[5723] | 1 | #region License Information
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| 2 | /* HeuristicLab
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[17180] | 3 | * Copyright (C) Heuristic and Evolutionary Algorithms Laboratory (HEAL)
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[5723] | 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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[5931] | 31 | /// and the dissimilarities becomes minimal.
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[5723] | 32 | /// </summary>
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[5871] | 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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[5931] | 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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[5871] | 37 | ///
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[5723] | 38 | /// <returns>A Nx2 matrix where the first column represents the x- and the second column the y coordinates.</returns>
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[5871] | 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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[5723] | 42 |
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[5871] | 43 | int dimension = dissimilarities.Rows;
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[5723] | 44 | if (dimension == 1) return new DoubleMatrix(new double[,] { { 0, 0 } });
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[5931] | 45 | else if (dimension == 2) return new DoubleMatrix(new double[,] { { 0, 0 }, { 0, dissimilarities[0, 1] } });
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[5723] | 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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[5871] | 54 | return KruskalShepard(dissimilarities, coordinates);
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[5723] | 55 | }
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| 56 |
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[5871] | 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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[5931] | 60 | /// and the dissimilarities is minimal.
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[5871] | 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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[5931] | 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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[5871] | 66 | /// <param name="coordinates">The Nx2 matrix of initial coordinates.</param>
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[5931] | 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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[5871] | 69 | /// <returns>A Nx2 matrix where the first column represents the x- and the second column the y coordinates.</returns>
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[6342] | 70 | public static DoubleMatrix KruskalShepard(DoubleMatrix dissimilarities, DoubleMatrix coordinates, int maximumIterations = 10) {
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[5871] | 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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[5723] | 73 |
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| 74 | double epsx = 0;
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[6342] | 75 | int maxits = 0;
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[5723] | 76 |
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[6342] | 77 | alglib.minlmstate state;
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| 78 | alglib.minlmreport rep;
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[5931] | 79 | for (int iterations = 0; iterations < maximumIterations; iterations++) {
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| 80 | bool changed = false;
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[5723] | 81 | for (int i = 0; i < dimension; i++) {
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| 82 | double[] c = new double[] { coordinates[i, 0], coordinates[i, 1] };
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| 83 |
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[5855] | 84 | try {
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[6342] | 85 | alglib.minlmcreatevj(dimension - 1, c, out state);
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[17931] | 86 | alglib.minlmsetcond(state, epsx, maxits);
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[6342] | 87 | alglib.minlmoptimize(state, StressFitness, StressJacobian, null, new Info(coordinates, dissimilarities, i));
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| 88 | alglib.minlmresults(state, out c, out rep);
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[5871] | 89 | } catch (alglib.alglibexception) { }
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[5855] | 90 | if (!double.IsNaN(c[0]) && !double.IsNaN(c[1])) {
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[5931] | 91 | changed = changed || (coordinates[i, 0] != c[0]) || (coordinates[i, 1] != c[1]);
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[5855] | 92 | coordinates[i, 0] = c[0];
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| 93 | coordinates[i, 1] = c[1];
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[5723] | 94 | }
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| 95 | }
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[5931] | 96 | if (!changed) break;
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[5723] | 97 | }
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| 98 | return coordinates;
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| 99 | }
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| 100 |
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[6342] | 101 | private static void StressFitness(double[] x, double[] fi, object obj) {
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[5723] | 102 | Info info = (obj as Info);
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[7416] | 103 | int idx = 0;
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[5723] | 104 | for (int i = 0; i < info.Coordinates.Rows; i++) {
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[7416] | 105 | if (i == info.Row) continue;
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| 106 | if (!double.IsNaN(info.Dissimilarities[info.Row, i]))
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| 107 | fi[idx++] = Stress(x, info.Dissimilarities[info.Row, i], info.Coordinates[i, 0], info.Coordinates[i, 1]);
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| 108 | else fi[idx++] = 0.0;
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[6342] | 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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[7416] | 116 | if (i == info.Row) continue;
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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 | if (!double.IsNaN(c)) {
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| 121 | fi[idx] = Stress(x, c, a, b); ;
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[6342] | 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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[7416] | 124 | } else {
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| 125 | fi[idx] = jac[idx, 0] = jac[idx, 1] = 0;
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[5723] | 126 | }
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[7416] | 127 | idx++;
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[5723] | 128 | }
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| 129 | }
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| 130 |
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| 131 | private static double Stress(double[] x, double distance, double xCoord, double yCoord) {
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| 132 | return Stress(x[0], x[1], distance, xCoord, yCoord);
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| 133 | }
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| 134 |
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| 135 | private static double Stress(double x, double y, double distance, double otherX, double otherY) {
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| 136 | double d = Math.Sqrt((x - otherX) * (x - otherX)
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| 137 | + (y - otherY) * (y - otherY));
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| 138 | return (d - distance) * (d - distance);
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| 139 | }
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| 140 |
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[5871] | 141 | /// <summary>
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| 142 | /// 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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| 143 | /// </summary>
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| 144 | /// <remarks>
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| 145 | /// Throws an ArgumentException when the <paramref name="dissimilarities"/> matrix is not symmetric.
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| 146 | /// </remarks>
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| 147 | ///
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| 148 | /// <param name="dissimilarities">The matrix with the dissimilarities.</param>
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| 149 | /// <param name="coordinates">The actual location of the points.</param>
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| 150 | /// <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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| 151 | public static double CalculateNormalizedStress(DoubleMatrix dissimilarities, DoubleMatrix coordinates) {
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| 152 | int dimension = dissimilarities.Rows;
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| 153 | 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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[5855] | 154 | double stress = 0, normalization = 0;
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[5723] | 155 | for (int i = 0; i < dimension - 1; i++) {
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| 156 | for (int j = i + 1; j < dimension; j++) {
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[7416] | 157 | if (dissimilarities[i, j] != dissimilarities[j, i] && !(double.IsNaN(dissimilarities[i, j]) && double.IsNaN(dissimilarities[j, i])))
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| 158 | throw new ArgumentException("Dissimilarities is not a symmetric matrix.", "dissimilarities");
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| 159 | if (!double.IsNaN(dissimilarities[i, j])) {
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[5871] | 160 | stress += Stress(coordinates[i, 0], coordinates[i, 1], dissimilarities[i, j], coordinates[j, 0], coordinates[j, 1]);
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| 161 | normalization += (dissimilarities[i, j] * dissimilarities[i, j]);
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[5723] | 162 | }
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| 163 | }
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| 164 | }
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[5871] | 165 | return stress / normalization;
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[5723] | 166 | }
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| 167 |
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| 168 | private class Info {
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| 169 | public DoubleMatrix Coordinates { get; set; }
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[5931] | 170 | public DoubleMatrix Dissimilarities { get; set; }
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[5723] | 171 | public int Row { get; set; }
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| 172 |
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| 173 | public Info(DoubleMatrix c, DoubleMatrix d, int r) {
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| 174 | Coordinates = c;
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[5931] | 175 | Dissimilarities = d;
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[5723] | 176 | Row = r;
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| 177 | }
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| 178 | }
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| 179 | }
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| 180 | } |
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