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

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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 modified 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 semidefinite 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 W Y, where Y = y (1) . . . y ( k ) Φ = ϕ T (1) . . . ϕ T ( k ) . Problem 10: (15 points) The loss function V ( θ, k ) = 1 2 k i =1 λ k - i y ( i ) - ϕ T ( i ) θ 2 contains a forgetting factor λ , where 0 < λ 1. By including the forgetting factor, data that is n time units old is weighted by λ n . Following 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) K ( k ) = P ( k ) ϕ ( k ) = P ( k - 1) ϕ ( k ) λI + ϕ T ( k ) P ( k - 1) ϕ ( k ) - 1 P ( k ) = I - K ( k ) ϕ T ( k ) P ( k - 1) /λ.
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