Observations as Time Series

Autocorrelation

Autocorrelation measures the level of correlation of the time series with a time-shifted version of itself. For example, autocorrelation at lag 2 would be a correlation between \(X_1, X_2, X_3, \ldots, X_{n - 3}, X_{n - 2}\) and \(X_3, X_4, \ldots, X_{n - 1}, X_n\). When the shift lag is 0, then the autocorrelation is just a correlation.

Let \(X_1, X_2, \ldots, X_n\) be a sample where the order of observations is important. The indices \(1, 2, \ldots, n\) may correspond to measurements taken at time points \(t, t + \Delta t, t + 2 \delta t, \ldots, t + (n - 1) \Delta t\), for some start time \(t\) and time increment \(\Delta t\). The autocovariance at lag \(0 \leq k \leq n - 1\) is defined as:

\[ \hat{\gamma}(k) = \frac{1}{n} \sum^{n - k}_{i = 1} \left(X_{i + k} - \bar{X} \right) \left( X_i - \bar{X} \right). \]

Note that the sum is normalized by a factor \(\frac{1}{n}\). The autocorrelation is defined as normalized autocovariance:

\[ \hat{\rho}(k) = \frac{\hat{\gamma}(k)}{\hat{\gamma}(0)}. \]