Initialization of Continuous Nonlinear Models Using Extended Kalman Filter

The two main objectives of Bayesian inference are to estimate parameters and states. In this thesis, we are interested in how this can be done in the framework of state-space models when there is a complete or partial lack of knowledge of the initial state of a continuous nonlinear dynamical system. In literature, similar problems have been referred to as diffuse initialization problems. This is achieved first by extending the previously developed diffuse initialization Kalman filtering techniques for discrete systems to continuous systems. The second objective is to estimate parameters using MCMC methods with a likelihood function obtained from the diffuse filtering. These methods are tried on the data collected from the 1995 Ebola outbreak in Kikwit, DRC in order to estimate the parameters of the system.

Bayesian analysis of SEIR epidemic models

This thesis concerns the analysis of epidemic models. We adopt the Bayesian paradigm and develop suitable Markov Chain Monte Carlo (MCMC) algorithms. This is done by considering an Ebola outbreak in the Democratic Republic of Congo, former Zaïre, 1995 as a case of SEIR epidemic models. We model the Ebola epidemic deterministically using ODEs and stochastically through SDEs to take into account a possible bias in each compartment. Since the model has unknown parameters, we use different methods to estimate them such as least squares, maximum likelihood and MCMC. The motivation behind choosing MCMC over other existing methods in this thesis is that it has the ability to tackle complicated nonlinear problems with large number of parameters. First, in a deterministic Ebola model, we compute the likelihood function by sum of square of residuals method and estimate parameters using the LSQ and MCMC methods. We sample parameters and then use them to calculate the basic reproduction number and to study the disease-free equilibrium. From the sampled chain from the posterior, we test the convergence diagnostic and confirm the viability of the model. The results show that the Ebola model fits the observed onset data with high precision, and all the unknown model parameters are well identified. Second, we convert the ODE model into a SDE Ebola model. We compute the likelihood function using extended Kalman filter (EKF) and estimate parameters again. The motivation of using the SDE formulation here is to consider the impact of modelling errors. Moreover, the EKF approach allows us to formulate a filtered likelihood for the parameters of such a stochastic model. We use the MCMC procedure to attain the posterior distributions of the parameters of the SDE Ebola model drift and diffusion parts. In this thesis, we analyse two cases: (1) the model error covariance matrix of the dynamic noise is close to zero , i.e. only small stochasticity added into the model. The results are then similar to the ones got from deterministic Ebola model, even if methods of computing the likelihood function are different (2) the model error covariance matrix is different from zero, i.e. a considerable stochasticity is introduced into the Ebola model. This accounts for the situation where we would know that the model is not exact. As a results, we obtain parameter posteriors with larger variances. Consequently, the model predictions then show larger uncertainties, in accordance with the assumption of an incomplete model.

Les modèles vectoriels et multiplicatifs avec erreurs non-négatives de séries chronologiques.

L'objectif du présent mémoire vise à présenter des modèles de séries chronologiques multivariés impliquant des vecteurs aléatoires dont chaque composante est non-négative. Nous considérons les modèles vMEM (modèles vectoriels et multiplicatifs avec erreurs non-négatives) présentés par Cipollini, Engle et Gallo (2006) et Cipollini et Gallo (2010). Ces modèles représentent une généralisation au cas multivarié des modèles MEM introduits par Engle (2002). Ces modèles trouvent notamment des applications avec les séries chronologiques financières. Les modèles vMEM permettent de modéliser des séries chronologiques impliquant des volumes d'actif, des durées, des variances conditionnelles, pour ne citer que ces applications. Il est également possible de faire une modélisation conjointe et d'étudier les dynamiques présentes entre les séries chronologiques formant le système étudié. Afin de modéliser des séries chronologiques multivariées à composantes non-négatives, plusieurs spécifications du terme d'erreur vectoriel ont été proposées dans la littérature. Une première approche consiste à considérer l'utilisation de vecteurs aléatoires dont la distribution du terme d'erreur est telle que chaque composante est non-négative. Cependant, trouver une distribution multivariée suffisamment souple définie sur le support positif est plutôt difficile, au moins avec les applications citées précédemment. Comme indiqué par Cipollini, Engle et Gallo (2006), un candidat possible est une distribution gamma multivariée, qui impose cependant des restrictions sévères sur les corrélations contemporaines entre les variables. Compte tenu que les possibilités sont limitées, une approche possible est d'utiliser la théorie des copules. Ainsi, selon cette approche, des distributions marginales (ou marges) peuvent être spécifiées, dont les distributions en cause ont des supports non-négatifs, et une fonction de copule permet de tenir compte de la dépendance entre les composantes. Une technique d'estimation possible est la méthode du maximum de vraisemblance. Une approche alternative est la méthode des moments généralisés (GMM). Cette dernière méthode présente l'avantage d'être semi-paramétrique dans le sens que contrairement à l'approche imposant une loi multivariée, il n'est pas nécessaire de spécifier une distribution multivariée pour le terme d'erreur. De manière générale, l'estimation des modèles vMEM est compliquée. Les algorithmes existants doivent tenir compte du grand nombre de paramètres et de la nature élaborée de la fonction de vraisemblance. Dans le cas de l'estimation par la méthode GMM, le système à résoudre nécessite également l'utilisation de solveurs pour systèmes non-linéaires. Dans ce mémoire, beaucoup d'énergies ont été consacrées à l'élaboration de code informatique (dans le langage R) pour estimer les différents paramètres du modèle. Dans le premier chapitre, nous définissons les processus stationnaires, les processus autorégressifs, les processus autorégressifs conditionnellement hétéroscédastiques (ARCH) et les processus ARCH généralisés (GARCH). Nous présentons aussi les modèles de durées ACD et les modèles MEM. Dans le deuxième chapitre, nous présentons la théorie des copules nécessaire pour notre travail, dans le cadre des modèles vectoriels et multiplicatifs avec erreurs non-négatives vMEM. Nous discutons également des méthodes possibles d'estimation. Dans le troisième chapitre, nous discutons les résultats des simulations pour plusieurs méthodes d'estimation. Dans le dernier chapitre, des applications sur des séries financières sont présentées. Le code R est fourni dans une annexe. Une conclusion complète ce mémoire.

Bayesian model comparison with un-normalised likelihoods

Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalizing constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes’ factors for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates.

abctools: an R package for tuning approximate Bayesian computation analyses

Approximate Bayesian computation (ABC) is a popular family of algorithms which perform approximate parameter inference when numerical evaluation of the likelihood function is not possible but data can be simulated from the model. They return a sample of parameter values which produce simulations close to the observed dataset. A standard approach is to reduce the simulated and observed datasets to vectors of summary statistics and accept when the difference between these is below a specified threshold. ABC can also be adapted to perform model choice. In this article, we present a new software package for R, abctools which provides methods for tuning ABC algorithms. This includes recent dimension reduction algorithms to tune the choice of summary statistics, and coverage methods to tune the choice of threshold. We provide several illustrations of these routines on applications taken from the ABC literature.

Inference for a skew extension of the Grubbs model

In this paper, we discuss inferential aspects for the Grubbs model when the unknown quantity x (latent response) follows a skew-normal distribution, extending early results given in Arellano-Valle et al. (J Multivar Anal 96:265-281, 2005b). Maximum likelihood parameter estimates are computed via the EM-algorithm. Wald and likelihood ratio type statistics are used for hypothesis testing and we explain the apparent failure of the Wald statistics in detecting skewness via the profile likelihood function. The results and methods developed in this paper are illustrated with a numerical example.

Improved testing inference in mixed linear models

Mixed linear models are commonly used in repeated measures studies. They account for the dependence amongst observations obtained from the same experimental unit. Often, the number of observations is small, and it is thus important to use inference strategies that incorporate small sample corrections. In this paper, we develop modified versions of the likelihood ratio test for fixed effects inference in mixed linear models. In particular, we derive a Bartlett correction to such a test, and also to a test obtained from a modified profile likelihood function. Our results generalize those in [Zucker, D.M., Lieberman, O., Manor, O., 2000. Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood. Journal of the Royal Statistical Society B, 62,827-838] by allowing the parameter of interest to be vector-valued. Additionally, our Bartlett corrections allow for random effects nonlinear covariance matrix structure. We report simulation results which show that the proposed tests display superior finite sample behavior relative to the standard likelihood ratio test. An application is also presented and discussed. (C) 2008 Elsevier B.V. All rights reserved.

Reduction of Models in the Presence of Nuisance Parameters

In many statistical inference problems, there is interest in estimation of only some elements of the parameter vector that defines the adopted model. In general, such elements are associated to measures of location and the additional terms, known as nuisance parameters, to control the dispersion and asymmetry of the underlying distributions. To estimate all the parameters of the model and to draw inferences only on the parameters of interest. Depending on the adopted model, this procedure can be both algebraically is common and computationally very costly and thus it is convenient to reduce it, so that it depends only on the parameters of interest. This article reviews estimation methods in the presence of nuisance parameters and consider some applications in models recently discussed in the literature.

Feasible estimation of generalized linear mixed models (GLMM) with weak dependency between groups

This paper presents a two-step pseudo likelihood estimation technique for generalized linear mixed models with the random effects being correlated between groups. The core idea is to deal with the intractable integrals in the likelihood function by multivariate Taylor's approximation. The accuracy of the estimation technique is assessed in a Monte-Carlo study. An application of it with a binary response variable is presented using a real data set on credit defaults from two Swedish banks. Thanks to the use of two-step estimation technique, the proposed algorithm outperforms conventional pseudo likelihood algorithms in terms of computational time.

On the statistical estimation of diffusion processes: a survey

Data available on continuos-time diffusions are always sampled discretely in time. In most cases, the likelihood function of the observations is not directly computable. This survey covers a sample of the statistical methods that have been developed to solve this problem. We concentrate on some recent contributions to the literature based on three di§erent approaches to the problem: an improvement of the Euler-Maruyama discretization scheme, the use of Martingale Estimating Functions and the application of Generalized Method of Moments (GMM).

On the statistical estimation of diffusion processes - a partial survey

Data available on continuous-time diffusions are always sampled discretely in time. In most cases, the likelihood function of the observations is not directly computable. This survey covers a sample of the statistical methods that have been developed to solve this problem. We concentrate on some recent contributions to the literature based on three di§erent approaches to the problem: an improvement of the Euler-Maruyama discretization scheme, the employment of Martingale Estimating Functions, and the application of Generalized Method of Moments (GMM).

Comparação de modelos de regressão aleatória para estimação de parâmetros genéticos em caprinos leiteiros

Objetivou-se avaliar a melhor modelagem para as variâncias genética aditiva, de ambiente permanente e residual da produção de leite no dia do controle (PLDC) de caprinos. Utilizaram-se modelos de regressão aleatória sobre polinômios ortogonais de Legendre com diferentes ordens de ajuste e variância residual heterogênea. Consideraram-se como efeitos fixos os efeitos de grupo de contemporâneos, a idade da cabra ao parto (co-variável) e a regressão fixa da PLDC sobre polinômios de Legendre, para modelar a trajetória média da população; e, como efeitos aleatórios, os efeitos genético aditivo e de ambiente permanente. O modelo com quatro classes de variâncias residuais foi o que proporcionou melhor ajuste. Os valores do logaritmo da função de verossimilhança, de AIC e BIC apontaram para seleção de modelos com ordens mais altas (cinco para o efeito genético e sete para o efeito de ambiente permanente). Entretanto, os autovalores associados às matrizes de co-variâncias entre os coeficientes de regressão indicaram a possibilidade de redução da dimensionalidade. As altas ordens de ajuste proporcionaram estimativas de variâncias genéticas e correlações genéticas e de ambiente permanente que não condizem com o fenômeno biológico estudado. O modelo de quinta ordem para a variância genética aditiva e de sétima ordem para o ambiente permanente foi indicado. Entretanto, um modelo mais parcimonioso, de quarta ordem para o efeito genético aditivo e de sexta ordem para o efeito de ambiente permanente, foi suficiente para ajustar as variâncias nos dados.

Influência da informação à priori na escolha de modelos para avaliação genética de suínos

Programas de melhoramento são atividades que se desenvolvem durante anos e, por isso, devem ser flexíveis ao ajuste às novas situações criadas por mudanças nas tendências de mercado, na situação econômica e aquelas causadas por aumento do volume e qualidade dos dados e, também, por novas técnicas propostas pela comunidade científica. O ajuste a essas últimas deve ser feito, principalmente, por meio da substituição e escolha do modelo mais adequado para a descrição do fenômeno, em um determinado cenário. Os dados de ganho de peso médio diário, de um programa de melhoramento de suínos, envolvendo as raças Duroc, Landrace e Large White, foram analisados por meio da teoria bayesiana, por meio de dois modelos candidatos. Foram simulados três níveis de informação à priori: informativa, pouco informativa e não informativa. O comportamento das curvas das distribuições à posteriori e as respectivas estimativas associadas a cada nível de informação à priori foram analisadas e comparadas. Os resultados indicam que no modelo mais simples, as amostras das três raças são suficientes para produzir estimativas que não são alteradas pela informação à priori. Com relação ao mais parametrizado, as estimativas, para a raça Duroc, são alteradas pelo conhecimento prévio e, nesse caso, deve se buscar a melhor representação possível da distribuição à priori para obtenção de estimativas que são mais adequadas, dado o estado de conhecimento atual do melhorista.

Efeito da interação reprodutor x rebanho sobre as produções de leite e de gordura na raça Pardo-Suíça

Dados de 4.959 lactações de 2.414 vacas da raça Pardo-Suíça, filhas de 70 reprodutores, distribuídos em 51 rebanhos, foram utilizados para se estimar o componente de variância para a interação reprodutor x rebanho das produções de leite e de gordura e verificar o efeito desta interação sobre a avaliação genética dos reprodutores, por meio de modelos que diferiam na presença e ausência do termo de interação. As produções de leite e de gordura foram ajustadas para duas ordenhas diárias, 305 dias de lactação e idade adulta da vaca. O teste da razão de verossimilhança foi utilizado na verificação da efetividade da inclusão da interação no modelo. As médias das produções de leite e de gordura foram 6085,79 ± 1629,73 kg e 225,61 ± 60,44 kg, respectivamente. A proporção da variância total decorrente da interação reprodutor x rebanho foi 0,4%, para a produção de leite, e 1%, para a produção de gordura. A estimativa de herdabilidade foi 0,38, para a produção de leite, utilizando-se ambos os modelos, e reduziu de 0,40 para 0,39, para a produção de gordura, quando o modelo com interação foi considerado. A função de verossimilhança aumentou significativamente com a inclusão da interação no modelo. A correlação de Spearman foi próxima de um para ambas as características, quando todos os reprodutores foram considerados. Houve redução de 1% na estimativa de acurácia dos valores genéticos preditos para ambas as características, porém, a correlação de Pearson estimada entre as acurácias obtidas para cada modelo estudado foi próxima à unidade. A interaçãoreprodutor x rebanho não afetou as estimativas de componentes de variâncias genética e residual e a ordem de classificação dos reprodutores para ambas as características.

Solving the Levins' paradox in the logistic model to the population growth

We introduce a new method to improve Markov maps by means of a Bayesian approach. The method starts from an initial map model, wherefrom a likelihood function is defined which is regulated by a temperature-like parameter. Then, the new constraints are added by the use of Bayes rule in the prior distribution. We applied the method to the logistic map of population growth of a single species. We show that the population size is limited for all ranges of parameters, allowing thus to overcome difficulties in interpretation of the concept of carrying capacity known as the Levins paradox. © Published under licence by IOP Publishing Ltd.