Creating a precision matrix for Gaussian markov random field - normal-distribution

Question

I am currently trying to create a precision matrix for a Gaussian markov random field. Lets say I have random variables in a spatial grid of 6x6. Then I will have a precision matrix of 36x36.
Now suppose that I have a neighbor hood of 3x3, then my precision matrix will be
Q= nnbs[1] -1 0 0 0 0 -1.......0
-1 nnbs[2] -1 0 0 0 0 ......0
0 -1 nnbs[3] -1 0 0 0 ......0
...................................................
...................................................
and so on. Can anyone suggest me how can I code this precision matrix. I mean if I change the window size/neighborhood size to 5x5, then I will have a new precision matrix. How can I code this? where nnbs is the number of neighbors of that element
rows=20;
columns=20;
%Random initialization
data=zeros(1000,3);
index=1;
value=-1;
%3x3 neighborhood
%For each element the neighbors are accessible within 1 hop so neighbors=1
neighbors=1;
for i=1:rows
for j=1:columns
for k=1:neighbors
%same row right
if j+k <= columns
data(index,1) = (i-1)*columns+j;
data(index,2) = ((i-1)*columns) + (j+k);
data(index,3) = value;
index=index+1;
end
%same row left
if j-k >= 1;
data(index,1) = (i-1)*columns+j;
data(index,2) = ((i-1)*columns) + (j-k);
data(index,3) = value;
index=index+1;
end
end
%row below -> bottom left right
for k=i+1:i+neighbors
if k <= rows
%bottom
data(index,1) = (i-1)*columns+j;
data(index,2) = (k-1)*columns + j;
data(index,3) = value;
index=index+1;
for l=1:neighbors
%right
if j+l <= columns
data(index,1) = (i-1)*columns+j;
data(index,2) = ((k-1)*columns) + (j+1);
data(index,3) = value;
index=index+1;
end
%left
if j-l >= 1;
data(index,1) = (i-1)*columns+j;
data(index,2) = ((k-1)*columns)+(j-1);
data(index,3) = value;
index=index+1;
end
end
end
end
%row above top left right
for k=i-1:i-neighbors
if k >= 1
%top
data(index,1) = (i-1)*columns+j;
data(index,2) = ((k-1)*columns) +j;
data(index,3) = value;
index=index+1;
for l=1:neighbors
%right
if j+l <= columns
data(index,1) = (i-1)*columns+j;
data(index,2) = ((k-1)*columns) + (j+1);
data(index,3) = value;
index=index+1;
end
%left
if j-k >= 1;
data(index,1) = (i-1)*columns+j;
data(index,2) = ((k-1)*columns) + (j-1);
data(index,3) = value;
index=index+1;
end
end
end
end
end
end
%Get the values for the diagonal elements(which is equal to the number of
%neighbors or absolute sum of the nondiagonal elements of the corresponding
%row)
diagonal_values = zeros(rows*columns,3);
for i=1:rows*columns
pointer=find(data(:,1) == i);
diag_value=abs(sum(data(pointer,3)));
diagonal_values(i,1) = i;
diagonal_values(i,2) = i;
diagonal_values(i,3) = diag_value;
end
data(index:index+rows*columns-1,:)=diagonal_values(:,:);
Q = sparse(data(:,1), data(:,2), data(:,3), rows*columns, rows*columns);
I tried something like this but I don't think this is the most efficient of ways. I think there should be a better way.

Answer 1

A bit too late but it may be useful for someone else :
Your precision matrix is a linear combination of kronecker product of symmetric Toeplitz matrix : to each neighbour type corresponds a kronecker product of 2 Toeplitz matrix.
More info about toeplitz Matrix
Example :
you want a precision matrix only with the horizontal link for each pixel
Writing I_n the identity matrix of size n and H_{n,p} the Symmetric Toeplitz matrix of dimension [n n] filled with 0 everywhere excepts on the p-th diagonals
H_{4,2} =
0 1 0 0
1 0 1 0
0 1 0 1
0 0 1 0
In Matlab :
H_nonSym_n_p = toeplitz(zeros(n,1), [[zeros(1,p-1) 1] zeros(1,n-p)]) ;
H_sym_n_p = H_nonSym + H_nonSym' ;
Then, if you have a [n m] image and if you want to code the horizontal neighbour of every pixel, you can express it through the kronecker product (hope the LaTeX-like code will work): Q = - I_n \otimes \H_{n,2}.
And finally, to get your nnbs : something like Q = Q - diag(sum(Q,2))...
Now, if you want other links, for instance 2 horizontal and 2 vertical links : Q = - I_n \otimes \H_{n,2} - I_n \otimes \H_{n,3} - \H_{n,2} \otimes I_{n} - \H_{n,3} \otimes I_{n}.
and again Q = Q - diag(sum(Q,2))
Note that the diagonal neighbours are a bit more difficult to produce but it is still represented by a kronecker product of toeplitz matrix (maybe non-symmetric this time).