An introduction to Statistical Analysis of Extreme Values

Frederico Caeiro (DM/FCT NOVA)

Frederico Caeiro is an Auxiliary Professor at the Mathematics Department of the Faculty of Sciences and Technology – Nova University of Lisbon (FCT NOVA) and is a member of the Mathematics and Application Research Center (Portugal). He has an MSc degree in Probability and Statistics (2001) and a PhD degree in Statistics (2006) from the University of Lisbon. His current research interests include Statistics of Extremes, Extreme Value Theory, Nonparametric Statistics and Computational Statistics Methods.

Abstract: Statistics of Extreme (values) is a subject that has gained considerable importance in the past few decades. This discipline provides the adequate methodology for the knowledge and prediction of extreme and rare events, that is, of events that occur irregularly with a very small probability. We can find applications in several fields, such as finance, geology, genetics, hydrology, insurance, meteorology, public health, sports or structural engineering. In this course we will study some of the probabilistic models and statistical methods used to study extreme events. Those methods are essentially based on the well-established limiting results for the sample maximum and for the excesses of a high threshold.

The application of the Extreme Value Theory in the traffic of the 25 de Abril Bridge

Maria da Conceição Almeida (FCT NOVA Alumni,  Millenium BCP)

Maria da Conceição Almeida works at Millennium BCP at Sales and Trading in treasury and international markets department. She has an MSc in Mathematics and Applications, specializing in Financial Mathematics (2019) from Nova University of Lisbon (FCT NOVA) and a degree in Management (2016) from ISEG - Lisbon School of Economics & Management – University of Lisbon. She was also an Erasmus student at Universitat Autònoma de Barcelona (2018).

Abstract: The Extreme Values Theory enables the study of extreme events that are possibly disastrous and of great impact for society. The behavior of the Extremes can be modelled by using one of three distributions – Gumbel, Fréchet and Weibull – even though they can be represented in a single expression, the Generalized Extreme-Value distribution (GEV).  In my dissertation, the numbers of vehicles crossing daily and in both directions in the 25 de Abril Bridge were analyzed. Two analyzes were carried out, as a result of these data, one based on the verified seasonality and another in relation to the tolls and revenues collected in the crossing of this Bridge. A parametric approach was used for statistical inference about rare events. To achieve this three methods it was used: the GEV Model (also known as the Annual Maximum Model), the Multivariate GEV Model (or r Largest order statistic Model) and the Generalized Pareto Model (GP or Peak Over Threshold Model). These models are widely used in various areas.  In my thesis I made a description of the traffic flow in the 25 de Abril Bridge and the Methods of the Extreme Values were used to make a prediction of the behavior of this traffic. Return levels, return periods and probabilities of exceedance were estimated. The Maximum Likelihood Method was used for the estimation of parameters and also the Profile Log-Likelihood Method when estimating Confidence Intervals.

Spatial analysis of bat abundance with adjustment for detection probability – Why does this matter for an environmental consultancy company?

Sandra Rodrigues (Bioinsight Environmental Consulting)

Sandra Rodrigues is an ecologist and works at Bioinsight – an environmental consultancy company specialised in biodiversity. She is currently undertaking a PhD in Statistics and Operational Research based on an R&D project developed at the company. She (and the company) intend to use the project’s outcome to greatly enhance the quality of the services provided to their clients.

Abstract: Spatial analysis of bats has become increasingly important for impact assessment studies since it allows to avoid constructing the project over areas where bats are more abundant. However, spatial analysis with bats have been performed without adjusting for detection bias of the detector. In this study we used an approach that allows to use spatial models and adjust for unknown detection probability. The case study is a project located in South Africa and it is shown how this approach provides a better estimate of bat abundance in the study area and how it enhances the company’s studies.