I am currently using the GPML toolbox to perform regression.
Generally, after learning the hyperparameters we can extract the posterior mean and variance by using the function in the toolbox as
[m s2] = gp(hyp2, @infExact, [], covfunc, likfunc, x, y, z); Here, we can say that m is the Posterior mean and s2 is the posterior variance.
But the code only gives out the self-variance or variance at the target points and not the full covariance matrix.
$s2 = diagonal(k(x_{target},x_{target}'))$
A vector with only the diagonal terms.
How can I get the full covariance matrix between all the target points i.e. $k(x_{target},x_{target}')$?
I tried writing something like:
kTT = feval(covFunc{:}, hyp2.cov, xTarget); kTI = feval(covFunc{:}, hyp2.cov, xTarget,x); kII = feval(covFunc{:}, hyp2.cov, x); sigma = exp(2*hypProd.lik)*eye(max(size(x))); % Posterior covariance matrix kPost = kTT - kTI/(kII+sigma)*kTI'; But the diagonal variance values in the kPost matrix don't match those with the s2 matrix.
What am I doing wrong?
Best Answer
As you already answered yourself, the m and s2 output variables are just the means and variances of prediction at the given points z.
However, I'd like to add that you can also get a posterior structure by calling the function as
[predmeans, predvars, latentmeans, latentvars, [], poststr] = gp(hyp, inffun, meanfun, covfun, likfun, xtrain, ytrain, xtest) In this case, poststr is going to be a structure consisting of three fields: alpha, sW and L which are supposed to be used to construct the posterior means and variances using the formula given in the GPML toolbox for Matlab manual, page 4.
Also of note: this posterior structure can also be returned when training (instead of predicting, as in your example as well as the one I wrote out above), using a function call like
[negloglik, derivnegloglik, poststr] = gp(hyp, inffun, meanfun, covfun, likfun, xtrain, ytrain)