Concept in probability and statistics
Part of a series on Statistics |
Correlation and covariance |
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![](//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/CorrelationIcon.svg/100px-CorrelationIcon.svg.png) |
For random vectors - Autocorrelation matrix
- Cross-correlation matrix
- Auto-covariance matrix
- Cross-covariance matrix
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For deterministic signals |
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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
Definition
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
![{\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})=\operatorname {cov} \left[X_{t_{1}},X_{t_{2}}\right]=\operatorname {E} [(X_{t_{1}}-\mu _{t_{1}})(X_{t_{2}}-\mu _{t_{2}})]=\operatorname {E} [X_{t_{1}}X_{t_{2}}]-\mu _{t_{1}}\mu _{t_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92c3c2da09a072e13e513b1bbe6933e25cde5645) | | (Eq.1) |
where
and
are two instances in time.
Definition for weakly stationary process
If
is a weakly stationary (WSS) process, then the following are true:[1]: p. 163
for all ![{\displaystyle t_{1},t_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e76daf1a59dca26c96dbca2863a1c236b15b5a1)
and
for all ![{\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
and
![{\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})=\operatorname {K} _{XX}(t_{2}-t_{1},0)\triangleq \operatorname {K} _{XX}(t_{2}-t_{1})=\operatorname {K} _{XX}(\tau ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70c338953af0513dfc62451b0d42f734a2f45601)
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
![{\displaystyle \operatorname {K} _{XX}(\tau )=\operatorname {E} [(X_{t}-\mu _{t})(X_{t-\tau }-\mu _{t-\tau })]=\operatorname {E} [X_{t}X_{t-\tau }]-\mu _{t}\mu _{t-\tau }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bebac457858097c963a0420549814469b9957a91) | | (Eq.2) |
which is equivalent to
.
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
.
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
Symmetry property
[3]: p.169
respectively for a WSS process:
[3]: p.173
Linear filtering
The autocovariance of a linearly filtered process
![{\displaystyle Y_{t}=\sum _{k=-\infty }^{\infty }a_{k}X_{t+k}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e6e48b511109956168c704db8951347c701f38)
is
![{\displaystyle K_{YY}(\tau )=\sum _{k,l=-\infty }^{\infty }a_{k}a_{l}K_{XX}(\tau +k-l).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c848ab818d0847ee4b900b072df0edc317cf46)
Calculating turbulent diffusivity
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):
![{\displaystyle U(x,t)=\langle U(x,t)\rangle +u'(x,t),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9de01b0f0e0869a827821afd0e96b847a61e23ba)
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:
![{\displaystyle J_{{\text{turbulence}}_{x}}=\langle u'c'\rangle \approx D_{T_{x}}{\frac {\partial \langle c\rangle }{\partial x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70bab75600baf80bdbdbdce847b5b81a2f429ae0)
The velocity autocovariance is defined as
or ![{\displaystyle K_{XX}\equiv \langle u'(x_{0})u'(x_{0}+r)\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/182d51c712578161252f8e82a2b5714e0e9397ca)
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:
![{\displaystyle D_{T_{x}}=\int _{\tau }^{\infty }u'(t_{0})u'(t_{0}+\tau )\,d\tau .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/993e8a71ccbc9f0eb24b926315ef7a66d1d07d23)
- If we have velocity data at one fixed (Eulerian) location[citation needed]:
![{\displaystyle D_{T_{x}}\approx [0.3\pm 0.1]\left[{\frac {\langle u'u'\rangle +\langle u\rangle ^{2}}{\langle u'u'\rangle }}\right]\int _{\tau }^{\infty }u'(t_{0})u'(t_{0}+\tau )\,d\tau .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec64e8eaa03768b3b941ad1e1a0ad878dd8384f7)
- If we have velocity information at two fixed (Eulerian) locations[citation needed]:
![{\displaystyle D_{T_{x}}\approx [0.4\pm 0.1]\left[{\frac {1}{\langle u'u'\rangle }}\right]\int _{r}^{\infty }u'(x_{0})u'(x_{0}+r)\,dr,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c27ff93a23a26b35510de18e610047c20baf639c)
where
is the distance separated by these two fixed locations.
Auto-covariance of random vectors
See also
References
- ^ 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
- Hoel, P. G. (1984). Mathematical Statistics (Fifth ed.). New York: Wiley. ISBN 978-0-471-89045-4.
- Lecture notes on autocovariance from WHOI