llseresult

llseresult - Boston University Department of Electrical and...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online