Matlab中三点相关函数的矢量化计算？

Question

I would like to compute the 2- and 3-point correlation functions R2, R3 of samples of a vector by appropriate histogramming of the elements of a vector (num_samples samples of length system_size), and the corresponding cluster functions T2, T3. For simplicity I am considering histogramming across uniform bins.

What is a good way to vectorize and/or speed up the following code?

n = length(mesh);
R2 = zeros(n, n);
R3 = zeros(n, n, n);
for sample_id=1:num_samples 
    s = samples(:, sample_id);
    d = mesh(2) - mesh(1);
    % Which bin does the ith sample s belong to?
    bins = ceil((s - mesh(1))/d);

    % Compute two-point correlation function
    for i = 1:system_size
        for j = 1:system_size
            if i ~= j
                R2(bins(i), bins(j))=R2(bins(i), bins(j))+1;
            end
        end
    end

    % Compute three-point correlation function
    for i = 1:system_size
        for j = 1:system_size
            if i ~= j
                for k = 1:system_size
                    if k ~= j && k ~= i
                        R3(bins(i), bins(j), bins(k))=R3(bins(i), bins(j), bins(k))+1;
                        T3(x1, x2, x3) = R3(x1,x2,x3)-R1(x1)*R2(x2,x3)-R1(x2)*R2(x1,x3)...
                             -R1(x3)*R2(x1,x2)+2*R1(x1)*R1(x2)*R1(x3);
                    end
                end
            end
        end
    end
end
R2 = R2/sum(R2(:));
R3 = R3/sum(R3(:));

T3 = zeros(n, n, n);
% Compute three-point cluster function
for i = 1:n
    for j = 1:n
        if i ~= j
            for k = 1:n
                if k ~= j && k ~= i
                    T3(x1, x2, x3) = R3(x1,x2,x3)-R1(x1)*R2(x2,x3)-R1(x2)*R2(x1,x3)...
                         -R1(x3)*R2(x1,x2)+2*R1(x1)*R1(x2)*R1(x3);
                end
            end
        end
    end
end

Naively I thought hist3(bins, bins...) or crosstab(bins, bins) would almost do what I want, which is to look for correlated occurrences of elements of the vector, but it doesn't.

---

Example:

If my inputs within the outermost loop are

s = [1.2 3.1 4.6 4.7 5.1]
mesh = 0:0.5:6

then the quantized data should be

bins = [3 7 10 10 11]

and R2 should be

>> R2

R2 =

     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     1     0     0     2     1     0
     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     0     0     0     0     0     0     2     1     0
     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     2     0     0     0     2     0     0     2     2     0
     0     0     1     0     0     0     1     0     0     2     0     0
     0     0     0     0     0     0     0     0     0     0     0     0

Answer 1

'R2 and R3` are easy:

R2 = R2 + 1 - diag(ones(size(R2, 1), 1); % you can replace the loop with this
eye3 = zeros(n, n, n);
eye3(linspace(1, numel(eye3), n)) = 1;
R3 = R3 + 1 - eye3; % can move R3 computation outside the loop

For T3 :

temp = repmat(R2, [1 1 n]).*permute(repmat(R1, [n, 1, n]), [1, 3, 2]);
T3 = R3 - temp - permute(temp, [2 3 1]) - permute(temp, [3 1 2]);
temp2 = repmat(R1'*R1, [1 1 n]).*permute(repmat(R1, [n, 1, n]), [1, 3, 2]);
T3 = T3 + temp2;

assuming R1 is a row vector.

You may have to play with this a little since there are some things still unclear from your code, but this should be pretty close to what you will eventually need.

EDIT after clarification:

For R2 :

ubins = unique(bins);
bincounts = histc(bins, ubins);
for i=1:max(bincounts)
    indices = find(bincounts == i);
    R2(indices, indices) = R2(indices, indices) + i
end

This will only be useful for large vectors and arrays. In effect you are vectorizing the computation of chunks of the matrix, not the entire matrix (because of potential repetition in bins ).

You can write something similar for R3 . The T3 should still look similar to my earlier answer.