Autocorrelation

Definition

When Error and Residuals are correlated over time.

Temporal autocorrelation

Important for Time Series Analysis, we can define data as being temporally autocorrelated as:

Plots

ACF

The ACF plot shows the correlation between the current value of the series at time t and its past values at various lags. In stationary data, the ACF plot should ideally show a rapid decline in correlation after a few lags.

Interpretation

Several features are evident from this plot: First, we observe positive autocorrelation at small lags. When time series show a trend, autocorrelations for small lags will often to be large and positive because observations nearby in time have similar values. So the ACF suggests there is evidence of a trend as autocorrelations tend to slowly decrease as the lags increase. Second, we see evidence of seasonality in captures. When data are seasonal, the autocorrelations will often be larger for the seasonal lags (at multiples of the seasonal period) than for other lags. We know this species shows evidence of torpor in the winter months, so it is not surprising to see a strong annual pattern in these autocorrelations.

PCF

The partial autocorrelation function. It's autocorrelation for lag k after removing the effect of the lower lag.

The partial autocorrelation function shows something similar, but has controlled for the short-term lagged correlations. These plots are often used for selecting orders of autoregressive models. For example, if we choose an AR2 model then we hope our AR2 parameter will capture remaining autocorrelations at lag 2 when controlling for any autocorrelations at lag 1.