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# Graduate Certificate Programs

Graduate Certificate Programs

The Department currently offers graduate certificates in Applied Statistics and Stochastic Systems. Each program consists of four courses, including one elective chosen with the consent of the departmental advisor. Most courses may be used toward a master's degree, as well as for the certificate. The required courses and list of approved electives are included below. Admission requirements are the same as for the master's programs.

Applied Statistics

Required Courses:

- MA 552 Axiomatic Linear Algebra
Fields and vector spaces; subspaces and quotient spaces; basis and dimension; linear transformations and matrices; determinants; and the theory of a single linear transformation.

- MA 611 Probability
Foundations of probability, random variables and their distributions, discrete and continuous random variables, independence, expectation and conditioning, generating functions, multivariate distributions, convergence of random variables, and classical limit theorems.

- MA 612 Mathematical Statistics
Point estimation, method of moments, maximum likelihood, and properties of point estimators; confidence intervals and hypothesis testing; sufficiency; Neyman-Pearson theorem, uniformly most powerful tests, and likelihood ratio tests; and Fisher information and the Cramer-Rao inequality. Additional topics may include nonparametric statistics, decision theory, and linear models.

Typical electives:

- CE 679 Regression and Stochastic Methods
An introduction to the applied nonlinear regression, multiple regression and time-series methods for modeling civil and environmental engineering processes. Topics include: coefficient estimation of linear and nonlinear models; construction of multivariate transfer function models; modeling of linear and nonlinear systems; forecast and prediction using multiple regression and time series models; statistical quality control techniques; ANOVA tables and analysis of model residuals. Applications include monitoring and control of wastewater treatment plants, hydrologic-climatic histories of watercourses, and curve-fitting of experimental and field data.

- MA 641 Time Series Analysis I
Scope and applications of time series analysis: process control, financial data analysis and forecasting, and signal processing. Exploratory data analysis: graphical analysis, trend and seasonality detection and removal, and moving-average filtering. Review of basic statistical concepts related to the characterization of stationary processes. ARMA models and prediction of stationary processes. Estimation of ARMA models and model building and forecasting with ARMA models. Spectral analysis: periodogram testing for seasonality and periodicities and the maximum entropy and maximum-likelihood estimators. Asymptotic convergence. Selected topics, such as multivariate time series, nonlinear models, Kalman filtering, econometric forecasting, and long-memory processes. Selected applications, such as the unit-root problem in economics, forecasting and testing for market efficiency in financial time series, process control, and quality control.

- MGT 718 Multivariate Analysis
Experimental design, statistical estimation, and hypothesis testing from multivariate distributions. Topics covered will include regression models, multivariate analysis of variance, canonical correlations, classification procedures, and factor analysis. Computer applications of these techniques will be examined.

- MGT 730 Design and Analysis of Experiments
This course starts with the design and analysis of one factor analysis of variance. Methods of testing specific questions using planned comparisons are stressed. Models with two or more factors are considered with detailed instruction on the analysis of interactions. Repeated-measures designs are also covered, as well as designs with random and fixed factors.

Stochastic Systems

Required courses:

- MA 611 Probability
Foundations of probability, random variables and their distributions, discrete and continuous random variables, independence, expectation and conditioning, generating functions, multivariate distributions, convergence of random variables, and classical limit theorems.

- MA 623 Stochastic Processes
Random walks and Markov chains; Brownian motions and Markov processes; and applications, stationary (wide sense) processes, infinite divisibility, and spectral decomposition.

- MA 629 Nonlinear Optimization
This course introduces the students to the foundation of optimization. The first part of the class focuses on basic results of convex analysis and their application to the development of necessary and sufficient conditions of optimality and Lagrangian duality theory. The main numerical methods of optimization and their convergence constitute the second portion of the class. Along with the theoretical results and methods, examples of optimization models in probability, statistics, and approximation theory will be discussed as well as some basic models from management, finance, and other practical situations will be introduced in order to illustrate the discussed notions and phenomena, and to demonstrate the scope of applications. Linear optimization techniques will be treated as a special case. Some attention will be paid to using optimization software such as AMPL and CPLEX in the numerical assignments.

Choose one elective:

- MA 612 Mathematical Statistics
Point estimation, method of moments, maximum likelihood, and properties of point estimators; confidence intervals and hypothesis testing; sufficiency; Neyman-Pearson theorem, uniformly most powerful tests, and likelihood ratio tests; and Fisher information and the Cramer-Rao inequality. Additional topics may include nonparametric statistics, decision theory, and linear models.

- MA 630 Advanced Optimization Methods
This course introduces the students to the several advanced topics in the theory and methods of optimization. The first portion of the class focuses on subgradient calculus for non-smooth convex functions, optimality conditions for non-smooth optimization problems, conjugate and Lagrangian convex duality. The second part of the class discusses numerical methods for non-smooth optimization as well as approaches to large-scale optimization problems. The latter include decomposition methods, design of distributed and parallel methods of optimization, as well as stochastic approximation methods. Along with the theoretical results and methods, examples of optimization models in statistical learning and data mining, compressed sensing and image reconstruction will be discussed in order to illustrate the challenges and the phenomena, and to demonstrate the scope of applications. Some attention will be paid to using optimization software such as AMPL, CPLEX and SNOPT in the numerical assignments.

- MA 661 Dynamic Programming & Stochastic Optimal Control
The main purpose of this course is to present the foundations of the stochastic control theory, the corresponding numerical methods, and some applications. The focus will be on the idea of dynamic programming which will be developed starting from deterministic models, through finite-horizon stochastic problems, to infinite-horizon stochastic problems of various types. Applications to queuing systems, network design, and routing; supply-chain management and others will be discussed in detail. Topics to be covered: basic concepts of control theory for stochastic dynamic systems; controlled Markov chains; dynamic programming for finite horizon problems; infinite horizon discounted problems; numerical methods for infinite horizon problems; linear stochastic dynamic systems in discrete time; tracking and Kalman filtering; linear quadratic models; controlled Markov processes in continuous time; and elements of stochastic control theory in continuous time and state space.

- MA 662 Stochastic Programming
This course introduces students to modeling and numerical techniques for optimization under uncertainty and risk. Topics include: generalized concavity of measures, optimization problems with probabilistic constraints (convexity, differentiability, optimality, and duality), numerical methods for solving problems with probabilistic constraints, two-stage and multi-stage models (structure, optimality, duality), decomposition methods for two-stage and multi-stage models, risk averse optimization models,

Contact

Prof. Doug Bauer
*Graduate Chair*

[email protected]

Ms. Linda Habermann
*Administrative Assistant*

[email protected]