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Course/Section

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The text will made available to enrolled students free of charge.

Software: Scientific Note Book (SNB) 5.5  (available through MacKichan Software, )

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.

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

Prerequisites: Graduate standing.

Instructor's prerequisite: Calculus 3 (multi-dimensional integrals), very minimal background in Probability. 

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.

Topics: The course introduces foundational ideas in real analysis, focusing on structure and behavior of functions on subsets of 鈩濃伩. Topics include:

• Open and closed sets, compactness, and convergence in 鈩濃伩
• Continuity, uniform continuity, and consequences on compact sets
• Differentiation and the Mean Value Theorem
• Riemann integration and the Fundamental Theorem of Calculus

Students will develop proof skills, explore counterexamples, and connect topological ideas with analytic results.

Text(s):

• Lawrence C. Evans, `Partial Differential Equations', Graduate studies in mathematics 19.2, 1998. (required)
• Robert McOwen, `Partial Differential Equations, Methods and Applications', 2nd Ed., 2004. (optional)

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鈥� 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.

Text(s):

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

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.

Software Used:

• Make sure to download (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:

1. Review of metric spaces, completeness, characterization of compactness, extreme value theorem.
2. Contraction mappings and fixed points. Applications of contractions mappings: integral equations, solutions to initial value problems. Local existence and uniqueness of solutions, stability.
3. Lp spaces as metric completions. Extending the Riemann integral to Lp spaces. Banach spaces.
4. Dual spaces. Uniform boundedness.
5. Consequences of uniform boundedness for Fourier series and polynomial interpolation.
6. Uniform convexity, best approximation property and duality for Lp-spaces. Bounded inverse, closed graph theorem.
7. Hilbert spaces. Orthonormal bases and their characterization. Characterization of best approximation by orthogonal projection. Fourier series.
8. Convergence in L2 and pointwise convergence. Weak convergence.
9. Nonlinear best approximations and (approximate) sparsity.
10. Relationships between weak and norm convergence. Weak compactness in Hilbert spaces. Linear and convex programming in Hilbert spaces.
11. Operators and bilinear forms. The Lax-Milgram theorem.
12. Linear inverse problems. Sparse recovery by norm minimization.The
13. Hilbert-Schmidt norm and Hilbert-Schmidt operators. Compact self-adjoint operators. The spectral theorem for compact, self-adjoint operators.
14. Diagonalizing normal operators. Solutions to Schrodinger's eigenvalue problem and compact integral operators.
    Introduction to the Calculus of Variations.
15. Other topics in coordination with faculty.

Prerequisites: 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.

Text(s):

• "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

Free online copy:  

Text(s):

• Required text book for some homework and reference: All of Statistics: A concise course in statistical inference. Larry Wasserman. Springer. 

• Other suggested reading material:
  • Shao. Mathematical Statistics (second edition).
  • Mathematical Statistics: Basic Ideas and Selected Topics, Volume I and II, Bickel and Doksum
  • P. MuCullagh and J.A. Nelder: Generealized Linear Models, 2nd ed. 1999 Chapman Hall/CRC

Text(s):

• Lectures will be based on lecture notes provided by the instructor.
• Suggested reading material:
  • TBA