I have $N$ time series $x_{1t},x_{2t},\dots,x_{Nt}$. I could measure the pair-wise correlations $\rho_{ij}=Corr[x_{it},x_{jt}]$, and get an $N\times N$ correlation matrix. I can also get its largest eigenvalue $\lambda_0$ to measure overall correlation of series. This is all good, but I want to measure how overall correlation changes with time. One approach would be to get a small moving window $h$, and measure the correlation matrix $\rho_{ij}(t)$ at different time periods $t\in[t-h,t]$.

However, in my case I don't want to do it, because I believe that the correlation matrix may experience an instantaneous jump in one period, then come back very quickly. So, the time window is very narrow or even $h=1$, and calculating even pair-wise correlation becomes pointless. In any case $h\ll N$.

So, I want to approach this differently. I'd like to come up with a measure of overall correlation of the series in a single period. Some measure of how the series moved together in one single period. Maybe half the series were positive and half negative, that would suggest correlation close to zero? Or 60% of series went up, and 40% down, which is 20% correlation? How to scale this, since the series may have different ranges? Any ideas?


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