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Perfect simulation of some spatial point processes
Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University.
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 [en]
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: urn:nbn:se:umu:diva-33757OAI: oai:DiVA.org:umu-33757DiVA: diva2:317868
Presentation
, Matematiskt centrum, Göteborg (English)
Supervisors
Note
Eg. Kajsa FröjdAvailable from: 2010-05-06 Created: 2010-05-05 Last updated: 2010-05-07Bibliographically approved
In thesis
1. On perfect simulation and EM estimation
Open this publication in new window or tab >>On perfect simulation and EM estimation
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
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:nbn:se:umu:diva-33779 (URN)978-91-7264-985-9 (ISBN)
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

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Citation style
  • apa
  • ieee
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  • Other style
More styles
Language
  • de-DE
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  • Other locale
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Output format
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