Autocovariance - crowlogic/arb4j GitHub Wiki

An autocovariance function is a tool used in the statistical analysis of stochastic processes. It describes the autocovariance between different points in time of a time series or stochastic process.

In other words, the autocovariance function quantifies the dependence or correlation between values of a variable at different time points. It is especially useful in understanding the predictability of a process over time and is used extensively in fields like signal processing, time series analysis, and control theory.

The autocovariance function, often denoted by $C(t, s)$, is defined as the expected value (or mean) of the product of the deviations of a random variable from its expected value at two different time points, $t$ and $s$.

Mathematically, it can be expressed as:

$$C(t, s) = E{[X(t) - m(t)][X(s) - m(s)]},$$

where:

• $X(t)$ and $X(s)$ are the values of the process at times $t$ and $s$ respectively,
• $m(t)$ and $m(s)$ are the expected values of the process at those times,
• $E$ represents the expectation operator.

The autocovariance function measures the linear dependence between two points on the same series observed at different times. An important characteristic of the autocovariance function is that it depends only on the time difference $h = t - s$, not on the actual time at which the covariance is computed. This property is known as stationarity.

In practical terms, the autocovariance function helps determine the extent to which an observation at one point in time can predict a subsequent observation. A high autocovariance implies that the values at one time point are useful for predicting the values at future time points, while a low autocovariance suggests little or no relationship between the values at different time points.