-
Notifications
You must be signed in to change notification settings - Fork 64
/
spm_DEM_int.m
183 lines (147 loc) · 6.29 KB
/
spm_DEM_int.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
function [V,X,Z,W] = spm_DEM_int(M,z,w,c)
% Integrates/evaluates a hierarchical model given innovations z{i} and w{i}
% FORMAT [V,X,Z,W] = spm_DEM_int(M,z,w,c);
%
% M{i} - model structure
% z{i} - innovations (causes)
% w{i} - innovations (states)
% c{i} - exogenous causes
%
% V{i} - causal states (V{1} = y = response)
% X{i} - hidden states
% Z{i} - fluctuations (causes)
% W{i} - fluctuations (states)
%
% The system is evaluated at the prior expectation of the parameters
%__________________________________________________________________________
% Copyright (C) 2008 Wellcome Trust Centre for Neuroimaging
% Karl Friston
% $Id: spm_DEM_int.m 6132 2014-08-06 19:59:46Z karl $
% set model indices and missing fields
%--------------------------------------------------------------------------
M = spm_DEM_M_set(M);
% innovations
%--------------------------------------------------------------------------
z = spm_cat(z(:)) + spm_cat(c(:));
w = spm_cat(w(:));
% number of states and parameters
%--------------------------------------------------------------------------
nt = size(z,2); % number of time steps
nl = size(M,2); % number of levels
nv = sum(spm_vec(M.l)); % number of v (casual states)
nx = sum(spm_vec(M.n)); % number of x (hidden states)
% order parameters (n= 1 for static models)
%==========================================================================
dt = M(1).E.dt; % time step
n = M(1).E.n + 1; % order of embedding
nD = M(1).E.nD; % number of iterations per sample
td = dt/nD; % integration time for D-Step
% initialize cell arrays for derivatives z{i} = (d/dt)^i[z], ...
%--------------------------------------------------------------------------
u.v = cell(n,1);
u.x = cell(n,1);
u.z = cell(n,1);
u.w = cell(n,1);
[u.v{:}] = deal(sparse(nv,1));
[u.x{:}] = deal(sparse(nx,1));
[u.z{:}] = deal(sparse(nv,1));
[u.w{:}] = deal(sparse(nx,1));
% hyperparameters
%--------------------------------------------------------------------------
ph.h = {M.hE};
ph.g = {M.gE};
% initialize with starting conditions
%--------------------------------------------------------------------------
vi = {M.v};
xi = {M.x};
u.v{1} = spm_vec(vi);
u.x{1} = spm_vec(xi);
% derivatives for Jacobian of D-step
%--------------------------------------------------------------------------
Dx = kron(spm_speye(n,n,1),spm_speye(nx,nx,0));
Dv = kron(spm_speye(n,n,1),spm_speye(nv,nv,0));
D = spm_cat(spm_diag({Dv,Dx,Dv,Dx}));
dfdw = kron(eye(n,n),eye(nx,nx));
% initialize conditional estimators of states to be saved (V and X)
%--------------------------------------------------------------------------
mnx = 0;
mnv = 0;
for i = 1:nl
V{i} = sparse(M(i).l,nt);
X{i} = sparse(M(i).n,nt);
Z{i} = sparse(M(i).l,nt);
W{i} = sparse(M(i).n,nt);
% check for state-dependent precision
%----------------------------------------------------------------------
mnx = mnx | length(M(i).pg);
mnv = mnv | length(M(i).ph);
end
% defaults for state-dependent precision
%--------------------------------------------------------------------------
Sz = 1;
Sw = 1;
% iterate over sequence (t) and within for static models
%==========================================================================
for t = 1:nt
for iD = 1:nD
% Get generalised motion of random fluctuations
%==================================================================
% sampling time
%------------------------------------------------------------------
ts = (t + (iD - 1)/nD)*dt;
% evaluate state-dependent precision
%------------------------------------------------------------------
if mnx || mnv
vi{nl} = vi{nl} + c{nl}(:,t);
pu.x = {spm_vec(xi(1:end - 1))};
pu.v = {spm_vec(vi(1 + 1:end))};
p = spm_LAP_eval(M,pu,ph);
if mnv, Sz = sparse(diag(exp(-p.h/2))); end
if mnx, Sw = sparse(diag(exp(-p.g/2))); end
end
% derivatives of innovations (and exogenous input)
%------------------------------------------------------------------
u.z = spm_DEM_embed(Sz*z,n,ts,dt);
u.w = spm_DEM_embed(Sw*w,n,ts,dt);
% Evaluate and update states
%==================================================================
% evaluate functions
%------------------------------------------------------------------
[u,dg,df] = spm_DEM_diff(M,u);
% tensor products for Jacobian
%------------------------------------------------------------------
dgdv = kron(spm_speye(n,n,1),dg.dv);
dgdx = kron(spm_speye(n,n,1),dg.dx);
dfdv = kron(spm_speye(n,n,0),df.dv);
dfdx = kron(spm_speye(n,n,0),df.dx);
% Save realization
%==================================================================
vi = spm_unvec(u.v{1},{M.v});
xi = spm_unvec(u.x{1},{M.x});
zi = spm_unvec(u.z{1},{M.v});
wi = spm_unvec(u.w{1},{M.x});
if iD == 1
for i = 1:nl
if M(i).l, V{i}(:,t) = spm_vec(vi{i}); end
if M(i).n, X{i}(:,t) = spm_vec(xi{i}); end
if M(i).l, Z{i}(:,t) = spm_vec(zi{i}); end
if M(i).n, W{i}(:,t) = spm_vec(wi{i}); end
end
end
% break for static models
%------------------------------------------------------------------
if nt == 1, break, end
% Jacobian for update
%------------------------------------------------------------------
J = spm_cat({dgdv dgdx Dv [] ;
dfdv dfdx [] dfdw;
[] [] Dv [] ;
[] [] [] Dx});
% update states u = {x,v,z,w}
%------------------------------------------------------------------
du = spm_dx(J,D*spm_vec(u),td);
% and unpack
%------------------------------------------------------------------
u = spm_unvec(spm_vec(u) + du,u);
end % iterations over iD
end % iterations over t