/
spm_DEM_eval.m
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spm_DEM_eval.m
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function [E,dE,f,g] = spm_DEM_eval(M,qu,qp)
% evaluates state equations and derivatives for DEM schemes
% FORMAT [E dE f g] = spm_DEM_eval(M,qu,qp)
%
% M - model structure
% qu - conditional mode of states
% qu.v{i} - casual states
% qu.x(i) - hidden states
% qu.y(i) - response
% qu.u(i) - input
% qp - conditional density of parameters
% qp.p{i} - parameter deviates for i-th level
% qp.u(i) - basis set
% qp.x(i) - expansion point ( = prior expectation)
%
% E - generalised errors (i.e.., y - g(x,v,P); x[1] - f(x,v,P))
%
% dE:
% dE.du - de[1:n]/du
% dE.dy - de[1:n]/dy[1:n]
% dE.dc - de[1:n]/dc[1:d]
% dE.dp - de[1:n]/dp
% dE.dup - d/dp[de[1:n]/du
% dE.dpu - d/du[de[1:n]/dp
%
% where u = x{1:d]; v[1:d]
%__________________________________________________________________________
% Copyright (C) 2008 Wellcome Trust Centre for Neuroimaging
% Karl Friston
% $Id: spm_DEM_eval.m 3695 2010-01-22 14:18:14Z karl $
% get dimensions
%==========================================================================
nl = size(M,2); % number of levels
ne = sum(spm_vec(M.l)); % number of e (errors)
nv = sum(spm_vec(M.m)); % number of x (causal states)
nx = sum(spm_vec(M.n)); % number of x (hidden states)
np = sum(spm_vec(M.p)); % number of p (parameters)
ny = M(1).l; % number of y (inputs)
nc = M(end).l; % number of c (prior causes)
% evaluate functions at each hierarchical level
%==========================================================================
% Get states {qu.v{1},qu.x{1}} in hierarchical form (v{i},x{i})
%--------------------------------------------------------------------------
v = spm_unvec(qu.v{1},{M(1 + 1:end).v});
x = spm_unvec(qu.x{1},{M(1:end - 1).x});
for i = 1:(nl - 1)
p = spm_unvec(spm_vec(M(i).pE) + qp.u{i}*qp.p{i},M(i).pE);
f{i,1} = feval(M(i).f,x{i},v{i},p);
g{i,1} = feval(M(i).g,x{i},v{i},p);
end
% Get Derivatives
%==========================================================================
persistent D
try
method = M(1).E.linear;
catch
method = 0;
end
switch method
% get derivatives at each iteration of D-step - full evaluation
%----------------------------------------------------------------------
case{0}
D = spm_DEM_eval_diff(x,v,qp,M);
% gradients w.r.t. states
%------------------------------------------------------------------
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdv = D.dfdv;
dfdx = D.dfdx;
dgdxp = D.dgdxp;
dfdxp = D.dfdxp;
dgdvp = D.dgdvp;
dfdvp = D.dfdvp;
% gradients w.r.t. parameters
%------------------------------------------------------------------
dgdp = D.dgdp;
dfdp = D.dfdp;
% linear: assume equations are linear in x and v
%----------------------------------------------------------------------
case{1}
% get derivatives and store expansion point (states)
%------------------------------------------------------------------
if isempty(D)
D = spm_DEM_eval_diff(x,v,qp,M);
D.x = x;
D.v = v;
% gradients w.r.t. states
%--------------------------------------------------------------
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdv = D.dfdv;
dfdx = D.dfdx;
% gradients w.r.t. parameters (state-dependent)
%--------------------------------------------------------------
dgdxp = D.dgdxp;
dfdxp = D.dfdxp;
dgdvp = D.dgdvp;
dfdvp = D.dfdvp;
% gradients w.r.t. parameters
%--------------------------------------------------------------
dgdp = D.dgdp;
dfdp = D.dfdp;
% linear expansion for derivatives w.r.t. parameters
%------------------------------------------------------------------
else
% gradients w.r.t. states
%--------------------------------------------------------------
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdv = D.dfdv;
dfdx = D.dfdx;
dgdxp = D.dgdxp;
dfdxp = D.dfdxp;
dgdvp = D.dgdvp;
dfdvp = D.dfdvp;
% gradients w.r.t. parameters
%--------------------------------------------------------------
dx = spm_vec(qu.x{1}) - spm_vec(D.x);
dv = spm_vec(qu.v{1}) - spm_vec(D.v);
dgdp = D.dgdp;
dfdp = D.dfdp;
for p = 1:np
dgdp(:,p) = D.dgdp(:,p) + D.dgdxp{p}*dx + D.dgdvp{p}*dv;
if nx
dfdp(:,p) = D.dfdp(:,p) + D.dfdxp{p}*dx + D.dfdvp{p}*dv;
end
end
end
% bilinear: assume equations are linear in x and v and x*v
%----------------------------------------------------------------------
case{2}
% get derivatives and store expansion point (states)
%------------------------------------------------------------------
if isempty(D)
% get high-order derivatives
%--------------------------------------------------------------
[Dv D] = spm_diff('spm_DEM_eval_diff',x,v,qp,M,2);
for i = 1:nv, Dv{i} = spm_unvec(Dv{i},D); end
D.x = x;
D.v = v;
D.Dv = Dv;
% gradients w.r.t. states
%--------------------------------------------------------------
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdv = D.dfdv;
dfdx = D.dfdx;
dgdxp = D.dgdxp;
dfdxp = D.dfdxp;
dgdvp = D.dgdvp;
dfdvp = D.dfdvp;
% gradients w.r.t. parameters
%--------------------------------------------------------------
dgdp = D.dgdp;
dfdp = D.dfdp;
% linear expansion for derivatives w.r.t. parameters
%------------------------------------------------------------------
else
% gradients w.r.t. causes and data
%--------------------------------------------------------------
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
% states (relative to expansion point)
%--------------------------------------------------------------
dv = spm_vec(qu.v{1}) - spm_vec(D.v);
% gradients w.r.t. states
%--------------------------------------------------------------
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdx = D.dfdx;
dfdv = D.dfdv;
for i = 1:nv; dgdx = dgdx + D.Dv{i}.dgdx*dv(i); end
for i = 1:nv; dgdv = dgdv + D.Dv{i}.dgdv*dv(i); end
for i = 1:nv; dfdx = dfdx + D.Dv{i}.dfdx*dv(i); end
for i = 1:nv; dfdv = dfdv + D.Dv{i}.dfdv*dv(i); end
% second-order derivatives
%--------------------------------------------------------------
dgdxp = D.dgdxp;
dgdvp = D.dgdvp;
dfdxp = D.dfdxp;
dfdvp = D.dfdvp;
for p = 1:np
for i = 1:nv; dgdxp{p} = dgdxp{p} + D.Dv{i}.dgdxp{p}*dv(i); end
for i = 1:nv; dgdvp{p} = dgdvp{p} + D.Dv{i}.dgdvp{p}*dv(i); end
for i = 1:nv; dfdxp{p} = dfdxp{p} + D.Dv{i}.dfdxp{p}*dv(i); end
for i = 1:nv; dfdvp{p} = dfdvp{p} + D.Dv{i}.dfdvp{p}*dv(i); end
end
% gradients w.r.t. parameters
%--------------------------------------------------------------
dgdp = D.dgdp;
dfdp = D.dfdp;
for p = 1:np
Dgdxp = (D.dgdxp{p} + dgdxp{p})/2;
Dgdvp = (D.dgdvp{p} + dgdvp{p})/2;
Dfdxp = (D.dfdxp{p} + dfdxp{p})/2;
Dfdvp = (D.dfdvp{p} + dfdvp{p})/2;
dgdp(:,p) = dgdp(:,p) + Dgdvp*dv;
dfdp(:,p) = dfdp(:,p) + Dfdvp*dv;
end
end
% nonlinear: assume equations are bilinear in x and v
%----------------------------------------------------------------------
case{3}
% get derivatives and store expansion point (states)
%------------------------------------------------------------------
if isempty(D)
% get high-order derivatives
%--------------------------------------------------------------
[Dx D] = spm_diff('spm_DEM_eval_diff',x,v,qp,M,1);
[Dv D] = spm_diff('spm_DEM_eval_diff',x,v,qp,M,2);
for i = 1:nx, Dx{i} = spm_unvec(Dx{i},D); end
for i = 1:nv, Dv{i} = spm_unvec(Dv{i},D); end
D.x = x;
D.v = v;
D.Dx = Dx;
D.Dv = Dv;
% gradients w.r.t. states
%--------------------------------------------------------------
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdv = D.dfdv;
dfdx = D.dfdx;
dgdxp = D.dgdxp;
dfdxp = D.dfdxp;
dgdvp = D.dgdvp;
dfdvp = D.dfdvp;
% gradients w.r.t. parameters
%--------------------------------------------------------------
dgdp = D.dgdp;
dfdp = D.dfdp;
% linear expansion for derivatives w.r.t. parameters
%------------------------------------------------------------------
else
% gradients w.r.t. causes and data
%--------------------------------------------------------------
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
% states (relative to expansion point)
%--------------------------------------------------------------
dx = spm_vec(qu.x{1}) - spm_vec(D.x);
dv = spm_vec(qu.v{1}) - spm_vec(D.v);
% gradients w.r.t. states
%--------------------------------------------------------------
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdx = D.dfdx;
dfdv = D.dfdv;
for i = 1:nx; dgdx = dgdx + D.Dx{i}.dgdx*dx(i); end
for i = 1:nv; dgdx = dgdx + D.Dv{i}.dgdx*dv(i); end
for i = 1:nx; dgdv = dgdv + D.Dx{i}.dgdv*dx(i); end
for i = 1:nv; dgdv = dgdv + D.Dv{i}.dgdv*dv(i); end
for i = 1:nx; dfdx = dfdx + D.Dx{i}.dfdx*dx(i); end
for i = 1:nv; dfdx = dfdx + D.Dv{i}.dfdx*dv(i); end
for i = 1:nx; dfdv = dfdv + D.Dx{i}.dfdv*dx(i); end
for i = 1:nv; dfdv = dfdv + D.Dv{i}.dfdv*dv(i); end
% second-order derivatives
%--------------------------------------------------------------
dgdxp = D.dgdxp;
dgdvp = D.dgdvp;
dfdxp = D.dfdxp;
dfdvp = D.dfdvp;
for p = 1:np
for i = 1:nx; dgdxp{p} = dgdxp{p} + D.Dx{i}.dgdxp{p}*dx(i); end
for i = 1:nv; dgdxp{p} = dgdxp{p} + D.Dv{i}.dgdxp{p}*dv(i); end
for i = 1:nx; dgdvp{p} = dgdvp{p} + D.Dx{i}.dgdvp{p}*dx(i); end
for i = 1:nv; dgdvp{p} = dgdvp{p} + D.Dv{i}.dgdvp{p}*dv(i); end
for i = 1:nx; dfdxp{p} = dfdxp{p} + D.Dx{i}.dfdxp{p}*dx(i); end
for i = 1:nv; dfdxp{p} = dfdxp{p} + D.Dv{i}.dfdxp{p}*dv(i); end
for i = 1:nx; dfdvp{p} = dfdvp{p} + D.Dx{i}.dfdvp{p}*dx(i); end
for i = 1:nv; dfdvp{p} = dfdvp{p} + D.Dv{i}.dfdvp{p}*dv(i); end
end
% gradients w.r.t. parameters
%--------------------------------------------------------------
dgdp = D.dgdp;
dfdp = D.dfdp;
for p = 1:np
Dgdxp = (D.dgdxp{p} + dgdxp{p})/2;
Dgdvp = (D.dgdvp{p} + dgdvp{p})/2;
Dfdxp = (D.dfdxp{p} + dfdxp{p})/2;
Dfdvp = (D.dfdvp{p} + dfdvp{p})/2;
dgdp(:,p) = dgdp(:,p) + Dgdxp*dx + Dgdvp*dv;
dfdp(:,p) = dfdp(:,p) + Dfdxp*dx + Dfdvp*dv;
end
end
% repeated evaluation of first order derivatives (for Laplace scheme)
%----------------------------------------------------------------------
case{4}
% get derivatives and store expansion point (states)
%------------------------------------------------------------------
if isempty(D)
D = spm_DEM_eval_diff(x,v,qp,M);
D.x = x;
D.v = v;
% gradients w.r.t. states
%--------------------------------------------------------------
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdv = D.dfdv;
dfdx = D.dfdx;
% gradients w.r.t. parameters (state-dependent)
%--------------------------------------------------------------
dgdxp = D.dgdxp;
dfdxp = D.dfdxp;
dgdvp = D.dgdvp;
dfdvp = D.dfdvp;
% gradients w.r.t. parameters
%--------------------------------------------------------------
dgdp = D.dgdp;
dfdp = D.dfdp;
% re-evaluate first-order derivatives
%------------------------------------------------------------------
else
% retain second-order gradients
%--------------------------------------------------------------
dgdxp = D.dgdxp;
dfdxp = D.dfdxp;
dgdvp = D.dgdvp;
dfdvp = D.dfdvp;
% re-evaluate first-order gradients
%--------------------------------------------------------------
D = spm_DEM_eval_diff(x,v,qp,M,0);
dedy = D.dedy;
dedc = D.dedc;
dfdy = D.dfdy;
dfdc = D.dfdc;
dgdx = D.dgdx;
dgdv = D.dgdv;
dfdv = D.dfdv;
dfdx = D.dfdx;
% replace second-order gradients
%--------------------------------------------------------------
D.dgdxp = dgdxp;
D.dfdxp = dfdxp;
D.dgdvp = dgdvp;
D.dfdvp = dfdvp;
% gradients w.r.t. parameters
%--------------------------------------------------------------
dx = spm_vec(qu.x{1}) - spm_vec(x);
dv = spm_vec(qu.v{1}) - spm_vec(v);
dgdp = D.dgdp;
dfdp = D.dfdp;
for p = 1:np
dgdp(:,p) = D.dgdp(:,p) + D.dgdxp{p}*dx + D.dgdvp{p}*dv;
if nx
dfdp(:,p) = D.dfdp(:,p) + D.dfdxp{p}*dx + D.dfdvp{p}*dv;
end
end
end
otherwise
disp('Unknown method')
end
% order parameters (d = n = 1 for static models)
%--------------------------------------------------------------------------
d = M(1).E.d + 1; % generalisation order of q(v)
n = M(1).E.n + 1; % embedding order (n >= d)
% Generalised prediction errors and derivatives
%==========================================================================
Ex = cell(n,1);
Ev = cell(n,1);
[Ex{:}] = deal(sparse(nx,1));
[Ev{:}] = deal(sparse(ne,1));
% prediction error (E) - causes
%--------------------------------------------------------------------------
for i = 1:n
qu.y{i} = spm_vec(qu.y{i});
end
Ev{1} = [qu.y{1}; qu.v{1}] - [spm_vec(g); qu.u{1}];
for i = 2:n
Ev{i} = dedy*qu.y{i} + dedc*qu.u{i} ... % generalised response
- dgdx*qu.x{i} - dgdv*qu.v{i}; % and prediction
end
% prediction error (E) - states
%--------------------------------------------------------------------------
try
Ex{1} = qu.x{2} - spm_vec(f);
end
for i = 2:n - 1
Ex{i} = qu.x{i + 1} ... % generalised motion
- dfdx*qu.x{i} - dfdv*qu.v{i}; % and prediction
end
% error
%--------------------------------------------------------------------------
E = spm_vec({Ev,Ex});
% Kronecker forms of derivatives for generalised motion
%==========================================================================
if nargout < 2, return, end
% dE.dp (parameters)
%--------------------------------------------------------------------------
dgdp = {dgdp};
dfdp = {dfdp};
for i = 2:n
dgdp{i,1} = dgdp{1};
dfdp{i,1} = dfdp{1};
for p = 1:np
dgdp{i,1}(:,p) = dgdxp{p}*qu.x{i} + dgdvp{p}*qu.v{i};
dfdp{i,1}(:,p) = dfdxp{p}*qu.x{i} + dfdvp{p}*qu.v{i};
end
end
% generalised temporal derivatives: dE.du (states)
%--------------------------------------------------------------------------
dedy = kron(spm_speye(n,n),dedy);
dedc = kron(spm_speye(n,d),dedc);
dfdy = kron(spm_speye(n,n),dfdy);
dfdc = kron(spm_speye(n,d),dfdc);
dgdx = kron(spm_speye(n,n),dgdx);
dgdv = kron(spm_speye(n,d),dgdv);
dfdv = kron(spm_speye(n,d),dfdv);
dfdx = kron(spm_speye(n,n),dfdx) - kron(spm_speye(n,n,1),speye(nx,nx));
% 1st error derivatives (states)
%--------------------------------------------------------------------------
dE.dy = spm_cat({dedy; dfdy});
dE.dc = spm_cat({dedc; dfdc});
dE.dp = -spm_cat({dgdp; dfdp});
dE.du = -spm_cat({dgdx, dgdv ;
dfdx, dfdv});
% bilinear derivatives
%--------------------------------------------------------------------------
for i = 1:np
dgdxp{i} = kron(spm_speye(n,n),dgdxp{i});
dfdxp{i} = kron(spm_speye(n,n),dfdxp{i});
dgdvp{i} = kron(spm_speye(n,d),dgdvp{i});
dfdvp{i} = kron(spm_speye(n,d),dfdvp{i});
dE.dup{i} = -spm_cat({dgdxp{i}, dgdvp{i};
dfdxp{i}, dfdvp{i}});
end
if np
dE.dpu = spm_cell_swap(dE.dup);
else
dE.dpu = {};
end