How to move larger values close to matrix diagonal in a correlation matrix - matlab

Question

I have a correlation matrix X of five elements(C1,C2,C3,C4,C5)
C1 C2 C3 C4 C5
C1 * 1 0 1 0
C2 1 * 0 0 1
C3 0 0 * 1 1
C4 1 0 1 * 0
C5 0 1 1 0 *
I want to use MatLab to move as many as non-zero cells close to diagonal, while keep the diagonal cells are "*".
For example, you may notice that the columns and rows is shifting in the following matrix, while the diagonal cells are "*".
C1 C4 C2 C5 C3
C1 * 1 1 0 0
C4 1 * 0 0 1
C2 1 0 * 1 0
C5 0 0 1 * 1
C3 0 1 0 1 *
Because I want to do clustering, so I want as many as non-zero cells get close to diagonal after shifting. It's an NP-hard problem.
Anyone know what functions in MatLab can realize this?

Answer 1

What you're looking for is probably the reverse Cuthill-McKee algorithm (RCM), which pretty much does what you want: for a given matrix it finds a permutation that tends to have its non-zero elements closer to the diagonal. There's a built-in function symrcm in MATLAB that does just that.
So assuming that X is your matrix, you can do the following:
p = symrcm(X);
Xnew = X(p, p);
Xnew is the new reordered matrix, and p is the new row/column order.
Example
Let's create a matrix first:
X = [10 0 0 7 0; 3 20 0 0 11; 0 0 30 0 29; 12 7 0 40 0; 0 33 0 0 50]
Now let's reorder it:
p = symrcm(X);
Xnew = X(p, p)
The result is:
Xnew =
40 12 7 0 0
7 10 0 0 0
0 3 20 11 0
0 0 33 50 0
0 0 0 29 30
Seems right.

Answer 2

A = [1 0 0 1 0;
0 1 0 0 1;
0 0 1 0 1;
1 1 0 1 0;
0 1 0 0 1];
N = length(A);
switched = false;
%%
% Calculate initial Global Energy
disp(A);
global_energy = 0;
for l = 1:N
for m = 1:N
if(A(l,m))
global_energy = global_energy + (l-m)^2/2;
end
end
end
disp(global_energy);
counter = 0;
counter_cutoff = 10000000000;
while(true)
switched = false;
counter = counter + 1;
for i = 1:N
for j = i+1:N
current_metric = 0; % Calculate metric of row i and j with columns i and j
permuted_metric = 0; % Calculate metric if they were permuted
% Row i
for k = 1:N
if(k ~= i && k ~= j && A(i,k))
current_metric = current_metric + (i-k)^2/2;
permuted_metric = permuted_metric + (j-k)^2/2;
end
end
% Row j
for k = 1:N
if(k ~= i && k ~= j && A(j,k))
current_metric = current_metric + (j-k)^2/2;
permuted_metric = permuted_metric + (i-k)^2/2;
end
end
% Col i
for k = 1:N
if(k ~= i && k ~= j && A(k,i))
current_metric = current_metric + (i-k)^2/2;
permuted_metric = permuted_metric + (j-k)^2/2;
end
end
% Col j
for k = 1:N
if(k ~= i && k ~= j && A(k,j))
current_metric = current_metric + (j-k)^2/2;
permuted_metric = permuted_metric + (i-k)^2/2;
end
end
% If permuted metric is less, swap columns and rows - set switched to true
if(permuted_metric < current_metric)
switched = true; % there was at least one switch
% Now switch rows and columns
% Switch columns first
A(:,[i j]) = A(:,[j i]);
% Now switch rows
A([i j],:) = A([j i],:);
end
end
end
if(~switched || counter > counter_cutoff)
% All permutations did not lead to a switching of rows and columns
break;
end
end
% Calculate final Global Energy
disp(A);
global_energy = 0;
for l = 1:N
for m = 1:N
if(A(l,m))
global_energy = global_energy + (l-m)^2/2;
end
end
end
disp(global_energy);
Terminal:
1 0 0 1 0
0 1 0 0 1
0 0 1 0 1
1 1 0 1 0
0 1 0 0 1
22
1 1 0 0 0
1 1 1 0 0
0 0 1 1 0
0 0 1 1 0
0 0 0 1 1
3