autocovariance

With the usual notation $\backslash operatorname\{E\}$ for the expectation operator, if the stochastic process $\backslash left\backslash \{X\_t\backslash right\backslash \}$ has the mean function $\backslash mu\_t\; =\; \backslash operatorname\{E\}[X\_t]$, then the autocovariance is given by{{cite book |first=Hwei |last=Hsu |year=1997 |title=Probability, random variables, and random processes |publisher=McGraw-Hill |isbn=978-0-07-030644-8 |url-access=registration |url=https://archive.org/details/schaumsoutlineof00hsuh }}{{rp|p. 162}}

{{Equation box 1

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|equation = {{NumBlk||$\backslash operatorname\{K\}\_\{XX\}(t\_1,t\_2)\; =\; \backslash operatorname\{cov\}\backslash left[X\_\{t\_1\},\; X\_\{t\_2\}\backslash right]\; =\; \backslash operatorname\{E\}[(X\_\{t\_1\}\; -\; \backslash mu\_\{t\_1\})(X\_\{t\_2\}\; -\; \backslash mu\_\{t\_2\})]\; =\; \backslash operatorname\{E\}[X\_\{t\_1\}\; X\_\{t\_2\}]\; -\; \backslash mu\_\{t\_1\}\; \backslash mu\_\{t\_2\}$|{{EquationRef|Eq.1}}}}

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where $t\_1$ and $t\_2$ are two instances in time.

If $\backslash left\backslash \{X\_t\backslash right\backslash \}$ is a weakly stationary (WSS) process, then the following are true:{{rp|p. 163}}

:$\backslash mu\_\{t\_1\}\; =\; \backslash mu\_\{t\_2\}\; \backslash triangleq\; \backslash mu$ for all $t\_1,t\_2$

and

: for all $t$

and

:$\backslash operatorname\{K\}\_\{XX\}(t\_1,t\_2)\; =\; \backslash operatorname\{K\}\_\{XX\}(t\_2\; -\; t\_1,0)\; \backslash triangleq\; \backslash operatorname\{K\}\_\{XX\}(t\_2\; -\; t\_1)\; =\; \backslash operatorname\{K\}\_\{XX\}(\backslash tau),$

where $\backslash tau\; =\; t\_2\; -\; t\_1$ 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:{{cite book |first=Amos |last=Lapidoth |year=2009 |title=A Foundation in Digital Communication |publisher=Cambridge University Press |isbn=978-0-521-19395-5}}{{rp|p. 517}}

{{Equation box 1

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|equation = {{NumBlk||$\backslash operatorname\{K\}\_\{XX\}(\backslash tau)\; =\; \backslash operatorname\{E\}[(X\_t\; -\; \backslash mu\_t)(X\_\{t-\; \backslash tau\}\; -\; \backslash mu\_\{t-\; \backslash tau\})]\; =\; \backslash operatorname\{E\}[X\_t\; X\_\{t-\backslash tau\}]\; -\; \backslash mu\_t\; \backslash mu\_\{t-\backslash tau\}$|{{EquationRef|Eq.2}}}}

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which is equivalent to

:$\backslash operatorname\{K\}\_\{XX\}(\backslash tau)\; =\; \backslash operatorname\{E\}[(X\_\{t+\; \backslash tau\}\; -\; \backslash mu\_\{t\; +\backslash tau\})(X\_\{t\}\; -\; \backslash mu\_\{t\})]\; =\; \backslash operatorname\{E\}[X\_\{t+\backslash tau\}\; X\_t]\; -\; \backslash mu^2$.

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

:$\backslash rho\_\{XX\}(t\_1,t\_2)\; =\; \backslash frac\{\backslash operatorname\{K\}\_\{XX\}(t\_1,t\_2)\}\{\backslash sigma\_\{t\_1\}\backslash sigma\_\{t\_2\}\}\; =\; \backslash frac\{\backslash operatorname\{E\}[(X\_\{t\_1\}\; -\; \backslash mu\_\{t\_1\})(X\_\{t\_2\}\; -\; \backslash mu\_\{t\_2\})]\}\{\backslash sigma\_\{t\_1\}\backslash sigma\_\{t\_2\}\}$.

If the function $\backslash rho\_\{XX\}$ is well-defined, its value must lie in the range $[-1,1]$, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

For a WSS process, the definition is

:$\backslash rho\_\{XX\}(\backslash tau)\; =\; \backslash frac\{\backslash operatorname\{K\}\_\{XX\}(\backslash tau)\}\{\backslash sigma^2\}\; =\; \backslash frac\{\backslash operatorname\{E\}[(X\_t\; -\; \backslash mu)(X\_\{t+\backslash tau\}\; -\; \backslash mu)]\}\{\backslash sigma^2\}$.

where

:$\backslash operatorname\{K\}\_\{XX\}(0)\; =\; \backslash sigma^2$.

:$\backslash operatorname\{K\}\_\{XX\}(t\_1,t\_2)\; =\; \backslash overline\{\backslash operatorname\{K\}\_\{XX\}(t\_2,t\_1)\}$Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3{{rp|p.169}}

respectively for a WSS process:

:$\backslash operatorname\{K\}\_\{XX\}(\backslash tau)\; =\; \backslash overline\{\backslash operatorname\{K\}\_\{XX\}(-\backslash tau)\}${{rp|p.173}}

The autocovariance of a linearly filtered process $\backslash left\backslash \{Y\_t\backslash right\backslash \}$

:$Y\_t\; =\; \backslash sum\_\{k=-\backslash infty\}^\backslash infty\; a\_k\; X\_\{t+k\}\backslash ,$

is

:$K\_\{YY\}(\backslash tau)\; =\; \backslash sum\_\{k,l=-\backslash infty\}^\backslash infty\; a\_k\; a\_l\; K\_\{XX\}(\backslash tau+k-l).\backslash ,$

Autocovariance can be used to calculate turbulent diffusivity.{{Cite journal|last=Taylor|first=G. I.|date=1922-01-01|title=Diffusion by Continuous Movements|journal=Proceedings of the London Mathematical Society|language=en|volume=s2-20|issue=1|pages=196–212|doi=10.1112/plms/s2-20.1.196|bibcode=1922PLMS..220S.196T |issn=1460-244X|url=https://zenodo.org/record/1433523}} 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|date=September 2020}}.

Reynolds decomposition is used to define the velocity fluctuations $u\text{'}(x,t)$ (assume we are now working with 1D problem and $U(x,t)$ is the velocity along $x$ direction):

:$U(x,t)\; =\; \backslash langle\; U(x,t)\; \backslash rangle\; +\; u\text{'}(x,t),$

where $U(x,t)$ is the true velocity, and $\backslash langle\; U(x,t)\; \backslash rangle$ is the expected value of velocity. If we choose a correct $\backslash langle\; U(x,t)\; \backslash rangle$, all of the stochastic components of the turbulent velocity will be included in $u\text{'}(x,t)$. To determine $\backslash langle\; U(x,t)\; \backslash rangle$, 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 $\backslash langle\; u\text{'}c\text{'}\; \backslash rangle$ ($c\text{'}\; =\; c\; -\; \backslash langle\; c\; \backslash rangle$, 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:

:$J\_\{\backslash text\{turbulence\}\_x\}\; =\; \backslash langle\; u\text{'}c\text{'}\; \backslash rangle\; \backslash approx\; D\_\{T\_x\}\; \backslash frac\{\backslash partial\; \backslash langle\; c\; \backslash rangle\}\{\backslash partial\; x\}.$

The velocity autocovariance is defined as

:$K\_\{XX\}\; \backslash equiv\; \backslash langle\; u\text{'}(t\_0)\; u\text{'}(t\_0\; +\; \backslash tau)\backslash rangle$ or $K\_\{XX\}\; \backslash equiv\; \backslash langle\; u\text{'}(x\_0)\; u\text{'}(x\_0\; +\; r)\backslash rangle,$

where $\backslash tau$ is the lag time, and $r$ is the lag distance.

The turbulent diffusivity $D\_\{T\_x\}$ can be calculated using the following 3 methods:

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|If we have velocity data along a Lagrangian trajectory:

:$D\_\{T\_x\}\; =\; \backslash int\_\backslash tau^\backslash infty\; u\text{'}(t\_0)\; u\text{'}(t\_0\; +\; \backslash tau)\; \backslash ,d\backslash tau.$

|If we have velocity data at one fixed (Eulerian) location{{Citation needed|date=September 2020}}:

:$D\_\{T\_x\}\; \backslash approx\; [0.3\; \backslash pm\; 0.1]\; \backslash left[\backslash frac\{\backslash langle\; u\text{'}u\text{'}\; \backslash rangle\; +\; \backslash langle\; u\; \backslash rangle^2\}\{\backslash langle\; u\text{'}u\text{'}\; \backslash rangle\}\backslash right]\; \backslash int\_\backslash tau^\backslash infty\; u\text{'}(t\_0)\; u\text{'}(t\_0\; +\; \backslash tau)\; \backslash ,d\backslash tau.$

|If we have velocity information at two fixed (Eulerian) locations{{Citation needed|date=September 2020}}:

:$D\_\{T\_x\}\; \backslash approx\; [0.4\; \backslash pm\; 0.1]\; \backslash left[\backslash frac\{1\}\{\backslash langle\; u\text{'}u\text{'}\; \backslash rangle\}\backslash right]\; \backslash int\_r^\backslash infty\; u\text{'}(x\_0)\; u\text{'}(x\_0\; +\; r)\; \backslash ,dr,$

where $r$ is the distance separated by these two fixed locations.

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{{main|Auto-covariance matrix}}

• Autoregressive process
• Correlation
• Cross-covariance
• Cross-correlation
• Noise covariance estimation (as an application example)

{{Reflist}}

• {{cite book |first=P. G. |last=Hoel |title=Mathematical Statistics |publisher=Wiley |location=New York |year=1984 |edition=Fifth |isbn=978-0-471-89045-4 }}
• [https://web.archive.org/web/20060428122150/http://w3eos.whoi.edu/12.747/notes/lect06/l06s02.html Lecture notes on autocovariance from WHOI]

Category:Fourier analysis

Category:Autocorrelation