In
probability theory
Probability theory 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 expressing it through a set ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, 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. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
, 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. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
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, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
of the process in question.
Auto-covariance of stochastic processes
Definition
With the usual notation
for the
expectation operator, if the stochastic process
has the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
function
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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
) to normalize the autocovariance function to get a time-dependent
Pearson correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the 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_(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
In statistics, there is a negative relationship or inverse relationship between two variables if higher values of one variable tend to be associated with lower values of the other. A negative relationship between two variables usually implies that ...
.
For a WSS process, the definition is
:\rho_(\tau) = \frac = \frac.
where
:\operatorname_(0) = \sigma^2.
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
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
. 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 derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
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 is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
* 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 statistic ...
* 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 the ...
* 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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
) to normalize the autocovariance function to get a time-dependent
Pearson correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the 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_(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
In statistics, there is a negative relationship or inverse relationship between two variables if higher values of one variable tend to be associated with lower values of the other. A negative relationship between two variables usually implies that ...
.
For a WSS process, the definition is
:\rho_(\tau) = \frac = \frac.
where
:\operatorname_(0) = \sigma^2.
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
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
. 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 derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
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 is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
* 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 statistic ...
* 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 the ...
* 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