Autocovariance
Part of a series on Statistics |
Correlation and covariance |
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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
[edit]Definition
[edit]With the usual notation for the expectation operator, if the stochastic process has the mean function , then the autocovariance is given by[1]: p. 162
(Eq.1) |
where and are two instances in time.
Definition for weakly stationary process
[edit]If is a weakly stationary (WSS) process, then the following are true:[1]: p. 163
- for all
and
- for all
and
where 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:[2]: p. 517
(Eq.2) |
which is equivalent to
- .
Normalization
[edit]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
- .
If the function is well-defined, its value must lie in the range , with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.
For a WSS process, the definition is
- .
where
- .
Properties
[edit]Symmetry property
[edit]- [3]: p.169
respectively for a WSS process:
- [3]: p.173
Linear filtering
[edit]The autocovariance of a linearly filtered process
is
Calculating turbulent diffusivity
[edit]Autocovariance can be used to calculate turbulent diffusivity.[4] 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[citation needed].
Reynolds decomposition is used to define the velocity fluctuations (assume we are now working with 1D problem and is the velocity along direction):
where is the true velocity, and is the expected value of velocity. If we choose a correct , all of the stochastic components of the turbulent velocity will be included in . To determine , a set of velocity measurements that are assembled from points in space, moments in time or repeated experiments is required.
If we assume the turbulent flux (, 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
- or
where is the lag time, and is the lag distance.
The turbulent diffusivity can be calculated using the following 3 methods:
- If we have velocity data along a Lagrangian trajectory:
- If we have velocity data at one fixed (Eulerian) location[citation needed]:
- If we have velocity information at two fixed (Eulerian) locations[citation needed]:
- where is the distance separated by these two fixed locations.
Auto-covariance of random vectors
[edit]See also
[edit]- Autoregressive process
- Correlation
- Cross-covariance
- Cross-correlation
- Noise covariance estimation (as an application example)
References
[edit]- ^ a b Hsu, Hwei (1997). Probability, random variables, and random processes. McGraw-Hill. ISBN 978-0-07-030644-8.
- ^ Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5.
- ^ a b Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3
- ^ Taylor, G. I. (1922-01-01). "Diffusion by Continuous Movements" (PDF). Proceedings of the London Mathematical Society. s2-20 (1): 196–212. doi:10.1112/plms/s2-20.1.196. ISSN 1460-244X.
Further reading
[edit]- Hoel, P. G. (1984). Mathematical Statistics (Fifth ed.). New York: Wiley. ISBN 978-0-471-89045-4.
- Lecture notes on autocovariance from WHOI