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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

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 whitewithzeromeanandvariance2, thecovariancematrix of theleast-squaresestimateequals2-1.
- 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.

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