Inflation-restriction exact sequence - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T09:50:56Z

WP:CHECKWIKI error fixes using AWB (10093)

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-{{Technical|date=April 2011}}
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Deltahedron: cite Gille & Szamuely (2006)

2012-10-31T21:08:36Z

cite Gille & Szamuely (2006)

← Older revision		Revision as of 23:08, 31 October 2012
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			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 <math>(X_t,Y_t)</math> represent a pair of [[stochastic process]]es that are jointly [[wide sense stationary]] with covariance functions <math>\gamma_{xx}</math> and <math>\gamma_{yy}</math> and [[Cross-correlation#Time_series_analysis\|cross-covariance function]] <math>\gamma_{xy}</math>. Then the cross spectrum <math>\Gamma_{xy}</math> is defined as the [[Fourier transform]] of <math>\gamma_{xy}</math> <ref>{{Cite book
			\| publisher = Cambridge Univ Pr
			\| isbn = 0-521-01230-9
			\| last = von Storch
			\| first = H.
			\| coauthors = F. W Zwiers
			\| title = Statistical analysis in climate research
			\| year = 2001
			}}</ref>

			: <math>
			\Gamma_{xy}(f)= \mathcal{F}\{\gamma_{xy}\}(f) = \sum_{\tau=-\infty}^\infty \,\gamma_{xy}(\tau) \,e^{-2\,\pi\,i\,\tau\,f} .
			</math>

			The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and its imaginary part (quadrature spectrum)
			: <math>
			\Gamma_{xy}(f)= \Lambda_{xy}(f) + i \Psi_{xy}(f) ,
			</math>

			and (ii) in polar coordinates
			: <math>
			\Gamma_{xy}(f)= A_{xy}(f) \,e^{i \phi_{xy}(f) } .
			</math>
			Here, the amplitude spectrum <math>A_{xy}</math> is given by
			: <math>A_{xy}(f)= (\Lambda_{xy}(f)^2 + \Psi_{xy}(f)^2)^\frac{1}{2} ,</math>
			and the phase spectrum <math>\Phi_{xy}</math> given by
			: <math>\begin{cases}
			\tan^{-1} ( \Psi_{xy}(f) / \Lambda_{xy}(f) ) & \text{if } \Psi_{xy}(f) \ne 0 \wedge \Lambda_{xy}(f) \ne 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}</math>

			== Squared coherency spectrum ==
			The squared [[Coherence (signal processing)\|coherency spectrum]] is given by
			: <math>
			\kappa_{xy}(f)= \frac{A_{xy}^2}{ \Gamma_{xx}(f) \Gamma_{yy}(f)} ,
			</math>

			which expresses the amplitude spectrum in dimensionless units.

			==See also==
			* [[Cross-correlation#Time_series_analysis\|Cross-correlation]]
			* [[Spectral_density#Power_spectral_density\|Power spectrum]]
			* [[Scaled Correlation]]

			==References==
			<references/>

			[[Category:Frequency domain analysis]]
			[[Category:Multivariate time series analysis]]

Deltahedron: General n, cite Koch

2012-07-03T17:04:10Z

General n, cite Koch

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