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I have a time series $S(t)$ for $t\in I $ where $I=[0,t_\text{max}]$ is an interval. The time series is very regular, $S(t) = \sin(t)$, although my following question is independent of the explicit analytic form (if any) of the series.

I want to select two sub-intervals of $I$ of same length, for example $I_0 = [t_0,t_1]$ and $I_1 = [t_1,2t_1-t_0]$ for some $t_0>0,t_1>t_0$. Now, I consider the two following series

$$ S_0 (t) = S(t) \,,\quad \text{for}\,\,t \in I_0\\ S_1 (t) = S(t_1-t_0+t) \,,\quad \text{for}\,\,t \in I_0 $$

and I am interested in computing the correlation between $S_0 (t)$ and $S_1 (t)$

$$ \text{corr}\left(S_0 (t),S_1 (t)\right)\,,\qquad (\text{Eq}.1) $$

Is this quantity somewhere defined in statistics? Does it have any statistical property?

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  • $\begingroup$ Is $t$ continuous or discrete time? The notation suggests it is continuous, but I just want to be sure. $\endgroup$ Apr 13 at 13:45
  • $\begingroup$ @RichardHardy It is continuous for $S(t)=\sin(t)$ but eventually I want to consider discrete time $\endgroup$
    – apt45
    Apr 13 at 16:31
  • $\begingroup$ That is auto-correlation at lag $t_1-t_0$. See here. $\endgroup$
    – frank
    Apr 14 at 7:49

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