[RP]Lecture Note VI

# [RP]Lecture Note VI - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VI-1 C1002900 RP Lecture Handout VI: Mean Square Estimation Reading: Chapter 7.3 The estimation problem is fundamental in the applications of probability. Let X be a random variable with some known distribution. Suppose X is not observed but we wish to estimate X . (Optimality Criterion) Minimizing the mean square value of estimation error: minimum mean square error (MMSE) estimation. In many applications, a random variable X is estimated based on observation of the other random variable Y . Thus, an estimator is a function of Y . The two most frequently considered classes are essentially: all functions, leading to the best unconstrained estimator all linear functions, leading to the best linear estimator. 6.1. MS Estimation without Information If we use a constant c to estimate X , then the estimation error is eXc  . What is the best guess c of the value of X ?    2 argmin . c cX c MS error:    X eX c xc f xd x   2 2 X de xcf xd x dc   20   . X cx f x d xX    Solution:   Optimal (least) MS error: Var .

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VI-2 6.2. MS Estimation with Information: Nonlinear MS Estimation Now, we observe that Yy . Then,    XcY y  2 is minimized by   . cX Y g y y  View the estimator as a function gy .   XY minimizes {( ( )) } Xg Y 2 over all estimators g . We wish to estimate X by a function gY of the random variable Y . Our problem now is to find the optimal (unconstrained) that minimizes the MS error      , ,. eX g Y x g y f x y dxdy        2 2 Since   , , Y fx y y f y , is minimized if it is minimized for every . ey Y ef y x g y f x y d x d y   2    XY y If Xh Y , then      . hY Y y hy  Optimal estimator . gy hy e 0 If X and Y are independent, then     . X Knowledge of Y have no effect on the estimate of X . Optimal (least) MS estimator:  
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VI-3 6.3. Linear MS Estimation Linear estimation problem: estimate X in term of a linear function aY b of Y .

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[RP]Lecture Note VI - Kyung Hee University Department of...

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