Transcript Extremes_CCT_v3 - Biomathematics and Statistics Scotland

Extreme values and risk
CCTC meeting, September 2007
Adam Butler
Biomathematics & Statistics Scotland
Extreme values and risk
• Extreme value theory (EVT) is a branch of statistics
concerned with the frequency & size of rare events
• EVT methods are widely used in finance, hydrology &
engineering, usually for risk assessment, but are not
yet widely used in the biological sciences
Extreme values and risk
Risk assessment:
What is the probability we will
have more than 100mm of
rain on a given day?
Risk management:
I need to build a flood
defense, and I want the
probability that it fails on
any particular day to be
less than 1-in-10000.
How high should it be?
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What is the chance of getting a log
daily return of less than –0.1?
(i.e. a drop in value of 9% or more
since the previous day)
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Common features
• We are interested in a process that can be quantified,
and for which we have some data
• …and we want to use these data to say something about
the probability that a rare or extreme event will occur
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• We will usually be interested in events that are beyond
the range of the data i.e. we want to extrapolate
• Extrapolation is rarely advisable, but it is sometimes
unavoidable, especially when doing risk assessment
• The standard approach would be to assume that the
data come from, for example, a normal distribution…
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P(X < –0.1)  10-20
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…but:
The extreme values don’t play much of a role when we
estimate the parameters, so the model that we end up
fitting might not describe the extreme values at all well…
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Empirical:
P(X < –0.05)  0.002
Normal:
P(X < –0.05)  0.000001
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…and, worse still, extrapolations beyond the range of the
data often differ radically between models that provide
a very similar fit to the bulk of the data…
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Cauchy:
P(X < –0.1)  0.02
Normal:
P(X < –0.1)  10-20
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In EVT we adopt the principle that we should only make
use of the most extreme data that we have observed
 we throw away almost all of the data
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Threshold
exceedances
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We consider exceedances of a high threshold
EVT tells us that a good statistical model for exceedances,
x, is the Generalised Pareto Distribution (GPD),
P(x) = 1 – [1 + (x / )]-1/
 = “scale parameter”
 = “shape parameter”
(x > 0)
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GPD: impact of the scale parameter
 = “scale parameter”
=1
=2
=3
=0
 = “shape parameter”
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GPD: impact of the shape parameter
 = “scale parameter”
=0
=1
 = -0.5
=1
 = “shape parameter”
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Threshold = u = 25mm
 and  estimated by maximum likelihood
to be 7.70 and 0.108 respectively
P(X > 100) estimated to be 0.0000209
(once in a 131 years)
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…but why is the GPD the “right model” to use?
• In theory: for almost any random variable X, the
exceedances of a high threshold u will tend towards the
GPD model as u tends towards infinity
• In practice: we use a threshold that is high but still finite:
we rely on the fact that if this level is sufficiently high
then the asymptotic result will still be approximately true
Extreme value methods
“Parameter stability plot”
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Other extreme value models
• A related approach involves analysing the maximum
values per day, per month or per year (block maxima)
• EVT suggests that a good model to use in this case is
the GEV (Generalised Extreme Value)
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Advantages
• Robust
Disadvantages
• Inefficient
Relies on weak assumptions
Most of the data are thrown away
Avoids bias
…we may over-estimate uncertainty
• Theoretically sound
Justified by asymptotic theory
• Quick & relatively easy to use
• Honest
…about the uncertainties involved
in making statements about very
rare events
…relies on having a large sample size
• Asymptotics
The theory only holds exactly for
infinitely extreme events
Difficult to extend to multivariate case
• Data quality
Sensitive to errors in extreme data
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Practicalities
Basic course: http://www.bioss.ac.uk/alarm/training/
• Software: routines in… R, Genstat, S-plus, Matlab
• Extremes toolkit:
http://www.isse.ucar.edu/extremevalues/evtk.html
• Recommended book: Coles (2001) An introduction to
statistical modeling of extreme values. Springer.
• Contact me: [email protected]