Method of Least Squares

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# Method of Least Squares - PowerPoint PPT Presentation

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))

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### Method of Least Squares

Least Squares
• Method of LeastSquares:
• Deterministicapproach
• Theinputs u(1), u(2), ..., u(N) areappliedtothesystem
• Theoutputs y(1), y(2), ..., y(N) areobserved
• Find a model whichfitstheinput-outputrelationto a (linear?) curve, f(n,u(n))
• ‘best’ fit byminimisingthesum of thesqures of thedifference f - y
Least Squares
• The curve fitting problem can be formulated as
• Error:
• Sum-of-error-squares:
• Minimum (least-squares of error) is achieved when the gradient is zero

observations

model

variable

Problem Statement
• Fortheinputstothesystem, u(i)
• Theobserveddesiredresponse is, d(i)
• Relation is assumedto be linear
• Unobservablemeasurementerror
• Zeromean
• White
Problem Statement
• Design a transversalfilterwhichfindstheleastsquaressolution
• Then, sum of errorsquares 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

Prewindowing Method

i1=1, i2=N

Postwindowing Method

i1=M, i2=N+M1

Autocorr. Method

i1=1, i2=N+M1

Principle of Orthogonality
• Error signal
• Least squares (minimum of sum of squares) is achieved when
• i.e., when
• 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.

!Time averaging!

(For Wiener filtering)

(this was ensemble average)

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:

• Hence,

Expanded system of the normal equations for linear least-squares filters.

z(-k), 0 ≤k ≤M-1

time-average

cross-correlation bw

the desired response

and the input

(t,k), 0≤(t,k) ≤M-1

time-average

autocorrelation function

of the input

Normal Equations (Matrix Formulation)
• Matrix form of the normal equations for linear least-squares filters:
• 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.

(if -1 exists!)

Minimum Sum of Error Squares
• Energy contained in the time series is
• Or,
• Then the minimum sum of error squares 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

which yields

• Substituting into the minimum sum of error squares expression gives

then

! Pseudo-inverse !

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

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: Theleast-squaresestimate is unbiased, providedthatthemeasurementerrorprocesseo(i) has zeromean.
• Property II: Whenthemeasurementerrorprocesseo(i) is whitewithzeromeanandvariance2, thecovariancematrix of theleast-squaresestimateequals2-1.
• Property III: Whenthemeasurementerrorprocesseo(i) is whitewithzeromean, theleastsquaresestimate is thebestlinearunbiasedestimate.
• Property IV: Whenthemeasurementerrorprocesseo(i) is whiteandGaussianwithzeromean, theleast-squaresestimateachievestheCramer-Raolowerboundforunbiasedestimates.
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) (non-singular) 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,
• Unique solution (if A is full-rank)
• Non-unique, infinitely many solutions (if A is rank-deficient)
• is underdetermined if K<N, more unknowns than eqn.s,
• 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, ( ) the unique solution is
• Otherwise , infinitely many solutions, but pseudo-inverse gives the minimum-norm solution to the least squares problem.
• Shortest length possible in the Euclidean norm sense.