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Autocovariance

The autocovariance of a real-valued stochastic process {X_t} between times s and t is the covariance between X_s and X_t, so

gamma(s,t)=cov(X_s,X_t)=E[(X_s-mu_s)(X_t-mu_t)],

where E[Y] denotes the expectation value of Y and mu_t=E[X_t] is the population mean at time t. For a weakly stationary process, the mean is constant and the autocovariance depends only on the lag h=t-s, so it can be written as

gamma(h)=E[(X_t-mu)(X_(t+h)-mu)].

The corresponding autocorrelation is rho(h)=gamma(h)/gamma(0).

See also

Autocorrelation, Covariance, Expectation Value, Stochastic Process, Weak Stationarity

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References

Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

Cite this as:

Weisstein, Eric W. "Autocovariance." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Autocovariance.html

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