# Suchergebnis: Katalogdaten im Herbstsemester 2020

Mathematik Master | ||||||

Wahlfächer Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen. | ||||||

Wahlfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten | ||||||

Auswahl: Wahrscheinlichkeitstheorie, Statistik | ||||||

Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
---|---|---|---|---|---|---|

401-4607-70L | A Medley of Advanced Probability | W | 4 KP | 2V | W. Werner | |

Kurzbeschreibung | We will review various topics of probability theory, with the goal to provide a short self-contained introduction to each of them, and try to describe the type of ideas and techniques that are used. Exact topics will include (small bits of) Lévy processes, continuous-state branching processes, large deviation theory, large random matrices. | |||||

Lernziel | The goal is for each of the topics that will be covered to provide: - A general introduction to the subject - An example of one of the main statements, and some of the ideas that go into the proof - A detailed proof of one statement | |||||

Voraussetzungen / Besonderes | Prerequisites: Martingales, Markov chains, Brownian motion, stochastic calculus. | |||||

401-3628-14L | Bayesian StatisticsFindet dieses Semester nicht statt. | W | 4 KP | 2V | ||

Kurzbeschreibung | 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. | |||||

Lernziel | Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis. | |||||

Inhalt | 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) | |||||

Skript | A script will be available in English. | |||||

Literatur | 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. | |||||

Voraussetzungen / Besonderes | 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 KP | 2V + 1U | L. Meier | |

Kurzbeschreibung | 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. | |||||

Lernziel | 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. | |||||

Inhalt | 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. | |||||

Literatur | G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000. | |||||

Voraussetzungen / Besonderes | 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 KP | 2V + 1U | M. Dettling | |

Kurzbeschreibung | 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. | |||||

Lernziel | The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling. | |||||

Inhalt | 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. | |||||

Skript | A script will be available. | |||||

Literatur | 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 | |||||

Voraussetzungen / Besonderes | 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. | |||||

401-4521-70L | Geometric Tomography - Uniqueness, Statistical Reconstruction and Algorithms | W | 4 KP | 2V | J. Hörrmann | |

Kurzbeschreibung | Self-contained course on the theoretical aspects of the reconstruction of geometric objects from tomographic projection and section data. | |||||

Lernziel | Introduction to geometric tomography and understanding of various theoretical aspects of reconstruction problems. | |||||

Inhalt | The problem of reconstruction of an object from geometric information like X-ray data is a classical inverse problem on the overlap between applied mathematics, statistics, computer science and electrical engineering. We focus on various aspects of the problem in the case of prior shape information on the reconstruction object. We will answer questions on uniqueness of the reconstruction and also cover statistical and algorithmic aspects. | |||||

Literatur | R. Gardner: Geometric Tomography F. Natterer: The Mathematics of Computerized Tomography A. Rieder: Keine Probleme mit inversen Problemen | |||||

Voraussetzungen / Besonderes | A sound mathematical background in geometry, analysis and probability is required though a repetition of relevant material will be included. The ability to understand and write mathematical proofs is mandatory. | |||||

401-4607-59L | Percolation Theory | W | 4 KP | 2V | V. Tassion | |

Kurzbeschreibung | An introduction to the percolation theory. | |||||

Lernziel | Percolation theory has many applications and is one of the most famous model to describe phase transition phenomena in physics. One reason for this success is the variety of mathematical tools, which allows for a precise and rigorous description of the models. The objective of this course is to gain familiarity with the methods of the percolation theory and to learn some of its important results. The students will develop their background and intuition in probability, and the course is particularly recommended to students with additional interests in physics or graph theory. | |||||

Inhalt | Definition of percolation. Standard tools: FKG, BK inequalities, Mixing property, Russo's formula. Sharpness of the phase transition. Correlation length and interpretations. Uniqueness of the infinite cluster. Critical percolation in dimension 2. Supercritical percolation in dimension d>2, Grimmett-Marstrand Theorem and consequences. | |||||

Literatur | B. Bollobas, O. Riordan: Percolation, CUP 2006 G. Grimmett: Percolation 2ed, Springer 1999 | |||||

Voraussetzungen / Besonderes | Preliminaries: 401-2604-00L Probability and Statistics (mandatory) 401-3601-00L Probability Theory (recommended) | |||||

401-4619-67L | Advanced Topics in Computational StatisticsFindet dieses Semester nicht statt. | W | 4 KP | 2V | keine Angaben | |

Kurzbeschreibung | This lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling. | |||||

Lernziel | Students learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes. | |||||

Inhalt | 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 | |||||

Voraussetzungen / Besonderes | We assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics. | |||||

401-3627-00L | High-Dimensional StatisticsFindet dieses Semester nicht statt. | W | 4 KP | 2V | P. L. Bühlmann | |

Kurzbeschreibung | "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. | |||||

Lernziel | Knowledge of methods and basic theory for high-dimensional statistical inference | |||||

Inhalt | 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 | |||||

Literatur | 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. | |||||

Voraussetzungen / Besonderes | 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 | W | 6 KP | 3G | F. Balabdaoui | |

Kurzbeschreibung | 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. | |||||

Lernziel | 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. | |||||

Inhalt | 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 | |||||

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

Voraussetzungen / Besonderes | Basic knowledge in probability and statistics | |||||

401-3612-00L | Stochastic Simulation | W | 5 KP | 3G | F. Sigrist | |

Kurzbeschreibung | This course introduces statistical Monte Carlo methods. This includes applications of stochastic simulation in various fields (statistics, statistical mechanics, operations research, financial mathematics), generating uniform and arbitrary random variables (incl. rejection and importance sampling), the accuracy of methods, variance reduction, quasi-Monte Carlo, and Markov chain Monte Carlo. | |||||

Lernziel | Students know the stochastic simulation methods introduced in this course. Students understand and can explain these methods, show how they are related to each other, know their weaknesses and strengths, apply them in practice, and proof key results. | |||||

Inhalt | Examples of simulations in different fields (statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (including rejection sampling and importance sampling), simulation of multivariate normal variables and stochastic differential equations. The accuracy of Monte Carlo methods. Methods for variance reduction and quasi-Monte Carlo. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). Algorithms introduced in the course are illustrated with the statistical software R. | |||||

Skript | A script will be available in English. | |||||

Literatur | 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). | |||||

Voraussetzungen / Besonderes | It is assumed that students have had an introduction to probability theory and statistics (random variables, joint and conditional distributions, law of large numbers, central limit theorem, basics of measure theory). The course resources (including script, slides, exercises) will be provided via the Moodle online learning platform. |

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