Search result: Catalogue data in Autumn Semester 2019
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
| Electives: Applied Mathematics and Further Application-Oriented Fields|
|Selection: Probability Theory, Statistics|
|401-4597-67L||Random Walks on Transitive Graphs||W||4 credits||2V||V. Tassion|
|Abstract||In this course, we will present modern topics at the interface between probability and geometric group theory. We will be mainly focused on the random walk, and discuss its behavior depending on the geometric properties of the underlying graph.|
|Prerequisites / Notice||- Probability Theory.|
- Basic properties of Markov Chains.
- No prerequisite on group theory, all the background will be introduced in class.
|401-4619-67L||Advanced Topics in Computational Statistics|
Does not take place this semester.
|W||4 credits||2V||not available|
|Abstract||This lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling.|
|Objective||Students learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes.|
|Content||The main focus will be on graphical models in various forms: |
Markov properties of undirected graphs; Belief propagation; Hidden Markov Models; Structure estimation and parameter estimation; inference for high-dimensional data; causal graphical models
|Prerequisites / Notice||We assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics.|
|401-3628-14L||Bayesian Statistics||W||4 credits||2V||F. Sigrist|
|Abstract||Introduction to the Bayesian approach to statistics: decision theory, prior distributions, hierarchical Bayes models, empirical Bayes, Bayesian tests and model selection, empirical Bayes, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods.|
|Objective||Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis.|
|Content||Topics that we will discuss are:|
Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, noninformative priors, Jeffreys prior), tests and model selection (Bayes factors, hyper-g priors for regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods)
|Lecture notes||A script will be available in English.|
|Literature||Christian Robert, The Bayesian Choice, 2nd edition, Springer 2007.|
A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013).
Additional references will be given in the course.
|Prerequisites / Notice||Familiarity with basic concepts of frequentist statistics and with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.|
|401-3619-69L||Mathematics Tools in Machine Learning||W||4 credits||2G||F. Balabdaoui|
|Abstract||The course reviews many essential mathematical tools used in statistical learning. The lectures will cover the notions of hypotheses classes, sample complexity, PAC learnability, model validation and selection as well as results on several well-known algorithms and their convergence.|
|Objective||In the exploding world of artifical intelligence and automated learning, there is an urgent need to go back to the basis of what is driving many of the well-establsihed methods in statistical learning. The students attending the lectures will get acquainted with the main theoretical results needed to establish the theory of statistical learning. We start with defining what is meant by learning a task, a training sample, the trade-off between choosing a big class of functions (hypotheses) to learn the task and the difficulty of estimating the unknown function (generating the observed sample). The course will also cover the notion of learnability and the conditions under which it is possible to learn a task. In a second part, the lectures will cover algoritmic apsects where some well-known algorithms will be described and their convergence proved. |
Through the exerices classes, the students will deepen their understanding using their knowledge of the learned theory on some new situations, examples or some counterexamples.
|Content||The course will cover the following subjects:|
(*) Definition of Learning and Formal Learning Models
(*) Uniform Convergence
(*) Linear Predictors
(*) The Bias-Complexity Trade-off
(*) VC-classes and the VC dimension
(*) Model Selection and Validation
(*) Convex Learning Problems
(*) Regularization and Stability
(*) Stochastic Gradient Descent
(*) Support Vector Machines
|Literature||The course will be based on the book|
"Understanding Machine Learning: From Theory to Algorithms"
by S. Shalev-Shwartz and S. Ben-David, which is available online through the ETH electronic library.
Other good sources can be also read. This includes
(*) the book "Neural Network Learning: Theoretical Foundations" de Martin Anthony and Peter L. Bartlett. This book can be borrowed from the ETH library.
(*) the lectures notes on "Mathematics of Machine Learning" taught by Philippe Rigollet available through the OpenCourseWare website of MIT
|Prerequisites / Notice||Being able to follow the lectures requires a solid background in Probability Theory and Mathematical Statistical. Notions in computations, convergence of algorithms can be helpful but are not required.|
|401-0625-01L||Applied Analysis of Variance and Experimental Design||W||5 credits||2V + 1U||L. Meier|
|Abstract||Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power.|
|Objective||Participants will be able to plan and analyze efficient experiments in the fields of natural sciences. They will gain practical experience by using the software R.|
|Content||Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power.|
|Literature||G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000.|
|Prerequisites / Notice||The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software R, for which an introduction will be held.|
|401-0649-00L||Applied Statistical Regression||W||5 credits||2V + 1U||M. Dettling|
|Abstract||This course offers a practically oriented introduction into regression modeling methods. The basic concepts and some mathematical background are included, with the emphasis lying in learning "good practice" that can be applied in every student's own projects and daily work life. A special focus will be laid in the use of the statistical software package R for regression analysis.|
|Objective||The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling.|
|Content||The course starts with the basics of linear modeling, and then proceeds to parameter estimation, tests, confidence intervals, residual analysis, model choice, and prediction. More rarely touched but practically relevant topics that will be covered include variable transformations, multicollinearity problems and model interpretation, as well as general modeling strategies. |
The last third of the course is dedicated to an introduction to generalized linear models: this includes the generalized additive model, logistic regression for binary response variables, binomial regression for grouped data and poisson regression for count data.
|Lecture notes||A script will be available.|
|Literature||Faraway (2005): Linear Models with R|
Faraway (2006): Extending the Linear Model with R
Draper & Smith (1998): Applied Regression Analysis
Fox (2008): Applied Regression Analysis and GLMs
Montgomery et al. (2006): Introduction to Linear Regression Analysis
|Prerequisites / Notice||The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software package R, for which an introduction will be held.|
In the Mathematics Bachelor and Master programmes, the two course units 401-0649-00L "Applied Statistical Regression" and 401-3622-00L "Statistical Modelling" are mutually exclusive. Registration for the examination of one of these two course units is only allowed if you have not registered for the examination of the other course unit.
|401-3627-00L||High-Dimensional Statistics||W||4 credits||2V||P. L. Bühlmann|
|Abstract||"High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed.|
|Objective||Knowledge of methods and basic theory for high-dimensional statistical inference|
|Content||Lasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling|
|Literature||Peter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag. |
|Prerequisites / Notice||Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics).|
|401-4623-00L||Time Series Analysis|
Does not take place this semester.
|W||6 credits||3G||N. Meinshausen|
|Abstract||Statistical analysis and modeling of observations in temporal order, which exhibit dependence. Stationarity, trend estimation, seasonal decomposition, autocorrelations,|
spectral and wavelet analysis, ARIMA-, GARCH- and state space models. Implementations in the software R.
|Objective||Understanding of the basic models and techniques used in time series analysis and their implementation in the statistical software R.|
|Content||This course deals with modeling and analysis of variables which change randomly in time. Their essential feature is the dependence between successive observations.|
Applications occur in geophysics, engineering, economics and finance. Topics covered: Stationarity, trend estimation, seasonal decomposition, autocorrelations,
spectral and wavelet analysis, ARIMA-, GARCH- and state space models. The models and techniques are illustrated using the statistical software R.
|Lecture notes||Not available|
|Literature||A list of references will be distributed during the course.|
|Prerequisites / Notice||Basic knowledge in probability and statistics|
Does not take place this semester.
|Abstract||This course provides an introduction to statistical Monte Carlo methods. This includes applications of simulations in various fields (Bayesian statistics, statistical mechanics, operations research, financial mathematics), algorithms for the generation of random variables (accept-reject, importance sampling), estimating the precision, variance reduction, introduction to Markov chain Monte Carlo.|
|Objective||Stochastic simulation (also called Monte Carlo method) is the experimental analysis of a stochastic model by implementing it on a computer. Probabilities and expected values can be approximated by averaging simulated values, and the central limit theorem gives an estimate of the error of this approximation. The course shows examples of the many applications of stochastic simulation and explains different algorithms used for simulation. These algorithms are illustrated with the statistical software R.|
|Content||Examples of simulations in different fields (computer science, statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (quantile transform, accept-reject, importance sampling), simulation of Gaussian processes and diffusions. The precision of simulations, methods for variance reduction. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC).|
|Lecture notes||A script will be available in English.|
|Literature||P. Glasserman, Monte Carlo Methods in Financial Engineering.|
B. D. Ripley. Stochastic Simulation. Wiley, 1987.
Ch. Robert, G. Casella. Monte Carlo Statistical Methods.
Springer 2004 (2nd edition).
|Prerequisites / Notice||Familiarity with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.|
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