Suboptimal LULU-estimators in measurements containing outliers

Astl, Stefan Ludwig (2013-12)

Thesis (MSc)--Stellenbosch University, 2013.

Thesis

ENGLISH ABSTRACT: Techniques for estimating a signal in the presence of noise which contains outliers are currently not well developed. In this thesis, we consider a constant signal superimposed by a family of noise distributions structured as a tunable mixture f(x) = α g(x) + (1 − α) h(x) between finitesupport components of “well-behaved” noise with small variance g(x) and of “impulsive” noise h(x) with a large amplitude and strongly asymmetric character. When α ≈ 1, h(x) can for example model a cosmic ray striking an experimental detector. In the first part of our work, a method for obtaining the expected values of the positive and negative pulses in the first resolution level of a LULU Discrete Pulse Transform (DPT) is established. Subsequent analysis of sequences smoothed by the operators L1U1 or U1L1 of LULU-theory shows that a robust estimator for the location parameter for g is achieved in the sense that the contribution by h to the expected average of the smoothed sequences is suppressed to order (1 − α)2 or higher. In cases where the specific shape of h can be difficult to guess due to the assumed lack of data, it is thus also shown to be of lesser importance. Furthermore, upon smoothing a sequence with L1U1 or U1L1, estimators for the scale parameters of the model distribution become easily available. In the second part of our work, the same problem and data is approached from a Bayesian inference perspective. The Bayesian estimators are found to be optimal in the sense that they make full use of available information in the data. Heuristic comparison shows, however, that Bayes estimators do not always outperform the LULU estimators. Although the Bayesian perspective provides much insight into the logical connections inherent in the problem, its estimators can be difficult to obtain in analytic form and are slow to compute numerically. Suboptimal LULU-estimators are shown to be reasonable practical compromises in practical problems.

AFRIKAANSE OPSOMMING: Tegnieke om ’n sein af te skat in die teenwoordigheid van geraas wat uitskieters bevat is tans nie goed ontwikkel nie. In hierdie tesis aanskou ons ’n konstante sein gesuperponeer met ’n familie van geraasverdelings wat as verstelbare mengsel f(x) = α g(x) + (1 − α) h(x) tussen eindige-uitkomsruimte geraaskomponente g(x) wat “goeie gedrag” en klein variansie toon, plus “impulsiewe” geraas h(x) met groot amplitude en sterk asimmetriese karakter. Wanneer α ≈ 1 kan h(x) byvoorbeeld ’n kosmiese straal wat ’n eksperimentele apparaat tref modelleer. In die eerste gedeelte van ons werk word ’n metode om die verwagtingswaardes van die positiewe en negatiewe pulse in die eerste resolusievlak van ’n LULU Diskrete Pulse Transform (DPT) vasgestel. Die analise van rye verkry deur die inwerking van die gladstrykers L1U1 en U1L1 van die LULU-teorie toon dat hul verwagte gemiddelde waardes as afskatters van die liggingsparameter van g kan dien wat robuus is in die sin dat die bydrae van h tot die gemiddeld van orde grootte (1 − α)2 of hoër is. Die spesifieke vorm van h word dan ook onbelangrik. Daar word verder gewys dat afskatters vir die relevante skaalparameters van die model maklik verkry kan word na gladstryking met die operatore L1U1 of U1L1. In die tweede gedeelte van ons werk word dieselfde probleem en data vanuit ’n Bayesiese inferensie perspektief benader. Die Bayesiese afskatters word as optimaal bevind in die sin dat hulle vol gebruikmaak van die beskikbare inligting in die data. Heuristiese vergelyking wys egter dat Bayesiese afskatters nie altyd beter vaar as die LULU afskatters nie. Alhoewel die Bayesiese sienswyse baie insig in die logiese verbindings van die probleem gee, kan die afskatters moeilik wees om analities af te lei en stadig om numeries te bereken. Suboptimale LULU-beramers word voorgestel as redelike praktiese kompromieë in praktiese probleme.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/85833
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