Thesis (MSc)--Stellenbosch University, 2013. / 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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/85833 |
| Date | 12 1900 |
| Creators | Astl, Stefan Ludwig |
| Contributors | Eggers, Hans C., Rohwer, Carl H., Stellenbosch University. Faculty of Science. Dept. of Physics. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | ix, 119 p. |
| Rights | Stellenbosch University |
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