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Method of Least Squares Least Squares Method of Least Squares: Deterministic approach The inputs u(1), u(2), ..., u(N) are applied to the system The outputs y(1), y(2), ..., y(N) are observed Find a model which fits the input-output relation to a (linear?) curve, f(n,u(n)) ‘best’ fit by minimising the sum of the squres of the difference f - y 50 45 40 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 Least Squares The curve fitting problem can be formulated as model variable observations Error: Sum-of-error-squares: Minimum (least-squares of error) is achieved when the gradient is zero Problem Statement For the inputs to the system, u(i) The observed desired response is, d(i) Relation is assumed to be linear Unobservable measurement error Zero mean White Problem Statement Design a transversal filter which finds the least squares solution Then, sum of error squares is Data Windowing We will express the input in matrix form Depending on the limits i1 and i2 this matrix changes Covariance Method i1=M, i2=N Postwindowing Method i1=M, i2=N+M1 Autocorr. Method i1=1, i2=N+M1 Prewindowing Method i1=1, i2=N Principle of Orthogonality Error signal Least squares (minimum of sum of squares) is achieved when i.e., when !Time averaging! (For Wiener filtering) (this was ensemble average) The minimum-error time series emin(i) is orthogonal to the time series of the input u(i-k) applied to tap k of a transversal filter of length M for k=0,1,...,M-1 when the filter is operating in its least-squares condition. Corollary of Principle of Orthogonality LS estimate of the desired response is Multiply principle of orthogonality by wk* and take summation over k Then When a transversal filter operates in its least-squares condition, the least-squares estimate of the desired response -produced at the output of the filter- and the minimum estimation error time series are orthogonal to each other over time i. Energy of Minimum Error Due to the principle of orthogonality, second and third terms are orthogonal, hence where , when eo(i)= 0 for all i, impossible , when the problem is underdetermined fewer data points than parameters infinitely many solutions (no unique soln.)! Normal Equations Principle of Orthogonality Minimum error: → (t,k), 0≤(t,k) ≤M-1 time-average autocorrelation function of the input z(-k), 0 ≤k ≤M-1 time-average cross-correlation bw the desired response and the input Hence, Expanded system of the normal equations for linear least-squares filters. Normal Equations (Matrix Formulation) Matrix form of the normal equations for linear least-squares filters: (if -1 exists!) Linear least-squares counterpart of the Wiener-Hopf eqn.s. Here and z are time averages, whereas in Wiener-Hopf eqn.s they were ensemble averages. Minimum Sum of Error Squares Energy contained in the time series Or, Then the minimum sum of error squares is is Properties of the Time-Average Correlation Matrix Property I: The correlation matrix is Hermitian symmetric, Property II: The correlation matrix is nonnegative definite, Property III: The correlation matrix is nonsingular iff det() is nonzero Property IV: The eigenvalues of the correlation matrix are real and non-negative. Properties of the Time-Average Correlation Matrix Property V: The correlation matrix is the product of two rectangular Toeplitz matrices that are Hermitian transpose of each other. Normal Equations (Reformulation) But we know that then which yields ! Pseudo-inverse ! Substituting into the minimum sum of error squares expression gives Projection The LS estimate of d is given by The matrix is a projection operator onto the linear space spanned by the columns of data matrix A i.e. the space Ui. The orthogonal complement projector is Projection - Example M=2 tap filter, N=4 → N-M+1=3 Let Then And orthogonal Projection - Example Uniqueness of the LS Solution LS always has a solution, is that solution unique? The least-squares estimate is unique if and only if the nullity (the dimension of the null space) of the data matrix A equals zero. AKxM, (K=N-M+1) Solution is unique when A is of full column rank, K≥M All columns of A are linearly independent Overdetermined system (more eqns. than variables (taps)) (AHA)-1 nonsingular → exists and unique Infinitely many solutions when A has linearly dependent columns, K<M (AHA)-1 is singular Properties of the LS Estimates Property I: The least-squares estimate is unbiased, provided that the measurement error process eo(i) has zero mean. Property II: When the measurement error process eo(i) is white with zero mean and variance 2, the covariance matrix of the leastsquares estimate equals 2-1. Property III: When the measurement error process eo(i) is white with zero mean, the least squares estimate is the best linear unbiased estimate. Property IV: When the measurement error process eo(i) is white and Gaussian with zero mean, the least-squares estimate achieves the Cramer-Rao lower bound for unbiased estimates. Computation of the LS Estimates The rank (W) of an KxN (K≥N or K<N) matrix A gives The number of linearly independent columns/rows The number of non-zero eigenvalues/singular values The matrix is said to be full rank (full column or row rank) if Otherwise, it is said to be rank-deficient Rank is an important parameter for matrix inversion If K=N (square matrix) and the matrix is full rank (W=K=N) (nonsingular) inverse of the matrix can be calculated, A-1=adj(A)/det(A) If the matrix is not square (K≠N), and/or it is rank-deficient (singular), A-1 does not exist, instead we can use the pseudo-inverse (a projection of the inverse), A+ SVD We can calculate the pseudo-inverse using SVD. Any KxN matrix (K≥N or K<N) can be decomposed using the Singular Value Decomposition (SVD) as follows: SVD The system of eqn.s, is overdetermined if K>N, more eqn.s than unknowns, is underdetermined if K<N, more unknowns than eqn.s, Unique solution (if A is full-rank) Non-unique, infinitely many solutions (if A is rank-deficient) Non-unique, infinitely many solutions In either case the solution(s) is(are) where Computation of the LS Estimates Find the solution of (A: KxM) If K>M and rank(A)=M, ( Otherwise , infinitely many solutions, but pseudo-inverse gives the minimum-norm solution to the least squares problem. ) the unique solution is Shortest length possible in the Euclidean norm sense.