autocorrelation function (ACF)
The autocorrelation function (ACF) is the normalized autocovariance
function defined as
The autocovariance function is defined as the second moment productfor all , . Note that for all time points s and t. The autocovariance measures the linear dependence between two points on the same series observed at different times. Recall from classical statistic , and are not linearly related, but there still may be some dependence structure between them. If, however, and are bivariate normal, ensures their independence. It is clear that, for , the autocovariance reduces to the (assumed finite) variance,
Example: The white noise serieshas and
The cross-covariance function scaled to live in [-1, 1] is called cross-correlation function and given by
The cross-covariance function between two series,
Differences of order d are defined as
where we may expand the operatoralgebraically to evaluate for higher integer values of d. When d = 1, we drop it from the notation.
Gaussian white noise
See white noise.
The mean function of a time series process is defined as
Becausefor all and is a constant we have which is a straight line with slope .
If the value of the time series at time
In time series analysis context, if the time series is one for which the probabilistic behavior of every collection of valuesis identical to that of the time shifted series for all , all time points and all time shifts , time series is said to be a strictly stationary.
It is difficult to assess strict stationarity from data. Rather than imposing conditions on all possible distributions of a time series, a milder version called weakly stationary is used that imposes conditions only on the first two moments of the series such as a) the mean value functionis constant and does not depend on time and b) the autocovariance function depends on and only through their difference .
Stationarity requires regularity in the mean and autocorrelation functions so that these quantities (at least) may be estimated by averaging. It should be clear that a strictly stationary, finite variance, time series is also stationary. The converse is not true in general. One important case where stationarity implies strict stationarity is if the time series is Gaussian [meaning all finite collections of the series are Gaussian].