Autocorrelation Function

Autocorrelation, or self correlation, is often used to detect periodicity in time series data. Autocorrelation measures the degree of association between numbers in a time series separated by different time lags. For example, the autocorrelation between the logarithm of population counts separated by one time lag is obtained by calculating the coefficient of correlation between ln N(t) and ln N(t-1) over the entire series. This is called the autocorrelation at lag 1. Thus, autocorrelation at lag d is obtained by calculating the correlation between n(t) = lnN(t) and n(t-d) = lnN(t-d). The autocorrelation function, ACF, is computed from the autocovariance function

where COV(d) is the autocovariance at lag d (d = 0,1, 2....), Y is the number of observations (usually years) in the series, n(t) is the logarithm of population density at time t, and is the mean of the log-transformed series. The autocorrelation function is then given by

and is often plotted as a histogram. For more information on calculating the ACF click here.

The ACF provides clues to

1. Stationarity. Stationary ACFs are balanced more or less evenly around zero correlation while nonstationary ones usually decline continuously with increasing lag.

• Correlation coefficients in a stationary ACF that damp out as the lag increases, indicate that population fluctuations are largely due to endogenous feedback.
• Correlation coefficients in a stationary series that do not damp out at increasing lags, imply that population fluctuations are caused by periodic exogenous forcing; e.g., seasonality, sun-spot cycles, and such.

2. Periodicity. The number of lags it between peaks (or troughs) of a stationary ACF indicates the period of fluctuation, or the time it takes for the oscillation to repeat itself. For example, the ACF in the figure has a period of about two because positive and negative spikes repeat themselves, on the average, every second lag.