ps3 - EE 530 Reading assignment: Ch 3 Problem 9: (15...

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EE 530 PROBLEM SET 3 DUE: 22 DEC 2006 Reading assignment: Ch 3 Problem 9: (15 points) The LS method can be modifed to enable different weighting oF the errors by using the loss Function V ( θ,k )= 1 2 E T W E where E = ε (1) . . . ε ( k ) = y (1) - ϕ T (1) θ . . . y ( k ) - ϕ T ( k ) θ and W is a positive semidefnite diagonal matrix with the weights in the diagonal. Show that the least squares estimate is given by ˆ θ = ( Φ T W Φ ) - 1 Φ T WY , where Y = y (1) . . . y ( k ) Φ= ϕ T (1) . . . ϕ T ( k ) . Problem 10: (15 points) The loss Function V ( 1 2 k ± i =1 λ k - i ² y ( i ) - ϕ T ( i ) θ ³ 2 contains a forgetting factor λ , where 0 1. By including the Forgetting Factor, data that is n time units old is weighted by λ n . ±ollowing the analysis in lecture that lead to Theorem 2.3, derive the Following recursive relations For the parameter estimate ˆ θ ( k ) that minimizes the loss Function with a Forgetting Factor ˆ θ ( k ˆ θ ( k - 1) + K ( k ) ´ y ( k ) - ϕ T ( k ) ˆ θ ( k - 1) µ K ( k P ( k ) ϕ ( k P ( k - 1) ϕ ( k ) ² λI + ϕ T ( k ) P ( k - 1) ϕ ( k ) ³ - 1 P ( k ² I - K ( k ) ϕ T ( k ) ³ P ( k - 1) /λ.
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This note was uploaded on 07/23/2008 for the course EE 530 taught by Professor Schiano during the Fall '06 term at Pennsylvania State University, University Park.

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ps3 - EE 530 Reading assignment: Ch 3 Problem 9: (15...

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