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.  By permission of instructor.

• MA 629 Convex Analysis and Optimization

  The objective of this course is to introduce the students to the basic results of convex analysis and optimization. The properties of nonlinear non-smooth optimization models will be analyzed. Topics include: separation and representation of convex sets, properties of convex functions, subgradients, optimality conditions, saddle points, constraint qualifications, Fenchel and Lagrange duality, and sensitivity analysis.  Examples of optimization models from probability, statistics, and approximation theory will be discussed, as well as some basic models from management, finance, telecommunications, and other fields.

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 Numerical Methods of Optimization

  The objective of this course is to introduce the students to the most popular numerical methods for solving nonlinear and non-smooth optimization problems. The techniques will be based on the properties of nonlinear non-smooth optimization models and optimality conditions. Linear optimization techniques will be treated as a special case. Some emphasis will be put on using optimization software. Examples using AMPL and CPLEX will be demonstrated in class. Topics include line search, non-derivative methods, basic decent methods, conjugate gradient methods, subgradient methods, Newton methods, projection methods, penalty, barrier, interior point methods, Lagrangian methods, bundle methods, trust-region method, numerical treatment of non-convex models, and decomposition methods.

• 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,