Autocovariance

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 ${\displaystyle \operatorname {E} }$ for the expectation operator, if the stochastic process ${\displaystyle \left\{X_{t}\right\}}$ has the mean function ${\displaystyle \mu _{t}=\operatorname {E} [X_{t}]}$, then the autocovariance is given by ^{[1] : p. 162}

${\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})=\operatorname {cov} \left[X_{t_{1}},X_{t_{2}}\right]=\operatorname {E} [(X_{t_{1}}-\mu _{t_{1}})(X_{t_{2}}-\mu _{t_{2}})]=\operatorname {E} [X_{t_{1}}X_{t_{2}}]-\mu _{t_{1}}\mu _{t_{2}}}$

(Eq.1)

where ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ are two instances in time.

Definition for weakly stationary process

If ${\displaystyle \left\{X_{t}\right\}}$ is a weakly stationary (WSS) process, then the following are true: ^{[1] : p. 163}

${\displaystyle \mu _{t_{1}}=\mu _{t_{2}}\triangleq \mu }$ for all ${\displaystyle t_{1},t_{2}}$

and

${\displaystyle \operatorname {E} [|X_{t}|^{2}]<\infty }$ for all ${\displaystyle t}$

and

${\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})=\operatorname {K} _{XX}(t_{2}-t_{1},0)\triangleq \operatorname {K} _{XX}(t_{2}-t_{1})=\operatorname {K} _{XX}(\tau ),}$

where ${\displaystyle \tau =t_{2}-t_{1}}$ 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: ^{[2] : p. 517}

${\displaystyle \operatorname {K} _{XX}(\tau )=\operatorname {E} [(X_{t}-\mu _{t})(X_{t-\tau }-\mu _{t-\tau })]=\operatorname {E} [X_{t}X_{t-\tau }]-\mu _{t}\mu _{t-\tau }}$

(Eq.2)

which is equivalent to

${\displaystyle \operatorname {K} _{XX}(\tau )=\operatorname {E} [(X_{t+\tau }-\mu _{t+\tau })(X_{t}-\mu _{t})]=\operatorname {E} [X_{t+\tau }X_{t}]-\mu ^{2}}$.

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

${\displaystyle \rho _{XX}(t_{1},t_{2})={\frac {\operatorname {K} _{XX}(t_{1},t_{2})}{\sigma _{t_{1}}\sigma _{t_{2}}}}={\frac {\operatorname {E} [(X_{t_{1}}-\mu _{t_{1}})(X_{t_{2}}-\mu _{t_{2}})]}{\sigma _{t_{1}}\sigma _{t_{2}}}}}$.

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

For a WSS process, the definition is

${\displaystyle \rho _{XX}(\tau )={\frac {\operatorname {K} _{XX}(\tau )}{\sigma ^{2}}}={\frac {\operatorname {E} [(X_{t}-\mu )(X_{t+\tau }-\mu )]}{\sigma ^{2}}}}$.

where

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

Properties

Symmetry property

${\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})={\overline {\operatorname {K} _{XX}(t_{2},t_{1})}}}$ ^{[3] : p.169}

respectively for a WSS process:

${\displaystyle \operatorname {K} _{XX}(\tau )={\overline {\operatorname {K} _{XX}(-\tau )}}}$ ^{[3] : p.173}

Linear filtering

The autocovariance of a linearly filtered process ${\displaystyle \left\{Y_{t}\right\}}$

${\displaystyle Y_{t}=\sum _{k=-\infty }^{\infty }a_{k}X_{t+k}\,}$

is

${\displaystyle K_{YY}(\tau )=\sum _{k,l=-\infty }^{\infty }a_{k}a_{l}K_{XX}(\tau +k-l).\,}$

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 ${\displaystyle u'(x,t)}$ (assume we are now working with 1D problem and ${\displaystyle U(x,t)}$ is the velocity along ${\displaystyle x}$ direction):

${\displaystyle U(x,t)=\langle U(x,t)\rangle +u'(x,t),}$

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

${\displaystyle J_{{\text{turbulence}}_{x}}=\langle u'c'\rangle \approx D_{T_{x}}{\frac {\partial \langle c\rangle }{\partial x}}.}$

The velocity autocovariance is defined as

${\displaystyle K_{XX}\equiv \langle u'(t_{0})u'(t_{0}+\tau )\rangle }$ or ${\displaystyle K_{XX}\equiv \langle u'(x_{0})u'(x_{0}+r)\rangle ,}$

where ${\displaystyle \tau }$ is the lag time, and ${\displaystyle r}$ is the lag distance.

The turbulent diffusivity ${\displaystyle D_{T_{x}}}$ can be calculated using the following 3 methods:

1. If we have velocity data along a Lagrangian trajectory :

   ${\displaystyle D_{T_{x}}=\int _{\tau }^{\infty }u'(t_{0})u'(t_{0}+\tau )\,d\tau .}$

2. If we have velocity data at one fixed (Eulerian) location^{[ citation needed ]}:

   ${\displaystyle D_{T_{x}}\approx [0.3\pm 0.1]\left[{\frac {\langle u'u'\rangle +\langle u\rangle ^{2}}{\langle u'u'\rangle }}\right]\int _{\tau }^{\infty }u'(t_{0})u'(t_{0}+\tau )\,d\tau .}$

3. If we have velocity information at two fixed (Eulerian) locations^{[ citation needed ]}:

   ${\displaystyle D_{T_{x}}\approx [0.4\pm 0.1]\left[{\frac {1}{\langle u'u'\rangle }}\right]\int _{r}^{\infty }u'(x_{0})u'(x_{0}+r)\,dr,}$

   where ${\displaystyle r}$ is the distance separated by these two fixed locations.

Auto-covariance of random vectors

Main article: Auto-covariance matrix

See also

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

References

1. Hsu, Hwei (1997). Probability, random variables, and random processes . McGraw-Hill. ISBN   978-0-07-030644-8.
2. Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN   978-0-521-19395-5.
3. Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3
4. 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