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spm_Markov_blanket.m
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spm_Markov_blanket.m
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function [x,u,y] = spm_Markov_blanket(J,z,m,mj)
% FORMAT [x,u,y] = spm_Markov_blanket(J,z,m,mj)
% Markovian partition
% J - Jacobian
% z - {1 x N} partition of states (indices)
% m - number of internal states [default: 3]
%
% x - {3 x n} particular partition of state indices
% x{1,j} - active states of j-th partition
% x{2,j} - sensory states of j-th partition
% x{3,j} - internal states of j-th partition
%
% u - location of partitions in scaling or embedding space
%
% y - {3 x n} particular partition of partition indices
% y{1,j} - active states of j-th partition
% y{2,j} - sensory states of j-th partition
% y{3,j} - internal states of j-th partition
%
% Partition or Grouping (coarse-scaling) operator
%__________________________________________________________________________
% Copyright (C) 2008 Wellcome Trust Centre for Neuroimaging
% Karl Friston
% $Id: spm_Markov_blanket.m 7655 2019-08-25 20:10:20Z karl $
% preliminaries
%--------------------------------------------------------------------------
GRAPHICS = 1; % Graphics switch
nz = length(z); % number of partitions
if nargin < 3
m = 3; % maximum size of internal states
end
if nargin < 4
mj = ones(nz,1); % eligible internal states
end
if isempty(mj)
mj = ones(nz,1); % eligible internal states
end
% Adjacency matrix (over z)
%--------------------------------------------------------------------------
for i = 1:nz
for j = 1:nz
Lij = J(z{i},z{j});
if any(any(Lij))
L(i,j) = abs(norm(full(Lij)) > 1/128);
else
L(i,j) = 0;
end
end
end
L = double(L);
% get Markov blanket
%--------------------------------------------------------------------------
B = L + L' + L'*L;
B = B - diag(diag(B));
% scaling space (defined by graph Laplacian)
%--------------------------------------------------------------------------
G = L + L';
G = G - diag(diag(G));
G = G - diag(sum(G));
G = expm(G);
% get principal dimensions of scaling space (u)
%--------------------------------------------------------------------------
if GRAPHICS
[u,v] = eig(G,'nobalance');
v = abs(diag(v));
for i = 1:nz
[p,h] = hist(real(u(:,i)),16);
dh = h(2) - h(1) + exp(-16);
p = p(:)/sum(p)/dh;
v(i) = log(v(i)) - p'*log(p + exp(-16))*dh;
end
[v,j] = sort(real(v),'descend');
u = real(u(:,j));
end
% recursive (particular) partition into internal, sensory and active states
%--------------------------------------------------------------------------
nn = zeros(nz,1);
for i = 1:nz
% internal states (defined by graph Laplacian)
%----------------------------------------------------------------------
jj = ~(B*nn) & ~nn & mj;
if any(jj)
% find densely coupled internal states (using the graph Laplacian)
%------------------------------------------------------------------
[g,j] = max(diag(G).*jj);
if m > 1
g = G(:,j);
g(j) = 0;
g(~jj) = 0;
[g,k] = sort(g,'descend');
try
j = [j; k(1:m - 1)];
end
end
jj = sparse(j,1,1,size(L,1),1) & jj; % internal states
bb = B*jj & ~jj & ~nn; % Markov blanket
ee = ~bb & ~jj & ~nn; % external states
b = find(bb);
e = find(ee);
s = b(find( any(L(b,e),2)));
a = b(find(~any(L(b,e),2)));
% indices of individual states in the i-th particle
%------------------------------------------------------------------
x{1,i} = spm_cat(z(a));
x{2,i} = spm_cat(z(s));
x{3,i} = spm_cat(z(j));
% states accounted for (nn)
%------------------------------------------------------------------
nn = nn | bb | jj;
else
% no internal states - find active states (not influenced by e)
%------------------------------------------------------------------
j = ~any(L(~nn,nn),2);
if any(j)
% sensory states connected with active states
%--------------------------------------------------------------
a = find(~nn);
a = a(find(j,1));
aa = sparse(a,1,1,size(L,1),1);
ss = (L*aa | L'*aa) & ~aa & ~nn;
a = find(aa);
s = find(ss);
j = [];
% indices of individual states in the i-th particle
%--------------------------------------------------------------
x{1,i} = spm_cat(z(a));
x{2,i} = spm_cat(z(s));
x{3,i} = [];
% states accounted for (nn)
%--------------------------------------------------------------
nn = nn | aa | ss;
elseif any(~nn)
% sensory states connected with sensory states
%--------------------------------------------------------------
s = find(~nn);
ss = sparse(s(1),1,1,nz,1);
ss = ss | B*ss & ~nn;
s = find(ss);
a = [];
j = [];
% indices of individual states in the i-th particle
%--------------------------------------------------------------
x{1,i} = [];
x{2,i} = spm_cat(z(s));
x{3,i} = [];
% states accounted for (nn)
%--------------------------------------------------------------
nn = nn | ss;
end
end
% indices of partitions (i.e., n-states) in the i-th particle
%----------------------------------------------------------------------
y{1,i} = a;
y{2,i} = s;
y{3,i} = j;
% plot
%----------------------------------------------------------------------
if all(nn) && numel(u) > 1
% remove isolated (internal) states
%--------------------------------------------------------------
j = [];
for n = 1:size(x,2)
if any(x{1,n}) || any(x{2,n})
j = [j,n];
end
end
x = x(:,j);
y = y(:,j);
if GRAPHICS,clf
% colours for different particles
%--------------------------------------------------------------
nx = size(x,2);
[col,bol,msz] = spm_MB_col(nx);
% plot partitions in embedding space (which particle)
%--------------------------------------------------------------
subplot(3,2,3)
for k = 1:nx
plot(u(y{1,k},1),u(y{1,k},2),'.','color',bol{k},'MarkerSize',msz), hold on
plot(u(y{2,k},1),u(y{2,k},2),'.','color',bol{k},'MarkerSize',msz), hold on
plot(u(y{3,k},1),u(y{3,k},2),'.','color',col{k},'MarkerSize',msz), hold on
end
axis square
title(sprintf('Particles [%i n-states]',nz),'Fontsize',16)
% plot particles in embedding space (which sort of state)
%--------------------------------------------------------------
subplot(3,2,4)
for k = 1:nx
plot(u(y{1,k},1),u(y{1,k},2),'.r','MarkerSize',msz), hold on
plot(u(y{2,k},1),u(y{2,k},2),'.m','MarkerSize',msz), hold on
plot(u(y{3,k},1),u(y{3,k},2),'.b','MarkerSize',msz), hold on
end
axis square
title(sprintf('Markov partition [%i particles]',nx),'Fontsize',16)
% plot particles in three embedding dimensions
%--------------------------------------------------------------
subplot(3,2,2)
try
for k = 1:nx
plot3(u(y{1,k},1),u(y{1,k},2),u(y{1,k},3),'.r','MarkerSize',msz), hold on
plot3(u(y{2,k},1),u(y{2,k},2),u(y{2,k},3),'.m','MarkerSize',msz), hold on
plot3(u(y{3,k},1),u(y{3,k},2),u(y{3,k},3),'.b','MarkerSize',msz), hold on
end
catch
for k = 1:nx
plot(u(y{1,k},1),u(y{1,k},2),'.r','MarkerSize',msz), hold on
plot(u(y{2,k},1),u(y{2,k},2),'.m','MarkerSize',msz), hold on
plot(u(y{3,k},1),u(y{3,k},2),'.b','MarkerSize',msz), hold on
end
end
axis square
title('Embedding space','Fontsize',16)
rotate3d(gca,'on')
% Jacobian (ordered by partition and type)
%--------------------------------------------------------------
j = spm_vec(x');
k = spm_vec(x );
subplot(3,2,5),imagesc(-log(abs(J(k,k)) + exp(-4))),axis square
subplot(3,2,6),imagesc(-log(abs(J(j,j)) + exp(-4))),axis square
% Colors
%--------------------------------------------------------------
nj = spm_length(x);
msz = fix(16 + 128/nj);
j = 1:nj;
k = spm_unvec(j,x')';
j = spm_unvec(j,x);
subplot(3,2,5),hold on
for q = 1:nx
plot(j{1,q},ones(size(x{1,q})),'.','color',bol{q}, 'MarkerSize',msz)
plot(j{2,q},ones(size(x{2,q})),'.','color',bol{q}, 'MarkerSize',msz)
plot(j{3,q},ones(size(x{3,q})),'.','color',col{q}, 'MarkerSize',msz)
plot(j{1,q},zeros(size(x{1,q})) + nj,'.','color','r','MarkerSize',msz)
plot(j{2,q},zeros(size(x{2,q})) + nj,'.','color','m','MarkerSize',msz)
plot(j{3,q},zeros(size(x{3,q})) + nj,'.','color','b','MarkerSize',msz)
end
title(sprintf('Jacobian (by %i particles)',nx),'Fontsize',16)
subplot(3,2,6),hold on
for q = 1:nx
plot(k{1,q},ones(size(x{1,q})),'.','color',bol{q}, 'MarkerSize',msz)
plot(k{2,q},ones(size(x{2,q})),'.','color',bol{q}, 'MarkerSize',msz)
plot(k{3,q},ones(size(x{3,q})),'.','color',col{q}, 'MarkerSize',msz)
plot(k{1,q},zeros(size(x{1,q})) + nj,'.','color','r','MarkerSize',msz)
plot(k{2,q},zeros(size(x{2,q})) + nj,'.','color','m','MarkerSize',msz)
plot(k{3,q},zeros(size(x{3,q})) + nj,'.','color','b','MarkerSize',msz)
end
title('Jacobian (by type)','Fontsize',16)
end
break
end
end
return