Cross-spectrum

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.

Definition

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

Γ x y ( f ) = F { γ x y } ( f ) = ∑ τ = − ∞ ∞ γ x y ( τ ) e − 2 π i τ f , {\displaystyle \Gamma _{xy}(f)={\mathcal {F}}\{\gamma _{xy}\}(f)=\sum _{\tau =-\infty }^{\infty }\,\gamma _{xy}(\tau )\,e^{-2\,\pi \,i\,\tau \,f},}

where

γ x y ( τ ) = E ⁡ [ ( x t − μ x ) ( y t + τ − μ y ) ] {\displaystyle \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)

Γ x y ( f ) = Λ x y ( f ) − i Ψ x y ( f ) , {\displaystyle \Gamma _{xy}(f)=\Lambda _{xy}(f)-i\Psi _{xy}(f),}

and (ii) in polar coordinates

Γ x y ( f ) = A x y ( f ) e i ϕ x y ( f ) . {\displaystyle \Gamma _{xy}(f)=A_{xy}(f)\,e^{i\phi _{xy}(f)}.}

Here, the amplitude spectrum A x y {\displaystyle A_{xy}} is given by

A x y ( f ) = ( Λ x y ( f ) 2 + Ψ x y ( f ) 2 ) 1 2 , {\displaystyle A_{xy}(f)=(\Lambda _{xy}(f)^{2}+\Psi _{xy}(f)^{2})^{\frac {1}{2}},}

and the phase spectrum Φ x y {\displaystyle \Phi _{xy}} is given by

{ tan − 1 ⁡ ( Ψ x y ( f ) / Λ x y ( f ) ) if  Ψ x y ( f ) ≠ 0  and  Λ x y ( f ) ≠ 0 0 if  Ψ x y ( f ) = 0  and  Λ x y ( f ) > 0 ± π if  Ψ x y ( f ) = 0  and  Λ x y ( f ) < 0 π / 2 if  Ψ x y ( f ) > 0  and  Λ x y ( f ) = 0 − π / 2 if  Ψ x y ( f ) < 0  and  Λ x y ( f ) = 0 {\displaystyle {\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

κ x y ( f ) = A x y 2 Γ x x ( f ) Γ y y ( f ) , {\displaystyle \kappa _{xy}(f)={\frac {A_{xy}^{2}}{\Gamma _{xx}(f)\Gamma _{yy}(f)}},}

which expresses the amplitude spectrum in dimensionless units.

See also

References

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