Daniel@0: function h=plotgauss2d(mu, Sigma)
Daniel@0: % PLOTGAUSS2D Plot a 2D Gaussian as an ellipse with optional cross hairs
Daniel@0: % h=plotgauss2(mu, Sigma)
Daniel@0: %
Daniel@0: 
Daniel@0: h = plotcov2(mu, Sigma);
Daniel@0: return;
Daniel@0: 
Daniel@0: %%%%%%%%%%%%%%%%%%%%%%%%
Daniel@0: 
Daniel@0: % PLOTCOV2 - Plots a covariance ellipse with major and minor axes
Daniel@0: %            for a bivariate Gaussian distribution.
Daniel@0: %
Daniel@0: % Usage:
Daniel@0: %   h = plotcov2(mu, Sigma[, OPTIONS]);
Daniel@0: % 
Daniel@0: % Inputs:
Daniel@0: %   mu    - a 2 x 1 vector giving the mean of the distribution.
Daniel@0: %   Sigma - a 2 x 2 symmetric positive semi-definite matrix giving
Daniel@0: %           the covariance of the distribution (or the zero matrix).
Daniel@0: %
Daniel@0: % Options:
Daniel@0: %   'conf'    - a scalar between 0 and 1 giving the confidence
Daniel@0: %               interval (i.e., the fraction of probability mass to
Daniel@0: %               be enclosed by the ellipse); default is 0.9.
Daniel@0: %   'num-pts' - the number of points to be used to plot the
Daniel@0: %               ellipse; default is 100.
Daniel@0: %
Daniel@0: % This function also accepts options for PLOT.
Daniel@0: %
Daniel@0: % Outputs:
Daniel@0: %   h     - a vector of figure handles to the ellipse boundary and
Daniel@0: %           its major and minor axes
Daniel@0: %
Daniel@0: % See also: PLOTCOV3
Daniel@0: 
Daniel@0: % Copyright (C) 2002 Mark A. Paskin
Daniel@0: 
Daniel@0: function h = plotcov2(mu, Sigma, varargin)
Daniel@0: 
Daniel@0: if size(Sigma) ~= [2 2], error('Sigma must be a 2 by 2 matrix'); end
Daniel@0: if length(mu) ~= 2, error('mu must be a 2 by 1 vector'); end
Daniel@0: 
Daniel@0: [p, ...
Daniel@0:  n, ...
Daniel@0:  plot_opts] = process_options(varargin, 'conf', 0.9, ...
Daniel@0: 					'num-pts', 100);
Daniel@0: h = [];
Daniel@0: holding = ishold;
Daniel@0: if (Sigma == zeros(2, 2))
Daniel@0:   z = mu;
Daniel@0: else
Daniel@0:   % Compute the Mahalanobis radius of the ellipsoid that encloses
Daniel@0:   % the desired probability mass.
Daniel@0:   k = conf2mahal(p, 2);
Daniel@0:   % The major and minor axes of the covariance ellipse are given by
Daniel@0:   % the eigenvectors of the covariance matrix.  Their lengths (for
Daniel@0:   % the ellipse with unit Mahalanobis radius) are given by the
Daniel@0:   % square roots of the corresponding eigenvalues.
Daniel@0:   if (issparse(Sigma))
Daniel@0:     [V, D] = eigs(Sigma);
Daniel@0:   else
Daniel@0:     [V, D] = eig(Sigma);
Daniel@0:   end
Daniel@0:   % Compute the points on the surface of the ellipse.
Daniel@0:   t = linspace(0, 2*pi, n);
Daniel@0:   u = [cos(t); sin(t)];
Daniel@0:   w = (k * V * sqrt(D)) * u;
Daniel@0:   z = repmat(mu, [1 n]) + w;
Daniel@0:   % Plot the major and minor axes.
Daniel@0:   L = k * sqrt(diag(D));
Daniel@0:   h = plot([mu(1); mu(1) + L(1) * V(1, 1)], ...
Daniel@0: 	   [mu(2); mu(2) + L(1) * V(2, 1)], plot_opts{:});
Daniel@0:   hold on;
Daniel@0:   h = [h; plot([mu(1); mu(1) + L(2) * V(1, 2)], ...
Daniel@0: 	       [mu(2); mu(2) + L(2) * V(2, 2)], plot_opts{:})];
Daniel@0: end
Daniel@0: 
Daniel@0: h = [h; plot(z(1, :), z(2, :), plot_opts{:})];
Daniel@0: if (~holding) hold off; end
Daniel@0: 
Daniel@0: %%%%%%%%%%%%
Daniel@0: 
Daniel@0: % CONF2MAHAL - Translates a confidence interval to a Mahalanobis
Daniel@0: %              distance.  Consider a multivariate Gaussian
Daniel@0: %              distribution of the form
Daniel@0: %
Daniel@0: %   p(x) = 1/sqrt((2 * pi)^d * det(C)) * exp((-1/2) * MD(x, m, inv(C)))
Daniel@0: %
Daniel@0: %              where MD(x, m, P) is the Mahalanobis distance from x
Daniel@0: %              to m under P:
Daniel@0: %
Daniel@0: %                 MD(x, m, P) = (x - m) * P * (x - m)'
Daniel@0: %
Daniel@0: %              A particular Mahalanobis distance k identifies an
Daniel@0: %              ellipsoid centered at the mean of the distribution.
Daniel@0: %              The confidence interval associated with this ellipsoid
Daniel@0: %              is the probability mass enclosed by it.  Similarly,
Daniel@0: %              a particular confidence interval uniquely determines
Daniel@0: %              an ellipsoid with a fixed Mahalanobis distance.
Daniel@0: %
Daniel@0: %              If X is an d dimensional Gaussian-distributed vector,
Daniel@0: %              then the Mahalanobis distance of X is distributed
Daniel@0: %              according to the Chi-squared distribution with d
Daniel@0: %              degrees of freedom.  Thus, the Mahalanobis distance is
Daniel@0: %              determined by evaluating the inverse cumulative
Daniel@0: %              distribution function of the chi squared distribution
Daniel@0: %              up to the confidence value.
Daniel@0: %
Daniel@0: % Usage:
Daniel@0: % 
Daniel@0: %   m = conf2mahal(c, d);
Daniel@0: %
Daniel@0: % Inputs:
Daniel@0: %
Daniel@0: %   c    - the confidence interval
Daniel@0: %   d    - the number of dimensions of the Gaussian distribution
Daniel@0: %
Daniel@0: % Outputs:
Daniel@0: %
Daniel@0: %   m    - the Mahalanobis radius of the ellipsoid enclosing the
Daniel@0: %          fraction c of the distribution's probability mass
Daniel@0: %
Daniel@0: % See also: MAHAL2CONF
Daniel@0: 
Daniel@0: % Copyright (C) 2002 Mark A. Paskin
Daniel@0: 
Daniel@0: function m = conf2mahal(c, d)
Daniel@0: 
Daniel@0: m = chi2inv(c, d); % matlab stats toolbox