Autocovariance

In probability theory and statistics, given a stochastic process $X=(X_{t})$ , the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. With the usual notation E for the expectation operator, if the process has the mean function $\mu _{t}=E[X_{t}]$ , then the autocovariance is given by

C_{XX}(t,s)={\text{cov}}(X_{t},X_{s})=E[(X_{t}-\mu _{t})(X_{s}-\mu _{s})]=E[X_{t}X_{s}]-\mu _{t}\mu _{s},\,

where t and s are two time periods or moments in time.

Autocovariance is closely related to the autocorrelation of the process in question.

In the case of a multivariate random vector $X=(X_{1},X_{2},...,X_{n})$ , the autocovariance becomes a square n × n matrix with entries given by $C_{X_{i}X_{j}}(t,s)={\text{cov}}(X_{i,t},X_{j,s})$ and commonly referred to as the autocovariance matrix associated with vectors $X_{t}$ and $X_{s}$ .

Weak stationarity

If X(t) is a weakly stationary process, then the following are true:

\mu _{t}=\mu _{s}=\mu

for all t, s

and

C_{XX}(t,s)=C_{XX}(s-t)=C_{XX}(\tau ),

where $\tau =|s-t|$ is the lag time, or the amount of time by which the signal has been shifted.

Normalization

When normalizing the autocovariance C of a weakly stationary process with its variance $C_{XX}(0)=\sigma ^{2}$ , one obtains the autocorrelation coefficient^[1]

\rho _{XX}(\tau )={\frac {C_{XX}(\tau )}{\sigma ^{2}}}

with $-1\leq \rho _{XX}(\tau )\leq 1$ .

Properties

The autocovariance of a linearly filtered process $Y_{t}$

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

is

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

Calculating turbulent diffusivity

Autocovariance can be used to calculate turbulent diffusivity.^[2] Turbulence in a flow can cause the fluctulation of velocity in space and time. Thus, we are able to identify turbulence through the statistics of those fluctuations.

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

U(x,t)=\langle U(x,t)\rangle +u'(x,t),

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

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

C_{XX}\equiv \langle u'(t_{0})u'(t_{0}+\tau )\rangle

or

C_{XX}\equiv \langle u'(x_{0})u'(x_{0}+r)\rangle ,

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

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

If we have velocity data along a Lagrangian trajectory:
$D_{T_{x}}=\int _{\tau }^{\infty }u'(t_{0})u'(t_{0}+\tau )\,d\tau .$
If we have velocity data at one fixed (Eulerian) location:
$D_{T_{x}}\approx [0.3\pm 0.1]\left[{\frac {\langle u'u'\rangle +\langle u\rangle ^{2}}{\langle u'u'\rangle }}\right]\int _{\tau }^{\infty }u'(t_{0})u'(t_{0}+\tau )\,d\tau .$
If we have velocity information at two fixed (Eulerian) locations:
$D_{T_{x}}\approx [0.4\pm 0.1]\left[{\frac {1}{\langle u'u'\rangle }}\right]\int _{r}^{\infty }u'(x_{0})u'(x_{0}+r)\,dr,$
where $r$ is the distance separated by these two fixed locations.

References

↑ Westwick, David T. (2003). Identification of Nonlinear Physiological Systems. IEEE Press. pp. 17–18. ISBN 0-471-27456-9.
↑ Taylor, G. I. (1922-01-01). "Diffusion by Continuous Movements". Proceedings of the London Mathematical Society. s2-20 (1): 196–212. doi:10.1112/plms/s2-20.1.196. ISSN 1460-244X.

Hoel, P. G. (1984). Mathematical Statistics (Fifth ed.). New York: Wiley. ISBN 0-471-89045-6.
Lecture notes on autocovariance from WHOI

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[nonlinSystems-1] Westwick, David T. (2003). Identification of Nonlinear Physiological Systems. IEEE Press. pp. 17–18. ISBN 0-471-27456-9.

[2] Taylor, G. I. (1922-01-01). "Diffusion by Continuous Movements". Proceedings of the London Mathematical Society. s2-20 (1): 196–212. doi:10.1112/plms/s2-20.1.196. ISSN 1460-244X.