Discrete weak-KAM methods for stationary uniquely ergodic setting
Eduardo Garibaldi, Samuel Petite, Philippe Thieullen
The Frenkel-Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost-periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We introduce, in this context, the notion of calibrated configuration (stronger than the standard minimizing condition) and, for continuous superlinear interaction energies, we prove its existence for some environment of the dynamical system. Furthermore, in one dimension, we give sufficient conditions on the family of interaction energies to ensure the existence of calibrated configurations for any environment when the underlying dynamics is uniquely ergodic. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry-Mather theory and spaces of Delone sets.
Diagnostics for censored mixed-effects models using the multivariate t-distribution
Dipankar Bandyopadhyay, Larissa A. Matos, Luis M. Castro, Víctor H. Lachos
In biomedical studies on HIV RNA dynamics, the viral loads generate repeated measures that are often subjected to (upper and lower) detection limits, and hence these responses are either left- or right-censored. Linear and non-linear mixed-effects censored (LMEC/NLMEC) models are routinely used to analyze these longitudinal data, with normality assumptions for the random effects and residual errors. However, the derived inference may not be robust when these underlying normality assumptions are questionable, specially presence of outliers and thick-tails. Motivated by this, Matos et al. (2013b) recently proposed an exact EM-type algorithm for LMEC/NLMEC models using a multivariate Student’s-t distribution, with closed-form expressions at the E-step. In this paper, we develop influence diagnostics for LMEC/NLMEC models using multivariate Student’s-t density, based on the conditional expectation of the complete data log-likelihood which eliminates the complexity associated with the approach of Cook (1977, 1986) for censored mixed-effects models. The new methodology is illustrated through an application to a longitudinal HIV dataset using the NLMEC framework. In addition, a simulation study is presented, which explores the accuracy of the proposed measures in detecting influential observations in heavy-tailed censored data under different perturbation schemes.
Estimation Methods for Multivariate Tobit Confirmatory Factor Analysis
D. R. Costa, Víctor H. Lachos, J. L. Bazan, Caio L. N. Azevedo
We propose two methods for estimating multivariate Tobit Confirmatory Factor Analysis (TCFA) with covariates, one from Bayesian and another from likelihood based perspectives. TCFA is particularly useful in analysis of multivariate data with censored information. In contrast with previous developments that utilize Monte Carlo simulations for maximum likelihood estimation, an exact EM-type algorithm is proposed, which uses closed form expressions at the E-step that rely on the mean and variance of a truncated multinormal distribution and can be computed using available software. Through simulation studies, we compare the performance of the proposed algorithm when the censored pattern is ignored for different levels of censoring. Our results suggest that this algorithm has excellent performance, since it recovered the true parameters of the TCFA model much better than did the traditional CFA model. In addition, by considering a hierarchical formulation of the models, we also explore the estimation of the parameters via MCMC techniques by using proper priors. A Bayesian case deletion influence diagnostic based on the q-divergence measure and model selection criteria is also developed and applied to analyze a real dataset from an education assessment. In addition, a simulation study is conducted to compare the performance of the proposed method with the traditional CFA model.
Models Applied to DNA Sequences with Multinomial Correlated Responses
Beatriz Cuyabano, Hildete P. Pinheiro, Aluísio Pinheiro
Multinomial multivariate models are proposed to describe the codon frequencies in DNA sequences, as well as the order and frequency that nucleotide bases have in each codon considering the dependence among the bases inside a codon. Logistic regressive models are used with diﬀerent dependence structures on the three codon positions. Also, multinomialextensions of the Bahadur’s representation are proposed to model correlated multinomial data. An application of these models to the NADH4 gene from human mitochondrial genome is presented. AIC, BIC and the leave-one-out cross validation are employed to compare the various models peformance.
Likelihood Based Inference for Quantile Regression Using the Asymmetric Laplace Distribution
Luis Benites Sánchez, Víctor H. Lachos, Filidor E. Vilca-Labra
To make inferences about the shape of a population distribution, the widely popular mean regression model, for example, is inadequate if the distribution is not approximately Gaussian (or symmetric). Compared to conventional mean regression (MR), quantile regression (QR)can characterize the entire conditional distribution of the outcome variable, and is more robust to outliers and misspecification of the error distribution. We present a likelihood-based approach to the estimation of the regression quantiles based on the asymmetric Laplace distribution (ALD), a choice that turns out to be natural in this context. The ALD has a nice hierarchical representation which facilitates the implementation of the EM algorithm for maximum-likelihood estimation of the parameters at the pth level with the observed information matrix as a byproduct. Inspired by the EM algorithm, we develop case-deletion diagnostics analysis for QR models, following the approach of Zhu et al. (2001). This is because the observed data log–likelihood function associated with the proposed model is somewhat complex (e.g., not differentiable at zero) and by using Cook’s well-known approach it can be very difficult to obtain case-deletion measures. The techniques are illustrated with both simulated and real data. In particular, in an empirical comparison, our approach out-performed other common classic estimators under a wide array of simulated data models and is flexible enough to easily accommodate changes in their assumed distribution. The proposed algorithm and methods are implemented in the R package ALDqr().
Bayesian Analysis Censored Linear Regression Models with Scale Mixtures of Normal Distributions
Aldo M. Garay, Heleno Bolfarine, Víctor H. Lachos, Celso R. B. Cabral
As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyses might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Studentt, Pearson type VII, slash and the contaminated normal distributions, asspecial cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo (MCMC) algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also todevelop Bayesian case-deletion influence diagnostics based on the q-divergence measures. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with an application and simulated data.
Linear Censored Regression Models with Scale Mixtures of Normal Distributions
Aldo M. Garay, Víctor H. Lachos, Heleno Bolfarine, Celso R. B. Cabral
In the framework of censored regression models the random errors are routinely assumed to have a normal distribution, mainly for mathematical convenience. However, this method has been criticized in the literature because of its sensitivity to deviations from the normality assumption. In practice, data such as income or viral load in AIDS studies, often violate this assumption because of heavy tails. Here, we first establish a new link between the cen-sored regression model and a recently studied class of symmetric distributions, which extend the normal one by the inclusion of kurtosis, called scale mixtures of normal (SMN) distributions. The Student-t, Pearson type VII, slash, contaminated normal, among others distributions, are contained in this class. Choosing a member of this class can be a good alternative to model this kind of data, because they have been shown its flexibility in several applications. In this work, we develop an analytically simple and efficient EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters, with standard errors as a by-product. The algorithm has closed-form expressions at the E-step, that rely on formulas for the mean and variance of certain truncated SMN distributions. The proposed algorithm is implemented in the R package SMNCensReg. Applications with simulated and a real data are reported, illustrating the usefulness of the new methodology.
On Sliding Periodic Solutions for Piecewise Continuous Systems Defined on the 2-Cylinder
Douglas D. Novaes, Mike R. Jeffrey, Marco A. Teixeira
This paper deals with discontinuous differential equations defined on the 2--dimensional cylinder. The main goal is to exhibit conditions for the existence of typical periodic solutions of such systems. An averaging method for computing sliding periodic solutions is developed, subject to convenient assumptions. We also apply the method to example problems. The main tools used are structural stability theory for discontinuous differential systems and Brouwer degree theory.
Birth of limit cycles bifurcating from a nonsmooth center
Cláudio A. Buzzi, Tiago de Carvalho, Marco A. Teixeira
This paper is concerned with a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate Σ-center). We prove that any nondegenerate Σ-center is Σ-equivalent to a particular normal form Z0 . Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from Z0 that have k hyperbolic limit cycles bifurcating from the nondegenerate Σ-center of Z0 (the same holds for k = ∞).Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from Z0 . As a consequence we prove that Z0 has infinite codimension.
Ruelle Operator Duality for Coupled Smooth Markov Maps of the Circle
Let TL and TR be two smooth surjective Markov maps of the circle, with TR expansive, coupled in such a way that there exists an extension (C, TC ) whose first factor is TL−1 and the second factor of TC is TR . Let AL piecewise continuous and AR piecewise absolutelycontinuous be two respective potentials. We show that, when those potentials are in involutionby a smooth kernel W on C, there is an explicit isomorphism between eigenfunctions of the Ruelle operator of (TL , AL ) and eigendistributions of the Ruelle operator of (TR , AR ) for the same eigenvalue. This gives a regularity result for eigendistributions of transfer operators associated with non-maximal eigenvalues.
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