, 0.2900, 0.0850, 0.1560, 0.1092, 0.3600, 0.4000, 0.3989, 0.0860, 0.1450
0.3900, 1.0000, 0.2400, 0.0897, 0.1900, 0.1136, 0.5200, 0.3650, 0.4804, 0.1010, 0.0740
0.3400, 1.0000, 0.2060, 0.0980, 0.1830, 0.0920, 0.8300, 0.2000, 0.3689, 0.0920, 0.0450
0.6500, 0.0000, 0.2630, 0.0700, 0.2440, 0.1662, 0.5100, 0.5000, 0.4898, 0.0980, 0.1150
0.6600, 1.0000, 0.3460, 0.1150, 0.2040, 0.1394, 0.3600, 0.6000, 0.4963, 0.1090, 0.2640
0.5100, 0.0000, 0.2340, 0.0870, 0.2200, 0.1088, 0.9300, 0.2000, 0.4511, 0.0820, 0.0870
0.5000, 1.0000, 0.2920, 0.1190, 0.1620, 0.0852, 0.5400, 0.3000, 0.4736, 0.0950, 0.2020
0.5900, 1.0000, 0.2720, 0.1070, 0.1580, 0.1020, 0.3900, 0.4000, 0.4443, 0.0930, 0.1270
0.5200, 0.0000, 0.2700, 0.0783, 0.1340, 0.0730, 0.4400, 0.3050, 0.4443, 0.0690, 0.1820
0.6900, 1.0000, 0.2450, 0.1080, 0.2430, 0.1364, 0.4000, 0.6000, 0.5808, 0.1000, 0.2410
0.5300, 0.0000, 0.2410, 0.1050, 0.1840, 0.1134, 0.4600, 0.4000, 0.4812, 0.0950, 0.0660
0.4700, 1.0000, 0.2530, 0.0980, 0.1730, 0.1056, 0.4400, 0.4000, 0.4762, 0.1080, 0.0940
0.5200, 0.0000, 0.2880, 0.1130, 0.2800, 0.1740, 0.6700, 0.4000, 0.5273, 0.0860, 0.2830
0.3900, 0.0000, 0.2090, 0.0950, 0.1500, 0.0656, 0.6800, 0.2000, 0.4407, 0.0950, 0.0640
0.6700, 1.0000, 0.2300, 0.0700, 0.1840, 0.1280, 0.3500, 0.5000, 0.4654, 0.0990, 0.1020
0.5900, 1.0000, 0.2410, 0.0960, 0.1700, 0.0986, 0.5400, 0.3000, 0.4466, 0.0850, 0.2000
0.5100, 1.0000, 0.2810, 0.1060, 0.2020, 0.1222, 0.5500, 0.4000, 0.4820, 0.0870, 0.2650
0.2300, 1.0000, 0.1800, 0.0780, 0.1710, 0.0960, 0.4800, 0.4000, 0.4905, 0.0920, 0.0940
0.6800, 0.0000, 0.2590, 0.0930, 0.2530, 0.1812, 0.5300, 0.5000, 0.4543, 0.0980, 0.2300
0.4400, 0.0000, 0.2150, 0.0850, 0.1570, 0.0922, 0.5500, 0.3000, 0.3892, 0.0840, 0.1810
0.6000, 1.0000, 0.2430, 0.1030, 0.1410, 0.0866, 0.3300, 0.4000, 0.4673, 0.0780, 0.1560
0.5200, 0.0000, 0.2450, 0.0900, 0.1980, 0.1290, 0.2900, 0.7000, 0.5298, 0.0860, 0.2330
0.3800, 0.0000, 0.2130, 0.0720, 0.1650, 0.0602, 0.8800, 0.2000, 0.4431, 0.0900, 0.0600
0.6100, 0.0000, 0.2580, 0.0900, 0.2800, 0.1954, 0.5500, 0.5000, 0.4997, 0.0900, 0.2190
0.6800, 1.0000, 0.2480, 0.1010, 0.2210, 0.1514, 0.6000, 0.4000, 0.3871, 0.0870, 0.0800
0.2800, 1.0000, 0.3150, 0.0830, 0.2280, 0.1494, 0.3800, 0.6000, 0.5313, 0.0830, 0.0680
0.6500, 1.0000, 0.3350, 0.1020, 0.1900, 0.1262, 0.3500, 0.5000, 0.4970, 0.1020, 0.3320
0.6900, 0.0000, 0.2810, 0.1130, 0.2340, 0.1428, 0.5200, 0.4000, 0.5278, 0.0770, 0.2480
0.5100, 0.0000, 0.2430, 0.0853, 0.1530, 0.0716, 0.7100, 0.2150, 0.3951, 0.0820, 0.0840
0.2900, 0.0000, 0.3500, 0.0983, 0.2040, 0.1426, 0.5000, 0.4080, 0.4043, 0.0910, 0.2000
0.5500, 1.0000, 0.2350, 0.0930, 0.1770, 0.1268, 0.4100, 0.4000, 0.3829, 0.0830, 0.0550
0.3400, 1.0000, 0.3000, 0.0830, 0.1850, 0.1072, 0.5300, 0.3000, 0.4820, 0.0920, 0.0850
0.6700, 0.0000, 0.2070, 0.0830, 0.1700, 0.0998, 0.5900, 0.3000, 0.4025, 0.0770, 0.0890
0.4900, 0.0000, 0.2560, 0.0760, 0.1610, 0.0998, 0.5100, 0.3000, 0.3932, 0.0780, 0.0310
0.5500, 1.0000, 0.2290, 0.0810, 0.1230, 0.0672, 0.4100, 0.3000, 0.4304, 0.0880, 0.1290
0.5900, 1.0000, 0.2510, 0.0900, 0.1630, 0.1014, 0.4600, 0.4000, 0.4357, 0.0910, 0.0830
0.5300, 0.0000, 0.3320, 0.0827, 0.1860, 0.1068, 0.4600, 0.4040, 0.5112, 0.1020, 0.2750
0.4800, 1.0000, 0.2410, 0.1100, 0.2090, 0.1346, 0.5800, 0.4000, 0.4407, 0.1000, 0.0650
0.5200, 0.0000, 0.2950, 0.1043, 0.2110, 0.1328, 0.4900, 0.4310, 0.4984, 0.0980, 0.1980
0.6900, 0.0000, 0.2960, 0.1220, 0.2310, 0.1284, 0.5600, 0.4000, 0.5451, 0.0860, 0.2360
0.6000, 1.0000, 0.2280, 0.1100, 0.2450, 0.1898, 0.3900, 0.6000, 0.4394, 0.0880, 0.2530
0.4600, 1.0000, 0.2270, 0.0830, 0.1830, 0.1258, 0.3200, 0.6000, 0.4836, 0.0750, 0.1240
0.5100, 1.0000, 0.2620, 0.1010, 0.1610, 0.0996, 0.4800, 0.3000, 0.4205, 0.0880, 0.0440
0.6700, 1.0000, 0.2350, 0.0960, 0.2070, 0.1382, 0.4200, 0.5000, 0.4898, 0.1110, 0.1720
0.4900, 0.0000, 0.2210, 0.0850, 0.1360, 0.0634, 0.6200, 0.2190, 0.3970, 0.0720, 0.1140
0.4600, 1.0000, 0.2650, 0.0940, 0.2470, 0.1602, 0.5900, 0.4000, 0.4935, 0.1110, 0.1420
0.4700, 0.0000, 0.3240, 0.1050, 0.1880, 0.1250, 0.4600, 0.4090, 0.4443, 0.0990, 0.1090
0.7500, 0.0000, 0.3010, 0.0780, 0.2220, 0.1542, 0.4400, 0.5050, 0.4779, 0.0970, 0.1800
0.2800, 0.0000, 0.2420, 0.0930, 0.1740, 0.1064, 0.5400, 0.3000, 0.4220, 0.0840, 0.1440
0.6500, 1.0000, 0.3130, 0.1100, 0.2130, 0.1280, 0.4700, 0.5000, 0.5247, 0.0910, 0.1630
0.4200, 0.0000, 0.3010, 0.0910, 0.1820, 0.1148, 0.4900, 0.4000, 0.4511, 0.0820, 0.1470
0.5100, 0.0000, 0.2450, 0.0790, 0.2120, 0.1286, 0.6500, 0.3000, 0.4522, 0.0910, 0.0970
0.5300, 1.0000, 0.2770, 0.0950, 0.1900, 0.1018, 0.4100, 0.5000, 0.5464, 0.1010, 0.2200
0.5400, 0.0000, 0.2320, 0.1107, 0.2380, 0.1628, 0.4800, 0.4960, 0.4913, 0.1080, 0.1900
0.7300, 0.0000, 0.2700, 0.1020, 0.2110, 0.1210, 0.6700, 0.3000, 0.4745, 0.0990, 0.1090
0.5400, 0.0000, 0.2680, 0.1080, 0.1760, 0.0806, 0.6700, 0.3000, 0.4956, 0.1060, 0.1910
0.4200, 0.0000, 0.2920, 0.0930, 0.2490, 0.1742, 0.4500, 0.6000, 0.5004, 0.0920, 0.1220
0.7500, 0.0000, 0.3120, 0.1177, 0.2290, 0.1388, 0.2900, 0.7900, 0.5724, 0.1060, 0.2300
0.5500, 1.0000, 0.3210, 0.1127, 0.2070, 0.0924, 0.2500, 0.8280, 0.6105, 0.1110, 0.2420
0.6800, 1.0000, 0.2570, 0.1090, 0.2330, 0.1126, 0.3500, 0.7000, 0.6057, 0.1050, 0.2480
0.5700, 0.0000, 0.2690, 0.0980, 0.2460, 0.1652, 0.3800, 0.7000, 0.5366, 0.0960, 0.2490
0.4800, 0.0000, 0.3140, 0.0753, 0.2420, 0.1516, 0.3800, 0.6370, 0.5568, 0.1030, 0.1920
0.6100, 1.0000, 0.2560, 0.0850, 0.1840, 0.1162, 0.3900, 0.5000, 0.4970, 0.0980, 0.1310
0.6900, 0.0000, 0.3700, 0.1030, 0.2070, 0.1314, 0.5500, 0.4000, 0.4635, 0.0900, 0.2370
0.3800, 0.0000, 0.3260, 0.0770, 0.1680, 0.1006, 0.4700, 0.4000, 0.4625, 0.0960, 0.0780
0.4500, 1.0000, 0.2120, 0.0940, 0.1690, 0.0968, 0.5500, 0.3000, 0.4454, 0.1020, 0.1350
0.5100, 1.0000, 0.2920, 0.1070, 0.1870, 0.1390, 0.3200, 0.6000, 0.4382, 0.0950, 0.2440
0.7100, 1.0000, 0.2400, 0.0840, 0.1380, 0.0858, 0.3900, 0.4000, 0.4190, 0.0900, 0.1990
0.5700, 0.0000, 0.3610, 0.1170, 0.1810, 0.1082, 0.3400, 0.5000, 0.5268, 0.1000, 0.2700
0.5600, 1.0000, 0.2580, 0.1030, 0.1770, 0.1144, 0.3400, 0.5000, 0.4963, 0.0990, 0.1640
0.3200, 1.0000, 0.2200, 0.0880, 0.1370, 0.0786, 0.4800, 0.3000, 0.3951, 0.0780, 0.0720
0.5000, 0.0000, 0.2190, 0.0910, 0.1900, 0.1112, 0.6700, 0.3000, 0.4078, 0.0770, 0.0960
0.4300, 0.0000, 0.3430, 0.0840, 0.2560, 0.1726, 0.3300, 0.8000, 0.5529, 0.1040, 0.3060
0.5400, 1.0000, 0.2520, 0.1150, 0.1810, 0.1200, 0.3900, 0.5000, 0.4701, 0.0920, 0.0910
0.3100, 0.0000, 0.2330, 0.0850, 0.1900, 0.1308, 0.4300, 0.4000, 0.4394, 0.0770, 0.2140
0.5600, 0.0000, 0.2570, 0.0800, 0.2440, 0.1516, 0.5900, 0.4000, 0.5118, 0.0950, 0.0950
0.4400, 0.0000, 0.2510, 0.1330, 0.1820, 0.1130, 0.5500, 0.3000, 0.4249, 0.0840, 0.2160
0.5700, 1.0000, 0.3190, 0.1110, 0.1730, 0.1162, 0.4100, 0.4000, 0.4369, 0.0870, 0.2630

Test data:

# diabetes_norm_test_100.txt
#
0.6400, 1.0000, 0.2840, 0.1110, 0.1840, 0.1270, 0.4100, 0.4000, 0.4382, 0.0970, 0.1780
0.4300, 0.0000, 0.2810, 0.1210, 0.1920, 0.1210, 0.6000, 0.3000, 0.4007, 0.0930, 0.1130
0.1900, 0.0000, 0.2530, 0.0830, 0.2250, 0.1566, 0.4600, 0.5000, 0.4719, 0.0840, 0.2000
0.7100, 1.0000, 0.2610, 0.0850, 0.2200, 0.1524, 0.4700, 0.5000, 0.4635, 0.0910, 0.1390
0.5000, 1.0000, 0.2800, 0.1040, 0.2820, 0.1968, 0.4400, 0.6000, 0.5328, 0.0950, 0.1390
0.5900, 1.0000, 0.2360, 0.0730, 0.1800, 0.1074, 0.5100, 0.4000, 0.4682, 0.0840, 0.0880
0.5700, 0.0000, 0.2450, 0.0930, 0.1860, 0.0966, 0.7100, 0.3000, 0.4522, 0.0910, 0.1480
0.4900, 1.0000, 0.2100, 0.0820, 0.1190, 0.0854, 0.2300, 0.5000, 0.3970, 0.0740, 0.0880
0.4100, 1.0000, 0.3200, 0.1260, 0.1980, 0.1042, 0.4900, 0.4000, 0.5412, 0.1240, 0.2430
0.2500, 1.0000, 0.2260, 0.0850, 0.1300, 0.0710, 0.4800, 0.3000, 0.4007, 0.0810, 0.0710
0.5200, 1.0000, 0.1970, 0.0810, 0.1520, 0.0534, 0.8200, 0.2000, 0.4419, 0.0820, 0.0770
0.3400, 0.0000, 0.2120, 0.0840, 0.2540, 0.1134, 0.5200, 0.5000, 0.6094, 0.0920, 0.1090
0.4200, 1.0000, 0.3060, 0.1010, 0.2690, 0.1722, 0.5000, 0.5000, 0.5455, 0.1060, 0.2720
0.2800, 1.0000, 0.2550, 0.0990, 0.1620, 0.1016, 0.4600, 0.4000, 0.4277, 0.0940, 0.0600
0.4700, 1.0000, 0.2330, 0.0900, 0.1950, 0.1258, 0.5400, 0.4000, 0.4331, 0.0730, 0.0540
0.3200, 1.0000, 0.3100, 0.1000, 0.1770, 0.0962, 0.4500, 0.4000, 0.5187, 0.0770, 0.2210
0.4300, 0.0000, 0.1850, 0.0870, 0.1630, 0.0936, 0.6100, 0.2670, 0.3738, 0.0800, 0.0900
0.5900, 1.0000, 0.2690, 0.1040, 0.1940, 0.1266, 0.4300, 0.5000, 0.4804, 0.1060, 0.3110
0.5300, 0.0000, 0.2830, 0.1010, 0.1790, 0.1070, 0.4800, 0.4000, 0.4788, 0.1010, 0.2810
0.6000, 0.0000, 0.2570, 0.1030, 0.1580, 0.0846, 0.6400, 0.2000, 0.3850, 0.0970, 0.1820
0.5400, 1.0000, 0.3610, 0.1150, 0.1630, 0.0984, 0.4300, 0.4000, 0.4682, 0.1010, 0.3210
0.3500, 1.0000, 0.2410, 0.0947, 0.1550, 0.0974, 0.3200, 0.4840, 0.4852, 0.0940, 0.0580
0.4900, 1.0000, 0.2580, 0.0890, 0.1820, 0.1186, 0.3900, 0.5000, 0.4804, 0.1150, 0.2620
0.5800, 0.0000, 0.2280, 0.0910, 0.1960, 0.1188, 0.4800, 0.4000, 0.4984, 0.1150, 0.2060
0.3600, 1.0000, 0.3910, 0.0900, 0.2190, 0.1358, 0.3800, 0.6000, 0.5421, 0.1030, 0.2330
0.4600, 1.0000, 0.4220, 0.0990, 0.2110, 0.1370, 0.4400, 0.5000, 0.5011, 0.0990, 0.2420
0.4400, 1.0000, 0.2660, 0.0990, 0.2050, 0.1090, 0.4300, 0.5000, 0.5580, 0.1110, 0.1230
0.4600, 0.0000, 0.2990, 0.0830, 0.1710, 0.1130, 0.3800, 0.4500, 0.4585, 0.0980, 0.1670
0.5400, 0.0000, 0.2100, 0.0780, 0.1880, 0.1074, 0.7000, 0.3000, 0.3970, 0.0730, 0.0630
0.6300, 1.0000, 0.2550, 0.1090, 0.2260, 0.1032, 0.4600, 0.5000, 0.5951, 0.0870, 0.1970
0.4100, 1.0000, 0.2420, 0.0900, 0.1990, 0.1236, 0.5700, 0.4000, 0.4522, 0.0860, 0.0710
0.2800, 0.0000, 0.2540, 0.0930, 0.1410, 0.0790, 0.4900, 0.3000, 0.4174, 0.0910, 0.1680
0.1900, 0.0000, 0.2320, 0.0750, 0.1430, 0.0704, 0.5200, 0.3000, 0.4635, 0.0720, 0.1400
0.6100, 1.0000, 0.2610, 0.1260, 0.2150, 0.1298, 0.5700, 0.4000, 0.4949, 0.0960, 0.2170
0.4800, 0.0000, 0.3270, 0.0930, 0.2760, 0.1986, 0.4300, 0.6420, 0.5148, 0.0910, 0.1210
0.5400, 1.0000, 0.2730, 0.1000, 0.2000, 0.1440, 0.3300, 0.6000, 0.4745, 0.0760, 0.2350
0.5300, 1.0000, 0.2660, 0.0930, 0.1850, 0.1224, 0.3600, 0.5000, 0.4890, 0.0820, 0.2450
0.4800, 0.0000, 0.2280, 0.1010, 0.1100, 0.0416, 0.5600, 0.2000, 0.4127, 0.0970, 0.0400
0.5300, 0.0000, 0.2880, 0.1117, 0.1450, 0.0872, 0.4600, 0.3150, 0.4078, 0.0850, 0.0520
0.2900, 1.0000, 0.1810, 0.0730, 0.1580, 0.0990, 0.4100, 0.4000, 0.4500, 0.0780, 0.1040
0.6200, 0.0000, 0.3200, 0.0880, 0.1720, 0.0690, 0.3800, 0.4000, 0.5784, 0.1000, 0.1320
0.5000, 1.0000, 0.2370, 0.0920, 0.1660, 0.0970, 0.5200, 0.3000, 0.4443, 0.0930, 0.0880
0.5800, 1.0000, 0.2360, 0.0960, 0.2570, 0.1710, 0.5900, 0.4000, 0.4905, 0.0820, 0.0690
0.5500, 1.0000, 0.2460, 0.1090, 0.1430, 0.0764, 0.5100, 0.3000, 0.4357, 0.0880, 0.2190
0.5400, 0.0000, 0.2260, 0.0900, 0.1830, 0.1042, 0.6400, 0.3000, 0.4304, 0.0920, 0.0720
0.3600, 0.0000, 0.2780, 0.0730, 0.1530, 0.1044, 0.4200, 0.4000, 0.3497, 0.0730, 0.2010
0.6300, 1.0000, 0.2410, 0.1110, 0.1840, 0.1122, 0.4400, 0.4000, 0.4935, 0.0820, 0.1100
0.4700, 1.0000, 0.2650, 0.0700, 0.1810, 0.1048, 0.6300, 0.3000, 0.4190, 0.0700, 0.0510
0.5100, 1.0000, 0.3280, 0.1120, 0.2020, 0.1006, 0.3700, 0.5000, 0.5775, 0.1090, 0.2770
0.4200, 0.0000, 0.1990, 0.0760, 0.1460, 0.0832, 0.5500, 0.3000, 0.3664, 0.0790, 0.0630
0.3700, 1.0000, 0.2360, 0.0940, 0.2050, 0.1388, 0.5300, 0.4000, 0.4190, 0.1070, 0.1180
0.2800, 0.0000, 0.2210, 0.0820, 0.1680, 0.1006, 0.5400, 0.3000, 0.4205, 0.0860, 0.0690
0.5800, 0.0000, 0.2810, 0.1110, 0.1980, 0.0806, 0.3100, 0.6000, 0.6068, 0.0930, 0.2730
0.3200, 0.0000, 0.2650, 0.0860, 0.1840, 0.1016, 0.5300, 0.4000, 0.4990, 0.0780, 0.2580
0.2500, 1.0000, 0.2350, 0.0880, 0.1430, 0.0808, 0.5500, 0.3000, 0.3584, 0.0830, 0.0430
0.6300, 0.0000, 0.2600, 0.0857, 0.1550, 0.0782, 0.4600, 0.3370, 0.5037, 0.0970, 0.1980
0.5200, 0.0000, 0.2780, 0.0850, 0.2190, 0.1360, 0.4900, 0.4000, 0.5136, 0.0750, 0.2420
0.6500, 1.0000, 0.2850, 0.1090, 0.2010, 0.1230, 0.4600, 0.4000, 0.5075, 0.0960, 0.2320
0.4200, 0.0000, 0.3060, 0.1210, 0.1760, 0.0928, 0.6900, 0.3000, 0.4263, 0.0890, 0.1750
0.5300, 0.0000, 0.2220, 0.0780, 0.1640, 0.0810, 0.7000, 0.2000, 0.4174, 0.1010, 0.0930
0.7900, 1.0000, 0.2330, 0.0880, 0.1860, 0.1284, 0.3300, 0.6000, 0.4812, 0.1020, 0.1680
0.4300, 0.0000, 0.3540, 0.0930, 0.1850, 0.1002, 0.4400, 0.4000, 0.5318, 0.1010, 0.2750
0.4400, 0.0000, 0.3140, 0.1150, 0.1650, 0.0976, 0.5200, 0.3000, 0.4344, 0.0890, 0.2930
0.6200, 1.0000, 0.3780, 0.1190, 0.1130, 0.0510, 0.3100, 0.4000, 0.5043, 0.0840, 0.2810
0.3300, 0.0000, 0.1890, 0.0700, 0.1620, 0.0918, 0.5900, 0.3000, 0.4025, 0.0580, 0.0720
0.5600, 0.0000, 0.3500, 0.0793, 0.1950, 0.1408, 0.4200, 0.4640, 0.4111, 0.0960, 0.1400
0.6600, 0.0000, 0.2170, 0.1260, 0.2120, 0.1278, 0.4500, 0.4710, 0.5278, 0.1010, 0.1890
0.3400, 1.0000, 0.2530, 0.1110, 0.2300, 0.1620, 0.3900, 0.6000, 0.4977, 0.0900, 0.1810
0.4600, 1.0000, 0.2380, 0.0970, 0.2240, 0.1392, 0.4200, 0.5000, 0.5366, 0.0810, 0.2090
0.5000, 0.0000, 0.3180, 0.0820, 0.1360, 0.0692, 0.5500, 0.2000, 0.4078, 0.0850, 0.1360
0.6900, 0.0000, 0.3430, 0.1130, 0.2000, 0.1238, 0.5400, 0.4000, 0.4710, 0.1120, 0.2610
0.3400, 0.0000, 0.2630, 0.0870, 0.1970, 0.1200, 0.6300, 0.3000, 0.4249, 0.0960, 0.1130
0.7100, 1.0000, 0.2700, 0.0933, 0.2690, 0.1902, 0.4100, 0.6560, 0.5242, 0.0930, 0.1310
0.4700, 0.0000, 0.2720, 0.0800, 0.2080, 0.1456, 0.3800, 0.6000, 0.4804, 0.0920, 0.1740
0.4100, 0.0000, 0.3380, 0.1233, 0.1870, 0.1270, 0.4500, 0.4160, 0.4318, 0.1000, 0.2570
0.3400, 0.0000, 0.3300, 0.0730, 0.1780, 0.1146, 0.5100, 0.3490, 0.4127, 0.0920, 0.0550
0.5100, 0.0000, 0.2410, 0.0870, 0.2610, 0.1756, 0.6900, 0.4000, 0.4407, 0.0930, 0.0840
0.4300, 0.0000, 0.2130, 0.0790, 0.1410, 0.0788, 0.5300, 0.3000, 0.3829, 0.0900, 0.0420
0.5500, 0.0000, 0.2300, 0.0947, 0.1900, 0.1376, 0.3800, 0.5000, 0.4277, 0.1060, 0.1460
0.5900, 1.0000, 0.2790, 0.1010, 0.2180, 0.1442, 0.3800, 0.6000, 0.5187, 0.0950, 0.2120
0.2700, 1.0000, 0.3360, 0.1100, 0.2460, 0.1566, 0.5700, 0.4000, 0.5088, 0.0890, 0.2330
0.5100, 1.0000, 0.2270, 0.1030, 0.2170, 0.1624, 0.3000, 0.7000, 0.4812, 0.0800, 0.0910
0.4900, 1.0000, 0.2740, 0.0890, 0.1770, 0.1130, 0.3700, 0.5000, 0.4905, 0.0970, 0.1110
0.2700, 0.0000, 0.2260, 0.0710, 0.1160, 0.0434, 0.5600, 0.2000, 0.4419, 0.0790, 0.1520
0.5700, 1.0000, 0.2320, 0.1073, 0.2310, 0.1594, 0.4100, 0.5630, 0.5030, 0.1120, 0.1200
0.3900, 1.0000, 0.2690, 0.0930, 0.1360, 0.0754, 0.4800, 0.3000, 0.4143, 0.0990, 0.0670
0.6200, 1.0000, 0.3460, 0.1200, 0.2150, 0.1292, 0.4300, 0.5000, 0.5366, 0.1230, 0.3100
0.3700, 0.0000, 0.2330, 0.0880, 0.2230, 0.1420, 0.6500, 0.3400, 0.4357, 0.0820, 0.0940
0.4600, 0.0000, 0.2110, 0.0800, 0.2050, 0.1444, 0.4200, 0.5000, 0.4533, 0.0870, 0.1830
0.6800, 1.0000, 0.2350, 0.1010, 0.1620, 0.0854, 0.5900, 0.3000, 0.4477, 0.0910, 0.0660
0.5100, 0.0000, 0.3150, 0.0930, 0.2310, 0.1440, 0.4900, 0.4700, 0.5252, 0.1170, 0.1730
0.4100, 0.0000, 0.2080, 0.0860, 0.2230, 0.1282, 0.8300, 0.3000, 0.4078, 0.0890, 0.0720
0.5300, 0.0000, 0.2650, 0.0970, 0.1930, 0.1224, 0.5800, 0.3000, 0.4143, 0.0990, 0.0490
0.4500, 0.0000, 0.2420, 0.0830, 0.1770, 0.1184, 0.4500, 0.4000, 0.4220, 0.0820, 0.0640
0.3300, 0.0000, 0.1950, 0.0800, 0.1710, 0.0854, 0.7500, 0.2000, 0.3970, 0.0800, 0.0480
0.6000, 1.0000, 0.2820, 0.1120, 0.1850, 0.1138, 0.4200, 0.4000, 0.4984, 0.0930, 0.1780
0.4700, 1.0000, 0.2490, 0.0750, 0.2250, 0.1660, 0.4200, 0.5000, 0.4443, 0.1020, 0.1040
0.6000, 1.0000, 0.2490, 0.0997, 0.1620, 0.1066, 0.4300, 0.3770, 0.4127, 0.0950, 0.1320
0.3600, 0.0000, 0.3000, 0.0950, 0.2010, 0.1252, 0.4200, 0.4790, 0.5130, 0.0850, 0.2200
0.3600, 0.0000, 0.1960, 0.0710, 0.2500, 0.1332, 0.9700, 0.3000, 0.4595, 0.0920, 0.0570

In machine learning, to train a linear model (linear regression, quadratic regression, etc.) you can use several techniques including stochastic gradient descent (SGD), L-BFGS optimization, and pseudo-inverse. There are two main categories of pseudo-inverse algorithms for machine learning scenarios where the training data has more rows than columns: relaxed Moore-Penrose pseudo-inverse, and left pseudo-inverse via normal equations.

For relaxed Moore-Penrose pseudo-inverse there are two main categories of techniques: using singular value decomposition (SVD) or QR decomposition. There are several SVD algorithms, including Jacobi, Householder+QR, Golub-Kahan-Reinsch, and Lanczos. There are several QR algorithms including Householder, modified Gram-Schmidt, and Givens.

For left pseudo-inverse via normal equations, the most common technique is Cholesky decomposition. There are several versions of Cholesky, but the most common is the Banachiewicz algorithm.

Each of these pseudo-inverse techniques has pros and cons related to numerical stability for large matrices, performance speed, and implementation complexity. For example, SVD Jacobi is the most stable (in theory) but the most complex. And QR modified Gram-Schmidt is relatively simple but less stable (in theory). And Cholesky is very stable but is vulnerable to poorly conditioned matrices.

I write code almost every day. Coding is a skill that must be practiced, and I enjoy coding. One morning before work, I figured I’d refactor my C# implementation of left pseudo-inverse via normal equations using Cholesky (Banachiewicz). So I did.

If you have a matrix A of training data (or a design matrix derived from the training data) with more rows than columns, the left pseudo-inverse can be computed as pinv = inv(At * A) * At, where At is the transpose of A, * is matrix multiplication, and inv() is any matrix inverse function. But At * A is square, symmetric, positive definite (assuming you can a small conditioning constant to the diagonal elements) and you can use a relatively simple matrix inverse — Cholesky decomposition.

This technique — left pseudo-inverse via normal equations using Cholesky — is dramatically simpler than any other form of pseudo-inverse for machine learning scenarios. But the technique can fail when the source matrix has many very small values or many large values. The problem is the At * A computation, which has lots of multiplication operations.

The key code in my demo is:

public class Cholesky
{
  public static double[][] MatPseudoInv(double[][] A)
  {
    // pinv = inv(At * A) * At
    double[][] At = MatTranspose(A);
    double[][] AtA = MatProduct(At, A);
    double[][] AtAinv = MatInvCholesky(AtA);
    double[][] result = MatProduct(AtAinv, At);
    return result;
  }

  private static double[][] MatInvCholesky(double[][] A)
  {
    // A must be square symmetric positive definite
    // A = L * Lt, inv(A) = inv(Lt) * inv(L)
    double[][] L = MatDecompCholesky(A); // lower tri
    double[][] Lt = MatTranspose(L); // upper tri
    double[][] invL = MatInvLowerTri(L);
    double[][] invLt = MatInvUpperTri(Lt);
    double[][] result = MatProduct(invLt, invL);
    return result;
  }

  private static double[][] MatDecompCholesky(double[][] A)
  {
    // lots of code
  }

  // lots of minor helpers like MatTranspose()

} // class Cholesky

I tested my refactored pseudo-inverse implementation by generating 10,000 random matrices, computing the pseudo-inverse, then verifying that A * Apinv * A = A to within 1.0e-8 in each cell. Each matrix has between 100 and 1,000 rows, between 2 and 20 columns, and each value is between -10.0 and +10.0. All 10,000 tests passed. This is by no means thorough testing, but it does indicate that the implementation is basically correct for non-pathological matrices.

Good fun.

I miss the days of newspapers.

Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operators (my blog editor chokes on symbols).

using System.IO;

namespace MatrixPseudoInverseCholesky
{
  internal class MatrixPseudoInverseCholeskyProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin left pseudo-inverse" +
        " via normal equations (Cholesky) ");

      int minRows = 100; int maxRows = 1000;
      int minCols = 2; int maxCols = 20;
      Random rnd = new Random(0);
      int nTrials = 10000;
      int nPass = 0; int nFail = 0;
      for (int i = 0; i "lt" nTrials; ++i)
      {
        Console.WriteLine("\n==========");
        Console.WriteLine("trial " + i);
        double[][] A =
          MakeRandomMatrix(minRows, maxRows,
          minCols, maxCols, rnd);
        double[][] Apinv = Cholesky.MatPseudoInv(A);

        // check (A * Apinv) * A = A
        double[][] AApinv = MatProduct(A, Apinv);
        double[][] C = MatProduct(AApinv, A);

        //Console.WriteLine("\nA = ");
        //MatShow(A, 4, 9);
        //Console.WriteLine("\nApinv = ");
        //MatShow(Apinv, 4, 9);
        //Console.WriteLine("\n(A * Apinv) * A = ");
        //MatShow(C, 4, 9);

        if (MatAreEqual(C, A, 1.0e-8) == true)
        {
          Console.WriteLine("pass");
          ++nPass;
        }
        else
        {
          Console.WriteLine("FAIL");
          Console.WriteLine("nRows = " + A.Length +
            " nCols = " + A[0].Length);
          Console.ReadLine();
          ++nFail;
        }
        Console.WriteLine("==========");
     } // nTrials

      Console.WriteLine("\nNumber pass = " + nPass);
      Console.WriteLine("Number fail = " + nFail);

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

    // ------------------------------------------------------
    // helpers: MakeRandomMatrix(), MatProduct(), 
    // MatShow(), MatAreEqual()
    // ------------------------------------------------------

    public static double[][] MakeRandomMatrix(int minRows,
      int maxRows, int minCols, int maxCols, Random rnd)
    {
      int nRows = rnd.Next(minRows, maxRows);  // [2,maxN)
      int nCols = rnd.Next(minCols, maxCols);
      double[][] result = new double[nRows][];
      for (int i = 0; i "lt" nRows; ++i)
        result[i] = new double[nCols];

      double lo = -10.0; double hi = 10.0;
      for (int r = 0; r "lt" nRows; ++r)
        for (int c = 0; c "lt" nCols; ++c)
          result[r][c] = (hi - lo) * rnd.NextDouble() + lo;
      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 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 bool MatAreEqual(double[][] A, double[][] B,
      double epsilon)
    {
      int nRows = A.Length; int nCols = A[0].Length;
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          if (Math.Abs(A[i][j] - B[i][j]) "gt" epsilon)
            return false;
      return true;
    }

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

  } // Program

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

  public class Cholesky
  {
    public static double[][] MatPseudoInv(double[][] A)
    {
      // pinv = inv(At * A) * At
      double[][] At = MatTranspose(A);
      double[][] AtA = MatProduct(At, A);
      double[][] AtAinv = MatInvCholesky(AtA);
      double[][] result = MatProduct(AtAinv, At);
      return result;
    }

    private static double[][] MatInvCholesky(double[][] A)
    {
      // A must be square symmetric positive definite
      // A = L * Lt, inv(A) = inv(Lt) * inv(L)
      double[][] L = MatDecompCholesky(A); // lower tri
      double[][] Lt = MatTranspose(L); // upper tri
      double[][] invL = MatInvLowerTri(L);
      double[][] invLt = MatInvUpperTri(Lt);
      double[][] result = MatProduct(invLt, invL);
      return result;
    }

    private static double[][] MatDecompCholesky(double[][] A)
    {
      // A must be square, symmetric, positive definite
      int m = A.Length; int n = A[0].Length;  // m == n
      double[][] L = new double[n][];
      for (int i = 0; i "lt" n; ++i)
        L[i] = new double[n];

      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 += L[i][k] * L[j][k];
          if (i == j)
          {
            double tmp = A[i][i] - sum;
            if (tmp "lt" 0.0)
              throw new
                Exception("decomp Cholesky fatal");
            L[i][j] = Math.Sqrt(tmp);
          }
          else
          {
            if (L[j][j] == 0.0)
              throw new
                Exception("decomp Cholesky fatal ");
            L[i][j] = (A[i][j] - sum) / L[j][j];
          }
        } // j
      } // i

      return L;
    }

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

    private static double[][] MatInvLowerTri(double[][] A)
    {
      int n = A.Length;  // must be square matrix
      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] * A[k][i];
          }
          result[k][j] /= (A[k][k] + 1.0e-8);
        }
      }
      return result;
    }

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

    private static double[][] MatInvUpperTri(double[][] A)
    {
      int n = A.Length;  // must be square matrix
      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[j][k] -= result[j][i] * A[i][k];
          }
          result[j][k] /= (A[k][k] + 1.0e-8); // avoid 0
        }
      }
      return result;
    }

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

    private 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;
    }

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

    private 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;
    }

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

    private static double[][] MatTranspose(double[][] M)
    {
      int nRows = M.Length; int nCols = M[0].Length;
      double[][] result = MatMake(nCols, nRows);  // note
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          result[j][i] = M[i][j];
      return result;
    }

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

    private 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 = 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] += A[i][k] * B[k][j];

      return result;
    }

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

  } // class Cholesky

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

} // ns

Bottom line: When using linear regression on data that has categorical predictor variables, you must use drop-first encoding instead of one-hot encoding if you train the model using a closed form pseudo-inverse technique (left pseudo-inverse, Moore-Penrose pseudo-inverse). If you use iterative techniques SGD or L-BFGS to train, you can use one-hot encoding or drop-first encoding, but drop-first is preferred because it gives a more interpretable model than one-hot encoding. In short, for linear regression, drop-first encoding of categorical predictors is the preferred encoding technique.

The goal of a machine learning regression model is to predict a single numeric value. For example, suppose you have data that looks like:

F, 24, michigan, 29500.00, liberal
M, 39, oklahoma, 51200.00, moderate
F, 63, nebraska, 75800.00, conservative
M, 36, virginia, 44500.00, moderate
F, 27, nebraska, 28600.00, liberal
. . .

The fields are sex, age, State (Michigan or Nebraska or Oklahoma or Virginia), annual income, and political leaning. And suppose you want to predict income from the other four variables.

The most fundamental regression technique is linear regression.

In most scenarios, the best way to normalize and encode the data for linear regression is to use divide-by-constant normalization and drop-first encoding:

1, 0.24, 0 0, 0 0 0, 0.29500, 0 1
0, 0.39, 0 1, 0 1 0, 0.51200, 1 0
1, 0.63, 1 0, 1 0 0, 0.75800, 0 0
0, 0.36, 0 1, 0 0 1, 0.44500, 1 0
1, 0.27, 0 0, 1 0 0, 0.28600, 0 1
. . .

The age values are divided by 100, and the income values are divided by 100,000. The binary sex values are encoded as M = 0, F = 1.

If the four State values were one-hot encoded, they would be Michigan = 1000, Nebraska = 0100, Oklahoma = 0010, Virginia = 0001, but drop-first gives Michigan = 000, Nebraska = 100, Oklahoma = 010, Virginia = 001.

If the three political leaning values were one-hot encoded, they would be conservative = 100, moderate = 010, liberal = 001, but drop-first drops the first digit, giving conservative = 00, moderate = 10, liberal = 01.

If you use one-hot encoding, you include redundant information in a sense, in part because both one-hot encoding and drop-first encoding unambiguously identify a variable’s value, but drop-first does so using one less bit of information.

Now, if you train a linear regression model using a pseudo-inverse (either left-pseudo inverse via normal equations, or relaxed Moore-Penrose pseudo-inverse via SVD or QR decomposition), one-hot encoding creates multicollinearity in the data and the inverse operation will usually fail. So, when using a pseudo-inverse training technique, drop-first encoding is pretty much required.

If you train a linear regression model using stochastic gradient descent (SGD), you don’t run into the matrix inverse failure issue, and so one-hot encoding will work. However, because one-hot encoding has redundant information, SGD can produce wildly different results in the model weights and bias, but which give the same prediction accuracy. So, if you use one-hot encoding and SGD training, you get a model that predicts well, but one where the weights and bias can’t be interpreted well. Therefore, you might as well just use drop-first encoding.

Note that for neural networks, one-hot encoding is always used with SGD training because neural networks can’t be interpreted easily, so drop-first encoding provides no advantage.

It’s important to pay attention to details when working with machine learning. I love old science fiction movies from the 1950s. Minor characters can make a big difference in movies.

Right: In “20 Million Miles to Earth”, Zaremba plays Dr. Judson Uhl, a scientist involved with a secret manned mission to Venus. The spacecraft brings back a small Venusian creature who grows to giant size and causes havoc.

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 LinearRegressionCategorical
{
  internal class LinearRegressionProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin C# linear regression ");
      Console.WriteLine("Predict salary from sex, age," +
        " State, politics ");

      // 1. load data
      Console.WriteLine("\nLoading people train" +
        " (200) and test (40) data");
      string trainFile =
        "C:\\VSM\\LinearRegressionCategorical\\" +
        "Data\\people_train_df.txt";
      int[] colsX = new int[] { 0, 1, 2, 3, 5, 6 };
      int colY = 4;
      double[][] trainX =
        MatLoad(trainFile, colsX, ',', "#");
      double[] trainY =
        MatToVec(MatLoad(trainFile,
        new int[] { colY }, ',', "#"));

      string testFile =
        "C:\\VSM\\LinearRegressionCategorical\\" +
        "Data\\people_test_df.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("F5").
          PadLeft(8));

      // 2. create and train model
      double lrnRate = 0.001;
      int maxEpochs = 3000;
      int seed = 0;
      Console.WriteLine("\nSetting lrnRate = " +
        lrnRate.ToString("F4"));
      Console.WriteLine("Setting maxEpohcs = " +
        maxEpochs);

      Console.WriteLine("\nCreating and training" +
        " Linear Regression model using SGD ");
      LinearRegressor model =
        new LinearRegressor(seed);
      model.Train(trainX, trainY, lrnRate, maxEpochs);
      Console.WriteLine("Done ");

      // 2b. show model parameters
      Console.WriteLine("\nCoefficients/weights: ");
      for (int i = 0; i "lt" model.weights.Length; ++i)
        Console.Write(model.weights[i].ToString("F4") + "  ");
      Console.WriteLine("\nBias/constant: " +
        model.bias.ToString("F4"));

      // 3. 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"));

      double mseTrain = model.MSE(trainX, trainY);
      Console.WriteLine("\nMSE train = " +
        mseTrain.ToString("F6"));
      double mseTest = model.MSE(testX, testY);
      Console.WriteLine("MSE test = " +
        mseTest.ToString("F6"));

      // 4. use model to predict first training item
      double[] x = trainX[0];
      Console.WriteLine("\nPredicting for x = ");
      VecShow(x, 4, 9);
      double predY = model.Predict(x);
      Console.WriteLine("\nPredicted y = " +
        predY.ToString("F5"));

      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;
    public double bias;
    private Random rnd;

    public LinearRegressor(int seed = 0)
    {
      this.weights = new double[0];
      this.bias = 0;
      this.rnd = new Random(seed);
    }

    public void Train(double[][] trainX, double[] trainY,
      double lrnRate, int maxEpochs)
    {
      int n = trainX.Length;
      int dim = trainX[0].Length;
      this.weights = new double[dim];
      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[n];
      for (int i = 0; i "lt" n; ++i)
        indices[i] = i;

      for (int epoch = 0; epoch "lt" maxEpochs; ++epoch)
      {
        Shuffle(indices, this.rnd);
        for (int i = 0; i "lt" n; ++i)
        {
          int idx = indices[i];
          double[] x = trainX[idx];
          double predY = this.Predict(x);
          double actualY = trainY[idx];
          for (int j = 0; j "lt" dim; ++j)
            this.weights[j] -= lrnRate *
              (predY - actualY) * x[j];
          this.bias -= lrnRate * (predY - actualY);
        }
        if (epoch % (int)(maxEpochs / 5) == 0)
        {
          double mse = this.MSE(trainX, trainY);
          string s1 = "epoch = " + 
            epoch.ToString().PadLeft(5);
          string s2 = "  MSE = " + 
            mse.ToString("F6").PadLeft(8);
          Console.WriteLine(s1 + s2);
        }
      }
    }

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

# people_train_df.txt
# sex (0 = male, 1 = female)
# age, state (michigan, nebraska, oklahoma), income,
# politics type (conservative, moderate, liberal)
#
1, 0.24, 0, 0, 0.29500, 0, 1
0, 0.39, 0, 1, 0.51200, 1, 0
1, 0.63, 1, 0, 0.75800, 0, 0
0, 0.36, 0, 0, 0.44500, 1, 0
1, 0.27, 1, 0, 0.28600, 0, 1
1, 0.50, 1, 0, 0.56500, 1, 0
1, 0.50, 0, 1, 0.55000, 1, 0
0, 0.19, 0, 1, 0.32700, 0, 0
1, 0.22, 1, 0, 0.27700, 1, 0
0, 0.39, 0, 1, 0.47100, 0, 1
1, 0.34, 0, 0, 0.39400, 1, 0
0, 0.22, 0, 0, 0.33500, 0, 0
1, 0.35, 0, 1, 0.35200, 0, 1
0, 0.33, 1, 0, 0.46400, 1, 0
1, 0.45, 1, 0, 0.54100, 1, 0
1, 0.42, 1, 0, 0.50700, 1, 0
0, 0.33, 1, 0, 0.46800, 1, 0
1, 0.25, 0, 1, 0.30000, 1, 0
0, 0.31, 1, 0, 0.46400, 0, 0
1, 0.27, 0, 0, 0.32500, 0, 1
1, 0.48, 0, 0, 0.54000, 1, 0
0, 0.64, 1, 0, 0.71300, 0, 1
1, 0.61, 1, 0, 0.72400, 0, 0
1, 0.54, 0, 1, 0.61000, 0, 0
1, 0.29, 0, 0, 0.36300, 0, 0
1, 0.50, 0, 1, 0.55000, 1, 0
1, 0.55, 0, 1, 0.62500, 0, 0
1, 0.40, 0, 0, 0.52400, 0, 0
1, 0.22, 0, 0, 0.23600, 0, 1
1, 0.68, 1, 0, 0.78400, 0, 0
0, 0.60, 0, 0, 0.71700, 0, 1
0, 0.34, 0, 1, 0.46500, 1, 0
0, 0.25, 0, 1, 0.37100, 0, 0
0, 0.31, 1, 0, 0.48900, 1, 0
1, 0.43, 0, 1, 0.48000, 1, 0
1, 0.58, 1, 0, 0.65400, 0, 1
0, 0.55, 1, 0, 0.60700, 0, 1
0, 0.43, 1, 0, 0.51100, 1, 0
0, 0.43, 0, 1, 0.53200, 1, 0
0, 0.21, 0, 0, 0.37200, 0, 0
1, 0.55, 0, 1, 0.64600, 0, 0
1, 0.64, 1, 0, 0.74800, 0, 0
0, 0.41, 0, 0, 0.58800, 1, 0
1, 0.64, 0, 1, 0.72700, 0, 0
0, 0.56, 0, 1, 0.66600, 0, 1
1, 0.31, 0, 1, 0.36000, 1, 0
0, 0.65, 0, 1, 0.70100, 0, 1
1, 0.55, 0, 1, 0.64300, 0, 0
0, 0.25, 0, 0, 0.40300, 0, 0
1, 0.46, 0, 1, 0.51000, 1, 0
0, 0.36, 0, 0, 0.53500, 0, 0
1, 0.52, 1, 0, 0.58100, 1, 0
1, 0.61, 0, 1, 0.67900, 0, 0
1, 0.57, 0, 1, 0.65700, 0, 0
0, 0.46, 1, 0, 0.52600, 1, 0
0, 0.62, 0, 0, 0.66800, 0, 1
1, 0.55, 0, 1, 0.62700, 0, 0
0, 0.22, 0, 1, 0.27700, 1, 0
0, 0.50, 0, 0, 0.62900, 0, 0
0, 0.32, 1, 0, 0.41800, 1, 0
0, 0.21, 0, 1, 0.35600, 0, 0
1, 0.44, 1, 0, 0.52000, 1, 0
1, 0.46, 1, 0, 0.51700, 1, 0
1, 0.62, 1, 0, 0.69700, 0, 0
1, 0.57, 1, 0, 0.66400, 0, 0
0, 0.67, 0, 1, 0.75800, 0, 1
1, 0.29, 0, 0, 0.34300, 0, 1
1, 0.53, 0, 0, 0.60100, 0, 0
0, 0.44, 0, 0, 0.54800, 1, 0
1, 0.46, 1, 0, 0.52300, 1, 0
0, 0.20, 1, 0, 0.30100, 1, 0
0, 0.38, 0, 0, 0.53500, 1, 0
1, 0.50, 1, 0, 0.58600, 1, 0
1, 0.33, 1, 0, 0.42500, 1, 0
0, 0.33, 1, 0, 0.39300, 1, 0
1, 0.26, 1, 0, 0.40400, 0, 0
1, 0.58, 0, 0, 0.70700, 0, 0
1, 0.43, 0, 1, 0.48000, 1, 0
0, 0.46, 0, 0, 0.64400, 0, 0
1, 0.60, 0, 0, 0.71700, 0, 0
0, 0.42, 0, 0, 0.48900, 1, 0
0, 0.56, 0, 1, 0.56400, 0, 1
0, 0.62, 1, 0, 0.66300, 0, 1
0, 0.50, 0, 0, 0.64800, 1, 0
1, 0.47, 0, 1, 0.52000, 1, 0
0, 0.67, 1, 0, 0.80400, 0, 1
0, 0.40, 0, 1, 0.50400, 1, 0
1, 0.42, 1, 0, 0.48400, 1, 0
1, 0.64, 0, 0, 0.72000, 0, 0
0, 0.47, 0, 0, 0.58700, 0, 1
1, 0.45, 1, 0, 0.52800, 1, 0
0, 0.25, 0, 1, 0.40900, 0, 0
1, 0.38, 0, 0, 0.48400, 0, 0
1, 0.55, 0, 1, 0.60000, 1, 0
0, 0.44, 0, 0, 0.60600, 1, 0
1, 0.33, 0, 0, 0.41000, 1, 0
1, 0.34, 0, 1, 0.39000, 1, 0
1, 0.27, 1, 0, 0.33700, 0, 1
1, 0.32, 1, 0, 0.40700, 1, 0
1, 0.42, 0, 1, 0.47000, 1, 0
0, 0.24, 0, 1, 0.40300, 0, 0
1, 0.42, 1, 0, 0.50300, 1, 0
1, 0.25, 0, 1, 0.28000, 0, 1
1, 0.51, 1, 0, 0.58000, 1, 0
0, 0.55, 1, 0, 0.63500, 0, 1
1, 0.44, 0, 0, 0.47800, 0, 1
0, 0.18, 0, 0, 0.39800, 0, 0
0, 0.67, 1, 0, 0.71600, 0, 1
1, 0.45, 0, 1, 0.50000, 1, 0
1, 0.48, 0, 0, 0.55800, 1, 0
0, 0.25, 1, 0, 0.39000, 1, 0
0, 0.67, 0, 0, 0.78300, 1, 0
1, 0.37, 0, 1, 0.42000, 1, 0
0, 0.32, 0, 0, 0.42700, 1, 0
1, 0.48, 0, 0, 0.57000, 1, 0
0, 0.66, 0, 1, 0.75000, 0, 1
1, 0.61, 0, 0, 0.70000, 0, 0
0, 0.58, 0, 1, 0.68900, 1, 0
1, 0.19, 0, 0, 0.24000, 0, 1
1, 0.38, 0, 1, 0.43000, 1, 0
0, 0.27, 0, 0, 0.36400, 1, 0
1, 0.42, 0, 0, 0.48000, 1, 0
1, 0.60, 0, 0, 0.71300, 0, 0
0, 0.27, 0, 1, 0.34800, 0, 0
1, 0.29, 1, 0, 0.37100, 0, 0
0, 0.43, 0, 0, 0.56700, 1, 0
1, 0.48, 0, 0, 0.56700, 1, 0
1, 0.27, 0, 1, 0.29400, 0, 1
0, 0.44, 0, 0, 0.55200, 0, 0
1, 0.23, 1, 0, 0.26300, 0, 1
0, 0.36, 1, 0, 0.53000, 0, 1
1, 0.64, 0, 1, 0.72500, 0, 0
1, 0.29, 0, 1, 0.30000, 0, 1
0, 0.33, 0, 0, 0.49300, 1, 0
0, 0.66, 1, 0, 0.75000, 0, 1
0, 0.21, 0, 1, 0.34300, 0, 0
1, 0.27, 0, 0, 0.32700, 0, 1
1, 0.29, 0, 0, 0.31800, 0, 1
0, 0.31, 0, 0, 0.48600, 1, 0
1, 0.36, 0, 1, 0.41000, 1, 0
1, 0.49, 1, 0, 0.55700, 1, 0
0, 0.28, 0, 0, 0.38400, 0, 0
0, 0.43, 0, 1, 0.56600, 1, 0
0, 0.46, 1, 0, 0.58800, 1, 0
1, 0.57, 0, 0, 0.69800, 0, 0
0, 0.52, 0, 1, 0.59400, 1, 0
0, 0.31, 0, 1, 0.43500, 1, 0
0, 0.55, 0, 0, 0.62000, 0, 1
1, 0.50, 0, 0, 0.56400, 1, 0
1, 0.48, 1, 0, 0.55900, 1, 0
0, 0.22, 0, 1, 0.34500, 0, 0
1, 0.59, 0, 1, 0.66700, 0, 0
1, 0.34, 0, 0, 0.42800, 0, 1
0, 0.64, 0, 0, 0.77200, 0, 1
1, 0.29, 0, 1, 0.33500, 0, 1
0, 0.34, 1, 0, 0.43200, 1, 0
0, 0.61, 0, 0, 0.75000, 0, 1
1, 0.64, 0, 1, 0.71100, 0, 0
0, 0.29, 0, 0, 0.41300, 0, 0
1, 0.63, 1, 0, 0.70600, 0, 0
0, 0.29, 1, 0, 0.40000, 0, 0
0, 0.51, 0, 0, 0.62700, 1, 0
0, 0.24, 0, 1, 0.37700, 0, 0
1, 0.48, 1, 0, 0.57500, 1, 0
1, 0.18, 0, 0, 0.27400, 0, 0
1, 0.18, 0, 0, 0.20300, 0, 1
1, 0.33, 1, 0, 0.38200, 0, 1
0, 0.20, 0, 1, 0.34800, 0, 0
1, 0.29, 0, 1, 0.33000, 0, 1
0, 0.44, 0, 1, 0.63000, 0, 0
0, 0.65, 0, 1, 0.81800, 0, 0
0, 0.56, 0, 0, 0.63700, 0, 1
0, 0.52, 0, 1, 0.58400, 1, 0
0, 0.29, 1, 0, 0.48600, 0, 0
0, 0.47, 1, 0, 0.58900, 1, 0
1, 0.68, 0, 0, 0.72600, 0, 1
1, 0.31, 0, 1, 0.36000, 1, 0
1, 0.61, 1, 0, 0.62500, 0, 1
1, 0.19, 1, 0, 0.21500, 0, 1
1, 0.38, 0, 1, 0.43000, 1, 0
0, 0.26, 0, 0, 0.42300, 0, 0
1, 0.61, 1, 0, 0.67400, 0, 0
1, 0.40, 0, 0, 0.46500, 1, 0
0, 0.49, 0, 0, 0.65200, 1, 0
1, 0.56, 0, 0, 0.67500, 0, 0
0, 0.48, 1, 0, 0.66000, 1, 0
1, 0.52, 0, 0, 0.56300, 0, 1
0, 0.18, 0, 0, 0.29800, 0, 0
0, 0.56, 0, 1, 0.59300, 0, 1
0, 0.52, 1, 0, 0.64400, 1, 0
0, 0.18, 1, 0, 0.28600, 1, 0
0, 0.58, 0, 0, 0.66200, 0, 1
0, 0.39, 1, 0, 0.55100, 1, 0
0, 0.46, 0, 0, 0.62900, 1, 0
0, 0.40, 1, 0, 0.46200, 1, 0
0, 0.60, 0, 0, 0.72700, 0, 1
1, 0.36, 1, 0, 0.40700, 0, 1
1, 0.44, 0, 0, 0.52300, 1, 0
1, 0.28, 0, 0, 0.31300, 0, 1
1, 0.54, 0, 1, 0.62600, 0, 0

Test data:

# people_test_df.txt
#
0, 0.51, 0, 0, 0.61200, 1, 0
0, 0.32, 1, 0, 0.46100, 1, 0
1, 0.55, 0, 0, 0.62700, 0, 0
1, 0.25, 0, 1, 0.26200, 0, 1
1, 0.33, 0, 1, 0.37300, 0, 1
0, 0.29, 1, 0, 0.46200, 0, 0
1, 0.65, 0, 0, 0.72700, 0, 0
0, 0.43, 1, 0, 0.51400, 1, 0
0, 0.54, 1, 0, 0.64800, 0, 1
1, 0.61, 1, 0, 0.72700, 0, 0
1, 0.52, 1, 0, 0.63600, 0, 0
1, 0.30, 1, 0, 0.33500, 0, 1
1, 0.29, 0, 0, 0.31400, 0, 1
0, 0.47, 0, 1, 0.59400, 1, 0
1, 0.39, 1, 0, 0.47800, 1, 0
1, 0.47, 0, 1, 0.52000, 1, 0
0, 0.49, 0, 0, 0.58600, 1, 0
0, 0.63, 0, 1, 0.67400, 0, 1
0, 0.30, 0, 0, 0.39200, 0, 0
0, 0.61, 0, 1, 0.69600, 0, 1
0, 0.47, 0, 1, 0.58700, 1, 0
1, 0.30, 0, 1, 0.34500, 0, 1
0, 0.51, 0, 1, 0.58000, 1, 0
0, 0.24, 0, 0, 0.38800, 1, 0
0, 0.49, 0, 0, 0.64500, 1, 0
1, 0.66, 0, 1, 0.74500, 0, 0
0, 0.65, 0, 0, 0.76900, 0, 0
0, 0.46, 1, 0, 0.58000, 0, 0
0, 0.45, 0, 1, 0.51800, 1, 0
0, 0.47, 0, 0, 0.63600, 0, 0
0, 0.29, 0, 0, 0.44800, 0, 0
0, 0.57, 0, 1, 0.69300, 0, 1
0, 0.20, 0, 0, 0.28700, 0, 1
0, 0.35, 0, 0, 0.43400, 1, 0
0, 0.61, 0, 1, 0.67000, 0, 1
0, 0.31, 0, 1, 0.37300, 1, 0
1, 0.18, 0, 0, 0.20800, 0, 1
1, 0.26, 0, 1, 0.29200, 0, 1
0, 0.28, 0, 0, 0.36400, 0, 1
0, 0.59, 0, 1, 0.69400, 0, 1

I write code almost every day. Like many activities, writing code is a skill that must be practiced, plus I just enjoy writing code. And in the emerging world of AI-generated software, it’s still important to write code by hand so that good prompts can be generated and the resulting code can be evaluated.

One morning before work, I figured I’d run the well-known Diabetes Dataset through a random forest regression model. Based on previous experiments with other models including linear regression, quadratic regression, neural network regression, and a scikit DecisionTreeRegressor model, I was pretty sure the C# random forest model would give poor prediction accuracy. And that’s exactly what happened, as expected.

The Diabetes Dataset looks like:

59, 2, 32.1, 101.00, 157,  93.2, 38, 4.00, 4.8598, 87, 151
48, 1, 21.6,  87.00, 183, 103.2, 70, 3.00, 3.8918, 69,  75
72, 2, 30.5,  93.00, 156,  93.6, 41, 4.00, 4.6728, 85, 141
. . .

Each line represents a patient. The first 10 values on each line are predictors. The last value on each line is the target value (a diabetes metric) to predict. The predictors are: age, sex, body mass index, blood pressure, serum cholesterol, low-density lipoproteins, high-density lipoproteins, total cholesterol, triglycerides, blood sugar. There are 442 data items.

The sex encoding isn’t explained anywhere but I suspect male = 1, female = 2 because there are 235 1 values and 206 2 values).

Note that this is the diabetes dataset that is included in the scikit-learn demo datasets, not the similar Pima Diabetes dataset.

I converted the sex values encoding from 1,2 into 0,1. Then I applied divide-by-constant normalization by dividing the 10 predictor columns by (100, 1, 100, 1000, 1000, 1000, 100, 10, 10, 1000) and the target y values by 1000. The resulting encoded and normalized data looks like:

0.5900, 1.0000, 0.3210, . . . 0.1510
0.4800, 0.0000, 0.2160, . . . 0.0750
0.7200, 1.0000, 0.3050, . . . 0.1410
. . .

Normalization isn’t necessary for decision tree models, but it doesn’t hurt. I split the 442-items into a 342-item training set and a 100-item test set.

I implemented a random forest tree regression model, from scratch, using the C# language. The idea of a random forest model is to create a collection of basic decision trees, each of which is trained on a different subset of the training data. This reduces model overfitting which is the main problem with decision trees.

I used 100 trees for the random forest. I set the depth of each tree to 5. I specified that a random 300 out of the 342 rows of training data would be used to train each tree. I specified that a random 9 out of the 10 columns would be used during each tree split operation during training — this is a secondary regularization technique.

The output of the demo is:

Begin C# Random Forest regression demo

Loading diabetes train (342) and test (100) data
Done

First three train X:
  0.5900  1.0000  0.3210  . . .  0.0870
  0.4800  0.0000  0.2160  . . .  0.0690
  0.7200  1.0000  0.3050  . . .  0.0850

First three train y:
  0.1510
  0.0750
  0.1410

Setting nTrees = 100
Setting maxDepth = 5
Setting minSamples = 2
Setting minLeaf = 1
Setting nCols = 9
Setting nRows = 300

Creating and training random forest regression model
Done

Accuracy train (within 0.10) = 0.2456
Accuracy test (within 0.10) = 0.2400

MSE train = 0.0016
MSE test = 0.0032

Predicting for trainX[0] =
   0.5900   1.0000   0.3210   . . .   0.0870
Predicted y = 0.1923

End demo

These results were essentially the same as the results I got using other regression techniques such as quadratic regression and neural network regression.

I have done a lot of experiments with the Diabetes Dataset and I’ve concluded the the default target value in the last column (a patient diabetes score) simply cannot be predicted well. But the variables in columns [4], [5], [6], [7], and [8] can be meaningfully predicted from the other columns.

Good fun on a rainy morning in the Pacific Northwest.

I see beauty in machine learning code. Physical beauty in people is very subjective. Actresses of one era might be considered beautiful, but only average at best in another era.

Right: Louise Brooks (1906-1985) acted in 24 films from 1925-1938. She was well-known for popularizing a bob hair style in the 1920s. The one movie that she appeared in that I am most familiar with is “Overland Stage Raiders” (1938), with actor John Wayne. Like many actresses of the 1920s through 1950s, she ended her days as a serial mistress passed around from one movie executive to another.

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 RandomForestRegression
{
  internal class RandomForestRegressionProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin C# Random Forest" +
        " regression demo ");

      // 1. load data
      Console.WriteLine("\nLoading diabetes train (342)" +
        " and test (100) data");
      string trainFile =
        "..\\..\\..\\Data\\diabetes_norm_train_342.txt";
      int[] colsX = new int[] { 0, 1, 2, 3, 4,
        5, 6, 7, 8, 9};
      int colY = 10;
      double[][] trainX =
        MatLoad(trainFile, colsX, ',', "#");
      double[] trainY =
        MatToVec(MatLoad(trainFile,
        new int[] { colY }, ',', "#"));

      string testFile =
        "..\\..\\..\\Data\\diabetes_norm_test_100.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));

      int nTrees = 100;
      int maxDepth = 5;
      int minSamples = 2;
      int minLeaf = 1;
      int nCols = 9;  // if use all 10 == bagging tree
      int nRows = 300;  // of 342 train data for each tree

      Console.WriteLine("\nSetting nTrees = " + nTrees);
      Console.WriteLine("Setting maxDepth = " + maxDepth);
      Console.WriteLine("Setting minSamples = " + minSamples);
      Console.WriteLine("Setting minLeaf = " + minLeaf);
      Console.WriteLine("Setting nCols = " + nCols);
      Console.WriteLine("Setting nRows = " + nRows);

      Console.WriteLine("\nCreating and training" +
        " random forest regression model ");
      RandomForestRegressor rfr =
        new RandomForestRegressor(nTrees, maxDepth,
        minSamples, minLeaf, nCols, nRows, seed: 0);
      rfr.Train(trainX, trainY);
      Console.WriteLine("Done ");

      // 3. evaluate model
      double accTrain = rfr.Accuracy(trainX, trainY, 0.10);
      Console.WriteLine("\nAccuracy train (within 0.10) = " +
        accTrain.ToString("F4"));
      double accTest = rfr.Accuracy(testX, testY, 0.10);
      Console.WriteLine("Accuracy test (within 0.10) = " +
        accTest.ToString("F4"));

      double mseTrain = rfr.MSE(trainX, trainY);
      Console.WriteLine("\nMSE train = " +
        mseTrain.ToString("F4"));
      double mseTest = rfr.MSE(testX, testY);
      Console.WriteLine("MSE test = " +
        mseTest.ToString("F4"));

      // 4. use model
      double[] x = trainX[0];
      Console.WriteLine("\nPredicting for x = ");
      VecShow(x, 4, 9);
      double predY = rfr.Predict(x);
      Console.WriteLine("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[][] 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 RandomForestRegressor
  {
    public int nTrees;
    public int maxDepth;
    public int minSamples;
    public int minLeaf;
    public int nSplitCols;  // number cols to use each split
    public int nRows;
    public List"lt"DecisionTreeRegressor"gt" trees;
    public Random rnd;

    public RandomForestRegressor(int nTrees, int maxDepth,
      int minSamples, int minLeaf, int nCols, int nRows,
      int seed = 0)
    {
      this.nTrees = nTrees;
      this.maxDepth = maxDepth;
      this.minSamples = minSamples;
      this.minLeaf = minLeaf;
      this.nRows = nRows;
      this.nSplitCols = nCols;  // pass to DecisionTree
      this.trees = new List"lt"DecisionTreeRegressor"gt"();
      this.rnd = new Random(seed);
    }

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

    public void Train(double[][] trainX, double[] trainY)
    {
      int nr = trainX.Length;
      // int nc = trainX[0].Length;

      for (int i = 0; i "lt" this.nTrees; ++i) // each tree
      {
        // construct data rows-subset 
        int[] rndRows =
          GetRandomRowIdxs(nr, this.nRows, this.rnd);
        double[][] subsetX =
          MakeRowsSubsetX(trainX, rndRows);
        double[] subsetY =
          MakeSubsetY(trainY, rndRows);

        // make and train curr tree
        DecisionTreeRegressor dtr =
          new DecisionTreeRegressor(this.maxDepth,
          this.minSamples, this.minLeaf, this.nSplitCols);
        // random cols will be used during splits
        dtr.Train(subsetX, subsetY);
        this.trees.Add(dtr);
      } // i
    }

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

    public double Predict(double[] x)
    {
      double sum = 0.0;
      for (int t = 0; t "lt" this.nTrees; ++t)
        sum += this.trees[t].Predict(x);
      return sum / this.nTrees;
    }

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

    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 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 int[] GetRandomRowIdxs(int N, int n,
      Random rnd)
    {
      // pick n rows from N with replacement
      int[] result = new int[n];
      for (int i = 0; i "lt" n; ++i)
        result[i] = rnd.Next(0, N);
      Array.Sort(result);
      return result;
    }

    private static double[][] MakeRowsSubsetX(double[][] trainX,
      int[] rndRows)
    {
      int nCols = trainX[0].Length;  // use all cols
      double[][] result = new double[rndRows.Length][];
      for (int i = 0; i "lt" rndRows.Length; ++i)
        result[i] = new double[nCols];

      for (int i = 0; i "lt" rndRows.Length; ++i)
      {
        for (int j = 0; j "lt" nCols; ++j)
        {
          int r = rndRows[i]; // random row in trainX
          result[i][j] = trainX[r][j];
        }
      }
      return result;
    }

    private static double[] MakeSubsetY(double[] trainY,
      int[] rndRows)
    {
      double[] result = new double[rndRows.Length];
      for (int i = 0; i "lt" rndRows.Length; ++i)
        result[i] = trainY[rndRows[i]];
      return result;
    }

  } // class RandomForestRegressor

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

  public class DecisionTreeRegressor
  {
    public int maxDepth;
    public int minSamples;  // aka min_samples_split
    public int minLeaf;  // min number of values in a leaf
    public int numSplitCols; // mostly for random forest
    public List"lt"Node"gt" tree = new List"lt"Node"gt"();
    public Random rnd;  // order in which cols are searched

    public double[][] trainX;  // store data by ref
    public double[] trainY;

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

    public class Node
    {
      public int id;
      public int colIdx;  // aka featureIdx
      public double thresh;
      public int left;  // index into List
      public int right;
      public double value;
      public bool isLeaf;
      public List"lt"int"gt" rows;  // assoc rows in train data

      public Node()
      {
        this.id = -1;
        this.colIdx = -1;
        this.thresh = 0.0;  // aka split value
        this.left = -1;
        this.right = -1;
        this.value = 0.0;  // aka pred y
        this.isLeaf = false;
        this.rows = null;
      }
    } // class Node

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

    public DecisionTreeRegressor(int maxDepth = 2,
      int minSamples = 2, int minLeaf = 1,
      int numSplitCols = -1, int seed = 0)
    {
      // if maxDepth = 0, tree has just a root node
      // if maxDepth = 1, at most 3 nodes (root, l, r)
      // if maxDepth = n, at most 2^(n+1) - 1 nodes
      this.maxDepth = maxDepth;
      this.minSamples = minSamples;
      this.minLeaf = minLeaf;
      this.numSplitCols = numSplitCols;  // for ran. forest

      // create full tree List with null nodes
      int numNodes = (int)Math.Pow(2, (maxDepth + 1)) - 1;
      for (int i = 0; i "lt" numNodes; ++i)
      {
        this.tree.Add(null);  // empty nodes
      }
      this.rnd = new Random(seed);
    }

    // ------------------------------------------------------
    // public: Train(), Predict()
    // helpers: MakeTree(), BestSplit(), TreeTargetMean(),
    //   TreeTargetVariance().
    // ------------------------------------------------------

    public void Train(double[][] trainX, double[] trainY)
    {
      this.trainX = trainX; // 
      this.trainY = trainY;
      this.MakeTree();

      // delete rows in each Node to save space
      // when tree is part of an ensemble
       for (int i = 0; i "lt" this.tree.Count; ++i)
        if (this.tree[i] != null)
          this.tree[i].rows = null;
    }

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

    public double Predict(double[] x)
    {
      int p = 0;
      Node currNode = this.tree[p];
      while (currNode != null && 
        currNode.isLeaf == false &&
        p "lt" this.tree.Count)
      {
        if (x[currNode.colIdx] "lte" currNode.thresh)
          p = currNode.left;
        else
          p = currNode.right;
        currNode = this.tree[p];
      }
      return this.tree[p].value;
    }

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

    // helpers: MakeTree(), BestSplit(),
    // TreeTargetMean(), TreeTargetVariance()

    private void MakeTree()
    {
      // no recursion, no pointers, List storage, no stack
      if (this.numSplitCols == -1) // use all cols
        this.numSplitCols = this.trainX[0].Length;

      // prepare root node
      List"lt"int"gt" allRows = new List"lt"int"gt"();
      for (int i = 0; i "lt" this.trainX.Length; ++i)
        allRows.Add(i);
      double grandMean = this.TreeTargetMean(allRows);
     
      // wait to supply colIdx and thresh in loop
      Node root = new Node();
      root.id = 0;
      root.left = 1;
      root.right = 2;
      root.value = grandMean;
      root.isLeaf = false; // already set
      root.rows = allRows;
      this.tree[0] = root;

      for (int i = 0; i "lt" this.tree.Count; ++i) 
      {
        Node currNode = this.tree[i];
        // curr node has values for everything
        // except colIdx and thresh

        // curr node too deep to have children OR
        // curr node not enough rows to split then
        // leave both children as null
        if (currNode == null ||
          currNode.rows.Count == 0) { continue; }

        // if parent cannot be split, make parent a leaf
        if (currNode.id "gte" (int)Math.Pow(2,
          (this.maxDepth)) - 1 ||
          currNode.rows.Count "lt" this.minSamples)
        {
          currNode.isLeaf = true;
          continue;
        }

        // parent has enough rows to try to split
        double[] splitInfo = this.BestSplit(currNode.rows);
        int colIdx = (int)splitInfo[0];
        double splitVal = splitInfo[1]; //split value

        if (colIdx == -1)  // parent is a leaf
        {
          currNode.isLeaf = true;
          continue;
        }

        // complete the fields for curr node
        currNode.colIdx = colIdx;
        currNode.thresh = splitVal;
       
        // construct the children, except colIdx and thresh
        Node leftNode = new Node();
        Node rightNode = new Node();

        // construct the children rows using split info
        // all info except colIdx and thresh
        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);
        }

        // assign -1 to children if out of range
        leftNode.id = currNode.id * 2 + 1;
        if (leftNode.id "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) leftNode.id = -1;
        leftNode.left = leftNode.id * 2 + 1;
        if (leftNode.left "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) leftNode.left = -1;
        leftNode.right = leftNode.id * 2 + 2;
        if (leftNode.right "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) leftNode.right = -1;
        leftNode.rows = leftIdxs;
        leftNode.value = 
          this.TreeTargetMean(leftNode.rows);
        this.tree[leftNode.id] = leftNode;

        rightNode.id = currNode.id * 2 + 2;
        if (rightNode.id "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) rightNode.id = -1;
        rightNode.left = rightNode.id * 2 + 1;
        if (rightNode.left "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) rightNode.left = -1;
        rightNode.right = rightNode.id * 2 + 2;
        if (rightNode.right "gt" (int)Math.Pow(2,
          (maxDepth + 1)) - 2) rightNode.right = -1;
        rightNode.rows = rightIdxs;
        rightNode.value = 
          this.TreeTargetMean(rightNode.rows);
        this.tree[rightNode.id] = rightNode;

      } // i
      return;
    }

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

    private double[] BestSplit(List"lt"int"gt" rows)
    {
      // implicit params numSplitCols, minLeaf, numsplitCols
      // result[0] = best col idx (as double)
      // result[1] = best split value
      rows.Sort();

      int bestColIdx = -1;  // indicates bad split
      double bestThresh = 0.0;
      double bestVar = double.MaxValue;  // smaller better

      int nRows = rows.Count;  // or dataY.Length
      int nCols = this.trainX[0].Length;

      if (nRows == 0)
      {
        throw new Exception("empty data in BestSplit()");
      }

      // process cols in scrambled order
      int[] colIndices = new int[nCols];
      for (int k = 0; k "lt" nCols; ++k)
        colIndices[k] = k;
      // shuffle, inline Fisher-Yates
      int n = colIndices.Length;
      for (int i = 0; i "lt" n; ++i)
      {
        int ri = rnd.Next(i, n);  // be careful
        int tmp = colIndices[i];
        colIndices[i] = colIndices[ri];
        colIndices[ri] = tmp;
      }

      // numSplitCols is usually all columns (-1)
      for (int j = 0; j "lt" this.numSplitCols; ++j)
      {
        int colIdx = colIndices[j];
        HashSet"lt"double"gt" examineds = 
          new HashSet"lt"double"gt"();

        for (int i = 0; i "lt" nRows; ++i) // each row
        {
          // if curr thresh been seen, skip it
          double thresh = this.trainX[rows[i]][colIdx];
          if (examineds.Contains(thresh)) continue;
          examineds.Add(thresh);

          // get row idxs where x is lte, gt thresh
          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" nRows; ++k)
          {
            if (this.trainX[rows[k]][colIdx] "lte" thresh)
              leftIdxs.Add(rows[k]);
            else
              rightIdxs.Add(rows[k]);
          }

          // Check if proposed split has too few values
          if (leftIdxs.Count "lt" this.minLeaf ||
            rightIdxs.Count "lt" this.minLeaf)
            continue;  // to next row

          double leftVar = 
            this.TreeTargetVariance(leftIdxs);
          double rightVar = 
            this.TreeTargetVariance(rightIdxs);
          double weightedVar = (leftIdxs.Count * leftVar +
            rightIdxs.Count * rightVar) / nRows;

          if (weightedVar "lt" bestVar)
          {
            // if this never happens, bestColIdx remains -1
            // which means a bad split. used in MakeTree()
            bestColIdx = colIdx;
            bestThresh = thresh;
            bestVar = weightedVar;
          }

        } // each row
      } // j each col

      double[] result = new double[2];  // out params ugly
      result[0] = 1.0 * bestColIdx;
      result[1] = bestThresh;
      return result;

    } // BestSplit()

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

    private double TreeTargetMean(List"lt"int"gt" rows)
    {
      // mean of rows items in trainY: for node prediction
      double sum = 0.0;
      for (int i = 0; i "lt" rows.Count; ++i)
      {
        int r = rows[i];
        sum += this.trainY[r];
      }
      return sum / rows.Count;
    }

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

    private double TreeTargetVariance(List"lt"int"gt" rows)
    {
      double mean = this.TreeTargetMean(rows);
      double sum = 0.0;
      for (int i = 0; i "lt" rows.Count; ++i)
      {
        int r = rows[i];
        sum += (this.trainY[r] - mean) *
          (this.trainY[r] - mean);
      }
      return sum / rows.Count;
    }

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

  } // class DecisionTreeRegressor

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

} // ns

Training data:

# diabetes_norm_train_342.txt
# cols [0] to [9] predictors. col [10] target
# norm division constants:
# 100, -1, 100, 1000, 1000, 1000, 100, 10, 10, 1000, 1000
#
0.5900, 1.0000, 0.3210, 0.1010, 0.1570, 0.0932, 0.3800, 0.4000, 0.4860, 0.0870, 0.1510
0.4800, 0.0000, 0.2160, 0.0870, 0.1830, 0.1032, 0.7000, 0.3000, 0.3892, 0.0690, 0.0750
0.7200, 1.0000, 0.3050, 0.0930, 0.1560, 0.0936, 0.4100, 0.4000, 0.4673, 0.0850, 0.1410
0.2400, 0.0000, 0.2530, 0.0840, 0.1980, 0.1314, 0.4000, 0.5000, 0.4890, 0.0890, 0.2060
0.5000, 0.0000, 0.2300, 0.1010, 0.1920, 0.1254, 0.5200, 0.4000, 0.4291, 0.0800, 0.1350
0.2300, 0.0000, 0.2260, 0.0890, 0.1390, 0.0648, 0.6100, 0.2000, 0.4190, 0.0680, 0.0970
0.3600, 1.0000, 0.2200, 0.0900, 0.1600, 0.0996, 0.5000, 0.3000, 0.3951, 0.0820, 0.1380
0.6600, 1.0000, 0.2620, 0.1140, 0.2550, 0.1850, 0.5600, 0.4550, 0.4249, 0.0920, 0.0630
0.6000, 1.0000, 0.3210, 0.0830, 0.1790, 0.1194, 0.4200, 0.4000, 0.4477, 0.0940, 0.1100
0.2900, 0.0000, 0.3000, 0.0850, 0.1800, 0.0934, 0.4300, 0.4000, 0.5385, 0.0880, 0.3100
0.2200, 0.0000, 0.1860, 0.0970, 0.1140, 0.0576, 0.4600, 0.2000, 0.3951, 0.0830, 0.1010
0.5600, 1.0000, 0.2800, 0.0850, 0.1840, 0.1448, 0.3200, 0.6000, 0.3584, 0.0770, 0.0690
0.5300, 0.0000, 0.2370, 0.0920, 0.1860, 0.1092, 0.6200, 0.3000, 0.4304, 0.0810, 0.1790
0.5000, 1.0000, 0.2620, 0.0970, 0.1860, 0.1054, 0.4900, 0.4000, 0.5063, 0.0880, 0.1850
0.6100, 0.0000, 0.2400, 0.0910, 0.2020, 0.1154, 0.7200, 0.3000, 0.4291, 0.0730, 0.1180
0.3400, 1.0000, 0.2470, 0.1180, 0.2540, 0.1842, 0.3900, 0.7000, 0.5037, 0.0810, 0.1710
0.4700, 0.0000, 0.3030, 0.1090, 0.2070, 0.1002, 0.7000, 0.3000, 0.5215, 0.0980, 0.1660
0.6800, 1.0000, 0.2750, 0.1110, 0.2140, 0.1470, 0.3900, 0.5000, 0.4942, 0.0910, 0.1440
0.3800, 0.0000, 0.2540, 0.0840, 0.1620, 0.1030, 0.4200, 0.4000, 0.4443, 0.0870, 0.0970
0.4100, 0.0000, 0.2470, 0.0830, 0.1870, 0.1082, 0.6000, 0.3000, 0.4543, 0.0780, 0.1680
0.3500, 0.0000, 0.2110, 0.0820, 0.1560, 0.0878, 0.5000, 0.3000, 0.4511, 0.0950, 0.0680
0.2500, 1.0000, 0.2430, 0.0950, 0.1620, 0.0986, 0.5400, 0.3000, 0.3850, 0.0870, 0.0490
0.2500, 0.0000, 0.2600, 0.0920, 0.1870, 0.1204, 0.5600, 0.3000, 0.3970, 0.0880, 0.0680
0.6100, 1.0000, 0.3200, 0.1037, 0.2100, 0.0852, 0.3500, 0.6000, 0.6107, 0.1240, 0.2450
0.3100, 0.0000, 0.2970, 0.0880, 0.1670, 0.1034, 0.4800, 0.4000, 0.4357, 0.0780, 0.1840
0.3000, 1.0000, 0.2520, 0.0830, 0.1780, 0.1184, 0.3400, 0.5000, 0.4852, 0.0830, 0.2020
0.1900, 0.0000, 0.1920, 0.0870, 0.1240, 0.0540, 0.5700, 0.2000, 0.4174, 0.0900, 0.1370
0.4200, 0.0000, 0.3190, 0.0830, 0.1580, 0.0876, 0.5300, 0.3000, 0.4466, 0.1010, 0.0850
0.6300, 0.0000, 0.2440, 0.0730, 0.1600, 0.0914, 0.4800, 0.3000, 0.4635, 0.0780, 0.1310
0.6700, 1.0000, 0.2580, 0.1130, 0.1580, 0.0542, 0.6400, 0.2000, 0.5293, 0.1040, 0.2830
0.3200, 0.0000, 0.3050, 0.0890, 0.1820, 0.1106, 0.5600, 0.3000, 0.4344, 0.0890, 0.1290
0.4200, 0.0000, 0.2030, 0.0710, 0.1610, 0.0812, 0.6600, 0.2000, 0.4234, 0.0810, 0.0590
0.5800, 1.0000, 0.3800, 0.1030, 0.1500, 0.1072, 0.2200, 0.7000, 0.4644, 0.0980, 0.3410
0.5700, 0.0000, 0.2170, 0.0940, 0.1570, 0.0580, 0.8200, 0.2000, 0.4443, 0.0920, 0.0870
0.5300, 0.0000, 0.2050, 0.0780, 0.1470, 0.0842, 0.5200, 0.3000, 0.3989, 0.0750, 0.0650
0.6200, 1.0000, 0.2350, 0.0803, 0.2250, 0.1128, 0.8600, 0.2620, 0.4875, 0.0960, 0.1020
0.5200, 0.0000, 0.2850, 0.1100, 0.1950, 0.0972, 0.6000, 0.3000, 0.5242, 0.0850, 0.2650
0.4600, 0.0000, 0.2740, 0.0780, 0.1710, 0.0880, 0.5800, 0.3000, 0.4828, 0.0900, 0.2760
0.4800, 1.0000, 0.3300, 0.1230, 0.2530, 0.1636, 0.4400, 0.6000, 0.5425, 0.0970, 0.2520
0.4800, 1.0000, 0.2770, 0.0730, 0.1910, 0.1194, 0.4600, 0.4000, 0.4852, 0.0920, 0.0900
0.5000, 1.0000, 0.2560, 0.1010, 0.2290, 0.1622, 0.4300, 0.5000, 0.4779, 0.1140, 0.1000
0.2100, 0.0000, 0.2010, 0.0630, 0.1350, 0.0690, 0.5400, 0.3000, 0.4094, 0.0890, 0.0550
0.3200, 1.0000, 0.2540, 0.0903, 0.1530, 0.1004, 0.3400, 0.4500, 0.4533, 0.0830, 0.0610
0.5400, 0.0000, 0.2420, 0.0740, 0.2040, 0.1090, 0.8200, 0.2000, 0.4174, 0.1090, 0.0920
0.6100, 1.0000, 0.3270, 0.0970, 0.1770, 0.1184, 0.2900, 0.6000, 0.4997, 0.0870, 0.2590
0.5600, 1.0000, 0.2310, 0.1040, 0.1810, 0.1164, 0.4700, 0.4000, 0.4477, 0.0790, 0.0530
0.3300, 0.0000, 0.2530, 0.0850, 0.1550, 0.0850, 0.5100, 0.3000, 0.4554, 0.0700, 0.1900
0.2700, 0.0000, 0.1960, 0.0780, 0.1280, 0.0680, 0.4300, 0.3000, 0.4443, 0.0710, 0.1420
0.6700, 1.0000, 0.2250, 0.0980, 0.1910, 0.1192, 0.6100, 0.3000, 0.3989, 0.0860, 0.0750
0.3700, 1.0000, 0.2770, 0.0930, 0.1800, 0.1194, 0.3000, 0.6000, 0.5030, 0.0880, 0.1420
0.5800, 0.0000, 0.2570, 0.0990, 0.1570, 0.0916, 0.4900, 0.3000, 0.4407, 0.0930, 0.1550
0.6500, 1.0000, 0.2790, 0.1030, 0.1590, 0.0968, 0.4200, 0.4000, 0.4615, 0.0860, 0.2250
0.3400, 0.0000, 0.2550, 0.0930, 0.2180, 0.1440, 0.5700, 0.4000, 0.4443, 0.0880, 0.0590
0.4600, 0.0000, 0.2490, 0.1150, 0.1980, 0.1296, 0.5400, 0.4000, 0.4277, 0.1030, 0.1040
0.3500, 0.0000, 0.2870, 0.0970, 0.2040, 0.1268, 0.6400, 0.3000, 0.4190, 0.0930, 0.1820
0.3700, 0.0000, 0.2180, 0.0840, 0.1840, 0.1010, 0.7300, 0.3000, 0.3912, 0.0930, 0.1280
0.3700, 0.0000, 0.3020, 0.0870, 0.1660, 0.0960, 0.4000, 0.4150, 0.5011, 0.0870, 0.0520
0.4100, 0.0000, 0.2050, 0.0800, 0.1240, 0.0488, 0.6400, 0.2000, 0.4025, 0.0750, 0.0370
0.6000, 0.0000, 0.2040, 0.1050, 0.1980, 0.0784, 0.9900, 0.2000, 0.4635, 0.0790, 0.1700
0.6600, 1.0000, 0.2400, 0.0980, 0.2360, 0.1464, 0.5800, 0.4000, 0.5063, 0.0960, 0.1700
0.2900, 0.0000, 0.2600, 0.0830, 0.1410, 0.0652, 0.6400, 0.2000, 0.4078, 0.0830, 0.0610
0.3700, 1.0000, 0.2680, 0.0790, 0.1570, 0.0980, 0.2800, 0.6000, 0.5043, 0.0960, 0.1440
0.4100, 1.0000, 0.2570, 0.0830, 0.1810, 0.1066, 0.6600, 0.3000, 0.3738, 0.0850, 0.0520
0.3900, 0.0000, 0.2290, 0.0770, 0.2040, 0.1432, 0.4600, 0.4000, 0.4304, 0.0740, 0.1280
0.6700, 1.0000, 0.2400, 0.0830, 0.1430, 0.0772, 0.4900, 0.3000, 0.4431, 0.0940, 0.0710
0.3600, 1.0000, 0.2410, 0.1120, 0.1930, 0.1250, 0.3500, 0.6000, 0.5106, 0.0950, 0.1630
0.4600, 1.0000, 0.2470, 0.0850, 0.1740, 0.1232, 0.3000, 0.6000, 0.4644, 0.0960, 0.1500
0.6000, 1.0000, 0.2500, 0.0897, 0.1850, 0.1208, 0.4600, 0.4020, 0.4511, 0.0920, 0.0970
0.5900, 1.0000, 0.2360, 0.0830, 0.1650, 0.1000, 0.4700, 0.4000, 0.4500, 0.0920, 0.1600
0.5300, 0.0000, 0.2210, 0.0930, 0.1340, 0.0762, 0.4600, 0.3000, 0.4078, 0.0960, 0.1780
0.4800, 0.0000, 0.1990, 0.0910, 0.1890, 0.1096, 0.6900, 0.3000, 0.3951, 0.1010, 0.0480
0.4800, 0.0000, 0.2950, 0.1310, 0.2070, 0.1322, 0.4700, 0.4000, 0.4935, 0.1060, 0.2700
0.6600, 1.0000, 0.2600, 0.0910, 0.2640, 0.1466, 0.6500, 0.4000, 0.5568, 0.0870, 0.2020
0.5200, 1.0000, 0.2450, 0.0940, 0.2170, 0.1494, 0.4800, 0.5000, 0.4585, 0.0890, 0.1110
0.5200, 1.0000, 0.2660, 0.1110, 0.2090, 0.1264, 0.6100, 0.3000, 0.4682, 0.1090, 0.0850
0.4600, 1.0000, 0.2350, 0.0870, 0.1810, 0.1148, 0.4400, 0.4000, 0.4710, 0.0980, 0.0420
0.4000, 1.0000, 0.2900, 0.1150, 0.0970, 0.0472, 0.3500, 0.2770, 0.4304, 0.0950, 0.1700
0.2200, 0.0000, 0.2300, 0.0730, 0.1610, 0.0978, 0.5400, 0.3000, 0.3829, 0.0910, 0.2000
0.5000, 0.0000, 0.2100, 0.0880, 0.1400, 0.0718, 0.3500, 0.4000, 0.5112, 0.0710, 0.2520
0.2000, 0.0000, 0.2290, 0.0870, 0.1910, 0.1282, 0.5300, 0.4000, 0.3892, 0.0850, 0.1130
0.6800, 0.0000, 0.2750, 0.1070, 0.2410, 0.1496, 0.6400, 0.4000, 0.4920, 0.0900, 0.1430
0.5200, 1.0000, 0.2430, 0.0860, 0.1970, 0.1336, 0.4400, 0.5000, 0.4575, 0.0910, 0.0510
0.4400, 0.0000, 0.2310, 0.0870, 0.2130, 0.1264, 0.7700, 0.3000, 0.3871, 0.0720, 0.0520
0.3800, 0.0000, 0.2730, 0.0810, 0.1460, 0.0816, 0.4700, 0.3000, 0.4466, 0.0810, 0.2100
0.4900, 0.0000, 0.2270, 0.0653, 0.1680, 0.0962, 0.6200, 0.2710, 0.3892, 0.0600, 0.0650
0.6100, 0.0000, 0.3300, 0.0950, 0.1820, 0.1148, 0.5400, 0.3000, 0.4190, 0.0740, 0.1410
0.2900, 1.0000, 0.1940, 0.0830, 0.1520, 0.1058, 0.3900, 0.4000, 0.3584, 0.0830, 0.0550
0.6100, 0.0000, 0.2580, 0.0980, 0.2350, 0.1258, 0.7600, 0.3000, 0.5112, 0.0820, 0.1340
0.3400, 1.0000, 0.2260, 0.0750, 0.1660, 0.0918, 0.6000, 0.3000, 0.4263, 0.1080, 0.0420
0.3600, 0.0000, 0.2190, 0.0890, 0.1890, 0.1052, 0.6800, 0.3000, 0.4369, 0.0960, 0.1110
0.5200, 0.0000, 0.2400, 0.0830, 0.1670, 0.0866, 0.7100, 0.2000, 0.3850, 0.0940, 0.0980
0.6100, 0.0000, 0.3120, 0.0790, 0.2350, 0.1568, 0.4700, 0.5000, 0.5050, 0.0960, 0.1640
0.4300, 0.0000, 0.2680, 0.1230, 0.1930, 0.1022, 0.6700, 0.3000, 0.4779, 0.0940, 0.0480
0.3500, 0.0000, 0.2040, 0.0650, 0.1870, 0.1056, 0.6700, 0.2790, 0.4277, 0.0780, 0.0960
0.2700, 0.0000, 0.2480, 0.0910, 0.1890, 0.1068, 0.6900, 0.3000, 0.4190, 0.0690, 0.0900
0.2900, 0.0000, 0.2100, 0.0710, 0.1560, 0.0970, 0.3800, 0.4000, 0.4654, 0.0900, 0.1620
0.6400, 1.0000, 0.2730, 0.1090, 0.1860, 0.1076, 0.3800, 0.5000, 0.5308, 0.0990, 0.1500
0.4100, 0.0000, 0.3460, 0.0873, 0.2050, 0.1426, 0.4100, 0.5000, 0.4673, 0.1100, 0.2790
0.4900, 1.0000, 0.2590, 0.0910, 0.1780, 0.1066, 0.5200, 0.3000, 0.4575, 0.0750, 0.0920
0.4800, 0.0000, 0.2040, 0.0980, 0.2090, 0.1394, 0.4600, 0.5000, 0.4771, 0.0780, 0.0830
0.5300, 0.0000, 0.2800, 0.0880, 0.2330, 0.1438, 0.5800, 0.4000, 0.5050, 0.0910, 0.1280
0.5300, 1.0000, 0.2220, 0.1130, 0.1970, 0.1152, 0.6700, 0.3000, 0.4304, 0.1000, 0.1020
0.2300, 0.0000, 0.2900, 0.0900, 0.2160, 0.1314, 0.6500, 0.3000, 0.4585, 0.0910, 0.3020
0.6500, 1.0000, 0.3020, 0.0980, 0.2190, 0.1606, 0.4000, 0.5000, 0.4522, 0.0840, 0.1980
0.4100, 0.0000, 0.3240, 0.0940, 0.1710, 0.1044, 0.5600, 0.3000, 0.3970, 0.0760, 0.0950
0.5500, 1.0000, 0.2340, 0.0830, 0.1660, 0.1016, 0.4600, 0.4000, 0.4522, 0.0960, 0.0530
0.2200, 0.0000, 0.1930, 0.0820, 0.1560, 0.0932, 0.5200, 0.3000, 0.3989, 0.0710, 0.1340
0.5600, 0.0000, 0.3100, 0.0787, 0.1870, 0.1414, 0.3400, 0.5500, 0.4060, 0.0900, 0.1440
0.5400, 1.0000, 0.3060, 0.1033, 0.1440, 0.0798, 0.3000, 0.4800, 0.5142, 0.1010, 0.2320
0.5900, 1.0000, 0.2550, 0.0953, 0.1900, 0.1394, 0.3500, 0.5430, 0.4357, 0.1170, 0.0810
0.6000, 1.0000, 0.2340, 0.0880, 0.1530, 0.0898, 0.5800, 0.3000, 0.3258, 0.0950, 0.1040
0.5400, 0.0000, 0.2680, 0.0870, 0.2060, 0.1220, 0.6800, 0.3000, 0.4382, 0.0800, 0.0590
0.2500, 0.0000, 0.2830, 0.0870, 0.1930, 0.1280, 0.4900, 0.4000, 0.4382, 0.0920, 0.2460
0.5400, 1.0000, 0.2770, 0.1130, 0.2000, 0.1284, 0.3700, 0.5000, 0.5153, 0.1130, 0.2970
0.5500, 0.0000, 0.3660, 0.1130, 0.1990, 0.0944, 0.4300, 0.4630, 0.5730, 0.0970, 0.2580
0.4000, 1.0000, 0.2650, 0.0930, 0.2360, 0.1470, 0.3700, 0.7000, 0.5561, 0.0920, 0.2290
0.6200, 1.0000, 0.3180, 0.1150, 0.1990, 0.1286, 0.4400, 0.5000, 0.4883, 0.0980, 0.2750
0.6500, 0.0000, 0.2440, 0.1200, 0.2220, 0.1356, 0.3700, 0.6000, 0.5509, 0.1240, 0.2810
0.3300, 1.0000, 0.2540, 0.1020, 0.2060, 0.1410, 0.3900, 0.5000, 0.4868, 0.1050, 0.1790
0.5300, 0.0000, 0.2200, 0.0940, 0.1750, 0.0880, 0.5900, 0.3000, 0.4942, 0.0980, 0.2000
0.3500, 0.0000, 0.2680, 0.0980, 0.1620, 0.1036, 0.4500, 0.4000, 0.4205, 0.0860, 0.2000
0.6600, 0.0000, 0.2800, 0.1010, 0.1950, 0.1292, 0.4000, 0.5000, 0.4860, 0.0940, 0.1730
0.6200, 1.0000, 0.3390, 0.1010, 0.2210, 0.1564, 0.3500, 0.6000, 0.4997, 0.1030, 0.1800
0.5000, 1.0000, 0.2960, 0.0943, 0.3000, 0.2424, 0.3300, 0.9090, 0.4812, 0.1090, 0.0840
0.4700, 0.0000, 0.2860, 0.0970, 0.1640, 0.0906, 0.5600, 0.3000, 0.4466, 0.0880, 0.1210
0.4700, 1.0000, 0.2560, 0.0940, 0.1650, 0.0748, 0.4000, 0.4000, 0.5526, 0.0930, 0.1610
0.2400, 0.0000, 0.2070, 0.0870, 0.1490, 0.0806, 0.6100, 0.2000, 0.3611, 0.0780, 0.0990
0.5800, 1.0000, 0.2620, 0.0910, 0.2170, 0.1242, 0.7100, 0.3000, 0.4691, 0.0680, 0.1090
0.3400, 0.0000, 0.2060, 0.0870, 0.1850, 0.1122, 0.5800, 0.3000, 0.4304, 0.0740, 0.1150
0.5100, 0.0000, 0.2790, 0.0960, 0.1960, 0.1222, 0.4200, 0.5000, 0.5069, 0.1200, 0.2680
0.3100, 1.0000, 0.3530, 0.1250, 0.1870, 0.1124, 0.4800, 0.4000, 0.4890, 0.1090, 0.2740
0.2200, 0.0000, 0.1990, 0.0750, 0.1750, 0.1086, 0.5400, 0.3000, 0.4127, 0.0720, 0.1580
0.5300, 1.0000, 0.2440, 0.0920, 0.2140, 0.1460, 0.5000, 0.4000, 0.4500, 0.0970, 0.1070
0.3700, 1.0000, 0.2140, 0.0830, 0.1280, 0.0696, 0.4900, 0.3000, 0.3850, 0.0840, 0.0830
0.2800, 0.0000, 0.3040, 0.0850, 0.1980, 0.1156, 0.6700, 0.3000, 0.4344, 0.0800, 0.1030
0.4700, 0.0000, 0.3160, 0.0840, 0.1540, 0.0880, 0.3000, 0.5100, 0.5199, 0.1050, 0.2720
0.2300, 0.0000, 0.1880, 0.0780, 0.1450, 0.0720, 0.6300, 0.2000, 0.3912, 0.0860, 0.0850
0.5000, 0.0000, 0.3100, 0.1230, 0.1780, 0.1050, 0.4800, 0.4000, 0.4828, 0.0880, 0.2800
0.5800, 1.0000, 0.3670, 0.1170, 0.1660, 0.0938, 0.4400, 0.4000, 0.4949, 0.1090, 0.3360
0.5500, 0.0000, 0.3210, 0.1100, 0.1640, 0.0842, 0.4200, 0.4000, 0.5242, 0.0900, 0.2810
0.6000, 1.0000, 0.2770, 0.1070, 0.1670, 0.1146, 0.3800, 0.4000, 0.4277, 0.0950, 0.1180
0.4100, 0.0000, 0.3080, 0.0810, 0.2140, 0.1520, 0.2800, 0.7600, 0.5136, 0.1230, 0.3170
0.6000, 1.0000, 0.2750, 0.1060, 0.2290, 0.1438, 0.5100, 0.4000, 0.5142, 0.0910, 0.2350
0.4000, 0.0000, 0.2690, 0.0920, 0.2030, 0.1198, 0.7000, 0.3000, 0.4190, 0.0810, 0.0600
0.5700, 1.0000, 0.3070, 0.0900, 0.2040, 0.1478, 0.3400, 0.6000, 0.4710, 0.0930, 0.1740
0.3700, 0.0000, 0.3830, 0.1130, 0.1650, 0.0946, 0.5300, 0.3000, 0.4466, 0.0790, 0.2590
0.4000, 1.0000, 0.3190, 0.0950, 0.1980, 0.1356, 0.3800, 0.5000, 0.4804, 0.0930, 0.1780
0.3300, 0.0000, 0.3500, 0.0890, 0.2000, 0.1304, 0.4200, 0.4760, 0.4927, 0.1010, 0.1280
0.3200, 1.0000, 0.2780, 0.0890, 0.2160, 0.1462, 0.5500, 0.4000, 0.4304, 0.0910, 0.0960
0.3500, 1.0000, 0.2590, 0.0810, 0.1740, 0.1024, 0.3100, 0.6000, 0.5313, 0.0820, 0.1260
0.5500, 0.0000, 0.3290, 0.1020, 0.1640, 0.1062, 0.4100, 0.4000, 0.4431, 0.0890, 0.2880
0.4900, 0.0000, 0.2600, 0.0930, 0.1830, 0.1002, 0.6400, 0.3000, 0.4543, 0.0880, 0.0880
0.3900, 1.0000, 0.2630, 0.1150, 0.2180, 0.1582, 0.3200, 0.7000, 0.4935, 0.1090, 0.2920
0.6000, 1.0000, 0.2230, 0.1130, 0.1860, 0.1258, 0.4600, 0.4000, 0.4263, 0.0940, 0.0710
0.6700, 1.0000, 0.2830, 0.0930, 0.2040, 0.1322, 0.4900, 0.4000, 0.4736, 0.0920, 0.1970
0.4100, 1.0000, 0.3200, 0.1090, 0.2510, 0.1706, 0.4900, 0.5000, 0.5056, 0.1030, 0.1860
0.4400, 0.0000, 0.2540, 0.0950, 0.1620, 0.0926, 0.5300, 0.3000, 0.4407, 0.0830, 0.0250
0.4800, 1.0000, 0.2330, 0.0893, 0.2120, 0.1428, 0.4600, 0.4610, 0.4754, 0.0980, 0.0840
0.4500, 0.0000, 0.2030, 0.0743, 0.1900, 0.1262, 0.4900, 0.3880, 0.4304, 0.0790, 0.0960
0.4700, 0.0000, 0.3040, 0.1200, 0.1990, 0.1200, 0.4600, 0.4000, 0.5106, 0.0870, 0.1950
0.4600, 0.0000, 0.2060, 0.0730, 0.1720, 0.1070, 0.5100, 0.3000, 0.4249, 0.0800, 0.0530
0.3600, 1.0000, 0.3230, 0.1150, 0.2860, 0.1994, 0.3900, 0.7000, 0.5472, 0.1120, 0.2170
0.3400, 0.0000, 0.2920, 0.0730, 0.1720, 0.1082, 0.4900, 0.4000, 0.4304, 0.0910, 0.1720
0.5300, 1.0000, 0.3310, 0.1170, 0.1830, 0.1190, 0.4800, 0.4000, 0.4382, 0.1060, 0.1310
0.6100, 0.0000, 0.2460, 0.1010, 0.2090, 0.1068, 0.7700, 0.3000, 0.4836, 0.0880, 0.2140
0.3700, 0.0000, 0.2020, 0.0810, 0.1620, 0.0878, 0.6300, 0.3000, 0.4025, 0.0880, 0.0590
0.3300, 1.0000, 0.2080, 0.0840, 0.1250, 0.0702, 0.4600, 0.3000, 0.3784, 0.0660, 0.0700
0.6800, 0.0000, 0.3280, 0.1057, 0.2050, 0.1164, 0.4000, 0.5130, 0.5493, 0.1170, 0.2200
0.4900, 1.0000, 0.3190, 0.0940, 0.2340, 0.1558, 0.3400, 0.7000, 0.5398, 0.1220, 0.2680
0.4800, 0.0000, 0.2390, 0.1090, 0.2320, 0.1052, 0.3700, 0.6000, 0.6107, 0.0960, 0.1520
0.5500, 1.0000, 0.2450, 0.0840, 0.1790, 0.1058, 0.6600, 0.3000, 0.3584, 0.0870, 0.0470
0.4300, 0.0000, 0.2210, 0.0660, 0.1340, 0.0772, 0.4500, 0.3000, 0.4078, 0.0800, 0.0740
0.6000, 1.0000, 0.3300, 0.0970, 0.2170, 0.1256, 0.4500, 0.5000, 0.5447, 0.1120, 0.2950
0.3100, 1.0000, 0.1900, 0.0930, 0.1370, 0.0730, 0.4700, 0.3000, 0.4443, 0.0780, 0.1010
0.5300, 1.0000, 0.2730, 0.0820, 0.1190, 0.0550, 0.3900, 0.3000, 0.4828, 0.0930, 0.1510
0.6700, 0.0000, 0.2280, 0.0870, 0.1660, 0.0986, 0.5200, 0.3000, 0.4344, 0.0920, 0.1270
0.6100, 1.0000, 0.2820, 0.1060, 0.2040, 0.1320, 0.5200, 0.4000, 0.4605, 0.0960, 0.2370
0.6200, 0.0000, 0.2890, 0.0873, 0.2060, 0.1272, 0.3300, 0.6240, 0.5434, 0.0990, 0.2250
0.6000, 0.0000, 0.2560, 0.0870, 0.2070, 0.1258, 0.6900, 0.3000, 0.4111, 0.0840, 0.0810
0.4200, 0.0000, 0.2490, 0.0910, 0.2040, 0.1418, 0.3800, 0.5000, 0.4796, 0.0890, 0.1510
0.3800, 1.0000, 0.2680, 0.1050, 0.1810, 0.1192, 0.3700, 0.5000, 0.4820, 0.0910, 0.1070
0.6200, 0.0000, 0.2240, 0.0790, 0.2220, 0.1474, 0.5900, 0.4000, 0.4357, 0.0760, 0.0640
0.6100, 1.0000, 0.2690, 0.1110, 0.2360, 0.1724, 0.3900, 0.6000, 0.4812, 0.0890, 0.1380
0.6100, 1.0000, 0.2310, 0.1130, 0.1860, 0.1144, 0.4700, 0.4000, 0.4812, 0.1050, 0.1850
0.5300, 0.0000, 0.2860, 0.0880, 0.1710, 0.0988, 0.4100, 0.4000, 0.5050, 0.0990, 0.2650
0.2800, 1.0000, 0.2470, 0.0970, 0.1750, 0.0996, 0.3200, 0.5000, 0.5380, 0.0870, 0.1010
0.2600, 1.0000, 0.3030, 0.0890, 0.2180, 0.1522, 0.3100, 0.7000, 0.5159, 0.0820, 0.1370
0.3000, 0.0000, 0.2130, 0.0870, 0.1340, 0.0630, 0.6300, 0.2000, 0.3689, 0.0660, 0.1430
0.5000, 0.0000, 0.2610, 0.1090, 0.2430, 0.1606, 0.6200, 0.4000, 0.4625, 0.0890, 0.1410
0.4800, 0.0000, 0.2020, 0.0950, 0.1870, 0.1174, 0.5300, 0.4000, 0.4419, 0.0850, 0.0790
0.5100, 0.0000, 0.2520, 0.1030, 0.1760, 0.1122, 0.3700, 0.5000, 0.4898, 0.0900, 0.2920
0.4700, 1.0000, 0.2250, 0.0820, 0.1310, 0.0668, 0.4100, 0.3000, 0.4754, 0.0890, 0.1780
0.6400, 1.0000, 0.2350, 0.0970, 0.2030, 0.1290, 0.5900, 0.3000, 0.4318, 0.0770, 0.0910
0.5100, 1.0000, 0.2590, 0.0760, 0.2400, 0.1690, 0.3900, 0.6000, 0.5075, 0.0960, 0.1160
0.3000, 0.0000, 0.2090, 0.1040, 0.1520, 0.0838, 0.4700, 0.3000, 0.4663, 0.0970, 0.0860
0.5600, 1.0000, 0.2870, 0.0990, 0.2080, 0.1464, 0.3900, 0.5000, 0.4727, 0.0970, 0.1220
0.4200, 0.0000, 0.2210, 0.0850, 0.2130, 0.1386, 0.6000, 0.4000, 0.4277, 0.0940, 0.0720
0.6200, 1.0000, 0.2670, 0.1150, 0.1830, 0.1240, 0.3500, 0.5000, 0.4788, 0.1000, 0.1290
0.3400, 0.0000, 0.3140, 0.0870, 0.1490, 0.0938, 0.4600, 0.3000, 0.3829, 0.0770, 0.1420
0.6000, 0.0000, 0.2220, 0.1047, 0.2210, 0.1054, 0.6000, 0.3680, 0.5628, 0.0930, 0.0900
0.6400, 0.0000, 0.2100, 0.0923, 0.2270, 0.1468, 0.6500, 0.3490, 0.4331, 0.1020, 0.1580
0.3900, 1.0000, 0.2120, 0.0900, 0.1820, 0.1104, 0.6000, 0.3000, 0.4060, 0.0980, 0.0390
0.7100, 1.0000, 0.2650, 0.1050, 0.2810, 0.1736, 0.5500, 0.5000, 0.5568, 0.0840, 0.1960
0.4800, 1.0000, 0.2920, 0.1100, 0.2180, 0.1516, 0.3900, 0.6000, 0.4920, 0.0980, 0.2220
0.7900, 1.0000, 0.2700, 0.1030, 0.1690, 0.1108, 0.3700, 0.5000, 0.4663, 0.1100, 0.2770
0.4000, 0.0000, 0.3070, 0.0990, 0.1770, 0.0854, 0.5000, 0.4000, 0.5338, 0.0850, 0.0990
0.4900, 1.0000, 0.2880, 0.0920, 0.2070, 0.1400, 0.4400, 0.5000, 0.4745, 0.0920, 0.1960
0.5100, 0.0000, 0.3060, 0.1030, 0.1980, 0.1066, 0.5700, 0.3000, 0.5148, 0.1000, 0.2020
0.5700, 0.0000, 0.3010, 0.1170, 0.2020, 0.1396, 0.4200, 0.5000, 0.4625, 0.1200, 0.1550
0.5900, 1.0000, 0.2470, 0.1140, 0.1520, 0.1048, 0.2900, 0.5000, 0.4511, 0.0880, 0.0770
0.5100, 0.0000, 0.2770, 0.0990, 0.2290, 0.1456, 0.6900, 0.3000, 0.4277, 0.0770, 0.1910
0.7400, 0.0000, 0.2980, 0.1010, 0.1710, 0.1048, 0.5000, 0.3000, 0.4394, 0.0860, 0.0700
0.6700, 0.0000, 0.2670, 0.1050, 0.2250, 0.1354, 0.6900, 0.3000, 0.4635, 0.0960, 0.0730
0.4900, 0.0000, 0.1980, 0.0880, 0.1880, 0.1148, 0.5700, 0.3000, 0.4394, 0.0930, 0.0490
0.5700, 0.0000, 0.2330, 0.0880, 0.1550, 0.0636, 0.7800, 0.2000, 0.4205, 0.0780, 0.0650
0.5600, 1.0000, 0.3510, 0.1230, 0.1640, 0.0950, 0.3800, 0.4000, 0.5043, 0.1170, 0.2630
0.5200, 1.0000, 0.2970, 0.1090, 0.2280, 0.1628, 0.3100, 0.8000, 0.5142, 0.1030, 0.2480
0.6900, 0.0000, 0.2930, 0.1240, 0.2230, 0.1390, 0.5400, 0.4000, 0.5011, 0.1020, 0.2960
0.3700, 0.0000, 0.2030, 0.0830, 0.1850, 0.1246, 0.3800, 0.5000, 0.4719, 0.0880, 0.2140
0.2400, 0.0000, 0.2250, 0.0890, 0.1410, 0.0680, 0.5200, 0.3000, 0.4654, 0.0840, 0.1850
0.5500, 1.0000, 0.2270, 0.0930, 0.1540, 0.0942, 0.5300, 0.3000, 0.3526, 0.0750, 0.0780
0.3600, 0.0000, 0.2280, 0.0870, 0.1780, 0.1160, 0.4100, 0.4000, 0.4654, 0.0820, 0.0930
0.4200, 1.0000, 0.2400, 0.1070, 0.1500, 0.0850, 0.4400, 0.3000, 0.4654, 0.0960, 0.2520
0.2100, 0.0000, 0.2420, 0.0760, 0.1470, 0.0770, 0.5300, 0.3000, 0.4443, 0.0790, 0.1500
0.4100, 0.0000, 0.2020, 0.0620, 0.1530, 0.0890, 0.5000, 0.3000, 0.4249, 0.0890, 0.0770
0.5700, 1.0000, 0.2940, 0.1090, 0.1600, 0.0876, 0.3100, 0.5000, 0.5333, 0.0920, 0.2080
0.2000, 1.0000, 0.2210, 0.0870, 0.1710, 0.0996, 0.5800, 0.3000, 0.4205, 0.0780, 0.0770
0.6700, 1.0000, 0.2360, 0.1113, 0.1890, 0.1054, 0.7000, 0.2700, 0.4220, 0.0930, 0.1080
0.3400, 0.0000, 0.2520, 0.0770, 0.1890, 0.1206, 0.5300, 0.4000, 0.4344, 0.0790, 0.1600
0.4100, 1.0000, 0.2490, 0.0860, 0.1920, 0.1150, 0.6100, 0.3000, 0.4382, 0.0940, 0.0530
0.3800, 1.0000, 0.3300, 0.0780, 0.3010, 0.2150, 0.5000, 0.6020, 0.5193, 0.1080, 0.2200
0.5100, 0.0000, 0.2350, 0.1010, 0.1950, 0.1210, 0.5100, 0.4000, 0.4745, 0.0940, 0.1540
0.5200, 1.0000, 0.2640, 0.0913, 0.2180, 0.1520, 0.3900, 0.5590, 0.4905, 0.0990, 0.2590
0.6700, 0.0000, 0.2980, 0.0800, 0.1720, 0.0934, 0.6300, 0.3000, 0.4357, 0.0820, 0.0900
0.6100, 0.0000, 0.3000, 0.1080, 0.1940, 0.1000, 0.5200, 0.3730, 0.5347, 0.1050, 0.2460
0.6700, 1.0000, 0.2500, 0.1117, 0.1460, 0.0934, 0.3300, 0.4420, 0.4585, 0.1030, 0.1240
0.5600, 0.0000, 0.2700, 0.1050, 0.2470, 0.1606, 0.5400, 0.5000, 0.5088, 0.0940, 0.0670
0.6400, 0.0000, 0.2000, 0.0747, 0.1890, 0.1148, 0.6200, 0.3050, 0.4111, 0.0910, 0.0720
0.5800, 1.0000, 0.2550, 0.1120, 0.1630, 0.1106, 0.2900, 0.6000, 0.4762, 0.0860, 0.2570
0.5500, 0.0000, 0.2820, 0.0910, 0.2500, 0.1402, 0.6700, 0.4000, 0.5366, 0.1030, 0.2620
0.6200, 1.0000, 0.3330, 0.1140, 0.1820, 0.1140, 0.3800, 0.5000, 0.5011, 0.0960, 0.2750
0.5700, 1.0000, 0.2560, 0.0960, 0.2000, 0.1330, 0.5200, 0.3850, 0.4318, 0.1050, 0.1770
0.2000, 1.0000, 0.2420, 0.0880, 0.1260, 0.0722, 0.4500, 0.3000, 0.3784, 0.0740, 0.0710
0.5300, 1.0000, 0.2210, 0.0980, 0.1650, 0.1052, 0.4700, 0.4000, 0.4159, 0.0810, 0.0470
0.3200, 1.0000, 0.3140, 0.0890, 0.1530, 0.0842, 0.5600, 0.3000, 0.4159, 0.0900, 0.1870
0.4100, 0.0000, 0.2310, 0.0860, 0.1480, 0.0780, 0.5800, 0.3000, 0.4094, 0.0600, 0.1250
0.6000, 0.0000, 0.2340, 0.0767, 0.2470, 0.1480, 0.6500, 0.3800, 0.5136, 0.0770, 0.0780
0.2600, 0.0000, 0.1880, 0.0830, 0.1910, 0.1036, 0.6900, 0.3000, 0.4522, 0.0690, 0.0510
0.3700, 0.0000, 0.3080, 0.1120, 0.2820, 0.1972, 0.4300, 0.7000, 0.5342, 0.1010, 0.2580
0.4500, 0.0000, 0.3200, 0.1100, 0.2240, 0.1342, 0.4500, 0.5000, 0.5412, 0.0930, 0.2150
0.6700, 0.0000, 0.3160, 0.1160, 0.1790, 0.0904, 0.4100, 0.4000, 0.5472, 0.1000, 0.3030
0.3400, 1.0000, 0.3550, 0.1200, 0.2330, 0.1466, 0.3400, 0.7000, 0.5568, 0.1010, 0.2430
0.5000, 0.0000, 0.3190, 0.0783, 0.2070, 0.1492, 0.3800, 0.5450, 0.4595, 0.0840, 0.0910
0.7100, 0.0000, 0.2950, 0.0970, 0.2270, 0.1516, 0.4500, 0.5000, 0.5024, 0.1080, 0.1500
0.5700, 1.0000, 0.3160, 0.1170, 0.2250, 0.1076, 0.4000, 0.6000, 0.5958, 0.1130, 0.3100
0.4900, 0.0000, 0.2030, 0.0930, 0.1840, 0.1030, 0.6100, 0.3000, 0.4605, 0.0930, 0.1530
0.3500, 0.0000, 0.4130, 0.0810, 0.1680, 0.1028, 0.3700, 0.5000, 0.4949, 0.0940, 0.3460
0.4100, 1.0000, 0.2120, 0.1020, 0.1840, 0.1004, 0.6400, 0.3000, 0.4585, 0.0790, 0.0630
0.7000, 1.0000, 0.2410, 0.0823, 0.1940, 0.1492, 0.3100, 0.6260, 0.4234, 0.1050, 0.0890
0.5200, 0.0000, 0.2300, 0.1070, 0.1790, 0.1237, 0.4250, 0.4210, 0.4159, 0.0930, 0.0500
0.6000, 0.0000, 0.2560, 0.0780, 0.1950, 0.0954, 0.9100, 0.2000, 0.3761, 0.0870, 0.0390
0.6200, 0.0000, 0.2250, 0.1250, 0.2150, 0.0990, 0.9800, 0.2000, 0.4500, 0.0950, 0.1030
0.4400, 1.0000, 0.3820, 0.1230, 0.2010, 0.1266, 0.4400, 0.5000, 0.5024, 0.0920, 0.3080
0.2800, 1.0000, 0.1920, 0.0810, 0.1550, 0.0946, 0.5100, 0.3000, 0.3850, 0.0870, 0.1160
0.5800, 1.0000, 0.2900, 0.0850, 0.1560, 0.1092, 0.3600, 0.4000, 0.3989, 0.0860, 0.1450
0.3900, 1.0000, 0.2400, 0.0897, 0.1900, 0.1136, 0.5200, 0.3650, 0.4804, 0.1010, 0.0740
0.3400, 1.0000, 0.2060, 0.0980, 0.1830, 0.0920, 0.8300, 0.2000, 0.3689, 0.0920, 0.0450
0.6500, 0.0000, 0.2630, 0.0700, 0.2440, 0.1662, 0.5100, 0.5000, 0.4898, 0.0980, 0.1150
0.6600, 1.0000, 0.3460, 0.1150, 0.2040, 0.1394, 0.3600, 0.6000, 0.4963, 0.1090, 0.2640
0.5100, 0.0000, 0.2340, 0.0870, 0.2200, 0.1088, 0.9300, 0.2000, 0.4511, 0.0820, 0.0870
0.5000, 1.0000, 0.2920, 0.1190, 0.1620, 0.0852, 0.5400, 0.3000, 0.4736, 0.0950, 0.2020
0.5900, 1.0000, 0.2720, 0.1070, 0.1580, 0.1020, 0.3900, 0.4000, 0.4443, 0.0930, 0.1270
0.5200, 0.0000, 0.2700, 0.0783, 0.1340, 0.0730, 0.4400, 0.3050, 0.4443, 0.0690, 0.1820
0.6900, 1.0000, 0.2450, 0.1080, 0.2430, 0.1364, 0.4000, 0.6000, 0.5808, 0.1000, 0.2410
0.5300, 0.0000, 0.2410, 0.1050, 0.1840, 0.1134, 0.4600, 0.4000, 0.4812, 0.0950, 0.0660
0.4700, 1.0000, 0.2530, 0.0980, 0.1730, 0.1056, 0.4400, 0.4000, 0.4762, 0.1080, 0.0940
0.5200, 0.0000, 0.2880, 0.1130, 0.2800, 0.1740, 0.6700, 0.4000, 0.5273, 0.0860, 0.2830
0.3900, 0.0000, 0.2090, 0.0950, 0.1500, 0.0656, 0.6800, 0.2000, 0.4407, 0.0950, 0.0640
0.6700, 1.0000, 0.2300, 0.0700, 0.1840, 0.1280, 0.3500, 0.5000, 0.4654, 0.0990, 0.1020
0.5900, 1.0000, 0.2410, 0.0960, 0.1700, 0.0986, 0.5400, 0.3000, 0.4466, 0.0850, 0.2000
0.5100, 1.0000, 0.2810, 0.1060, 0.2020, 0.1222, 0.5500, 0.4000, 0.4820, 0.0870, 0.2650
0.2300, 1.0000, 0.1800, 0.0780, 0.1710, 0.0960, 0.4800, 0.4000, 0.4905, 0.0920, 0.0940
0.6800, 0.0000, 0.2590, 0.0930, 0.2530, 0.1812, 0.5300, 0.5000, 0.4543, 0.0980, 0.2300
0.4400, 0.0000, 0.2150, 0.0850, 0.1570, 0.0922, 0.5500, 0.3000, 0.3892, 0.0840, 0.1810
0.6000, 1.0000, 0.2430, 0.1030, 0.1410, 0.0866, 0.3300, 0.4000, 0.4673, 0.0780, 0.1560
0.5200, 0.0000, 0.2450, 0.0900, 0.1980, 0.1290, 0.2900, 0.7000, 0.5298, 0.0860, 0.2330
0.3800, 0.0000, 0.2130, 0.0720, 0.1650, 0.0602, 0.8800, 0.2000, 0.4431, 0.0900, 0.0600
0.6100, 0.0000, 0.2580, 0.0900, 0.2800, 0.1954, 0.5500, 0.5000, 0.4997, 0.0900, 0.2190
0.6800, 1.0000, 0.2480, 0.1010, 0.2210, 0.1514, 0.6000, 0.4000, 0.3871, 0.0870, 0.0800
0.2800, 1.0000, 0.3150, 0.0830, 0.2280, 0.1494, 0.3800, 0.6000, 0.5313, 0.0830, 0.0680
0.6500, 1.0000, 0.3350, 0.1020, 0.1900, 0.1262, 0.3500, 0.5000, 0.4970, 0.1020, 0.3320
0.6900, 0.0000, 0.2810, 0.1130, 0.2340, 0.1428, 0.5200, 0.4000, 0.5278, 0.0770, 0.2480
0.5100, 0.0000, 0.2430, 0.0853, 0.1530, 0.0716, 0.7100, 0.2150, 0.3951, 0.0820, 0.0840
0.2900, 0.0000, 0.3500, 0.0983, 0.2040, 0.1426, 0.5000, 0.4080, 0.4043, 0.0910, 0.2000
0.5500, 1.0000, 0.2350, 0.0930, 0.1770, 0.1268, 0.4100, 0.4000, 0.3829, 0.0830, 0.0550
0.3400, 1.0000, 0.3000, 0.0830, 0.1850, 0.1072, 0.5300, 0.3000, 0.4820, 0.0920, 0.0850
0.6700, 0.0000, 0.2070, 0.0830, 0.1700, 0.0998, 0.5900, 0.3000, 0.4025, 0.0770, 0.0890
0.4900, 0.0000, 0.2560, 0.0760, 0.1610, 0.0998, 0.5100, 0.3000, 0.3932, 0.0780, 0.0310
0.5500, 1.0000, 0.2290, 0.0810, 0.1230, 0.0672, 0.4100, 0.3000, 0.4304, 0.0880, 0.1290
0.5900, 1.0000, 0.2510, 0.0900, 0.1630, 0.1014, 0.4600, 0.4000, 0.4357, 0.0910, 0.0830
0.5300, 0.0000, 0.3320, 0.0827, 0.1860, 0.1068, 0.4600, 0.4040, 0.5112, 0.1020, 0.2750
0.4800, 1.0000, 0.2410, 0.1100, 0.2090, 0.1346, 0.5800, 0.4000, 0.4407, 0.1000, 0.0650
0.5200, 0.0000, 0.2950, 0.1043, 0.2110, 0.1328, 0.4900, 0.4310, 0.4984, 0.0980, 0.1980
0.6900, 0.0000, 0.2960, 0.1220, 0.2310, 0.1284, 0.5600, 0.4000, 0.5451, 0.0860, 0.2360
0.6000, 1.0000, 0.2280, 0.1100, 0.2450, 0.1898, 0.3900, 0.6000, 0.4394, 0.0880, 0.2530
0.4600, 1.0000, 0.2270, 0.0830, 0.1830, 0.1258, 0.3200, 0.6000, 0.4836, 0.0750, 0.1240
0.5100, 1.0000, 0.2620, 0.1010, 0.1610, 0.0996, 0.4800, 0.3000, 0.4205, 0.0880, 0.0440
0.6700, 1.0000, 0.2350, 0.0960, 0.2070, 0.1382, 0.4200, 0.5000, 0.4898, 0.1110, 0.1720
0.4900, 0.0000, 0.2210, 0.0850, 0.1360, 0.0634, 0.6200, 0.2190, 0.3970, 0.0720, 0.1140
0.4600, 1.0000, 0.2650, 0.0940, 0.2470, 0.1602, 0.5900, 0.4000, 0.4935, 0.1110, 0.1420
0.4700, 0.0000, 0.3240, 0.1050, 0.1880, 0.1250, 0.4600, 0.4090, 0.4443, 0.0990, 0.1090
0.7500, 0.0000, 0.3010, 0.0780, 0.2220, 0.1542, 0.4400, 0.5050, 0.4779, 0.0970, 0.1800
0.2800, 0.0000, 0.2420, 0.0930, 0.1740, 0.1064, 0.5400, 0.3000, 0.4220, 0.0840, 0.1440
0.6500, 1.0000, 0.3130, 0.1100, 0.2130, 0.1280, 0.4700, 0.5000, 0.5247, 0.0910, 0.1630
0.4200, 0.0000, 0.3010, 0.0910, 0.1820, 0.1148, 0.4900, 0.4000, 0.4511, 0.0820, 0.1470
0.5100, 0.0000, 0.2450, 0.0790, 0.2120, 0.1286, 0.6500, 0.3000, 0.4522, 0.0910, 0.0970
0.5300, 1.0000, 0.2770, 0.0950, 0.1900, 0.1018, 0.4100, 0.5000, 0.5464, 0.1010, 0.2200
0.5400, 0.0000, 0.2320, 0.1107, 0.2380, 0.1628, 0.4800, 0.4960, 0.4913, 0.1080, 0.1900
0.7300, 0.0000, 0.2700, 0.1020, 0.2110, 0.1210, 0.6700, 0.3000, 0.4745, 0.0990, 0.1090
0.5400, 0.0000, 0.2680, 0.1080, 0.1760, 0.0806, 0.6700, 0.3000, 0.4956, 0.1060, 0.1910
0.4200, 0.0000, 0.2920, 0.0930, 0.2490, 0.1742, 0.4500, 0.6000, 0.5004, 0.0920, 0.1220
0.7500, 0.0000, 0.3120, 0.1177, 0.2290, 0.1388, 0.2900, 0.7900, 0.5724, 0.1060, 0.2300
0.5500, 1.0000, 0.3210, 0.1127, 0.2070, 0.0924, 0.2500, 0.8280, 0.6105, 0.1110, 0.2420
0.6800, 1.0000, 0.2570, 0.1090, 0.2330, 0.1126, 0.3500, 0.7000, 0.6057, 0.1050, 0.2480
0.5700, 0.0000, 0.2690, 0.0980, 0.2460, 0.1652, 0.3800, 0.7000, 0.5366, 0.0960, 0.2490
0.4800, 0.0000, 0.3140, 0.0753, 0.2420, 0.1516, 0.3800, 0.6370, 0.5568, 0.1030, 0.1920
0.6100, 1.0000, 0.2560, 0.0850, 0.1840, 0.1162, 0.3900, 0.5000, 0.4970, 0.0980, 0.1310
0.6900, 0.0000, 0.3700, 0.1030, 0.2070, 0.1314, 0.5500, 0.4000, 0.4635, 0.0900, 0.2370
0.3800, 0.0000, 0.3260, 0.0770, 0.1680, 0.1006, 0.4700, 0.4000, 0.4625, 0.0960, 0.0780
0.4500, 1.0000, 0.2120, 0.0940, 0.1690, 0.0968, 0.5500, 0.3000, 0.4454, 0.1020, 0.1350
0.5100, 1.0000, 0.2920, 0.1070, 0.1870, 0.1390, 0.3200, 0.6000, 0.4382, 0.0950, 0.2440
0.7100, 1.0000, 0.2400, 0.0840, 0.1380, 0.0858, 0.3900, 0.4000, 0.4190, 0.0900, 0.1990
0.5700, 0.0000, 0.3610, 0.1170, 0.1810, 0.1082, 0.3400, 0.5000, 0.5268, 0.1000, 0.2700
0.5600, 1.0000, 0.2580, 0.1030, 0.1770, 0.1144, 0.3400, 0.5000, 0.4963, 0.0990, 0.1640
0.3200, 1.0000, 0.2200, 0.0880, 0.1370, 0.0786, 0.4800, 0.3000, 0.3951, 0.0780, 0.0720
0.5000, 0.0000, 0.2190, 0.0910, 0.1900, 0.1112, 0.6700, 0.3000, 0.4078, 0.0770, 0.0960
0.4300, 0.0000, 0.3430, 0.0840, 0.2560, 0.1726, 0.3300, 0.8000, 0.5529, 0.1040, 0.3060
0.5400, 1.0000, 0.2520, 0.1150, 0.1810, 0.1200, 0.3900, 0.5000, 0.4701, 0.0920, 0.0910
0.3100, 0.0000, 0.2330, 0.0850, 0.1900, 0.1308, 0.4300, 0.4000, 0.4394, 0.0770, 0.2140
0.5600, 0.0000, 0.2570, 0.0800, 0.2440, 0.1516, 0.5900, 0.4000, 0.5118, 0.0950, 0.0950
0.4400, 0.0000, 0.2510, 0.1330, 0.1820, 0.1130, 0.5500, 0.3000, 0.4249, 0.0840, 0.2160
0.5700, 1.0000, 0.3190, 0.1110, 0.1730, 0.1162, 0.4100, 0.4000, 0.4369, 0.0870, 0.2630

Test data:

# diabetes_norm_test_100.txt
#
0.6400, 1.0000, 0.2840, 0.1110, 0.1840, 0.1270, 0.4100, 0.4000, 0.4382, 0.0970, 0.1780
0.4300, 0.0000, 0.2810, 0.1210, 0.1920, 0.1210, 0.6000, 0.3000, 0.4007, 0.0930, 0.1130
0.1900, 0.0000, 0.2530, 0.0830, 0.2250, 0.1566, 0.4600, 0.5000, 0.4719, 0.0840, 0.2000
0.7100, 1.0000, 0.2610, 0.0850, 0.2200, 0.1524, 0.4700, 0.5000, 0.4635, 0.0910, 0.1390
0.5000, 1.0000, 0.2800, 0.1040, 0.2820, 0.1968, 0.4400, 0.6000, 0.5328, 0.0950, 0.1390
0.5900, 1.0000, 0.2360, 0.0730, 0.1800, 0.1074, 0.5100, 0.4000, 0.4682, 0.0840, 0.0880
0.5700, 0.0000, 0.2450, 0.0930, 0.1860, 0.0966, 0.7100, 0.3000, 0.4522, 0.0910, 0.1480
0.4900, 1.0000, 0.2100, 0.0820, 0.1190, 0.0854, 0.2300, 0.5000, 0.3970, 0.0740, 0.0880
0.4100, 1.0000, 0.3200, 0.1260, 0.1980, 0.1042, 0.4900, 0.4000, 0.5412, 0.1240, 0.2430
0.2500, 1.0000, 0.2260, 0.0850, 0.1300, 0.0710, 0.4800, 0.3000, 0.4007, 0.0810, 0.0710
0.5200, 1.0000, 0.1970, 0.0810, 0.1520, 0.0534, 0.8200, 0.2000, 0.4419, 0.0820, 0.0770
0.3400, 0.0000, 0.2120, 0.0840, 0.2540, 0.1134, 0.5200, 0.5000, 0.6094, 0.0920, 0.1090
0.4200, 1.0000, 0.3060, 0.1010, 0.2690, 0.1722, 0.5000, 0.5000, 0.5455, 0.1060, 0.2720
0.2800, 1.0000, 0.2550, 0.0990, 0.1620, 0.1016, 0.4600, 0.4000, 0.4277, 0.0940, 0.0600
0.4700, 1.0000, 0.2330, 0.0900, 0.1950, 0.1258, 0.5400, 0.4000, 0.4331, 0.0730, 0.0540
0.3200, 1.0000, 0.3100, 0.1000, 0.1770, 0.0962, 0.4500, 0.4000, 0.5187, 0.0770, 0.2210
0.4300, 0.0000, 0.1850, 0.0870, 0.1630, 0.0936, 0.6100, 0.2670, 0.3738, 0.0800, 0.0900
0.5900, 1.0000, 0.2690, 0.1040, 0.1940, 0.1266, 0.4300, 0.5000, 0.4804, 0.1060, 0.3110
0.5300, 0.0000, 0.2830, 0.1010, 0.1790, 0.1070, 0.4800, 0.4000, 0.4788, 0.1010, 0.2810
0.6000, 0.0000, 0.2570, 0.1030, 0.1580, 0.0846, 0.6400, 0.2000, 0.3850, 0.0970, 0.1820
0.5400, 1.0000, 0.3610, 0.1150, 0.1630, 0.0984, 0.4300, 0.4000, 0.4682, 0.1010, 0.3210
0.3500, 1.0000, 0.2410, 0.0947, 0.1550, 0.0974, 0.3200, 0.4840, 0.4852, 0.0940, 0.0580
0.4900, 1.0000, 0.2580, 0.0890, 0.1820, 0.1186, 0.3900, 0.5000, 0.4804, 0.1150, 0.2620
0.5800, 0.0000, 0.2280, 0.0910, 0.1960, 0.1188, 0.4800, 0.4000, 0.4984, 0.1150, 0.2060
0.3600, 1.0000, 0.3910, 0.0900, 0.2190, 0.1358, 0.3800, 0.6000, 0.5421, 0.1030, 0.2330
0.4600, 1.0000, 0.4220, 0.0990, 0.2110, 0.1370, 0.4400, 0.5000, 0.5011, 0.0990, 0.2420
0.4400, 1.0000, 0.2660, 0.0990, 0.2050, 0.1090, 0.4300, 0.5000, 0.5580, 0.1110, 0.1230
0.4600, 0.0000, 0.2990, 0.0830, 0.1710, 0.1130, 0.3800, 0.4500, 0.4585, 0.0980, 0.1670
0.5400, 0.0000, 0.2100, 0.0780, 0.1880, 0.1074, 0.7000, 0.3000, 0.3970, 0.0730, 0.0630
0.6300, 1.0000, 0.2550, 0.1090, 0.2260, 0.1032, 0.4600, 0.5000, 0.5951, 0.0870, 0.1970
0.4100, 1.0000, 0.2420, 0.0900, 0.1990, 0.1236, 0.5700, 0.4000, 0.4522, 0.0860, 0.0710
0.2800, 0.0000, 0.2540, 0.0930, 0.1410, 0.0790, 0.4900, 0.3000, 0.4174, 0.0910, 0.1680
0.1900, 0.0000, 0.2320, 0.0750, 0.1430, 0.0704, 0.5200, 0.3000, 0.4635, 0.0720, 0.1400
0.6100, 1.0000, 0.2610, 0.1260, 0.2150, 0.1298, 0.5700, 0.4000, 0.4949, 0.0960, 0.2170
0.4800, 0.0000, 0.3270, 0.0930, 0.2760, 0.1986, 0.4300, 0.6420, 0.5148, 0.0910, 0.1210
0.5400, 1.0000, 0.2730, 0.1000, 0.2000, 0.1440, 0.3300, 0.6000, 0.4745, 0.0760, 0.2350
0.5300, 1.0000, 0.2660, 0.0930, 0.1850, 0.1224, 0.3600, 0.5000, 0.4890, 0.0820, 0.2450
0.4800, 0.0000, 0.2280, 0.1010, 0.1100, 0.0416, 0.5600, 0.2000, 0.4127, 0.0970, 0.0400
0.5300, 0.0000, 0.2880, 0.1117, 0.1450, 0.0872, 0.4600, 0.3150, 0.4078, 0.0850, 0.0520
0.2900, 1.0000, 0.1810, 0.0730, 0.1580, 0.0990, 0.4100, 0.4000, 0.4500, 0.0780, 0.1040
0.6200, 0.0000, 0.3200, 0.0880, 0.1720, 0.0690, 0.3800, 0.4000, 0.5784, 0.1000, 0.1320
0.5000, 1.0000, 0.2370, 0.0920, 0.1660, 0.0970, 0.5200, 0.3000, 0.4443, 0.0930, 0.0880
0.5800, 1.0000, 0.2360, 0.0960, 0.2570, 0.1710, 0.5900, 0.4000, 0.4905, 0.0820, 0.0690
0.5500, 1.0000, 0.2460, 0.1090, 0.1430, 0.0764, 0.5100, 0.3000, 0.4357, 0.0880, 0.2190
0.5400, 0.0000, 0.2260, 0.0900, 0.1830, 0.1042, 0.6400, 0.3000, 0.4304, 0.0920, 0.0720
0.3600, 0.0000, 0.2780, 0.0730, 0.1530, 0.1044, 0.4200, 0.4000, 0.3497, 0.0730, 0.2010
0.6300, 1.0000, 0.2410, 0.1110, 0.1840, 0.1122, 0.4400, 0.4000, 0.4935, 0.0820, 0.1100
0.4700, 1.0000, 0.2650, 0.0700, 0.1810, 0.1048, 0.6300, 0.3000, 0.4190, 0.0700, 0.0510
0.5100, 1.0000, 0.3280, 0.1120, 0.2020, 0.1006, 0.3700, 0.5000, 0.5775, 0.1090, 0.2770
0.4200, 0.0000, 0.1990, 0.0760, 0.1460, 0.0832, 0.5500, 0.3000, 0.3664, 0.0790, 0.0630
0.3700, 1.0000, 0.2360, 0.0940, 0.2050, 0.1388, 0.5300, 0.4000, 0.4190, 0.1070, 0.1180
0.2800, 0.0000, 0.2210, 0.0820, 0.1680, 0.1006, 0.5400, 0.3000, 0.4205, 0.0860, 0.0690
0.5800, 0.0000, 0.2810, 0.1110, 0.1980, 0.0806, 0.3100, 0.6000, 0.6068, 0.0930, 0.2730
0.3200, 0.0000, 0.2650, 0.0860, 0.1840, 0.1016, 0.5300, 0.4000, 0.4990, 0.0780, 0.2580
0.2500, 1.0000, 0.2350, 0.0880, 0.1430, 0.0808, 0.5500, 0.3000, 0.3584, 0.0830, 0.0430
0.6300, 0.0000, 0.2600, 0.0857, 0.1550, 0.0782, 0.4600, 0.3370, 0.5037, 0.0970, 0.1980
0.5200, 0.0000, 0.2780, 0.0850, 0.2190, 0.1360, 0.4900, 0.4000, 0.5136, 0.0750, 0.2420
0.6500, 1.0000, 0.2850, 0.1090, 0.2010, 0.1230, 0.4600, 0.4000, 0.5075, 0.0960, 0.2320
0.4200, 0.0000, 0.3060, 0.1210, 0.1760, 0.0928, 0.6900, 0.3000, 0.4263, 0.0890, 0.1750
0.5300, 0.0000, 0.2220, 0.0780, 0.1640, 0.0810, 0.7000, 0.2000, 0.4174, 0.1010, 0.0930
0.7900, 1.0000, 0.2330, 0.0880, 0.1860, 0.1284, 0.3300, 0.6000, 0.4812, 0.1020, 0.1680
0.4300, 0.0000, 0.3540, 0.0930, 0.1850, 0.1002, 0.4400, 0.4000, 0.5318, 0.1010, 0.2750
0.4400, 0.0000, 0.3140, 0.1150, 0.1650, 0.0976, 0.5200, 0.3000, 0.4344, 0.0890, 0.2930
0.6200, 1.0000, 0.3780, 0.1190, 0.1130, 0.0510, 0.3100, 0.4000, 0.5043, 0.0840, 0.2810
0.3300, 0.0000, 0.1890, 0.0700, 0.1620, 0.0918, 0.5900, 0.3000, 0.4025, 0.0580, 0.0720
0.5600, 0.0000, 0.3500, 0.0793, 0.1950, 0.1408, 0.4200, 0.4640, 0.4111, 0.0960, 0.1400
0.6600, 0.0000, 0.2170, 0.1260, 0.2120, 0.1278, 0.4500, 0.4710, 0.5278, 0.1010, 0.1890
0.3400, 1.0000, 0.2530, 0.1110, 0.2300, 0.1620, 0.3900, 0.6000, 0.4977, 0.0900, 0.1810
0.4600, 1.0000, 0.2380, 0.0970, 0.2240, 0.1392, 0.4200, 0.5000, 0.5366, 0.0810, 0.2090
0.5000, 0.0000, 0.3180, 0.0820, 0.1360, 0.0692, 0.5500, 0.2000, 0.4078, 0.0850, 0.1360
0.6900, 0.0000, 0.3430, 0.1130, 0.2000, 0.1238, 0.5400, 0.4000, 0.4710, 0.1120, 0.2610
0.3400, 0.0000, 0.2630, 0.0870, 0.1970, 0.1200, 0.6300, 0.3000, 0.4249, 0.0960, 0.1130
0.7100, 1.0000, 0.2700, 0.0933, 0.2690, 0.1902, 0.4100, 0.6560, 0.5242, 0.0930, 0.1310
0.4700, 0.0000, 0.2720, 0.0800, 0.2080, 0.1456, 0.3800, 0.6000, 0.4804, 0.0920, 0.1740
0.4100, 0.0000, 0.3380, 0.1233, 0.1870, 0.1270, 0.4500, 0.4160, 0.4318, 0.1000, 0.2570
0.3400, 0.0000, 0.3300, 0.0730, 0.1780, 0.1146, 0.5100, 0.3490, 0.4127, 0.0920, 0.0550
0.5100, 0.0000, 0.2410, 0.0870, 0.2610, 0.1756, 0.6900, 0.4000, 0.4407, 0.0930, 0.0840
0.4300, 0.0000, 0.2130, 0.0790, 0.1410, 0.0788, 0.5300, 0.3000, 0.3829, 0.0900, 0.0420
0.5500, 0.0000, 0.2300, 0.0947, 0.1900, 0.1376, 0.3800, 0.5000, 0.4277, 0.1060, 0.1460
0.5900, 1.0000, 0.2790, 0.1010, 0.2180, 0.1442, 0.3800, 0.6000, 0.5187, 0.0950, 0.2120
0.2700, 1.0000, 0.3360, 0.1100, 0.2460, 0.1566, 0.5700, 0.4000, 0.5088, 0.0890, 0.2330
0.5100, 1.0000, 0.2270, 0.1030, 0.2170, 0.1624, 0.3000, 0.7000, 0.4812, 0.0800, 0.0910
0.4900, 1.0000, 0.2740, 0.0890, 0.1770, 0.1130, 0.3700, 0.5000, 0.4905, 0.0970, 0.1110
0.2700, 0.0000, 0.2260, 0.0710, 0.1160, 0.0434, 0.5600, 0.2000, 0.4419, 0.0790, 0.1520
0.5700, 1.0000, 0.2320, 0.1073, 0.2310, 0.1594, 0.4100, 0.5630, 0.5030, 0.1120, 0.1200
0.3900, 1.0000, 0.2690, 0.0930, 0.1360, 0.0754, 0.4800, 0.3000, 0.4143, 0.0990, 0.0670
0.6200, 1.0000, 0.3460, 0.1200, 0.2150, 0.1292, 0.4300, 0.5000, 0.5366, 0.1230, 0.3100
0.3700, 0.0000, 0.2330, 0.0880, 0.2230, 0.1420, 0.6500, 0.3400, 0.4357, 0.0820, 0.0940
0.4600, 0.0000, 0.2110, 0.0800, 0.2050, 0.1444, 0.4200, 0.5000, 0.4533, 0.0870, 0.1830
0.6800, 1.0000, 0.2350, 0.1010, 0.1620, 0.0854, 0.5900, 0.3000, 0.4477, 0.0910, 0.0660
0.5100, 0.0000, 0.3150, 0.0930, 0.2310, 0.1440, 0.4900, 0.4700, 0.5252, 0.1170, 0.1730
0.4100, 0.0000, 0.2080, 0.0860, 0.2230, 0.1282, 0.8300, 0.3000, 0.4078, 0.0890, 0.0720
0.5300, 0.0000, 0.2650, 0.0970, 0.1930, 0.1224, 0.5800, 0.3000, 0.4143, 0.0990, 0.0490
0.4500, 0.0000, 0.2420, 0.0830, 0.1770, 0.1184, 0.4500, 0.4000, 0.4220, 0.0820, 0.0640
0.3300, 0.0000, 0.1950, 0.0800, 0.1710, 0.0854, 0.7500, 0.2000, 0.3970, 0.0800, 0.0480
0.6000, 1.0000, 0.2820, 0.1120, 0.1850, 0.1138, 0.4200, 0.4000, 0.4984, 0.0930, 0.1780
0.4700, 1.0000, 0.2490, 0.0750, 0.2250, 0.1660, 0.4200, 0.5000, 0.4443, 0.1020, 0.1040
0.6000, 1.0000, 0.2490, 0.0997, 0.1620, 0.1066, 0.4300, 0.3770, 0.4127, 0.0950, 0.1320
0.3600, 0.0000, 0.3000, 0.0950, 0.2010, 0.1252, 0.4200, 0.4790, 0.5130, 0.0850, 0.2200
0.3600, 0.0000, 0.1960, 0.0710, 0.2500, 0.1332, 0.9700, 0.3000, 0.4595, 0.0920, 0.0570

The goal of a machine learning regression model is to predict a single numeric value. For example, you might want to predict a person’s credit score based on age, annual income, bank account balance, and so on. Common regression techniques include: linear regression, nearest neighbors regression, quadratic regression, kernel ridge regression, neural network regression, random forest regression, and gradient boosting regression. Each technique has pros and cons.

The form of a linear regression model is y’ = (w0 * x0) + (w1 * x1) + . . + b, where y’ is the predicted value, the xi are predictor variables, the wi are constants called weights (or coefficients), and b is a constant called the bias (or intercept). Training is the process of finding good values for the weights and bias so that predicted y values closely match known correct target y values in a set of training data.

There are several techniques that can be used to train a linear regression model. I put together a demo, using the C# language, of four techniques:

1.) left pseudo-inverse via normal equations with Cholesky inverse.
2.) basic stochastic gradient descent (SGD).
3.) relaxed Moore-Penrose pseudo-inverse with QR modified Gram-Schmidt inverse.
4.) relaxed Moore-Penrose pseudo-inverse with SVD Jacobi inverse.

Technique #1 works well for most scenarios, but can fail when the training data has an unfortunate combination of very small and/or very large values. Technique #2 is iterative and works well for most training datasets, even huge datasets, but requires you to specify a maximum number of training iterations and a constant called a learning rate, both of which must be determined by trial and error. Technique #3 can fail for very large datasets but is relatively easy to implement. Technique #4 is very similar to #3, but is less likely to fail for large datasets, but is very complicated to implement.

The output of my demo is:

Begin C# linear regression demo

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

==========================

1. Training model using left pseudo-inv via
   normal equations (Cholesky)
Done

Weights/coefficients:
-0.2656  0.0333  -0.0454  0.0358  -0.1146
Bias/constant: 0.3619

==========================

2. Training model using SGD
Setting lrnRate = 0.0010
Setting maxEpochs = 1000
epoch =     0  MSE =   0.1132
epoch =   200  MSE =   0.0026
epoch =   400  MSE =   0.0026
epoch =   600  MSE =   0.0026
epoch =   800  MSE =   0.0026
Done

Weights/coefficients:
-0.2656  0.0333  -0.0453  0.0358  -0.1146
Bias/constant: 0.3619

==========================

3. Training model using relaxed Moore-Penrose
   pseudo-inv via QR (Gram-Schmidt)
Done

Weights/coefficients:
-0.2656  0.0333  -0.0454  0.0358  -0.1146
Bias/constant: 0.3619

==========================

4. Training model using relaxed Moore-Penrose
   pseudo-inv via SVD (Jacobi)
Done

Weights/coefficients:
-0.2656  0.0333  -0.0454  0.0358  -0.1146
Bias/constant: 0.3619

==========================

End demo

The demo data is synthetic. There are just 200 training items. There are five predictor variables, so the regression model has five weights and a bias. The output shows that all four training techniques give the same model, meaning the values of the weights and the bias are the same (to four decimal places).

Good fun for me.

Machine learning regression is sort of like finding hidden patterns in datasets. Every cover of Playboy magazine (except for the very first issue in December 1953) has a logo of a bunny head. Sometimes the logo is obvious but sometimes the logo is very cleverly hidden in the cover photo. It’s kind of like a “Where’s Waldo” for adults.

Right: On the June 1980 issue, the bunny logo is disguised as one of the butterflies.

Demo program. Very long. 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 LinearRegressionFourTrainingTechniques
{
  internal class LinearRegressionProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin C# linear regression demo ");

      // 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 model
      //Console.WriteLine("\nCreating linear " +
      //  "regression model ");
      //LinearRegressor model = new LinearRegressor();
      //Console.WriteLine("Done");

      // 3a. train model using left pseudo-inv
      Console.WriteLine("\n========================== ");
      Console.WriteLine("\n1. Training model using" +
        " left pseudo-inv via normal equations (Cholesky) ");
      LinearRegressor model1 = new LinearRegressor();
      model1.TrainClosed(trainX, trainY);
      Console.WriteLine("Done ");
      Console.WriteLine("\nWeights/coefficients: ");
      for (int i = 0; i "lt" model1.weights.Length; ++i)
        Console.Write(model1.weights[i].ToString("F4") +
          "  ");
      Console.WriteLine("\nBias/constant: " +
        model1.bias.ToString("F4"));

      // 3b. train model using SGD
      Console.WriteLine("\n========================== ");
      Console.WriteLine("\n2. Training model using SGD ");
      LinearRegressor model2 = new LinearRegressor();
      double lrnRate = 0.001;
      int maxEpochs = 1000;
      Console.WriteLine("Setting lrnRate = " +
        lrnRate.ToString("F4"));
      Console.WriteLine("Setting maxEpochs = " +
        maxEpochs);
      model2.TrainSGD(trainX, trainY, lrnRate, maxEpochs);
      Console.WriteLine("Done ");
      Console.WriteLine("\nWeights/coefficients: ");
      for (int i = 0; i "lt" model2.weights.Length; ++i)
        Console.Write(model2.weights[i].ToString("F4") +
          "  ");
      Console.WriteLine("\nBias/constant: " +
        model2.bias.ToString("F4"));

      // 3c. train model using MP pseudo-inv QR
      Console.WriteLine("\n========================== ");
      Console.WriteLine("\n3. Training model using" +
        " relaxed Moore-Penrose pseudo-inv via" +
        " QR (Gram-Schmidt) ");
      LinearRegressor model3 = new LinearRegressor();
      model3.TrainPinvQR(trainX, trainY);
      Console.WriteLine("Done ");
      Console.WriteLine("\nWeights/coefficients: ");
      for (int i = 0; i "lt" model3.weights.Length; ++i)
        Console.Write(model3.weights[i].ToString("F4") +
          "  ");
      Console.WriteLine("\nBias/constant: " +
        model3.bias.ToString("F4"));

      // 3d. train model using MP pseudo-inv SVD
      Console.WriteLine("\n========================== ");
      Console.WriteLine("\n4. Training model using" +
        " relaxed Moore-Penrose pseudo-inv via" +
        " SVD (Jacobi) ");
      LinearRegressor model4 = new LinearRegressor();
      model4.TrainPinvSVD(trainX, trainY);
      Console.WriteLine("Done ");
      Console.WriteLine("\nWeights/coefficients: ");
      for (int i = 0; i "lt" model4.weights.Length; ++i)
        Console.Write(model4.weights[i].ToString("F4") +
          "  ");
      Console.WriteLine("\nBias/constant: " +
        model4.bias.ToString("F4"));

      //// 4. 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"));

      //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("Predicted y = " +
      //  predY.ToString("F4"));

      // 6. TODO: save model weights, bias to text file

      Console.WriteLine("\n========================== ");
      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[][] 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;
    public double bias;
    private Random rnd;

    public LinearRegressor(int seed = 0)
    {
      this.weights = new double[0]; // quasi-null
      this.bias = 0.0;
      this.rnd = new Random(seed);
    }

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

    public double Predict(double[] x)
    {
      double result = 0.0;
      for (int j = 0; j "lt" x.Length; ++j)
        result += x[j] * this.weights[j];
      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 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 void TrainClosed(double[][] trainX,
      double[] trainY)
    {
      // train using left pinv via normal equations
      // can fail for very large trainX (then use SGD)
      int dim = trainX[0].Length;
      this.weights = new double[dim];

      double[][] X = Cholesky.MatToDesign(trainX);
      double[][] Xpinv = Cholesky.MatPseudoInv(X);
      double[] biasAndWts = 
        Cholesky.MatVecProduct(Xpinv, trainY);
      this.bias = biasAndWts[0];
      for (int i = 1; i "lt" biasAndWts.Length; ++i)
        this.weights[i - 1] = biasAndWts[i];
      return;
    }

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

    public void TrainSGD(double[][] trainX, double[] trainY,
      double lrnRate, int maxEpochs)
    {
      int n = trainX.Length;
      int dim = trainX[0].Length;
      this.weights = new double[dim];
      double lo = -0.01; double hi = 0.01;
      for (int i = 0; i "lt" dim; ++i)
        this.weights[i] = (hi - lo) *
          this.rnd.NextDouble() + lo;
      this.bias = (hi - lo) *
          this.rnd.NextDouble() + lo;

      int[] indices = new int[n];
      for (int i = 0; i "lt" n; ++i)
        indices[i] = i;

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

          for (int j = 0; j "lt" dim; ++j)
            this.weights[j] -= lrnRate *
              (predY - actualY) * x[j];
          this.bias -= lrnRate * (predY - actualY);
        }
        if (epoch % (int)(maxEpochs / 5) == 0)
        {
          double mse = this.MSE(trainX, trainY);
          string s1 = "epoch = " + 
            epoch.ToString().PadLeft(5);
          string s2 = "  MSE = " + 
            mse.ToString("F4").PadLeft(8);
          Console.WriteLine(s1 + s2);
        }
      }
    } // TrainSGD()

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

    private static void Shuffle(int[] indices, Random rnd)
    {
      // helper for TrainSGD()
      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;
      }
    }

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

    public void TrainPinvQR(double[][] trainX, double[] trainY)
    {
      // train using relaxed Moore-Penrose pinv, via
      // QR inverse, via QR decomp, via modified Gram-Schmidt

      int dim = trainX[0].Length;
      this.weights = new double[dim];

      double[][] X = QR_GramSchmidt.MatToDesign(trainX);
      double[][] Xpinv = QR_GramSchmidt.MatPseudoInv(X);
      double[] biasAndWts = 
        QR_GramSchmidt.MatVecProduct(Xpinv, trainY);
      this.bias = biasAndWts[0];
      for (int i = 1; i "lt" biasAndWts.Length; ++i)
        this.weights[i - 1] = biasAndWts[i];
      return;
    }

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

    public void TrainPinvSVD(double[][] trainX, double[] trainY)
    {
      // train using relaxed Moore-Penrose pinv, via
      // SVD inverse, via SVD decomp, via Jacobi

      int dim = trainX[0].Length;
      this.weights = new double[dim];

      double[][] X = SVD_Jacobi.MatToDesign(trainX);
      double[][] Xpinv = SVD_Jacobi.MatPseudoInv(X);
      double[] biasAndWts = 
        SVD_Jacobi.MatVecProduct(Xpinv, trainY);
      this.bias = biasAndWts[0];
      for (int i = 1; i "lt" biasAndWts.Length; ++i)
        this.weights[i - 1] = biasAndWts[i];
      return;
    }

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

  } // class LinearRegressor

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

  public class Cholesky
  {
    // container class for MatPseudoInv(), used in Train()
    // left pseudo-inv via normal equations

    public static double[][] MatPseudoInv(double[][] A)
    {
      // left pseudo-inverse via normal equations
      // A is design matrix where nRows must be gte nCols
      // pinv = inv(At * A) * A
      // calls MatInvCholesky, calls MatDecompCholesky
      double[][] At = MatTranspose(A);
      double[][] AtA = MatProduct(At, A);
      for (int i = 0; i "lt" AtA.Length; ++i)
        AtA[i][i] += 1.0e-8; /// condition before inv
      double[][] AtAinv = MatInvCholesky(AtA);
      double[][] pinv = MatProduct(AtAinv, At);
      return pinv;
    }

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

    public static double[][] MatToDesign(double[][] X)
    {
      // utility function
      // construct design matrix from X (add col of 1.0s)
      int nRows = X.Length; // aka m
      int nCols = X[0].Length; // n
      
      double[][] result = MatMake(nRows, nCols + 1);
      for (int i = 0; i "lt" nRows; ++i)
      {
        result[i][0] = 1.0;
        for (int j = 1; j "lt" nCols + 1; ++j)
          result[i][j] = X[i][j - 1];
      }
      return result;
    }

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

    public static double[] MatVecProduct(double[][] M,
      double[] v)
    {
      // M * v. return a regular vector
      int nRows = M.Length;
      int nCols = M[0].Length;
      int n = v.Length;
      if (nCols != n)
        throw new Exception("non-conform 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;
    }

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

    private static double[][] MatInvCholesky(double[][] A)
    {
      // A must be square, symmetric, positive definite
      // calls MatDecompCholesky()
      int m = A.Length;
      int n = A[0].Length;  // m == n
      double[][] L = MatDecompCholesky(A); // L * Lt = A

      double[][] result = MatMake(n, n); // Identity
      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[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;
    }

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

    private static double[][] MatDecompCholesky(double[][] A)
    {
      // decompose A to L such that L * Lt = A
      // A must be square
      int m = A.Length; int n = A[0].Length;  // m == n
      double[][] L = MatMake(n, n);  // or m

      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 += L[i][k] * L[j][k];
          if (i == j)
          {
            double tmp = A[i][i] - sum;
            if (tmp "lt" 0.0)
              throw new
                Exception("decomp Cholesky fatal");
            L[i][j] = Math.Sqrt(tmp);
          }
          else
          {
            if (L[j][j] == 0.0)
              throw new
                Exception("decomp Cholesky fatal ");
            L[i][j] = (A[i][j] - sum) / L[j][j];
          }
        } // j
      } // i

      return L;
    }

    // ----------------------------------------------------
    // helpers: MatMake, MatTranspose, MatProduct
    // ----------------------------------------------------

    private 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;
    }

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

    private static double[][] MatTranspose(double[][] M)
    {
      int nr = M.Length; int nc = M[0].Length;
      double[][] result = new double[nc][]; // note
      for (int i = 0; i "lt" nc; ++i)
        result[i] = new double[nr];
      for (int i = 0; i "lt" nr; ++i)
        for (int j = 0; j "lt" nc; ++j)
          result[j][i] = M[i][j]; // note
      return result;
    }

    // ````````````````````````````````````````````````````

    private 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;
    }

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

  } // class Cholesky

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

  public class QR_GramSchmidt
  {
    public static double[][] MatPseudoInv(double[][] A)
    {
      // relaxed Moore-Penrose pseudo-inv via QR
      // decomposition, modified Gram-Schmidt algorithm
      // A is design matrix where nRows must be gte nCols
      // pinv = inv(R) * inv(Q)

      int nr = A.Length;
      int nc = A[0].Length; // aka m, n
      if (nr "lt" nc)
        Console.WriteLine("ERROR: Works only m "gte" n");

      double[][] Q; double[][] R;
      MatDecompQR(A, out Q, out R); // Gram-Schmidt
      double[][] Rinv = MatInvUpperTri(R); // std algo
      double[][] Qinv = MatTranspose(Q);  // is inv(Q)
      double[][] pinv = MatProduct(Rinv, Qinv);
      return pinv;
    }

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

    public static double[][] MatToDesign(double[][] X)
    {
      // utility function
      // construct design matrix from X (add col of 1.0s)
      int nRows = X.Length; // aka m
      int nCols = X[0].Length; // n
      
      double[][] result = MatMake(nRows, nCols + 1);
      for (int i = 0; i "lt" nRows; ++i)
      {
        result[i][0] = 1.0;
        for (int j = 1; j "lt" nCols + 1; ++j)
          result[i][j] = X[i][j - 1];
      }
      return result;
    }

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

    public static double[] MatVecProduct(double[][] M,
      double[] v)
    {
      // M * v. return a regular vector
      int nRows = M.Length;
      int nCols = M[0].Length;
      int n = v.Length;
      if (nCols != n)
        throw new Exception("non-conform 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;
    }

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

    private static void MatDecompQR(double[][] A,
      out double[][] Q, out double[][] R)
    {
      // QR decomposition, modified Gram-Schmidt
      // 'reduced' mode (for machine learning scenarios)
      int m = A.Length; 
      int n = A[0].Length;
      if (m "lt" n)
        throw new Exception("m must be gte n ");

      double[][] QQ = new double[m][];  // working Q mxn
      for (int i = 0; i "lt" m; ++i)
        QQ[i] = new double[n];

      double[][] RR = new double[n][];  // working R nxn
      for (int i = 0; i "lt" n; ++i)
        RR[i] = new double[n];

      for (int k = 0; k "lt" n; ++k) // main loop each col
      {
        double[] v = new double[m];
        for (int i = 0; i "lt" m; ++i)  // col k
          v[i] = A[i][k];
        for (int j = 0; j "lt" k; ++j)  // inner loop
        {
          double[] colj = new double[QQ.Length];
          for (int i = 0; i "lt" colj.Length; ++i)
            colj[i] = QQ[i][j];

          double vecdot = 0.0;
          for (int i = 0; i "lt" colj.Length; ++i)
            vecdot += colj[i] * v[i];
          RR[j][k] = vecdot;

          // v = v - (R[j, k] * Q[:, j])
          for (int i = 0; i "lt" v.Length; ++i)
            v[i] = v[i] - (RR[j][k] * QQ[i][j]);
        } // j

        double normv = 0.0;
        for (int i = 0; i "lt" v.Length; ++i)
          normv += v[i] * v[i];
        normv = Math.Sqrt(normv);
        RR[k][k] = normv;

        // Q[:, k] = v / R[k, k]
        for (int i = 0; i "lt" QQ.Length; ++i)
          QQ[i][k] = v[i] / (RR[k][k] + 1.0e-12);

      } // k

      Q = QQ;
      R = RR;
    }

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

    private static double[][] MatInvUpperTri(double[][] A)
    {
      // used to invert R from QR
      int n = A.Length;  // must be square matrix
      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[j][k] -= result[j][i] * A[i][k];
          }
          result[j][k] /= (A[k][k] + 1.0e-8); // avoid 0
        }
      }
      return result;
    }

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

    private static double[][] MatTranspose(double[][] M)
    {
      int nRows = M.Length; int nCols = M[0].Length;
      double[][] result = MatMake(nCols, nRows);  // note
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          result[j][i] = M[i][j];
      return result;
    }

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

    private 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;
    }

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

    private 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;
    }

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

    private 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 = 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] += A[i][k] * B[k][j];

      return result;
    }

  } // class QR_GramSchmidt

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

  public class SVD_Jacobi
  {
    public static double[][] MatPseudoInv(double[][] A)
    {
      double[][] U; double[] s; double[][] Vh;
      MatDecompSVD_Jacobi(A, out U, out s, out Vh);

      double[][] Uh = MatTranspose(U);
      double[][] V = MatTranspose(Vh);

      double[][] S = MatMake(s.Length, s.Length);
      for (int i = 0; i "lt" s.Length; ++i)
        S[i][i] = 1.0 / (s[i] + 1.0e-12); // avoid div by 0
 
      double[][] result = MatProduct(V, MatProduct(S, Uh));
      return result;
    }

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

    public static double[][] MatToDesign(double[][] X)
    {
      // utility function
      // construct design matrix from X (add col of 1.0s)
      int nRows = X.Length; // aka m
      int nCols = X[0].Length; // n
      
      double[][] result = MatMake(nRows, nCols + 1);
      for (int i = 0; i "lt" nRows; ++i)
      {
        result[i][0] = 1.0;
        for (int j = 1; j "lt" nCols + 1; ++j)
          result[i][j] = X[i][j - 1];
      }
      return result;
    }

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

    public static double[] MatVecProduct(double[][] M,
      double[] v)
    {
      // M * v. return a regular vector
      int nRows = M.Length;
      int nCols = M[0].Length;
      int n = v.Length;
      if (nCols != n)
        throw new Exception("non-conform 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;
    }

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

    public static void MatDecompSVD_Jacobi(double[][] M,
      out double[][] U, out double[] s, out double[][] Vh)
    {
      // see references above
      double epsilon = 1.0e-15;

      double[][] A = MatCopyOf(M); // working U
      int m = A.Length; int n = A[0].Length;
      double[][] Q = MatIdentity(n); // working V
      double[] sv = new double[n];  // working s

      // initialize counters
      int count = 1;
      int pass = 0;
      double tolerance = 10 * m * epsilon; // heuristic

      // always do at least 12 sweeps
      int passMax = Math.Max(5 * n, 12); // heuristic

      // store the column error estimates for use
      // during orthogonalization

      for (int j = 0; j "lt" n; ++j)
      {
        double[] cj = MatGetColumn(A, j);
        double sj = VecNorm(cj);
        sv[j] = epsilon * sj;
      }

      // orthogonalize A using plane rotations
      while (count "gt" 0 && pass "lte" passMax)
      {
        // initialize rotation counter
        count = n * (n - 1) / 2;

        for (int j = 0; j "lt" n - 1; ++j)
        {
          for (int k = j + 1; k "lt" n; ++k)
          {
            double cosine, sine;

            double[] cj = MatGetColumn(A, j);
            double[] ck = MatGetColumn(A, k);

            double p = 2.0 * VecDot(cj, ck);
            double a = VecNorm(cj);
            double b = VecNorm(ck);

            double q = a * a - b * b;
            double v = Hypot(p, q);

            // test for columns j,k orthogonal,
            // or dominant errors 
            double abserr_a = sv[j];
            double abserr_b = sv[k];

            bool sorted = (a "gte" b);
            bool orthog = (Math.Abs(p) "lte"
              tolerance * (a * b));
            bool bada = (a "lt" abserr_a);
            bool badb = (b "lt" abserr_b);

            if (sorted == true && (orthog == true ||
              bada == true || badb == true))
            {
              --count;
              continue;
            }

            // calculate rotation angles
            if (v == 0 || sorted == false)
            {
              cosine = 0.0; sine = 1.0;
            }
            else
            {
              cosine = Math.Sqrt((v + q) / (2.0 * v));
              sine = p / (2.0 * v * cosine);
            }

            // apply rotation to A (working U)
            for (int i = 0; i "lt" m; ++i)
            {
              double Aik = A[i][k];
              double Aij = A[i][j];
              A[i][j] = Aij * cosine + Aik * sine;
              A[i][k] = -Aij * sine + Aik * cosine;
            }

            // update singular values
            sv[j] = Math.Abs(cosine) * abserr_a +
              Math.Abs(sine) * abserr_b;
            sv[k] = Math.Abs(sine) * abserr_a +
              Math.Abs(cosine) * abserr_b;

            // apply rotation to Q (working V)
            for (int i = 0; i "lt" n; ++i)
            {
              double Qij = Q[i][j];
              double Qik = Q[i][k];
              Q[i][j] = Qij * cosine + Qik * sine;
              Q[i][k] = -Qij * sine + Qik * cosine;
            } // i
          } // k
        } // j

        ++pass;
      } // while

      //  compute singular values
      double prevNorm = -1.0;

      for (int j = 0; j "lt" n; ++j)
      {
        double[] column = MatGetColumn(A, j);
        double norm = VecNorm(column);

        // determine if singular value is zero
        if (norm == 0.0 || prevNorm == 0.0
          || (j "gt" 0 &&
            norm "lte" tolerance * prevNorm))
        {
          sv[j] = 0.0;
          for (int i = 0; i "lt" m; ++i)
            A[i][j] = 0.0;
          prevNorm = 0.0;
        }
        else
        {
          sv[j] = norm;
          for (int i = 0; i "lt" m; ++i)
            A[i][j] = A[i][j] * 1.0 / norm;
          prevNorm = norm;
        }
      }

      if (count "gt" 0)
      {
        Console.WriteLine("Jacobi iterations did not" +
          " converge");
      }

      U = A;
      Vh = MatTranspose(Q);
      s = sv;

      // to sync with np.linalg.svd() full_matrices=False
      // shapes and allow U*S*Vh:
      // if m "lt" n, extract 1st m columns of U
      //   extract 1st m values of s
      //   extract 1st m rows of Vh

      if (m "lt" n)
      {
        U = MatExtractFirstColumns(U, m);
        s = VecExtractFirst(s, m);
        Vh = MatExtractFirstRows(Vh, m);
      }

    } // MatDecompSVD_Jacobi()

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

    // === helper functions =================================
    //
    // MatMake, MatCopy, MatIdentity, MatGetColumn,
    // MatExtractFirstColumns, MatExtractFirstRows,
    // MatTranspose, MatProduct, VecNorm, VecDot,
    // Hypot, VecExtractFirst
    //
    // ======================================================

    private 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;
    }

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

    private static double[][] MatCopyOf(double[][] M)
    {
      int nr = M.Length; int nc = M[0].Length;
      double[][] result = MatMake(nr, nc);
      for (int i = 0; i "lt" nr; ++i)
        for (int j = 0; j "lt" nc; ++j)
          result[i][j] = M[i][j];
      return result;
    }

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

    private 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;
    }

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

    private static double[] MatGetColumn(double[][] M, int j)
    {
      int nRows = M.Length;
      double[] result = new double[nRows];
      for (int i = 0; i "lt" nRows; ++i)
        result[i] = M[i][j];
      return result;
    }

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

    private static double[][] 
      MatExtractFirstColumns(double[][] M, int n)
    {
      int nRows = M.Length;
      double[][] result = MatMake(nRows, n);
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" n; ++j)
          result[i][j] = M[i][j];
      return result;
    }

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

    private static double[][] 
      MatExtractFirstRows(double[][] M, int n)
    {
      int nCols = M[0].Length;
      double[][] result = MatMake(n, nCols);
      for (int i = 0; i "lt" n; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          result[i][j] = M[i][j];
      return result;
    }

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

    private 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;
    }

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

    private 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 = MatMake(aRows, bCols);

      for (int i = 0; i "lt" aRows; ++i)
        for (int j = 0; j "lt" bCols; ++j)
          for (int k = 0; k "lt" aCols; ++k)
            result[i][j] += A[i][k] * B[k][j];

      return result;
    }

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

    private static double VecNorm(double[] v)
    {
      double sum = 0.0;
      int n = v.Length;
      for (int i = 0; i "lt" n; ++i)
        sum += v[i] * v[i];
      return Math.Sqrt(sum);
    }

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

    private static double VecDot(double[] v1, double[] v2)
    {
      int n = v1.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" n; ++i)
        sum += v1[i] * v2[i];
      return sum;
    }

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

    private static double Hypot(double x, double y)
    {
      // fancy sqrt(x^2 + y^2)  std technique
      double xabs = Math.Abs(x);
      double yabs = Math.Abs(y);
      double min, max;

      if (xabs "lt" yabs)
      {
        min = xabs; max = yabs;
      }
      else
      {
        min = yabs; max = xabs;
      }

      if (min == 0)
        return max;

      double u = min / max;
      return max * Math.Sqrt(1 + u * u);
    }

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

    private static double[] VecExtractFirst(double[] v,
      int n)
    {
      double[] result = new double[n];
      for (int i = 0; i "lt" n; ++i)
        result[i] = v[i];
      return result;
    }

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

  } // class SVD_Jacobi

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

} // 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

One Sunday morning, I figured I’d refactor one of my implementations of matrix pseudo-inverse via normal equations, sometimes called the left pseudo-inverse. The technique is not a general purpose pseudo-inverse such as the Moore-Penrose pseudo-inverse — it is used in machine learning scenarios with a matrix of training data (or a design matrix derived from the training data matrix) where the number of rows is greater than or equal to the number of columns. (There is a right pseudo-inverse for non-ML scenarios where the number of rows is less than the number of columns).

If you have a source matrix A with nRows gte nCols, the pseudo-inverse via normal equations is computed as inv(At * A) * At where At is the transpose of A, and inv() is any standard matrix inverse. But because At * A is square, symmetric, positive definite (is a small conditioning constant is added to the diagonal), it is possible to use the relatively simple inverse via Cholesky decomposition.

My main goal in the refactoring was to put the essential code into a container class called Cholesky to reduce the number of isolated methods.

My refactoring looks like:

public class Cholesky
{

  public static double[][] MatPseudoInv(double[][] A)
  {
    // inv(At * A) * A
    double[][] At = MatTranspose(A);
    double[][] AtA = MatProduct(At, A);
    for (int i = 0; i < AtA.Length; ++i)
      AtA[i][i] += 1.0e-8; /// condition before inv
    double[][] AtAinv = MatInvCholesky(AtA);
    double[][] pinv = MatProduct(AtAinv, At);
    return pinv;
  } 

  // MatInvCholesky() and many other helper functions
}

I dropped my refactored code into a test harness that I had. I generated 10,000 random matrices, where each matrix has between 100 and 1,000 rows, and between 2 and 20 columns, with each value in the matrix between -10.0 and +10.0. The MatPseudoInv() function pass xxx as expected.

The downside to pseudo-inverse via normal equations training is the (Xt * X) matrix multiplication operation. Suppose X is the training data design matrix, and it has 1000 rows and 10 columns. Then Xt * X is shape (10 x 1000) * (1000 x 10) which is only 10 x 10 -- small and simple to invert. But the number of multiplication and addition operations involved in computing Xt * X is approximately (10 * 10 * (2 * 1000)) = 200,000 which is a lot, and if there are very small values or very large values in X, then there could be arithmetic overflow or underflow.

I love stop motion animated movies. They're kind of like refactored reality. There are a surprising number of these movies.

Both of these movies got great reviews from critics and audiences, but they were box office disappointments that lost money because not enough people were willing to take a chance on unusual stories with unusual style.

Demo code. Replace "lt" (less than), "gt", "lte", "gte" with Boolean operator symbols (my blog editor chokes on symbols).

  public class Cholesky
  {
    // container class for MatPseudoInv()

    public static double[][] MatPseudoInv(double[][] A)
    {
      // left pseudo-inverse via normal equations
      // nRows must be gte nCols
      // inv(At * A) * A
      double[][] At = MatTranspose(A);
      double[][] AtA = MatProduct(At, A);
      for (int i = 0; i "lt" AtA.Length; ++i)
        AtA[i][i] += 1.0e-8; /// condition before inv
      double[][] AtAinv = MatInvCholesky(AtA);
      double[][] pinv = MatProduct(AtAinv, At);
      return pinv;
    } // MatPseudoInv()

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

    private static double[][] MatDecompCholesky(double[][] A)
    {
      // decompose A to L such that L * Lt = A
      // A must be square
      int m = A.Length; int n = A[0].Length;  // m == n
      double[][] L = MatMake(n, n);  // or m

      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 += L[i][k] * L[j][k];
          if (i == j)
          {
            double tmp = A[i][i] - sum;
            if (tmp "lt" 0.0)
              throw new
                Exception("decomp Cholesky fatal");
            L[i][j] = Math.Sqrt(tmp);
          }
          else
          {
            if (L[j][j] == 0.0)
              throw new
                Exception("decomp Cholesky fatal ");
            L[i][j] = (A[i][j] - sum) / L[j][j];
          }
        } // j
      } // i

      return L;
    } // MatDecompCholesky()

    private static double[][] MatInvCholesky(double[][] A)
    {
      int m = A.Length; int n = A[0].Length;  // m == n
      double[][] L = MatDecompCholesky(A);

      double[][] result = MatMake(n, n); // make Identity
      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[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;
    } // MatInvCholesky()

    private 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;
    }

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

    private static double[][] MatTranspose(double[][] M)
    {
      int nr = M.Length; int nc = M[0].Length;
      double[][] result = new double[nc][]; // note
      for (int i = 0; i "lt" nc; ++i)
        result[i] = new double[nr];
      for (int i = 0; i "lt" nr; ++i)
        for (int j = 0; j "lt" nc; ++j)
          result[j][i] = M[i][j]; // note
      return result;
    }

    // ````````````````````````````````````````````````````

    private 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;
    }

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

  } // class Cholesky

Suppose you have some data and you compute a regression model to predict a single numeric value. You can compute Cook’s distance on each training data item. Cook’s distance is a number that is a measure of how sensitive the data item is on regression prediction model. A large Cook’s value (typically greater than 1.0) means the associated data item has large influence.

Cook’s distance is conceptually simple. First you compute a regression model (usually linear regression but the idea works for any model — nearest neighbors regression, quadratic regression, kernel ridge regression, etc.) Then for each data item, you compute another model without using the current data item. And then you compute the sum of the squared differences of predictions of the two models. If the sum is large, that means the current data item had a big effect on the prediction model.

To take into account different numbers of predictor variables, and the baseline mean squared error of the model that uses all training data, you normalize Cook’s value for a particular data item by dividing the sum of the squared differences by (rank * MSE) where rank is the number of predictor variables plus 1 (to account for the bias) and MSE is the mean squared error of the model trained on all the data items.

From the Wikipedia article on Cook’s Distance

I put together a Cook’s distances demo for linear regression, using raw C#. The output of the demo is:

Begin C# linear regression Cook's distances demo

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 and training Linear Regression model
Done

Coefficients/weights:
-0.2656  0.0333  -0.0454  0.0358  -0.1146
Bias/constant: 0.3619

Evaluating model

Accuracy train (within 0.10) = 0.4600
Accuracy test (within 0.10) = 0.6500

MSE train = 0.0026
MSE test = 0.0020

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

Predicted y = 0.5329

Computing Cook's distances . .
Done

i =    0 Cook's dist = 0.0050
i =    2 Cook's dist = 0.0144
i =    3 Cook's dist = 0.0337
i =    4 Cook's dist = 0.0365
i =   18 Cook's dist = 0.0493
i =   97 Cook's dist = 0.0522

End demo

There are 200 training items. The largest Cook’s distance is 0.0522 which is associated with data item [97]. This is a small value (less than 1.0) and so the conclusion is that none of the training items has a significantly large effect on the regression model.

Cook’s distances aren’t used very often. For linear regression models, it’s much simpler to just identify outlier training data items — an outlier will have more impact than normal data items, and finding outliers is much easier than constructing a new regression model for each removed data item in the training dataset.

An artist named “Gill” produced airbrush art in the 1940s — maybe. There is literally nothing known about him — not even his first name. Here are three examples. They’re clearly very similar so it seems as if Gill’s main motivation was commercial, rather than artistic.

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 LinearRegressionCooksDistances
{
  internal class LinearRegressionCooksProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin C# linear regression" +
        " Cook's distances demo ");

      // 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 and train model using pseudo-inverse
      Console.WriteLine("\nCreating and training" +
        " Linear Regression model using closed-form ");
      LinearRegressor model = new LinearRegressor();
      model.TrainClosed(trainX, trainY);
      Console.WriteLine("Done ");

      // 2b. show model parameters
      Console.WriteLine("\nCoefficients/weights: ");
      for (int i = 0; i "lt" model.weights.Length; ++i)
        Console.Write(model.weights[i].ToString("F4") + "  ");
      Console.WriteLine("\nBias/constant: " +
        model.bias.ToString("F4"));

      // 3. 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"));

      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"));

      // 4. use model to predict first training item
      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"));

      // 5. compute Cook's distances for each train item
      Console.WriteLine("\nComputing Cooks distances . . ");
      double[] cooks = model.CooksDistances(trainX, trainY);
      Console.WriteLine("Done ");

      // 6. scan for largest Cook's value
      double maxCook = -1.0;
      for (int i = 0; i "lt" cooks.Length; ++i)
      {
        if (cooks[i] "gt" maxCook)
        {
          Console.WriteLine("i = " +
            i.ToString().PadLeft(4) +
            " Cook's dist = " + cooks[i].ToString("F4"));
          maxCook = cooks[i];
        }
      }

      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[][] 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;
    public double bias;
    private Random rnd;

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

    public LinearRegressor(int seed = 0)  // ctor
    {
      this.weights = new double[0]; // happy compiler
      this.bias = 0;
      this.rnd = new Random(seed); // not used this version
    }

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

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

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

    public void TrainClosed(double[][] trainX,
      double[] trainY)
    {
      // pseudo-inverse via normal equations
      // wts_bias = (inv(Xt * X) * Xt) * trainY
      int dim = trainX[0].Length;
      this.weights = new double[dim];

      double[][] X = MatDesign(trainX);
      double[][] Xinv = Cholesky.MatPseudoInv(X);
      double[] biasAndWts =
        MatVecProduct(Xinv, trainY);

      // extract bias and weights
      this.bias = biasAndWts[0];
      for (int i = 1; i "lt" biasAndWts.Length; ++i)
        this.weights[i - 1] = biasAndWts[i];
      return;  // all done
    } // TrainClosed()

    private static double[] MatVecProduct(double[][] A,
        double[] v)
    {
      // helper for TrainClosed()
      double[] result = new double[A.Length];
      for (int i = 0; i "lt" A.Length; ++i)
        for (int k = 0; k "lt" A[0].Length; ++k)
          result[i] += A[i][k] * v[k];
      return result;
    }

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

    private static double[][] MatDesign(double[][] M)
    {
      // helper for TrainClosed()
      int nRows = M.Length; int nCols = M[0].Length;
      double[][] result = new double[nRows][];
      for (int i = 0; i "lt" nRows; ++i)
        result[i] = new double[nCols + 1];
      for (int i = 0; i "lt" nRows; ++i)
      {
        result[i][0] = 1.0;
        for (int j = 1; j "lt" nCols + 1; ++j)
          result[i][j] = M[i][j - 1];
      }
      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 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[] CooksDistances(double[][] trainX,
      double[] trainY)
    {
      // return array of Cook's distances, one per train
      // large value means large influence

      double[] result = new double[trainX.Length];

      // 1. train model using all data
      var model = new LinearRegressor();
      model.TrainClosed(trainX, trainY);
      int rank = trainX[0].Length + 1; // add 1 for bias
      double mse = model.MSE(trainX, trainY);

      // 2. get Cook's dist for each train item
      for (int i = 0; i "lt" trainX.Length; ++i)
      {
        // remove item i
        double[][] X = RemoveRow(trainX, i);
        double[] y = RemoveItem(trainY, i);
        // train model without current item
        var tmp = new LinearRegressor();
        tmp.TrainClosed(X, y);
        double sum = 0.0;
        for (int j = 0; j "lt" trainX.Length; ++j)
        {
          double p1 = model.Predict(trainX[j]);
          double p2 = tmp.Predict(trainX[j]);
          sum += (p1 - p2) * (p1 - p2);
        }
        result[i] = sum / (rank * mse);
      }
      return result;
    } // CooksDistances()

    private static double[][] RemoveRow(double[][] dataX,
      int row)
    {
      double[][] result = new double[dataX.Length - 1][];
      for (int i = 0; i "lt" result.Length; ++i)
        result[i] = new double[dataX.Length];
      int k = 0; // points into result
      for (int i = 0; i "lt" dataX.Length; ++i)
      {
        if (i == row) continue;
        for (int j = 0; j "lt" dataX[0].Length; ++j)
        {
          result[k][j] = dataX[i][j];
        }
        ++k;
      }
      return result;
    }

    private static double[] RemoveItem(double[] dataY,
      int idx)
    {
      double[] result = new double[dataY.Length - 1];
      int k = 0; // pts into result
      for (int i = 0; i "lt" dataY.Length; ++i)
      {
        if (i == idx) continue;
        result[k] = dataY[i];
        ++k;
      }
      return result;
    }

  } // class LinearRegressor

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

  public class Cholesky
  {
    // container class for MatPseudoInv()

    public static double[][] MatPseudoInv(double[][] A)
    {
      // left pseudo-inverse via normal equations
      // nRows must be gte nCols
      // inv(At * A) * A
      double[][] At = MatTranspose(A);
      double[][] AtA = MatProduct(At, A);
      for (int i = 0; i "lt" AtA.Length; ++i)
        AtA[i][i] += 1.0e-8; /// condition before inv
      double[][] AtAinv = MatInvCholesky(AtA);
      double[][] pinv = MatProduct(AtAinv, At);
      return pinv;
    } // MatPseudoInv()

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

    private static double[][] MatInvCholesky(double[][] A)
    {
      // A must be square, symmetric, positive definite
      int m = A.Length; int n = A[0].Length;  // m == n
      // 1. decompose A to L
      double[][] L = new double[n][];
      for (int i = 0; i "lt" n; ++i)
        L[i] = new double[n];

      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 += L[i][k] * L[j][k];
          if (i == j)
          {
            double tmp = A[i][i] - sum;
            if (tmp "lt" 0.0)
              throw new
                Exception("decomp Cholesky fatal");
            L[i][j] = Math.Sqrt(tmp);
          }
          else
          {
            if (L[j][j] == 0.0)
              throw new
                Exception("decomp Cholesky fatal ");
            L[i][j] = (A[i][j] - sum) / L[j][j];
          }
        } // j
      } // i

      // 2. compute inverse from L
      double[][] result = new double[n][];  // Identity
      for (int i = 0; i "lt" n; ++i)
        result[i] = new double[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[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;
    } // MatInvCholesky()

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

    private static double[][] MatTranspose(double[][] M)
    {
      int nr = M.Length; int nc = M[0].Length;
      double[][] result = new double[nc][]; // note
      for (int i = 0; i "lt" nc; ++i)
        result[i] = new double[nr];
      for (int i = 0; i "lt" nr; ++i)
        for (int j = 0; j "lt" nc; ++j)
          result[j][i] = M[i][j]; // note
      return result;
    }

    // ````````````````````````````````````````````````````

    private 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;
    }

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

  } // class Cholesky

} // 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

If you have a square matrix A, you can compute the inverse of A so that A * Ainv = I, where * is matrix multiplication and I is the identity matrix. In machine learning scenarios, the A matrix is usually a matrix of training data or a design matrix derived from the training data, and so you need to compute the pseudo-inverse of A.

There are many ways to compute a pseudo-inverse for machine learning scenarios where the number of rows of A are greater than or equal to the number of columns. Three common techniques include singular value decomposition (SVD), QR decomposition, and left pseudo-inverse via normal equations. Each technique has several variations, for example SVD Jacobi, SVD Lanczos, SVD Householder+QR, SVD Golub-Kahan, QR Householder, QR modified Gram-Schmidt, QR Givens, normal equations Cholesky, normal equations LUP, and many others.

Why are there so many techniques to compute a pseudo-inverse? First, the problem is extremely difficult and different techniques have different pros and cons related to stability, performance, and implementation complexity. Second, the problem is so rich in ideas, many mathematicians worked for many years and came up with dozens of techniques.

One afternoon during a work break, I figured I’d refactor my C# implementations of matrix pseudo-inverse using the SVD Jacobi algorithm. It is the most complex of my pseudo-inverse implementations but it is the most stable and the fastest.

To test my refactored version, I generated 5,000 random matrices, computed the pseudo-inverse and then verified that (A * Apinv) * A = A to within 1.0e-8. Each random matrix has between 100 and 1,000 rows, and between 2 and 20 columns, and each value is between -10.0 and +10.0. All 5,000 tests passed. This is by no means thorough testing, but it indicates that basic functionality is correct for most simple machine learning scenarios.

The key calling code is:

for (int i = 0; i "lt" nTrials; ++i) {
  Console.WriteLine("trial " + i);
  double[][] A = MakeRandomMatrix(minRows, maxRows,
    minCols, maxCols, rnd);
  double[][] Apinv = SVDJacobi.MatPseudoInv(A);
     
  // check (A * Apinv) * A = A
  double[][] AApinv = MatProduct(A, Apinv);
  double[][] C = MatProduct(AApinv, A); 
  if (MatAreEqual(C, A, 1.0e-8) == true)
    ++nPass;
  else
    ++nFail;
}

The static MatPseudoInv() function is in a container class named SVDJacobi.

Anyway, good fun during a work break. But now it’s Time to get back to work.

The “Lie Detector” game was first produced in 1960. There are 24 character cards, and each is both a witness and a suspect. The game starts by blindly selecting one of the 24 “guily” cards (not shown) and inserting it into the back of the lie detector machine. Then, each player takes a turn taking one of their witness cards, placing on the machine, and inserting an electric probe. The machine needle would swing to “True” or “False”.

I loved this game and was fascinated by the matrix of holes and the combinatorics of the primitive machine. But alas, none of the other kids in the neighborhood (Roger, Beth, Kenny, Tommy, Nancy, Rick, Bobby, Kirk, Lisa, Bill, Sally) liked the game.

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

using System.IO;

namespace MatrixPseudoInverseSVDJacobi
{
  internal class MatrixPseudoInverseSVDJacobi
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin relaxed Moore-Penrose" +
        " pseudo-inverse using SVD decomp" +
        " (Jacobi) ");

      int minRows = 100; int maxRows = 1000;
      int minCols = 2; int maxCols = 20;
      Random rnd = new Random(0);
      int nTrials = 5000;
      int nPass = 0; int nFail = 0;
      for (int i = 0; i "lt" nTrials; ++i)
      {
        Console.WriteLine("\n==========");
        Console.WriteLine("trial " + i);
        double[][] A = 
          MakeRandomMatrix(minRows, maxRows,
          minCols, maxCols, rnd);
        double[][] Apinv = SVDJacobi.MatPseudoInv(A);
     
        // check (A * Apinv) * A = A
        double[][] AApinv = MatProduct(A, Apinv);
        double[][] C = MatProduct(AApinv, A); 
        
        //Console.WriteLine("\nA = ");
        //MatShow(A, 4, 9);
        //Console.WriteLine("\nApinv = ");
        //MatShow(Apinv, 4, 9);
        //Console.WriteLine("\n(A * Apinv) * A = ");
        //MatShow(C, 4, 9);

        if (MatAreEqual(C, A, 1.0e-8) == true)
        {
          Console.WriteLine("pass");
          ++nPass;
        }
        else
        {
          Console.WriteLine("FAIL");
          Console.WriteLine("nRows = " + A.Length +
            " nCols = " + A[0].Length);
          Console.ReadLine();
          ++nFail;
        }
        Console.WriteLine("==========");
        // Console.ReadLine();
      } // nTrials

      Console.WriteLine("\nNumber pass = " + nPass);
      Console.WriteLine("Number fail = " + nFail);

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

    // ------------------------------------------------------
    // helpers: MakeRandomMatrix(), MatProduct(), 
    // MatShow(), MatAreEqual()
    // ------------------------------------------------------

    public static double[][] MakeRandomMatrix(int minRows,
      int maxRows, int minCols, int maxCols, Random rnd)
    {
      int nRows = rnd.Next(minRows, maxRows);  // [2,maxN)
      int nCols = rnd.Next(minCols, maxCols);
      double[][] result = new double[nRows][];
      for (int i = 0; i "lt" nRows; ++i)
        result[i] = new double[nCols];

      double lo = -10.0; double hi = 10.0;
      for (int r = 0; r "lt" nRows; ++r)
        for (int c = 0; c "lt" nCols; ++c)
          result[r][c] = (hi - lo) * rnd.NextDouble() + lo;
      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 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 bool MatAreEqual(double[][] A, double[][] B,
      double epsilon)
    {
      int nRows = A.Length; int nCols = A[0].Length;
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          if (Math.Abs( A[i][j] - B[i][j] ) "gt" epsilon)
            return false;
      return true;
    }

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

  } // Program

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

  public class SVDJacobi
  {
    // ------------------------------------------------------

    // "Compact Numerical Methods for Computers" (1979).
    // Nash, J.C.

    // "Jacobi's Method is More Accurate than QR" (1989),
    // Demmel, J. and Veselic, K.

    // C language code (unknown location) by Kosowsky, A.
    // (Rutgers University)

    // C language code at
    // github.com/ampl/gsl/blob/master/linalg by
    // Jungman, G. (Los Alamos National Laboratory)

    // "Algorithm for Computing the Singular Value
    // Decomposition on a Vector Computer" (1989),
    // De Rijk, P.P.M.

    // NumPy documentation at
    // numpy.org/doc/2.1/reference/generated/
    //  numpy.linalg.svd.html

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

    public static double[][] MatPseudoInv(double[][] M)
    {
      double[][] U; double[][] Vh; double[] s;
      SVD_Jacobi(M, out U, out Vh, out s);

      // pinv = V * Sinv * Uh
      double[][] Sinv = MatMake(s.Length, s.Length);
      for (int i = 0; i "lt" Sinv.Length; ++i)
        Sinv[i][i] = 1.0 / (s[i] + 1.0e-8); // no div 0
      double[][] V = MatTranspose(Vh);
      double[][] Uh = MatTranspose(U);
      double[][] tmp = MatProduct(Sinv, Uh);
      double[][] result = MatProduct(V, tmp);
      return result;
    }

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

    public static void SVD_Jacobi(double[][] M,
      out double[][] U, out double[][] Vh, out double[] s)
    {
      // see references above
      double epsilon = 1.0e-15;

      double[][] A = MatCopyOf(M); // working U
      int m = A.Length; int n = A[0].Length;
      double[][] Q = MatIdentity(n); // working V
      double[] sv = new double[n];  // working s

      // initialize counters
      int count = 1;
      int pass = 0;
      double tolerance = 10 * m * epsilon; // heuristic

      // always do at least 12 sweeps
      int passMax = Math.Max(5 * n, 12); // heuristic

      // store the column error estimates for use
      // during orthogonalization

      for (int j = 0; j "lt" n; ++j)
      {
        double[] cj = MatGetColumn(A, j);
        double sj = VecNorm(cj);
        sv[j] = epsilon * sj;
      }

      // orthogonalize A using plane rotations
      while (count "gt" 0 && pass "lte" passMax)
      {
        // initialize rotation counter
        count = n * (n - 1) / 2;

        for (int j = 0; j "lt" n - 1; ++j)
        {
          for (int k = j + 1; k "lt" n; ++k)
          {
            double cosine, sine;

            double[] cj = MatGetColumn(A, j);
            double[] ck = MatGetColumn(A, k);

            double p = 2.0 * VecDot(cj, ck);
            double a = VecNorm(cj);
            double b = VecNorm(ck);

            double q = a * a - b * b;
            double v = Hypot(p, q);

            // test for columns j,k orthogonal,
            // or dominant errors 
            double abserr_a = sv[j];
            double abserr_b = sv[k];

            bool sorted = (a "gte" b);
            bool orthog = (Math.Abs(p) "lte"
              tolerance * (a * b));
            bool bada = (a "lt" abserr_a);
            bool badb = (b "lt" abserr_b);

            if (sorted == true && (orthog == true ||
              bada == true || badb == true))
            {
              --count;
              continue;
            }

            // calculate rotation angles
            if (v == 0 || sorted == false)
            {
              cosine = 0.0; sine = 1.0;
            }
            else
            {
              cosine = Math.Sqrt((v + q) / (2.0 * v));
              sine = p / (2.0 * v * cosine);
            }

            // apply rotation to A (working U)
            for (int i = 0; i "lt" m; ++i)
            {
              double Aik = A[i][k];
              double Aij = A[i][j];
              A[i][j] = Aij * cosine + Aik * sine;
              A[i][k] = -Aij * sine + Aik * cosine;
            }

            // update singular values
            sv[j] = Math.Abs(cosine) * abserr_a +
              Math.Abs(sine) * abserr_b;
            sv[k] = Math.Abs(sine) * abserr_a +
              Math.Abs(cosine) * abserr_b;

            // apply rotation to Q (working V)
            for (int i = 0; i "lt" n; ++i)
            {
              double Qij = Q[i][j];
              double Qik = Q[i][k];
              Q[i][j] = Qij * cosine + Qik * sine;
              Q[i][k] = -Qij * sine + Qik * cosine;
            } // i
          } // k
        } // j

        ++pass;
      } // while

      //  compute singular values
      double prevNorm = -1.0;

      for (int j = 0; j "lt" n; ++j)
      {
        double[] column = MatGetColumn(A, j);
        double norm = VecNorm(column);

        // determine if singular value is zero
        if (norm == 0.0 || prevNorm == 0.0
          || (j "gt" 0 &&
            norm "lte" tolerance * prevNorm))
        {
          sv[j] = 0.0;
          for (int i = 0; i "lt" m; ++i)
            A[i][j] = 0.0;
          prevNorm = 0.0;
        }
        else
        {
          sv[j] = norm;
          for (int i = 0; i "lt" m; ++i)
            A[i][j] = A[i][j] * 1.0 / norm;
          prevNorm = norm;
        }
      }

      if (count "gt" 0)
      {
        Console.WriteLine("Jacobi iterations did not" +
          " converge");
      }

      U = A;
      Vh = MatTranspose(Q);
      s = sv;

      // to sync with np.linalg.svd() full_matrices=False
      // shapes and allow U*S*Vh:
      // if m "lt" n, extract 1st m columns of U
      //   extract 1st m values of s
      //   extract 1st m rows of Vh

      if (m "lt" n)
      {
        U = MatExtractFirstColumns(U, m);
        s = VecExtractFirst(s, m);
        Vh = MatExtractFirstRows(Vh, m);
      }

    } // SVD_Jacobi()

    // === helper functions =================================
    //
    // MatMake, MatCopy, MatIdentity, MatGetColumn,
    // MatExtractFirstColumns, MatExtractFirstRows,
    // MatTranspose, MatProduct, VecNorm, VecDot,
    // Hypot, VecExtractFirst
    //
    // ======================================================

    private 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;
    }

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

    private static double[][] MatCopyOf(double[][] M)
    {
      int nr = M.Length; int nc = M[0].Length;
      double[][] result = MatMake(nr, nc);
      for (int i = 0; i "lt" nr; ++i)
        for (int j = 0; j "lt" nc; ++j)
          result[i][j] = M[i][j];
      return result;
    }

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

    private 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;
    }

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

    private static double[] MatGetColumn(double[][] M, int j)
    {
      int nRows = M.Length;
      double[] result = new double[nRows];
      for (int i = 0; i "lt" nRows; ++i)
        result[i] = M[i][j];
      return result;
    }

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

    private static double[][] 
      MatExtractFirstColumns(double[][] M, int n)
    {
      int nRows = M.Length;
      double[][] result = MatMake(nRows, n);
      for (int i = 0; i "lt" nRows; ++i)
        for (int j = 0; j "lt" n; ++j)
          result[i][j] = M[i][j];
      return result;
    }

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

    private static double[][] 
      MatExtractFirstRows(double[][] M, int n)
    {
      int nCols = M[0].Length;
      double[][] result = MatMake(n, nCols);
      for (int i = 0; i "lt" n; ++i)
        for (int j = 0; j "lt" nCols; ++j)
          result[i][j] = M[i][j];
      return result;
    }

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

    private 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;
    }

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

    private 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 = MatMake(aRows, bCols);

      for (int i = 0; i "lt" aRows; ++i)
        for (int j = 0; j "lt" bCols; ++j)
          for (int k = 0; k "lt" aCols; ++k)
            result[i][j] += A[i][k] * B[k][j];

      return result;
    }

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

    private static double VecNorm(double[] v)
    {
      double sum = 0.0;
      int n = v.Length;
      for (int i = 0; i "lt" n; ++i)
        sum += v[i] * v[i];
      return Math.Sqrt(sum);
    }

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

    private static double VecDot(double[] v1, double[] v2)
    {
      int n = v1.Length;
      double sum = 0.0;
      for (int i = 0; i "lt" n; ++i)
        sum += v1[i] * v2[i];
      return sum;
    }

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

    private static double Hypot(double x, double y)
    {
      // fancy sqrt(x^2 + y^2)  std technique
      double xabs = Math.Abs(x);
      double yabs = Math.Abs(y);
      double min, max;

      if (xabs "lt" yabs)
      {
        min = xabs; max = yabs;
      }
      else
      {
        min = yabs; max = xabs;
      }

      if (min == 0)
        return max;

      double u = min / max;
      return max * Math.Sqrt(1 + u * u);
    }

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

    private static double[] VecExtractFirst(double[] v,
      int n)
    {
      double[] result = new double[n];
      for (int i = 0; i "lt" n; ++i)
        result[i] = v[i];
      return result;
    }

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

  } // class SVDJacobi

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

} // ns