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# H15 - H15 HW7(due Mar 20 Exercises 12.1 12.2(a d from the...

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H15 HW7 (due Mar. 20). Exercises 12.1, 12.2 (a, d) from the textbook In many cases, at each time t, several related quantities are observed and, therefore, we want to study these quantities simultaneously by grouping them to form a vector. By doing so we have a vector or multivariate process . We consider first the case of bivariate processes. Suppose we are given two processes, . ,1 ,2 {} , { tt XX } } } } } μ We say that is a stationary bivariate process (or that are jointly stationary ) if ,1 {, ,1 , { (a) and are each (univariate) stationary processes, and ,1 { t X { t X (b) is a function of h only. cov{ , } th t + The individual ACVFs are defined in the usual way, The corresponding ACFs are then 11 1 1 22 , 2 2 , 2 2 () { } { } { } { } t t hE X X X X γμ + + ⎡⎤ =− ⎣⎦ 11 11 11 22 22 22 / ( 0 ) / ( 0 ) hh ρ γγ ργ γ = = The cross-covariance function describes the correlation structure between the processes, and is defined by 12 1 2 () c o v { , } { } { } t t hX X E X X ++ == . Its normalized version is the cross-correlation function [] 12 12 1/2 11 22 (0) (0) h h = .

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H15 - H15 HW7(due Mar 20 Exercises 12.1 12.2(a d from the...

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