在Matlab中优化成对距离函数的参数

Question

This question is related to matlab: find the index of common values at the same entry from two arrays .

Suppose that I have an 1000 by 10000 matrix that contains value 0 , 1 ,and 2 . Each row are treated as a sample. I want to calculate the pairwise distance between those samples according to the formula d = 1-1/(2p)sum(a/c+b/d) where a , b , c , d can treated as as the row vector of length 10000 according to some definition and p=10000 . c and d are probabilities such that c+d=1 .

An example of how to find the values of a , b , c , d : suppose we want to find d between sample i and b j , then I look at row i and j .

If k th entry of row i and j has value 2 and 2 , then a=2,b=0,c=1,d=0 (I guess I will assign 0/0=0 in this case).

If k th entry of row i and j has value 2 and 1 or vice versa, then a=1,b=0,c=3/4,d=1/4 .

The similar assignment will give to the case for 2,0 ( a=0,b=0,c=1/2,d=1/2 ), 1,1 ( a=1,b=1,c=1/2,d=1/2 ), 1,0 ( a=0,b=1,c=1/4,d=3/4 ), 0,0 ( a=0,b=2,c=0,d=1 ).

The matlab code I have so far is using for loops for i and j , then find the cases above by using find , then create two arrays for a/c and b/d . This is extremely slow, is there a way that I can improve the efficiency?

Edit: the distance d is the formula given in this paper on page 13.

Answer 1

Provided those coefficients are fixed, then I think I've successfully vectorised the distance function. Figuring out the formulae was fun. I flipped things around a bit to minimise division, and since I wasn't aware of pdist until @horchler's comment, you get it wrapped in loops with the constants factored out:

% m is the data
[n p] = size(m, 1);
distance = zeros(n);
for ii=1:n
    for jj=ii+1:n
        a = min(m(ii,:), m(jj,:));
        b = 2 - max(m(ii,:), m(jj,:));
        c = 4 ./ (m(ii,:) + m(jj,:));
        c(c == Inf) = 0;
        d = 1 - c;

        distance(ii,jj) = sum(a.*c + b.*d);
        % distance(jj,ii) = distance(ii,jj); % optional for the full matrix
    end
end
distance = 1 - (1 / (2 * p)) * distance;