Generalised Random Fields and the De Wijs Process: Theory and Implementation

Fitzhugh, Nicholas

4. term

Mathematics, Master

2019

2019-06-06

66

I denne afhandling opbygger vi den matematiske baggrund for generaliserede funktioner som grundlag for generaliserede stokastiske processer. Generaliserede funktioner udvider den almindelige funktionsidé, så meget ujævne signaler eller målinger kan beskrives på en konsistent måde. Vi introducerer derefter generaliserede stokastiske processer, en abstraktion af konventionelle stokastiske processer. Et vigtigt tilfælde er det generaliserede stokastiske felt, dvs. en proces indekseret over rum snarere end tid. Vores hovedfokus er en særlig konform model kendt som De Wijs plus white noise-processen. Vi præsenterer teori og metoder til at estimere modellens parametre og implementerer to estimeringsmetoder i statistikprogrammeringssproget R. Til sidst forsøger vi at anvende disse implementeringer på bcicov-datasættet, som indeholder målinger fra jordprøver indsamlet på Barro Colorado Island i Panama.

This thesis develops the mathematical background of generalized functions as a basis for generalized stochastic processes. Generalized functions extend the usual idea of a function so that very irregular signals or measurements can be described in a consistent way. We then introduce generalized stochastic processes, an abstraction of conventional stochastic processes. A key case is the generalized random field, meaning a process indexed over space rather than time. Our main focus is a particular conformal model known as the De Wijs plus white noise process. We present theory and methods for estimating the parameters of this model and implement two estimation approaches in the statistical programming language R. Finally, we attempt to apply these implementations to the bcicov dataset, which contains measurements from soil samples collected on Barro Colorado Island in Panama.

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