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Re: fitting 2d normal distribution


From: John W. Eaton
Subject: Re: fitting 2d normal distribution
Date: Wed, 22 Oct 2008 10:42:44 -0400

On 22-Oct-2008, Søren Hauberg wrote:

| tir, 21 10 2008 kl. 23:33 +0200, skrev Francesco Potorti`:
| > >> What about 2d data?  Are there ready-made functions for computing the
| > >> necessary parameters and plotting the 2d normal pdf?
| > >
| > >'mean' and 'cov' ?
| > 
| > How do I use them?  If my data are in the 2d matrix X, how do I plot the
| > fitting 2d Gaussian pdf or -- better for me -- the contour ellipses?
| > 
| > More precisely, I can do that "by hand", with a little effort, but I
| > wondered if there were simple ready-made functions to do that.
| 
| I don't think there are any functions for plotting ellipses from
| covariance matrices.

Does the following function do (at least part of) what you want?  For
example:

  tmp = rand (2);
  amat = tmp'*tmp;  %% matrix must be positive definite

  [x, y, major, minor, bbox] = ellipse (amat, 0.95);

  plot (x, y, major(:,1), major(:,2), minor(:,1), minor(:,2), bbox(:,1), 
bbox(:,2));
  axis ("square");

jwe

%% Copyright (C) 2001, James B. Rawlings and John W. Eaton
%%
%% This program 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 2, or (at
%% your option) any later version.
%%
%% This program 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 this program; see the file COPYING.  If not, write to
%% the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
%% MA 02111-1307, USA.

%% [x, y, major, minor, bbox] = ellipse (amat, level, n, shift)
%%
%% Given a 2x2 matrix, generate ellipse data for plotting.  The
%% arguments N and SHIFT are optional.  If N is an empty matrix, a
%% default value of 100 is used.

function [x, y, major, minor, bbox] = ellipse (amat, level, n, shift)

  if (nargin < 3)
    n = 100;
  end

  if (isempty (n))
    n = 100;
  end

  if (nargin < 4)
    shift = [0, 0];
  end

  ss = size (shift);

  if (any (ss ~= [1, 2]))
    if (ss == [2, 1])
      shift = shift';
    else
      error ('shift must be a 2-element row vector');
    end
  end

  if (nargin > 1)

    [v, l] = eig (amat / level);

    dl = diag(l);
    if (any (imag (dl)) || any (dl <= 0))
      error ('ellipse: amat must be positive definite');
    end

    %% Generate contour data.

    a = 1 / sqrt (l(1,1));
    b = 1 / sqrt (l(2,2));

    t = linspace (0, 2*pi, n)';

    xt = a * cos (t);
    yt = b * sin (t);

    %% Rotate the contours.

    ra = atan2 (v(2,1), v(1,1));

    cos_ra = cos (ra);
    sin_ra = sin (ra);

    x = xt * cos_ra - yt * sin_ra + shift(1);
    y = xt * sin_ra + yt * cos_ra + shift(2);

    %% Endpoints of the major and minor axes.

    minor = (v * diag ([a, b]))';
    major = minor;

    major(2,:) = -major(1,:);
    minor(1,:) = -minor(2,:);

    t = [1; 1] * shift;

    major = major + t;
    minor = minor + t;

    %% Bounding box for the ellipse using magic formula.

    ainv = inv (amat);
    xbox = sqrt (level * ainv(1,1));
    ybox = sqrt (level * ainv(2,2));

    bbox = [xbox ybox; xbox -ybox; -xbox -ybox; -xbox ybox; xbox ybox];

    t = [1; 1; 1; 1; 1] * shift;
    bbox = bbox + t;

  else
    error ('usage: ellipse (amat, level, n, shift)');
  end

%% endfunction

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