Binding of transcription factors on specific sites of DNA is central to the regulation of gene expression. ChIP-seq technology is a novel tool that combines the method of chromatin immunoprecipitation (ChIP) with the next generation DNA sequencing (seq) to identify the transcription factor binding loci on DNA. ChIP-seq has revolutionized the process of biological data acquisition for elucidating fundamental gene regulation mechanisms. However, the acquired large dataset on transcription factor-DNA binding calls for analyses using statistical tools, which will provide predictions that guide the wet-lab biological research. This research is part of statistical modeling of patterns of transcription factor-DNA binding which serves to analyze the various patterns of transcription factor co-clustering on DNA in a ChIP-seq dataset obtained in the mouse embryonic stem cells for 15 transcription factors/coregulators. First, we used the Chi-square goodness of fit test to determine whether the location of binding sites for each transcription factor constitute a Poisson process. The results indicated that it is unlikely to be a homogenous Poisson process. Second, we studied the correlation among the bindings by various transcription factors. Third, the patterns of various clustered sites containing three transcription factors were analyzed. It is found that there are a total of 3353 such sites. The transcription factors Smad1, Tcfcp2l1, Stat3, Klf4 and Esrrb and the coregulator p300 are preferentially co-localized with Nanog, Oct4, Sox2, while E2f1 and Zfx are preferentially colocalized with n-Myc and c-Myc.

Modern portfolio theory is a statistical framework to allocate investment assets properly, with the aim of reducing risk by diversification. In the past decades, a variety of index and group models (with different covariance assumption) have been proposed to optimize the portfolio, including Single Index Model, Constant Correlation Model, Multi-Group Model, and Multi-Index Model. An R package "stockPortfolio" is developed by Drs. Christou and Diez, and fully implemented Single Index Model, Constant Correlation Model. Besides, this package also includes functions to download historical data from Yahoo Finance, build models, estimate optimal portfolios, and test portfolios. However, stockPortfolio package does not include the optimization functions for Multi-Group & Multi-Index Models with & without short selling. Besides, corresponding covariance matrix calculation functions are also not completed for these two models. My thesis will implement all these missing elements and aims to make a complete stockPorfolio package. The performance of these functions will be evaluated with multiple different historical data sets.

In recent years, Social Network Service (SNS) is a novel, popular way to make friends and

convey information online. Therefore, the analysis of network data has attracted a lot of

attention. It is an area that is rapidly growing, both with Statistics and Computer Science.

This paper rst provides a summary of statistical methods used in network data analysis,

including basic denitions, measurements, and descriptive statistics. We then introduce the

Exponential Random Graph Model to t network data. Secondly, we dig into a more specic

area of network analysis: Community Detection. We discuss two dierent methods to explore

the community stucture, one is Louvain algorithm and the other is Mixed Membership

Stochastic Blockmodels. After that, we combine the community identication with a two-

stage user similarity algorithm to build a friend recommendation method. In the empirical

study section, we apply this method to a real-world dataset and evaluate its performace

through specic measurements.

Directed Acyclic Graphs (DAGs) are a powerful tool to model the network of dependencies among variables. They provide a basis for causal discovery, and have been widely used in many fields, especially biology. Unfortunately, structure learning is quite non-trivial for DAG. One major difficulty is that some DAGs are unidentifiable with observational data only, and undirected edges cannot be resolved to directed edges.

The opportunity to apply interventions motivates interest in the smaller interventional Markov equivalence class. In this dissertation, we discuss how to modify the classic PC algorithm for causal discovery so that it can be used safely on interventional data. We introduce invariance relations on conditional distributions with different intervention targets that provide a powerful rule for edge orientation. There are several advantages of this rule: first, it does not require the Gaussian distribution assumption, instead a general structural equation model (SEM) of DAG is sufficient; second, it works for both (structural) intervention and soft intervention. Finally, we can merge some data blocks with different interventions for edge orientation.

A new constraint-based method is proposed to recover the interventional essential graph from the CPDAG (or called as observational essential graph) based on the invariance rule. We also establish consistency guarantees for both an interventional PC and an edge orientation algorithm under a sparse high-dimensional setting. Such high-dimensional consistency results are rarely seen in this area. It is also worthwhile to emphasize that the constraints on the family of interventions throughout this dissertation are mild. Finally, simulations are used to show the effectiveness of our method.

Transcription factors bind sequence-specific sites in DNA to regulate gene transcription. Identifying transcription factor binding sites (TFBS’s) is an important step for understanding gene regulation. Although sophisticated in modeling TFBS’s and their combinatorial patterns, computational methods for TFBS detection and motif finding often make oversimplified homogeneous model assumptions for background sequences. Since nucleotide base composition varies across genomic regions, it is expected to be helpful for motif finding to incorporate the heterogeneity into background modeling. When sequences from multiple species are utilized, variation in evolutionary conservation violates the common assumption of an identical conservation level in multiple alignments. To handle both types of heterogeneity, we propose a generative model in which a segmented Markov chain is used to partition a multiple alignment into regions of homogeneous nucleotide base composition and a hidden Markov model (HMM) is employed to account for different conservation levels. Bayesian inference on the model is developed via Gibbs sampling with dynamic programming recursions. Simulation studies and empirical evidence from biological data sets reveal the dramatic effect of background modeling on motif finding, and demonstrate that the proposed approach is able to achieve substantial improvements over commonly used background models.

Bayesian networks, with structure given by a directed acyclic graph (DAG), are a popular class of graphical models. In this dissertation, we develop two structure learning methods for Bayesian networks.

First we propose a score-based algorithm for discrete data that can incorporate experimental intervention for causal learning. Learning Bayesian networks from discrete or categorical data is particularly challenging, due to the large parameter space and the difficulty in searching for a sparse structure. In this thesis, we develop a maximum penalized likelihood method to tackle this problem. Instead of the commonly used multinomial distribution, we model the conditional distribution of a node given its parents by multi-logit regression, in which an edge is parameterized by a set of coefficient vectors with dummy variables encoding the levels of a node. To obtain a sparse DAG, a group norm penalty is employed, and a blockwise coordinate descent algorithm is developed to maximize the penalized likelihood subject to the acyclicity constraint of a DAG. When interventional data are available, our method constructs a causal network, in which a directed edge represents a causal relation. We apply our method to various simulated and real data sets. The results show that our method is very competitive, compared to many existing methods, in DAG estimation from both interventional and high-dimensional observational data. In addition, an R package discretecdAlgorithm for this algorithm is published on CRAN, with a user friendly interface.

The second method we propose is a framework for fast learning of extremely large Bayesian networks.

The size of the DAG space grows super-exponentially in the number of nodes, and it is challenging to develop an efficient structure learning method for massive networks. We propose a three step divide-and-conquer framework for Gaussian observational data:

1) partition the full DAG into several sub-networks (P-step),

2) estimate each disconnected sub-network individually (E-setp),

3) fuse sub-networks by adding edges between them (F-step).

To partition the full DAG into some sub-networks, a modified hierarchical clustering with average linkage is proposed to automatically choose the number of sub-networks. For the estimation step, users can apply any proper structure learning algorithm; in our implementation we choose to use the Concave penalized Coordinate Descent with reparameterization (CCDr) algorithm (Aragam and Zhou, 2015) as an example. Finally we develop a hybrid fusion step that uses both conditional independence tests and a penalized likelihood scoring function to recover the structure of the full DAG. The simulation results show that our three step framework had significant improvement in both speed and accuracy compared to estimating the DAG as a whole with the algorithm used in the second estimation step.

In this study, we analyzed the TF ChIP-Seq data of 105 (i.e., 15 choose 2) pairs. Each pair is based on two TF and three binding-dependent (BD) sequence datasets. The BD were generated from the two TF ChIP-Seq datasets in each pair. That is, the three scenario datasets are containing TFBS sequences of type 1, 2 or both (i.e., 1 and 2) TF.

The objective is to identify motif 1, 2 or even both (i.e., interactive motifs) by contrasting two of the three BD datasets at a time by using the contrast-motif-finder (CMF) algorithm. Each of the CMF's output not only provides estimated consensus motifs based on its full name PWM but also provides likelihood ratios (LRs) as a measure of the enrichment of an identified motif. Using this idea, we

construct a dataset where the first column lists the locations of identified enriched motif in the genome, column 2 to n+1 contains the estimated consensus motifs and the last column shows a binary (i.e., 0/1) of which set it is from and n is the number of consensus motifs.

Once these datasets are obtained, we use statistical model such as logistics regression, support vector machine (SVM) and classification tree models to determine their performance (i.e., error rates) and selection power. We have shownthat the SVM Radial kernel seems to have the best performance when using all the motifs in the dataset whereas classification tree selects the fewest motifs in almost every analyzed datasets but at the same time, the error rates and selection power do not drop as much. As a result, we believe the classification tree model is a better model since it not only provides a competitive predictive power with simpler models but also takes far less computational time than the other two models.

Diagnostic Classification Models (DCM) are a class of models popular in the social sciences such as Education that can be used to measure the unobservable attributes of students over multiple time points. Hidden Markov Diagnostic Classification Models (HMDCM) are an implementation of DCMs using Hidden Markov Models (HMM). The HMDCM from Yamaguchi and Martinez (2021) is a special case of HMMs where the observable and hidden states are assumed to be binary, and the models for the transition probabilities, emissions probabilities, and initial hidden state probabilities are Multinomial-Dirichlet Bayesian models. In this paper, we propose a modification to this HMDCM and introduce a Bayesian Logistic Regression Model for the emissions probabilities, where the independent variables of the logistic regression model are the hidden states and observable covariates. Our model requires that all time points have the same items, but our model omits the need for a design matrix. We adapt the VB algorithm from Yamaguchi and Martinez (2021) for this modified model, and we show, using simulations, that the algorithm can recover the true parameters with reasonable accuracy.

Recent development in ChIP-Seq technology has generated binding data for many transcription factors (TFs) in various cell types and cellular conditions. This opens great opportunities for studying combinatorial binding patterns among a set of TFs active in a particular cellular condition, which is a key component for understanding the interaction between TFs in gene regulation. As a first step to the identification of combinatorial binding patterns, we develop statistical methods to detect clustering and ordering patterns among binding sites (BSs) of a pair of TFs. Testing procedures based on Ripley's K-function and its generalizations are developed to identify binding patterns from large collections of BSs in ChIP-Seq data. We have applied our methods to the ChIP-Seq data of 91 pairs of TFs in mouse embryonic stem cells (ESCs) and 210 pairs in human ESCs. Our methods have detected clustering binding patterns between most TF pairs, which is consistent with the findings in the literature, and have identified significant ordering preferences, relative to the direction of target gene transcription, among the BSs of seven TFs in mouse ESCs . More interestingly, our results demonstrate that the identified clustering and ordering binding patterns between TFs are associated with the expression of the target genes. Lastly we study the effects of binding of multiple TFs and histone modifications (HMs) on the clustering patterns of the TF pairs in mouse and human ESCs using multiple linear regression analysis. The results suggest that the role of TFs or HMs that are significant in regression models are consistent to the clustering patterns of different pairs of TFs. These findings provide new insights into the co-regulation among TFs and HMs.