comparison toolboxes/FullBNT-1.0.7/KPMtools/plotgauss2d.m @ 0:e9a9cd732c1e tip

first hg version after svn

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