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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))

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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 squares1
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
Problem Statement

  • Fortheinputstothesystem, u(i)

  • Theobserveddesiredresponse is, d(i)

  • Relation is assumedto be linear

  • Unobservablemeasurementerror

    • Zeromean

    • White


Problem statement1
Problem Statement

  • Design a transversalfilterwhichfindstheleastsquaressolution

  • Then, sum of errorsquares is


Data windowing
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
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
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
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
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
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
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
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 matrix1
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
Normal Equations (Reformulation)

  • But we know that

    which yields

  • Substituting into the minimum sum of error squares expression gives

then

! Pseudo-inverse !


Projection
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
Projection - Example

  • M=2 tap filter, N=4 → N-M+1=3

  • Let

  • Then

  • And

orthogonal



Uniqueness of the ls solution
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
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
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 estimates1
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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