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ch08 - 8 Bivariate Processes 8.1 Time-domain Quantities...

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Unformatted text preview: 8 Bivariate Processes 8.1 Time-domain Quantities Suppose we have N observations recorded on two variables, ( x 1 , y 1 ) , . . . , ( x N , y N ) . This can be thought of as a finite realization of a discrete bivariate stochastic process ( X t , Y t ). Definition 8.1.1: The cross-covariance function is defined as γ XY ( t, τ ) = cov( X t , Y t + τ ) . (8.1) For stationary processes we may abbreviate this to γ XY ( τ ) = γ XY ( t, τ ) . (8.2) Proposition 8.1: γ XY (- τ ) = γ Y X ( τ ) . Definition 8.1.2: We define the cross-correlation function as ρ XY ( τ ) = γ XY ( τ ) radicalbig γ XX (0) γ Y Y (0) (8.3) Suppose that { X t } and { Y t } are both formed from the same purely random process { Z t } with mean 0 and variance σ 2 Z by X t = Z t , Y t = 0 . 5 · ( Z t- 1 + Z t- 2 ) . Then after some calculations ρ XY ( τ ) = braceleftBigg 1 √ 2 , τ = 1 , 2 , otherwise 8.2 Estimation Basically uses obvious quantities similar to quantities in Section 2.7 which are sensibleBasically uses obvious quantities similar to quantities in Section 2....
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ch08 - 8 Bivariate Processes 8.1 Time-domain Quantities...

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