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Autocovariance - Wikipedia

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Concept in probability and statistics

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

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

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

μ t 1 = μ t 2 ≜ μ {\displaystyle \mu _{t_{1}}=\mu _{t_{2}}\triangleq \mu } for all t 1 , t 2 {\displaystyle t_{1},t_{2}}

and

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

and

K X X ⁡ ( t 1 , t 2 ) = K X X ⁡ ( t 2 − t 1 , 0 ) ≜ K X X ⁡ ( t 2 − t 1 ) = K X X ⁡ ( τ ) , {\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 τ = t 2 − t 1 {\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}

which is equivalent to

K X X ⁡ ( τ ) = E ⁡ [ ( X t + τ − μ t + τ ) ( X t − μ t ) ] = E ⁡ [ X t + τ X t ] − μ 2 {\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}} .

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

ρ X X ( t 1 , t 2 ) = K X X ⁡ ( t 1 , t 2 ) σ t 1 σ t 2 = E ⁡ [ ( X t 1 − μ t 1 ) ( X t 2 − μ t 2 ) ] σ t 1 σ t 2 {\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 ρ X X {\displaystyle \rho _{XX}} is well-defined, its value must lie in the range [ − 1 , 1 ] {\displaystyle [-1,1]} , with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

For a WSS process, the definition is

ρ X X ( τ ) = K X X ⁡ ( τ ) σ 2 = E ⁡ [ ( X t − μ ) ( X t + τ − μ ) ] σ 2 {\displaystyle \rho _{XX}(\tau )={\frac {\operatorname {K} _{XX}(\tau )}{\sigma ^{2}}}={\frac {\operatorname {E} [(X_{t}-\mu )(X_{t+\tau }-\mu )]}{\sigma ^{2}}}} .

where

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

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

respectively for a WSS process:

K X X ⁡ ( τ ) = K X X ⁡ ( − τ ) ¯ {\displaystyle \operatorname {K} _{XX}(\tau )={\overline {\operatorname {K} _{XX}(-\tau )}}} ^[3]: p.173

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

Y t = ∑ k = − ∞ ∞ a k X t + k {\displaystyle Y_{t}=\sum _{k=-\infty }^{\infty }a_{k}X_{t+k}\,}

is

K Y Y ( τ ) = ∑ k , l = − ∞ ∞ a k a l K X X ( τ + k − l ) . {\displaystyle K_{YY}(\tau )=\sum _{k,l=-\infty }^{\infty }a_{k}a_{l}K_{XX}(\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^{[citation needed]}.

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

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

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

J turbulence x = ⟨ u ′ c ′ ⟩ ≈ D T x ∂ ⟨ c ⟩ ∂ x . {\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

K X X ≡ ⟨ u ′ ( t 0 ) u ′ ( t 0 + τ ) ⟩ {\displaystyle K_{XX}\equiv \langle u'(t_{0})u'(t_{0}+\tau )\rangle } or K X X ≡ ⟨ u ′ ( x 0 ) u ′ ( x 0 + r ) ⟩ , {\displaystyle K_{XX}\equiv \langle u'(x_{0})u'(x_{0}+r)\rangle ,}

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

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

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

• Hoel, P. G. (1984). Mathematical Statistics (Fifth ed.). New York: Wiley. ISBN 978-0-471-89045-4.
• Lecture notes on autocovariance from WHOI

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