Monday, June 24, 2013

Gaussian Process GP



https://www.youtube.com/watch?v=16oPvgOd3UI

function GP_1d

kernel=5;
switch kernel
    case 1; k = @(x,y) 1*x'*y; % linear
    case 2; k = @(x,y) 1*min(x,y); % brownian motion
    case 3; k = @(x,y) exp(-100*(x-y)'*(x-y)); % squared
    case 4; k = @(x,y) exp(-1*sqrt(x-y)'*(x-y)); % Ornistin   
    case 5; k = @(x,y) exp(-1*sin(5*pi*(x-y))^2); % periodic     
end

% choose points at which to sample
x = (0:.005:1);
n = length(x);


% covariance matrix
C = zeros(n,n);
for i=1:n
   for j=1:n
      C(i,j) = k (x(i), x(j)) ;
   end
end

% sample from gaussian process at this points
u = randn(n,1);
[A, S , B ] = svd(C);
z = A *sqrt(S)*u;


% plot
figure(2); hold on;
plot(x, z, '.-');
axis([0, 1, -2, 2]);



end

============ IN 2D ========

function GP_2d

kernel=3;
switch kernel
    case 1; k = @(x,y) 1*x'*y; % linear
    case 2; k = @(x,y) 1*min(x,y); % brownian motion
    case 3; k = @(x,y) exp(-100*(x-y)'*(x-y)); % squared
    case 4; k = @(x,y) exp(-1*sqrt(x-y)'*(x-y)); % Ornistin   
    case 5; k = @(x,y) exp(-1*sin(5*pi*(x-y))^2 ); % periodic     
end

% choose points at which to sample
points = (0:0.05:1)';
[U,V] = meshgrid(points,points);
x = [U(:) V(:)]';
n = size(x,2);


% covariance matrix
C = zeros(n,n);
for i=1:n 
   for j=1:n
      C(i,j) = k (x(:,i), x(:,j)) ;
   end
end

% sample from gaussian process at this points
u = randn(n,1);
[A, S , B ] = svd(C);
z = A*sqrt(S)*u;


% plot
figure(2); clf;
Z = reshape(z,sqrt(n) , sqrt(n));
surf(U,V,Z);


end