code: ///////////////////////////////////////////////////////////////////////////////
  // For information as to what this class does, see the Javadoc, below.       //
  // Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006,       //
  // 2007, 2008, 2009, 2010, 2014, 2015 by Peter Spirtes, Richard Scheines, Joseph   //
  // Ramsey, and Clark Glymour.                                                //
  //                                                                           //
  // This program is free software; you can redistribute it and/or modify      //
  // it under the terms of the GNU General Public License as published by      //
  // the Free Software Foundation; either version 2 of the License, or         //
  // (at your option) any later version.                                       //
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  // This program is distributed in the hope that it will be useful,           //
  // but WITHOUT ANY WARRANTY; without even the implied warranty of            //
  // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the             //
  // GNU General Public License for more details.                              //
  //                                                                           //
  // You should have received a copy of the GNU General Public License         //
  // along with this program; if not, write to the Free Software               //
  // Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA //
  ///////////////////////////////////////////////////////////////////////////////
  
  package edu.cmu.tetrad.search;
  
  import edu.cmu.tetrad.data.*;
  import edu.cmu.tetrad.graph.Node;
  import edu.cmu.tetrad.util.TetradMatrix;
  import org.apache.commons.math3.distribution.ChiSquaredDistribution;
  import org.apache.commons.math3.linear.SingularMatrixException;
  
  import java.util.*;
  
  /**
   * Implements a test for simultaneously zero sextads in the style of Bollen, K. (1990).
   * Sociological Methods and Research 19, 80-92 and Bollen and Ting, Confirmatory Tetrad
   * Analysis.
   *
   * @author Joseph Ramsey
   */
  public class DeltaSextadTest {
      static final long serialVersionUID = 23L;
  
      private double[][] data;
      private int N;
      private ICovarianceMatrix cov;
      private List<Node> variables;
  
      // As input we require a data set and a list of non-redundant Tetrads.
  
      // Need a method to remove Tetrads from the input list until what's left is
      // non-redundant. Or at least to check for non-redundancy. Maybe simply
      // checking to see whether a matrix exception is thrown is sufficient.
      // Peter suggests looking at Modern Factor Analysis by Harmon, at triplets.
  
      /**
       * Constructs a test using a given data set. If a data set is provided (that is, a tabular data set), fourth moment
       * statistics can be calculated (p. 160); otherwise, it must be assumed that the data are multivariate Gaussian.
       */
      public DeltaSextadTest(DataSet dataSet) {
          if (dataSet == null) {
              throw new NullPointerException();
          }
  
          if (!dataSet.isContinuous()) {
              throw new IllegalArgumentException();
          }
  
          this.cov = new CovarianceMatrixOnTheFly(dataSet);
  
          TetradMatrix centered = DataUtils.centerData(dataSet.getDoubleData());
          this.data = centered.transpose().toArray();
          this.N = dataSet.getNumRows();
          this.variables = dataSet.getVariables();
      }
  
      /**
       * Constructs a test using the given covariance matrix. Fourth moment statistics are not caculated; it is assumed
       * that the data are distributed as multivariate Gaussian.
       */
      public DeltaSextadTest(ICovarianceMatrix cov) {
          if (cov == null) {
              throw new NullPointerException();
          }
  
          this.cov = cov;
          this.N = cov.getSampleSize();
          this.variables = cov.getVariables();
      }
  
      /**
       * Generates a simple exemplar of this class to test serialization.
       */
      public static DeltaSextadTest serializableInstance() {
          return new DeltaSextadTest(ColtDataSet.serializableInstance());
      }
  
      /**
       * Takes a list of tetrads for the given data set and returns the chi square value for the test. We assume that the
       * tetrads are non-redundant; if not, a matrix exception will be thrown.
       * <p>
       * Calculates the T statistic (Bollen and Ting, p. 161). This is significant if tests as significant using the Chi
       * Square distribution with degrees of freedom equal to the number of nonredundant tetrads tested.
       */
      public double getPValue(IntSextad... sextads) {
  //        int df = sextads.length;
          int df = dofHarman(sextads.length);
          double chisq = calcChiSquare(sextads);
  //        double cdf = ProbUtils.chisqCdf(chisq, df);
          double cdf = new ChiSquaredDistribution(df).cumulativeProbability(chisq);
          return 1.0 - cdf;
      }
  
      /**
       * Takes a list of tetrads for the given data set and returns the chi square value for the test. We assume that the
       * tetrads are non-redundant; if not, a matrix exception will be thrown.
       * <p>
       * Calculates the T statistic (Bollen and Ting, p. 161). This is significant if tests as significant using the Chi
       * Square distribution with degrees of freedom equal to the number of nonredundant tetrads tested.
       */
      public double calcChiSquare(IntSextad[] sextads) {
          Set<Sigma> boldSigmaSet = new HashSet<>();
  
          for (IntSextad sextad : sextads) {
              List<Integer> _nodes = sextad.getNodes();
  
  //            for (int i = 0; i < _nodes.size(); i++) {
  //                for (int j = i; j < _nodes.size(); j++) {
  //                    boldSigmaSet.add(new Sigma(_nodes.get(i), _nodes.get(j)));
  //                }
  //            }
  //
              for (int k1 = 0; k1 < 3; k1++) {
                  for (int k2 = 0; k2 < 3; k2++) {
                      boldSigmaSet.add(new Sigma(_nodes.get(k1), _nodes.get(3 + k2)));
                  }
              }
          }
  
          List<Sigma> boldSigma = new ArrayList<>(boldSigmaSet);
  
          // Need a matrix of variances and covariances of sample covariances.
          TetradMatrix sigma_ss = new TetradMatrix(boldSigma.size(), boldSigma.size());
  
          for (int i = 0; i < boldSigma.size(); i++) {
              for (int j = i; j < boldSigma.size(); j++) {
                  Sigma sigmaef = boldSigma.get(i);
                  Sigma sigmagh = boldSigma.get(j);
  
                  int e = sigmaef.getA();
                  int f = sigmaef.getB();
                  int g = sigmagh.getA();
                  int h = sigmagh.getB();
  
                  if (cov != null && cov instanceof CorrelationMatrix) {
  
  //                Assumes multinormality. Using formula 23. (Not implementing formula 22 because that case
  //                does not come up.)
                      double rr = 0.5 * (r(e, f) * r(g, h))
                              * (r(e, g) * r(e, g) + r(e, h) * r(e, h) + r(f, g) * r(f, g) + r(f, h) * r(f, h))
                              + r(e, g) * r(f, h) + r(e, h) * r(f, g)
                              - r(e, f) * (r(f, g) * r(f, h) + r(e, g) * r(e, h))
                              - r(g, h) * (r(f, g) * r(e, g) + r(f, h) * r(e, h));
  
                      // General.
                      double rr2 = r(e, f, g, h) + 0.25 * r(e, f) * r(g, h) *
                              (r(e, e, g, g) * r(f, f, g, g) + r(e                        , e, h, h) + r(f, f, h, h))
                              - 0.5 * r(e, f) * (r(e, e, g, h) + r(f, f, g, h))
                              - 0.5 * r(g, h) * (r(e, f, g, g) + r(e, f, h, h));
  
                      sigma_ss.set(i, j, rr);
                      sigma_ss.set(j, i, rr);
                  } else if (cov != null && data == null) {
  
                      // Assumes multinormality--see p. 160.
  //                    double _ss = r(e, g) * r(f, h) + r(e, h) * r(f, g); // + or -? Different advise. + in the code.
                      double _ss = r(e, g) * r(f, h) + r(e, h) * r(f, g);
                      sigma_ss.set(i, j, _ss);
                      sigma_ss.set(j, i, _ss);
                  } else {
                      double _ss = r(e, f, g, h) - r(e, f) * r(g, h);
                      sigma_ss.set(i, j, _ss);
                      sigma_ss.set(j, i, _ss);
                  }
              }
          }
  
          // Need a matrix of of population estimates of partial derivatives of tetrads
          // with respect to covariances in boldSigma.
          TetradMatrix del = new TetradMatrix(boldSigma.size(), sextads.length);
  
          for (int j = 0; j < sextads.length; j++) {
              IntSextad sextad = sextads[j];
  
              for (int i = 0; i < boldSigma.size(); i++) {
                  Sigma sigma = boldSigma.get(i);
                  double derivative = getDerivative(sextad, sigma);
                  del.set(i, j, derivative);
              }
          }
  
          // Need a vector of population estimates of the sextads.
          TetradMatrix t = new TetradMatrix(sextads.length, 1);
  
          for (int i = 0; i < sextads.length; i++) {
              IntSextad sextad = sextads[i];
              List<Integer> nodes = sextad.getNodes();
              TetradMatrix m = new TetradMatrix(3, 3);
  
              for (int k1 = 0; k1 < 3; k1++) {
                  for (int k2 = 0; k2 < 3; k2++) {
                      m.set(k1, k2, r(nodes.get(k1), nodes.get(3 + k2)));
                  }
              }
  
              double det = m.det();
              t.set(i, 0, det);
          }
  
          TetradMatrix sigma_tt = del.transpose().times(sigma_ss).times(del);
          double chisq;
          try {
              chisq = N * t.transpose().times(sigma_tt.inverse()).times(t).get(0, 0);
          } catch (SingularMatrixException e) {
              throw new RuntimeException("Singularity problem.", e);
          }
          return chisq;
      }
  
       /**
       * If using a covariance matrix or a correlation matrix, just returns the lookups. Otherwise calculates the
       * covariance.
       */
      private double r(int i, int j) {
          if (cov != null) {
              return cov.getValue(i, j);
          } else {
              double[] arr1 = data[i];
              double[] arr2 = data[j];
              return r(arr1, arr2, arr1.length);
          }
      }
  
      private double getDerivative(IntSextad sextad, Sigma sigma) {
          int a = sigma.getA();
          int b = sigma.getB();
  
          int n1 = sextad.getI();
          int n2 = sextad.getJ();
          int n3 = sextad.getK();
          int n4 = sextad.getL();
          int n5 = sextad.getM();
          int n6 = sextad.getN();
  
          double x1 = derivative(a, b, n1, n2, n3, n4, n5, n6);
  //        double x2 = derivative(a, b, n4, n5, n6, n1, n2, n3);
          double x2 = derivative(b, a, n1, n2, n3, n4, n5, n6);
  
          if (x1 == 0) return x2;
          if (x2 == 0) return x1;
          throw new IllegalStateException("Both nonzero at the same time: x1 = " + x1 + " x2 = " + x2);
      }
  
      private double derivative(int a, int b, int n1, int n2, int n3, int n4, int n5, int n6) {
          if (a == n1) {
              if (b == n4) {
                  return r(n2, n5) * r(n3, n6) - r(n2, n6) * r(n3, n5);
              } else if (b == n5) {
                  return -r(n2, n4) * r(n3, n6) + r(n3, n4) * r(n2, n6);
              } else if (b == n6) {
                  return r(n2, n4) * r(n3, n5) - r(n3, n4) * r(n2, n5);
              }
  
          } else if (a == n2) {
              if (b == n4) {
                  return r(n3, n5) * r(n1, n6) - r(n1, n5) * r(n3, n6);
              } else if (b == n5) {
                  return r(n1, n4) * r(n3, n6) - r(n3, n4) * r(n1, n6);
              } else if (b == n6) {
                  return -r(n1, n4) * r(n3, n5) + r(n3, n4) * r(n1, n5);
              }
  
          } else if (a == n3) {
              if (b == n4) {
                  return r(n1, n5) * r(n2, n6) - r(n2, n5) * r(n1, n6);
              } else if (b == n5) {
                  return -r(n1, n4) * r(n2, n6) + r(n2, n4) * r(n1, n6);
              } else if (b == n6) {
                  return r(n1, n4) * r(n2, n5) - r(n2, n4) * r(n1, n5);
              }
  
          }
  
          return 0.0;
      }
  
      public List<Node> getVariables() {
          return variables;
      }
  
      // Represents a single covariance symbolically.
      private static class Sigma {
          private int a;
          private int b;
  
          public Sigma(int a, int b) {
              this.a = a;
              this.b = b;
          }
  
          public int getA() {
              return a;
          }
  
          public int getB() {
              return b;
          }
  
          public boolean equals(Object o) {
              if (!(o instanceof Sigma)) {
                  throw new IllegalArgumentException();
              }
  
              Sigma _o = (Sigma) o;
              return (_o.getA() == (getA()) && _o.getB() == (getB())) || (_o.getB() == (getA()) && _o.getA() == (getB()));
          }
  
          public int hashCode() {
              return a + b;
          }
  
          public String toString() {
              return "Sigma(" + getA() + ", " + getB() + ")";
          }
      }
  
      // Assumes data are mean-centered.
      private double r(int x, int y, int z, int w) {
          double sxyzw = 0.0;
  
          double[] _x = data[x];
          double[] _y = data[y];
          double[] _z = data[z];
          double[] _w = data[w];
  
          int N = _x.length;
  
          for (int j = 0; j < N; j++) {
              sxyzw += _x[j] * _y[j] * _z[j] * _w[j];
          }
  
          return (1.0 / N) * sxyzw;
      }
  
      // Assumes data are mean-centered.
      private double r(double array1[], double array2[], int N) {
          int i;
          double sum = 0.0;
  
          for (i = 0; i < N; i++) {
              sum += array1[i] * array2[i];
          }
  
          return (1.0 / N) * sum;
      }
  
      private int dofDrton(int n) {
          int dof = ((n - 2) * (n - 3)) / 2 - 2;
          if (dof < 1) dof = 1;
          return dof;
      }
  
      private int dofHarman(int n) {
          int dof = n * (n - 5) / 2 + 1;
          if (dof < 1) dof = 1;
          return dof;
      }
  
  }
  
  
  
  