Program in Applied Mathematics




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MATHEMATICAL METHODS IN NATURAL LANGUAGE PROCESSING


  • Prerequisites: A proper understanding of elementary probability theory is necessary. Familiarity with statistics and formal languages might be helpful, but not required.




  • Books:
    1. D. Jurafsky and J.H. Martin: Speech and Language Processing, Second Edition, Prentice Hall Inc., to appear in 2008, available online.
    2. C.D. Manning and H. Schűtze: Foundations of Statistical Natural Language Processing, MIT Press, 1999.
    3. A. Kornai: Mathematical Linguistics, Springer, 2007.




  • Commitment: 3 hours/week, 3 credits




  • Contents:


1. Finite-state automata and transducers, rules in phonology and morphology
2. Counting words in corpora, Zipf's law
3. Hidden Markov Models, training and decoding algorithms
4. Speech recognition architecture, low-level processing, feature extraction
5. Discriminative training of Hidden Markov Models
6. Language modeling: n-gram and factored language models
7. Maximum entropy modeling
8. Document classification


STOCHASTICS PROCESSES AND APPLICATIONS


  • No. of Credits: 3, and no. of ECTS credits: 6

  • Semester or Time Period of the course: Fall Semester

  • Prerequisites: Probability and Statistics

  • Course Level: advanced

  • Brief introduction to the course:


The most common classes of stochastic processes are presented that are important in applications an stochastic modeling. Several real word applications are shown. Emphasis is put on learning the methods and the tricks of stochastic modeling.


  • The goals of the course:


The main goal of the course is to learn the basic tricks of stochastic modeling via studying many applications. It is also important to understand the theoretical background of the methods.


  • The learning outcomes of the course


The students will learn the most common methods in stochastic processes and their applications.


  • More detailed display of contents:




  1. Stochastic processes: Kolmogorov theorem, classes of stochastic processes, branching processes

  2. Poisson processes: properties, arrival times; compound, non-homogeneous and rarefied Poisson process; application to queuing

  3. Martingales: conditional expectation, martingales, stopping times, Wald's equation, convergence of martingales

  4. Applications of martingales: applications to risk processes, log-optimal portfolio

  5. Martingales and Barabási-Albert graph model: preferential attachment (BA model), degree distribution

  6. Renewal processes: renewal function, renewal equation, limit theorems, Elementary Renewal Theorem,

  7. Renewal processes: Blackwell's theorem, key renewal theorem, excess life and age distribution, delayed renewal processes

  8. Renewal processes: applications to queuing, renewal reward processes, age dependent branching process

  9. Markov chains: classification of states, limit theorems, stationary distribution

  10. Markov chains: transition among classes, absorption, applications

  11. Coupling: geometrically ergodic Markov chains, proof of renewal theorem

  12. Regenerative processes: limit theorems, application to queuing, Little's law


Books:

1. S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970.

2. S. Asmussen, Applied Probability and Queues, Wiley, 1987.


STATISTICS OF STOCHASTIC PROCESSES



  • Prerequisites: Probability and Statistics




  • Books:
    1. T. W. Anderson, The Statistical Analysis of Time Series, Wiley, 1971.
    2. S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, 1970.
    3. O. Cappé, E. Moulines, T. Rydén, Inference in Hidden Markov Models, Springer, 2005.
    4. H. Jaeger, Discrete-time, discrete-valued observable operator models: a tutorial, on-line notes, 2000.




  • Commitment: 3 hours/week, 3 credits




  • Contents:


1. Stationary processes, ARMA processes
2. Time series, trend and seasonality analysis
3. Spectrum analysis, parameter estimation of stationary processes
4. Markov decision processes, semi-Markov decision processes
5.Inventory theory, continuous time optimization models
6. Hidden Markov Models and their applications
7. Observable Operator Models


MULTIVARIATE STATISTICAL INFERENCE


  • No. of Credits: 3,  no. of ECTS credits: 6




  • Pre-requisite:  Probability and Statistics




  • Course Level: intermediate




  • Brief introduction to the course:


The course is based on the Probability and  Statistics course,  and generalizes the concepts studied there to multivariate observations and multidimensional parameter spaces. Students will be introduced to basic models of multivariate analysis with applications. We also aim at developing skills to work with real-world data.



  • The goals of the course:


The first part of the course gives an introduction to the multivariate normal distribution
and deals with spectral techniques to reveal the covariance structure of the data. In the second part dimension reduction methods will be introduced (factor analysis and canonical correlation analysis) together with linear models, regression analysis and analysis of variance. In the third part students will learn classification and clustering methods to establish connections between the observations. Finally, algorithmic models are introduced for large data sets. Applications are also discussed, mainly on a theoretical basis, but we make the students capable of using statistical program packages.



  • The learning outcomes of the course:


Students will be able to identify multivariate statistical models, analyze the results and make further inferences on them. Students will gain familiarity with basic methods of dimension reduction and classification (applied to scale, ordinal or nominal data). They will become familiar with applications to real-world data sets, and will be able to choose the most convenient method for given real-life problems. 


  • More detailed display of contents:


1. Multivariate normal distribution, conditional distributions, multiple and partial correlations.
2. The Wishart distribution and distribution of eigenvalues of sample covariance matrices.
3. Multidimensional Central Limit Theorem. Multinomial sampling and the chi-square test.
4. Parameter estimation and Fisher information matrix.
5. Likelihood ratio tests and testing hypotheses about the mean. Hotelling’s T-square distribution.
6. Multivariate statistical methods for reduction of dimensionality: principal components and factor analysis, canonical correlation analysis.
7. Theory of least squares. Multivariate regression, Gauss-Markov theory.
8. Fisher-Cochran Theorem. Analysis of variance.
9. Classification and clustering. Discriminant analysis, k-means and hierarchical clustering methods.
10. Factoring and classifying categorical data. Contingency tables, correspondence analysis.
11. Algorithmic models: EM-algorithm for missing data, ACE-algorithm for generalized regression, Kaplan-Meier algorithm for censored data.
12. Resampling methods: jackknife and bootstrap. Statistical graph theory.

Literature:
1. R.A. Johnson, G.K. Bhattacharyya, Statistics. Principles and Methods. Wiley, New York, 1992.
2. C.R. Rao, Linear statistical inference and its applications. Wiley, New York, 1973.
3. K.V. Mardia, J.T. Kent, M. Bibby, Multivariate analysis. Academic Press, New York, 1979.

Handouts: ANOVA tables and outputs of the BMDP Program Package, while processing real-world data.


SURVEY METHODOLOGY


  • Prerequisites: Probability and Statistics




  • Books:
    1. E.K. Foreman, Survey Sampling Principles, Marcel Dekker, 1991.
    2. D. Freedman, R. Pisani, R. Purves, A. Adhikari, Statistics 2nd ed, Norton 1991.
    3. M.H. Hansen, W.G. Hurwitz, W.G. Madow, Sample Survey Methods and Theory, Vol 1, Wiley, 1993.




  • Commitment: 3 hours/week, 3 credits




  • Course Description:


Every empirical investigation in the social sciences requires valid and reliable data, and the application of carefully selected statistical methods. The typical form of data collection is conducting a survey, and well designed surveys can provide the researcher with good data, even based on surprisingly small sample sizes. The course will discuss the most important concepts and techniques in survey design. A clear understanding of these methods is necessary for any scientist  who is engaged in data collection, but it is also useful for the researcher who analyses or interprets data. 
Topics:
•    Surveys and censuses
•    Probability versus non-probability samples
•    Role of the sample size, accuracy of estimates
•    Sampling and nonsampling errors
•    Sample-based and model-based approaches to surveys
•    Questionnaire design
•    Sample survey design
•    Main sampling techniques:
-simple random sampling
-stratified sampling
-cluster sampling
•    Handling of missing data


Facilities and Infrastructure


The Central European University (CEU) has a rich library which offers books and journal collections in many fields of interest, including mathematics and its applications. Lately, an additional departmental library has been organized to offer our students some of the most important books, including several copies of textbooks which are frequently used by our faculty. Furthermore, the Renyi Institute (which is close to our departmental offices) has one of the richest mathematical library in the region, with about 40,000 volumes and 350 periodicals. There are many other libraries in Budapest which are freely available.

CEU has classrooms, study rooms, and computer labs. They are equipped with the usual facilities, including blackboards, computers, overhead projectors, printers, scanners. There is an IT department which is responsible for the overall soft- and hardware development, maintenance, and acquisitions. A significant portion of the university budget is used every year to maintain and develop IT facilities in accordance with the standards of research-intensive universities. In addition to CEU's computer resources, our students and faculty have free use of the Renyi Institute computers.

Our students can benefit from the very rich cultural and scientific life of Budapest. There are several universities in Budapest, in particular the Eötvös University (ELTE) and the Budapest University of Technology and Economics (BME). They organize frequently seminars and conferences. Our department is already integrated into this scientific environment. We also organize seminars and workshops.


CEU has a Residence Center where students may live. This is essentially a modern hotel, with a restaurant as well as conference rooms, swimming pool, sauna, fitness room - all freely available.


Faculty


An important issue is attracting quality faculty to participate in our M.S. program. CEU has an agreement with the Renyi Institute of the Hungarian Academy of Sciences, so some of the courses are delivered by Renyi professors. To increase our teaching force, we invite specialists from other local Hungarian institutions (the Eotvos University (ELTE), the Budapest University of Technology and Economics (BME), the Computer and Automation Research Institute (SZTAKI) and the Research Institute for Particle Nuclear Physics (KFKI) of the Hungarian Academy of Sciences) who are able to cover various applied fields of the program. Furthermore, we invite frequently foreign specialists, depending on the interests of our students.


MS Teaching Program for AY 2009-2010



Mandatory Courses for First Year Students

Fall Term

Winter Term

Basic Algebra 1 
(Pal Hegedus)

Basic Algebra 2
(Pal Hegedus)

Introduction to Computer Science 
(Istvan Miklos)

Complex Function Theory
(Robert Szoke)

Probability and Statistics
(Marianna Bolla)

Functional Analysis and Differential Equations
(TBA)

Real Analysis
(Laszlo Csirmaz)





     

Elective Courses for all MS Students

Fall Term

Winter Term

Spring Term

Introduction to Discrete Mathematics
(Ervin Gyori)

Combinatorial Optimization
(Ervin Gyori)

TBA,  distinguished lecture series, 1 credit or audit (Carsten Carstensen, Humboldt University, Berlin)

Optimization in Economics
(Alexandru Kristaly)

Cryptology
(Laszlo Csirmaz)

TBA, distinguished lecture series, 1 credit or audit
(Eduard Feireisl, Czech Academy of Sciences, Prague)

Stochastic Processes and Applications
(Balazs Szekely)

Quantitative Financial Risk Analysis
(Balazs Janecsko and Imre Kondor)







Statistics of Stochastic Processes
(Balazs Szekely)







Topics in Financial Mathematics
(Alexandru Kristaly)



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