Course Descriptions
MATH 5051: Probability Theory & Application
Prereq: None
Description: This is a MS level introduction to probability theory. Topics include probability measures, independence and conditional probability, discrete and continuous random variables and their properties, joint distributions, moment generating functions, notions of convergence, Laws of Large Numbers, and the Central Limit Theorem. A working knowledge of multiple integrals and partial derivatives is essential for this course. Some previous exposure to elementary probability and statistics, at least at the level of Math 1040, is recommended. This course is not based on measure theory.
MATH 5052: Deterministic Math Models
Prereq: None
Description: This is a MS level course in basic applied mathematical modeling with emphasis on derivation and analysis of models via linear algebra and differential equation methods. Topics in linear algebra include eigenvalues and eigenvectors, matrix decompositions such as singular value decomposition and their applications. Students develop modeling skills through example problems from various fields of applications which are interpreted into problems of differential equations. Some differential equations will be solved. Methods for analyzing/visualizing differential equations’ solution behaviors will be provided, with emphasis on their significance on the applications.
MATH 5060: Survey Sampling
Prereq: None
Description: This is a MS level course that covers design and analysis of sample surveys. Sample designs include simple random sampling, systematic, stratified, cluster, double, and multistage sampling. Analytical methods include sample size determination, ratio and regression estimation, imputation for missing data, and nonsampling error adjustment.
MATH 5061: Design of Experiments
Prereq: None
Description: This MS level course emphasizes applied and practical aspects of experimental design and analysis. Both design and analysis of experiments as well as software implementation (SAS and R) are discussed. The class mainly covers: Randomization, Analysis of Variance (ANOVA), Completely Randomized Designs, Multiple Comparisons and Contrasts, Power and Sample Size, Factorial Treatment Structure, Mixed Effects Models, Split-Plot Design (Cross-Over Design and Repeated Measures) and Analysis of Covariance (ANCOVA). Applications refer to clinical trials, dose-response modeling, bioequivalence assessment and quality control problems among others.
MATH 5070: Intro: Non-Parametric Stats
Prereq: MATH 5151
Description: This MS level course provides a survey of nonparametric and rank based methods for data analysis. Frequently, common assumptions about the distributional form of data, required for many hypothesis testing or model building techniques, are violated. Ignoring these violations can lead to misleading or incorrect conclusions. Nonparametric and rank based methods provide statistically valid methods of analysis while requiring minimal assumptions about the data to be analyzed. Some topics covered will include one, two, and K-sample tests of hypotheses including, but not limited to, permutation tests, Wilcoxon Rank-Sum test, Mann-Whitney test, and the Kruskal Wallis test. Further topics including paired comparisons, blocked design, multivariate tests, and nonparametric bootstrap methods, among others, will also be covered. Students should expect a mix of theory and application.
MATH 5151: Statistical Inference & Modeling (Stat Inference & Modeling)
Prereq: MATH 5051
Description: This is a MS level course.This course provides a rigorous introduction to the theory and applications of statistical inference. Topics to be covered include methods of estimation (optimization, EM algorithm, and MCMC), properties of point estimators, interval estimators, and likelihood-based tests (score, Wald, and LRT). The course will also explore application of these concepts in various statistical models.
MATH 5152: Numerical Methods For Data Science (Numerical Methods For Data Sci)
Prereqs: MATH 5052 & MATH 5200; familiarity with programming and basic data science concepts would be beneficial but not mandatory.
Description: This MS level course provides a deep dive into the numerical techniques and algorithms that form the foundation of modern data science. Students will explore the mathematical principles and computational methods essential for solving large-scale, data-driven problems in science, engineering, and beyond. Key topics include (1) numerical linear algebra and matrix computations, critical for handling high-dimensional datasets; (2) optimization techniques for model training and tuning; (3) numerical solutions to differential equations in predictive modeling; (4) probabilistic and statistical numerical methods for uncertainty quantification; (5) applications in machine learning, big data analytics, and scientific computing. Hands-on programming exercises using Python or R will be integrated throughout the course, allowing students to implement and experiment with algorithms on real-world datasets. By the end of the course, participants will gain a robust understanding of numerical methods, enabling them to tackle complex problems in data science with precision and efficiency.
MATH 5200: Computing Using R & Python
Prereq: None
Description: The goal of this MS level course is to provide students with a programming background sufficient for graduate level study in mathematics and statistics. The course gives an introduction to R and Python. R is widely used by practicing statisticians and data scientists. This portion of the course will be structured around statistical methods and examples will be worked out using both computing environments. Statistical topics to be covered include data management, simulation, descriptive statistics, graphical displays, hypothesis testing, correlation, regression models, and simple multivariate analysis methods. The introduction to Python will cover the basic structure of the language, commands, scripts and graphing.
MATH 5210: Cloud Computing
Prereqs: MATH 5051 AND MATH 5200 or Equivalents
Description: The topics covered in this MS level course are all concerned with analytics and computation involving data sets which may be very large, e.g. big data. The term analytics will refer to modeling and computation within specific mathematical frameworks such as large matrices and other definite file types, and tools that enable measuring, parsing, understanding and visualizing the data. The term big data refers to an analytical context that requires a distributed computing framework for effective processing. The course goals are to learn about advanced computational tools and methods utilized in the data science field. We will learn about how to use and administer cloud computing resources, useful utilities in the linux environment, and the AWS suite of cloud services. Among the AWS services we will focus on general use services (e.g. EC2, S3), but also the Hadoop distributed computing ecosystem (as implemented in AWS EMR), the map reduce framework, and the application of these concepts to working with and understanding large datasets. Prospective students should study the prerequisites listed below that include knowledge of the Linux command line language and some use of Python.
MATH 5310: Deep Learning
Prereqs: MATH 5051 AND MATH 5200 or Equivalents
Description: This MS level course is an introduction to Neural Networks and Deep Learning. This is a first course on the mathematical foundations of neural networks with practical applications in R or Python. This course will begin with review of preliminaries in linear algebra, machine learning, and numerical computation. Then this course will move on to several neural network architectures such as feed forward networks, recurrent neural networks (RNN), convolutional neural networks (CNN), long short-term memory (LSTM) networks, autoencoders, generative adversarial networks (GAN), and transformers. Students will learn the mathematics of these architectures, their real-world applications, optimization techniques, and implementations in R or Python, depending on the student’s choice.
MATH 5320: Supervised Statistical Learning (Supervised Stat Learning)
Prereq: None
Description: This is a MS level course. Machine or statistical learning is concerned with algorithms that automatically improve their performance through experience and active feedback. This course covers topics such as neural networks, probabilistic networks, statistical learning methods (logistic regression, decision trees, random forests, among others), and reinforcement learning. This course is designed to provide a solid mathematical and statistical background in the theory and applications of supervised modeling and algorithms.
MATH 5330: Data Mining
Prereq: MATH 5151
Description: This MS level course presents an introduction to computational and statistical methods for exploring large data sets and discovering patterns in them. Visualization and other exploratory methods will be used throughout the course. The course will integrate data science theory, application and hands-on demo to guide students explore big data with different formats. The course surveys methods in predictive modeling (classification) including tree-based models, neural network, nearest neighbor methods, support vector machine, and etc. In the process, we will study discretization, data normalization and attribute selection as well as sampling methods, model evaluation, validation, bagging and boosting. Other topics will include cluster analysis, anomaly detection, and popular data mining applications in text mining and sequence mining. For all topics studied, students will work with various real and synthetic data sets to see the impact of different distributions on the performance of the algorithms. A variety of performance metrics will also be studied. This course will integrate dynamic elements in lectures and promote interactions with students in class, hands-on programming practice, student presentations and group projects. The software R and R-Studio will be used in the course. Basic knowledge in programming code in R is helpful, though there will be an introductory/refresh session for using basic R codes.
MATH 5340: Social Network Analysis
Prereq: None
Description: This MS level course will cover the mathematical concepts used in Social Network Analysis (SNA), in particular those drawn from graph theory and linear algebra. The primary focus of the applications of these methods is the analysis of relational data measured on groups of social agents or graph nodes. Topics to be discussed include graph theory, link analysis, centrality measures, estimation, sampling, large-scale analysis, functional granulation, visualization of network data including issues of validity and representation, and diffusions on networks. After identifying an area of interest, students will prepare a Research Paper and Final Project that uses tools from network theory to quantify the structure of their system and provide a meaningful interpretation of their findings.
MATH 5410: Operations Research
Prereq: None
Description: This is a MS level course covering the foundations of operations research with emphasis on mathematical modeling, linear optimization, and metaheuristics. Linear optimization (or linear programming, LP) is a fundamental branch of optimization, with applications to several areas such as physical sciences, health care, manufacturing, logistics, computer science, and finance. This course will provide an integrated view of the theory, solution techniques and applications of LP. Metaheuristics are algorithms that can be used to find quality solutions to a variety of optimization problems, where analytic solutions are impractical.
MATH 5420: Financial Mathematics
Prereqs: – MATH 5051 & MATH 5052
Description: This is a MS level course on the mathematics of financial derivatives. It covers the modeling of underlying assets as Browning motions and the pricing of derivatives by the Black-Scholes analysis. Topics covered include present value, risk free rates, drift terms, asset price volatility, stochastic differential equations and applications both within finance and elsewhere, Merton Firm Value, partial differential equations (PDE) and risk analysis including value at risk. Both the PDE and binomial Black-Scholes pricing models are covered.
MATH 5500: Regression Models
Prereq: MATH 5051
Description: This MS level course will focus on the theory and application of regression methods for statistical modeling and data analysis. Emphasis will be in the following areas: simple and multiple regression, inference and prediction, model building and diagnostics, model selection and validation, penalized regression (ridge, LASSO, elastic net), and selected topics in non-Gaussian and non-linear models. Practical issues involved in implementation of these methods will be presented using statistical software R based on example problems from a wide range of applications.
MATH 5510: Generalized Linear Models (Generalized Lin Models)
Prereqs: MATH 5151 And/Or MATH 5500
Description: This is a MS level course. Generalized Linear Models (GLM) provide a unifying statistical framework for analyzing data with a response variable in the exponential family distribution. This course will focus on the theory and application of GLMs. It will cover commonly encountered GLMs, including a review of linear models for normally distributed responses, models for binary outcomes (logistic regression and probit models), models for multinomial outcomes (polytomous regression), models for ordinal responses (proportional odds model), models for count data (Poisson regression, negative binomial regression, zero-inflated models). The statistical software packages SAS and R will be used to illustrate the implementation of these models to data from a wide range of applications.
MATH 5520: Time Series
Prereqs: MATH-5151 Mathematical Statistics or equivalent; MATH-5200 is optional but highly encouraged (Note: basic familiarity with R is expected).
Description: This MS level course will focus on the theory and application of methods for time series data. This course discusses the modeling and forecasting of univariate time series. Course topics include exploratory methods, white noise and random walk models, smoothing techniques, regression methods for time series data, time series models for stationary and non-stationary data including autoregressive (AR), moving average (MA) and autoregressive moving average(ARMA) models as well as advanced models (ARCH, GARCH, etc.), and advanced topics in machine learning for time series and multivariate time series (vector autoregressive models). Applications to real-world data sets will be explored using the software package R. Examples are drawn from a variety of areas including finance, business, economics, public policy, health, environment and ecology.
MATH 5600: Bayesian Statistics
Prereq: MATH 5051
Description: This MS level course provides a practical introduction to Bayesian statistical methods. It assumes familiarity with concepts in probability theory and statistical inference, as well as some programming experience. Students will learn the fundamentals of Bayesian inference and will be exposed to Monte Carlo simulation methods. The first part of the course will focus on the specification of prior distributions, the evaluation of posterior and predictive distributions, and the theory of Bayesian estimation and hypothesis testing. The second part of the course will focus on Monte Carlo simulation with an emphasis on Markov chain Monte Carlo methods, including the Gibbs sampler and the Metropolis-Hastings algorithm. A variety of statistical models will be considered and illustrated with examples from a wide range of applications. The open source software R will be used to carry out Bayesian analysis.
MATH 5700: Numerical Optimization for AI and Data Science
Prereqs: – MATH 5200 & MATH 5152
Description: This is a MS level course. Optimization is central to many problems involving decision making in data science and artificial intelligence (AI), economics, finance, medical research, among others. This course delves into the numerical techniques and optimization algorithms that are the backbone of AI and data science. Students will learn to solve unconstrained and constrained optimization problems, focusing on applications in training machine learning models, hyperparameter tuning, and deep learning. Key topics covered include foundations of optimization in AI (loss function minimization, feature selection, and hyperparameter tuning); unconstrained optimization (gradient descent variants, stochastic methods, convergence and regularization techniques; constrained optimization (formulating and solving constrained problems using penalty methods and augmented Lagrangians, applied to resource allocation and fairness-aware ML), convex optimization (principles of convex functions and sets, subgradient methods). Advanced topics that will be covered in this course as time permits include optimization for deep learning (addressing challenges like non-convexity and vanishing gradients), probabilistic and Bayesian optimization (techniques for hyperparameter tuning, Gaussian processes, and automating ML pipelines), scaling optimization (Parallel and distributed methods and GPU-accelerated techniques for scalability). Through hands-on projects students will apply these methods to real-world challenges in computer vision, NLP, and predictive analytics. Familiarity with machine learning concepts is recommended but not mandatory.
MATH 5710: Unsupervised Learning
Prereqs: MATH 5151 & MATH 5200
Description: This MS level course covers a range of methods in unsupervised learning with a focus on latent feature inference and learning as well as generative models. We will start with classical methods such as PCA, then consider learning and inference for different statistical models, with an emphasis on connections to clustering and several examples of Bayes nets. In this context we’ll cover MCMC and variational approaches. Roughly the last third of the course will cover deep learning methods for unsupervised learning including autoencoders, variational autoencoders and, time permitting, generative diffusions and transformers. Coding in Python and R will be a central component of the course.
MATH 5925: Internship
Prereq: None
Description: Students who are simultaneously engaging in an internship in a private, non-profit, or governmental organization may enroll in MATH -5925 to gain credit. International students who would like to obtain CPT approval must enroll in MATH-5925. Qualifying internship opportunities must involve substantial application of the mathematical, statistical and computational skills acquired in the program courses for solving real world problems. All students in the Mathematics/Statistics (MAST) graduate program who are eligible to engage in an internship must receive Program Director approval to enroll in this course. International students must complete a full academic year in the program (Fall and Spring semesters) to be able to apply for CPT. Students must submit a proposal of the topics that they will be working on during the internship and a final oral and/or written presentation is required.