Английская Википедия:Autocovariance

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Шаблон: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 <math>\operatorname{E}</math> for the expectation operator, if the stochastic process <math>\left\{X_t\right\}</math> has the mean function <math>\mu_t = \operatorname{E}[X_t]</math>, then the autocovariance is given byШаблон:Rp Шаблон:Equation box 1 where <math>t_1</math> and <math>t_2</math> are two instances in time.

Definition for weakly stationary process

If <math>\left\{X_t\right\}</math> is a weakly stationary (WSS) process, then the following are true:Шаблон:Rp

<math>\mu_{t_1} = \mu_{t_2} \triangleq \mu</math> for all <math>t_1,t_2</math>

and

<math>\operatorname{E}[|X_t|^2] < \infty</math> for all <math>t</math>

and

<math>\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),</math>

where <math>\tau = t_2 - t_1</math> 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:Шаблон:Rp

which is equivalent to

<math>\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 </math>.

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

<math>\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}}</math>.

If the function <math>\rho_{XX}</math> is well-defined, its value must lie in the range <math>[-1,1]</math>, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

For a WSS process, the definition is

<math>\rho_{XX}(\tau) = \frac{\operatorname{K}_{XX}(\tau)}{\sigma^2} = \frac{\operatorname{E}[(X_t - \mu)(X_{t+\tau} - \mu)]}{\sigma^2}</math>.

where

<math>\operatorname{K}_{XX}(0) = \sigma^2</math>.

Properties

Symmetry property

<math>\operatorname{K}_{XX}(t_1,t_2) = \overline{\operatorname{K}_{XX}(t_2,t_1)}</math>Шаблон:Rp

respectively for a WSS process:

<math>\operatorname{K}_{XX}(\tau) = \overline{\operatorname{K}_{XX}(-\tau)}</math>Шаблон:Rp

Linear filtering

The autocovariance of a linearly filtered process <math>\left\{Y_t\right\}</math>

<math>Y_t = \sum_{k=-\infty}^\infty a_k X_{t+k}\,</math>

is

<math>K_{YY}(\tau) = \sum_{k,l=-\infty}^\infty a_k a_l K_{XX}(\tau+k-l).\,</math>

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 fluctuationsШаблон:Citation needed.

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

<math>U(x,t) = \langle U(x,t) \rangle + u'(x,t),</math>

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

<math>J_{\text{turbulence}_x} = \langle u'c' \rangle \approx D_{T_x} \frac{\partial \langle c \rangle}{\partial x}.</math>

The velocity autocovariance is defined as

<math>K_{XX} \equiv \langle u'(t_0) u'(t_0 + \tau)\rangle</math> or <math>K_{XX} \equiv \langle u'(x_0) u'(x_0 + r)\rangle,</math>

where <math>\tau</math> is the lag time, and <math>r</math> is the lag distance.

The turbulent diffusivity <math>D_{T_x}</math> can be calculated using the following 3 methods: Шаблон:Numbered list

See also

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

Further reading

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