Latent Stick-Breaking Processes.

We develop a model for stochastic processes with random marginal distributions. Our model relies on a stick-breaking construction for the marginal distribution of the process, and introduces dependence across locations by using a latent Gaussian copula model as the mechanism for selecting the atoms. The resulting latent stick-breaking process (LaSBP) induces a random partition of the index space, with points closer in space having a higher probability of being in the same cluster. We develop an efficient and straightforward Markov chain Monte Carlo (MCMC) algorithm for computation and discuss applications in financial econometrics and ecology. This article has supplementary material online.

Stochastic monotonicity and duality for continuous time Markov chains with general q-Matrix

The key problems in discussing stochastic monotonicity and duality for continuous time Markov chains are to give the criteria for existence and uniqueness and to construct the associated monotone processes in terms of their infinitesimal q -matrices. In their recent paper, Chen and Zhang [6] discussed these problems under the condition that the given q-matrix Q is conservative. The aim of this paper is to generalize their results to a more general case, i.e., the given q-matrix Q is not necessarily conservative. New problems arise 'in removing the conservative assumption. The existence and uniqueness criteria for this general case are given in this paper. Another important problem, the construction of all stochastically monotone Q-processes, is also considered.

Existence, uniqueness and constructions for stochastically monotone Q-processes

The key problems in discussing duality and monotonicity for continuous-time Markov chains are to find conditions for existence and uniqueness and then to construct corresponding processes in terms of infinitesimal characteristics, i.e., q-matrices. Such problems are solved in this paper under the assumption that the given q-matrix is conservative. Some general properties of stochastically monotone Q-process ( Q is not necessarily conservative) are also discussed.

A Markov modulated Poisson process model for rainfall increments

The problems encountered when using traditional rectangular pulse hierarchical point processmodels for fine temporal resolution and the growing number of available tip-time records suggest that rainfall increments from tipping-bucket gauges be modelled directly. Poisson processes are used with an arrival rate modulated by a Markov chain in Continuous time. The paper shows how, by using two or three states for this chain, much of the structure of the rainfall intensity distribution and the wet/dry sequences can be represented for time-scales as small as 5 minutes.

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

Problèmes de premier passage et de commande optimale pour des chaînes de Markov à temps discret.

Nous considérons des processus de diffusion, définis par des équations différentielles stochastiques, et puis nous nous intéressons à des problèmes de premier passage pour les chaînes de Markov en temps discret correspon- dant à ces processus de diffusion. Comme il est connu dans la littérature, ces chaînes convergent en loi vers la solution des équations différentielles stochas- tiques considérées. Notre contribution consiste à trouver des formules expli- cites pour la probabilité de premier passage et la durée de la partie pour ces chaînes de Markov à temps discret. Nous montrons aussi que les résultats ob- tenus convergent selon la métrique euclidienne (i.e topologie euclidienne) vers les quantités correspondantes pour les processus de diffusion. En dernier lieu, nous étudions un problème de commande optimale pour des chaînes de Markov en temps discret. L’objectif est de trouver la valeur qui mi- nimise l’espérance mathématique d’une certaine fonction de coût. Contraire- ment au cas continu, il n’existe pas de formule explicite pour cette valeur op- timale dans le cas discret. Ainsi, nous avons étudié dans cette thèse quelques cas particuliers pour lesquels nous avons trouvé cette valeur optimale.

Parameter Estimation in Minification Processes

In this article it is proved that the stationary Markov sequences generated by minification models are ergodic and uniformly mixing. These results are used to establish the optimal properties of estimators for the parameters in the model. The problem of estimating the parameters in the exponential minification model is discussed in detail.

Identification of plasmodium vivax proteins with potential role in invasion using sequence redundancy reduction and profile hidden markov models

Background: This study describes a bioinformatics approach designed to identify Plasmodium vivax proteins potentially involved in reticulocyte invasion. Specifically, different protein training sets were built and tuned based on different biological parameters, such as experimental evidence of secretion and/or involvement in invasion-related processes. A profile-based sequence method supported by hidden Markov models (HMMs) was then used to build classifiers to search for biologically-related proteins. The transcriptional profile of the P. vivax intra-erythrocyte developmental cycle was then screened using these classifiers. Results: A bioinformatics methodology for identifying potentially secreted P. vivax proteins was designed using sequence redundancy reduction and probabilistic profiles. This methodology led to identifying a set of 45 proteins that are potentially secreted during the P. vivax intra-erythrocyte development cycle and could be involved in cell invasion. Thirteen of the 45 proteins have already been described as vaccine candidates; there is experimental evidence of protein expression for 7 of the 32 remaining ones, while no previous studies of expression, function or immunology have been carried out for the additional 25. Conclusions: The results support the idea that probabilistic techniques like profile HMMs improve similarity searches. Also, different adjustments such as sequence redundancy reduction using Pisces or Cd-Hit allowed data clustering based on rational reproducible measurements. This kind of approach for selecting proteins with specific functions is highly important for supporting large-scale analyses that could aid in the identification of genes encoding potential new target antigens for vaccine development and drug design. The present study has led to targeting 32 proteins for further testing regarding their ability to induce protective immune responses against P. vivax malaria.

Probabilidad de ruina con un modelo de riesgo Markov-modulado y propiedades del proceso telegráfico

En esta tesis se entiende la intuición y las matemáticas del modelo generalizado de la teoría de la ruina, expuesto por Li y Lu quienes se remiten a los cálculos de la probabilidad de ruina de Reinhard. Teniendo en cuenta las definiciones, la media y la varianza del proceso telegráfico con saltos encontradas en la tesis doctoral de Óscar López. Luego se simula el proceso de riesgo y finalmente con un ejemplo se calcula la probabilidad de ruina numéricamente. Todo el proceso se va a realizar teniendo en cuenta reclamaciones con distribución exponencial.

A Wiener-Hopf type factorization for the exponential functional of Levy processes

For a Lévy process ξ=(ξt)t≥0 drifting to −∞, we define the so-called exponential functional as follows: Formula Under mild conditions on ξ, we show that the following factorization of exponential functionals: Formula holds, where × stands for the product of independent random variables, H− is the descending ladder height process of ξ and Y is a spectrally positive Lévy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of Iξ for a large class of Lévy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein–Uhlenbeck processes, which is itself of independent interest. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process.

The uniqueness of signature problem in the non-Markov setting

We establish a general framework for a class of multidimensional stochastic processes over [0,1] under which with probability one, the signature (the collection of iterated path integrals in the sense of rough paths) is well-defined and determines the sample paths of the process up to reparametrization. In particular, by using the Malliavin calculus we show that our method applies to a class of Gaussian processes including fractional Brownian motion with Hurst parameter H>1/4, the Ornstein–Uhlenbeck process and the Brownian bridge.

QUASISTATIONARY DISTRIBUTIONS AND FLEMING-VIOT PROCESSES IN FINITE SPACES

Consider a continuous-time Markov process with transition rates matrix Q in the state space Lambda boolean OR {0}. In In the associated Fleming-Viot process N particles evolve independently in A with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When Lambda is finite, we show that the empirical distribution of the particles at a fixed time converges as N -> infinity to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N -> infinity to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1/N.

Adaptive control charts: A Markovian approach for processes subject to independent disturbances

Recent studies have shown that adaptive X control charts are quicker than traditional X charts in detecting small to moderate shifts in a process. In this article, we propose a joint statistical design of adaptive X and R charts having all design parameters varying adaptively. The process is subjected to two independent assignable causes. One cause changes the process mean and the other changes the process variance. However, the occurrence of one kind of assignable cause does not preclude the occurrence of the other. It is assumed that the quality characteristic is normally distributed and the time that the process remains in control has exponential distribution. Performance measures of these adaptive control charts are obtained through a Markov chain approach. (c) 2005 Elsevier B.V. All rights reserved.

A Markov random field model for combining optimum-path forest classifiers using decision graphs and game strategy approach

The research on multiple classifiers systems includes the creation of an ensemble of classifiers and the proper combination of the decisions. In order to combine the decisions given by classifiers, methods related to fixed rules and decision templates are often used. Therefore, the influence and relationship between classifier decisions are often not considered in the combination schemes. In this paper we propose a framework to combine classifiers using a decision graph under a random field model and a game strategy approach to obtain the final decision. The results of combining Optimum-Path Forest (OPF) classifiers using the proposed model are reported, obtaining good performance in experiments using simulated and real data sets. The results encourage the combination of OPF ensembles and the framework to design multiple classifier systems. © 2011 Springer-Verlag.

A Bayesian changepoint analysis on spatio-temporal point processes

Changepoint analysis is a well established area of statistical research, but in the context of spatio-temporal point processes it is as yet relatively unexplored. Some substantial differences with regard to standard changepoint analysis have to be taken into account: firstly, at every time point the datum is an irregular pattern of points; secondly, in real situations issues of spatial dependence between points and temporal dependence within time segments raise. Our motivating example consists of data concerning the monitoring and recovery of radioactive particles from Sandside beach, North of Scotland; there have been two major changes in the equipment used to detect the particles, representing known potential changepoints in the number of retrieved particles. In addition, offshore particle retrieval campaigns are believed may reduce the particle intensity onshore with an unknown temporal lag; in this latter case, the problem concerns multiple unknown changepoints. We therefore propose a Bayesian approach for detecting multiple changepoints in the intensity function of a spatio-temporal point process, allowing for spatial and temporal dependence within segments. We use Log-Gaussian Cox Processes, a very flexible class of models suitable for environmental applications that can be implemented using integrated nested Laplace approximation (INLA), a computationally efficient alternative to Monte Carlo Markov Chain methods for approximating the posterior distribution of the parameters. Once the posterior curve is obtained, we propose a few methods for detecting significant change points. We present a simulation study, which consists in generating spatio-temporal point pattern series under several scenarios; the performance of the methods is assessed in terms of type I and II errors, detected changepoint locations and accuracy of the segment intensity estimates. We finally apply the above methods to the motivating dataset and find good and sensible results about the presence and quality of changes in the process.