Writing a summation code where i≠j

Hi, I want to write a summation code to calculate Getis ord index as below:

.

Here, x is a 1xn vector, and w is a square matrix where w_ii=0

I wrote the code to calculate the denominator, but it's quite complicated to calculate the nominator.

I tried with the below code, but I'm not sure it's right...

How can I do that?

x = [1,2,3,4]; % feature vector
w= rand(4,2); D=pdist(w); 
W= squareform(D); % weight matrix

% Calculate Denominator
denom=0;
for i = 1:4
   
    for j = 1:4       
        if i~=j
            denom = denom + x(j)*x(i);
        end    
    end
end

% Calculate nominator
Wu = triu(W,1);
Wd = tril(W,-1);

nom = (sum(sum(Wu)) + sum(sum(Wd))) * denom;

Answer 1

[Voss]

The weight matrix is zeros along the diagonal. Thus you can use it as-is in the summation, since the terms where i==j (which correspond to the diagonal elements of W) will be 0.

To get the same effect in the denominator, you can set the diagonal elements of x.*x.' to zero and use that.

x = [1,2,3,4]; % feature vector
w = rand(4,2);
D = pdist(w);
W = squareform(D) % weight matrix

W = 4×4

0 0.4616 0.3697 0.5919 0.4616 0 0.8026 0.7224 0.3697 0.8026 0 0.8581 0.5919 0.7224 0.8581 0

denom_matrix = x.*x.';
denom_matrix(1:numel(x)+1:end) = 0

denom_matrix = 4×4

0 2 3 4 2 0 6 8 3 6 0 12 4 8 12 0

G = sum(W.*x.*x.','all')/sum(denom_matrix,'all')

G = 0.7226

For reference, here is a way to do the same thing using for loops:

denominator = 0;
numerator = 0;
for ii = 1:numel(x)
    for jj = 1:numel(x)
        if ii == jj % skip the ii == jj terms
            continue
        end
        denominator = denominator + x(ii)*x(jj);
        numerator = numerator + W(ii,jj)*x(ii)*x(jj);
    end
end
G_loop = numerator/denominator

G_loop = 0.7226

Check that the results are identical:

isequal(G,G_loop)

ans = logical
   1

[aposta95]

Many thanks for your elaboration :)

[Voss]

You're welcome!

Answer 2

[Abhishek Chakram]

Hi,

As I understood, you want to calculate the numerator of Getis ord index.

You can do it the same way as you calculated the denominator. The code will look like :

numerator = 0;
for i = 1:4
    for j = 1:4
        if i~=j
            numerator = numerator + W(i,j)*x(i)*x(j);
        end
    end
end

[aposta95]

Thanks it works :)

Answer 3

[David Hill]

No loops needed

G=sum(w.*x.*x','all')/sum(x.*x','all');

[Jan]

Isn't the "for all i~=j" missing?