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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, given a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
, the autocovariance is a function that gives the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
of the process with itself at pairs of time points. Autocovariance is closely related to the
autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...
of the process in question.


Auto-covariance of stochastic processes


Definition

With the usual notation \operatorname for the expectation operator, if the stochastic process \left\ has the
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
function \mu_t = \operatorname _t/math>, then the autocovariance is given by where t_1 and t_2 are two instances in time.


Definition for weakly stationary process

If \left\ is a weakly stationary (WSS) process, then the following are true: :\mu_ = \mu_ \triangleq \mu for all t_1,t_2 and :\operatorname X_ - \mu_)(X_ - \mu_)= \operatorname _ X_t- \mu^2 .


Normalization

It is common practice in some disciplines (e.g. statistics and
time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
) to normalize the autocovariance function to get a time-dependent
Pearson correlation coefficient In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviatio ...
. 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_(t_1,t_2) = \frac = \frac. If the function \rho_ is well-defined, its value must lie in the range 1,1/math>, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. For a WSS process, the definition is :\rho_(\tau) = \frac = \frac. where :\operatorname_(0) = \sigma^2.


Properties


Symmetry property

:\operatorname_(t_1,t_2) = \overlineKun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3 respectively for a WSS process: :\operatorname_(\tau) = \overline


Linear filtering

The autocovariance of a linearly filtered process \left\ :Y_t = \sum_^\infty a_k X_\, is :K_(\tau) = \sum_^\infty a_k a_l K_(\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. Reynolds decomposition is used to define the velocity fluctuations u'(x,t) (assume we are now working with 1D problem and U(x,t) is the velocity along x direction): :U(x,t) = \langle U(x,t) \rangle + u'(x,t), where U(x,t) is the true velocity, and \langle U(x,t) \rangle is the expected value of velocity. If we choose a correct \langle U(x,t) \rangle, all of the stochastic components of the turbulent velocity will be included in u'(x,t). To determine \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 \langle u'c' \rangle (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_ = \langle u'c' \rangle \approx D_ \frac. The velocity autocovariance is defined as :K_ \equiv \langle u'(t_0) u'(t_0 + \tau)\rangle or K_ \equiv \langle u'(x_0) u'(x_0 + r)\rangle, where \tau is the lag time, and r is the lag distance. The turbulent diffusivity D_ can be calculated using the following 3 methods:


Auto-covariance of random vectors


See also

*
Autoregressive process In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregre ...
*
Correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
*
Cross-covariance In probability and statistics, given two stochastic processes \left\ and \left\, the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation \operatorname E for th ...
*
Cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
* Noise covariance estimation (as an application example)


References


Further reading

* {{cite book , first=P. G. , last=Hoel , title=Mathematical Statistics , publisher=Wiley , location=New York , year=1984 , edition=Fifth , isbn=978-0-471-89045-4
Lecture notes on autocovariance from WHOI
Fourier analysis Autocorrelation>X_t, ^2< \infty for all t and :\operatorname_(t_1,t_2) = \operatorname_(t_2 - t_1,0) \triangleq \operatorname_(t_2 - t_1) = \operatorname_(\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: which is equivalent to :\operatorname_(\tau) = \operatorname X_ - \mu_)(X_ - \mu_)= \operatorname _ X_t- \mu^2 .


Normalization

It is common practice in some disciplines (e.g. statistics and
time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
) to normalize the autocovariance function to get a time-dependent
Pearson correlation coefficient In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviatio ...
. 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_(t_1,t_2) = \frac = \frac. If the function \rho_ is well-defined, its value must lie in the range 1,1/math>, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. For a WSS process, the definition is :\rho_(\tau) = \frac = \frac. where :\operatorname_(0) = \sigma^2.


Properties


Symmetry property

:\operatorname_(t_1,t_2) = \overlineKun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3 respectively for a WSS process: :\operatorname_(\tau) = \overline


Linear filtering

The autocovariance of a linearly filtered process \left\ :Y_t = \sum_^\infty a_k X_\, is :K_(\tau) = \sum_^\infty a_k a_l K_(\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. Reynolds decomposition is used to define the velocity fluctuations u'(x,t) (assume we are now working with 1D problem and U(x,t) is the velocity along x direction): :U(x,t) = \langle U(x,t) \rangle + u'(x,t), where U(x,t) is the true velocity, and \langle U(x,t) \rangle is the expected value of velocity. If we choose a correct \langle U(x,t) \rangle, all of the stochastic components of the turbulent velocity will be included in u'(x,t). To determine \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 \langle u'c' \rangle (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_ = \langle u'c' \rangle \approx D_ \frac. The velocity autocovariance is defined as :K_ \equiv \langle u'(t_0) u'(t_0 + \tau)\rangle or K_ \equiv \langle u'(x_0) u'(x_0 + r)\rangle, where \tau is the lag time, and r is the lag distance. The turbulent diffusivity D_ can be calculated using the following 3 methods:


Auto-covariance of random vectors


See also

*
Autoregressive process In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregre ...
*
Correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
*
Cross-covariance In probability and statistics, given two stochastic processes \left\ and \left\, the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation \operatorname E for th ...
*
Cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
* Noise covariance estimation (as an application example)


References


Further reading

* {{cite book , first=P. G. , last=Hoel , title=Mathematical Statistics , publisher=Wiley , location=New York , year=1984 , edition=Fifth , isbn=978-0-471-89045-4
Lecture notes on autocovariance from WHOI
Fourier analysis Autocorrelation