Autocorrelación

Autocorrelación refers to the similarity of a signal or a time series with a delayed version of itself as a function of the delay. In simpler terms, it measures how correlated a sequence is with itself at different points in time. It is a mathematical tool used to identify repeating patterns.

The autocorrelation function (ACF) quantifies this relationship. For a discrete-time signal x[n], the autocorrelation Rxx[m] at a lag m is calculated as the expected value of the product of the signal at time n and the signal at time n-m, averaged over all possible n. For a continuous-time signal x(t), it's the integral of x(t)x(t-τ) over all t, where τ is the time lag.

High autocorrelation values at a particular lag indicate that the signal strongly resembles itself when shifted by that lag. This is useful for detecting periodicity in data. For example, in signal processing, autocorrelation can reveal hidden periodicities or identify the fundamental frequency of a signal. In statistics, it helps understand the dependence structure of time series data, such as in econometrics or climate science, to model trends and seasonality. Low or zero autocorrelation suggests a lack of linear relationship between the signal and its lagged version.