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Jun Ji Hansung University ( visiting the University of Texas at Austin )

Robust Inversion using Biweight norm. Jun Ji Hansung University ( visiting the University of Texas at Austin ). Contents. Introduction IRLS Review Robust norms and their weightings in IRLS l 1 norm Huber norm Hybrid norm Biweight norm Examples Conclusions. Introduction.

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Jun Ji Hansung University ( visiting the University of Texas at Austin )

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  1. Robust Inversion using Biweight norm Jun Ji Hansung University ( visiting the University of Texas at Austin )

  2. Contents • Introduction • IRLS Review • Robust norms and their weightings in IRLS • l 1 norm • Huber norm • Hybrid norm • Biweight norm • Examples • Conclusions

  3. Introduction • Least-squares ( l 2 ) inversion • Sensitive to outliers • Least-absolute ( l 1 ) inversion • Resistant to outliers (Robust) • Variants of l 1 norm inversion • (Huber norm, Hybrid norm, Etc.) l 2 line fitting l 1 line fitting

  4. IRLS Review d : a data vector, m : a model vector, L : an operator matrix • A linear system • LS solution • Weighted LS solution

  5. IRLS Review • WLS solution ( ) requires a nonlinear inversion • such as IRLS. • IRLS (Iteratively Reweighted LS) Compute residual Compute weighting Solve WLS to find model Iterate until satisfy

  6. IRLS Review • IRLS algorithm using a Nonlinear Conjugate Gradient (NCG) method (Claerbout, 1991)

  7. Robust norm : l 1 norm • l 1norm function : • Weighting • e.g. :

  8. Robust norm : Huber norm (Huber, 1981) • Huber norm function : • Weighting : ε= 1.345 x MAD/0.6746 ( ~95% of efficiency for Gaussian Noise) (Holland & Welsch, 1977)

  9. Robust norm : Hybrid norm (Bube & Langan, 1997) • Hybrid l 1 / l 2norm function : • Weighting : (Bube &Langan, 1977)

  10. Robust norm : Tukey’sBiweight norm (Beaton & Tukey, 1974) • Biweight norm function : • Weighting : ε= 4.685 x MAD/0.6745 ( ~95% of efficiency for Gaussian Noise) (Holland & Welsch, 1977)

  11. Robust norm : Tukey’sBiweight norm (Beaton & Tukey, 1974) • Problems for Biweight norm IRLS • Local minimum (due to noncovex measure) good initial guess(e.g. l 1 /Huber norm solution) would be helpful • Carefully choose thethreshold (ε) and do not change during iteration (Holland & Welsch, 1977)

  12. Various norms and weightings

  13. Property of different norms • Single 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

  14. Example : Line fitting BG noise : N(µ, σ) = (0, 0.02) Outliers (20% of data) 2 spikes(4.5,5) + 8 points with N(3,0.1)

  15. Example : Hyperbola fitting • BG noise : N(0,0.4) • Outliers • Three spikes : 10 times of signal amplitude • A bad trace with N(0,1)

  16. Example : Hyperbola fitting • BG noise : N(0,0.4) • Outliers • Three spikes : 10 times of signal amplitude • A bad trace with N(0,1) • 12 bad traces with U(10,2) • ~ 10 % of data

  17. Real data Example • This year’s technical committee comprised of 25 key contacts in different technical areas. • Each key contact utilized a network of reviewers to evaluate papers. • Online review using TPO.

  18. Real data Example (with strong additional noise) • This year’s technical committee comprised of 25 key contacts in different technical areas. • Each key contact utilized a network of reviewers to evaluate papers. • Online review using TPO.

  19. Conclusions • IRLS using Biweight norm provides a robust inversion method like the variants of l 1 norm approaches such as l 1, Huber, & 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 the noise distribution and do not change during iteration.

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