In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process in question.

Auto-covariance of stochastic processes

Definition

for the expectation has the mean, then the autocovariance is given by are two instances in time.

Definition for weakly stationary process

is a weakly stationary (WSS) process, then the following are true:

and

and

is the lag time, or the amount of time by which the signal has been shifted.

The autocovariance function of a WSS process is therefore given by:

which is equivalent to

1. .

Normalization

It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably.

The definition of the normalized auto-correlation of a stochastic process is

1. .

, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

For a WSS process, the definition is

1. .

where

1. .

Properties

Symmetry property

respectively for a WSS process:

Linear filtering

is

Calculating turbulent diffusivity

Autocovariance can be used to calculate turbulent diffusivity. Turbulence in a flow can cause the fluctuation of velocity in space and time. Thus, we are able to identify turbulence through the statistics of those fluctuations.

Reynolds decomposition direction):

is the expected value of velocity, a set of velocity measurements that are assembled from points in space, moments in time or repeated experiments is required.

, and c is the concentration term) can be caused by a random walk, we can use Fick's laws of diffusion to express the turbulent flux term:

The velocity autocovariance is defined as

is the lag distance.

can be calculated using the following 3 methods:

Auto-covariance of random vectors

See also

• Autoregressive process

• Correlation

• Cross-covariance

• Cross-correlation

• Noise covariance estimation (as an application example)

References

Further reading

• Lecture notes on autocovariance from WHOI