Download Extreme Value Theory

Extreme Value Theory
Richard L. Smith
Department of Statistics and Operations Research
University of North Carolina, Chapel Hill
[email protected]
www.unc.edu/∼rls
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Background
This is intended as a preliminary version of the presentation to be given at the AMS conference
Statistics of Extreme Events, January 11 2009. The actual presentation will be posted, when it is
available, at http://www.stat.unc.edu/faculty/rs/talks/talks.html.
Extreme Value Theory is the branch of statistics that is used to model extreme events. The topic
is of interest to meteorologists because much of the recent literature on climate change has focussed
on the possibility that extreme events (very high or low temperatures, high precipitation events,
droughts, hurricanes etc.) may be changing in parallel with global warming. As a specific example,
the paper by Stott, Stone and Allen (2004) used the generalized Pareto distribution (see Section
2) to estimate the probability of the European heatwave event of 2003 under two conditions, (a)
based on climate model data without an anthropogenic signal, (b) including anthropogenic effects
(greenhouse gases etc.). They estimated a probability of about 1/1000 under (a) but about 1/250
under (b). Although even the probability under (b) is low, the increase in probability compared with
(a) led them to conclude that the fraction of attributable risk due to the anthropogenic influence is
about 75%. Another example of the use of statistics to examine trends in probabilities of extreme
events is the recent paper by Elsner et al. (2008), which is highly relevant to the question of whether
there is an increasing trend in severe hurricanes that may possibly be associated with anthropogenic
global warming.
However, although recent attention has naturally focussed on these and similar climatic events,
there is in fact a long history, stretching back to Gumbel’s work of the 1930s, for the use of extreme
value theory to characterize extremes in meteorological and hydrological events. Examples include
use of the annual maximum method to estimate a 100-year windspeed or flood based on a long
time series of annual maximum events. Over the past 30 years, attention has shifted more towards
methods based on exceedances over high thresholds, of which the generalized Pareto distribution is
one example. There is also an increasing focus given to extreme value theory in more complicated
settings, in particular, extremes of spatial processes. The intention of this lecture is to review some
of the current leading methods. Further details may be found in Coles (2001) and Smith (2003),
among other recent references on extreme value theory.
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Basics of Extreme Value Theory
The foundation of extreme value theory is a family of probability distributions known as extreme
value distributions. The can be written in various ways, but one popular representation, known as
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the generalized extreme value or GEV distribution, is the formula
(
µ
y−µ
Pr {Y ≤ y} = G(y ; µ, ψ, ξ) = exp − 1 + ξ
ψ
¶−1/ξ )
(1)
+
where Y is some random quantity and the parameters µ, ψ, ξ represent respectively a location
parameter, scale parameter and shape parameter for the distribution. All three parameters are
important but ξ is especially critical for extrapolation of extreme events: a value ξ ≤ 0 is generally
interpreted as a short-tailed distribution but ξ > 0 is interpreted as long-tailed, and this is the
most interesting case in many situations. In (1), the notation x+ means the maximum of x and 0;
this is used to restrict the range of Y under certain circumstances.
The distribution (1) was originally derived as the limiting distribution of maxima of independent
and identically distributed (IID) random variables, under certain (mild) restrictions of the sampling
distribution. From a modern perspective, particularly in meteorological applications, it is most
often applied through the annual maximum method, in which the data points Y1 , Y2 , ..., are annual
maxima of some meteorological variable. The annual maximum method is also sometimes called
the block maximum method, to signify that the observations may also be taken as maxima over
blocks of length other than one year (for instance, monthly maxima), though in the latter case it
would be usual to adjust (1) to account for seasonality.
In estimating the parameters µ, ψ and ξ from observations, there are a number of popular
methods including maximum likelihood estimation (MLE), Bayesian methods, and the L-moments
technique that has been popularized by Hosking and Wallis. Since I tend to favor MLE as the
default method of estimation of any probability distribution, I shall focus on that in this talk.
Alternative approaches may be based on exceedances over thresholds. The basic method here
is to choose some threshold, usually written u, and form exceedances over the threshold u. That
is, if the original data are written Xt , t ∈ T (T is a time index set), then we essentially ignore
observations for which Xt < u, but the exceeses Xt − u, given Xt > u, are modeled with suitable
probability distributions. The most common distribution here is the generalized Pareto distribution
(GPD), given by
µ
y
H(y ; φ, ξ) = 1 − 1 + ξ
φ
¶−1/ξ
.
(2)
+
In (2), Y is the excess over the threshold (so Y ≥ 0 by definition), φ is another scale parameter and
ξ has the same interpretation as in (1) — in particular, cases when ξ > 0 correspond to long-tailed
distributions. Usually, some preliminary declustering of the data is done to ensure that separate
exceedances come from independent events, and it may also be necessary to modify (2) to account
for seasonality.
Two alternative methods are
• The r largest order statistics approach, in which, instead of just taking annual maxima, we
take the r largest values per year, where r is usually some fairly small number, e.g. 3 or 5.
This approach has been used in cases where individual data points are not readily available,
but since this is only rarely the case in meteorology, it is not widely applied to meteorological
data.
• The point process approach, in which a construction based on the statistical theory of point
processes is used to define a distribution for threshold exceedances that is based on (1) instead
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of (2). The advantage of this approach tends to be seen in problems involving covariates, which
are often more conveniently represented in the GEV parameters than the GPD parameters.
Apart from these different representations of the probability distributions, there are also numerous different methods for applying them to meteorological and other kinds of data. In my
presentation I shall particularly emphasize two aspects of practical application:
• Using covariates to represent trends, seasonal factors and other external variables that may
affect the distribution of extremes,
• Diagnostics including probability plots, mean excess over threshold plots, residual plots and
others, that are used to decide whether a model fits adequately to the data. In the case of
threshold methods, these methods are also used to assist in selection of a suitable threshold.
If time permits I may also include some brief discussion of Bayesian methods. In a Bayesian
approach, we start with a prior distribution on the unknown parameters and apply Bayes’ Theorem
to calculate a posterior distribution based on the data. The posterior distribution can then be used
to calculate predictive distributions for future events. The advantage of this over an MLE or Lmoments approach is that we can incorporate uncertainty over the estimated parameters directly
into the predictive distribution.
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Extensions: Spatial Statistics
In recent years, many extensions of extreme value theory have been proposed, but there is particular
interest in extensions involving spatial statistics. A canonical example is the precipitation extremes
problem: we have data on daily precipitation at many meteorological stations, but how should we
best combine the information spatially? Two particular ideas have been explored:
• Estimate an extreme value distribution for each station, then use spatial statistics methods
to combine the information across stations,
• Use multivariate extreme value theory and its extension, the theory of max-stable processes,
to characterize joint probabilities of extreme events at multiple sites.
Most likely, I shall not have time to discuss both these problems in detail during the lecture.
But I shall at least say a few words about ongoing research in these topics.
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References
Coles, S.G. (2001), An Introduction to Statistical Modeling of Extreme Values. Springer Verlag,
New York.
Elsner, J.B., Kossin, J.P and Jagger, T.H. (2008), The increasing intensity of the strongest
tropical cyclones. Nature 455, 92–95 (September 4 2008).
Smith, R.L. (2003), Statistics of extremes, with applications in environment, insurance and
finance. Chapter 1 of Extreme Values in Finance, Telecommunications and the Environment, edited
by B. Finkenstadt and H. Rootzen, Chapman and Hall/CRC Press, London, pp. 1-78.
(See http://www.stat.unc.edu/postscript/rs/semstatrls.pdf)
Stott, P.A., Stone, D.A. and Allen, M.R. (2004), Human contribution to the European heatwave
of 2003. Nature 432, 610–614 (December 2 2004).
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