function [a,fval,P] = nolin_a(M,y,hyp)

% NOLIN_A       Calculates the hyperparameter value of 'a', exact final
% function evaluation value of the log-likelihood function and the
% normalised covariance matrix.
%
% [A,FVAL,P] = NOLIN_A(M,y,hyp)
%
% This function calculates the hyper-paramter value of 'a' from the
% optimised lengthscales {g1,g2,...,gd} and noise {v} hyperparameters. In
% addition, it also returns the function evaluation of the negative 
% log-likelihood function of the Gaussian process prior models, given by,
%
%                   fval = log(det(Q)) + y'*inv(Q)*y
%
% and the normalised covariance matrix, P, where Q = a*P, with the prior
% covariance function (of Q) is of the form,
%
%                   C(zi,zj) = a*{exp([zi-zj]'*D*[zi-zj]) + v}
%
% Inputs:
%       M   = Input explanatory variable, of dimension {N0 x D0}
%       y   = Outcome (vector) from every input of the explantory variable
%       hyp = Hyperparameters, [a g1 g2 ... gd v], D=diag{g1,g2,...,gd}
%
% Outputs:
%       A       = Hyperparameter value of "a"
%       FVAL    = Function value evaluation
%       P       = Covariance matrix
%
%
% (C) Gaussian Process Toolbox 1.0
% (C) Copyright 2005-2007, Keith Neo <keith.neo@nuim.ie>
%                          http://www.hamilton.ie

[N0,L0]=size(M);
g=hyp(1:L0);
v=hyp(L0+1);
expT=repmat(permute(M,[1,3,2]),[1,N0,1])-repmat(permute(M',[3,2,1]),[N0,1,1]);
P=diag_add(exp(-0.5*sum((expT.^2).*repmat(permute(g,[3 2 1]),[N0,N0,1]),3)),v,N0);
a=y'*(P\y)/N0;
clear expT

[cholC,pp]=chol(P);
if pp==0, logdet=2*sum(log(diag(cholC)));
else logdet=sum(log(svd(P)));
end
fval=N0+N0*log(a)+logdet;

function Q=diag_add(Q,v,N0)
for pp=1:N0
    Q(pp,pp)=Q(pp,pp)+v;
end