Estimation CH8 Least Squares
description
Transcript of Estimation CH8 Least Squares
Wireless Information Transmission System Lab.Institute of Communications EngineeringNational Sun Yat-sen University
11
EstimationCH8 Least Squares
2011/08/6劉永富
22
CH8 Least Square : 8.1 Introduction
◊ Advantage◊ NO probabilistic assumptions are made about the
data, only a signal model is assumed. ◊ Easy to implement.
◊ Disadvantage◊ NO claims about optimality can be made. ◊ The statistical performance cannot be assessed
without some specific assumptions about the probabilistic structure of the data.
33
8.3 The Least Square Approach
◊ In the Least Square (LS) approach, we attempt to minimize the squared difference between the given data x[n] and the assumed signal or noiseless data.
44
◊ The least squares estimator (LSE) chooses the value that makes s[n] closest to the observed data x[n]. Closeness is measured by the LS error criterion
◊ The value of that minimizes is the LSE.
◊ Note that, NO probabilistic assumptions have been made about the data x[n].
55
◊ The method is equally valid for Gaussian as well as non-Gaussian noise.
◊ The performance of the LSE will undoubtedly depend upon the properties of the corrupting noise as well as any modeling errors.
◊ LSEs are usually applied in situations where a precise statistical characterization of the data is unknown or where an optimal estimator cannot be found or may be too complicated to apply in practice.
66
Example 8.1
◊ Assume that the signal model in Fig. 8.1 is s[n]=A and we observe x[n] for n=0,1,…,N-1. By LS approach
◊ Differentiating with respect to A and setting the result equal to zero
◊ Our estimator, cannot be claimed to be optimal in the MVU sense but only in that it minimizes the LS error.
77
◊ If , where w[n] is zero mean WGN, then the LSE will also be the MVU estimator, but otherwise not.
◊ Suppose the noise is not zero mean
◊ The observed data are composed of a deterministic signal and zero mean noise.
◊ This modeling error would also cause the LSE to be biased.
88
Example 8.2
◊ Consider the signal model
◊ The LSE is found by minimizing
◊ The problem is a nonlinear least squares problem.
◊ Nonlinear LS problems are solved via grid searches or iterative minimization methods as described in section 8.9.
99
Example 8.3
◊ Consider the signal model , where f0 is known and A is to be estimated. Then the LSE minimizes
◊ This is easily accomplished by differentiation since J(A) is quadratic in A.
◊ If A were known and the frequency were to be estimated, the problem would be equivalent to that in example 8.2.
1010
◊ In the vector parameter case, both A and f0 might need to be estimated. Then, the error criterion
is quadratic in A but non-quadratic in f0.◊ J can be minimized in closed form with
respect to A for a given f0, reducing the minimization of J to one over f0 only.
◊ This type of problem is termed a separable least squares problem, which will discussed in section 8.9.
1111
8.4 Linear Least squares
◊ In applying the linear LS approach for a scalar parameter we must assume that
where is a known sequence.
◊ The LS error criterion becomes
1212
8.4 Linear Least squares
◊
◊ 1 1 12
0 0 0
1
1 12 0
120 0
0
ˆ ˆ[ ]( [ ] [ ]) [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] 0
[ ]
N N N
n n n
N
N NnN
n n
n
S h n x n h n h n x n h n
x n h nh n x n h n
h n
1313
8.4 Linear Least squares
◊
◊ For Example 8.1 and
Original
Reduction due to signal fitting
1414
8.4 Linear Least squares
◊
◊ If the data is noiseless (x[n] = A) Jmin = 0
◊ If The minimum LS error would then be
1515
Extension
◊ S: p x 1 H: N x p θ:p x 1
◊ The H is referred to as the observation matrix (P.84).
◊
1616
Compare with Ch 4 and Ch 6
◊ Setting the gradient equal to zero yields the LSE
◊ For it to be the BLUE would require
and , and to be efficient would in addition to these properties require x to be Gaussian.
◊ Ch 6 :
1717
Jmin
◊
◊ IdempotentA2 = A
1818
8.5 Geometrical interpretations
◊ Recall the general signal model . If we denote the columns of H by hi, we have
1919
Example 8.4 – Fourier Analysis
◊ Referring to Example 4.2 (P.88), we suppose the signal model to be
◊
2020
Example 8.4 – Fourier Analysis
◊ The LS error was defined to be
2121
2222
2323
Example 8.5
◊
(p.89)
∵
◊ And also
1
0
1 2 ( ) 2 ( )sin sin2
N
n
n i j n i jN N
-1 2
0
sin and 0, 1 12
njx jx N j mN
n
e ex e N m Nj
2424
Example 8.5
◊ Therefore,
2525
8.6 Order-Recursive Least Square
◊ In many cases the signal model is unknown and must be assumed. Ex.
2626
8.6 Order-Recursive Least Square
◊ The following models might be assumed
◊ Using a LSE with
would produce the estimate for the intercept and slope as
2727
8.6 Order-Recursive Least Square
and
◊ We plotted and where T=100.
2828
8.6 Order-Recursive Least Square
◊ The fit using two parameter is better, as expect. We will later show that the minimum LS error must decrease as we add more parameters.
◊ The data are subject to error, we may very well be fitting the noise.
2929
8.6 Order-Recursive Least Square
◊ In practice, we increase the order of the polynomial until the minimum LS error decrease only slightly as the order is increased.
3030
8.6 Order-Recursive Least Square
◊ In this example the signal was actually
and the noise was WGN with Note that if A, B had been estimated perfectly, the minimum LS error:
◊ When true order is reached, This is verified in Fig8.6 and increase our confidence in the chosen model.
3131
8.7 Sequential least squares
◊ LSE (vector form) : .◊ Assume we have determined the LSE
based on . If we now observe , we can update (in time) without having to resolve the linear equations .
◊ The procedure is termed Sequential Least Squares.
3232
Sequential least squares
◊ Consider example 8.1 in which the DC signal level is to be estimated. The LSE is
◊ If we now observe the new data sample , then the LSE becomes
3333
Sequential least squares
◊ The new LSE is found by using the previous one and the new observation
◊ The new estimate is equal to the old one plus a correction term
◊ If the error is zero, then no correction takes place for that update.
3434
Sequential least squares
3535
Sequential least squares
3636
Sequential least squares
3737
Sequential least squares
3838
Variance
3939
Gain
4040
Estimator
◊ The estimate appears to be converging to the true value of . This is in agreement with the variance approaching zero.
4141
Sequential least squares