Modelação de valores extremos e sua importância na

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1 Modelação de valores extremos e sua importância na segurança e saúde Margarida Brito Departamento de Matemática FCUP (FCUP) Valores Extremos - DemSSO 1 / 12

2 Motivation Consider the following events Occurance of floods, heat waves, strong winds, high wave heights. Exposure of workers to abnormal low or high temperatures (outdoor work; glass and ceramic production and other hot industrial processes, mines,...), high noise levels, organic solvents or chemicals, toxic gases and vapors, dangerous radiation, abnormally high or low air pressure; Occupational exposure to organic solvents or chemicals, toxic gases and vapors exceeding workplace exposure limits (ELs); Dangerous levels of air pollutants such as ozone, carbon monoxide, fine particles. These examples have in common that they concern unusually large or small values of a variable (extreme values). The consequences of these events for population, environment and economy are often enormous. (FCUP) Valores Extremos - DemSSO 2 / 12

3 In many situations, the analysis of the extreme values of the sample is very important. For example, in the monitoring of industrial processes, systems, patients an alarm is generated if one or more relevant characteristics, like concentrations of toxic gases, temperature, heart and respiration rates, blood pressure of a patient, exceed fixed upper or/and lower thresholds. Certain of these levels are defined heuristically, not providing adequate alarms. Several procedures for generating more robust alarms or detecting abnormal values have been proposed in the literature making use of extreme value models. (FCUP) Valores Extremos - DemSSO 3 / 12

4 River floods Based on a dataset concerning the flows of a river over a long period of time, we want to address questions such as What is the probability that the level x will be exceeded next year? What is the expected time between successive exceedances of the level x? How big is the expected 100-year flood (level exceeded by the annual maximum in any particular year with probability 0.01)? Statistical model It is important to look at the upper part of the sample, model the upper tail. Floods are the result of complex events and situations and other variables should be taken into account (such as rainfall, soil types,...). For illustration, a very simple model will be considered here. (FCUP) Valores Extremos - DemSSO 4 / 12

5 Simple Extreme Value Models Underlying key idea: consider the large values of the sample instead of all the data. Two general approaches: block maxima and exceedances of a high threshold. Theoretical foundations of extreme value theory (EVT): Fisher and Tippet (1928); Gnedenko (1943). Suppose the observations are independent with a common distribution (i.i.d.), M n denoting the maximum of n values. How to model the distribution of M n? (FCUP) Valores Extremos - DemSSO 5 / 12

6 If there exists a nondegenerate limit distribution for the linearly normalized maximum M n, then it can only belong to one of the three families Type I or Gumbel: exponential-tail decay; Type II or Fréchet: polynomial-tail decay, Type III or Weibull: finite upper endpoint. This type of result for maxima is comparable to the central limit theorem, which concerns the limiting distribution for partial sums. (FCUP) Valores Extremos - DemSSO 6 / 12

7 These families can be combined into a single family, Generalized Extreme Value family (GEV), with distribution function (df) ( G(y) = exp (1+γ( y µ ) σ )) 1/γ, where for γ = 0, G(y) = exp( exp( ( y µ σ ))), and µ: location parameter; σ > 0: scale parameter; γ: shape parameter. Correspondance: γ = 0 Gumbel df; γ > 0 Fréchet df; γ < 0 Weibull df. (FCUP) Valores Extremos - DemSSO 7 / 12

8 Correspondance γ = 0 Gumbel df; γ > 0 Fréchet df; γ < 0 Weibull df. Figure: fdp GEV(-0.5), GEV(0.5) e GEV(0) (FCUP) Valores Extremos - DemSSO 8 / 12

9 How to apply? Block the data into groups of equal length, obtaining a sequence of block maxima (such as annual maximum flows, temperatures,...); Fit a GEV to this sequence of maxima. Notice that for the T-year level, u(t), for annual maxima u(t) G 1 (1 1/T), ( (1 1/T)-quantile of G). Models for Minima Some situations require models for unusual low values. Models for minima can be easily derived from models for maxima, using the duality between the distribution for maxima and minima. (FCUP) Valores Extremos - DemSSO 9 / 12

10 Threshold Models Theoretical basis: If block maxima have approximate distribution G, then, for large enough u, the distribution of the excesses X u given that X > u, is approximately ( H(y) = 1 1 γ y ) 1/γ, σ(u) the Generalized Pareto family (GPD). Application: for a dataset {x 1,x 2,...,x n }, identify the extreme events {x i : x i > u}; record these exceedances as y 1,... and corresponding excesses by z i = y i u; fit GPD to {z i } data. Other main approaches: top order statistics models; point processes representations. (FCUP) Valores Extremos - DemSSO 10 / 12

11 Parameter Estimation in Extreme Value Models Several approaches and methods, including graphical procedures; moment-based methods; likelihood-based methods, direct estimation of the shape parameter via functions of order statistics. Non i.i.d. Data In most applications, data have a more complex structure. Several more realistic models have been proposed, taking into account specific features of the data, for instance, time-dependency, clustering (temporal and/or spatial),..., and also relevant information from other sources. Often, the construction of an adequate model requires the combination of different types of models and methods. (FCUP) Valores Extremos - DemSSO 11 / 12

12 Some References Beirlant J., Y. Goegebeur and J. Teugels (2004), Statistics of Extremes: Theory and Applications, Wiley. Coles, S. (2001). An introduction to statistical modeling of extreme values. Springer-Verlag. Coles S.G. and L.R. Pericchi (2003), Anticipating catastrophes through extreme value modelling, Applied Statistics 52, M. H. Feinberga, M. H. and Leblanca, J. Ch. (2004), Probabilistic exposure assessment to food chemicals based on extreme value theory. Application to heavy metals from fish and sea products, Food and Chemical Toxicology 42, Hugueny, S., Clifton, D.A., and Tarassenko, L. (2011), Probabilistic Patient Monitoring with Multivariate, Multimodal Extreme Value Theory, Biomedical Engineering Systems and Technologies, Communications in Computer Science 127, Roberts, S. J. (1999). Novelty detection using extreme value statistics. IEE Proceedings on Vision, Image and Signal Processing, 146(3), Smith R.L. (1989), Extreme value analysis of environmental time series: An example based on ozone data (with discussion), Statistical Science 4, (FCUP) Valores Extremos - DemSSO 12 / 12