Correlations in time series are sensitive to timescale

... and it's something that perhaps we don't look at quite often enough!

What am I talking about? Well, let's imagine that we're interested in the relationship between two signals, X_1(t), and X_2(t). One of the most basic analyses we might do is ask "are they correlated?" But perhaps the correlation depends on the timescale that we focus on. Could a signal be positively correlated at once timescale and negatively correlated at another scale?


Since I've shown you an example, hopefully you believe that a signal could be positively correlated at one timescale and negatively correlated at another! Here, X_1 and X_2 are positively correlated on a long timescale but negatively correlated on a short timescale.

Can we characterize this sort of relationship? Yes, and I'll outline one way of characterizing this sort of time-scale dependent correlation below.

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