Autocovariance

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In statistics, given a time series or continuous signal X_t, the autocovariance is simply the covariance of the signal against a time-shifted version of itself. If each state of the series has a mean, E[X_t] = μ_t, then the autocovariance is given by

$\, \gamma(i,j) = E[(X_i - \mu_i)(X_j - \mu_j)] = E[X_i\cdot X_j]-\mu_i\cdot\mu_j.\,$

Where E is the expectation operator. If X_t is second-order stationary then the following definition becomes the more familiar:

$\, \gamma(k) = E[(X_i - \mu)(X_{i-k} - \mu)]= E[X_i\cdot X_{i-k}]-\mu^2,\,$

with $μ = μ i = μ j$ , for all $i, j$ (because of second-order stationarity).

The k is the amount the signal has been shifted and is usually referred to as the lag. When normalised by dividing by the variance σ² then the autocovariance becomes the autocorrelation R(k). That is

$R(k) = \frac{\gamma(k)}{\sigma^2}.\,$

Note, however, that some disciplines use the terms autocovariance and autocorrelation interchangeably.

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 normalisation with the variance will put this into the range [−1, 1].