Autocorrelation

Quantifying Autocorrelation

Autocorrelation is easiest to demonstrate mathematically using time series data, where it is often represented in the form of a linear regression. The following equation shows the autoregressive (AR) model of autocorrelation, which shows the current observation (Yt) as a linear function of previous observations plus a residual error:

$$Y_t={\beta }_0+{\beta }_1{Y}_{t-1}+{\beta }_2{Y}_{t-2}+\dots {\beta }_p{Y}_{t-p}+e_t.$$

Here is the familiar structure of an ordinary least squares regression with βs as constant coefficients, Y values at p previous intervals or lags, and a random error series et. This model assumes stationarity, such that the mean, variance, and covariance of the observations remain constant for all time periods. However, in many nonstationary cases, the residual error itself may be an autocorrelated series that requires more advanced AR moving average or AR integrated moving average models. First-order autocorrelation, in which an observation (Yt) is correlated only with the observation immediately preceding it (Yt-1), is commonly observed in time series data. Larger orders of autocorrelation include more lags and imply longer decays or greater inertia in temporal and spatial processes.

The most common way to measure the strength of autocorrelation is by computing the autocorrelation coefficient, symbolized as ρ. For the common case of the first-order autocorrelation, ρ is a two-variable correlation coefficient ranging between −1 and 1. Positive values of ρ are most frequently observed, indicating similarity between successive observations. Higher values of the coefficient indicate stronger dependency on previous values and milder decay. Although positive autocorrelation suggests that successive observations will move in the same direction, negative autocorrelation characterizes processes that oscillate in direction. A negative value of the autocorrelation coefficient can be exemplified by imagining the task of cutting equally sized pieces of ribbon into two unequal pieces. The smaller the first piece is cut, the larger the second piece will be. Thus, successive observations oscillate in size and are negatively correlated.

...