# Complex covariance?

1. Nov 4, 2015

### yumyumyum

1) Let's say I have some process e.g. an gravitational orbit or something that results in x = sin(w t) and y = cos (w t)

2) a. Clearly x and y are related, but using a simple correlation <x|y>/(<x^2><y^2>)**0.5 will result in 0. That is, x and y are not correlated.
b. My question is, what non parametric techniques (e.g. SVD, PCA, TLS) are there to extract the nature of the relationship between x and y?

3) I could extract some relation by doing total least squares / SVD on a matrix of time series for column vectors of [x, x^2, y, y^2], but that would only result in relating the the x^2 and y^2 components.

4) a. Alternatively, I could construct the the complex 'x_complex' = x + i*hilbert_transform(x), do the same for y.
b. now 'x_complex' and 'y_complex' have a correlation of one. (hilbert transform and kramers kronig transform are the same thing)
c. but this isn't the case b/c the definition of correlation is <x_complex|y_complex_conjugate> which evaluates to zero.

5) The approach in 4b is promising, but I don't know what it's called, so I can't even figure out what to google to see what's been done on this .

2. Nov 5, 2015

### andrewkirk

There is a 100% correlation of x lagged by 90 degrees with y.
Analysing lagged correlations between time series is a standard non-parametric statistical analysis technique.