Autocovariance

Template:Short description Template:Correlation and covariance 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 $\mathrm{E}$ for the expectation operator, if the stochastic process $\left\{X_t\right\}$ has the mean function ${\mu }_t=\mathrm{E}[X_t]$, then the autocovariance is given byTemplate:Rp Template:Equation box 1 where $t_1$ and $t_2$ are two instances in time.

Definition for weakly stationary process

If $\left\{X_t\right\}$ is a weakly stationary (WSS) process, then the following are true:Template:Rp

${\mu }_{t_1}={\mu }_{t_2}\triangleq \mu$ for all $t_1,t_2$

and

$\mathrm{E}[|X_t{|}^2]<\mathrm{\infty }$ for all $t$

and

${\mathrm{K}}_{XX}(t_1,t_2)={\mathrm{K}}_{XX}(t_2-t_1,0)\triangleq {\mathrm{K}}_{XX}(t_2-t_1)={\mathrm{K}}_{XX}(\tau ),$

where $\tau =t_2-t_1$ 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:Template:Rp

which is equivalent to

${\mathrm{K}}_{XX}(\tau )=\mathrm{E}[(X_{t+\tau }-{\mu }_{t+\tau })(X_t-{\mu }_t)]=\mathrm{E}[X_{t+\tau }{X}_t]-{\mu }^2$.

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

${\rho }_{XX}(t_1,t_2)=\frac{{\mathrm{K}}_{XX}(t_1,t_2)}{{\sigma }_{t_1}{\sigma }_{t_2}}=\frac{\mathrm{E}[(X_{t_1}-{\mu }_{t_1})(X_{t_2}-{\mu }_{t_2})]}{{\sigma }_{t_1}{\sigma }_{t_2}}$.

If the function ${\rho }_{XX}$ 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

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

where

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

Properties

Symmetry property

${\mathrm{K}}_{XX}(t_1,t_2)=\overline{{\mathrm{K}}_{XX}(t_2,t_1)}$Template:Rp

respectively for a WSS process:

${\mathrm{K}}_{XX}(\tau )=\overline{{\mathrm{K}}_{XX}(-\tau )}$Template:Rp

Linear filtering

The autocovariance of a linearly filtered process $\left\{Y_t\right\}$

$Y_t=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_k{X}_{t+k}$

is

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

Calculating turbulent diffusivity

Autocovariance can be used to calculate turbulent diffusivity. 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 fluctuationsTemplate:Citation needed.

Reynolds decomposition is used to define the velocity fluctuations $u^{\prime }(x,t)$ (assume we are now working with 1D problem and $U(x,t)$ is the velocity along $x$ direction):

$U(x,t)=⟨U(x,t)⟩+u^{\prime }(x,t),$

where $U(x,t)$ is the true velocity, and $⟨U(x,t)⟩$ is the expected value of velocity. If we choose a correct $⟨U(x,t)⟩$, all of the stochastic components of the turbulent velocity will be included in $u^{\prime }(x,t)$. To determine $⟨U(x,t)⟩$, 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 $⟨u^{\prime }{c}^{\prime }⟩$ ($c^{\prime }=c-⟨c⟩$, 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:

$J_{{\text{turbulence}}_x}=⟨u^{\prime }{c}^{\prime }⟩\approx D_{T_x}\frac{\partial ⟨c⟩}{\partial x}.$

The velocity autocovariance is defined as

$K_{XX}\equiv ⟨u^{\prime }(t_0)u^{\prime }(t_0+\tau )⟩$ or $K_{XX}\equiv ⟨u^{\prime }(x_0)u^{\prime }(x_0+r)⟩,$

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: Template:Numbered list

See also

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

Further reading

• Template:Cite book
• Lecture notes on autocovariance from WHOI