1 | #region License Information
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2 | /* HeuristicLab
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3 | * Copyright (C) 2002-2011 Heuristic and Evolutionary Algorithms Laboratory (HEAL)
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4 | *
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5 | * This file is part of HeuristicLab.
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6 | *
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7 | * HeuristicLab is free software: you can redistribute it and/or modify
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8 | * it under the terms of the GNU General Public License as published by
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9 | * the Free Software Foundation, either version 3 of the License, or
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10 | * (at your option) any later version.
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11 | *
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12 | * HeuristicLab is distributed in the hope that it will be useful,
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13 | * but WITHOUT ANY WARRANTY; without even the implied warranty of
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14 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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15 | * GNU General Public License for more details.
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16 | *
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17 | * You should have received a copy of the GNU General Public License
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18 | * along with HeuristicLab. If not, see <http://www.gnu.org/licenses/>.
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19 | */
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20 | #endregion
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21 |
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22 | using System;
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23 | using HeuristicLab.Data;
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24 |
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25 | namespace HeuristicLab.Analysis {
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26 | public static class MultidimensionalScaling {
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27 |
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28 | /// <summary>
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29 | /// Performs the Kruskal-Shepard algorithm and applies a gradient descent method
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30 | /// to fit the coordinates such that the difference between the fit distances
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31 | /// and the dissimilarities becomes minimal.
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32 | /// </summary>
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33 | /// <remarks>
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34 | /// It will initialize the coordinates in a deterministic fashion such that all initial points are equally spaced on a circle.
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35 | /// </remarks>
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36 | /// <param name="dissimilarities">A symmetric NxN matrix that specifies the dissimilarities between each element i and j. Diagonal elements are ignored.</param>
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37 | ///
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38 | /// <returns>A Nx2 matrix where the first column represents the x- and the second column the y coordinates.</returns>
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39 | public static DoubleMatrix KruskalShepard(DoubleMatrix dissimilarities) {
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40 | if (dissimilarities == null) throw new ArgumentNullException("dissimilarities");
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41 | if (dissimilarities.Rows != dissimilarities.Columns) throw new ArgumentException("Dissimilarities must be a square matrix.", "dissimilarities");
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42 |
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43 | int dimension = dissimilarities.Rows;
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44 | if (dimension == 1) return new DoubleMatrix(new double[,] { { 0, 0 } });
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45 | else if (dimension == 2) return new DoubleMatrix(new double[,] { { 0, 0 }, { 0, dissimilarities[0, 1] } });
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46 |
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47 | DoubleMatrix coordinates = new DoubleMatrix(dimension, 2);
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48 | double rad = (2 * Math.PI) / coordinates.Rows;
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49 | for (int i = 0; i < dimension; i++) {
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50 | coordinates[i, 0] = 10 * Math.Cos(rad * i);
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51 | coordinates[i, 1] = 10 * Math.Sin(rad * i);
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52 | }
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53 |
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54 | return KruskalShepard(dissimilarities, coordinates);
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55 | }
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56 |
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57 | /// <summary>
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58 | /// Performs the Kruskal-Shepard algorithm and applies a gradient descent method
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59 | /// to fit the coordinates such that the difference between the fit distances
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60 | /// and the dissimilarities is minimal.
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61 | /// </summary>
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62 | /// <remarks>
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63 | /// It will use a pre-initialized x,y-coordinates matrix as a starting point of the gradient descent.
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64 | /// </remarks>
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65 | /// <param name="dissimilarities">A symmetric NxN matrix that specifies the dissimilarities between each element i and j. Diagonal elements are ignored.</param>
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66 | /// <param name="coordinates">The Nx2 matrix of initial coordinates.</param>
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67 | /// <param name="maximumIterations">The number of iterations for which the algorithm should run.
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68 | /// In every iteration it tries to find the best location for every item.</param>
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69 | /// <returns>A Nx2 matrix where the first column represents the x- and the second column the y coordinates.</returns>
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70 | public static DoubleMatrix KruskalShepard(DoubleMatrix dissimilarities, DoubleMatrix coordinates, int maximumIterations = 10) {
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71 | int dimension = dissimilarities.Rows;
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72 | if (dimension != dissimilarities.Columns || coordinates.Rows != dimension) throw new ArgumentException("The number of coordinates and the number of rows and columns in the dissimilarities matrix do not match.");
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73 |
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74 | double epsg = 1e-7;
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75 | double epsf = 0;
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76 | double epsx = 0;
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77 | int maxits = 0;
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78 |
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79 | alglib.minlmstate state;
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80 | alglib.minlmreport rep;
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81 | for (int iterations = 0; iterations < maximumIterations; iterations++) {
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82 | bool changed = false;
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83 | for (int i = 0; i < dimension; i++) {
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84 | double[] c = new double[] { coordinates[i, 0], coordinates[i, 1] };
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85 |
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86 | try {
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87 | alglib.minlmcreatevj(dimension - 1, c, out state);
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88 | alglib.minlmsetcond(state, epsg, epsf, epsx, maxits);
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89 | alglib.minlmoptimize(state, StressFitness, StressJacobian, null, new Info(coordinates, dissimilarities, i));
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90 | alglib.minlmresults(state, out c, out rep);
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91 | } catch (alglib.alglibexception) { }
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92 | if (!double.IsNaN(c[0]) && !double.IsNaN(c[1])) {
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93 | changed = changed || (coordinates[i, 0] != c[0]) || (coordinates[i, 1] != c[1]);
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94 | coordinates[i, 0] = c[0];
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95 | coordinates[i, 1] = c[1];
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96 | }
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97 | }
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98 | if (!changed) break;
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99 | }
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100 | return coordinates;
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101 | }
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102 |
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103 | private static void StressFitness(double[] x, double[] fi, object obj) {
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104 | Info info = (obj as Info);
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105 | for (int i = 0; i < info.Coordinates.Rows; i++) {
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106 | double f = Stress(x, info.Dissimilarities[info.Row, i], info.Coordinates[i, 0], info.Coordinates[i, 1]);
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107 | if (i < info.Row) fi[i] = f;
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108 | else if (i > info.Row) fi[i - 1] = f;
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109 | }
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110 | }
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111 |
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112 | private static void StressJacobian(double[] x, double[] fi, double[,] jac, object obj) {
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113 | Info info = (obj as Info);
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114 | int idx = 0;
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115 | for (int i = 0; i < info.Coordinates.Rows; i++) {
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116 | if (i != info.Row) {
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117 | double c = info.Dissimilarities[info.Row, i];
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118 | double a = info.Coordinates[i, 0];
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119 | double b = info.Coordinates[i, 1];
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120 | double f = Stress(x, c, a, b);
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121 | fi[idx] = f;
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122 | jac[idx, 0] = 2 * (x[0] - a) * (Math.Sqrt((a - x[0]) * (a - x[0]) + (b - x[1]) * (b - x[1])) - c) / Math.Sqrt((a - x[0]) * (a - x[0]) + (b - x[1]) * (b - x[1]));
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123 | jac[idx, 1] = 2 * (x[1] - b) * (Math.Sqrt((a - x[0]) * (a - x[0]) + (b - x[1]) * (b - x[1])) - c) / Math.Sqrt((a - x[0]) * (a - x[0]) + (b - x[1]) * (b - x[1]));
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124 | idx++;
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125 | }
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126 | }
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127 | }
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128 |
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129 | private static double Stress(double[] x, double distance, double xCoord, double yCoord) {
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130 | return Stress(x[0], x[1], distance, xCoord, yCoord);
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131 | }
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132 |
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133 | private static double Stress(double x, double y, double distance, double otherX, double otherY) {
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134 | double d = Math.Sqrt((x - otherX) * (x - otherX)
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135 | + (y - otherY) * (y - otherY));
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136 | return (d - distance) * (d - distance);
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137 | }
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138 |
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139 | /// <summary>
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140 | /// This method computes the normalized raw-stress value according to Groenen and van de Velden 2004. "Multidimensional Scaling". Technical report EI 2004-15.
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141 | /// </summary>
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142 | /// <remarks>
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143 | /// Throws an ArgumentException when the <paramref name="dissimilarities"/> matrix is not symmetric.
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144 | /// </remarks>
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145 | ///
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146 | /// <param name="dissimilarities">The matrix with the dissimilarities.</param>
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147 | /// <param name="coordinates">The actual location of the points.</param>
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148 | /// <returns>The normalized raw-stress value that describes the goodness-of-fit between the distances in the points and the size of the dissimilarities. If the value is < 0.1 the fit is generally considered good. If between 0.1 and 0.2 it is considered acceptable, but the usefulness of the scaling with higher values is doubtful.</returns>
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149 | public static double CalculateNormalizedStress(DoubleMatrix dissimilarities, DoubleMatrix coordinates) {
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150 | int dimension = dissimilarities.Rows;
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151 | if (dimension != dissimilarities.Columns || dimension != coordinates.Rows) throw new ArgumentException("The number of coordinates and the number of rows and columns in the dissimilarities matrix do not match.");
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152 | double stress = 0, normalization = 0;
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153 | for (int i = 0; i < dimension - 1; i++) {
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154 | for (int j = i + 1; j < dimension; j++) {
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155 | if (dissimilarities[i, j] != dissimilarities[j, i]) throw new ArgumentException("Dissimilarities is not a symmetric matrix.", "dissimilarities");
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156 | if (dissimilarities[i, j] != 0) {
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157 | stress += Stress(coordinates[i, 0], coordinates[i, 1], dissimilarities[i, j], coordinates[j, 0], coordinates[j, 1]);
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158 | normalization += (dissimilarities[i, j] * dissimilarities[i, j]);
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159 | }
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160 | }
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161 | }
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162 | return stress / normalization;
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163 | }
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164 |
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165 | private class Info {
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166 | public DoubleMatrix Coordinates { get; set; }
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167 | public DoubleMatrix Dissimilarities { get; set; }
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168 | public int Row { get; set; }
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169 |
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170 | public Info(DoubleMatrix c, DoubleMatrix d, int r) {
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171 | Coordinates = c;
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172 | Dissimilarities = d;
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173 | Row = r;
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174 | }
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175 | }
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176 | }
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177 | } |
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