Advances in Dynamic Modeling and Computation for Count Data

A class of multi-process models is developed for collections of time indexed count data. Autocorrelation in counts is achieved with dynamic models for the natural parameter of the binomial distribution. In addition to modeling binomial time series, the framework includes dynamic models for multinomial and Poisson time series. Markov chain Monte Carlo (MCMC) and Po ́lya-Gamma data augmentation (Polson et al., 2013) are critical for fitting multi-process models of counts. To facilitate computation when the counts are high, a Gaussian approximation to the P ́olya- Gamma random variable is developed.

Three applied analyses are presented to explore the utility and versatility of the framework. The first analysis develops a model for complex dynamic behavior of themes in collections of text documents. Documents are modeled as a “bag of words”, and the multinomial distribution is used to characterize uncertainty in the vocabulary terms appearing in each document. State-space models for the natural parameters of the multinomial distribution induce autocorrelation in themes and their proportional representation in the corpus over time.

The second analysis develops a dynamic mixed membership model for Poisson counts. The model is applied to a collection of time series which record neuron level firing patterns in rhesus monkeys. The monkey is exposed to two sounds simultaneously, and Gaussian processes are used to smoothly model the time-varying rate at which the neuron’s firing pattern fluctuates between features associated with each sound in isolation.

The third analysis presents a switching dynamic generalized linear model for the time-varying home run totals of professional baseball players. The model endows each player with an age specific latent natural ability class and a performance enhancing drug (PED) use indicator. As players age, they randomly transition through a sequence of ability classes in a manner consistent with traditional aging patterns. When the performance of the player significantly deviates from the expected aging pattern, he is identified as a player whose performance is consistent with PED use.

All three models provide a mechanism for sharing information across related series locally in time. The models are fit with variations on the P ́olya-Gamma Gibbs sampler, MCMC convergence diagnostics are developed, and reproducible inference is emphasized throughout the dissertation.

Bayesian Emulation for Sequential Modeling, Inference and Decision Analysis

The advances in three related areas of state-space modeling, sequential Bayesian learning, and decision analysis are addressed, with the statistical challenges of scalability and associated dynamic sparsity. The key theme that ties the three areas is Bayesian model emulation: solving challenging analysis/computational problems using creative model emulators. This idea defines theoretical and applied advances in non-linear, non-Gaussian state-space modeling, dynamic sparsity, decision analysis and statistical computation, across linked contexts of multivariate time series and dynamic networks studies. Examples and applications in financial time series and portfolio analysis, macroeconomics and internet studies from computational advertising demonstrate the utility of the core methodological innovations.

Chapter 1 summarizes the three areas/problems and the key idea of emulating in those areas. Chapter 2 discusses the sequential analysis of latent threshold models with use of emulating models that allows for analytical filtering to enhance the efficiency of posterior sampling. Chapter 3 examines the emulator model in decision analysis, or the synthetic model, that is equivalent to the loss function in the original minimization problem, and shows its performance in the context of sequential portfolio optimization. Chapter 4 describes the method for modeling the steaming data of counts observed on a large network that relies on emulating the whole, dependent network model by independent, conjugate sub-models customized to each set of flow. Chapter 5 reviews those advances and makes the concluding remarks.

Predicting Transcript Production Rates in Yeast With Sparse Linear Models

To provide biological insights into transcriptional regulation, a couple of groups have recently presented models relating the promoter DNA-bound transcription factors (TFs) to downstream gene’s mean transcript level or transcript production rates over time. However, transcript production is dynamic in response to changes of TF concentrations over time. Also, TFs are not the only factors binding to promoters; other DNA binding factors (DBFs) bind as well, especially nucleosomes, resulting in competition between DBFs for binding at same genomic location. Additionally, not only TFs, but also some other elements regulate transcription. Within core promoter, various regulatory elements influence RNAPII recruitment, PIC formation, RNAPII searching for TSS, and RNAPII initiating transcription. Moreover, it is proposed that downstream from TSS, nucleosomes resist RNAPII elongation.

Here, we provide a machine learning framework to predict transcript production rates from DNA sequences. We applied this framework in the S. cerevisiae yeast for two scenarios: a) to predict the dynamic transcript production rate during the cell cycle for native promoters; b) to predict the mean transcript production rate over time for synthetic promoters. As far as we know, our framework is the first successful attempt to have a model that can predict dynamic transcript production rates from DNA sequences only: with cell cycle data set, we got Pearson correlation coefficient Cp = 0.751 and coefficient of determination r2 = 0.564 on test set for predicting dynamic transcript production rate over time. Also, for DREAM6 Gene Promoter Expression Prediction challenge, our fitted model outperformed all participant teams, best of all teams, and a model combining best team’s k-mer based sequence features and another paper’s biologically mechanistic features, in terms of all scoring metrics.

Moreover, our framework shows its capability of identifying generalizable fea- tures by interpreting the highly predictive models, and thereby provide support for associated hypothesized mechanisms about transcriptional regulation. With the learned sparse linear models, we got results supporting the following biological insights: a) TFs govern the probability of RNAPII recruitment and initiation possibly through interactions with PIC components and transcription cofactors; b) the core promoter amplifies the transcript production probably by influencing PIC formation, RNAPII recruitment, DNA melting, RNAPII searching for and selecting TSS, releasing RNAPII from general transcription factors, and thereby initiation; c) there is strong transcriptional synergy between TFs and core promoter elements; d) the regulatory elements within core promoter region are more than TATA box and nucleosome free region, suggesting the existence of still unidentified TAF-dependent and cofactor-dependent core promoter elements in yeast S. cerevisiae; e) nucleosome occupancy is helpful for representing +1 and -1 nucleosomes’ regulatory roles on transcription.

Distributed Feature Selection in Large n and Large p Regression Problems

Fitting statistical models is computationally challenging when the sample size or the dimension of the dataset is huge. An attractive approach for down-scaling the problem size is to first partition the dataset into subsets and then fit using distributed algorithms. The dataset can be partitioned either horizontally (in the sample space) or vertically (in the feature space), and the challenge arise in defining an algorithm with low communication, theoretical guarantees and excellent practical performance in general settings. For sample space partitioning, I propose a MEdian Selection Subset AGgregation Estimator ({\em message}) algorithm for solving these issues. The algorithm applies feature selection in parallel for each subset using regularized regression or Bayesian variable selection method, calculates the `median' feature inclusion index, estimates coefficients for the selected features in parallel for each subset, and then averages these estimates. The algorithm is simple, involves very minimal communication, scales efficiently in sample size, and has theoretical guarantees. I provide extensive experiments to show excellent performance in feature selection, estimation, prediction, and computation time relative to usual competitors.

While sample space partitioning is useful in handling datasets with large sample size, feature space partitioning is more effective when the data dimension is high. Existing methods for partitioning features, however, are either vulnerable to high correlations or inefficient in reducing the model dimension. In the thesis, I propose a new embarrassingly parallel framework named {\em DECO} for distributed variable selection and parameter estimation. In {\em DECO}, variables are first partitioned and allocated to m distributed workers. The decorrelated subset data within each worker are then fitted via any algorithm designed for high-dimensional problems. We show that by incorporating the decorrelation step, DECO can achieve consistent variable selection and parameter estimation on each subset with (almost) no assumptions. In addition, the convergence rate is nearly minimax optimal for both sparse and weakly sparse models and does NOT depend on the partition number m. Extensive numerical experiments are provided to illustrate the performance of the new framework.

For datasets with both large sample sizes and high dimensionality, I propose a new "divided-and-conquer" framework {\em DEME} (DECO-message) by leveraging both the {\em DECO} and the {\em message} algorithm. The new framework first partitions the dataset in the sample space into row cubes using {\em message} and then partition the feature space of the cubes using {\em DECO}. This procedure is equivalent to partitioning the original data matrix into multiple small blocks, each with a feasible size that can be stored and fitted in a computer in parallel. The results are then synthezied via the {\em DECO} and {\em message} algorithm in a reverse order to produce the final output. The whole framework is extremely scalable.