Asymmetric sliding-window cross-correlation
Alex Bäcker · 2002
Sliding-window cross-correlation is a common method to esimate time-varying correlations between signals (Laurent and Davidowitz, 1994; Laurent et al., 1996; Macleod and Laurent, 1996; Stopfer and Laurent, 1997; Wehr, 1999 (p. 96)). It produces a correlation value betwen two signals (positive or negative) for every (time,lag) pair of values. In principle, the expected value of the correlation for any pair of (time,lag) values must be computed by averaging x(t).y(t+lag) over many realizations of the stochastic process. This is called an ``ensemble average'' across realizations of a stochastic process. Lacking a large enough number of realization over which to average, one must resort to other methods. If the correlations are stationary (i.e. time-invariant), then we may average across time to estimate the expected value. If the correlations are not stationary, we may divide the signals into sliding windows of a size such that the correlations can be considered stationary on the timescale of the window width, and calculate the cross-correlation, as a function of lag, for each window, sliding the window along the signal to obtain correlations for different time values. This is what is called a sliding-window cross-correlogram.