# 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: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::p. 163

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

and

$\displaystyle{ \operatorname{E}[|X_t|^2] \lt \infty }$ 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::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

$\displaystyle{ \operatorname{K}_{XX}(t_1,t_2) = \overline{\operatorname{K}_{XX}(t_2,t_1)} }$:p.169

respectively for a WSS process:

$\displaystyle{ \operatorname{K}_{XX}(\tau) = \overline{\operatorname{K}_{XX}(-\tau)} }$: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. 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.