Statistics of Weather and Climate Extremes

Background | History | Links | Quotes | References | Software| Toolkit
Hurricane Andrew

Hurricane Andrew 		Hurricane damage

Quotes

Emil Gumbel:

"It seems that the rivers know the theory. It only remains to convince the engineers of the validity of this analysis."

"Il est impossible que l'improbable n'arrive jamais."

Nassim Taleb:

"Extremistan: the province where the total can be conceivably impacted by a single observation."

John Tukey:

"As I am sure almost every geophysicist knows, distributions of actual errors and fluctuations have much more straggling extreme values than would correspond to the magic bell-shaped distribution of Gauss and Laplace."

President's Water Commission (1950):

"However big floods get, there will always be a bigger one coming; so says one theory of extremes, and experience suggests it is true."


Background

Purpose

 This web page is intended to serve as a resource for the use of the statistical theory of extreme values in the analysis of weather and climate extremes and their impacts. It also serves as a gateway to the Extremes Toolkit (extRemes). It was originally developed in conjunction with NCAR's Weather and Climate Impact Assessment Science Program (WCIAP), concerned with improving the scientific basis of assessments of the impacts of weather and climate on society (e.g., those of the U.N. Intergovernmental Panel on Climate Change, IPCC). See list of current research projects on extremes sponsored by WCIAP.

Statistical theory of extreme values
(1) Block Maxima

This theory has been well developed for quite a while. One important theorem states that the maximum of a sequence of observations, under very general conditions, is approximately distributed as the generalized extreme value (GEV) distribution. This distribution has three forms (click on image below to enlarge):
(i) Gumbel
A distribution with a light upper tail and positively skewed.

(ii) Frechet
A distribution with a heavy upper tail and infinite higher order moments.

(iii) Weibull
A distribution with a bounded upper tail. 		GEV distribution
(2) Peaks Over Threshold

In terms of the tail of a distribution, the corresponding theorem states that the observations exceeding a high threshold, under very general conditions, are approximately distributed as the generalized Pareto (GP) distribution. This distribution has three forms (click on image below to enlarge):
(i) Exponential
A light-tailed distribution with a "memoryless" property.

(ii) Pareto
A heavy-tailed distribution (sometimes called "power law").

(iii) Beta
A bounded distribution. 		GP distribution
The modern approach to extreme value analysis is based on a point process representation, equivalent to: (i) a Poisson process governing the rate of occurrence of exceedance of a high threshold; and (ii) a generalized Pareto distribution for the excess over the threshold. Through a reflection principle, the above theory can be converted into an equivalent form when the maximum is replaced by the minimum or an upper tail by a lower tail.

Weather and climate extremes

 With the computational advances and software developed in recent years, the application of the statistical theory of extreme values to weather and climate has become relatively straightforward. Annual and diurnal cycles, trends (e.g., reflecting climate change), and physically-based covariates (e.g., El Nino events) all can be incorporated in a straightforward manner. Consistent with the point process representation, the "peaks over threshold" (or "partial duration series") approach enables the use of more of the information available about the upper tail of the distribution (e.g., not just the annual maxima).

Return levels and return periods
The concepts of return level and return period are commonly used to convey information about the likelihood of rare events such as floods. A return level with a return period of T = 1/p years is a high threshold x(p) (e.g., annual peak flow of a river) whose probability of exceedance is p. For example, if p = 0.01, then the return period is T = 100 years.

Two common interpretations of a return level with a return period of T years are:

Return level
(i) Waiting time: Average waiting time until next occurrence of event is T years

(ii) Number of events: Average number of events occurring within a T-year time period is one


History


The story behind some of the names that are attached to extreme value distributions:
Maurice Frechet Maurice Frechet: 1878-1973

French mathematician who made major contributions to pure mathematics as well as probability and statistics. He also collected empirical examples of heavy-tailed distributions. The Frechet type of extreme value distribution is named after him (this distribution has a heavy tail).
Emil Gumbel Emil Gumbel: 1891-1966

Born and trained as a statistician in Germany, he was forced to move to France and then the U.S. because of his pacifist and socialist views. He was a pioneer in the application of extreme value theory, particularly to climate and hydrology. The Gumbel distribution is named after him.
Vilfredo Pareto Vilfredo Pareto: 1848-1923

Born in France, he was trained as an engineer in Italy, but turned to the social sciences and ended his career in Switzerland. He formulated the power-law distribution (or "Pareto's Law"), as a model for how income or wealth is distributed across society. The Pareto distribution is named after him.
Waloddi Weibull Waloddi Weibull: 1887-1979

Swedish engineer famous for his pioneering work on reliability, providing a statistical treatment of fatigue, strength, and lifetime in engineering design. The Weibull distribution is named after him. Today it is a popular distribution for use in modeling lifetimes.

Other names in extreme value theory:
Ladislaus von Bortkiewicz Ladislaus von Bortkiewicz: 1868-1931

Russian-born mathematician who spent most of his life in Berlin. He coined the term "law of small numbers" and is best known for showing that the Poisson distribution provides a good fit to the number of Prussian officers killed by horse kicks. He also made significant contributions to the field of economics.
Richard von Mises Richard von Mises: 1883-1953

Trained in mathematics in Vienna, he worked in Austria and Germany, until forced to move to Turkey and then the U.S. because of being part Jewish. Well known for fundamental contributions to probability theory, he obtained sufficient conditions for the three types of generalized extreme value distribution.
Simeon Denis Poisson Simeon Denis Poisson: 1781-1840

French mathematician who derived what is now called the Poisson distribution as an approximation to the binomial distribution for rare events. Several other concepts in mathematics and physics are also named after him.

Toolkit


The Extremes Toolkit (extRemes) consists of functions written in "R" (The R Project for Statistical Computing) to perform extreme value analysis. Knowledge of R is not necessarily required, as a graphical user interface is provided. A tutorial explains how the toolkit can be used to treat weather and climate extremes in a realistic manner (e.g., taking into account diurnal and annual cycles, trends, physically-based covariates). This toolkit is being developed and maintained by Eric Gilleland (it was originally started by Greg Young). We gratefully acknowledge permission from Stuart Coles to make use of his "S" functions.
Extremes Toolkit

Software


Software for Extreme Value Theory

Web page on software for extremes maintained by Eric Gilleland.

Stephenson, A., and E. Gilleland, 2006: "Software for the analysis of extreme events: The current state and future directions." Extremes, 8, 87-109.

Review paper on software for extreme value analysis.

evd

"R" functions for statistical analysis of extremes, including multivariate extremes and Bayesian methods. Maintained by Alec Stephenson.

extRemes

Graphical user interface based on ismev. Written and maintained by Eric Gilleland.

ismev

"R" functions including classical extreme value analysis of fitting generalized extreme value distribution to "block maxima," as well as generalized Pareto distribution to excesses over a high threshold. Original functions written in "S" by Stuart Coles, converted to "R" by Alec Stephenson, and maintained by Eric Gilleland.

XTREMES

Menu-driven system with both block maxima and peaks over threshold options. Includes hydrology component. Companion to book Statistical Analysis of Extreme Values by R.-D. Reiss and M. Thomas.

References


Bibliography: Statistics of extremes in climate change

Click here to access a bibliography of recent papers, appearing in the atmospheric science and related literature, applying the statistics of extremes to climate change


Books and lecture notes on statistics of extremes

An Introduction to Statistical Modeling of Extreme Values by Stuart Coles (Springer, 2001)
Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology, and Other Fields by R.-D. Reiss and M. Thomas (Birkhauser, Third Edition, 2007)
Lecture notes on environmental statistics by Richard Smith (see Chapter 8 on extremes)


Reviews on applying statistics of extremes to weather/climate/related areas

IPCC Workshop on Changes in Extreme Weather and Climate Events, Beijing, China, 11-13 June 2002 (see Section 7 on statistical methods, pp. 31-32).
Weather and Climate Extremes in a Changing Climate, U.S. Climate Change Science Program, June 2008 (see Chapter 2).
Engeland, K., H. Hisdal, and A. Frigessi, 2004: "Practical extreme value modelling of hydrological floods and droughts: A case study." Extremes, 7, 5-30.
Garrett, C., and P. Muller, 2008: "Extreme events." Bulletin of the American Meteorological Society, 89, ES45-ES56. (Online Supplement)
Katz, R.W., G.S. Brush, and M.B. Parlange, 2005: "Statistics of extremes: Modeling ecological disturbances." Ecology, 86, 1124-1134.
Katz, R.W., M.B. Parlange, and P. Naveau, 2002: "Statistics of extremes in hydrology." Advances in Water Resources, 25, 1287-1304.
Naveau, P., M. Nogaj, C. Ammann, P. Yiou, D. Cooley, and V. Jomelli, 2005: "Statistical methods for the analysis of climate extremes." Comptes Rendus Geoscience, 337, 1013-1022.
Palutikof, J.P., B.B. Brabson, D.H. Lister, and S.T. Adcock, 1999: "A review of methods to calculate extreme wind speeds." Meteorological Applications, 6, 119-132.


Simulation of extremes by stochastic weather generators

Furrer, E.M., and R.W. Katz, 2008: "Improving the simulation of extreme precipitation events by stochastic weather generators." Water Resources Research, 44, W12439, doi:10.1029/2008WR007316.
Kysely, K., and M. Dubrovsky, 2005: "Simulation of extreme temperature events by a stochastic weather generator: Effects of interdiurnal and interannual variability reproduction." International Journal of Climatology, 25, 251-269.
Qian, B., S. Gameda, and H. Hayhoe, 2008: "Performance of stochastic weather generators LARS-WG and AAFC-WG for reproducing daily extremes of diverse Canadian climates." Climate Research, 37, 17-33.
Semenov, M.A., 2008: "Simulation of extreme weather events by a stochastic weather generator." Climate Research, 35, 203-212.


Verification of forecasts of extreme weather events

Web page by Barbara Casati on Analysis and Verification of Weather and Climate Extremes
Ferro, C.A.T., 2007: "A probability model for verifying deterministic forecasts of extreme events." Weather and Forecasting, 22, 1089-1100.

Katz, R.W., 2006: "Forecast verification of extremes: Use of extreme value theory." American Meteorological Society, 18th Conference on Probability and Statistics in the Atmospheric Sciences, Atlanta, GA.

Stephenson, D.B., B. Casati, C.A.T. Ferro and C.A. Wilson, 2008: "The extreme dependency score: A non-vanishing measure for forecasts of rare events." Meteorological Applications, 15, 41-50. 		Yakima temperature

Links


CCES - EXTREMES

Spatial Extremes and Environmental Sustainability: Statistical Methods and Applications in Geophysics and the Environment

Extremes: Statistical Theory and Applications in Science, Engineering and Economics

Journal published by Kluwer Academic Publishers, now Springer, since 1998

Statistics of Extreme Weather and Climate Events

Climate Analysis Team, University of Exeter

Background | History | Links | Quotes | References | Software| Toolkit
For more information, contact Richard Katz (email: rwk@ucar.edu).

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