An Introduction to Stochastic Modeling" by Mark Pinsky, Samuel Karlin. Academic Press, Fourth Edition.
ISBN-10: 9780123814166
ISBN-13: 978-0123814166

We study the theory and applications of stochastic processes. Topics include discrete-time and continuous-time Markov chains, Poisson process, branching process, Brownian motion. Considerable emphasis will be given to applications and examples.

While lecture notes will serve as the main source of material for the course, the following book constitutes a great reference:
”An Introduction to Statistical Learning (with applications in R)” by James, Witten et al. ISBN: 978-1461471370
”Neural Networks with R” by G. Ciaburro. ISBN: 978-1788397872

Course will deal with theory and applications for such statistical learning techniques as linear and logistic regression, classification and regression trees, random forests, neural networks. Other topics might include: fit quality assessment, model validation, resampling methods. R Statistical programming will be used throughout the course.

Learning Objectives: By the end of the course a successful student should:

• Have a solid conceptual grasp on the described statistical learning methods.
• Be able to correctly identify the appropriate techniques to deal with particular data sets.
• Have a working knowledge of R programming software in order to apply those techniques and subse- quently assess the quality of fitted models.
• Demonstrate the ability to clearly communicate the results of applying selected statistical learning methods to the data.

Software: Make sure to download R and RStudio (which can’t be installed without R) before the course starts. Use the link https://www.rstudio.com/products/rstudio/download/ to download it from the mirror appropriate for your platform. Let me know via email in case you encounter difficulties.

Course Outline:
Introduction: What is Statistical Learning?

Supervised and unsupervised learning. Regression and classification.
Linear and Logistic Regression. Continuous response: simple and multiple linear regression. Binary response: logistic regression. Assessing quality of fit.
Model Validation. Validation set approach. Cross-validation.
Tree-based Models. Decision and regression trees: splitting algorithm, tree pruning. Random forests: bootstrap, bagging, random splitting.
Neural Networks. Single-layer perceptron: neuron model, learning weights. Multi-Layer Perceptron: backpropagation, multi-class discrimination

This first course in the sequence Math 4331-4332 provides a solid introduction to deeper properties of the real numbers, continuous functions, differentiability and integration needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333, including an introduction to the real numbers, basic properties of continuous and differentiable functions on the real line, and an ability to do epsilon-delta proofs.

Topics: Open and closed sets, compact and connected sets, convergence of sequences, Cauchy sequences and completeness, properties of continuous functions, fixed points and the contraction mapping principle, differentiation and integration.

MATH 3331 or equivalent, and three additional hours of 3000-4000 level Mathematics. Previous exposure to Partial Differential Equations (Math 3363) is recommended.

"Partial Differential Equations: An Introduction (second edition)," by Walter A. Strauss, published by Wiley, ISBN-13 978-0470-05456-7  

Description:Initial and boundary value problems, waves and diffusions, reflections, boundary values, Fourier series.

Instructor's Description: will cover the first 6 chapters of the textbook. See the departmental course description.

MATH 3349

• Applied Multivariate Statistical Analysis (6th Edition), Pearson. Richard A. Johnson, Dean W. Wichern. ISBN: 978-0131877153 (Required)
• An Introduction to Statistical Learning, with Applications in R. by G. James, D. Witten, T. Hastie and R. Tibshirani, Springer. ISBN: 978-1461471370 (Recommended)

Course Description: Multivariate analysis is a set of techniques used for analysis of data sets that contain more than one variable, and the techniques are especially valuable when working with correlated variables. The techniques provide a method for information extraction, regression, or classification. This includes applications of data sets using statistical software.

• Understand how to use R and R Markdown
• Understand matrix algebra using R
• Understand the geometry of a sample and random sampling
• Understand the properties of multivariate normal distribution
• Make inferences about a mean vector
• Compare several multivariate means
• Identify and interpret multivariate linear regression models

• Introduction to R Markdown, Review of R commands (Notes)
• Introduction to Multivariate Analysis (Ch.1)
• Matrix Algebra, R Matrix Commands (Ch.2)
• Sample Geometry and Random Sampling (Ch.3)
• Multivariate Normal Distribution (Ch.4)
• MANOVA (Ch.6)
• Multiple Regression (Ch.7)
• Logistic Regression (Notes)
• Classification (Ch.11)

MATH 3349

- Applied Multivariate Statistical Analysis (6th Edition), Pearson. Richard A. Johnson, Dean W. Wichern. ISBN: 978-0131877153 (Required)

- Using R With Multivariate Statistics (1st Edition). Schumacker, R. E. SAGE Publications. ISBN: 978-1483377964 (recommended)

Course Description: Multivariate analysis is a set of techniques used for analysis of data sets that contain more than one variable, and the techniques are especially valuable when working with correlated variables. The techniques provide a method for information extraction, regression, or classification. This includes applications of data sets using statistical software.

Course Objectives:

• Understand how to use R and R Markdown
• Understand matrix algebra using R
• Understand the geometry of a sample and random sampling
• Understand the properties of multivariate normal distribution
• Make inferences about a mean vector
• Compare several multivariate means
• Identify and interpret multivariate linear regression models

Course Topics:

• Introduction to R Markdown, Review of R commands (Notes)
• Introduction to Multivariate Analysis (Ch.1)
• Matrix Algebra, R Matrix Commands (Ch.2)
• Sample Geometry and Random Sampling (Ch.3)
• Multivariate Normal Distribution (Ch.4)
• MANOVA (Ch.6)
• Multiple Regression (Ch.7)
• Logistic Regression (Notes)
• Classification (Ch.11)

MATH 3331 and COSC 1410 or equivalent. (2017—2018 Catalog)

MATH 3331 or MATH 3321 or equivalent, and three additional hours of 3000-4000 level Mathematics (2018—2019 Catalog)

*Ability to do computer assignments in FORTRAN, C, Matlab, Pascal, Mathematica or Maple.

This is an one semester course which introduces core areas of numerical analysis and scientific computing along with basic themes such as solving nonlinear equations, interpolation and splines fitting, curve fitting, numerical differentiation and integration, initial value problems of ordinary differential equations, direct methods for solving linear systems of equations, and finite-difference approximation to a two-points boundary value problem. This is an introductory course and will be a mix of mathematics and computing.

MATH 2331, or equivalent, and six additional hours of 3000-4000 level Mathematics.

Conditioning and stability of linear systems, matrix factorizations, direct and iterative methods for solving linear systesm, computing eigenvalues and eigenvectors, introduction to linear and nonlinear optimization 

Catalog Description: Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors.

Instructor's Description: The course covers the following topics: vector spaces, subspaces, linear combinations,systems of linear equations, linear dependence and linear independence, bases and dimension,linear transformations, null spaces, ranges, matrix rank, matrix inverse and invertibility,determinants and their properties, eigenvalues and eigenvectors, diagonalizability.

Instructor Notes.

Recommended texts:

• “Beginning Number Theory” Edition: 2, by Neville Robbins. ISBN: 9780763737689; Jones & Bartlett Learning
• "Elementary Number Theory" Edition 7, by David Burton. ISBN: 9780073383149; McGraw-Hill Higher Education

Catalog Description: Divisibility theory, primes and their distribution, theory of congruences and application in security, integer representations, Fermat’s Little Theorem and Euler’s Theorem, primitive roots, quadratic reciprocity, and introduction to cryptography

Catalog Prerequisite: MATH 3333 or consent of instructor.

• • Required Textbook: Mathematical Methods in the Physical Sciences by Mary Boas, Third edition, Wiley. 9780471198260
  • Reference texts:
    • Mathematics for Physicists, by Susan M. Lea, ISBN: 9780534379971; Cengage Learning
    • Advanced Engineering Mathematics, by Greenberg, ISBN: 9780133214314, Pearson
    • Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence: ISBN: 9780521679718, Cambridge University Press
    • Mathematical Physics; A Modern Introduction to Its Foundations by Sadri Hassani; ISBN: 9780387985794, Springer
    • An Introduction to Mathematical Methods of Physics by Lorella Jones; ISBN: 9780805351309, Benjamin-Cummings Publishing Company

• • Software: You may use any mathematical assistant to check your work, but my recommendation is that you NEVER as a computer to do anything that you do not know how to do. You will have to do the work by hand on the examinations. UH recently purchased some licenses for Maple, and I recommend that you use it. You may purchase Maple at a 25% discount at the Maplesoft web store.

Course Content: 

• A. Vector algebra review and vector calculus: Coordinate system construction, nonorthogonal coordinate systems, covariant and contravariant vector components, rotations, vector operators (divergence, curl, gradient, and Laplacian), integral theorems (Stokes’ theorem and divergence theorem), line integrals and surface integrals, curvilinear coordinates, and dual vector spaces.

• B. Introduction to partial differential equations, boundary value problems, and special functions: Sturm-Liouville systems, boundary value problems, solutions to Laplace’s equation, wave equation, Helmholtz equation, and Schrödinger equation in Cartesian, cylindrical coordinate systems, and spherical coordinate systems, sines and cosine functions and Legendre polynomials.

• C. Advanced techniques for solving partial differential equations, Bessel functions, modified Bessel functions, Hankel functions, spherical harmonics, Green functions, and applications of these topics to basic physics.

Graduate standing.

Linear Algebra Using MATLAB, Selected material from the text Linear Algebra and Differential Equations Using Matlab by Martin Golubitsky and Michael Dellnitz)
The text  will made available to enrolled students free of charge.

Software: Scientific Note Book (SNB) 5.5  (available through MacKichan Software, http://www.mackichan.com/)

Syllabus: Chapter 1 (1.1, 1.3, 1.4), Chapter 2 (2.1-2.5), Chapter 3 (3.1-3.8), Chapter 4 (4.1-4.4), Chapter 5 (5.1-5.2, 5.4-5-6), Chapter 6 (6.1-6.4), Chapter 7 (7.1-7.4), Chapter 8 (8.1)

Project: Applications of linear algebra to demographics. To be completed by the end of the semester as part of the final.

Course Description: Solving Linear Systems of Equations, Linear Maps and Matrix Algebra, Determinants and Eigenvalues, Vector Spaces, Linear Maps, Orthogonality, Symmetric Matrices, Spectral Theorem

Students will also learn how to use the computer algebra portion of SNB for completing the project. 

Homework: Weekly assignments to be emailed as SNB file.

There will be two tests and a Final.

Grading: Tests count for 90% (25+25+40), HW 10%

TBD

TBD

• Applied Multivariate Statistical Analysis (6th Edition), Pearson. Richard A. Johnson, Dean W. Wichern. ISBN: 978-0131877153 (Required)
• An Introduction to Statistical Learning, with Applications in R. by G. James, D. Witten, T. Hastie and R. Tibshirani, Springer. ISBN: 978-1461471370 (Recommended)

Required Text: Abstract Algebra by David S. Dummit and Richard M. Foote, ISBN: 9780471433347

This book is encyclopedic with good examples and it is one of the few books that includes material for all of the four main topics we will cover: groups, rings, field, and modules.  While some students find it difficult to learn solely from this book, it does provide a nice resource to be used in parallel with class notes or other sources.

Catalog Prerequisite: Graduate standing, MATH 2331 and a minimum of 3 semester hours transformations, eigenvalues and eigenvectors.

Instructor's Prerequisite: MATH 2331, or equivalent, and a minimum of three semester hours of 3000-4000 level Mathematics.

Catalog Description: An expository paper or talk on a subject related to the course content is required.

Instructor's Description: The course covers the following topics: vector spaces, subspaces, linear combinations,systems of linear equations, linear dependence and linear independence, bases and dimension,linear transformations, null spaces, ranges, matrix rank, matrix inverse and invertibility,determinants and their properties, eigenvalues and eigenvectors, diagonalizability.

Graduate standing and MATH 3334.

In depth knowledge of Math 3325 and Math 3333 is required.

This first course in the sequence Math 4331-4332 provides a solid introduction to deeper properties of the real numbers, continuous functions, differentiability and integration needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333, including an introduction to the real numbers, basic properties of continuous and differentiable functions on the real line, and an ability to do epsilon-delta proofs.

Topics: Open and closed sets, compact and connected sets, convergence of sequences, Cauchy sequences and completeness, properties of continuous functions, fixed points and the contraction mapping principle, differentiation and integration.

Graduate Standing and MATH 4331.

Required: The instructor will provide notes on much of the material for the course and there is no prescribed text.

Recommended Texts: The course will cover the material in the first three chapters of ”Hilbert Space Methods in Partial Differential Equations” by Ralph E. Showalter (Dover or free online) and some of the text ”Elliptic Equations: An Introductory Course”,by Michel Chipot, Birkhauser, 2009. The Universitext ”Functional Analysis, Sobolev Spaces and Partial Differential Equations” by Haim Brezis, Springer 2011 provides a thorough treatment of the functional analysis used. These three texts may be good reference texts for the material treated in the course.

Course Outline: This course will provide an introduction to the modern theory of elliptic partial differential equations using Sobolev space methods. The prerequisite is competence in multivariable calculus and real analysis. Ideally a student should have done well in M6320- 21 and having a working knowledge of Lebesgue integration and some Fourier analysis. The basic constructions of linear analysis in Hilbert and Banach spaces will be assumed known.

The first lectures will provide an introduction to boundary value problems for elliptic PDEs and describe the types of results that will be obtained. In particular weak formulations of various boundary problems and the background to finite element modeling of the problems will be described. Some results about the formulations typically used for numerical simulations will be treated in the course.

To do this the calculus of weak derivatives and the associated Sobolev function spaces needed for weak formulations of boundary value problems will be studied. Various results needed for the existence and properties of solutions of partial differential equations will be proved.

There will be an emphasis on important examples from applications involving equations posed on bounded subsets of RN with 1 ≤ N ≤ 3.

(Required) Topology, A First Course, J. R. Munkres, Second Edition, Prentice-Hall Publishers.

link to text

Catalog Description: Point-set topology: compactness, connectedness, quotient spaces, separation properties, Tychonoff’s theorem, the Urysohn lemma, Tietze’s theorem, and the characterization of separable metric spaces

Instructor's Description: Topology is a foundational pillar supporting the study of advanced mathematics. It is an elegant subject with deep links to algebra and analysis. We will study general topology as well as elements of algebraic topology (the fundamental group and homology theories).

Though traditionally viewed as a pure subject, algebraic topology has experienced a renaissance in recent years with the emergence of applied algebraic topology. To wit, SIAM has recently launched a new journal on applied algebra and geometry.

Recommended text: Reading assignments will be a set of selected chapters extracted from the following reference text:

• Introduction to Statistical Learning w/Applications in R, by James , Witten, Hastie, Tibshirani (This book is freely available online). ISBN: 9781461471370
• "Neural Networks with R” by G. Ciaburro. ISBN: 978-1788397872

Summary: A typical task of Machine Learning is to automatically classify observed "cases" or "individuals" into one of several "classes", on the basis of a fixed and possibly large number of features describing each "case". Machine Learning Algorithms (MLAs) implement computationally intensive algorithmic exploration of large set of observed cases. In supervised learning, adequate classification of cases is known for many training cases, and the MLA goal is to generate an accurate Automatic Classification of any new case. In unsupervised learning, no known classification of cases is provided, and the MLA goal is Automatic Clustering, which partitions the set of all cases into disjoint categories (discovered by the MLA).

Numerous MLAs have been developed and applied to images and faces identification, speech understanding, handwriting recognition, texts classification, stock prices anticipation, biomedical data in proteomics and genomics, Web traffic monitoring, etc.

This MSDSfall 2019 course will successively study :

1) Quick Review (Linear Algebra) : multi dimensional vectors, scalar products, matrices,  matrix eigenvectors and eigenvalues, matrix  diagonalization, positive definite matrices

2) Dimension Reduction for  Data Features : Principal Components Analysis (PCA)

3) Automatic Clustering of Data Sets by K-means algorithmics

3) Quick Reviev (Empirical Statistics) : Histograms, Quantiles, Means, Covariance Matrices

4) Computation of Data Features Discriminative Power

5) Automatic Classification by Support Vector Machines (SVMs)  

Emphasis will be on concrete algorithmic implementation and  testing on  actual data sets, as well as on  understanding importants  concepts.

Required Text: ”Neural Networks with R” by G. Ciaburro. ISBN: 9781788397872

• Required: Probability with Applications in Engineering, Science, and Technology, by Matthew A. Carlton and Jay L. Devore, 2014.
• Recommended: Introductory Statistics in R, Peter Dalgaard, 2nd ed., Springer, 2008
• Recommended: Introduction to Probability Models by Sheldon Axler 11th edition
• Lecture Notes

Course Description: Probability, independence, Markov property, Law of Large Numbers, major discrete and continuous distributions, joint distributions and conditional probability, models of convergence, and computational techniques based on the above.

Topics Covered: 

• Probability spaces, random variables, axioms of probability.
• Combinatorial analysis (sampling with, without replacement etc)
• Independence and the Markov property. Markov chains- stochastic processes, Markov property, first step analysis, transition probability matrices. Longterm behavior of Markov chains: communicating classes, transience/recurrence, criteria for transience/recurrence, random walks on the integers.
• Distribution of a random variable, distribution functions, probability density function. Independence.
• Strong law of large numbers and the central limit theorem.
• Major discrete distributions- Bernoulli, Binomial, Poisson, Geometric. Modeling with the major discrete distributions.
• Important continuous distributions- Normal, Exponential. Beta and Gamma.
• Jointly distributed random variables, joint distribution function, joint probability density function, marginal distribution.
• Conditional probability- Bayes theorem. Discrete conditional distributions, continuous conditional distributions, conditional expectations and conditional probabilities. Applications of conditional probability.

• Make sure to download R and RStudio (which can't be installed without R) before the course starts. Use the link RStudio download to download it from the mirror appropriate for your platform.
• **New: Rstudio is in the cloud: RStudio.cloud.

J.K. Hunter and B. Nachtergaele, Applied Analysis, World Scientific, (2005). ISBN: 9789812705433

 A.W. Naylor and G.R. Sell, Linear Operator Theory in Engineering and Science,  Springer. ISBN: 9780387950013

Graduate standing and MATH 4331 and MATH 4377

Students are expected to have a good grounding in basic real analysis and linear algebra.

"Convex Optimization", Stephen Boyd, Lieven Vandenberghe, Cambridge University Press, ISBN: 9780521833783 (This text is available online. Speak to the instructor for more details)

Graduate Standing and must be in the MSDS Program.

Instructor prerequisites: The course is essentially self-contained. The necessary material from statistics and linear algebra is integrated into the course. Background in writing computer programs is preferred but not required.

• "Python for Data Analysis: Data Wrangling with Pandas, NumPy, and IPython", by Wes McKinney, 2 edition, 2017, O'Reilly. (PD) Paper Book. ISBN 13: 9781491957660. Available for free on Safari through UH library.

• "Python for Everybody (Exploring Data in Python3)",  by Dr. Charles Russell Severance, 2016, 1 edition, CreateSpace Independent Publishing Platform (PE) Paper Book. ISBN 13: 9781530051120

Instructor's Description: The course provides essential foundations of Python programming language for developing powerful and reusable data analysis models. The students will get hands-on training on writing programs to facilitate discoveries from data. The topics include data import/export, data types, control statements, functions, basic data processing, and data visualization.

Recommended Texts :
- A First Look at Rigorous Probability Theory, by Jeffrey Rosenthal, 2000. ISBN: 9789812703705
- An Intermediate Course in Probability Theory, by Allan Gut, Springer 2009 (any edition)

Review of Undergraduate Probability:
- A First Course in Probability, 6th Edit. by Sheldon Ross, 2002, Prentice Hall. ISBN: 9780321794772

Complementary Texts for further reading:
- Probability: Theory and Examples, 3rd Ed., Richard Durrett, Duxbury Press. ISBN: 9787506283403
- An Introduction to Probability Theory and Its Applications, Vol 1,  by William Feller. ISBN: 9780471257080
- Probability, by Leo Breiman, 1968, Addison-Wesley. ISBN: 9780898712964

General Background (A).
(1) Combinatorial analysis and axioms of probability
(2) Elementary random variables theory: expectation, variance, moments, distribution
function, probability density functions, impact of change of variable on density functions
(3) Major discrete probability distributions: Bernoulli, Binomial, Poisson, Geometric
Major continuous probability distributions: Uniform, Normal, Exponential
(4) Basic Modelling Applications
(5) Conditional probability: Bayes formula, Independence, Conditional Expectation,
Conditional density function, Conditional Probability distribution, Independent identically distributed random variables
(6) Joint distributions, joint density functions, marginal distributions, marginal densities, covariances and correlation coefficients
(7) Moment generating functions, Characteristic functions,

Measure theory (B).
(1) Elementary measure theory : Boolean algebras, probability spaces , continuity of probabilities, Borel-Cantelli lemma, Chebychevs inequality,
(2) Convergence of random variables: Almost sure convergence, Convergence in distribution, Law of Large Numbers, Central Limit theorem

Markov chains and random walks (C).
Markov chain theory for finite or countable state spaces
(1) Markov property and Transition matrix, Irreducibility
(2) First hitting times, Transience, Recurrence ,
(3) Stationary distributions : existence theorems and computation
(4) Random walks on Z and Z2 as Markov chains; Gambler's ruin problem

Introduction to Mathematical Finance: Discrete-time Models, by Stanley Pliska, Blackwell, 1997. ISBN: 9781557869456

Graduate standing.

 - "Banach spaces for Analysts", P. Wojtaszczyk

 - "Functional Analysis", by W. Rudin

Instructor's description: The course is designed to give an introduction to various concepts of mathematical analysis. The course is divided onto two parts. First parts covers classical functional analysis and spectral theory. In the second part we focus on some specific topics in analysis.

Course Content:

• 3.1 First Part The topics in functional analysis will include: Open Mappin Theorem, Closed Graph Theorem, Banach-Steinhaus Theorem, Hahn-Banach Theorem, Riesz-Thorin Theorem, Marcinkiewicz Theorem, Kintchine’s inequality, type and cotype inequalities, spectral theorem, functional calculus and other basic concepts in functional analysis.

• 3.2 Second Part In the second part we cover some topics in analysis. The following list contains some propositions of the topics. 1. Absolutely summing operators. 2. Bounded approximation property. 3. Bases in Banach spaces. 4. Expanders and non-linear spectral gap. 5. Coarse embeddings.

Graduate Standing and must be in the MSDS Program.

Much of the material is drawn from these works:

• "An Introduction to Statistical Learning: with Applications in R", by James, Witten, Hastie, and Tibshirani
• "Advances in Financial Machine Learning" by Lopez de Prado
• "The Econometrics of Financial Markets", by Campbell, Lo, and MacKinlay
• "Data Analysis: A Bayesian Tutorial", by Sivia

This is an applied data analysis course focusing on financial and economic data. We will cover various kinds of analyses common in the field and, as much as possible, use multiple approaches to each case in order to demonstrate the strengths, weaknesses, and advantages of each technique. This is not intended to be a programming course. There are many examples done in R and you are welcome to use that language. If you are, or aspire to be a strong Python programmer, you are welcome to use that language also. Proficiency in basic probability and linear algebra is assumed. By the end of the course you may find your skills in those areas strengthened as well.

The goals for the course are to familiarize students with common types of economic and financial data, some of the statistical properties of this kind of data which usually involves time series, and to equip everyone with a thorough enough understanding of the techniques available for them to make the best decision on the approach to take in an analysis depending on the nature of the data and the specific purpose of the study.

Catalog Description: 

Instructor's Description: This course will introduce the qualitative study of dynamical systems via historically and scientifically important examples such as the three-body problem, nonlinear oscillators, the Lorenz system, and the logistic map.

These examples illustrate how systems evolving in time according simple deterministic rules can exhibit complicated "chaotic" behavior.  Each of these systems depends on one or more parameters, and a central part of the course will be to study how the qualitative behavior of the system changes (or does not change) as the parameters vary. This will lead us to discuss topics such as KAM theory, phase locking, bifurcations, period-doubling cascades, renormalization, and parameter exclusion.

The topics to be covered in this course are largely distinct from the topics covered in previous edition of Math 7326 that I taught in Spring 2017, so students who took that class would benefit from taking this one as well.