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Summary
Summary
The most powerful earthquake in 40 years occurred on 26th December 2004 off the west coast of Sumatra, Indonesia. The tsunami it generated turned into one of the worst known natural disasters when walls of water crashed across the Indian Ocean, causing waves to reach Somalia in Africa. The death toll, mainly in Indonesia and Sri Lanka, exceeded 200,000. Nine months later, hurricane Katrina devastated the southern coast of USA along the Gulf coast. Winds reached 281 kilometers per hour and the storm surge of over nine meters was the highest recorded in the United States. It brought destruction to New Orleans when portions of the 563 kilometers of levees surrounding the city were suddenly breached. Nearly 1700 people died and damages are currently estimated at $100 billion, the costliest natural disaster in the United States. Within days hurricane Rita, another maximum category hurricane, struck the same coastal region damaging Texas and other states, followed soon aft- wards by hurricane Wilma. Then on October 8th 2005 an earthquake in Kashmir, part of northern Pakistan and India, killed 75,000 inhabitants when innumerable buildings collapsed. Simultaneously, hurricane Stan led to costly landslides and more than 2000 deaths in Central America. To highlight the major catastrophes of nature during the previous decade, Cyclone Gorky and its storm surge caused 139,000 deaths in coastal Bangladesh during 1991.
Table of Contents
Preface | p. ix |
1 Univariate Extreme Value theory | p. 1 |
1.1 Order Statistics | p. 1 |
1.1.1 Distribution of the smallest value | p. 2 |
1.1.2 Distribution of the largest value | p. 4 |
1.1.3 General distributions of order statistics | p. 5 |
1.1.4 Plotting positions | p. 9 |
1.2 Extreme Value theory | p. 11 |
1.2.1 "Block" model | p. 12 |
1.2.2 "Threshold" model | p. 31 |
1.2.3 Scaling of extremes | p. 39 |
1.2.4 Contagious Extreme Value distributions | p. 50 |
1.3 Hazard, return period, and risk | p. 53 |
1.4 Natural Hazards | p. 58 |
1.4.1 Earthquakes | p. 58 |
1.4.2 Volcanic eruptions | p. 66 |
1.4.3 Tsunamis | p. 68 |
1.4.4 Landslides | p. 72 |
1.4.5 Avalanches | p. 77 |
1.4.6 Windstorms | p. 83 |
1.4.7 Extreme sea levels and high waves | p. 86 |
1.4.8 Low flows and droughts | p. 89 |
1.4.9 Floods | p. 94 |
1.4.10 Wildfires | p. 106 |
2 Multivariate Extreme Value theory | p. 113 |
2.1 Multivariate Extreme Value distributions | p. 114 |
2.2 Characterization of the domain of attraction | p. 120 |
2.3 Multivariate dependence | p. 123 |
2.4 Multivariate return periods | p. 126 |
3 Bivariate Analysis via Copulas | p. 131 |
3.1 2-Copulas | p. 131 |
3.2 Archimedean copulas | p. 142 |
3.3 Return periods via copulas | p. 148 |
3.3.1 Univariate vs | p. 150 |
3.3.2 The "Or" case | p. 155 |
3.3.3 The "And" case | p. 157 |
3.3.4 Conditional return periods | p. 159 |
3.3.5 Secondary return period | p. 161 |
3.4 Tail dependence | p. 170 |
4 Multivariate Analysis via copulas | p. 177 |
4.1 Multivariate copulas | p. 178 |
4.2 Archimedean copulas | p. 183 |
4.3 Conditional mixtures | p. 186 |
4.3.1 The 3-Dimensional case | p. 186 |
4.3.2 The 4-Dimensional Case | p. 189 |
4.3.3 The general case | p. 190 |
5 Extreme Value Analysis via Copulas | p. 191 |
5.1 Extreme Value copulas | p. 191 |
5.2 Dependence function | p. 201 |
5.3 Tail dependence | p. 207 |
Appendix A Simulation of Copulas | p. 209 |
A.1 The 2-Dimensional case | p. 209 |
A.2 The general case | p. 212 |
Appendix B Dependence | p. 219 |
B.1 Bivariate concepts of dependence | p. 219 |
B.1.1 Quadrant dependence | p. 219 |
B.1.2 Tail monotonicity | p. 221 |
B.1.3 Stochastic monotonicity | p. 223 |
B.1.4 Corner set monotonicity | p. 224 |
B.1.5 Dependence orderings | p. 225 |
B.1.6 Measure of dependence | p. 225 |
B.2 Measures of association | p. 227 |
B.2.1 Measures of concordance | p. 227 |
B.2.2 Kendall's ¿k | p. 228 |
B.2.3 Spearman's ¿s | p. 230 |
Appendix C Families of Copulas | p. 233 |
C.1 The Frank family | p. 233 |
C.2 The Gumbel-Hougaard family | p. 236 |
C.3 The Clayton family | p. 237 |
C.4 The Ali-Mikhail-Haq (AMH) family | p. 240 |
C.5 The Joe family | p. 242 |
C.6 The Farlie-Gumbel-Morgenstern (FGM) family | p. 244 |
C.7 The Plackett family | p. 247 |
C.8 The Raftery family | p. 248 |
C.9 The Galambos family | p. 250 |
C.10 The Hüsler-Reiss family | p. 252 |
C.11 The Elliptical family | p. 254 |
C.12 The Fréchet family | p. 256 |
C.13 The Marshall-Olkin family | p. 257 |
C.14 The Archimax family | p. 264 |
C.15 Construction methods for copulas | p. 264 |
C.15.1 Transformation of copulas | p. 265 |
C.15.2 Composition of copulas | p. 266 |
C.15.3 Copulas with given diagonal section | p. 268 |
References | p. 271 |
Index | p. 285 |
Glossary | p. 291 |