The fact that the basic dot product can be seen to underlie all these similarity measures turns out to be convenient. If you stack all the vectors in your space on top of each other to create a matrix, you can produce all the inner products simply by multiplying the matrix by it’s transpose. Furthermore, the extra ingredient in every similarity measure I’ve looked at so far involves the magnitudes (or squared magnitudes) of the individual vectors. These drop out of this matrix multiplication as well. Just extract the diagonal.

Because of it’s exceptional utility, I’ve dubbed the symmetric matrix that results from this product the base similarity matrix. I haven’t been able to find many other references which formulate these metrics in terms of this matrix, or the inner product as you’ve done. Known mathematics is both broad and deep, so it seems likely that I’m stumbling upon something that’s already been investigated.

Do you know of other work that explores this underlying structure of similarity measures? Is the construction of this base similarity matrix a standard technique in the calculation of these measures? Does it have a common name?

Thanks again for sharing your explorations of this topic.

P.S. Here’s the other reference I’ve found that does similar work:
http://arxiv.org/pdf/1308.3740.pdf

By “invariant to shift in input”, I mean, if you *add* to the input. That is,
f(x, y) = f(x+a, y) for any scalar ‘a’.

By “scale invariant”, I mean, if you *multiply* the input by something.

For (1-corr), the problem is negative correlations. I think maximizing the squared correlation is the same thing as minimizing squared error .. that’s why it’s called R^2, the explained variance ratio.

I don’t understand your question about OLSCoef and have not seen the papers you’re talking about.

I have a few questions (i am pretty new to that field). You say correlation is invariant of shifts.

i guess you just mean if the x-axis is not 1 2 3 4 but 10 20 30 or 30 20 10.. then it doesn’t change anything.

but you doesn’t mean that if i shift the signal i will get the same correlation right?

ex: [1 2 1 2 1] and [1 2 1 2 1], corr = 1
but if i cyclically shift [1 2 1 2 1] and [2 1 2 1 2], corr = -1
or if i just shift by padding zeros [1 2 1 2 1 0] and [0 1 2 1 2 1] then corr = -0.0588

Please elaborate on that.

Also could we say that distance correlation (1-correlation) can be considered as norm_1 or norm_2 distance somehow? for example when we want to minimize the squared errors, usually we need to use euclidean distance, but could pearson’s correlation also be used?

Ans last, OLSCoef(x,y) can be considered as scale invariant? is very correlated to cosine similarity which is not scale invariant (Pearson’s correlation is right?). Look at: “Patterns of Temporal Variation in Online Media” and “Fast time-series searching with scaling and shifting”. That confuses me.. but maybe i am missing something.

Here’s a link, http://data.psych.udel.edu/laurenceau/PSYC861Regression%20Spring%202012/READINGS/rodgers-nicewander-1988-r-13-ways.pdf

Have you seen – ‘Thirteen Ways to Look at the Correlation Coefficient’ by Joseph Lee Rodgers; W. Alan Nicewander, The American Statistician, Vol. 42, No. 1. (Feb., 1988), pp. 59-66. It covers a related discussion.