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On perfect simulation and EM estimation
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Perfect simulation  and the EM algorithm are the main topics in this thesis. In paper I, we present coupling from the past (CFTP) algorithms that generate perfectly distributed samples from the multi-type Widom--Rowlin-son (W--R) model and some generalizations of it. The classical W--R model is a point process in the plane or the  space consisting of points of several different types. Points of different types are not allowed to be closer than some specified distance, whereas points of the same type can be arbitrary close. A stick-model and soft-core generalizations are also considered. Further, we  generate samples without edge effects, and give a bound on sufficiently small intensities (of the points) for the algorithm to terminate.

In paper II, we consider the  forestry problem on how to estimate  seedling dispersal distributions and effective plant fecundities from spatially data of adult trees  and seedlings, when the origin of the seedlings are unknown.   Traditional models for fecundities build on allometric assumptions, where the fecundity is related to some  characteristic of the adult tree (e.g.\ diameter). However, the allometric assumptions are generally too restrictive and lead to nonrealistic estimates. Therefore we present a new model, the unrestricted fecundity (UF) model, which uses no allometric assumptions. We propose an EM algorithm to estimate the unknown parameters.   Evaluations on real and simulated data indicates better performance for the UF model.

In paper III, we propose  EM algorithms to  estimate the passage time distribution on a graph.Data is obtained by observing a flow only at the nodes -- what happens on the edges is unknown. Therefore the sample of passage times, i.e. the times it takes for the flow to stream between two neighbors, consists of right censored and uncensored observations where it sometimes is unknown which is which.       For discrete passage time distributions, we show that the maximum likelihood (ML) estimate is strongly consistent under certain  weak conditions. We also show that our propsed EM algorithm  converges to the ML estimate if the sample size is sufficiently large and the starting value is sufficiently close to the true parameter. In a special case we show that it always converges.  In the continuous case, we propose an EM algorithm for fitting  phase-type distributions to data.

Place, publisher, year, edition, pages
Umeå: Print & Media , 2010. , 29 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300
Keyword [en]
Perfect simulation, coupling from the past, Markov chain Monte Carlo, point process, Widom-Rowlinson model, EM algorithm, dispersal distribution, fecundity, first-passage percolation
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-33779ISBN: 978-91-7264-985-9 (print)OAI: oai:DiVA.org:umu-33779DiVA: diva2:318076
Public defence
2010-06-01, N420, Umeå universitet, Umeå, 13:01 (English)
Opponent
Supervisors
Available from: 2010-05-07 Created: 2010-05-06 Last updated: 2010-05-07Bibliographically approved
List of papers
1. Perfect simulation of some spatial point processes
Open this publication in new window or tab >>Perfect simulation of some spatial point processes
1999 (English)Licentiate thesis, monograph (Other academic)
Abstract [en]

Coupling from the past (CFTP) algorithms are presented that generate perfectly distributed samples from the multi-type Widom--Rowlinson (W--R) model and some generalizations of it. The classical W--R model is a point process in the plane or the  space consisting of points of several different types. Points of different types are not allowed to be closer than some specified distance, whereas points of the same type can be arbitrary close. An application can be to describe certain gases consisting of several types of particles.

 We also consider a soft-core W--R model, where points of different types are not completely forbidden to be close to each other, just inhibited in various degrees. Furthermore, we allow the hindrance between two points of different types to be  explained by more than the Euclidean  distance between them. In particular we consider a  stick-model where the hindrance is defined by imaginary sticks, with centers at the associated points, and  where sticks are not allowed to cross each other. The  different directions of the sticks (a finite number), represent the different types of the points. A CFTP algorithm is also given for a soft-core version  of the stick-model.

 

Simulation studies indicate that the runtime of the CFTP algorithm for the multi-type W--R model   in the symmetric case (i.e.\ equal intensities),  first grows exponentially with the intensity, but then suddenly, when the intensity becomes larger seems to be superexponential. This change in growth  may be explained by a phase transition.

 We also present a CFTP algorithm that yields samples without edge effects from  the multi-type W--R model.  The underlying idea behind this algorithm is to not only simulate backwards in time, but also outwards in space. This algorithm does not always  terminate for large intensities of the points. A bound on sufficiently small intensities for the algorithm to terminate is given.

Place, publisher, year, edition, pages
Göteborg: Göteborgs universitet, 1999. 63 p.
Series
Department of Mathematics, Chalmers University of Technology and Göteborg University, ISSN 0347-2809 ; 59
Keyword
Perfect simulering; coupling from the past; Markov shain Monte Carlo; point process; Widom-Rowlinson model
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-33757 (URN)
Presentation
, Matematiskt centrum, Göteborg (English)
Supervisors
Note
Eg. Kajsa FröjdAvailable from: 2010-05-06 Created: 2010-05-05 Last updated: 2010-05-07Bibliographically approved
2. Inverse modeling for effective dispersal: do we need tree size to estimate fecundity?
Open this publication in new window or tab >>Inverse modeling for effective dispersal: do we need tree size to estimate fecundity?
(English)Manuscript (preprint) (Other (popular science, discussion, etc.))
Abstract [en]

Inverse modeling methods made possible the estimation of the dispersal kernel and of plant fecundity for the seedling and sapling stages of the recruitment process. Current models for the fecundities of adult trees are build on allometric assumptions where the number of successfully estab¬lished offspring produced by an adult is assumed to be in relation to some (easily) measured characteristic of the specific tree (usually the tree’s basal area). However, the allometric assumption relating tree size to reproduc¬tive success in the sapling (or seedling) stage should be questionable when numerous, well-documented, post-dispersal processes such as safe-site limitation for recruitment or negative density-dependent seedling mortality can cancel out the presumably strong relationship between tree size and seed set. In this paper we hypothesize that when the relationship between tree size and reproductive success is not strong enough then its use in in¬verse modeling is counter-productive and may lead to poor model fits and/or unstable solutions for the parameters of the model. We present a new model for effective dispersal termed the unrestricted fecundity (UF) model, which makes no allometric assumptions on the fecundities; instead they are allowed to vary freely and even to be zero. Based on this model, we examine the hypothesis that when fecundities are estimated indepen¬dently of tree size (or any other tree characteristic), the goodness-of-fit and the ecological meaning of dispersal models (in the seedling or sapling stage) may be enhanced. Parameters of the UF model are estimated through the EM algorithm and their standard errors are approximated via the observed information matrix. We fit the UF model to a dataset from an expanding European beech population of central Spain as well as to a set of simulated data. In comparisons with an allometric model, the UF model fitted the data better and the parameter estimates were less biased. The ecological meaning of the UF model results was also superior. We sug¬gest using this new approach for modeling dispersal in the seedling and sapling stages when tree size is not deemed to be in strong relation to the reproductive success of adults.

Keyword
EM algorithm, European beech, dispersal kernel, recruitment, reproductive process
Identifiers
urn:nbn:se:umu:diva-33759 (URN)
Available from: 2010-05-05 Created: 2010-05-05 Last updated: 2010-05-07Bibliographically approved
3. Estimation of the passage time distribution on a graph via the EM algorithm
Open this publication in new window or tab >>Estimation of the passage time distribution on a graph via the EM algorithm
2010 (English)Report (Other (popular science, discussion, etc.))
Abstract [en]

e propose EM algorithms to  estimate the passage time distribution on a graph.  Data is obtained by observing a flow only at the nodes -- what happens on the edges is unknown. Therefore the sample of passage times, i.e. the times it takes for the flow to stream between two neighbors, consists of right censored and uncensored observations where it sometimes is unknown which is which. For discrete passage time distributions, we show that the maximum likelihood (ML) estimate is strongly consistent under certain  weak conditions. We also show that the EM algorithm  converges to the ML estimate if the sample size is sufficiently large and the starting value is sufficiently close to the true parameter. In a special case we show that it always converges.  In the continuous case, we propose an EM algorithm for fitting  phase-type distributions to data.

Publisher
61 p.
Series
Research report in mathematical statistics, ISSN 1653-0829 ; 1
Keyword
EM algorithm, maximum likelihood, first-passage percolation, phase-type dsitribution
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-33760 (URN)
Available from: 2010-05-05 Created: 2010-05-05 Last updated: 2010-05-07Bibliographically approved

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