% regression demo, Barnett 3/6/06, requires stats toolbox

% create `fake' data, using noisy linear model...
n = 10;
x = rand(1,n);
ao = 1.5;  bo = 0.7; so = 0.3;
y = ao*x + bo + normrnd(0, so, size(x));
figure; plot(x, y, '+', 'markersize', 10); xlabel x; ylabel y; title 'data'


% likelihood plot for linear model Y = aX + b
a = -1:0.1:3; b = -1:0.05:2;
[aa,bb] = meshgrid(a, b);
L = ones(size(aa));
s = so;                % assume noise sigma known
for i=1:n
  %L = L .* exp(-(y(i) - aa*x(i) - bb).^2/(2*s^2)) / s;   % normal noise model
  L = L .* exp(-abs(y(i) - aa*x(i) - bb)/s) / s;       % or... exp noise model
end
figure; contourf(a, b, L); xlabel a; ylabel b;
title('L(a,b), noise \sigma known');
where = find(L==max(L(:)));             % locate the max in 2-param space
aml = aa(where); bml = bb(where);
figure; plot(x, y, '+', 'markersize', 10); xlabel x; ylabel y; title 'data'
xs = [0 1]; hold on; plot(xs, aml*xs + bml, '-');

% MCMC for a,b,s
opt.nits = 2000;             % how many iterations.
opt.e = 0.5;
opt.hess = eye(3);
opt.verbosity = 1;
X_in = ones(3, 1);
% normal noise model, or...
[X_chain, P_chain, info] = mcmc_run('nlp_linear_gauss', X_in, opt, x, y);
% exp noise model...
%[X_chain, P_chain, info] = mcmc_run('nlp_linear_exp', X_in, opt, x, y);
figure; plot(X_chain', '-');               % show the Markov chain.

% samples in parameter space (marginals onto 2 out of 3 dimensions)
ns = 400:opt.nits;
figure; plot(X_chain(1,ns), X_chain(2,ns), '.'); xlabel a; ylabel b;
figure; plot(X_chain(2,ns), X_chain(3,ns), '.'); xlabel b; ylabel('\sigma');

% samples of model fits (straight lines)...
ns = 400:50:opt.nits; nss = numel(ns); as = X_chain(1,ns)';bs = X_chain(2,ns)';
figure; plot(repmat(xs, [nss 1])', repmat(as, [1 2])' + kron(bs, xs)', '-');
hold on; plot(x, y, '+', 'markersize', 10); xlabel x; ylabel y;


% make one point an outlier, to test robustness (normal noise fails, exp not)
y(find(x==min(x))) = 5.0;
figure; plot(x, y, '+', 'markersize', 10); xlabel x; ylabel y; title 'data'