Ristikorrelaatiofunktio

Ristikorrelaatiofunktio, commonly known as the cross‑correlation function, is a statistical tool that measures the similarity between two time series or spatial signals as a function of a lag or shift applied to one of them. For discrete sequences \(x[n]\) and \(y[n]\) the cross‑correlation at lag \(k\) is defined as \(R_{xy}[k] = \sum_{n} x[n]\,y[n+k]\), where the sum runs over all indices for which both sequences are defined. For continuous signals the analogous definition uses an integral over time. Often a normalization factor is applied to restrict the values to the interval \([-1,1]\).

The cross‑correlation function is related to convolution: \(R_{xy}[k] = (x * y_{rev})[k]\), where \(y_{rev}\) denotes the time‑reversed version of \(y\). This relationship allows fast computation via the Fourier transform; the cross‑correlation can be obtained by multiplying the Fourier transforms of the two signals, one of which is complex conjugated, and then applying an inverse transform.

Applications of the ristikorrelaatiofunktio span signal processing, where it is used to detect time delays, template matching, and system identification; neuroscience, for measuring the relationship between neuronal spike trains or local field potentials; and geophysics, for comparing seismic traces. In statistical hypothesis testing, the cross‑correlation coefficient at zero lag serves as a measure of linear dependence between variables. The function is also used in image analysis for phase correlation methods that estimate relative motion or alignment.