# Search result: Catalogue data in Autumn Semester 2021

Mathematics Master | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Electives 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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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401-4607-67L | Schramm-Loewner Evolutions | W | 4 credits | 2V | W. Werner | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This advanced course will be an introduction to SLE (Schramm-Loewner Evolutions), which are a class of conformally invariant random curves in the plane. We will discuss their construction and some of their main properties. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Knowledge of Brownian motion and stochastic calculus and basic knowledge of complex analysis (Riemann's mapping theorem). Familiarity of lattice models such as percolation or the Ising model can be useful but not necessary. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3822-17L | Ising Model | W | 4 credits | 2V | V. Tassion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | - Probability Theory. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-0625-01L | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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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. ISBN 978-3-642-20191-2. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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 AnalysisDoes not take place this semester. | W | 6 credits | 3G | F. Balabdaoui | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The course offers an introduction into analyzing times series, that is observations which occur in time. The material will cover Stationary Models, ARMA processes, Spectral Analysis, Forecasting, Nonstationary Models, ARIMA Models and an introduction to GARCH models. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The goal of the course is to have a a good overview of the different types of time series and the approaches used in their statistical analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | This course treats modeling and analysis of time series, that is random variables which change in time. As opposed to the i.i.d. framework, the main feature exibited by time series is the dependence between successive observations. The key topics which will be covered as: Stationarity Autocorrelation Trend estimation Elimination of seasonality Spectral analysis, spectral densities Forecasting ARMA, ARIMA, Introduction into GARCH models | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | The main reference for this course is the book "Introduction to Time Series and Forecasting", by P. J. Brockwell and R. A. Davis | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Basic knowledge in probability and statistics | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3612-00L | Stochastic SimulationDoes not take place this semester. | W | 5 credits | 3G | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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. Springer 2004. 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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