2023
Modelling and Analysis of Extreme Events
Name: Modelling and Analysis of Extreme Events
Code: MAT13607M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese, English
Regime de Frequência: Presencial
Sustainable Development Goals
Learning Goals
The main goal of this course is to provide students with the probabilistic and statistical tools related with the modeling of rare events. The Extreme Value Analysis (EVA) is a very important research area with great development in the last decades that is used to model the risk of extreme and rare events. EVA is widely used in many disciplines, such as earth sciences, geological and structural engineering, biological sciences, finance and insurance, among others. This course also aims to develop technical and analytical skills in order to enable each student to be able to develop a project for analysis of data from extreme events, using the appropriate methodologies for this purpose.
Contents
1. Why Extreme Value Analysis?;
2. Graphical Tools for data analysis
2.1 QQ-plots, PP-plots; Excess plots;
2.2 Domains of applications;
3. The probabilistic side of EVA
3.1 Exact distribution and limit distribution of central order statistics;
3.2 Stable laws for maxima and minimum; Gnedenkos theorem;
3.3 Domains of attraction: Fréchet, Max-Weibull e Gumbel;
3.4 Limit distribution of exceedances;
3.5 Pickands-Balkema-de Haan theorem;
4. The statistical side of EVA
4.1 Parameters of extreme events: extreme value index (EVI), return period, exceedance probabilities, extremal quantiles (VaR) and extremal index;
4.2 Parametric approaches in EVA: Block maxima method; Gumbel, Fréchet, Max-Weibull and GEV models; the peaks over threshold (POT) methodology and the generalized Pareto distribution (GP);
4.3. Semi-parametric approach in EVA
5. Case Studies in the fields of earth and biological sciences, and finance and insurance.
2. Graphical Tools for data analysis
2.1 QQ-plots, PP-plots; Excess plots;
2.2 Domains of applications;
3. The probabilistic side of EVA
3.1 Exact distribution and limit distribution of central order statistics;
3.2 Stable laws for maxima and minimum; Gnedenkos theorem;
3.3 Domains of attraction: Fréchet, Max-Weibull e Gumbel;
3.4 Limit distribution of exceedances;
3.5 Pickands-Balkema-de Haan theorem;
4. The statistical side of EVA
4.1 Parameters of extreme events: extreme value index (EVI), return period, exceedance probabilities, extremal quantiles (VaR) and extremal index;
4.2 Parametric approaches in EVA: Block maxima method; Gumbel, Fréchet, Max-Weibull and GEV models; the peaks over threshold (POT) methodology and the generalized Pareto distribution (GP);
4.3. Semi-parametric approach in EVA
5. Case Studies in the fields of earth and biological sciences, and finance and insurance.
Teaching Methods
The theorical-practical classes will be based on the exposition and discussion of the syllabus and the analysis of data sets using the software R Development Core Team, 2011 and a set of statistical packages that address the different types of statistical modeling of extreme events.
The assessment will consist of the accomplishment of 1 written exam (A); the elaboration of 1 group data analysis project (B); and the scientific review, in group, of an article in a particular area of interest (C). The final result (RF) will be obtained by the formula: RF = 0.40 A + 0.30 B + 0.30 C.
The assessment will consist of the accomplishment of 1 written exam (A); the elaboration of 1 group data analysis project (B); and the scientific review, in group, of an article in a particular area of interest (C). The final result (RF) will be obtained by the formula: RF = 0.40 A + 0.30 B + 0.30 C.
Assessment
Continuous Assessment:
In this modality, knowledge assessment consists of the following elements:
• Mini-tests (number n to be defined), with the final grade being the average of the (n-1) best grades (A);
• Development of 1 data analysis project (in groups) consisting of two parts and its respective presentation (B);
• The final grade (NF) is the arithmetic mean of grades A and B.
• The student will be approved if the NF is greater than or equal to 9.5 points, provided they obtain a minimum score of 50% on the project and 40% on the mini-tests.
• As an optional element, students may also conduct a scientific review of an article in a specific area of interest. The final grade for this component ranges from 0 to 2, and is added to the NF, provided it is greater than or equal to 8 points.
Assessment by Exam:
Normal exam period and/or a resit exam. The student is approved if they obtain a grade in the exam greater than or equal to 9.5 points.
In this modality, knowledge assessment consists of the following elements:
• Mini-tests (number n to be defined), with the final grade being the average of the (n-1) best grades (A);
• Development of 1 data analysis project (in groups) consisting of two parts and its respective presentation (B);
• The final grade (NF) is the arithmetic mean of grades A and B.
• The student will be approved if the NF is greater than or equal to 9.5 points, provided they obtain a minimum score of 50% on the project and 40% on the mini-tests.
• As an optional element, students may also conduct a scientific review of an article in a specific area of interest. The final grade for this component ranges from 0 to 2, and is added to the NF, provided it is greater than or equal to 8 points.
Assessment by Exam:
Normal exam period and/or a resit exam. The student is approved if they obtain a grade in the exam greater than or equal to 9.5 points.
Teaching Staff
- Lígia Carla Pinto Henriques Jorge Rodrigues [responsible]
