Lecture14 - Mean Square Estimation Given X 1 , X 2 , , X n...

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1 Mean Square Estimation Given as a sequence of observed random Variables Y represents an unknown random variable to be estimated in terms of observations Note that can be a linear or a nonlinear function of represents the error in the above estimate , is a quadratic error n X X X , , , 2 1 " ) ( ϕ . , , , 2 1 n X X X " ) ( ˆ ) ( X Y Y Y X ε = = 2 | | represents the mean square error One strategy to obtain a good estimator would be to minimize the mean square error by varying This procedure gives rise to the M inimization of the M ean S quare E rror (MMSE) criterion for estimation . Theorem1: Under MMSE criterion, the best estimator for the unknown Y in terms of is given by the conditional mean of Y given X . Proof : Let represent an estimate of Y in terms of Error is given by } | | { 2 E ), ( n X X X , , , 2 1 " }. | { ) ( ˆ X Y E X Y = = ) ( ˆ X Y = ). , , , ( 2 1 n X X X X " = , ˆ Y Y = } | ) ( | { } | ˆ | { } | | { 2 2 2 2 X Y E Y Y E E σ = = = Using we can rewrite Note: the inner expectation is with respect to Y , and the outer one is with respect to ¾ Thus To obtain the best estimator we need to minimize with respect to Since and the variable appears in integrand minimizing w.r.t is equivalent to min w.r.t }] | { [ ] [ X z E E z E z X = }] | ) ( | { [ } | ) ( | { z 2 z 2 2 X X Y E E X Y E Y X ±² ±³ ´ ´ = = . X 22 2 X E[E{ |Y (x) | X }] E{ |Y (X) | X }f (X)dx. ε +∞ −∞ σ= −ϕ =− ϕ , 2 . , 0 ) ( X f X , 0 } | ) ( | { 2 X X Y E 2 } | ) ( | { 2 X X Y E . X being fixed at some value, is no longer random, and hence minimization of is equivalent to This gives or But Since when is a fixed number ) ( X } | ) ( | { 2 X X Y E . 0 } | ) ( | { 2 = X X Y E 0 } | ) ( {| = X X Y E ), ( } | ) ( { X X X E = ) ( , X x X = ). ( x . 0 } | ) ( { } | { = X X E X Y E
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2 We hence get Thus the conditional mean of Y given represents the best estimator for Y that minimizes the MSE The minimal MSE is given by Example: As an example, suppose is the unknown. Then the best MMSE estimator is given by Clearly if then indeed is best estimator }. , , , | { } | { ) ( ˆ 2 1 n X X X Y E X Y E X Y " = = = ϕ n X X X , , , 2 1 " . 0 )} | {var( ] } | ) | ( | { [ } | ) | ( | { ) var( 2 2 2 min = = = X Y E X X Y E Y E E X Y E Y E X Y ± ± ± ±² ± ± ± ±³ ´ σ 3 X Y = , 3 X Y = 3 ˆ X Y = . } | { } | { ˆ 3 3 X X X E X Y E Y = = = Example : Let where k > 0 is a suitable normalization constant. To determine best estimate for Y in terms of X , we need Thus Hence the best MMSE estimator is given by < < < = otherwise, 0 1 0 , ) , ( , y x kxy y x f Y X ). | ( | x y f X Y 1. x 0 , 2 ) 1 ( 2 ) , ( ) ( 2 1 2 1 1 , < < = = = = ∫∫ x kx kxy kxydy dy y x f x f x xx Y X X y x 1 1 . 1 0 ; 1 2 2 / ) 1 ( ) ( ) , ( ) | ( 2 2 , < < < = = = y x x y x kx kxy x f y x f x y f X Y X X Y Once again the best estimator is nonlinear.
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This note was uploaded on 09/24/2009 for the course ECE 514 taught by Professor Krim during the Fall '08 term at N.C. State.

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Lecture14 - Mean Square Estimation Given X 1 , X 2 , , X n...

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