Autocorrelation (also called serial correlation) adapts the concept of correlation to the time series context: just as the correlation coefficient measures the strength of a linear relationship between two variables, the autocorrelation coefficient, ρk, measures the extent of a linear relationship between time series values separated by a given lag, k:
Hence, we can calculate one autocorrelation coefficient for each of the T-1 lags in a time series; T is the length of the series. The autocorrelation function (ACF) computes the correlation coefficients as a function of the lag.
The autocorrelation for a lag larger than 1 (that is, between observations more than one time step apart) reflects both the direct correlation between these observations and the indirect influence of the intervening data points. The partial autocorrelation removes this influence and only measures the linear dependence between data points at the given lag distance. The partial autocorrelation function (PACF) provides all the correlations that result once the effects of a correlation at shorter lags have been removed.
There are algorithms that estimate the partial autocorrelation from the sample autocorrelation based on the exact theoretical relationship between the PACF and the ACF.
A correlogram is simply a plot of the ACF or PACF for sequential lags, k=0,1,...,n. It allows us to inspect the correlation structure across lags at one glance. The main usage of correlograms is to detect any autocorrelation after the removal of the effects of deterministic trend or seasonality. Both the ACF and the PACF are key diagnostic tools for the design of linear time series models and we will review examples of ACF and PACF plots in the following section on time series transformations.