correlation of time series

Say I have $$n$$ time series $$y_1, y_2,...y_n$$. Each $$y_i$$ being a vector of size $$m$$.

Given $$y_i$$ and $$y_j$$ the correlation $$\rho(y_i,y_j)$$ can be computed.

Say I want to scale the correlation up or down, that is change $$\rho(y_i,y_j)$$ to $$\rho(y_i,y_j) + 10\% of \rho(y_i,y_j)$$ (i.e change the current level of correlation to 10 percent more than what it is. ).

My question: Is there a way to achieve this by scaling the time series $$y_i$$ and $$y_j$$ ?

Meaning transform $$f: y_i-> z_i$$ for all i. such that

$$\rho(z_i,z_j) = \rho(y_i,y_j) + 10\% of \rho(y_i,y_j)$$

• what if $\rho(y_i,y_j)=0.95$ ? then you will have a hard time getting a correlation of $1.1$ since correlation is always between $-1$ and $1$...
– Surb
Oct 12, 2019 at 20:08
• that can be addressed by a bounding the correlation between 1 and -1 Oct 13, 2019 at 12:52

A linear $$f$$ will not change the correlation.
A non-linear function might do that, but depends very much on the characteristics of the signals $$y_i$$.
Suppose for instance that the $$y_i$$ be quite similar for the lower values , while differing in the higher range. Then a compressing function (log for instance) would do the job.