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1 % PLOTCOV3 - Plots a covariance ellipsoid with axes for a trivariate
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2 % Gaussian distribution.
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3 %
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4 % Usage:
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5 % [h, s] = plotcov3(mu, Sigma[, OPTIONS]);
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6 %
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7 % Inputs:
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8 % mu - a 3 x 1 vector giving the mean of the distribution.
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9 % Sigma - a 3 x 3 symmetric positive semi-definite matrix giving
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10 % the covariance of the distribution (or the zero matrix).
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11 %
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12 % Options:
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13 % 'conf' - a scalar between 0 and 1 giving the confidence
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14 % interval (i.e., the fraction of probability mass to
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15 % be enclosed by the ellipse); default is 0.9.
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16 % 'num-pts' - if the value supplied is n, then (n + 1)^2 points
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17 % to be used to plot the ellipse; default is 20.
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18 % 'plot-opts' - a cell vector of arguments to be handed to PLOT3
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19 % to contol the appearance of the axes, e.g.,
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20 % {'Color', 'g', 'LineWidth', 1}; the default is {}
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21 % 'surf-opts' - a cell vector of arguments to be handed to SURF
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22 % to contol the appearance of the ellipsoid
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23 % surface; a nice possibility that yields
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24 % transparency is: {'EdgeAlpha', 0, 'FaceAlpha',
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25 % 0.1, 'FaceColor', 'g'}; the default is {}
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26 %
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27 % Outputs:
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28 % h - a vector of handles on the axis lines
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29 % s - a handle on the ellipsoid surface object
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30 %
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31 % See also: PLOTCOV2
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32
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33 % Copyright (C) 2002 Mark A. Paskin
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34 %
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35 % This program is free software; you can redistribute it and/or modify
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36 % it under the terms of the GNU General Public License as published by
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37 % the Free Software Foundation; either version 2 of the License, or
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38 % (at your option) any later version.
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39 %
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40 % This program is distributed in the hope that it will be useful, but
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41 % WITHOUT ANY WARRANTY; without even the implied warranty of
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42 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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43 % General Public License for more details.
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44 %
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45 % You should have received a copy of the GNU General Public License
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46 % along with this program; if not, write to the Free Software
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47 % Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
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48 % USA.
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49 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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50
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51 function [h, s] = plotcov3(mu, Sigma, varargin)
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52
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53 if size(Sigma) ~= [3 3], error('Sigma must be a 3 by 3 matrix'); end
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54 if length(mu) ~= 3, error('mu must be a 3 by 1 vector'); end
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55
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56 [p, ...
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57 n, ...
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58 plot_opts, ...
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59 surf_opts] = process_options(varargin, 'conf', 0.9, ...
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60 'num-pts', 20, ...
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61 'plot-opts', {}, ...
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62 'surf-opts', {});
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63 h = [];
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64 holding = ishold;
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65 if (Sigma == zeros(3, 3))
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66 z = mu;
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67 else
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68 % Compute the Mahalanobis radius of the ellipsoid that encloses
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69 % the desired probability mass.
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70 k = conf2mahal(p, 3);
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71 % The axes of the covariance ellipse are given by the eigenvectors of
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72 % the covariance matrix. Their lengths (for the ellipse with unit
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73 % Mahalanobis radius) are given by the square roots of the
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74 % corresponding eigenvalues.
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75 if (issparse(Sigma))
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76 [V, D] = eigs(Sigma);
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77 else
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78 [V, D] = eig(Sigma);
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79 end
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80 if (any(diag(D) < 0))
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81 error('Invalid covariance matrix: not positive semi-definite.');
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82 end
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83 % Compute the points on the surface of the ellipsoid.
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84 t = linspace(0, 2*pi, n);
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85 [X, Y, Z] = sphere(n);
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86 u = [X(:)'; Y(:)'; Z(:)'];
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87 w = (k * V * sqrt(D)) * u;
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88 z = repmat(mu(:), [1 (n + 1)^2]) + w;
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89
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90 % Plot the axes.
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91 L = k * sqrt(diag(D));
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92 h = plot3([mu(1); mu(1) + L(1) * V(1, 1)], ...
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93 [mu(2); mu(2) + L(1) * V(2, 1)], ...
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94 [mu(3); mu(3) + L(1) * V(3, 1)], plot_opts{:});
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95 hold on;
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96 h = [h; plot3([mu(1); mu(1) + L(2) * V(1, 2)], ...
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97 [mu(2); mu(2) + L(2) * V(2, 2)], ...
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98 [mu(3); mu(3) + L(2) * V(3, 2)], plot_opts{:})];
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99 h = [h; plot3([mu(1); mu(1) + L(3) * V(1, 3)], ...
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100 [mu(2); mu(2) + L(3) * V(2, 3)], ...
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101 [mu(3); mu(3) + L(3) * V(3, 3)], plot_opts{:})];
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102 end
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103
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104 s = surf(reshape(z(1, :), [(n + 1) (n + 1)]), ...
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105 reshape(z(2, :), [(n + 1) (n + 1)]), ...
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106 reshape(z(3, :), [(n + 1) (n + 1)]), ...
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107 surf_opts{:});
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108
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109 if (~holding) hold off; end
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