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.
With the usual notation
\operatorname{E}
\left\{Xt\right\}
\mut=\operatorname{E}[Xt]
t1
t2
If
\left\{Xt\right\}
\mu | |
t1 |
=
\mu | |
t2 |
\triangleq\mu
t1,t2
and
2] | |
\operatorname{E}[|X | |
t| |
<infty
t
and
\operatorname{K}XX(t1,t2)=\operatorname{K}XX(t2-t1,0)\triangleq\operatorname{K}XX(t2-t1)=\operatorname{K}XX(\tau),
where
\tau=t2-t1
The autocovariance function of a WSS process is therefore given by:[2]
which is equivalent to
\operatorname{K}XX(\tau)=\operatorname{E}[(Xt+-\mut)(Xt-\mut)]=\operatorname{E}[Xt+\tauXt]-\mu2
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
\rhoXX(t1,t2)=
\operatorname{K | |
XX |
(t1,t2)}{\sigma
t1 |
\sigma | |
t2 |
If the function
\rhoXX
[-1,1]
For a WSS process, the definition is
\rhoXX(\tau)=
\operatorname{K | |
XX |
(\tau)}{\sigma2}=
\operatorname{E | |
[(X |
t-\mu)(Xt+\tau-\mu)]}{\sigma2}
where
\operatorname{K}XX(0)=\sigma2
\operatorname{K}XX(t1,t2)=\overline{\operatorname{K}XX(t2,t1)}
\operatorname{K}XX(\tau)=\overline{\operatorname{K}XX(-\tau)}
The autocovariance of a linearly filtered process
\left\{Yt\right\}
Yt=
infty | |
\sum | |
k=-infty |
akXt+k
KYY(\tau)=
infty | |
\sum | |
k,l=-infty |
akalKXX(\tau+k-l).
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.
Reynolds decomposition is used to define the velocity fluctuations
u'(x,t)
U(x,t)
x
U(x,t)=\langleU(x,t)\rangle+u'(x,t),
where
U(x,t)
\langleU(x,t)\rangle
\langleU(x,t)\rangle
u'(x,t)
\langleU(x,t)\rangle
If we assume the turbulent flux
\langleu'c'\rangle
c'=c-\langlec\rangle
J | |
turbulencex |
=\langleu'c'\rangle ≈
D | |
Tx |
\partial\langlec\rangle | |
\partialx |
.
The velocity autocovariance is defined as
KXX\equiv\langleu'(t0)u'(t0+\tau)\rangle
KXX\equiv\langleu'(x0)u'(x0+r)\rangle,
where
\tau
r
The turbulent diffusivity
D | |
Tx |
See main article: Auto-covariance matrix.