I'm a computational biologist trying to interpret this equation correctly.

$$CI(x)= 1- \frac{2}{(N(N-1))} \sum_i^N\sum_j^N [ S(x_i,x_j) / \sqrt{S(x_i,x_i)S(x_j,x_j)}\ ] $$

i=1 ; j=i+1

I'm confused about the part before the summation

This is how I under stand this:

This is a nested summation, so if N=3 then you sum the partitions like[(1+2)+(1+3)]+[(2+3)]

Then do you multiply by $ 1- \frac{2}{(N(N-1))}$ ? or are you multiplying the sum of i (and j?) for each summation cycle, such as multiplying for 1 and 2 instead of multiplying 1+2?

Thanks a lot, it's been a while since I worked with stuff like this. This is a conservation index metric for multiple sequence alignments using a BLOSUM62 substitution matrix (The S(x,x) part).

  • 1
    $\begingroup$ First, calculate the double sum. Then you multiply by $2/(N(N-1))$, which is a fixed number, since $N$ is fixed. Finally subtract all that from $1$. $\endgroup$ – angryavian Feb 27 '14 at 4:07
  • $\begingroup$ THANK YOU SO MUCH! I feel so dumb now. $\endgroup$ – QVINTVS FABIVS MAXIMVS Feb 27 '14 at 4:11

Just focus first on the summation; it write $$\sum _{i=1}^N \sum _{j=i+1}^N \frac{S(x(i),x(j))}{\sqrt{S(x(i),x(i)) S(x(j),x(j))}}$$ For clarity, I shall expand this sum for $N=4$ in order you see how the terms appear $$\frac{S(x(1),x(2))}{\sqrt{S(x(1),x(1)) S(x(2),x(2))}}+\frac{S(x(1),x(3))}{\sqrt{S(x(1),x(1)) S(x(3),x(3))}}+\frac{S(x(2),x(3))}{\sqrt{S(x(2),x(2)) S(x(3),x(3))}}+\frac{S(x(1),x(4))}{\sqrt{S(x(1),x(1)) S(x(4),x(4))}}+\frac{S(x(2),x(4))}{\sqrt{S(x(2),x(2)) S(x(4),x(4))}}+\frac{S(x(3),x(4))}{\sqrt{S(x(3),x(3)) S(x(4),x(4))}}$$ Once this summation is done, just multiply the sum by $ \frac{2}{(N(N-1))}$ and substract the result from $1$.


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