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Unformatted text preview: Boston University
Department of Electrical and Computer Engineering EC505 STOCHASTIC PROCESSES Notes on a LLSE Result
Consider the following problem. We want to ﬁnd the Bayes linear least squares estimate of the random vector Z based on observation of the random vector Y , with mean mY and covariance ΛY . Further, suppose the random vector Z is related to the random vectors X and W as follows: Z = F X + HW + b (1) where F and H are deterministic matrices, b is a deterministic vector, E [X ] = mX , Cov(X , X ) = λX , Cov(X , Y ) = λXY , E [W ] = 0, Cov(W , W ) = λw , and W is uncorrelated with both X and Y . Show that z L (y ), the LLSE of Z based on Y , can be written as a linear function of xL and w L (y ), the LLSE estimates of X and W based on Y , respectively. Further show that the corresponding error covariance matrix ΛLz , can be written as a ΛLx and ΛLw . This result will be important later, as it shows that we may use the estimate of one random variable to easily obtain the estimate of another if the two are linearly related! Solution: First note that w L (y ) = mw = 0 and ΛLw = Λw , since W is uncorrelated with Y . Now to ﬁnd the LLSE of Z we need the quantities mZ , ΛZ , and ΛZY . Lets ﬁnd them: mZ ΛZ = E [Z Z T ] − mZ mT Z = = F E [X X T ]F T + F E [X W T ]H T + F E [X ]bT + HE [W X T ]F T + HE [W W T ]H T +HE [W ]bT + bE [X T ]F T + bE [W T ]H T + bbT − F mX mT F T − F mX bT − bmT F T − bbT X X F ΛX F T + H Λw H T ΛZY = = = Putting these pieces together we ﬁnd: z L (y ) = = = = Λ Lz = = mZ + ΛZY Λ−1 (y − mY ) Y F mX + b + F ΛXY Λ−1 (y − mY ) Y F mX + ΛXY Λ−1 (y − mY ) + b Y F xL + b E [Z Y T ] − mZ mT Y E [(F X + HW + b)Y T ] − (F mX + b)mT Y F ΛXY = E [Z ] = F mX + Hmw + b = F mX + b T T ΛZ − ΛZY Λ−1 ΛT = F ΛX F T + H Λw H T − F ΛXY Λ−1 ΛT F T = F ΛX − ΛXY Λ−1 ΛT ZY XY XY F + H Λw H Y Y Y F Λ Lx F T + H Λ Lw H T 1 ...
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 Fall '04
 Karl

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