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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

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