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package de.lmu.ifi.dbs.elki.math.statistics.dependence;
/*
This file is part of ELKI:
Environment for Developing KDD-Applications Supported by Index-Structures
Copyright (C) 2015
Ludwig-Maximilians-Universität München
Lehr- und Forschungseinheit für Datenbanksysteme
ELKI Development Team
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU Affero General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
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 Affero General Public License for more details.
You should have received a copy of the GNU Affero General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
import java.util.List;
import de.lmu.ifi.dbs.elki.utilities.datastructures.arraylike.NumberArrayAdapter;
import de.lmu.ifi.dbs.elki.utilities.documentation.Reference;
import de.lmu.ifi.dbs.elki.utilities.optionhandling.AbstractParameterizer;
/**
* Distance correlation.
*
* The value returned is the square root of the dCor² value. This matches the R
* implementation by the original authors.
*
* Reference:
* <p>
* Székely, G. J., Rizzo, M. L., & Bakirov, N. K.<br />
* Measuring and testing dependence by correlation of distances<br />
* The Annals of Statistics, 35(6), 2769-2794
* </p>
*
* Implementation notice: we exploit symmetry, and thus use diagonal matrixes.
* While initially the diagonal is zero, after double-centering the matrix these
* values can become non-zero!
*
* @author Marie Kiermeier
* @author Erich Schubert
*/
@Reference(authors = "Székely, G. J., Rizzo, M. L., & Bakirov, N. K.", //
title = "Measuring and testing dependence by correlation of distances", //
booktitle = "The Annals of Statistics, 35(6), 2769-2794", //
url = "http://dx.doi.org/10.1214/009053607000000505")
public class DistanceCorrelationDependenceMeasure extends AbstractDependenceMeasure {
/**
* Static instance.
*/
public static final DistanceCorrelationDependenceMeasure STATIC = new DistanceCorrelationDependenceMeasure();
/**
* Constructor - use {@link #STATIC} instance instead!
*/
protected DistanceCorrelationDependenceMeasure() {
super();
}
@Override
public <A, B> double dependence(NumberArrayAdapter<?, A> adapter1, A data1, NumberArrayAdapter<?, B> adapter2, B data2) {
final int len = size(adapter1, data1, adapter2, data2);
double[] dMatrixA = computeDistances(adapter1, data1);
double[] dMatrixB = computeDistances(adapter2, data2);
// distance variance
double dVarA = computeDCovar(dMatrixA, dMatrixA, len);
if(!(dVarA > 0.)) {
return 0.;
}
double dVarB = computeDCovar(dMatrixB, dMatrixB, len);
if(!(dVarB > 0.)) {
return 0.;
}
double dCovar = computeDCovar(dMatrixA, dMatrixB, len);
// distance correlation
return Math.sqrt(dCovar / Math.sqrt(dVarA * dVarB));
}
@Override
public <A> double[] dependence(NumberArrayAdapter<?, A> adapter, List<? extends A> data) {
final int dims = data.size();
final int len = size(adapter, data);
double[][] dMatrix = new double[dims][];
for(int i = 0; i < dims; i++) {
dMatrix[i] = computeDistances(adapter, data.get(i));
}
double[] dVar = new double[dims];
for(int i = 0; i < dims; i++) {
dVar[i] = computeDCovar(dMatrix[i], dMatrix[i], len);
}
double[] dCor = new double[(dims * (dims - 1)) >> 1];
for(int y = 1, c = 0; y < dims; y++) {
for(int x = 0; x < y; x++) {
if(!(dVar[x] * dVar[y] > 0.)) {
dCor[c++] = 0.;
continue;
}
double dCovar = computeDCovar(dMatrix[x], dMatrix[y], len);
dCor[c++] = Math.sqrt(dCovar / Math.sqrt(dVar[x] * dVar[y]));
}
}
return dCor;
}
/**
* Compute the double-centered delta matrix.
*
* @param adapter Data adapter
* @param data Input data
* @return Double-centered delta matrix.
*/
protected static <A> double[] computeDistances(NumberArrayAdapter<?, A> adapter, A data) {
final int size = adapter.size(data);
double[] dMatrix = new double[(size * (size + 1)) >> 1];
for(int i = 0, c = 0; i < size; i++) {
for(int j = 0; j < i; j++) {
double dx = adapter.getDouble(data, i) - adapter.getDouble(data, j);
dMatrix[c++] = (dx < 0) ? -dx : dx; // Absolute difference.
}
c++; // Diagonal entry: zero
}
doubleCenterMatrix(dMatrix, size);
return dMatrix;
}
/**
* Computes the distance variance matrix of one axis.
*
* @param dMatrix distance matrix of the axis
* @param size Dimensionality
*/
public static void doubleCenterMatrix(double[] dMatrix, int size) {
double[] rowMean = new double[size];
// row sum
for(int i = 0, c = 0; i < size; i++) {
for(int j = 0; j < i; j++) {
double v = dMatrix[c++];
rowMean[i] += v;
rowMean[j] += v;
}
assert (dMatrix[c] == 0.);
c++; // Diagonal entry. Must be zero!
}
// Normalize averages:
double matrixMean = 0.;
for(int i = 0; i < size; i++) {
matrixMean += rowMean[i];
rowMean[i] /= size;
}
matrixMean /= size * size;
for(int o = 0, c = 0; o < size; o++) {
// Including row mean!
for(int p = 0; p <= o; p++) {
dMatrix[c++] -= rowMean[o] + rowMean[p] - matrixMean;
}
}
}
/**
* Computes the distance covariance for two axis. Can also be used to compute
* the distance variance of one axis (dVarMatrixA = dVarMatrixB).
*
* @param dVarMatrixA distance variance matrix of the first axis
* @param dVarMatrixB distance variance matrix of the second axis
* @param n number of points
* @return distance covariance
*/
protected double computeDCovar(double[] dVarMatrixA, double[] dVarMatrixB, int n) {
double result = 0.;
for(int i = 0, c = 0; i < n; i++) {
for(int j = 0; j < i; j++) {
result += 2. * dVarMatrixA[c] * dVarMatrixB[c];
c++;
}
// Diagonal entry.
result += dVarMatrixA[c] * dVarMatrixB[c];
c++;
}
return result / (n * n);
}
/**
* Parameterization class
*
* @author Marie Kiermeier
*
* @apiviz.exclude
*/
public static class Parameterizer extends AbstractParameterizer {
@Override
protected DistanceCorrelationDependenceMeasure makeInstance() {
return STATIC;
}
}
}
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