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Unformatted text preview: 8 Bivariate Processes 8.1 Timedomain 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 crosscovariance function is defined as XY ( t, ) = cov( X t , Y t + ) . (8.1) For stationary processes we may abbreviate this to XY ( ) = XY ( t, ) . (8.2) Proposition 8.1: XY ( ) = Y X ( ) . Definition 8.1.2: We define the crosscorrelation function as XY ( ) = XY ( ) radicalbig XX (0) 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 2 Z by X t = Z t , Y t = 0 . 5 ( Z t 1 + Z t 2 ) . Then after some calculations XY ( ) = braceleftBigg 1 2 , = 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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This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.
 Spring '09
 jsdkasj

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