Initial commit
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Sam Perry
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function [x_class, Perr] = classify()
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phoneme_1 = load('MoGVarsK3.mat');
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phoneme_2 = load('MoGVarsK3_p2.mat');
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J = load('PB_data.mat');
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% x = load('PB12.mat.mat');
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x = [J.f1(J.phno == 1), J.f2(J.phno == 1)];
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x1_min = min(J.f1)
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x1_max = max(J.f1)
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x2_min = min(J.f2)
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x2_max = max(J.f2)
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x = [linspace(x1_min, x1_max, 20)', linspace(x2_min, x2_max, 20)'];
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[p1, d1] = liklihood(x, phoneme_1);
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[p2, d2]= liklihood(x, phoneme_2);
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x_class = mean(p1, 2) > mean(p2, 2)
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y_class = mean(p1, 2) <= mean(p2, 2)
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Perr = (0.5*sum(p1(x_class).*d1(x_class)))+(0.5*sum(p2(~x_class).*d2(~x_class)));
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x_class + y_class'
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function [Px, d] = liklihood(x, phoneme)
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[n D] = size(x);
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k = size(phoneme.mu, 2);
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Px = zeros(n, k);
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d = zeros(n, k);
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p = phoneme.p;
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mu = phoneme.mu;
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s2 = phoneme.s2;
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for i=1:k
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Px(:,i) = p(i)*det(s2(:,:,i))^(-0.5)*exp(-0.5*sum((x'-repmat(mu(:,i),1,n))'*inv(s2(:,:,i)).*(x'-repmat(mu(:,i),1,n))',2));
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d(:,i) = abs(sum((x'-repmat(mu(:,i),1,n))));
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end
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%Z = sum(Z, 2) / k;
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%Px = mean(Px);
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@@ -0,0 +1,105 @@
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function demo_classification
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% Generative classification demo: The Gaussian classifier with shared diagonal covariance.
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% That is, a Gaussian Naive Bayes with shared covariance.
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% Note, whenever the covariance is shared (i.e. the same for both classes) then the decision function is linear.
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% This code is provided to complete Assignment 1 for the module Machine Learning (Extended)
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% It (akong with Assignment 1) also serves as a starting point for you if you want to experiment with variations
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% of the Gaussian Classifier such as non-diagonal class-covariance, non-shared class covariances etc.
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% Generate some data in 2 classes
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nInputs = 2; % dimensionality of data (number of attributes)
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nClasses = 2; % number of classes
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nSamples(1) = 200; nSamples(2) = 200; % number of points in each class
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mu1 = [1;-1]; sigma1 = 1.2;
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mu2 = [-1;1]; sigma2 = 1.8;
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X = zeros(nInputs,max(nSamples),nClasses);
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X(:,1:nSamples(1),1) = sigma1*randn(nInputs,nSamples(1)) + ...
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repmat(mu1,1,nSamples(1)); % class 1 training data
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X(:,1:nSamples(2),2) = sigma2*randn(nInputs,nSamples(2)) + ...
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repmat(mu2,1,nSamples(2)); % class 2 training data
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% Alternatively, load data
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% targets
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Y = [1,0]; % targets for class 1,2.
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% Plot the data
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clf, figure(1)
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plot(X(1,1:nSamples(1),1),X(2,1:nSamples(1),1),'b+'); hold on % class 1
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plot(X(1,1:nSamples(1),2),X(2,1:nSamples(1),2),'ro'); % class 2
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title('2-CLASS DATA')
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disp('Press a key to continue training the GAUSSIAN CLASSIFIER model ...'); pause
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N = sum(nSamples); % total number of points
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Sigma = zeros(nInputs);
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for c = 1:nClasses
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Nc = nSamples(c); Xc = X(:,1:Nc,c);
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alpha(c) = Nc/N; % class prior
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% estimate the mean of class c
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m(:,c) = sum(Xc,2)/Nc; % Equivalent to: mean(Xc')
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% estimate covariance of diagonal form for class c
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% Sigma(:,:,c) = diag(diag((Xc-repmat(m(:,c),1,Nc))*(Xc-repmat(m(:,c),1,Nc))'))/Nc; % each class with its own covariance
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% Equiv. to: cov(Xc')
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Sigma = Sigma + (Xc-repmat(m(:,c),1,Nc))*(Xc-repmat(m(:,c),1,Nc))'; % instead, here we take a shared covariance
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% this is estimated from all the points, not just from class c.
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end%for c
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% ML if all classes have same variance
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Sigma = diag(diag(Sigma))/N; % diagonal approximation of the shared covariance estimate.
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inv_Sigma = inv(Sigma);
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figure(1)
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plotgaus(m(:,1)',Sigma,'b');
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plotgaus(m(:,2)',Sigma,'r');
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% Computing & plotting the estimated decision boundary -- this is linear because the class covariances
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% were taken to be equal in the two classes. In case you decide to experiment with each class having its own covariance then
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% remove the following three lines.
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delta_m = m(:,1)-m(:,2); mean_m = (m(:,1)+m(:,2))/2;
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m
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m(:,1)
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m(:,2)
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w = inv_Sigma*delta_m;
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b = -delta_m'*inv_Sigma*mean_m + log(alpha(1)/(1-alpha(1)));
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xmin = -3; xmax = 3;
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hl2 = line([xmin,xmax],[(-b-w(1)*xmin)/w(2),(-b-w(1)*xmax)/w(2)]);
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set(hl2,'color','g','linewidth',2);
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drawnow
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function [h] = plotgaus( mu, sigma, colspec );
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% PLOTGAUS Plotting of a Gaussian contour
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%
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% PLOTGAUS(MU,SIGMA,COLSPEC) plots the mean and standard
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% deviation ellipsis of the Gaussian process that has mean MU
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% variance SIGMA, with color COLSPEC = [R,G,B].
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%
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% If you use only PLOTGAUS(MU,SIGMA), the default color is
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% [0 1 1] (cyan).
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%
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if nargin < 3; colspec = [0 1 1]; end;
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npts = 100;
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stdev = sqrtm(sigma);
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t = linspace(-pi, pi, npts);
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t=t(:);
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X = [cos(t) sin(t)] * stdev + repmat(mu,npts,1);
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h(1) = line(X(:,1),X(:,2),'color',colspec,'linew',2);
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h(2) = line(mu(1),mu(2),'marker','+','markersize',10,'color',colspec,'linew',2);
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@@ -0,0 +1,17 @@
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function main()
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J = load('PB_data.mat');
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% x = load('PB12.mat.mat');
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x = [J.f1(J.phno == 2), J.f2(J.phno == 2)]
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for i=1:10
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[mu, s2, p] = mog(x, 6);
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mu
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s2
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p
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disp('Press a key to continue')
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pause;
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save('MoGVarsK6_p2.mat', 'mu', 's2', 'p')
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end
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% plot(J(:,1),J(:,2),'.')
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@@ -0,0 +1,60 @@
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% Simple script to do EM for a mixture of Gaussians.
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% -------------------------------------------------
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% based on code from Rasmussen and Ghahramani
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% (http://www.gatsby.ucl.ac.uk/~zoubin/course02/)
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% Initialise parameters
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function [mu, s2, p] = mog(x, k)
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[n D] = size(x); % number of observations (n) and dimension (D)
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p = ones(1,k)/k; % mixing proportions
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mu = x(ceil(n.*rand(1,k)),:)'; % means picked randomly from data
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s2 = zeros(D,D,k); % covariance matrices
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niter=100; % number of iterations
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% initialize covariances
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for i=1:k
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s2(:,:,i) = cov(x)./k; % initially set to fraction of data covariance
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end
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set(gcf,'Renderer','zbuffer');
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clear Z;
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try
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% run EM for niter iterations
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for t=1:niter,
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fprintf('t=%d\r',t);
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% Do the E-step:
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for i=1:k
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Z(:,i) = p(i)*det(s2(:,:,i))^(-0.5)*exp(-0.5*sum((x'-repmat(mu(:,i),1,n))'*inv(s2(:,:,i)).*(x'-repmat(mu(:,i),1,n))',2));
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end
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Z = Z./repmat(sum(Z,2),1,k);
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% Do the M-step:
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for i=1:k
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mu(:,i) = (x'*Z(:,i))./sum(Z(:,i));
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% We will fit Gaussians with diagonal covariances:
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s2(:,:,i) = diag((x'-repmat(mu(:,i),1,n)).^2*Z(:,i)./sum(Z(:,i)));
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% To fit general Gaussians use the line:
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% s2(:,:,i) =
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% (x'-repmat(mu(:,i),1,n))*(repmat(Z(:,i),1,D).*(x'-repmat(mu(:,i),1,n))')./sum(Z(:,i));
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p(i) = mean(Z(:,i));
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end
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clf
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hold on
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plot(x(:,1),x(:,2),'.');
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for i=1:k
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plot_gaussian(2*s2(:,:,i),mu(:,i),i,11);
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end
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drawnow;
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end
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catch
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disp('Numerical Error in Loop - Possibly Singular Matrix');
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end;
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@@ -0,0 +1,73 @@
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% Plots a 2D or 3D 1 s.d. frame.
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% 2D: 'n-1' divisions polar-wise,
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% 3D: 'n-1' divisions each azimuthally and polar-wise,
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% for a Gaussian with covariance 'covar' and mean 'mu',
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% 'colour' can be any integer. M.Beal GMLC 26/03/99
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%
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% function plot_gaussian(covar,mu,col,n)
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function [hh] = plot_gaussian(covar,mu,col,n)
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if ~isempty(find(covar-covar == 0))
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if size(mu,1) < size(mu,2), mu = mu'; end
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if size(covar,1) == 3
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theta = (0:1:n-1)'/(n-1)*pi;
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phi = (0:1:n-1)/(n-1)*2*pi;
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sx = sin(theta)*cos(phi);
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sy = sin(theta)*sin(phi);
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sz = cos(theta)*ones(1,n);
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svect = [reshape(sx,1,n*n); reshape(sy,1,n*n); reshape(sz,1,n*n)];
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epoints = sqrtm(covar) * svect + mu*ones(1,n*n);
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ex = reshape(epoints(1,:),n,n);
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ey = reshape(epoints(2,:),n,n);
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ez = reshape(epoints(3,:),n,n);
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colourset = [1 0 0; 0 1 0; 0 0 1; 1 1 0; 1 0 1; 0 1 1];
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colour = colourset(mod(col-1,size(colourset,1))+1,:);
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hh = mesh(ex,ey,ez, reshape(ones(n*n,1)*colour,n,n,3) );
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hidden off
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light
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% for conversion to .eps figures, the 'mesh' command screws up the
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% resolution. Therefore use these 4 lines insteasd of the previous 5.
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% colourset = ['r'; 'g'; 'b'; 'y'; 'm'; 'c'];
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% colour = colourset(mod(col-1,size(colourset,1))+1,:);
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% plot3(epoints(1,:),epoints(2,:),epoints(3,:),colour)
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% plot3(reshape(ex',1,n*n),reshape(ey',1,n*n),reshape(ez',1,n*n),colour)
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else
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theta = (0:1:n-1)/(n-1)*2*pi;
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epoints = sqrtm(covar) * [cos(theta); sin(theta)] + mu*ones(1,n);
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colourset = ['r'; 'g'; 'b'; 'y'; 'm'; 'c'];
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colour = colourset(mod(col-1,size(colourset,1))+1,:);
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hh = plot(epoints(1,:),epoints(2,:),colour,'LineWidth',3);
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plot(mu(1,:),mu(2,:),[colour '.'])
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end
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else
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fprintf('\nVery ill covariance matrix - not plotting this one\n')
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end
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