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Least Squares Estimation

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Alexandra Oborina. This presentation is based on 3rd chapter of the book Dan Simon, ... Least Square Estimation Alexandra Oborina. 5. Weighted LS Estimation ... – PowerPoint PPT presentation

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Title: Least Squares Estimation


1
Least Squares Estimation
  • S-88.4221 Postgraduate Seminar on Signal
    Processing

Alexandra Oborina
This presentation is based on 3rd chapter of the
book Dan Simon, Optimal State Estimation
Kalman, Hinf, and Nonlinear Approaches
2
Content
  • Estimation of a constant vector
  • Weighted least square estimation
  • Recursive least square estimation
  • Wiener filtering
  • Homework

3
Estimation of a constant
  • Given x-constant, unknown vector, y-noisy
    measurement vector. Problem find best estimate
    of x.

4
Example
5
Weighted LS Estimation
  • Given x-constant, unknown vector, y-noisy
    measurement vector. The variance of the
    measurement noise may be different for each
    element of y. Problem find best estimate of x.

6
Example
7
Recursive LS Estimation
  • What if we get measurements sequentially?
  • Suppose is given after k-1 measurements. So
    new yk is obtained. How to update the estimate
    ?

8
Recursive LS Estimation
9
Recursive LS Estimation
10
Recursive LS EstimationAlternative forms
  • Using matrix inversion lemma, substitution and
    inversion alternative forms for Kk and Pk can be
    obtained.
  • Alternative forms are mathematically identical,
    but can be beneficial from computational point of
    view.

11
Recursive LS EstimationAlgorithm
  • Initialize the estimator as
  • For k1,2,.. perform
  • Obtain the measurements with white noise

12
Recursive LS EstimationAlgorithm
  • Update the estimate of x and estimation-error
    covariance

13
Recursive LS Estimation
14
Example 1
15
Example 1
  • By induction
  • If x is known perfectly a priori

16
Example 1
  • If x is completely unknown a priory

17
Example 1
18
Example 2 - Linear data fitting
  • Suppose we want to fit a straight line to a set
    of data points
  • Problem find linear relation between yk and tk,
    that means estimate the constants x1 and x2

19
Example 2 - Linear data fitting
  • Recursive LS initialization
  • Using equations (1), (3), (4) perform recursion

20
Example 3 - Quadratic data fitting
  • Suppose, a priory is known that the data is a
    quadratic function of time

21
Wiener Filtering
  • Problem design a stable LTI filter to extract a
    signal from noise.

22
Wiener Filtering - parametric filter optimization
  • Lets find optimal G(w) as a first order, stable,
    causal filter with 1/T BW
  • Suppose also the following forms for
  • Recall
  • Substitute everything to E(e2(t)) and
    differentiate with respect to t

23
Wiener Filtering - general filter optimization
  • Problem find filter g(t) that minimize E(e2(t))
  • Replace g(t) with

24
Wiener Filtering - noncausal filter optimization
  • Noncausal filter means
  • So,
  • Thus, quantity inside squire brackets must be zero

25
Wiener Filtering - causal filter optimization
  • Causal filter means g(t)0 for tlt0. So,
  • Let denote some function a(t), that is 0 for tgt0
    and arbitrary for tlt0.

26
Example
  • The signal and noise power spectra are given as

27
Homework
  • For recursive LS estimation decide is estimator
    unbiased or not.
  • For recursive LS estimation show explicitly
    calculations of
  • For Wiener filtering prove that
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