WLS solution Weighted LS robust weights require a nonlinear inversion such as IRLS IRLS Iteratively Reweighted LS ID: 805726
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Slide1
A linear system
LS solution WLS solution (Weighted LS) robust weights require a nonlinear inversion such as IRLS. IRLS (Iteratively Reweighted LS)
Robust Inversion using Biweight normJun Ji, Hansung University ( visiting the University of Texas at Austin )
SEG 2011 San Antonio
IRLS Review
Least-squares (
l
2
)
inversion:
Sensitive to outliers
Introduction
Least-absolute (
l
1
) inversion
Resistant to outliers (i.e. Robust)
Variants of
l
1
: - Huber norm - Hybrid norm etc.
Compute residual
Compute weighting
Solve
WLS to find
model
Iterate
until satisfy
IRLS algorithm implementation using
nonlinear Conjugate Gradient (NCG)
method (
Claerbout
, 1991)
Slide2l
1
norm function :Weighting : Robust norm : l 1 norm Huber norm function : Weighting :
Hybrid l 1 / l 2 norm function : Weighting :
Tukey’s
Biweight (
Bisquare Weight) norm function :
Weighting :
Robust norm : Huber norm
(Huber, 1981)
Robust norm : Hybrid norm
(
Bube
&
Langan
, 1977)
Robust norm :
Biweight
norm (Beaton & Tukey, 1974) ε = 1.345 x MAD/0.6746 ( ~95% of efficiency for Gaussian Noise) (Holland & Welsch
, 1977)
ε
= 4.685 x MAD/0.6745 ( ~95% of efficiency for Gaussian Noise) (Holland &
Welsch, 1977) Problems for Biweight norm IRLS Local minimum (due to noncovex measure) good initial guess (e.g. Huber norm sol.) would be helpful Carefully choose the threshold (ε) and do not change during iteration ε ~ 0.6 x σ (Bube &
Langan, 1977)
Slide3Single parameter estimation problem with N observations di Minimize squares of error (l 2 norm) : Minimize absolute of error (l 1 norm):Example data : ( 2, 3, 4, 5, 66 ) => Mean : 16, Median : 4, More robust estimation : ~ 3.5
Properties of different norms BG noise : N(µ,σ)=(0, 0.02) Outliers (20% of data) : 2 spikes(4.5,5) +
8 points with N(3,0.1)
Examples - Line fitting
BG noise :
N(0,0.4)Outliers
1) Three spikes of 10 times of signal amplitude
2) A bad trace with N(0,1)
BG noise :
N(0,0.4)Outliers 1) Three spikes : 10 times
of signal amplitude 2) A bad trace with N(0,1) 3) 12 bad traces with U(10,2) ~ 10 % of data
Example : Hyperbola fitting
Slide4Real data Example
IRLS using
Biweight norm provides a robust inversion method like the variants of l 1 norm approaches such as l 1 , Huber, and Hybrid norms. Biweight norm inversion sometimes demonstrates better estimation than the one of l 1 norm
variants when outliers are not simple. For optimum performance need a good initial guess (e.g. Huber norm solution) to converge to the global minimum carefully choose threshold (ε) based on noise distribution and do not change during iteration
Conclusions