524 research outputs found
An ergodicity result for adaptive Langevin algorithms
Get PDFWe consider a class of adaptive MCMC algorithms using a Langevin-type
proposal density. We prove that these are algorithms are ergodic
when the target density has exponential tail behaviour. Unlike previous
results, our approach does not require bounding the drift function
Discussion of "Modern Statistics for Spatial Point Processes"
No full textThe paper ‘Modern statistics for spatial point processes' by Jesper Møller and Rasmus P. Waagepetersen is based on a special invited lecture given by the authors at the 21st Nordic Conference on Mathematical Statistics, held at Rebild, Denmark, in June 2006. At the conference, Antti Penttinen and Eva B. Vedel Jensen were invited to discuss the paper. We here present the comments from the two invited discussants and from a number of other scholars, as well as the authors' responses to these comments. Below Figure 1, Figure 2, etc., refer to figures in the paper under discussion, while Figure A, Figure B, etc., refer to figures in the current discussion. All numbered sections and formulas refer to the paper.The paper ‘Modern statistics for spatial point processes' by Jesper Møller and Rasmus P. Waagepetersen is based on a special invited lecture given by the authors at the 21st Nordic Conference on Mathematical Statistics, held at Rebild, Denmark, in June 2006. At the conference, Antti Penttinen and Eva B. Vedel Jensen were invited to discuss the paper. We here present the comments from the two invited discussants and from a number of other scholars, as well as the authors' responses to these comments. Below Figure 1, Figure 2, etc., refer to figures in the paper under discussion, while Figure A, Figure B, etc., refer to figures in the current discussion. All numbered sections and formulas refer to the paper
Generalised Random Fields and the De Wijs Process: Theory and Implementation
No full textIn this thesis, we present the theory of generalised functions as a foundation of generalised stochastic processes. Afterwards, we introduced the generalised stochastic process, which serves as an abstraction of the conventional notion of stochastic processes. A particular case of generalised stochastic processes is the generalised random field, where a special case of these is of particular interest. Specifically, a so-called conformal model, called the De Wijs plus white noise process, is the centre of attention in the thesis. We present theory on parameter estimation for this process and seek to apply this to a particular dataset. To do this, we implement two different estimation method in the statistical programming language R. We then attempt to utilise these implemented functions on the so-called bcicov-dataset, which contains measurements of soil samples, from Barro Colorado Island in Panama
Quasi-likelihood functions on the sphere
No full textDette speciales formål er at introducere punktprocesteori på kuglen med særligt fokus på estimation af intensiteten/intensitetsfunktionen. Denne approksimation implementeres i R som en quasi-likelihoodfunktion og testes i det tilfælde, hvor intensitetsfunktionen er konstant. For at kunne behandle teorien for punktprocesser, Palm fordelingen og punktproces karakteristika på kuglen introduceres overflademålet. Desuden præsenteres nogle klynget punktprocesser, hvor den log Gaussiske Cox proces anvendes i testen af quasi-likelihoodfunktionen. For at udlede quasi-likelihoodfunktionen præsenteres teorien om estimationsfunktioner. Her anvendes Nyström approksimationen på den optimale første ordens estimationsfunktion med henblik på at løse den numerisk. Derudover anvendes Mercers repræsentation til at løse den optimale første ordens estimationsfunktion numerisk. I tilfældet, hvor denne metode kan anvendes, er løsningen til estimationsfunktionen konstant, hvorfor estimatet for intensiteten er det intuitive estimat, men også det optimale estimat. Testen af implementeringen af quasi-likelihoodfunktionen til estimation af intensitetsfunktionen stemmer i tilfældet med konstant intensitet og observationvindue W=S2 overens med det intuitive estimat, som er vist at være optimalt i dette tilfælde.This thesis treats the theory of point processes on the sphere with a focus on quasi-likelihood estimation. The purpose is to introduce theory for point processes on the sphere and investigate a new numeric approximation method for estimating functions and compare it to the Nyström approximation method and implement both in R.To treat the general theory of point processes we define the surface measure, which is used on the sphere instead of the Lebesgue measure. We introduce some general properties and formulas, which are the foundation of the thesis. Furthermore summary statistics for point processes on the sphere and there non-parametric estimates are presented. These are deduced by applying the theory of the Palm distribution, hence the Palm distribution, and its properties are introduced for point processes on the sphere.In addition to that some clustered point processes are presented, where the log Gaussian Cox process is used to test the implementation of the quasi-likelihood function.The quasi-likelihood function is a estimation function, hence we present the theory of estimation functions, where the Godambe information criteria is used to obtain a sufficient condition for a optimal estimation function. This condition is used to obtain a Fredholm integral equation of the second kind, where the solution is a function φ depending on the points of the point process and a parameter vector. The solution φ is the function solving the estimation function optimally. The exact solution of φ is generally not explicit. Therefore two numeric approximation methods are introduced. The first method is based on the theory of Mercer's representation for complex covariance functions. This method does only apply, when the intensity function is constant, i.e. the point process is assumed isotropic, and the observation window is the sphere. When applying Mercer's representation to numerically solve φ, we discover that the numeric solution of φ is a constant, hence the optimal first order estimating function is constant. This implies that the optimal estimate for the intensity is the intuitive estimate.The second approximation method, we introduce, is the Nyström approximation method. This method applies in a more general context and can be written as a quasi-likelihood function, i.e. a optimal estimating function. The quasi-likelihood function obtain by using Nyström approximation is implemented in R as a function s2quasi which is available in appendix.In hindsight it is discussed, whether a constant function is a exact solution to φ, when the point process is assumed isotrpoic, and the observation window is the sphere. Therefore the optimal estimate of the intensity is the intuitive estimate. The implemented R function s2quasi is used to fit a log Gaussian Cox process to the dataset "galaxies", when assuming constant intensity and isotropic covariance function with observation window W=S2. The estimate of the intensity function using s2quasi corresponds in this setting to the intuitive estimate of the intensity. Many of the properties and formulas for point process theory on Rd can be adapted to point processes on the sphere with clear differences in the non-parametric estimate of the pair correlation function and the relation between the pair correlation function and the K-function.Estimation of the intensity function when assumed a isotropic point process with the sphere as observation window gives that the optimal estimate of the intensity function is the intuitive estimate. Further work would involve simulation studies of the quasi-likelihood function compared to a composite likelihood function, examples with quasi-likelihood estimation for inhomogeneous intensity function and estimation of the pair correlation function in practice using either the K-function or the non-parametric estimate of the pair correlation function
Growth Mixture models - Longitudinal data with latent groups
No full textDette kandidatspeciale tager udgangspunkt i et datasæt bestående af folkeskoleelever, der over fire peroder er blevet undervist i forskellige emner inden for matematik. Specialet indeholder en analyse af, hvordan disse elever har udviklet deres brøkregningskompetencer over disse perioder, og yderligere hvordan eleverne kan grupperes samt den relevante underliggende teori. Datasættet indeholder dog manglende værdier i responsvariablen, hvorfor vi indledningsvist undersøger, om disse værdier er missing at random, således at de kan udelades fra analysen. Ved at inddele eleverne i højt, lavt og middel præsterende grupper, baseret på hvordan de har præsteret i de danske nationale tests, udføres en analyse af om elevgrupperne udvikler deres brøkregningskompetencer forskelligt i de fire perioder, og om dette stemmer overens med, hvad de er blevet undervist i. Dette gør vi ved at benytte en mixed model, hvorfor den underbyggende teori herfor beskrives. En anden inddeling kan ske ved brug af Growth Mixture modeller (GMM), som udover at modellere elevernes vækstkurver også finder en passende gruppeinddeling af eleverne for det antal latente grupper, der ønskes. Parameterestimation for en GMM kan dog være vanskeligt, hvorfor vi introducerer numeriske metoder til dette som Marquardt- og EM-algoritmen. Brugen af en GMM muliggør yderligere undersøgelsen af, hvilket antal grupper der er mest passende til data, hvortil informationskriterier og en likelihood ratio-test beskrives
Emerging Methods in Progression Modelling of Alzheimer's Disease: A Comparative Analysis
No full textThis thesis is a study of the progression models for repeated measures (PMRMs) introduced in[Raket, 2020]. Data for the thesis is provided by Novo Nordisk A/S and obtained from the Critical Path for Alzheimer's Disease database. The initial part of the thesis is preliminary theory regarding mixed models as well as a presentation of the PMRMs and the conventionally used constrained longitudinal data analysis (cLDA) model. Furthermore, we present modifications of the PMRMs, including the removal of a correlation structure in the errors, as well as the addition of a random effect. Moreover, a way of implementing the PMRMs in an analysis of heterogeneity of treatment effect between subgroups is explored.We conduct a simulation study, where the performance of the PMRMs is examined in different scenarios, and compared to each other as well as the cLDA model. Here, we find that there are both pros and cons of using the PMRMs, and extensions thereof, compared to the cLDA model. Overall, while the cLDA model offers robust performance and a controlled type I error rate, PMRMs provide better interpretability and higher statistical power in specific scenarios, highlighting their potential applicability in clinical trials.Lastly, we present a way of implementing the PMRMs in health economic modelling, utilising a Markov Model and the assumption of a constant treatment effect over time. Here, a brief overview of how they can be used in a cost-effectiveness analysis is presented. The concluding elements of the thesis discuss the interpretability of the models' estimates and their applicability in health economic modelling and decision-making
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