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Estimation of the passage time distribution on a graph via the EM algorithm
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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.

Place, publisher, year, edition, pages
2010. , 61 p.
Series
Research report in mathematical statistics, ISSN 1653-0829 ; 1
Keyword [en]
EM algorithm, maximum likelihood, first-passage percolation, phase-type dsitribution
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-33760OAI: oai:DiVA.org:umu-33760DiVA: diva2:317878
Available from: 2010-05-05 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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