Hierarchical modelling of timber reference properties using probabilistic methods: Maximum Likelihood Method, Bayesian Methods and Probability Networks

In recent decades, an increased interest has been evidenced in the research on multi-scale hierarchical modelling in the field of mechanics, and also in the field of wood products and timber engineering. One of the main motivations for hierar-chical modelling is to understand how properties, composition and structure at lower scale levels may influence and be used to predict the material properties on a macroscopic and structural engineering scale. This chapter presents the applicability of statistic and probabilistic methods, such as the Maximum Likelihood method and Bayesian methods, in the representation of timber’s mechanical properties and its inference accounting to prior information obtained in different importance scales. These methods allow to analyse distinct timber’s reference properties, such as density, bending stiffness and strength, and hierarchically consider information obtained through different non, semi or destructive tests. The basis and fundaments of the methods are described and also recommendations and limitations are discussed. The methods may be used in several contexts, however require an expert’s knowledge to assess the correct statistic fitting and define the correlation arrangement between properties.

Optimal designs for the prediction in hierarchical random coefficient regression models

Magdeburg, Univ., Fak. für Mathematik, Diss., 2015

Generalized factor models: a bayesian approach

There is recent interest in the generalization of classical factor models in which the idiosyncratic factors are assumed to be orthogonal and there are identification restrictions on cross-sectional and time dimensions. In this study, we describe and implement a Bayesian approach to generalized factor models. A flexible framework is developed to determine the variations attributed to common and idiosyncratic factors. We also propose a unique methodology to select the (generalized) factor model that best fits a given set of data. Applying the proposed methodology to the simulated data and the foreign exchange rate data, we provide a comparative analysis between the classical and generalized factor models. We find that when there is a shift from classical to generalized, there are significant changes in the estimates of the structures of the covariance and correlation matrices while there are less dramatic changes in the estimates of the factor loadings and the variation attributed to common factors.

On Identification of Bayesian DSGE Models

In recent years there has been increasing concern about the identification of parameters in dynamic stochastic general equilibrium (DSGE) models. Given the structure of DSGE models it may be difficult to determine whether a parameter is identified. For the researcher using Bayesian methods, a lack of identification may not be evident since the posterior of a parameter of interest may differ from its prior even if the parameter is unidentified. We show that this can even be the case even if the priors assumed on the structural parameters are independent. We suggest two Bayesian identification indicators that do not suffer from this difficulty and are relatively easy to compute. The first applies to DSGE models where the parameters can be partitioned into those that are known to be identified and the rest where it is not known whether they are identified. In such cases the marginal posterior of an unidentified parameter will equal the posterior expectation of the prior for that parameter conditional on the identified parameters. The second indicator is more generally applicable and considers the rate at which the posterior precision gets updated as the sample size (T) is increased. For identified parameters the posterior precision rises with T, whilst for an unidentified parameter its posterior precision may be updated but its rate of update will be slower than T. This result assumes that the identified parameters are pT-consistent, but similar differential rates of updates for identified and unidentified parameters can be established in the case of super consistent estimators. These results are illustrated by means of simple DSGE models.

Hierarchical conditional random fields for parts based models matching

A parts based model is a parametrization of an object class using a collection of landmarks following the object structure. The matching of parts based models is one of the problems where pairwise Conditional Random Fields have been successfully applied. The main reason of their effectiveness is tractable inference and learning due to the simplicity of involved graphs, usually trees. However, these models do not consider possible patterns of statistics among sets of landmarks, and thus they sufffer from using too myopic information. To overcome this limitation, we propoese a novel structure based on a hierarchical Conditional Random Fields, which we explain in the first part of this memory. We build a hierarchy of combinations of landmarks, where matching is performed taking into account the whole hierarchy. To preserve tractable inference we effectively sample the label set. We test our method on facial feature selection and human pose estimation on two challenging datasets: Buffy and MultiPIE. In the second part of this memory, we present a novel approach to multiple kernel combination that relies on stacked classification. This method can be used to evaluate the landmarks of the parts-based model approach. Our method is based on combining responses of a set of independent classifiers for each individual kernel. Unlike earlier approaches that linearly combine kernel responses, our approach uses them as inputs to another set of classifiers. We will show that we outperform state-of-the-art methods on most of the standard benchmark datasets.

Bayesian nash equilibrium in ''linear'' cournot models with private information about costs

Calculating explicit closed form solutions of Cournot models where firms have private information about their costs is, in general, very cumbersome. Most authors consider therefore linear demands and constant marginal costs. However, within this framework, the nonnegativity constraint on prices (and quantities) has been ignored or not properly dealt with and the correct calculation of all Bayesian Nash equilibria is more complicated than expected. Moreover, multiple symmetric and interior Bayesianf equilibria may exist for an open set of parameters. The reason for this is that linear demand is not really linear, since there is a kink at zero price: the general ''linear'' inverse demand function is P (Q) = max{a - bQ, 0} rather than P (Q) = a - bQ.

Further insights on the French WISC-IV factor structure through Bayesian structural equation modeling

The interpretation of the Wechsler Intelligence Scale for Children-Fourth Edition (WISC-IV) is based on a 4-factor model, which is only partially compatible with the mainstream Cattell-Horn-Carroll (CHC) model of intelligence measurement. The structure of cognitive batteries is frequently analyzed via exploratory factor analysis and/or confirmatory factor analysis. With classical confirmatory factor analysis, almost all crossloadings between latent variables and measures are fixed to zero in order to allow the model to be identified. However, inappropriate zero cross-loadings can contribute to poor model fit, distorted factors, and biased factor correlations; most important, they do not necessarily faithfully reflect theory. To deal with these methodological and theoretical limitations, we used a new statistical approach, Bayesian structural equation modeling (BSEM), among a sample of 249 French-speaking Swiss children (8-12 years). With BSEM, zero-fixed cross-loadings between latent variables and measures are replaced by approximate zeros, based on informative, small-variance priors. Results indicated that a direct hierarchical CHC-based model with 5 factors plus a general intelligence factor better represented the structure of the WISC-IV than did the 4-factor structure and the higher order models. Because a direct hierarchical CHC model was more adequate, it was concluded that the general factor should be considered as a breadth rather than a superordinate factor. Because it was possible for us to estimate the influence of each of the latent variables on the 15 subtest scores, BSEM allowed improvement of the understanding of the structure of intelligence tests and the clinical interpretation of the subtest scores.

Hierarchical location-allocation models for congested systems

In this paper we address the issue of locating hierarchical facilities in the presence of congestion. Two hierarchical models are presented, where lower level servers attend requests first, and then, some of the served customers are referred to higher level servers. In the first model, the objective is to find the minimum number of servers and theirlocations that will cover a given region with a distance or time standard. The second model is cast as a Maximal Covering Location formulation. A heuristic procedure is then presented together with computational experience. Finally, some extensions of these models that address other types of spatial configurations are offered.

Disentangling the formation of contrasting tree line physiognomies combining model selection and Bayesian parameterization for simulation models.

Alpine tree-line ecotones are characterized by marked changes at small spatial scales that may result in a variety of physiognomies. A set of alternative individual-based models was tested with data from four contrasting Pinus uncinata ecotones in the central Spanish Pyrenees to reveal the minimal subset of processes required for tree-line formation. A Bayesian approach combined with Markov chain Monte Carlo methods was employed to obtain the posterior distribution of model parameters, allowing the use of model selection procedures. The main features of real tree lines emerged only in models considering nonlinear responses in individual rates of growth or mortality with respect to the altitudinal gradient. Variation in tree-line physiognomy reflected mainly changes in the relative importance of these nonlinear responses, while other processes, such as dispersal limitation and facilitation, played a secondary role. Different nonlinear responses also determined the presence or absence of krummholz, in agreement with recent findings highlighting a different response of diffuse and abrupt or krummholz tree lines to climate change. The method presented here can be widely applied in individual-based simulation models and will turn model selection and evaluation in this type of models into a more transparent, effective, and efficient exercise.

Adaptive Markov chain Monte Carlo and Bayesian filtering for state space models

This thesis is concerned with the state and parameter estimation in state space models. The estimation of states and parameters is an important task when mathematical modeling is applied to many different application areas such as the global positioning systems, target tracking, navigation, brain imaging, spread of infectious diseases, biological processes, telecommunications, audio signal processing, stochastic optimal control, machine learning, and physical systems. In Bayesian settings, the estimation of states or parameters amounts to computation of the posterior probability density function. Except for a very restricted number of models, it is impossible to compute this density function in a closed form. Hence, we need approximation methods. A state estimation problem involves estimating the states (latent variables) that are not directly observed in the output of the system. In this thesis, we use the Kalman filter, extended Kalman filter, Gauss–Hermite filters, and particle filters to estimate the states based on available measurements. Among these filters, particle filters are numerical methods for approximating the filtering distributions of non-linear non-Gaussian state space models via Monte Carlo. The performance of a particle filter heavily depends on the chosen importance distribution. For instance, inappropriate choice of the importance distribution can lead to the failure of convergence of the particle filter algorithm. In this thesis, we analyze the theoretical Lᵖ particle filter convergence with general importance distributions, where p ≥2 is an integer. A parameter estimation problem is considered with inferring the model parameters from measurements. For high-dimensional complex models, estimation of parameters can be done by Markov chain Monte Carlo (MCMC) methods. In its operation, the MCMC method requires the unnormalized posterior distribution of the parameters and a proposal distribution. In this thesis, we show how the posterior density function of the parameters of a state space model can be computed by filtering based methods, where the states are integrated out. This type of computation is then applied to estimate parameters of stochastic differential equations. Furthermore, we compute the partial derivatives of the log-posterior density function and use the hybrid Monte Carlo and scaled conjugate gradient methods to infer the parameters of stochastic differential equations. The computational efficiency of MCMC methods is highly depend on the chosen proposal distribution. A commonly used proposal distribution is Gaussian. In this kind of proposal, the covariance matrix must be well tuned. To tune it, adaptive MCMC methods can be used. In this thesis, we propose a new way of updating the covariance matrix using the variational Bayesian adaptive Kalman filter algorithm.

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.

Bayesian estimation and selection of nonlinear vector error correction models: The case of the sugar-ethanol-oil nexus in Brazil

Nonlinear adjustment toward long-run price equilibrium relationships in the sugar-ethanol-oil nexus in Brazil is examined. We develop generalized bivariate error correction models that allow for cointegration between sugar, ethanol, and oil prices, where dynamic adjustments are potentially nonlinear functions of the disequilibrium errors. A range of models are estimated using Bayesian Monte Carlo Markov Chain algorithms and compared using Bayesian model selection methods. The results suggest that the long-run drivers of Brazilian sugar prices are oil prices and that there are nonlinearities in the adjustment processes of sugar and ethanol prices to oil price but linear adjustment between ethanol and sugar prices.

Detecting recombination in 4-taxa DNA sequence alignments with Bayesian hidden Markov models and Markov chain Monte Carlo

This article presents a statistical method for detecting recombination in DNA sequence alignments, which is based on combining two probabilistic graphical models: (1) a taxon graph (phylogenetic tree) representing the relationship between the taxa, and (2) a site graph (hidden Markov model) representing interactions between different sites in the DNA sequence alignments. We adopt a Bayesian approach and sample the parameters of the model from the posterior distribution with Markov chain Monte Carlo, using a Metropolis-Hastings and Gibbs-within-Gibbs scheme. The proposed method is tested on various synthetic and real-world DNA sequence alignments, and we compare its performance with the established detection methods RECPARS, PLATO, and TOPAL, as well as with two alternative parameter estimation schemes.