% [d,w,rx,cv,wx] = best_dir( x, [a , sx ] ) % % Some points x, are observed and one assumes that they belong to % parallel planes. There is an unknown direction d s.t. for each % point x(i,:), one has : % % x(i,:)*d == w(j(i)) + noise % % where j is known(given by the matrix a ), but w is unknown. % % Under the assumption that the error on x are i.i.d. gaussian, % best_dir returns the maximum likelihood estimate of d and w. % % This function is slower when cv is returned. % % INPUT : % ------- % x : D x P P points. Each one is the sum of a point that belongs % to a plane and a noise term. % % a : P x W 0-1 matrix describing association of points (rows of % x) to planes : % % a(p,i) == 1 iff point x(p,:) belongs to the i'th plane. % % Default is ones(P,1) % % sx : P x 1 Covariance of x(i,:) is sx(i)*eye(D). % Default is ones(P,1) % OUTPUT : % -------- % d : D x 1 All the planes have the same normal, d. d has unit % norm. % % w : W x 1 The i'th plane is { y | y*d = w(i) }. % % rx : P x 1 Residuals of projection of points to corresponding plane. % % % Assuming that the covariance of x (i.e. sx) was known % only up to a scale factor, an estimate of the % covariance of x and [w;d] are % % sx * mean(rx.^2)/mean(sx) and % cv * mean(rx.^2)/mean(sx), respectively. % % cv : (D+W)x(D+W) % Covariance of the estimator at [d,w] ( assuming that % diag(covariance(vec(x))) == sx ). % % wx : (D+W)x(D*P) % Derivatives of [w;d] wrt to x. % % Author : Etienne Grossmann <etienne@isr.ist.utl.pt> % Created : March 2000 %

- best_dir % [d,w,rx,cv,wx] = best_dir( x, [a , sx ] )
- best_dir_cov % [cv,wx] = best_dir_cov(x,a,sx,wd)

- best_dir % [d,w,rx,cv,wx] = best_dir( x, [a , sx ] )

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