Uç değerler yöntemi ile ani su baskınlarının istatiksel modellenmesi
Statistical modelling of sudden floods by extreme values method
Yazar: HALİL İBRAHİM ALTUN
Danışman: PROF. DR. MUSTAFA NADAR
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Mühendisliği Ana Bilim Dalı / Matematik Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2018
Konu: İstatistik = Statistics

Özet:
Ani su baskınları kıyı şehirlerimizin maruz kaldığı en önemli sorunlardan biridir ve baskınlarının yaşandığı şehirlerden birisi de İzmir'dir. İzmir Körfezi gibi kıyı ve haliç bölgelerindeki taşkınlar, uygun yönden esen kuvvetli rüzgar ve hava basıncında belirgin düşüş ve genelde bununla ilişkili yağış nedeni ile deniz seviyesinin yükselmesinden kaynaklanır. Bu tez çalışmasında amaç kıyı yönetim planlarının geliştirilmesinde katkıda bulunacak ani su baskını riskini öngörebilme, belirlenmiş bir deniz seviyesinin tekrarlama periyodu bilgilerini elde edebileceğimiz modeller geliştirmektir. Menteş istasyonundan alınan deniz seviyesi ve atmosfer basıncı ölçümleri verilerini kullanarak elde edilen modeller:
a) Deniz seviyesi ve atmosfer basıncı değerleri için marjinal kuyruk dağılım yapıları her bir değişken için yıllık blok maksimumları alınarak tek değişkenli GEV ve Gumbel dağılımlarıyla oturtulmuştur. 
b) Deniz seviyesi ve atmosfer basıncı değişkenleri için uç eşik değerleri saptanarak POT yöntemi ile Genelleştirilmiş Pareto dağılımları oturtulmuştur. 
c) Deniz seviyesi ve atmosfer basıncı değişkenleri için belli bir eşik seviyesini aşan değerlerin koşullu saptanmış önceki (k-1) değerinin o seviyeyi aşmaması şartıyla ACER yöntemiyle oturtulmuştur.
d) Deniz seviyesi ve atmosfer basıncı ölçümlerini birlikte değerlendirerek aralarındaki bağımlılık ölçüsünü de dikkate alarak iki değişkenli blok maksimum yöntemine göre blok büyülüğü aylık seçilerek hem Gumbel kopula hem de iki değişkenli LOG uç değer dağılımı ile modellenmiştir. Bağımlılık ilişkisi parametrik olmayan yöntemlerden Kendall tau yaklaşımıyla kopula ve Pickand bağımlılık fonksiyonu aracılığı ile tespit edilmiştir. 
e) Son olarak LOG modeli baz alınarak yüzdelik dilim yardımıyla saptanan eşik değer için iki değişkenli POT modeli oturtulmuş ve 2, 20 ve 100 yıllık tekrar seviyeleri öngörüleri kontor eğrileriyle tespit edilmiştir. 
Tek ve iki değişkenli blok maksimum ve POT yöntemleri ile tek değişkenli ACER yöntemini incelediğimizde en yüksek risk seviyesi öngörüsü tek değişkenli blok maksimum metoduyla diğer yandan en düşük risk seviyesi öngörüsü ise iki değişkenli blok maksimum metoduyla gözlemlenmiştir.
Farklı yöntemler farklı risk seviyeleri öngörmektedir. Ancak iki değişkenli POT yöntemi hem deniz ve basınç değişkenleri arasındaki bağımlılığı dikkate aldığından hem de seçilen bir eşik seviyesine bağlı olarak blok maksimum yöntemleriyle kıyaslandığında daha çok veriden yararlanabildiğinden ve ayrıca mevsimsellik etkisinden arındırdığından daha ön plana çıkmakta

Abstract:
One of the most important problem of coastal cities is the sudden flooding and İzmir is one of the cities that exposure to the sudden flooding. As Izmir bay, flooding in the region of coastal and estuarine is usually caused by strong winds blowing from the appropriate direction, significant decrease in air pressure and rising sea levels due to rainfall associated with it. 
The aim of this thesis is to develop a statistical model which may estimate the risk of sudden flooding, and the return levels. This model can contribute to improvement of the coastal management plans. Using the sea level and atmospheric pressure measurements obtained from Mentes station we obtained the following models: 
a) For both the see level and atmospheric pressure, we fited univariate tail distribution, based on annual block maxima, with GEV and Gumbel distributions. 
b) For both the see level and atmospheric pressure, after obtaining a threshold value Generalized Pareto distribution was fitted using POT method.
c) The ACER method has been fitted, provided that the conditionally determined previous (k-1) values of values exceeding a certain threshold level for sea level and atmospheric pressure variables do not exceed that level.
d) The asymptoptic joint distribution of the see level and atmospheric pressure were fitted by using Gumbel Copula and LOG distribution based on monthly blok size based on two parameter block maximum approach. The dependency was studied by copulas, which was expresed interms of Kendal tau, and Pickand dependency functions. 
e) Finally, using the LOG distribution the joint exceedance probability of the see level and atmospheric pressure was studied after determining the threshold value (POT) based on quantiles of the data. The return level of 2, 20 and 100 years expressed as contour plots. 
When we consider all five methods univariate block max predicts the higest level of risks, on the other hand bivariate block maximum method predicts the least level of risks. 
Different methods leads to different level of risks. However, bivariate methods takes dependency between variables which may lead to a better representation of the true problem. Morover, based on a good choice of a treshold, POT approach has advantage of using a lot more data then block maximum and deminish the seasonality effects. 

The thesis consists of five chapters, in the first chapters the topic of the thesis and its purpose are mentioned.
In the second part, Block Maximum and POT methods are described as univariate extreme value methods and these methods are asymptotically representing GEV and GPD distributions. Furthermore, the ACER method is explained in detail in the context of the cascade of conditioning approximations, and this method is explained. Return level is defined to describe univariate states.
In the third chapter, it is mentioned how Block Maximum and POT model generated for univariate extreme value methods are derived for two variable states. Here we talk about the exponential measures, spectral metrics, the functions reflecting stable tail dependency, copula method and the nonparametric methods which are used to represent the dependency structure between two variant states. Models such as LOG, ALOG, NEGLOG, ANEGLOG and their extreme value distributions that are to be used for two variable states throughout the study have been defined. The two variable extreme value distribution functions, relationship between the Pickand dependency function and the copula has been mentioned and the copula approach are discussed in detail. Extreme value copula functions, parameter spaces for Gumbel, Galambos, Husler Reiss, TEV and Tawn, which are extreme value families, are defined and the Pickands dependency functions of these extreme values are given. The joint return levels of the models representing these two variant states are specified.
In the fourth chapter, we give the related definitions, theorems and properties for the univariate and bivariate extreme value methods that we will use as a tool throughout the application. Definition of stationarity and KPSS, Augmented-Dickey Fuller test and Box-Ljung test for stationarity are given. Q-Q drawings are used as diagnostic tools. Chi-Plot is defined for graphical analysis of tail dependency and tail dependency. The Sklar Theorem, the extreme value copula and the Pickand dependency function, which are frequently used in the application, are given short descriptions. Kendall tau and Spearman rho dependent criteria are also defined. The R program are used for statistical analysis throughout the application. Firstly, stationary conditions are provided when annual blocks for sea level and atmospheric pressure data are taken using the univariate Block Maximum method. For models fitted to Gumbel and GEV distributions, a risk prediction for 2, 20 and 100 year return levels are made. With the help of the likelihood ratio test statistic, these models are tested to have no superiority between them and the model can be reduced to the Gumbel distribution with two parameters for two variables. Secondly, the POT method is used for sea level and atmospheric pressure data, and after the stationarity condition was satisfied, the model is fitted to the GPD and the risk prediction for 2, 20 and 100 year return levels is made. Model compatibility graphs for both Block Maximum and POT method are depicted with the help of Q-Q drawings. Finally, the ACER method is applied to univariate extreme value methods. Since the method can also be applied to non-stationary data, no stationarity condition is recognized and with this method, risks for 2, 20, and 100 year return levels are predicted. In the application phase of bivariate extreme value methods, the correlations between the sea level and atmospheric pressure variables are given by Kendal tau, Spearman Rho and Pearson. Dependency relationship between sea level and atmospheric pressure is modeled by using the extreme value copula approach. Since it is natural to expect upper queue dependency between these two variables, this is looked at with Chi-Plot graphs. The goodness of fit tests between the extreme value copulas are calculated by the Cramer-Von Mises test statistic. Bivariate extreme value distributions are derived with the appropriate selection of copulas and bivariate extreme value distributions are tried to be decided with the aid of Pickands dependency function and AIC information criterion. The LOG model is chosen for the bivariate block maximum method. Likewise, the LOG model is selected for the bivariate POT methods and the threshold values are determined. For both the Block Maximum and the POT methods, the joint return levels are calculated with contour curves in the bivariate extreme value method. 
In the fifth section, the joint return levels of 2, 20, and 100 year return levels obtained using univariate extreme value methods are interpreted in bivariate cases. The reasons for preferring the univariate and bivariate extreme value methods have been discussed and the appropriate method has been decided in the modeling of sudden floods in İzmir region.