Nessa dissertação estudamos Sistemas Dinâmicos do ponto de vista da Otimização Ergódica. Analizamos o problema da maximização da integral de potenciais com respeito a probabilidades invariantes pela dinâmica. Mostramos que toda medida ergódica e unicamente maximizante para algum potencial. Verificamos que o conjunto de potenciais com exatamente uma medida maximizadora e residual. Esses resultados são obtidos atrav es de técnicas da Teoria Ergódica e Análise Convexa. / In this thesis we study dynamical systems trough the viewpoint of ergodic optimization. We analyze the problem of maximizing integrals of potentials with respect to invariant probabilities. We show that every ergodic measure is uniquely maximizing for some potential. We also verify that the set of potentials with exactly one maximizing measure is residual. This results are obtained through techniques of ergodic theory and convex analysis.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: leaves 106-109. / by Rae Michael Andrew Shortt. / Ph.D.

Typescript (photocopy). / Digitized by Kansas Correctional Industries

The role of diagrammatic representations and visual reasoning in mathematics problem solving has been extensively studied. Prior research on visual reasoning and problem solving has provided evidence that the format of a diagram can modulate solvers’ interpretations of the structure and concept of the represented problem information, and influence their problem solving outcomes. In this dissertation, two studies investigated how different types of diagrams influence solvers’ choice of solution strategy and their success rate in solving probability word problems. Participants’ solution strategies suggested that problem solvers tended to construct solutions that reflect the structure of a provided diagram, resulting in different representations of the mathematical structure of the problem. For the present set of problems, a binary tree or a binary table tends to steer solvers to use a sequential-sampling strategy, which defines simple or conditional probabilities for each selection stage and calculates the intersection of these probabilities as the final probability value, using the multiplication rule of probability. This strategy choice is structurally matched with the diagrammatic structure of a binary tree or a binary table, which represents unequally-likely outcomes at the event level. In contrast, an N-by-N (outcome) table steers solvers to use of an outcome-search strategy, which involves searching for the total number of target outcomes and all the possible outcomes at the equally-likely outcome level, and calculates the part-over-the-whole value as the final probability, using the classical definition of probability. This strategy is strongly cued by the N-by-N (outcome) table, because the table structure represents all equally-likely outcomes for a probability problem, and organizes the information so that the target outcomes can be seen as a subset embedded in the whole outcome space. When an N-ary (outcome) tree was provided, choices were split between the two solutions, because the N-ary tree structure not only cues searching for equally-likely outcomes but also organizes the problem information in a sequential-sampling, stage-by-stage way. Furthermore, different diagrams seem to be associated with different patterns of characteristic errors. For example, solving a combinations problem with an N-by-N table tended to elicit erroneous solutions involving miscounting those self-repeated combinations represented by the table’s diagonal cells as valid outcomes. Typical errors associated with the use of a binary tree involved incorrect value definitions of the conditional probability of the outcome of a selection. And the N-ary tree may lead to less successful coordination of all the target outcomes for the studied problems, because the target outcomes were dispersed in the outcome space depicted by the tree, thus not salient. The findings support arguments (e.g., Tversky, Morrison, & Betrancourt, 2002) that in order to promote problem solving success, a diagrammatic representation must be carefully selected or designed so that its structure and content can be well-matched to the problem structure and content. And for computational efficiency, information should be spatially organized so that it can be processed readily and accurately. In addition to the implications for effective diagram design for problem solving activities, the findings also offer important insights for probability education. It is suggested that a variety of diagram types be utilized in the educational activities for novice learners of probability, because they tend to highlight different probability concepts and structures even for the same probability topic.

This dissertation focuses on developing novel model selection techniques, the process by which a statistician selects one of a number of competing models of varying dimensions, under an array of different statistical assumptions on observed data. Traditionally, two main reasons have been advocated by researchers for performing model selection strategies over classical maximum likelihood estimates (MLEs). The first reason is prediction accuracy, where by shrinking or setting to zero some model parameters, one sacrifices the unbiasedness of MLEs for a reduced variance, which in turn leads to an overall improvement in predictive performance. The second reason relates to interpretability of the selected models in the presence of a large number of predictors, where in order to obtain a parsimonious representation exhibiting the relationship between the response and covariates, we are willing to sacrifice some of the smaller details brought in by spurious predictors. In the first part of this work, we revisit the family of variable selection techniques known as sure independence screening procedures for generalized linear models and the Cox proportional hazards model. After clever combination of some of its most powerful variants, we propose new extensions based on the idea of sample splitting, data-driven thresholding, and combinations thereof. A publicly available package developed in the R statistical software demonstrates considerable improvements in terms of model selection and competitive computational time between our enhanced variable selection procedures and traditional penalized likelihood methods applied directly to the full set of covariates. Next, we develop model selection techniques within the framework of statistical network analysis for two frequent problems arising in the context of stochastic blockmodels: community number selection and change-point detection. In the second part of this work, we propose a composite likelihood based approach for selecting the number of communities in stochastic blockmodels and its variants, with robustness consideration against possible misspecifications in the underlying conditional independence assumptions of the stochastic blockmodel. Several simulation studies, as well as two real data examples, demonstrate the superiority of our composite likelihood approach when compared to the traditional Bayesian Information Criterion or variational Bayes solutions. In the third part of this thesis, we extend our analysis on static network data to the case of dynamic stochastic blockmodels, where our model selection task is the segmentation of a time-varying network into temporal and spatial components by means of a change-point detection hypothesis testing problem. We propose a corresponding test statistic based on the idea of data aggregation across the different temporal layers through kernel-weighted adjacency matrices computed before and after each candidate change-point, and illustrate our approach on synthetic data and the Enron email corpus. The matrix completion problem consists in the recovery of a low-rank data matrix based on a small sampling of its entries. In the final part of this dissertation, we extend prior work on nuclear norm regularization methods for matrix completion by incorporating a continuum of penalty functions between the convex nuclear norm and nonconvex rank functions. We propose an algorithmic framework for computing a family of nonconvex penalized matrix completion problems with warm-starts, and present a systematic study of the resulting spectral thresholding operators. We demonstrate that our proposed nonconvex regularization framework leads to improved model selection properties in terms of finding low-rank solutions with better predictive performance on a wide range of synthetic data and the famous Netflix data recommender system.

This dissertation contains two parts. The first part introduces the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues in which the arrival process is a general renewal process and the service times are independent and identically distributed (iid); the first-in-first-out FIFO GI/GI/c queue with 2 <= c < 1. Two main simulation algorithms are given in this context, where both of them are built on the classical dominated coupling from the past (DCFTP) protocol. In particular, the first algorithm uses a coupled multi-server vacation system as the upper bound process and it manages to simulate the vacation system backward in time from stationarity at time zero. The second algorithm utilizes the DCFTP protocol as well as the Random Assignment (RA) service discipline. Both algorithms have finite expected termination time with mild moment assumptions on the interarrival time and service time distributions. Our methods are also extended to produce exact simulation algorithms for Fork-Join queues and infinite server systems. The second part presents general principles for the design and analysis of unbiased Monte Carlo estimators in a wide range of settings. The estimators possess finite work-normalized variance under mild regularity conditions. We apply the estimators to various applications including unbiased steady-state simulation of regenerative processes, unbiased optimization in Sample Average Approximations and distribution quantile estimation.

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Um Jogo Markoviano Alternado (Alternating Markov Game - AMG) é uma extensão de um Processo de Decisão Markoviano (Markov Decision Process - MDP) para ambientes multiagentes. O modelo AMG é utilizado na tomada de decisão sequencial de n agentes quando são conhecidas as probabilidades de transição das ações a serem tomadas por cada agente. Nesse trabalho estamos interessados em AMGs com probabilidades de transição de estados imprecisas, por exemplo, quando elas são dadas na forma de intervalos de probabilidades. Apresentamos um novo modelo de AMG, que chamamos de Jogo Markoviano Alternado com Probabilidades Imprecisas (Alternate Markov Game with Imprecise Probabilities - AMGIP) que permite que as imprecisões nas probabilidades de transições de estados sejam dadas na forma de parâmetros sujeitos a restrições lineares que estende trabalhos anteriores em que a imprecisão é dada por intervalos de probabilidades (AMG-INTERVAL). Dizemos que a imprecisão representa escolhas da Natureza. A imprecisão desses modelos implica no valor do jogo ser dado por uma função intervalar. Existem diversas formas de calcular a solução do jogo, que depende do comportamento da Natureza e dos critérios de preferência dos jogadores diante das escolhas da Natureza. Assim, neste trabalho discutimos diversas soluções para o AMG-IP e AMG-INTERVAL. Também como resultado do estudo das relações existentes entre os MDPs e os AMGs, propomos um novo modelo chamado de AMG-ST (Alternating Markov Game with Set-valued Transition), capaz de modelar a incerteza do modelo MDP-ST (Markovian Decision Process with Set-valued Transition) como um jogo entre o agente e a Natureza, isto é, um jogo em que a Natureza faz o papel de um dos jogadores. / An Alternating Markov Game (AMG) is an extension of a Markov Decision Process (MDP) for multiagent environments. This model is used on sequencial decision making for n agents when we know the state transition probabilities of actions being taken by each agent. In this work we are interested in AMGs with imprecise probabilities on state transition function, for example, when they are given by probabilities intervals. We present a new AMG model, which we call Alternating Markov Game with Imprecise Probabilities (AMG-IP) that allows imprecision on state transition probabilities given by parameters subject to linear constraints that extend previous works which the imprecision is given by probabilities intervals (AMG-INTERVAL). We say that the imprecision represents the Nature choices. The imprecision of these models implies the game value is given by interval function. There are several ways to calculate the solution of the game, that depend on the behavior of the Nature and the preference criteria of the players on the choices of Nature. Therefore, in this work we discuss various solutions to AMG-IP and AMG-INTERVAL. Also from our study on the relationship among the MDPs and AMGs, we propose a new model called Alternating Markov Game with Set-valued Transition (AMG-ST), that can be used to model the uncertainty of an MDP-ST (Markovian Decision Process with Set-valued Transition) as a result of the match between the agent and the Nature, i.e., a game where the Nature is seen as one of the players.

Reinforcement menus were used to dhange response probabilities while maintaining control over two ''trainable," female, mentally retarded children. An empirically determined reinforcement menu representing high probability behaviors, five for S1 and four for S2, was used in a contingency management system. Instructions were given concerning the contingencies for obtaining reinforcement. Subjects were allowed the opportunity to engage in a high probability behavior only after successful completion of fixed units of reading or arithmetic tasks. After stable performance was established, four additional menus were prepared to approximate in increasing degree, low probability behavior. Measurements were taken of task time and response duration, the time spent traveling to and from the reinforcement area. Task time and response duration reached asymptotic values and remained at baseline values throughout the menu fading procedures. At the completion of the menu fading, subjects were doing units of work involving reading and mathematics in order to have the opportunity to do some reinforcing arithmetic.

Bibliography: leaves 397-406.