Statistical Inference for Dynamical Systems

Modul MA5612

Dieses Modul wird durch Fakultät für Mathematik bereitgestellt.

Diese Modulbeschreibung enthält neben den eigentlichen Beschreibungen der Inhalte, Lernergebnisse, Lehr- und Lernmethoden und Prüfungsformen auch Verweise auf die aktuellen Lehrveranstaltungen und Termine für die Modulprüfung in den jeweiligen Abschnitten.

Modulversion vom SS 2016 (aktuell)

Von dieser Modulbeschreibung gibt es historische Versionen. Eine Modulbeschreibung ist immer so lange gültig, bis sie von einer neuen abgelöst wird.

verfügbare Modulversionen
SS 2016WS 2013/4SS 2013

Basisdaten

MA5612 ist ein Semestermodul in Englisch auf Master-Niveau das unregelmäßig angeboten wird.

Das Modul ist Bestandteil der folgenden Kataloge in den Studienangeboten der Physik.

  • Spezialisierung im Elitemasterstudiengang Theoretische und Mathematische Physik (TMP)
GesamtaufwandPräsenzveranstaltungenUmfang (ECTS)
180 h 60 h 6 CP

Inhalte, Lernergebnisse und Voraussetzungen

Inhalt

Mathematical models are nowadays essential for the quantitative assessment of technical, physical, chemical, and biological processes. While a broad class of models is used in the different field, almost all models share one common property: the need for accurate parameter values. Due to experimental constraints, many parameters cannot be measured directly, but have to be estimated from the available experimental data.
In this course, we will introduce deterministic modeling approaches for biochemical reaction networks. These modeling approaches can be used to describe, e.g., signal transduction and metabolic processes. For these models the respective parameter estimation problem will be formulated and methods will be presented to solve these problems. As parameter estimates carry uncertainties due to limited amounts of data and measurement noise, we furthermore provide methods for a rigorous analysis of parameter uncertainties. This is crucial to evaluate the model uncertainties as well as the predictive power of models.
The participants will gather hands-on experiences with parameter estimation and uncertainty analysis, including the implementation of own models and estimation procedures in MATLAB. The estimation methods are presented in the context of biological processes, but the approaches are applicable in many other fields.

Content:
1) Modeling of biochemical reaction networks in a nutshell ( 2 lectures)
1.1) Introduction of biochemical reaction networks (including several examples)
1.2) Mass action and Michaelis-Menten kinetics
1.3) Reaction rate equation (RRE)
(The RRE is a system of ordinary differential equations which can be used describe the dynamics of reaction networks. It is widely used in chemistry, biochemistry and biology. In this lecture we develop methods for systems of ordinary differential equations and illustrate them using different RRE models.)

2) Maximum likelihood estimation for RREs (4 lectures)
2.1) Likelihood function
2.2) Maximum likelihood estimation as optimization problem
2.3) Local and global optimization
a. Gradient descent and interior point methods
b. Multi-start optimization

3) Identifiability and uncertainty analysis for RREs (3 lectures)
3.1) Structural and practical identifiability
3.2) Uncertainty analysis of and confidence intervals for parameters
a. Asymptotic confidence intervals
b. Bootstrapping confidence intervals
c. Profile likelihoods

4) Bayesian parameter estimation for RREs (3 lectures)
1) Bayes theorem and Bayesian statistics
2) Markov chain Monte-Carlo sampling
3) Bayesian confidence intervals for parameter estimates and predictions

5) Properties of estimators (e.g. bias and variance) (1 lectures)

6) Summary and Outlook (1 lectures)

Lernergebnisse

After successful completion of the module, the participants can:
1. model biochemical reaction networks using ODEs.
2. solve parameter estimation problems for ODEs using MATLAB.
3. analyze the uncertainty of parameter estimates using MATLAB.
4. critically evaluate parameter estimation procedures.

Voraussetzungen

Bachelor in mathematics, bioinformatics, statistics or related fields.
Basic MATLAB programming skills.

Lehrveranstaltungen, Lern- und Lehrmethoden und Literaturhinweise

Lehrveranstaltungen und Termine

Lern- und Lehrmethoden

lecture + exercise module
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

Medienformen

blackboard and slides

Literatur

Will be announced during the course.

Modulprüfung

Beschreibung der Prüfungs- und Studienleistungen

The exam will be in written form (90 min) if more than 15 students participate, otherwise the exam will be in oral form (20 min). The students will be asked to apply parameter optimization and uncertainty analysis techniques discussed in the course. In addition, the students will be expected to be able to analyze estimation results (e.g. check the plausibility and consistency) and to transfer the acquired knowledge to similar problems. To evaluate the programming skills, the students will be asked to write pseudo code for the implementation of an algorithm / a method. The students will be allowed to use any printed material (lecture notes, books, etc.)

Wiederholbarkeit

Eine Wiederholungsmöglichkeit wird am Semesterende angeboten.

Kondensierte Materie

Wenn Atome sich zusammen tun, wird es interessant: Grundlagenforschung an Festkörperelementen, Nanostrukturen und neuen Materialien mit überraschenden Eigenschaften treffen auf innovative Anwendungen.

Kern-, Teilchen-, Astrophysik

Ziel der Forschung ist das Verständnis unserer Welt auf subatomarem Niveau, von den Atomkernen im Zentrum der Atome bis hin zu den elementarsten Bausteinen unserer Welt.

Biophysik

Biologische Systeme, vom Protein bis hin zu lebenden Zellen und deren Verbänden, gehorchen physikalischen Prinzipien. Unser Forschungsbereich Biophysik ist deutschlandweit einer der größten Zusammenschlüsse in diesem Bereich.