Modelling and Estimation of Extremes in Space and Time

Masterarbeit von Sven Buhl, WS 2012/13
Betreuerin: Prof. Dr. Claudia Klüppelberg

In recent years, several damaging catastrophes have caught our attention. The pictures of Japan (2011) after the sea earthquake causing a devastating tsunami which in turn brought about a core meltdown in the nuclear power station of Fukushima are still in our minds. Furthermore, we remember the residents of New Orleans (2005) who had to be evacuated and brought to the local football arena after hurricane Katrina had caused disastrous damages. Last but not least, suffering from two 100-year-floodings within the last eleven years, we have to face extreme natural catastrophes in Germany as well.

Clearly, mathematicians and statisticians feel motivated to try and find ways to model rarely occurring events in order to be better prepared in the future. This results in Extreme Value Theory, the theory of extreme events. It deals with the distribution of the maximum of a set of random variables, a time series or a random field. Throughout my thesis, I consider stochastic processes in space and time, which can be, for instance, measurements of rainfall at certain spatial locations given in the state space $${\mathbb{R}}^d$$ at consecutive time points. In real world examples, the spatial dimension $$d$$ often satisfies $$d=2$$.The general concept in Extreme Value Theory is max-stability. A stochastic process $$\boldsymbol{\eta}:=\{\eta_{{\boldsymbol{s}},t}: {\boldsymbol{s}} \in {\mathbb{R}}^d, t \in [0,\infty)\}$$ is called max-stable if there exist sequences $$\left(a_{\boldsymbol{s},t}^{(n)}\right)$$, $$\left(b_{\boldsymbol{s},t}^{(n)}\right)$$ with $$a_{\boldsymbol{s},t}^{(n)} > 0$$ for all $$n \in {\mathbb{N}}$$, such that $$\frac{\bigvee\limits_{i=1}^n \eta_{{\boldsymbol{s}},t}^{(i)}-b_{\boldsymbol{s},t}^{(n)}}{a_{\boldsymbol{s},t}^{(n)}} \stackrel{\mathcal{D}}{=} \eta_{\boldsymbol{s},t}, \hspace{1cm} \boldsymbol{s} \in {\mathbb{R}}^2, \text{ } t \in [0,\infty),$$ where $$\boldsymbol{\eta}^{(i)}$$ are independent replications of $$\boldsymbol{\eta}$$. The operator $$\bigvee$$ takes the componentwise maximum of the processes it is applied to and $$\stackrel{\mathcal{D}}{=}$$ stands for equality in distribution.

Fig. 1: Perspective plots of a simulated anisotropic Brown-Resnick process. Four consecutive time steps from the left to the right (click to enlarge).
Max-stable processes can be used as models for extremes - for example daily rainfall maxima at various locations - which I do in this thesis. In the literature, there exist a large variety of constructions of max-stable processes. One example and of special interest is the Brown-Resnick process, which specifies extreme dependence by its tail dependence coefficient. When thinking, for example, of heavy rainfall, extreme dependence affects the probability of high rainfall at some location given that there is high rainfall in the neighbourhood. For the construction of the Brown-Resnick process, one can choose spatially isotropic or anisotropic models. Assuming isotropy suggests that, staying in the context of the rainfall example, the extreme rainfall weather fronts spread equally in all spatial directions. Since this is hardly true, I focus on anisotropic models.

The Brown-Resnick process depends on parameters $$C_m \in (0, \infty)$$ and $$\alpha_m \in (0,2]$$ for $$m=1, \ldots, d+1$$. That is, there is one pair of parameters $$C_m$$ and $$\alpha_m$$ for each spatial dimension and one for the time dimension. Those parameters determine the spatially isotropic or anisotropic behaviour. In the case of $$d=2$$ dimensions, the model is spatially isotropic if $$C_1=C_2$$ and $$\alpha_1=\alpha_2$$ and anisotropic if one of the equalities is hurt. When dealing with real data - in the case of my thesis with (suitably transformed) hourly accumulated values and daily maxima of radar rainfall measurements in the US state of Florida from 1999-2004 - I consider observations lying on a regular spatial grid and assume them to be realizations of a Brown-Resnick process $$\boldsymbol{\eta}$$ with spatial dimension $$d=2$$.

Fig. 2: Predicted conditional probability fields based on hourly measurement for reference locations (5,6) (left row) and (10,7) (right row) for four consecutive time points (from top to bottom). The levels $$z$$ and $$z^{\star}$$ are chosen as 1 (click to enlarge).
The task is then to estimate the model parameters mentioned above. For this purpose I use a pairwise likelihood method and show that, under certain regularity conditions, the resulting estimates are asymptotically normal and strongly consistent, that is, they converge almost surely to the respective true parameter values. The estimates can in turn be used to simulate the model and compute and plot conditional probability fields: for two (high) rainfall levels $$z^{\star}$$ and $$z$$ and for a chosen reference location $$(i_1^{\star},i_2^{\star})$$ and a reference time $$t^{\star}$$, I plot the probabilities $${\mathbb{P}}\left(\eta_{(i_1,i_2),t}>z \hspace{0.2cm}| \hspace{0.2cm}\eta_{(i_1^{\star},i_2^{\star}),t^{\star}}>z^{\star}\right)$$for all other locations $$(i_1,i_2)$$ and time points $$t$$, i.e., the probability that given there is extreme rainfall at $$(i_1^{\star},i_2^{\star})$$ at time $$t^{\star}$$, there is extreme rainfall at location $$(i_1,i_2)$$ at time $$t$$.

At the end of my thesis, I treat the question of spatial (an-)isotropy of the extremal rainfall weather fronts in Florida. The assumption of spatial isotropy suggests that the corresponding spatial parameters equal each other, i.e. $$C_1=C_2$$ and $$\alpha_1=\alpha_2$$. I perform a statistical test, testing the multiple null hypothesis $$H_0: C_1=C_2 \text{ and } \alpha_1=\alpha_2$$ versus the alternative hypothesis $$H_A: C_1 \neq C_2 \text{ or } \alpha_1 \neq \alpha_2.$$In order to obtain asymptotically correct confidence intervals for $$C_2-C_1$$ and $$\alpha_2-\alpha_1$$, one can use subsampling or bootstrap methods. I show that the $$2.5\%$$-confidence interval of $$C_2-C_1$$ excludes $$0$$ and, using Bonferroni's inequality, conclude that $$H_0$$ and therefore the assumption that the extremal rainfall weather fronts behave isotropically, that is, spread equally in both directions, can be rejected at a confidence level of $$5\%$$.

Zur vollständigen Master-Arbeit von Sven Buhl hier (mediaTUM).

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Steckbrief des Autors

• Sven Buhl
• Student an der TU von WS 2008/09 bis WS 2012/13
• B.Sc. in Mathematik im SS 2011
• M.Sc. in Mathematical Finance and Actuarial Science im WS 2012/13
• Mathematische Interessen: Math. Statistik, Extremwerttheorie
• nach dem Master: tätig in der aktuariellen Bilanzierung und mathematischen Statistik der Generali Lebensversicherung AG