O presente estudo tem como objetivo analisar o fenômeno do desmatamento no Brasil. Para este efeito, utilizou-se de instrumentais econométricos e matemáticos. O estudo se divide em três ensaios. No primeiro ensaio investigam-se os possíveis efeitos adversos da política governamental devido à existência de fracas instituições na maior parte da região da Amazônia legal. Neste primeiro ensaio também é analisado empiricamente a relação entre corrupção, desmatamento e Produto Interno Bruto (PIB) para os municípios de Mato Grosso. No segundo ensaio utiliza-se de jogos diferenciais para analisar teoricamente o efeito da corrupção no nível de desmatamento ilegal. Finalmente o terceiro ensaio, focalizando numa análise regional, faz-se uma análise empírica, através de modelos não paramétricos, para a relação entre corrupção, desmatamento, e PIB. No terceiro ensaio, também, utilizando-se de modelos não paramétricos, estima-se, numa análise internacional, a existência da curva de Kuznets. / The present study aims to analyze the phenomenon of deforestation in Brazil. For this purpose, we used econometrics and mathematical tools. The study is divided into three essays. In the first essay, through the standard game theory, we investigated the adverse effects of the government policy due the existence of weak institutions in the Amazon region. In this first essay it is also studied empirically, for the municipalities of Mato-grosso, the relationship between corruption, deforestation and Gross Domestic Product (GDP). In the second essay we used differential game theory to analyze the effect of corruption on the level of illegal logging. Finally on the third essay, we focused on a regional and international analysis. For the regional analysis, we used nonparametric models to test the relationship between corruption, deforestation, and GDP. We used the same methods to perform an international analysis related with the Kuznets curve.

O presente estudo tem como objetivo analisar o fenômeno do desmatamento no Brasil. Para este efeito, utilizou-se de instrumentais econométricos e matemáticos. O estudo se divide em três ensaios. No primeiro ensaio investigam-se os possíveis efeitos adversos da política governamental devido à existência de fracas instituições na maior parte da região da Amazônia legal. Neste primeiro ensaio também é analisado empiricamente a relação entre corrupção, desmatamento e Produto Interno Bruto (PIB) para os municípios de Mato Grosso. No segundo ensaio utiliza-se de jogos diferenciais para analisar teoricamente o efeito da corrupção no nível de desmatamento ilegal. Finalmente o terceiro ensaio, focalizando numa análise regional, faz-se uma análise empírica, através de modelos não paramétricos, para a relação entre corrupção, desmatamento, e PIB. No terceiro ensaio, também, utilizando-se de modelos não paramétricos, estima-se, numa análise internacional, a existência da curva de Kuznets. / The present study aims to analyze the phenomenon of deforestation in Brazil. For this purpose, we used econometrics and mathematical tools. The study is divided into three essays. In the first essay, through the standard game theory, we investigated the adverse effects of the government policy due the existence of weak institutions in the Amazon region. In this first essay it is also studied empirically, for the municipalities of Mato-grosso, the relationship between corruption, deforestation and Gross Domestic Product (GDP). In the second essay we used differential game theory to analyze the effect of corruption on the level of illegal logging. Finally on the third essay, we focused on a regional and international analysis. For the regional analysis, we used nonparametric models to test the relationship between corruption, deforestation, and GDP. We used the same methods to perform an international analysis related with the Kuznets curve.

O presente estudo tem como objetivo analisar o fenômeno do desmatamento no Brasil. Para este efeito, utilizou-se de instrumentais econométricos e matemáticos. O estudo se divide em três ensaios. No primeiro ensaio investigam-se os possíveis efeitos adversos da política governamental devido à existência de fracas instituições na maior parte da região da Amazônia legal. Neste primeiro ensaio também é analisado empiricamente a relação entre corrupção, desmatamento e Produto Interno Bruto (PIB) para os municípios de Mato Grosso. No segundo ensaio utiliza-se de jogos diferenciais para analisar teoricamente o efeito da corrupção no nível de desmatamento ilegal. Finalmente o terceiro ensaio, focalizando numa análise regional, faz-se uma análise empírica, através de modelos não paramétricos, para a relação entre corrupção, desmatamento, e PIB. No terceiro ensaio, também, utilizando-se de modelos não paramétricos, estima-se, numa análise internacional, a existência da curva de Kuznets. / The present study aims to analyze the phenomenon of deforestation in Brazil. For this purpose, we used econometrics and mathematical tools. The study is divided into three essays. In the first essay, through the standard game theory, we investigated the adverse effects of the government policy due the existence of weak institutions in the Amazon region. In this first essay it is also studied empirically, for the municipalities of Mato-grosso, the relationship between corruption, deforestation and Gross Domestic Product (GDP). In the second essay we used differential game theory to analyze the effect of corruption on the level of illegal logging. Finally on the third essay, we focused on a regional and international analysis. For the regional analysis, we used nonparametric models to test the relationship between corruption, deforestation, and GDP. We used the same methods to perform an international analysis related with the Kuznets curve.

Machine learning (ML) has become one of the most powerful classes of tools for artificial intelligence, personalized web services and data science problems across fields. Within the field of machine learning itself, there had been quite a number of paradigm shifts caused by the explosion of data size, computing power, modeling tools, and the new ways people collect, share, and make use of data sets. Data privacy, for instance, was much less of a problem before the availability of personal information online that could be used to identify users in anonymized data sets. Images, videos, as well as observations generated over a social networks, often have highly localized structures, that cannot be captured by standard nonparametric models. Moreover, the “common task framework” that is adopted by many sub- disciplines of AI has made it possible for many people to collaboratively and repeated work on the same data set, leading to implicit overfitting on public benchmarks. In addition, data collected in many internet services, e.g., web search and targeted ads, are not iid, but rather feedbacks specific to the deployed algorithm. This thesis presents technical contributions under a number of new mathematical frameworks that are designed to partially address these new paradigms. • Firstly, we consider the problem of statistical learning with privacy constraints. Under Vapnik’s general learning setting and the formalism of differential privacy (DP), we establish simple conditions that characterizes the private learnability, which reveals a mixture of positive and negative insight. We then identify generic methods that reuses existing randomness to effectively solve private learning in practice; and discuss weaker notions of privacy that allows for more favorable privacy-utility tradeoff. • Secondly, we develop a few generalizations of trend filtering, a locally-adaptive nonparametric regression technique that is minimax in 1D, to the multivariate setting and to graphs. We also study specific instances of the problems, e.g., total variation denoising on d-dimensional grids more closely and the results reveal interesting statistical computational trade-offs. • Thirdly, we investigate two problems in sequential interactive learning: a) off- policy evaluation in contextual bandits, that aims to use data collected from one algorithm to evaluate the performance of a different algorithm; b) the problem of adaptive data analysis, that uses randomization to prevent adversarial data analysts from a form of “p-hacking” through multiple steps of sequential data access. In the above problems, we will provide not only performance guarantees of algorithms but also certain notions of optimality. Whenever applicable, careful empirical studies on synthetic and real data are also included.

Profile monitoring is a relatively new approach in quality control best used when the process data follow a profile (or curve). The majority of previous studies in profile monitoring focused on the parametric modeling of either linear or nonlinear profiles under the assumption of the correct model specification. Our work considers those cases where the parametric model for the family of profiles is unknown or, at least uncertain. Consequently, we consider monitoring Poisson profiles via three methods, a nonparametric (NP) method using penalized splines, a nonparametric (NP) method using wavelets and a semi parametric (SP) procedure that combines both parametric and NP profile fits. Our simulation results show that SP method is robust to the common problem of model misspecification of the user's proposed parametric model. We also showed that Haar wavelets are a better choice than the penalized splines in situations where a sudden jump happens or the jump is edgy. In addition, we showed that the penalized splines are better than wavelets when the shape of the profiles are smooth. The proposed novel techniques have been applied to a real data set and compare with some state-of-the arts.

Functional data analysis (FDA) studies data in the form of measurements over a domain as whole entities. Our first focus is on the post-hoc analysis with pairwise and contrast comparisons of the popular functional ANOVA model comparing groups of functional data. Existing contrast tests assume independent functional observations within group. In reality, this assumption may not be satisfactory since functional data are often collected continually overtime on a subject. In this work, we introduce a new linear contrast test that accounts for time dependency among functional group members. For a significant contrast test, it can be beneficial to identify the region of significant difference. In the second part, we propose a non-parametric regression procedure to obtain a locally sparse estimate of functional contrast. Our work is motivated by a biomedical study using Raman spectroscopy to monitor hemodialysis treatment near real-time. With contrast test and sparse estimation, practitioners can monitor the progress of the hemodialysis within session and identify important chemicals for dialysis adequacy monitoring. In the third part, we propose a functional data model for degradation analysis of functional data. Motivated by degradation analysis application of rechargeable Li-ion batteries, we combine state-of-the-art functional linear models to produce fully functional prediction for curves on heterogenous domains. Simulation studies and data analysis demonstrate the advantage of the proposed method in predicting degradation measure than existing method using aggregation method. / Doctor of Philosophy / Functional data analysis (FDA) studies complex data structure in the form of curves and shapes. Our work is motivated by two applications concerning data from Raman spectroscopy and battery degradation study. Raman spectra of a liquid sample are curves with measurements over a domain of wavelengths that can identify chemical composition and whose values signify the constituent concentrations in the sample. We first propose a statistical procedure to test the significance of a functional contrast formed by spectra collected at beginning and at later time points during a dialysis session. Then a follow-up procedure is developed to produce a sparse representation of the contrast functional contrast with clearly identified zero and nonzero regions. The use of this method on contrast formed by Raman spectra of used dialysate collected at different time points during hemodialysis sessions can be adapted for evaluating the treatment efficacy in real time. In a third project, we apply state-of-the-art methodologies from FDA to a degradation study of rechargeable Li-ion batteries. Our proposed methods produce fully functional prediction of voltage discharge curves allowing flexibility in monitoring battery health.

Large scale Gaussian process (GP) regression is infeasible for larger data sets due to cubic scaling of flops and quadratic storage involved in working with covariance matrices. Remedies in recent literature focus on divide-and-conquer, e.g., partitioning into sub-problems and inducing functional (and thus computational) independence. Such approximations can speedy, accurate, and sometimes even more flexible than an ordinary GPs. However, a big downside is loss of continuity at partition boundaries. Modern methods like local approximate GPs (LAGPs) imply effectively infinite partitioning and are thus pathologically good and bad in this regard. Model averaging, an alternative to divide-and-conquer, can maintain absolute continuity but often over-smooth, diminishing accuracy. Here I propose putting LAGP-like methods into a local experts-like framework, blending partition-based speed with model-averaging continuity, as a flagship example of what I call precision aggregated local models (PALM). Using N_C LAGPs, each selecting n from N data pairs, I illustrate a scheme that is at most cubic in n, quadratic in N_C, and linear in N, drastically reducing computational and storage demands. Extensive empirical illustration shows how PALM is at least as accurate as LAGP, can be much faster in terms of speed, and furnishes continuous predictive surfaces. Finally, I propose sequential updating scheme which greedily refines a PALM predictor up to a computational budget, and several variations on the basic PALM that may provide predictive improvements. / Doctor of Philosophy / Occasionally, when describing the relationship between two variables, it may be helpful to use a so-called ``non-parametric" regression that is agnostic to the function that connects them. Gaussian Processes (GPs) are a popular method of non-parametric regression used for their relative flexibility and interpretability, but they have the unfortunate drawback of being computationally infeasible for large data sets. Past work into solving the scaling issues for GPs has focused on ``divide and conquer" style schemes that spread the data out across multiple smaller GP models. While these model make GP methods much more accessible to large data sets they do so either at the expense of local predictive accuracy of global surface continuity. Precision Aggregated Local Models (PALM) is a novel divide and conquer method for GP models that is scalable for large data while maintaining local accuracy and a smooth global model. I demonstrate that PALM can be built quickly, and performs well predictively compared to other state of the art methods. This document also provides a sequential algorithm for selecting the location of each local model, and variations on the basic PALM methodology.

Nowadays, Gaussian Process (GP) has been recognized as an indispensable statistical tool in computer experiments. Due to its computational complexity and storage demand, its application in real-world problems, especially in "big data" settings, is quite limited. Among many strategies to tailor GP to such settings, Gramacy and Apley (2015) proposed local approximate GP (laGP), which constructs approximate predictive equations by constructing small local designs around the predictive location under certain criterion. In this dissertation, several methodological extensions based upon laGP are proposed. One methodological contribution is the multilevel global/local modeling, which deploys global hyper-parameter estimates to perform local prediction. The second contribution comes from extending the laGP notion of "locale" to a set of predictive locations, along paths in the input space. These two contributions have been applied in the satellite drag emulation, which is illustrated in Chapter 3. Furthermore, the multilevel GP modeling strategy has also been applied to synthesize field data and computer model outputs of solar irradiance across the continental United States, combined with inverse-variance weighting, which is detailed in Chapter 4. Last but not least, in Chapter 5, laGP's performance has been tested on emulating daytime land surface temperatures estimated via satellites, in the settings of irregular grid locations. / Doctor of Philosophy / In many real-life settings, we want to understand a physical relationship/phenomenon. Due to limited resources and/or ethical reasons, it is impossible to perform physical experiments to collect data, and therefore, we have to rely upon computer experiments, whose evaluation usually requires expensive simulation, involving complex mathematical equations. To reduce computational efforts, we are looking for a relatively cheap alternative, which is called an emulator, to serve as a surrogate model. Gaussian process (GP) is such an emulator, and has been very popular due to fabulous out-of-sample predictive performance and appropriate uncertainty quantification. However, due to computational complexity, full GP modeling is not suitable for “big data” settings. Gramacy and Apley (2015) proposed local approximate GP (laGP), the core idea of which is to use a subset of the data for inference and further prediction at unobserved inputs. This dissertation provides several extensions of laGP, which are applied to several real-life “big data” settings. The first application, detailed in Chapter 3, is to emulate satellite drag from large simulation experiments. A smart way is figured out to capture global input information in a comprehensive way by using a small subset of the data, and local prediction is performed subsequently. This method is called “multilevel GP modeling”, which is also deployed to synthesize field measurements and computational outputs of solar irradiance across the continental United States, illustrated in Chapter 4, and to emulate daytime land surface temperatures estimated by satellites, discussed in Chapter 5.

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<p>Collections of large volumes of rich and complex data has become ubiquitous in recent years, posing new challenges in methodological and theoretical statistics alike. Today, statisticians are tasked with developing flexible methods capable of adapting to the degree of complexity and noise in increasingly rich data gathered across a variety of disciplines and settings. This has spurred the need for novel multivariate regression techniques that can efficiently capture a wide range of naturally occurring predictor-response relations, identify important predictors and their interactions and do so even when the number of predictors is large but the sample size remains limited. </p><p>Meanwhile, efficient model fitting tools must evolve quickly to keep pace with the rapidly growing dimension and complexity of data they are applied to. Aided by the tremendous success of modern computing, Bayesian methods have gained tremendous popularity in recent years. These methods provide a natural probabilistic characterization of uncertainty in the parameters and in predictions. In addition, they provide a practical way of encoding model structure that can lead to large gains in statistical estimation and more interpretable results. However, this flexibility is often hindered in applications to modern data which are increasingly high dimensional, both in the number of observations $n$ and the number of predictors $p$. Here, computational complexity and the curse of dimensionality typically render posterior computation inefficient. In particular, Markov chain Monte Carlo (MCMC) methods which remain the workhorse for Bayesian computation (owing to their generality and asymptotic accuracy guarantee), typically suffer data processing and computational bottlenecks as a consequence of (i) the need to hold the entire dataset (or available sufficient statistics) in memory at once; and (ii) having to evaluate of the (often expensive to compute) data likelihood at each sampling iteration. </p><p>This thesis divides into two parts. The first part concerns itself with developing efficient MCMC methods for posterior computation in the high dimensional {\em large-n large-p} setting. In particular, we develop an efficient and widely applicable approximate inference algorithm that extends MCMC to the online data setting, and separately propose a novel stochastic search sampling scheme for variable selection in high dimensional predictor settings. The second part of this thesis develops novel methods for structured sparsity in the high-dimensional {\em large-p small-n} regression setting. Here, statistical methods should scale well with the predictor dimension and be able to efficiently identify low dimensional structure so as to facilitate optimal statistical estimation in the presence of limited data. Importantly, these methods must be flexible to accommodate potentially complex relationships between the response and its associated explanatory variables. The first work proposes a nonparametric additive Gaussian process model to learn predictor-response relations that may be highly nonlinear and include numerous lower order interaction effects, possibly in different parts of the predictor space. A second work proposes a novel class of Bayesian shrinkage priors for multivariate regression with a tensor valued predictor. Dimension reduction is achieved using a low-rank additive decomposition for the latter, enabling a highly flexible and rich structure within which excellent cell-estimation and region selection may be obtained through state-of-the-art shrinkage methods. In addition, the methods developed in these works come with strong theoretical guarantees.</p> / Dissertation