1. Introduction to modelling extremes
Marian Scott
NERC Sept 2010

2. Introduction
what is an extreme?- examples of extremes in environmental contexts
Some statistical models for extremes
Block maxima, Peak over threshold
Including return levels
Return period
Statistical models for extremes are concerned with the tails of the distributions

3. problems
Normal distribution inappropriate- so we need some new distributions
Bulk of data not informing us about extremes
Extremes are rare, so not much data
Some statistical models for extremes
Block maxima, Peak over threshold
require parameter estimation which may prove difficult

4. Introduction
Modelling extremes, because we need to know about maxima and minima in many environmental systems to ensure that we know
How strong to make buildings
How high to make sea walls
How to plan for floods
etc

11. very common application
Flood Risk Management Act 2009 approved on 13th May 2009
River flow records studied to assess flood risk

12. Where do we start? Look at river flow records
Influenced by climate change in which way?
Interest in studying seasonal pattern
Is it constant?
Nature of its changes over the years
Changes in seasonal pattern might imply changes in extreme events

13. Data – River Nith South West of Scotland Catchment area of 799km2
18993 observations - 51 years (1/10/57 - 1/10/08) unusually long record!
Log transformed

14. Data(2)- consider the monthly maxima

15. Background
Assume typically that we have a time series of observations (eg maximum daily temperature for the last 20 years)
Assume that the data are independent and identically distributed (e.g might a Normal or Exponential be sensible or do we need other types of distributions?)
Interest is in predicting unusually high (or low) temperatures
Our statistical model needs to be good for the ‘tails’ of the distribution
Meet the distribution and cumulative distribution function

16. Background
The usual notation is to assume we have a series of random variables X1,X2,… each with cumulative distribution function G
Then G(x) is the probability (X<= x)
The inverse cumulative distribution function G-1(zp) is such that zp is the value of X such that Prob(X<=zp) =p

17. the pdf and cdf for X, where X is N(0,1)

18. the pdf and cdf for X, where X is exponential

19. Background
There exists a class of statistical models developed specifically for dealing with this situation
Generalised Extreme value (GEV) distribution, with three parameters and depending on the values of such parameters, can simplify to give Gumbel, Frechet and Weibull distributions for the maximum over particular blocks of time. Special distributions
Assumptions relating to the original time series: should be stationary (ie no trend)- see Richard’s section

20. G(z) =exp{-[1+ (z- )/ ]-1/ }
GEV distribution
Generalised Extreme value (GEV) distribution, has three parameters, location, scale and shape (usually written as , (>0) and . G(z) is the distribution function
G(z) =exp{-[1+ (z- )/ ]-1/ }
The Gumbel, Frechet and Weibull are all special cases depending on value of 

21. Block maxima
We can also break our time series X1, X2..into blocks of size n and only deal with the maximum or minimum in the block.
E.g if we have a daily series for 50 years, we could calculate the annual maximum and fit one of the statistical models mentioned earlier to the 50 realisations of the maxima.
GEV can then be applied to the block maxima etc
Quite wasteful of data (throws lots away)

22. Model fitting for GEV
estimates for the parameters of the distribution given observed ‘annual’ maxima X1,…., Xk
various approaches
graphical
likelihood based
for GEV it is possible to write down the likelihood, which is then numerically maximised
all this done in specialised software
how good is the model?

23. Model diagnostics for GEV
Probability plot
Quantile plot
Return level plot- will come back to this
Density plot
Probability and quantile plot should be straight lines.

24. POT modelling
There exists another type of statistical model developed specifically for dealing with this situation- known as Peak over threshold- (POT) modelling
Again we assume that we have a time series of observations, and define (somehow) a threshold u.
Typical distributions used here are Pareto, Beta and Exponential derived from the Generalised Pareto distribution (GPD)
How to define the threshold u is the big practical question

26. GPD model Asymptotic (so as u-> ) then G(y) = 1-(1+y/)-1/
 and  are shape and scale parameters
 =0 gives the exponential distribution with mean = 
How to define the threshold u is the big practical question
one answer is to try different values,

27. issues
Non-stationarity- eg in climate change there are trends in frequency and intensity of extreme weather events
There are cycles- annual, diurnal etc these are rather common
other.
What should be done|?
If there is a trend or cyclical component, then we need to de-trend/deseasonalise
Perhaps introduce covariates that can explain the non-stationarity

28. Issues for POT modelling
Often threshold exceedances are not independent.
Various ways to deal with this
Model the dependence
declustering
Another approach (depending on the application) might be to model the frequency and intensity of threshold excesses
Mean number of events in an interval [0, T] is T, where  is the frequency of occurrence of an event (so a rate)

29. How to communicate risk
Return level yp is the value associated with the return period 1/p. That is yp is the level expected to be exceeded on average once every 1/p years.
yp =G-1(1-p)
p=0.01 corresponds to the 100 year return period
The return level and return period are some of the most important quantities to derive from the fitted model (and as such are subject to uncertainty).

30. for the GEV, we can define G(yp)=1-p

31. Definition of return levels for POT
The level xm that is exceeded once every m observations is the solution of
u[1+ (x-u)/]-1/ = 1/m
where u is Pr(X>u)
Choose u such that GPD is a ‘good’ fit

32. Flood Estimation
AIM: to estimate the probability of an extreme event occurring in a given time period
eg the probability of Glasgow being flooded
In hydrology, there is a long history of methods designed to deal with extremes

33. Annual Floods
pq = the probability that discharge equals or exceeds q at least once in any given year; pq = annual exceedence probability
(1 – pq) = probability that this flood does NOT occur in a given year
Assume: stationarity; no long-memory

34. Recurrence Interval
Often refer to recurrence interval of floods (eg 1 in 200 year flood)
Recurrence interval: the average time between floods equaling or exceeding q
Recurrence interval (RIq) is the inverse of the exceedence probability (1/pq)

35. Extremes in space
Temperature network, many stations, at each station, we have a time series of annual maxima (for instance).
we need a spatio-temporal model (but this time it is extremes we are modelling)
as an exploratory analysis, fit a GEV or GPD to each station, map the parameters

36. Spatial extremes in river flows in Scotland
Exploratory analysis fitting GEV to each site
study the spatial pattern in GEV parameters- an alternative to max-stable processes

37. Summary
estimating extremes is inherently unreliable, even with large data sets
many environmental data sets are short,
various distributions may be used for estimation – which ones fit best in a particular situation is difficult to assess but diagnostic tools exist
data are assumed to be stationary – changing driving conditions, and long memory processes, may violate this assumption for many environmental data
but there are more complex models to deal with trends, seasonality etc

38. software some R packages available –ismev
Good book –Coles S, An introduction to modelling extremes.
Lots of very recent work looking at statistical models for extremes over space and time
try to work through the extremes.r script file and see handout in folder with some explanation of the results