Autocovariance

In probability and statistics, given a stochastic process X=(X_t), the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. With the usual notation E  for the expectation operator, if the process has the mean function \mu_t = E[X_t], then the autocovariance is given by

C_{XX}(t,s) = cov(X_t, X_s) = E[(X_t - \mu_t)(X_s - \mu_s)] = E[X_t X_s] - \mu_t \mu_s.\,

Autocovariance is related to the more commonly used autocorrelation of the process in question.

In the case of a random vector X=(X_1, X_2, ... , X_n), the autocovariance would be a square n by n matrix C_{XX} with entries C_{XX}(j,k) = cov(X_j, X_k).\, This is commonly known as the covariance matrix or matrix of covariances of the given random vector.

Stationarity

If X(t) is stationary process, then the following are true:

\mu_t = \mu_s = \mu \, for all t, s

and

C_{XX}(t,s) = C_{XX}(s-t) = C_{XX}(\tau)\,

where

\tau = s - t\,

is the lag time, or the amount of time by which the signal has been shifted.

As a result, the autocovariance becomes

C_{XX}(\tau) = E[(X(t) - \mu)(X(t+\tau) - \mu)]\,
= E[X(t) X(t+\tau)] - \mu^2\,
= \sigma^2 R_{XX}(\tau) - \mu^2,\,

where \sigma^2 is the variance and R_{XX}(\tau) is the autocorrelation of the signal.

Normalization

When normalized by dividing by the variance σ2, the autocovariance C becomes the autocorrelation coefficient function c,[1]

c_{XX}(\tau) = \frac{C_{XX}(\tau)}{\sigma^2}.\,

However, often the autocovariance is called autocorrelation even if this normalization has not been performed.

The autocovariance can be thought of as a measure of how similar a signal is to a time-shifted version of itself with an autocovariance of σ2 indicating perfect correlation at that lag. The normalization with the variance will put this into the range [−1, 1].

Properties

The autocovariance of a linearly filtered process Y_t

Y_t = \sum_{k=-\infty}^\infty a_k X_{t+k}\,
is C_{YY}(\tau) = \sum_{k,l=-\infty}^\infty a_k a^*_l C_{XX}(\tau+k-l).\,

See also

References

  1. ↑ Westwick, David T. (2003). Identification of Nonlinear Physiological Systems. IEEE Press. pp. 17–18. ISBN 0-471-27456-9.