# Relatórios de Pesquisa

 7/2012 Perturbed Damped Pendulum: Finding Periodic SolutionsDouglas D. NovaesUsing the equation of motion of the damped pendulum, we introduce the averaging method on the study of periodic solutions of dynamical systems with small perturbation. We provide sufficient conditions for the existence of periodic solutions of the perturbed damped pendulum with small oscillations having equations of motion$\ddot{\T}=-a\T-b\dot{\T}+\e f(t,\T,\dot{\T}),$where $a>0,\,b>0$ and $\e$ are real parameters, with $a=g/l$, $g$ the acceleration of the gravity, $l$ the length of the rod and $b$ the damping coefficient. Here the parameters $b$ and $\e$ are small and the smooth function $f$ is $T$--periodic in $t$. The averaging theory provides a useful means to study dynamical systems, accessible to Master and PhD students. 6/2012 On The Periodic Solutions of a Generalized Smooth and Non-Smooth Perturbed Planar Double Pendulum with Small OscillationsJaume Llibre, Douglas D. Novaes, Marco A. TeixeiraWe provide sufficient conditions for the existence of periodic solutions of the smooth and non-smooth perturbed planar double pendulum with small oscillations having equations of motion$\begin{array}{l}\ddot{\T}_{1}=-a\T_{1}+\T_{2}+\e \left(F_1(t,\T_1,\dot {\T}_1,\T_2, \dot{\T}_2)+F_2(t,\T_1,\dot {\T}_1,\T_2, \dot {\T}_2){\rmsgn}(\dot{\T_{1}})\right),\\\ddot{\T}_{2}=b\T_{1}-b\T_{2}+\e \left(F_3(t,\T_1,\dot {\T}_1,\T_2, \dot{\T}_2)+F_4(t,\T_1,\dot {\T}_1,\T_2, \dot {\T}_2){\rmsgn}(\dot{\T_{2}})\right),\end{array}$where $a>1,b>0$ and $\e$ are real parameters. Here the parameter $\e$ is small and the smooth functions $F_i$ for $i=1,2,3,4$ define the perturbation which are periodic functions in $t$ and in resonance $p_{i}$:$q_{i}$ with some of the periodic solutions of the unperturbed double pendulum, being $p_{i}$ and $q_{i}$ relatively prime positive integers. 5/2012 On The Periodic Solutions of a Perturbed Double PendulumJaume Llibre, Douglas D. Novaes, Marco A. TeixeiraWe provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations having equations of motion$\begin{array}{l}\ddot{\T}_{1}=-2a\T_{1}+a\T_{2}+\e F_1(t,\T_1,\dot \T_1,\T_2, \dot \T_2),\\\ddot{\T}_{2}=2a\T_{1}-2a\T_{2}+\e F_2(t,\T_1,\dot \T_1,\T_2, \dot\T_2),\end{array}$where $a$ and $\e$ are real parameters. The two masses of the unperturbed double pendulum are equal, and its two stems have the same length $l$. In fact $a=g/l$ where $g$ is the acceleration ofthe gravity. Here the parameter $\e$ is small and the smooth functions $F_1$ and $F_2$ define the perturbation which are periodic functions in $t$ and in resonance $p$:$q$ with some of the periodicsolutions of the unperturbed double pendulum, being $p$ and $q$ positive integers relatively prime. 4/2012 Minimização irrestrita usando gradientes conjugados e regiões de confiançaJohn Lenon C. Gardenghi, Sandra A. SantosThis work focus on the conjugate gradient method to solve the trust region sub-problem for unconstrained minimization. We aim to describe an intuitive and detailed study about this method, starting from an introduction to methods of conjugate directions, some necessary requisites and tools for understanding the conjugate gradient method and its integration with the trust region strategy for unconstrained minimization. The computational implementation of the method using the CAS Maxima enabled the numerical experiments, which validated the study and the implementation done and allowed a comparison between conjugate gradient and Leverberg-Marquardt for least squares problems. 3/2012 Bayesian general multivariate latent variable modeling of longitudinal item response dataCaio L. N. Azevedo, Jean-Paul Fox, Dalton F. AndradeLongitudinal item response data are characterized by examinees that are assessed at different time points or measurement occasions such that time-specific measurements are nested within examinees. Besides the usual nesting of response observations within examinees, the time-specific latent traits are also nested within examinees. In the well-known hierarchical modeling approach, the complex dependencies due to the nested structure of the data are commonly modeled by introducing random effects such that observations and latent traits are conditionally independently distributed. However, the implied compound symmetry structure is often not sufficient to model the complex time-heterogenous dependencies.Therefore, a Bayesian general multivariate item response modeling framework is proposed that accounts for the complex within-examinee latent trait dependencies. Flexible parametric covariance structures are considered to modelspecific within-examinee dependencies. Furthermore, it can handle many measurement occasions, different item response functions, and different latent trait population distributions, and generalizes some works of current literature. Due to identification rules and restricted parametric covariance structures, a conditional modeling approach is pursued to specify proper priors for the unrestricted parameters and to implement an efficient MCMC algorithm, by conditioning on baseline population parameters. The study is motivated by a large-scale longitudinal research program of the Brazilian Federal government to improve the teaching quality and general structure of schools for primary education. It is shown that the growth in math achievement can be accurately measured when accounting for complex dependencies over grades using time-heterogenous covariances structures. 2/2012 A note on identification and metric issues for skew IRT modelsCaio L. N. Azevedo, Heleno Bolfarine, Dalton F. AndradeThe skew-normal distribution (SND) is a flexible family of densities which preserves some useful properties of the original normal distribution. Some stochastic representations for the SND have beenproposed in the literature. The Henze (H) and Sahu, Branco and Dey (SBD) are the two most used ones. On the other hand, the centered parametrization is useful for inference purposes. The main goals ofthis article are: establish a link between the standard H and SDB skew-normal distributions and use this result to model the latent traits for IRT models. We proved that standard H and SDB distributions are related to each other through a function of the asymmetry parameter and also that they are exactly the same under centered parametrization (CP). Using these results, we showed that the common density obtained through the CP is useful to model the latent traits for unidimensional IRT models. This approach allows to represent asymmetric latent traits behavior and ensures the model identification as well. 1/2012 Likelihood Based Inference for Linear and Nonlinear Mixed-Effects Models with Censored Response Using the Multivariate-t DistributionLarissa A. Matos, Marcos O. Prates, Ming H. Chen, Víctor H. LachosMixed models are commonly used to represent longitudinal or repeated measures data. An additional complication arises when the response is censored, for example, due to limits of quantification of the assay used. Normal distributions for random effects and residual errors are usually assumed, but such assumptions make inferences vulnerable to the presence of outliers. Motivated by a concern of sensitivity to potential outliers or data with tails longerthan-normal, we aim to develop a likelihood based inference for linear and nonlinear mixed effects models with censored response (NLMEC/LMEC) based on the multivariate Student-t distribution, being a flexible alternative to the use of the corresponding normal distribution. We propose an ECM algorithm for computing the maximum likelihood estimates for NLMEC/LMEC with standard errors of the fixed effects and likelihood function as a by-product. This algorithm uses closed-form expressions at the E-step, which relies on formulas for the mean and variance of a truncated multivariate-t distribution, and can be computed using available software. The proposed algorithm is implemented in the R package tlmec. An appendix which includesfurther mathematical details, the R code, and datasets for examples and simulations are available as supplements. The newly developed procedures are illustrated with two case studies, involving the analysis of longitudinal HIV viral load in two recent AIDS studies. In addition, a simulation study is conducted to assess the performance of the proposed approach and its comparison with the approach by Vaida and Liu (2009). 11/2011 Generelized Skew- Normal/Independent Fields with ApplicationsMarcos O. Prates, Dipak K. Dey, Víctor H. LachosThe last decade has witnessed major developments in Geographical Information Systems (GIS) technology resulting in the need for Statisticians to develop models that account for spatial clustering and variation. Study of spatial patterns are very important in epidemiological and environmental problems. Due to spatial characteristics it is extremely important to correctly incorporate spatial dependence in modeling. This paper develops a novel spatial process using generalized skew-normal/independent distributions when the usual Gaussian process assumptions are invalid and transformation to a Gaussian random field is not appropriate. Our proposed method incorporates skewness as well as heavy tail behavior of the data while maintaining spatial dependence using a Conditional Auto Regressive (CAR)structure. We use Bayesian hierarchical methods to fit such models. Consequently we use a Bayesian model selection approach to choose appropriate models for a empirical data set. 10/2011 Influence Diagnostics in Linear and Nonlinear Mixed-Effects Models with Censored DataLarissa A. Matos, Víctor H. Lachos, N. Balakrishnan, Filidor E. Vilca-LabraHIV RNA viral load measures are often subjected to some upper and lower detection limits depending on the quantification assays, and consequently the responses are either left or right censored. Linear and nonlinear mixed-effects models with modifications to accommodate censoring (LMEC and NLMEC) are routinely used to analyze this type of data. Recently, Vaida and Liu (2009) proposed an exact EM-type algorithm for LMEC/NLMEC, called SAGE algorithm (Meng and Van Dyk, 1997), that uses closed-form expressions at the E-step, as opposed to Monte Carlo simulations. Motivated by this algorithm, we propose here an exact ECM algorithm (Meng and Rubin, 1993) for LMEC/NLMEC, which enable us to develop local influence analysis for mixed effects models on the basis of the conditional expectation of the complete-data log-likelihood function. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex that makes itdifficult to apply directly the approach of Cook (1977, 1986). Some useful perturbation schemes are discussed. Finally, the results obtained from the analyses of two HIV AIDS studies on viral loads are presented to illustrate the newly developed methodology. 9/2011 Bayesian Estimation of a Skew-t Stochastic Volatility ModelCarlos A. Abanto-Valle, Víctor H. Lachos, Dipak K. DeyIn this paper we present a stochastic volatility (SV) model assuming that the return shock has a skew-Student-t distribution. This allows a parsimonious, flexible treatment of asymmetry and heavy tails in the conditional distribution of returns. An efficient Markov chain Monte Carlo estimation method is described that exploits a skew-normal mixture representation of the error distribution with a gamma distribution as the mixing distribution. We apply the methodology to the NASDAQ daily index returns.