Dissertations / Theses on the topic 'Multilevel models (Statistics) Markov processes'

Thesis (Ph. D.)--University of Missouri-Columbia, 2007.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed Nov. 21, 2007). Vita. Includes bibliographical references.

Thesis (PhD)--Stellenbosch University, 2012.
ENGLISH ABSTRACT: Multi-state models are used in this dissertation to model panel data, also known as longitudinal or cross-sectional time-series data. These are data sets which include units that are observed across two or more points in time. These models have been used extensively in medical studies where the disease states of patients are recorded over time. A theoretical overview of the current multi-state Markov models when applied to panel data is presented and based on this theory, a simulation procedure is developed to generate panel data sets for given Markov models. Through the use of this procedure a simulation study is undertaken to investigate the properties of the standard likelihood approach when fitting Markov models and then to assess its shortcomings. One of the main shortcomings highlighted by the simulation study, is the unstable estimates obtained by the standard likelihood models, especially when fitted to small data sets. A Bayesian approach is introduced to develop multi-state models that can overcome these unstable estimates by incorporating prior knowledge into the modelling process. Two Bayesian techniques are developed and presented, and their properties are assessed through the use of extensive simulation studies. Firstly, Bayesian multi-state models are developed by specifying prior distributions for the transition rates, constructing a likelihood using standard Markov theory and then obtaining the posterior distributions of the transition rates. A selected few priors are used in these models. Secondly, Bayesian multi-state imputation techniques are presented that make use of suitable prior information to impute missing observations in the panel data sets. Once imputed, standard likelihood-based Markov models are fitted to the imputed data sets to estimate the transition rates. Two different Bayesian imputation techniques are presented. The first approach makes use of the Dirichlet distribution and imputes the unknown states at all time points with missing observations. The second approach uses a Dirichlet process to estimate the time at which a transition occurred between two known observations and then a state is imputed at that estimated transition time. The simulation studies show that these Bayesian methods resulted in more stable results, even when small samples are available.
AFRIKAANSE OPSOMMING: Meerstadium-modelle word in hierdie verhandeling gebruik om paneeldata, ook bekend as longitudinale of deursnee tydreeksdata, te modelleer. Hierdie is datastelle wat eenhede insluit wat oor twee of meer punte in tyd waargeneem word. Hierdie tipe modelle word dikwels in mediese studies gebruik indien verskillende stadiums van ’n siekte oor tyd waargeneem word. ’n Teoretiese oorsig van die huidige meerstadium Markov-modelle toegepas op paneeldata word gegee. Gebaseer op hierdie teorie word ’n simulasieprosedure ontwikkel om paneeldatastelle te simuleer vir gegewe Markov-modelle. Hierdie prosedure word dan gebruik in ’n simulasiestudie om die eienskappe van die standaard aanneemlikheidsbenadering tot die pas vanMarkov modelle te ondersoek en dan enige tekortkominge hieruit te beoordeel. Een van die hoof tekortkominge wat uitgewys word deur die simulasiestudie, is die onstabiele beramings wat verkry word indien dit gepas word op veral klein datastelle. ’n Bayes-benadering tot die modellering van meerstadiumpaneeldata word ontwikkel omhierdie onstabiliteit te oorkom deur a priori-inligting in die modelleringsproses te inkorporeer. Twee Bayes-tegnieke word ontwikkel en aangebied, en hulle eienskappe word ondersoek deur ’n omvattende simulasiestudie. Eerstens word Bayes-meerstadium-modelle ontwikkel deur a priori-verdelings vir die oorgangskoerse te spesifiseer en dan die aanneemlikheidsfunksie te konstrueer deur van standaard Markov-teorie gebruik te maak en die a posteriori-verdelings van die oorgangskoerse te bepaal. ’n Gekose aantal a priori-verdelings word gebruik in hierdie modelle. Tweedens word Bayesmeerstadium invul tegnieke voorgestel wat gebruik maak van a priori-inligting om ontbrekende waardes in die paneeldatastelle in te vul of te imputeer. Nadat die waardes ge-imputeer is, word standaard Markov-modelle gepas op die ge-imputeerde datastel om die oorgangskoerse te beraam. Twee verskillende Bayes-meerstadium imputasie tegnieke word bespreek. Die eerste tegniek maak gebruik van ’n Dirichletverdeling om die ontbrekende stadium te imputeer by alle tydspunte met ’n ontbrekende waarneming. Die tweede benadering gebruik ’n Dirichlet-proses om die oorgangstyd tussen twee waarnemings te beraam en dan die ontbrekende stadium te imputeer op daardie beraamde oorgangstyd. Die simulasiestudies toon dat die Bayes-metodes resultate oplewer wat meer stabiel is, selfs wanneer klein datastelle beskikbaar is.

In this thesis, player’s performance on ice hockey is modelled to create newmetricsby match and season for players. AD-trees have been used to summarize ice hockey matches using state variables, which combine context and action variables to estimate the impact of each action under that specific state using Markov Decision Processes. With that, an impact measure has been described and four player metrics have been derived by match for regular seasons 2007-2008 and 2008-2009. General analysis has been performed for these metrics and ARIMA models have been used to analyze and predict players performance. The best prediction achieved in the modelling is the mean of the previous matches. The combination of several metrics including the ones created in this thesis could be combined to evaluate player’s performance using salary ranges to indicate whether a player is worth hiring/maintaining/firing

Thesis (Ph. D.)--Ohio State University, 2005.
Title from first page of PDF file. Document formatted into pages; contains xiii, 108 p.; also includes graphics. Includes bibliographical references (p. 104-108). Available online via OhioLINK's ETD Center

My doctoral research dissertation focuses on two aspects of functional data analysis (FDA): FDA under spatial interdependence and FDA for multi-level data. The first part of my thesis focuses on developing modeling and inference procedure for functional data under spatial dependence. The methodology introduced in this part is motivated by a research study on inequities in accessibility to financial services. The first research problem in this part is concerned with a novel model-based method for clustering random time functions which are spatially interdependent. A cluster consists of time functions which are similar in shape. The time functions are decomposed into spatial global and time-dependent cluster effects using a semi-parametric model. We also assume that the clustering membership is a realization from a Markov random field. Under these model assumptions, we borrow information across curves from nearby locations resulting in enhanced estimation accuracy of the cluster effects and of the cluster membership. In a simulation study, we assess the estimation accuracy of our clustering algorithm under a series of settings: small number of time points, high noise level and varying dependence structures. Over all simulation settings, the spatial-functional clustering method outperforms existing model-based clustering methods. In the case study presented in this project, we focus on estimates and classifies service accessibility patterns varying over a large geographic area (California and Georgia) and over a period of 15 years. The focus of this study is on financial services but it generally applies to any other service operation. The second research project of this part studies an association analysis of space-time varying processes, which is rigorous, computational feasible and implementable with standard software. We introduce general measures to model different aspects of the temporal and spatial association between processes varying in space and time. Using a nonparametric spatiotemporal model, we show that the proposed association estimators are asymptotically unbiased and consistent. We complement the point association estimates with simultaneous confidence bands to assess the uncertainty in the point estimates. In a simulation study, we evaluate the accuracy of the association estimates with respect to the sample size as well as the coverage of the confidence bands. In the case study in this project, we investigate the association between service accessibility and income level. The primary objective of this association analysis is to assess whether there are significant changes in the income-driven equity of financial service accessibility over time and to identify potential under-served markets. The second part of the thesis discusses novel statistical methodology for analyzing multilevel functional data including a clustering method based on a functional ANOVA model and a spatio-temporal model for functional data with a nested hierarchical structure. In this part, I introduce and compare a series of clustering approaches for multilevel functional data. For brevity, I present the clustering methods for two-level data: multiple samples of random functions, each sample corresponding to a case and each random function within a sample/case corresponding to a measurement type. A cluster consists of cases which have similar within-case means (level-1 clustering) or similar between-case means (level-2 clustering). Our primary focus is to evaluate a model-based clustering to more straightforward hard clustering methods. The clustering model is based on a multilevel functional principal component analysis. In a simulation study, we assess the estimation accuracy of our clustering algorithm under a series of settings: small vs. moderate number of time points, high noise level and small number of measurement types. We demonstrate the applicability of the clustering analysis to a real data set consisting of time-varying sales for multiple products sold by a large retailer in the U.S. My ongoing research work in multilevel functional data analysis is developing a statistical model for estimating temporal and spatial associations of a series of time-varying variables with an intrinsic nested hierarchical structure. This work has a great potential in many real applications where the data are areal data collected from different data sources and over geographic regions of different spatial resolution.

Thesis (Ph. D.)--Ohio State University, 2005.
Title from first page of PDF file. Document formatted into pages; contains xiv, 155 p.; also includes graphics (some col.). Includes bibliographical references (p. 147-155). Available online via OhioLINK's ETD Center

Making decisions and predictions from noisy observations are two important and challenging problems in many areas of society. Some examples of applications are recommendation systems for online shopping and streaming services, connecting genes with certain diseases and modelling climate change. In this thesis, we make use of Bayesian statistics to construct probabilistic models given prior information and historical data, which can be used for decision support and predictions. The main obstacle with this approach is that it often results in mathematical problems lacking analytical solutions. To cope with this, we make use of statistical simulation algorithms known as Monte Carlo methods to approximate the intractable solution. These methods enjoy well-understood statistical properties but are often computational prohibitive to employ. The main contribution of this thesis is the exploration of different strategies for accelerating inference methods based on sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). That is, strategies for reducing the computational effort while keeping or improving the accuracy. A major part of the thesis is devoted to proposing such strategies for the MCMC method known as the particle Metropolis-Hastings (PMH) algorithm. We investigate two strategies: (i) introducing estimates of the gradient and Hessian of the target to better tailor the algorithm to the problem and (ii) introducing a positive correlation between the point-wise estimates of the target. Furthermore, we propose an algorithm based on the combination of SMC and Gaussian process optimisation, which can provide reasonable estimates of the posterior but with a significant decrease in computational effort compared with PMH. Moreover, we explore the use of sparseness priors for approximate inference in over-parametrised mixed effects models and autoregressive processes. This can potentially be a practical strategy for inference in the big data era. Finally, we propose a general method for increasing the accuracy of the parameter estimates in non-linear state space models by applying a designed input signal.
Borde Riksbanken höja eller sänka reporäntan vid sitt nästa möte för att nå inflationsmålet? Vilka gener är förknippade med en viss sjukdom? Hur kan Netflix och Spotify veta vilka filmer och vilken musik som jag vill lyssna på härnäst? Dessa tre problem är exempel på frågor där statistiska modeller kan vara användbara för att ge hjälp och underlag för beslut. Statistiska modeller kombinerar teoretisk kunskap om exempelvis det svenska ekonomiska systemet med historisk data för att ge prognoser av framtida skeenden. Dessa prognoser kan sedan användas för att utvärdera exempelvis vad som skulle hända med inflationen i Sverige om arbetslösheten sjunker eller hur värdet på mitt pensionssparande förändras när Stockholmsbörsen rasar. Tillämpningar som dessa och många andra gör statistiska modeller viktiga för många delar av samhället. Ett sätt att ta fram statistiska modeller bygger på att kontinuerligt uppdatera en modell allteftersom mer information samlas in. Detta angreppssätt kallas för Bayesiansk statistik och är särskilt användbart när man sedan tidigare har bra insikter i modellen eller tillgång till endast lite historisk data för att bygga modellen. En nackdel med Bayesiansk statistik är att de beräkningar som krävs för att uppdatera modellen med den nya informationen ofta är mycket komplicerade. I sådana situationer kan man istället simulera utfallet från miljontals varianter av modellen och sedan jämföra dessa mot de historiska observationerna som finns till hands. Man kan sedan medelvärdesbilda över de varianter som gav bäst resultat för att på så sätt ta fram en slutlig modell. Det kan därför ibland ta dagar eller veckor för att ta fram en modell. Problemet blir särskilt stort när man använder mer avancerade modeller som skulle kunna ge bättre prognoser men som tar för lång tid för att bygga. I denna avhandling använder vi ett antal olika strategier för att underlätta eller förbättra dessa simuleringar. Vi föreslår exempelvis att ta hänsyn till fler insikter om systemet och därmed minska antalet varianter av modellen som behöver undersökas. Vi kan således redan utesluta vissa modeller eftersom vi har en bra uppfattning om ungefär hur en bra modell ska se ut. Vi kan också förändra simuleringen så att den enklare rör sig mellan olika typer av modeller. På detta sätt utforskas rymden av alla möjliga modeller på ett mer effektivt sätt. Vi föreslår ett antal olika kombinationer och förändringar av befintliga metoder för att snabba upp anpassningen av modellen till observationerna. Vi visar att beräkningstiden i vissa fall kan minska ifrån några dagar till någon timme. Förhoppningsvis kommer detta i framtiden leda till att man i praktiken kan använda mer avancerade modeller som i sin tur resulterar i bättre prognoser och beslut.

En esta tesis se hace una mezcla del método de distancias con los modelos lineales generalizados mixtos tanto en lo espacial como en lo espacio-temporal. Con el empleo de las distancias se logran buenas predicciones y menores variabilidades en el espacio o espacio-tiempo de la región de estudio, provocando todo esto que se tomen mejores decisiones en los diferentes problemas de interés. Se propone un método alternativo para ajustar una variable respuesta tipo beta con dispersión variable usando distancias euclidianas entre los individuos. Se emplea el método de máxima verosimilitud para estimar los parámetros desconocidos del modelo propuesto y se presentan las principales propiedades de estos estimadores. Además, se realiza la inferencia estadística sobre los parámetros utilizando las aproximaciones obtenidas a partir de la normalidad asintótica del estimador de máxima verosimilitud; se desarrolla el diagnóstico y predicción de una nueva observación, y se estudia el problema de datos faltantes utilizando la metodología propuesta. Posteriormente, se propone una solución alterna para resolver problemas como el de prevalencia de Loa loa utilizando distancias euclidianas entre individuos; se describe un modelo lineal generalizado espacial mixto incorporando medidas generales de distancia o disimilaridad que se pueden aplicar a variables explicativas. En este caso, los parámetros involucrados en el modelo propuesto se estiman utilizando máxima verosimilitud mediante el método de Monte Carlo vía cadenas de Markov (MCMC). También se formula un modelo lineal beta espacial mixto con dispersión variable utilizando máxima verosimilitud mediante el método MCMC. El método propuesto se utiliza en situaciones donde la variable respuesta es una razón o proporción que esta relacionada con determinadas variables explicativas. Para este fin, se desarrolla una aproximación utilizando modelos lineales generalizados espaciales mixtos empleando la transformación Box-Cox en el modelo de precisión. Por lo tanto, se realiza el proceso de optimización de los parámetros tanto para modelo espacial de media como para el modelo espacial de dispersión variable. Además, se realiza la inferencia estadística sobre los parámetros utilizando las aproximaciones obtenidas a partir de la normalidad asintótica del estimador de máxima verosimilitud. También se desarrolla el diagnóstico del modelo y la predicción de nuevas observaciones. Por último, el método se ilustra a través de los contenidos de arcilla y magnesio. Adicionalmente, se describe el modelo basado en distancias para la predicción espacio-temporal usando modelos lineales generalizados. Se realiza el proceso de estimación de los parámetros involucrados en el modelo propuesto, mediante el método de ecuaciones de estimación generalizada y la inferencia estadística sobre los parámetros empleando las aproximaciones obtenidas a partir de la normalidad asintótica del estimador de máxima verosimilitud. Además, se desarrolla el diagnóstico del modelo y la predicción de nuevas observaciones. Se realiza una aplicación de la metodología propuesta para el número de acciones armadas estandarizada por cada 1000 km2 de los grupos irregulares FARC-EP y ELN en los diferentes departamentos de Colombia entre los años 2003 a 2009. Finalmente, se presenta un modelo autorregresivo espacial lineal generalizado mixto utilizando el método basado en distancias. Este modelo incluye retrasos tanto espaciales como temporales entre vectores de variables de estado estacionarias. Se utiliza la dinámica espacial de los datos econométricos tipo panel para estimar el modelo propuesto; los parámetros involucrados en el modelo se estiman utilizando el método MCMC mediante máxima verosimilitud. Además, se discute en este capítulo la interacción entre estacionariedad temporal y espacial, y se derivan las respuestas al impulso para el modelo propuesto, lo cual naturalmente depende de la dinámica temporal y espacial del modelo.
In the context of regression with a beta-type response variable, we propose a new method that links two methodologies: a distance-based model, and a beta regression with variable dispersion. The proposed model is useful for those situations where the response variable is a rate, a proportion or parts per million. This variable is related with a mixture between continuous and categorical explanatory variables. We present its main statistical properties and some measures for selection of the most predictive dimensions in the model. Furthermore, the prediction of a new observation and the problem of missing data are also developed. Using the proposed model, the mutual funds are analyzed employing the Gower distance for both the mean model and the variable dispersion model. Also, we present a new method based on distances, which allows the modeling of continuous and non-continuous random variables through distance-based spatial generalized linear mixed models (SGLMMs). The parameters are estimated using Markov chain Monte Carlo (MCMC) maximum likelihood. The method is illustrated through the analysis of the variation in the prevalence of Loa loa among a sample of village residents in Cameroon, where the explanatory variables included elevation, together with maximum normalized-difference vegetation index (NDVI) and the standard deviation of NDVI calculated from repeated satellite scans over time. Additionally, we propose a beta spatial linear mixed model with variable dispersion using MCMC. An approach to the SGLMMs using the Box-Cox transformation in the precision model is developed. Thus, the parameter optimization process is made for both the spatial mean model as the spatial variable dispersion model. Statistical inference over the parameters is performed using approximations obtained from the asymptotic normality of the maximum likelihood estimator. Diagnosis and prediction of a new observation are also developed. This model is illustrated through of the clay and magnesium contents. On the other hand, we present a solution to problems where the response variable is a count, a rate or a binary (dichotomous) using a refined distance-based generalized linear space-time-autoregressive model with space-time-autoregressive disturbances. This model may also contain additional spatial exogenous variables as well as time exogenous variables. The parameter estimation process is done by the space-time generalized estimating equations (GEE) method, and a measure of goodness-of-fit is presented. Also, the best linear unbiased predictor for prediction purposes is presented. An application for the standardized number of armed actions per 1000 km2 of rebel groups FARC-EP and ELN in different departments of Colombia from 2003 to 2009 is employed to illustrate the proposed methodology. Finally, a spatial generalized linear mixed autoregressive model using distance-based is defined including spatial as well as temporal lags between vectors of stationary state variables. Although the structural parameters are not fully identified in this model, contemporaneous spatial lag coefficients may be identified by exogenous state variables. Dynamic spatial panel data econometrics is used to estimate our proposed model. In this way, the parameters are estimated using MCMC maximum likelihood. We also discuss the interaction between temporal and spatial stationarity, and we derive the impulse responses for our model, which naturally depend upon the temporal and spatial dynamics of the model.

Cette thèse présente des contributions à la littérature des méthodes de Monte Carlo utilisé dans l'analyse des modèles complexes en statistique Bayésienne; l'accent est mis à la fois sur la complexité des modèles et sur les difficultés de calcul.Le premier chapitre élargit Delayed Acceptance, une variante computationellement efficace du Metropolis--Hastings, et agrandit son cadre théorique fournissant une justification adéquate pour la méthode, des limits pour sa variance asymptotique par rapport au Metropolis--Hastings et des idées pour le réglage optimal de sa distribution instrumentale.Nous allons ensuite développer une méthode Bayésienne pour analyser les processus environnementaux non stationnaires, appelées Expansion Dimension, qui considère le processus observé comme une projection depuis une dimension supérieure, où l'hypothèse de stationnarité pourrait etre acceptée. Le dernier chapitre sera finalement consacrée à l'étude des structures de dépendances conditionnelles par une formulation entièrement Bayésienne du modèle de Copule Gaussien graphique
This thesis presents contributions to the Monte Carlo literature aimed toward the analysis of complex models in Bayesian Statistics; the focus is on both complexity related to complicate models and computational difficulties.We will first expand Delayed Acceptance, a computationally efficient variant ofMetropolis--Hastings, to a multi-step procedure and enlarge its theoretical background, providing proper justification for the method, asymptotic variance bounds relative to its parent MH kernel and optimal tuning for the scale of its proposal.We will then develop a flexible Bayesian method to analyse nonlinear environmentalprocesses, called Dimension Expansion, that essentially consider the observed process as a projection from a higher dimension, where the assumption of stationarity could hold.The last chapter will finally be dedicated to the investigation of conditional (in)dependence structures via a fully Bayesian formulation of the Gaussian Copula graphical model

Sequence alignment is one of the most intensely studied problems in bioinformatics, and is an important step in a wide range of analyses. An issue that has gained much attention in recent years is the fact that downstream analyses are often highly sensitive to the specific choice of alignment. One way to address this is to jointly sample alignments along with other parameters of interest. In order to extend the range of applicability of this approach, the first chapter of this thesis introduces a probabilistic evolutionary model for protein structures on a phylogenetic tree; since protein structures typically diverge much more slowly than sequences, this allows for more reliable detection of remote homologies, improving the accuracy of the resulting alignments and trees, and reducing sensitivity of the results to the choice of dataset. In order to carry out inference under such a model, a number of new Markov chain Monte Carlo approaches are developed, allowing for more efficient convergence and mixing on the high-dimensional parameter space. The second part of the thesis presents a directed acyclic graph (DAG)-based approach for representing a collection of sampled alignments. This DAG representation allows the initial collection of samples to be used to generate a larger set of alignments under the same approximate distribution, enabling posterior alignment probabilities to be estimated reliably from a reasonable number of samples. If desired, summary alignments can then be generated as maximum-weight paths through the DAG, under various types of loss or scoring functions. The acyclic nature of the graph also permits various other types of algorithms to be easily adapted to operate on the entire set of alignments in the DAG. In the final part of this work, methodology is introduced for alignment-DAG-based sequence annotation using hidden Markov models, and RNA secondary structure prediction using stochastic context-free grammars. Results on test datasets indicate that the additional information contained within the DAG allows for improved predictions, resulting in substantial gains over simply analysing a set of alignments one by one.

Survival analysis is a well developed area which explores time to single event analysis. In some cases, however, such methods may not adequately capture the disease process as the disease progression may involve intermediate events of interest. Multistate models incorporate multiple events or states. This thesis proposes to demystify the theory of multistate models through an application based approach. We present the key components of multistate models, relevant derivations, model diagnostics and techniques for modeling the effect of covariates on transition intensities. The methods that are developed in the thesis are applied to HIV and TB data partly sourced from CAPRISA and the HPP programmes in the University of KwaZulu-Natal. HIV progression is investigated through the application of a five state Markov model with reversible transitions such that state 1: CD4 count 500, state 2: 350 CD4 count < 500, state 3: 200 CD4 count < 350, state 4: CD4 count < 200 and state 5: ARV initiation. The mean sojourn time in each state and transition probabilities are presented as well as the effect of covariates namely age, gender and baseline CD4 count on transition rates. A key finding, consistent with previous research, is that the rate of decline in CD4 count tends to decrease at lower levels of the marker. Further, patients enrolling with a CD4 count less than 350 had a far lower chance of immune recovery and a substantially higher chance of immune deterioration compared to patients with a higher CD4 count. We noted that older patients tend to progress more rapidly through the disease than younger patients.
Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

Leung Man-Kit.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.
Includes bibliographical references (leaves 37-41).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Basic concepts of survival data --- p.3
Chapter 1.2 --- Missing Complete at Random (MCAR) --- p.8
Chapter 1.3 --- Missing at Random (MAR) --- p.9
Chapter 2 --- The maximaization of the marginal likelihood --- p.11
Chapter 2.1 --- Survival function --- p.11
Chapter 2.2 --- Missing covariate pattern --- p.13
Chapter 2.3 --- Set of survival time with rank restrictions --- p.13
Chapter 2.4 --- Marginal likelihood --- p.14
Chapter 2.5 --- Score function --- p.15
Chapter 3 --- The MCMC stochastic approximation approach --- p.17
Chapter 4 --- Simulations Studies --- p.22
Chapter 4.1 --- MCAR : Simulation 1 --- p.23
Chapter 4.2 --- MCAR : Simulation 2 --- p.24
Chapter 4.3 --- MAR : Simulation 3 --- p.26
Chapter 4.4 --- MAR : Simulation 4 --- p.27
Chapter 5 --- Example --- p.30
Chapter 6 --- Discussion --- p.33
Appendix --- p.35
Bibliography --- p.37

Networks arise in nearly every branch of science, from biology and physics to sociology and economics. A signature of many network datasets is strong local dependence, which gives rise to phenomena such as sparsity, power law degree distributions, clustering, and structural heterogeneity. Statistical models of networks require a careful balance of flexibility to faithfully capture that dependence, and simplicity, to make analysis and inference tractable. In this dissertation, we introduce a class of models that insert one network edge at a time via a random walk, permitting the location of new edges to depend explicitly on the structure of the existing network, while remaining probabilistically and computationally tractable. Connections to graph kernels are made through the probability generating function of the random walk length distribution. The limiting degree distribution is shown to exhibit power law behavior, and the properties of the limiting degree sequence are studied analytically with martingale methods. In the second part of the dissertation, we develop a class of particle Markov chain Monte Carlo algorithms to perform inference for a large class of sequential random graph models, even when the observation consists only of a single graph. Using these methods, we derive a particle Gibbs sampler for random walk models. Fit to synthetic data, the sampler accurately recovers the model parameters; fit to real data, the model offers insight into the typical length scale of dependence in the network, and provides a new measure of vertex centrality. The arrival times of new vertices are the key to obtaining results for both theory and inference. In the third part, we undertake a careful study of the relationship between the arrival times, sparsity, and heavy tailed degree distributions in preferential attachment-type models of partitions and graphs. A number of constructive representations of the limiting degrees are obtained, and connections are made to exchangeable Gibbs partitions as well as to recent results on the limiting degrees of preferential attachment graphs.

While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.