[
]
Linear filtering
The autocovariance of a linearly filtered process
:
is
:
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
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity u the decomposition would be
u(x,y,z ...
is used to define the velocity fluctuations (assume we are now working with 1D problem and is the velocity along direction):
:
where is the true velocity, and is the expected value of velocity. If we choose a correct , all of the stochastic components of the turbulent velocity will be included in . To determine , 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 (, and ''c'' is the concentration term) can be caused by a random walk, we can use Fick's laws of diffusion
Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second ...
to express the turbulent flux term:
:
The velocity autocovariance is defined as
: or
where is the lag time, and is the lag distance.
The turbulent diffusivity 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
and
:
where 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
:.
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
:.
If the function is well-defined, its value must lie in the range
Properties
Symmetry property
:\operatorname_(t_1,t_2) = \overline[Kun 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
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity u the decomposition would be
u(x,y,z ...
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
Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second ...
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