{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

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

{[ snackBarMessage ]}