#region License Information /* HeuristicLab * Copyright (C) 2002-2015 Heuristic and Evolutionary Algorithms Laboratory (HEAL) * * This file is part of HeuristicLab. * * HeuristicLab 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 3 of the License, or * (at your option) any later version. * * HeuristicLab 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 HeuristicLab. If not, see . */ #endregion using System; using HeuristicLab.Data; namespace HeuristicLab.Problems.QuadraticAssignment.Views { public static class MultidimensionalScaling { /// /// Performs the Kruskal-Shepard algorithm and applies a gradient descent method /// to fit the coordinates such that the difference between the fit distances /// and the actual distances is minimal. /// /// A symmetric NxN matrix that specifies the distances between each element i and j. Diagonal elements are ignored. /// Returns the stress between the fit distances and the actual distances. /// A Nx2 matrix where the first column represents the x- and the second column the y coordinates. public static DoubleMatrix MetricByDistance(DoubleMatrix distances, out double stress) { if (distances == null) throw new ArgumentNullException("distances"); if (distances.Rows != distances.Columns) throw new ArgumentException("Distance matrix must be a square matrix.", "distances"); stress = 0.0; int dimension = distances.Rows; if (dimension == 1) return new DoubleMatrix(new double[,] { { 0, 0 } }); else if (dimension == 2) return new DoubleMatrix(new double[,] { { 0, distances[0, 1] } }); DoubleMatrix coordinates = new DoubleMatrix(dimension, 2); double rad = (2 * Math.PI) / coordinates.Rows; for (int i = 0; i < dimension; i++) { coordinates[i, 0] = 10 * Math.Cos(rad * i); coordinates[i, 1] = 10 * Math.Sin(rad * i); } double epsg = 1e-7; double epsf = 0; double epsx = 0; int maxits = 0; alglib.mincgstate state = null; alglib.mincgreport rep; for (int iterations = 0; iterations < 10; iterations++) { for (int i = 0; i < dimension; i++) { double[] c = new double[] { coordinates[i, 0], coordinates[i, 1] }; if (iterations == 0 && i == 0) { alglib.mincgcreate(c, out state); alglib.mincgsetcond(state, epsg, epsf, epsx, maxits); } else { alglib.mincgrestartfrom(state, c); } alglib.mincgoptimize(state, StressGradient, null, new Info(coordinates, distances, i)); alglib.mincgresults(state, out c, out rep); coordinates[i, 0] = c[0]; coordinates[i, 1] = c[1]; } } stress = CalculateNormalizedStress(dimension, distances, coordinates); return coordinates; } private static void StressGradient(double[] x, ref double func, double[] grad, object obj) { func = 0; grad[0] = 0; grad[1] = 0; Info info = (obj as Info); for (int i = 0; i < info.Coordinates.Rows; i++) { double c = info.Distances[info.Row, i]; if (i != info.Row) { double a = info.Coordinates[i, 0]; double b = info.Coordinates[i, 1]; func += Stress(x, c, a, b); grad[0] += ((2 * x[0] - 2 * a) * Math.Sqrt(x[1] * x[1] - 2 * b * x[1] + x[0] * x[0] - 2 * a * x[0] + b * b + a * a) - 2 * c * x[0] + 2 * a * c) / Math.Sqrt(x[1] * x[1] - 2 * b * x[1] + x[0] * x[0] - 2 * a * x[0] + b * b + a * a); grad[1] += ((2 * x[1] - 2 * b) * Math.Sqrt(x[1] * x[1] - 2 * b * x[1] + x[0] * x[0] - 2 * a * x[0] + b * b + a * a) - 2 * c * x[1] + 2 * b * c) / Math.Sqrt(x[1] * x[1] - 2 * b * x[1] + x[0] * x[0] - 2 * a * x[0] + b * b + a * a); } } } private static double Stress(double[] x, double distance, double xCoord, double yCoord) { return Stress(x[0], x[1], distance, xCoord, yCoord); } private static double Stress(double x, double y, double distance, double otherX, double otherY) { double d = Math.Sqrt((x - otherX) * (x - otherX) + (y - otherY) * (y - otherY)); return (d - distance) * (d - distance); } public static double CalculateNormalizedStress(int dimension, DoubleMatrix distances, DoubleMatrix coordinates) { double stress = 0; for (int i = 0; i < dimension - 1; i++) { for (int j = i + 1; j < dimension; j++) { if (distances[i, j] != 0) { stress += Stress(coordinates[i, 0], coordinates[i, 1], distances[i, j], coordinates[j, 0], coordinates[j, 1]) / (distances[i, j] * distances[i, j]); } } } return stress; } private class Info { public DoubleMatrix Coordinates { get; set; } public DoubleMatrix Distances { get; set; } public int Row { get; set; } public Info(DoubleMatrix c, DoubleMatrix d, int r) { Coordinates = c; Distances = d; Row = r; } } } }