Browsing by Subject "Gaussian process"

This dissertation is devoted to modeling complex data from the

Bayesian perspective via constructing priors with latent structures.

There are three major contexts in which this is done -- strategies for

the analysis of dynamic longitudinal data, estimating

shape-constrained functions, and identifying subgroups. The

methodology is illustrated in three different

interdisciplinary contexts: (1) adaptive measurement testing in

education; (2) emulation of computer models for vehicle crashworthiness; and (3) subgroup analyses based on biomarkers.

Chapter 1 presents an overview of the utilized latent structured

priors and an overview of the remainder of the thesis. Chapter 2 is

motivated by the problem of analyzing dichotomous longitudinal data

observed at variable and irregular time points for adaptive

measurement testing in education. One of its main contributions lies

in developing a new class of Dynamic Item Response (DIR) models via

specifying a novel dynamic structure on the prior of the latent

trait. The Bayesian inference for DIR models is undertaken, which

permits borrowing strength from different individuals, allows the

retrospective analysis of an individual's changing ability, and

allows for online prediction of one's ability changes. Proof of

posterior propriety is presented, ensuring that the objective

Bayesian analysis is rigorous.

Chapter 3 deals with nonparametric function estimation under

shape constraints, such as monotonicity, convexity or concavity. A

motivating illustration is to generate an emulator to approximate a computer

model for vehicle crashworthiness. Although Gaussian processes are

very flexible and widely used in function estimation, they are not

naturally amenable to incorporation of such constraints. Gaussian

processes with the squared exponential correlation function have the

interesting property that their derivative processes are also

Gaussian processes and are jointly Gaussian processes with the

original Gaussian process. This allows one to impose shape constraints

through the derivative process. Two alternative ways of incorporating derivative

information into Gaussian processes priors are proposed, with one

focusing on scenarios (important in emulation of computer

models) in which the function may have flat regions.

Chapter 4 introduces a Bayesian method to control for multiplicity

in subgroup analyses through tree-based models that limit the

subgroups under consideration to those that are a priori plausible.

Once the prior modeling of the tree is accomplished, each tree will

yield a statistical model; Bayesian model selection analyses then

complete the statistical computation for any quantity of interest,

resulting in multiplicity-controlled inferences. This research is

motivated by a problem of biomarker and subgroup identification to

develop tailored therapeutics. Chapter 5 presents conclusions and

some directions for future research.

The dissertation focuses on solving some important theoretical and methodological problems associated with Bayesian modeling of infinite dimensional `objects', popularly called nonparametric Bayes. The term `infinite dimensional object' can refer to a density, a conditional density, a regression surface or even a manifold. Although Bayesian density estimation as well as function estimation are well-justified in the existing literature, there has been little or no theory justifying the estimation of more complex objects (e.g. conditional density, manifold, etc.). Part of this dissertation focuses on exploring the structure of the spaces on which the priors for conditional densities and manifolds are supported while studying how the posterior concentrates as increasing amounts of data are collected.

With the advent of new acquisition devices, there has been a need to model complex objects associated with complex data-types e.g. millions of genes affecting a bio-marker, 2D pixelated images, a cloud of points in the 3D space, etc. A significant portion of this dissertation has been devoted to developing adaptive nonparametric Bayes approaches for learning low-dimensional structures underlying higher-dimensional objects e.g. a high-dimensional regression function supported on a lower dimensional space, closed curves representing the boundaries of shapes in 2D images and closed surfaces located on or near the point cloud data. Characterizing the distribution of these objects has a tremendous impact in several application areas ranging from tumor tracking for targeted radiation therapy, to classifying cells in the brain, to model based methods for 3D animation and so on.

The first three chapters are devoted to Bayesian nonparametric theory and modeling in unconstrained Euclidean spaces e.g. mean regression and density regression, the next two focus on Bayesian modeling of manifolds e.g. closed curves and surfaces, and the final one on nonparametric Bayes spatial point pattern data modeling when the sampling locations are informative of the outcomes.

The immune system is constantly faced with the daunting task of protecting the host from a large number of ever-evolving pathogens. In vertebrates, the immune response results from the interplay of two cellular systems: the innate immunity and the adaptive immunity. In the past decades, dendritic cells have emerged as major players in the modulation of the immune response, being one of the primary links between these two branches of the immune system.

Dendritic cells are pathogen-sensing cells that alert the rest of the immune system of the presence of infection. The signals sent by dendritic cells result in the recruitment of the appropriate cell types and molecules required for effectively clearing the infection. A question of utmost importance in our understanding of the immune response and our ability to manipulate it in the development of vaccines and therapies is: "How do dendritic cells translate the various cues they perceive from the environment into different signals that specifically activate the appropriate parts of the immune system that result in an immune response streamlined to clear the given pathogen?"

Here we have developed computational and statistical methods aimed to address specific aspects of this question. In particular, understanding how dendritic cells ultimately modulate the immune response requires an understanding of the subtleties of their maturation process in response to different environmental signals. Hence, the first part of this dissertation focuses on elucidating the changes in the transcriptional

program of dendritic cells in response to the detection of two common pathogen- associated molecules, LPS and CpG. We have developed a method based on Langevin and Dirichlet processes to model and cluster gene expression temporal data, and have used it to identify, on a large scale, genes that present unique and common transcriptional behaviors in response to these two stimuli. Additionally, we have also investigated a different, but related, aspect of dendritic cell modulation of the adaptive immune response. In the second part of this dissertation, we present a method to predict peptides that will bind to MHC molecules, a requirement for the activation of pathogen-specific T cells. Together, these studies contribute to the elucidation of important aspects of dendritic cell biology.

Our interest is the risk assessment of rare natural hazards, such as

large volcanic pyroclastic flows. Since catastrophic consequences of

volcanic flows are rare events, our analysis benefits from the use of

a computer model to provide information about these events under

natural conditions that may not have been observed in reality.

A common problem in the analysis of computer experiments, however, is the high computational cost associated with each simulation of a complex physical process. We tackle this problem by using a statistical approximation (emulator) to predict the output of this computer model at untried values of inputs. Gaussian process response surface is a technique commonly used in these applications, because it is fast and easy to use in the analysis.

We explore several aspects of the implementation of Gaussian process emulators in a Bayesian context. First, we propose an improvement for the implementation of the plug-in approach to Gaussian processes. Next, we also evaluate the performance of a spatial model for large data sets in the context of computer experiments.

Computer model data can also be combined to field observations in order to calibrate the emulator and obtain statistical approximations to the computer model that are closer to reality. We present an application where we learn the joint distribution of inputs from field data and then bind this auxiliary information to the emulator in a calibration process.

One of the outputs of our computer model is a surface of maximum volcanic flow height over some geographical area. We show how the topography of the volcano area plays an important role in determining the shape of this surface, and we propose methods

to incorporate geophysical information in the multivariate analysis of computer model output.

As a growing number of agents are deployed in complex environments for scientific research and human well-being, there are increasing demands for designing efficient learning algorithms for these agents to improve their control polices. Such policies must account for uncertainties, including those caused by environmental stochasticity, sensor noise and communication restrictions. These challenges exist in missions such as planetary navigation, forest firefighting, and underwater exploration. Ideally, good control policies should allow the agents to deal with all the situations in an environment and enable them to accomplish their mission within the budgeted time and resources. However, a correct model of the environment is not typically available in advance, requiring the policy to be learned from data. Model-free reinforcement learning (RL) is a promising candidate for agents to learn control policies while engaged in complex tasks, because it allows the control policies to be learned directly from a subset of experiences and with time efficiency. Moreover, to ensure persistent performance improvement for RL, it is important that the control policies be concisely represented based on existing knowledge, and have the flexibility to accommodate new experience. Bayesian nonparametric methods (BNPMs) both allow the complexity of models to be adaptive to data, and provide a principled way for discovering and representing new knowledge.

In this thesis, we investigate approaches for RL in centralized and decentralized sequential decision-making problems using BNPMs. We show how the control policies can be learned efficiently under model-free RL schemes with BNPMs. Specifically, for centralized sequential decision-making, we study Q learning with Gaussian processes to solve Markov decision processes, and we also employ hierarchical Dirichlet processes as the prior for the control policy parameters to solve partially observable Markov decision processes. For decentralized partially observable Markov decision processes, we use stick-breaking processes as the prior for the controller of each agent. We develop efficient inference algorithms for learning the corresponding control policies. We demonstrate that by combining model-free RL and BNPMs with efficient algorithm design, we are able to scale up RL methods for complex problems that cannot be solved due to the lack of model knowledge. We adaptively learn control policies with concise structure and high value, from a relatively small amount of data.

In many spatio-temporal applications a vector of covariates is measured alongside a spatio-temporal response. In such cases, the purpose of the statistical model is to quantify the change, in expectation or otherwise, in the response due to a change in the predictors while adequately accounting for the spatio-temporal structure of the response, the predictors, or both. The most common approach for building such a model is to confine the relationship between the response and the predictors to a single spatio-temporal coordinate. For spatio-temporal problems, however, the relationship between the response and predictors may not be so confined. For example, spatial models are often used to quantify the effect of pollution exposure on mortality. Yet, an unknown lag exists between time of exposure to pollutants and mortality. Furthermore, due to mobility and atmospheric movement, a spatial lag between pollution concentration and mortality may also exist (e.g. subjects may live in the suburbs where pollution levels are low but work in the city where pollution levels are high).

The contribution of this thesis is to propose a hierarchical modeling framework which captures complex spatio-temporal relationships between responses and covariates. Specifically, the models proposed here use kernels to capture spatial and/or temporal lagged effects. Several forms of kernels are proposed with varying degrees of complexity. In each case, however, the kernels are assumed to be parametric with parameters that are easily interpretable and estimable from the data. Full distributional results are given for the Gaussian setting along with consequences of model misspecification. The methods are shown to be effective in understanding the complex relationship between responses and covariates through various simulated examples and analyses of physical data sets.

Previous papers have elaborated formal gradient analysis for spatial processes, focusing on the distribution theory for directional derivatives associated with a response variable assumed to follow a Gaussian process model. In the current work, these ideas are extended to additionally accommodate one or more continuous covariate(s) whose directional derivatives are of interest and to relate the behavior of the directional derivatives of the response surface to those of the covariate surface(s). It is of interest to assess whether, in some sense, the gradients of the response follow those of the explanatory variable(s), thereby gaining insight into the local relationships between the variables. The joint Gaussian structure of the spatial random effects and associated directional derivatives allows for explicit distribution theory and, hence, kriging across the spatial region using multivariate normal theory. The gradient analysis is illustrated for bivariate and multivariate spatial models, non-Gaussian responses such as presence-absence and point patterns, and outlined for several additional spatial modeling frameworks that commonly arise in the literature. Working within a hierarchical modeling framework, posterior samples enable all gradient analyses to occur as post model fitting procedures.

This thesis presents data analysis and methodology for two prediction problems. The first problem is forecasting midlife credit ratings from personality information collected during early adulthood. The second problem is analysis of matrix multiplication in cloud computing.

The goal of the credit forecasting problem is to determine if there is a link between personality assessments of young adults with their propensity to develop credit in middle age. The data we use is from a long term longitudinal study of over 40 years. We do find an association between credit risk and personality in this cohort Such a link has obvious implications for lenders but also can be used to improve social utility via more efficient resource allocation

We analyze matrix multiplication in the cloud and model I/O and local computation for individual tasks. We established conditions for which the distribution of job completion times can be explicitly obtained. We further generalize these results to cases where analytic derivations are intractable.

We develop models that emulate the multiplication procedure, allowing job times for different deployment parameter settings to be emulated after only witnessing a subset of tasks, or subsets of tasks for nearby deployment parameter settings.

The modeling framework developed sheds new light on the problem of determining expected job completion time for sparse matrix multiplication.

Point process data are commonly observed in fields like healthcare and social science. Designing predictive models for such event streams is an under-explored problem, due to often scarce training data. In this thesis, a multitask point process model via a hierarchical Gaussian Process (GP) is proposed, to leverage statistical strength across multiple point processes. Nonparametric learning functions implemented by a GP, which map from past events to future rates, allow analysis of flexible arrival patterns. To facilitate efficient inference, a sparse construction for this hierarchical model is proposed, and a variational Bayes method is derived for learning and inference. Experimental results are shown on both synthetic data and as well as real electronic health-records data.

We present a scalable end-to-end classifier that uses streaming physiological and medication data to accurately predict the onset of sepsis, a life-threatening complication from infections that has high mortality and morbidity. Our proposed framework models the multivariate trajectories of continuous-valued physiological time series using multitask Gaussian processes, seamlessly accounting for the high uncertainty, frequent missingness, and irregular sampling rates typically associated with real clinical data. The Gaussian process is directly connected to a black-box classifier that predicts whether a patient encounter will become septic, chosen in our case to be a recurrent neural network to account for the extreme variability in the length of patient encounters. We show how several approximations scale the computations associated with the Gaussian process in a manner so that the entire system can be trained discriminatively end-to-end using backpropagation. In a large cohort of heterogeneous inpatient encounters at our university health system we find that it outperforms several baselines at predicting sepsis, and yields 33\% and 195\% improved areas under the Receiver Operating Characteristic and Precision Recall curves as compared to the NEWS score currently in use on our own hospital wards.

Capturing high dimensional complex ensembles of data is becoming commonplace in a variety of application areas. Some examples include

biological studies exploring relationships between genetic mutations and diseases, atmospheric and spatial data, and internet usage and online behavioral data. These large complex data present many challenges in their modeling and statistical analysis. Motivated by high dimensional data applications, in this thesis, we focus on building scalable Bayesian nonparametric regression algorithms and on developing models for joint distributions of complex object ensembles.

We begin with a scalable method for Gaussian process regression, a commonly used tool for nonparametric regression, prediction and spatial modeling. A very common bottleneck for large data sets is the need for repeated inversions of a big covariance matrix, which is required for likelihood evaluation and inference. Such inversion can be practically infeasible and even if implemented, highly numerically unstable. We propose an algorithm utilizing random projection ideas to construct flexible, computationally efficient and easy to implement approaches for generic scenarios. We then further improve the algorithm incorporating some structure and blocking ideas in our random projections and demonstrate their applicability in other contexts requiring inversion of large covariance matrices. We show theoretical guarantees for performance as well as substantial improvements over existing methods with simulated and real data. A by product of the work is that we discover hitherto unknown equivalences between approaches in machine learning, random linear algebra and Bayesian statistics. We finally connect random projection methods for large dimensional predictors and large sample size under a unifying theoretical framework.

The other focus of this thesis is joint modeling of complex ensembles of data from different domains. This goes beyond traditional relational modeling of ensembles of one type of data and relies on probability mixing measures over tensors. These models have added flexibility over some existing product mixture model approaches in letting each component of the ensemble have its own dependent cluster structure. We further investigate the question of measuring dependence between variables of different types and propose a very general novel scaled measure based on divergences between the joint and marginal distributions of the objects. Once again, we show excellent performance in both simulated and real data scenarios.

Bayesian nonparametric models, such as the Gaussian process and the Dirichlet process, have been extensively applied for target kinematics modeling in various applications including environmental monitoring, traffic planning, endangered species tracking, dynamic scene analysis, autonomous robot navigation, and human motion modeling. As shown by these successful applications, Bayesian nonparametric models are able to adjust their complexities adaptively from data as necessary, and are resistant to overfitting or underfitting. However, most existing works assume that the sensor measurements used to learn the Bayesian nonparametric target kinematics models are obtained a priori or that the target kinematics can be measured by the sensor at any given time throughout the task. Little work has been done for controlling the sensor with bounded field of view to obtain measurements of mobile targets that are most informative for reducing the uncertainty of the Bayesian nonparametric models. To present the systematic sensor planning approach to leaning Bayesian nonparametric models, the Gaussian process target kinematics model is introduced at first, which is capable of describing time-invariant spatial phenomena, such as ocean currents, temperature distributions and wind velocity fields. The Dirichlet process-Gaussian process target kinematics model is subsequently discussed for modeling mixture of mobile targets, such as pedestrian motion patterns.

Novel information theoretic functions are developed for these introduced Bayesian nonparametric target kinematics models to represent the expected utility of measurements as a function of sensor control inputs and random environmental variables. A Gaussian process expected Kullback Leibler divergence is developed as the expectation of the KL divergence between the current (prior) and posterior Gaussian process target kinematics models with respect to the future measurements. Then, this approach is extended to develop a new information value function that can be used to estimate target kinematics described by a Dirichlet process-Gaussian process mixture model. A theorem is proposed that shows the novel information theoretic functions are bounded. Based on this theorem, efficient estimators of the new information theoretic functions are designed, which are proved to be unbiased with the variance of the resultant approximation error decreasing linearly as the number of samples increases. Computational complexities for optimizing the novel information theoretic functions under sensor dynamics constraints are studied, and are proved to be NP-hard. A cumulative lower bound is then proposed to reduce the computational complexity to polynomial time.

Three sensor planning algorithms are developed according to the assumptions on the target kinematics and the sensor dynamics. For problems where the control space of the sensor is discrete, a greedy algorithm is proposed. The efficiency of the greedy algorithm is demonstrated by a numerical experiment with data of ocean currents obtained by moored buoys. A sweep line algorithm is developed for applications where the sensor control space is continuous and unconstrained. Synthetic simulations as well as physical experiments with ground robots and a surveillance camera are conducted to evaluate the performance of the sweep line algorithm. Moreover, a lexicographic algorithm is designed based on the cumulative lower bound of the novel information theoretic functions, for the scenario where the sensor dynamics are constrained. Numerical experiments with real data collected from indoor pedestrians by a commercial pan-tilt camera are performed to examine the lexicographic algorithm. Results from both the numerical simulations and the physical experiments show that the three sensor planning algorithms proposed in this dissertation based on the novel information theoretic functions are superior at learning the target kinematics with

little or no prior knowledge

Although quantile regression provides a comprehensive and robust replacement for the traditional mean regression, a complete estimation technique is in blank for a long time. Original separate estimation could cause severe problems, which obstructs its popularization in methodology and application. A novel complete Bayesian joint estimation of quantile regression is proposed and serves as a thorough solution to this historical challenge. In this thesis, we first introduce this modeling technique and propose some preliminary but important theoretical development on the posterior convergence rate of this novel joint estimation, which offers significant guidance to the ultimate results. We provide the posterior convergence rate for the density estimation model induced by this joint quantile regression model. Furthermore, the prior concentration condition of the truncated version of this joint quantile regression model is proved and the entropy condition of the truncated model with any sphere predictor plane centered at 0 is verified. An application on high school math achievement is also introduced, which reveals some deep association between math achievement and socio-economic status. Some further developments about the estimation technique, convergence rate and application are discussed. Furthermore, some suggestions on school choices for minority students are mentioned according to the application.

Linear quantile regression is a powerful tool to investigate how predictors may affect a response heterogeneously across different quantile levels. Unfortunately, existing approaches find it extremely difficult to adjust for any dependency between observation units, largely because such methods are not based upon a fully generative model of the data. In this dissertation, we address this difficulty for analyzing spatial point-referenced data and hierarchical data. Several models are introduced by generalizing the joint quantile regression model of Yang and Tokdar (2017) and characterizing different dependency structures via a copula model on the underlying quantile levels of the observation units. A Bayesian semiparametric approach is introduced to perform inference of model parameters and carry out prediction. Multiple copula families are discussed for modeling response data with tail dependence and/or tail asymmetry. An effective model comparison criterion is provided for selecting between models with different combinations of sets of predictors, marginal base distributions and copula models.

Extensive simulation studies and real applications are presented to illustrate substantial gains of the proposed models in inference quality, prediction accuracy and uncertainty quantification over existing alternatives. Through case studies, we highlight that the proposed models admit great interpretability and are competent in offering insightful new discoveries of response-predictor relationship at non-central parts of the response distribution. The effectiveness of the proposed model comparison criteria is verified with both empirical and theoretical evidence.

We address predictive modeling for spatial and spatiotemporal modeling in a variety of settings. First, we discuss spatial and spatiotemporal data and corresponding model types used in later chapters. Specifically, we discuss Markov random fields, Gaussian processes, and Bayesian inference. Then, we outline the dissertation.

In Chapter 2, we consider the setting where areal unit data are only partially observed. First, we consider setting where a portion of the areal units have been observed, and we seek prediction of the remainder. Second, we leverage these ideas for model comparison where we fit models of interest to a portion of the data and hold out the rest for model comparison.

In Chapters 3 and 4, we consider pollution data from Mexico City in 2017. In Chapter 3 we forecast pollution emergencies. Mexico City defines pollution emergencies using thresholds that rely on regional maxima for ozone and for particulate matter with diameter less than 10 micrometers (PM10). To predict local pollution emergencies and to assess compliance with Mexican ambient air quality standards, we analyze hourly ozone and PM10 measurements from 24 stations across Mexico City from 2017 using a bivariate spatiotemporal model. With this model, we predict future pollutant levels using current weather conditions and recent pollutant concentrations. Employing hourly pollutant projections, we predict regional maxima needed to estimate the probability of future pollution emergencies. We discuss how predicted compliance with legislated pollution limits varies across regions within Mexico City in 2017.

In Chapter 4, we propose a continuous spatiotemporal model for Mexico City ozone levels that accounts for distinct daily seasonality, as well as variation across the city and over the peak ozone season (April and May) of 2017. To account for these patterns, we use covariance models over space, circles, and time. We review relevant existing covariance models and develop new classes of nonseparable covariance models appropriate for seasonal data collected at many locations. We compare the predictive performance of a variety of models that utilize various nonseparable covariance functions. We use the best model to predict hourly ozone levels at unmonitored locations in April and May to infer compliance with Mexican air quality standards and to estimate respiratory health risk associated with ozone exposure.