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function g = mdngrad(net, x, t)
%MDNGRAD Evaluate gradient of error function for Mixture Density Network.
%
% Description
% G = MDNGRAD(NET, X, T) takes a mixture density network data
% structure NET, a matrix X of input vectors and a matrix T of target
% vectors, and evaluates the gradient G of the error function with
% respect to the network weights. The error function is negative log
% likelihood of the target data. Each row of X corresponds to one
% input vector and each row of T corresponds to one target vector.
%
% See also
% MDN, MDNFWD, MDNERR, MDNPROB, MLPBKP
%
% Copyright (c) Ian T Nabney (1996-2001)
% David J Evans (1998)
% Check arguments for consistency
errstring = consist(net, 'mdn', x, t);
if ~isempty(errstring)
error(errstring);
end
[mixparams, y, z] = mdnfwd(net, x);
% Compute gradients at MLP outputs: put the answer in deltas
ncentres = net.mdnmixes.ncentres;
dim_target = net.mdnmixes.dim_target;
nmixparams = net.mdnmixes.nparams;
ntarget = size(t, 1);
deltas = zeros(ntarget, net.mlp.nout);
e = ones(ncentres, 1);
f = ones(1, dim_target);
post = mdnpost(mixparams, t);
% Calculate prior derivatives
deltas(:,1:ncentres) = mixparams.mixcoeffs - post;
% Calculate centre derivatives
long_t = kron(ones(1, ncentres), t);
centre_err = mixparams.centres - long_t;
% Get the post to match each u_jk:
% this array will be (ntarget, (ncentres*dim_target))
long_post = kron(ones(dim_target, 1), post);
long_post = reshape(long_post, ntarget, (ncentres*dim_target));
% Get the variance to match each u_jk:
var = mixparams.covars;
var = kron(ones(dim_target, 1), var);
var = reshape(var, ntarget, (ncentres*dim_target));
% Compute centre deltas
deltas(:, (ncentres+1):(ncentres*(1+dim_target))) = ...
(centre_err.*long_post)./var;
% Compute variance deltas
dist2 = mdndist2(mixparams, t);
c = dim_target*ones(ntarget, ncentres);
deltas(:, (ncentres*(1+dim_target)+1):nmixparams) = ...
post.*((dist2./mixparams.covars)-c)./(-2);
% Now back-propagate deltas through MLP
g = mlpbkp(net.mlp, x, z, deltas);
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