Cross-spectrum

Last updated March 22, 2019

In time series analysis, the cross-spectrum is used as part of a frequency domain analysis of the cross-correlation or cross-covariance between two time series.

In electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.

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 for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology.

In probability and statistics, given two stochastic processes $and, the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation; for the expectation operator, if the processes have the mean functions and, then the cross-covariance is given by$

Definition

Let $(X_{t},Y_{t})$ represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions $\gamma _{xx}$ and $\gamma _{yy}$ and cross-covariance function $\gamma _{xy}$ . Then the cross-spectrum $\Gamma _{xy}$ is defined as the Fourier transform of $\gamma _{xy}$ ^[1]

Stochastic process mathematical object usually defined as a collection of random variables

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

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.

The Fourier transform (FT) decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest, one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time. In image processing the notion of a time domain is replaced by that of a spatial domain where the intensity of a signal is identified by its spatial position rather than at any point in time.

\Gamma _{xy}(f)={\mathcal {F}}\{\gamma _{xy}\}(f)=\sum _{\tau =-\infty }^{\infty }\,\gamma _{xy}(\tau )\,e^{-2\,\pi \,i\,\tau \,f},

where

\gamma _{xy}(\tau )=\operatorname {E} [(x_{t}-\mu _{x})(y_{t+\tau }-\mu _{y})]

.

The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and (ii) its imaginary part (quadrature spectrum)

\Gamma _{xy}(f)=\Lambda _{xy}(f)+i\Psi _{xy}(f),

and (ii) in polar coordinates

\Gamma _{xy}(f)=A_{xy}(f)\,e^{i\phi _{xy}(f)}.

Here, the amplitude spectrum $A_{xy}$ is given by

A_{xy}(f)=(\Lambda _{xy}(f)^{2}+\Psi _{xy}(f)^{2})^{\frac {1}{2}},

and the phase spectrum $\Phi _{xy}$ is given by

{\begin{cases}\tan ^{-1}(\Psi _{xy}(f)/\Lambda _{xy}(f))&{\text{if }}\Psi _{xy}(f)\neq 0{\text{ and }}\Lambda _{xy}(f)\neq 0\\0&{\text{if }}\Psi _{xy}(f)=0{\text{ and }}\Lambda _{xy}(f)>0\\\pm \pi &{\text{if }}\Psi _{xy}(f)=0{\text{ and }}\Lambda _{xy}(f)<0\\\pi /2&{\text{if }}\Psi _{xy}(f)>0{\text{ and }}\Lambda _{xy}(f)=0\\-\pi /2&{\text{if }}\Psi _{xy}(f)<0{\text{ and }}\Lambda _{xy}(f)=0\\\end{cases}}

Squared coherency spectrum

The squared coherency spectrum is given by

\kappa _{xy}(f)={\frac {A_{xy}^{2}}{\Gamma _{xx}(f)\Gamma _{yy}(f)}},

which expresses the amplitude spectrum in dimensionless units.

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References

↑ von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge Univ Pr. ISBN 0-521-01230-9.

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[1] von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge Univ Pr. ISBN 0-521-01230-9.

Cross-spectrum

Contents

Definition

Squared coherency spectrum

See also

Related Research Articles

References