EXTREMEVALUE ANALYSIS: FOCUSING ON THE FIT AND THE CONDITIONS, WITH HYDROLOGICAL APPLICATIONS

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EXTREME-VALUE ANALYSIS FOCUSING ON THE FIT AND
THE CONDITIONS, WITH HYDROLOGICAL APPLICATIONS
Dávid Bozsó, Pál Rakonczai, András
Zempléni Eötvös Loránd University, Budapest
4th Conference on Extreme Value
AnalysisProbabilistic and Statistical Models and
their Applications
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Table of contents

• Goodness of fit procedures
• Checking the conditions D, D(un) and D(un).
• Multivariate problems
• Copulas
• Simulations
• Goodness of fit tests for copulas
• Time dependence
• Hydrological applications

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Generalized Pareto distribution

• Peaks over a sufficiently high threshold u can be
  modeled by the generalized Pareto distribution
  (under mild conditions)
• 
  
• Appropriate threshold selection is very important

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Goodness of fit in univariate threshold models

• Usual goodness-of-fit tests (Chi-squared,
  Kolmogorov-Smirnov) are not sensitive for the
  tails
• A better alternative is the Anderson-Darling test
• ,
• where the discrepancies near the tails get
  larger weights. Its computation

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Goodness of fit - continued

• Modification often the focus is on one tail only
  
• For maximum
• (Zempléni, 2004)
• Computation
• Critical values can be simulated (like in
  Choulakian and Stevens, 2001)

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Finding thresholds

• Theoritical results related to GPD are doubly
  asymptotic, since not only the sample size but
  the threshold has to converge to infinity as well
• How can we find suitable thresholds?
• Suggestion
• Increase the threshold level step by step
• Fit the GPD (by ML method for example) and
  perform AD-type tests in all of the cases
• Select some levels, for which the fit is
  acceptable
• For more details, see Bozsó et al, 2005

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Hydrological applications

• Daily water level data from several stations
  along the river Tisza were given (time span more
  than 100 years)
• As an illustration we have chosen Szeged station,
  but in fact we have repeated the suggested
  procedures (almost) automatically for all the
  stations
• In later parts of the talk we shall also use data
  from Csenger (river Szamos)

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Finding thresholds
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Focusing on the conditions

• So far
• Threshold selection
• Fit a GPD model for data over the selected
  threshold
• for iid data
• Dependence is present
• Possible long range dependence?
• Are the return levels affected by it?

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Condition D and D(un)
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How to check condition D ?

• Set p1 and r1 in the definition of condition D
  and choose threshold u as the level of interest,
  e.g. 400 or 430 cm in our example
• Calculate
• for each lag l1,,1000

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Applications daily water level data

• 400 cm 80 quantile
• 430 cm 83 quantile
• Compare with d(l) for well-known sequences
• iid, normally distributed sequence
• AR(1) series

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Applications daily water level data

• Hydrological data (level 430
  cm)
• Normal iid sequences
• Sample mean
• 95 quantile
• AR(1) sequences
• Sample mean
• 95 quantile
• Simulation study confirms our hypothesis,
  empirical data is in the 95 confidence interval

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Condition D(un)

• Practical procedure select a sequence (un),
  calculate
• and plot it as a function of k

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Applications daily water level data

• Hydrological data
• Normal iid sequence
• Sample mean
• Ynmax(Xn,Xn1), where X2 has a standard normal
  distribution
• Sample mean

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Multivariate models

• Copulas are very useful tools for investigating
  dependence among the coordinates of multivariate
  observations
• The marginal distributions and the dependence
  structure can be modeled separately!
• Which parametric models to use for the
  hydrological applications? (in two dimensions)

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Hydrological applications

• Water level peaks measured in two different
  stations are shown (peaks were coupled to each
  other if occured nearer than one month)
• With the help of the earlier algorithm we can
  choose threshold levels (blue lines) and fit GPD
  to the marginals

Only those peaks are used, which are extremal in
both coordinates!
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QQ-Plot for marginals
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Empirical copula

• After transforming the data into uniform
  marginals the empirical copula is obtained
• Which parametric copula is the most adequate
  for the given application?

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Conceivable copulas in 2D

• Elliptical copulas
• Gauss
• Student-t
• Archimedian copulas
• Gumbel
• Clayton
• Other copulas
• Frechet

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Simulation - Gauss
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Simulation Student-t
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Simulation Clayton I.
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Simulation Clayton II.
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Simulation Gumbel
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Goodness of fit for copulas

• Cramér-von Mises and Kolmogorov-Smirnov
  functionals of might be used to test the
  null hypotesis
• A simple approach, which is based on the
  multivariate probability integral transformation
  of F, is defined by
• where (U1,...,Ud ) is a vector of uniform
  variables having C as their joint distribution

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Visual comparison

• Genest et al (2003) proposed a graphical
  procedure for model selection through the visual
  comparison of the non-parametric estimate Kn(.)
  of K to the parametric estimate K(?n,.)
• ,where
• The better the fit is, the closer the graphs of
  these functions are
• Question how to define the distance between the
  graphs?

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Weighted quadratic differences
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Which weights to use?

• In order to compare which test statistics
  performs better at detecting discrepancies in the
  upper tail we applied the following algorithm
• 1. Simulate a sample from a parametric copula
• 2. Randomly choose two not concordant points
  (x,y) near the right tail and permute their
  coordinates so that the new points x,y are
  concordant (the marginals do not change) but
  the copula changes
• 3. Perform the three versions of the test for the
  modified data set
• 4. Repeat steps 2 and 3, and investigate which
  statistics is faster in detecting the changes

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The data and its permutations
The number in the title gives the number
of changed pairs
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Detecting changes

• In general the tests based on weigthed squared
  deviation perform better than the original one..
• Among the two weighted tests, the modified
  version is more sensible!

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Simulation results

• We recorded how many steps the different tests
  needed to detect the changes during the
  replications
• As expected, the modified weights were the best!

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Time dependence

• Has the dependence structure of the observations
  changed in the last century?
• Windows of 80 years with a step size of 5 years
  were used to detect possible changes
• Firstly we have to decide which copula to use

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Time dependence

• In all of the three cases the Gumbel copula
  seems to be better than Frechet!

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Simulated critical values
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Applications for the hydrological data set time
dependence

• The only (marginally) significant value is marked
  with
• A simulation study may be used for detecting
  changes in the dependence
• structure

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References

• Bozsó, D., Rakonczai, P. and Zempléni, A. (2005).
  Floods on river Tisza and some of its affluents.
  Extreme-value modelling in practice. Statisztikai
  Szemle, accepted for publication. (In Hungarian.)
• Choulakian, V. and Stephens, M.A. (2001).
  Goodness-of-fit tests for the genaralized Pareto
  distribution. Technometrics 43, 478-484.
• DAgostino, R.B. and Stephens, M.A. (1986).
  Goodnes-of-fit Techniques. Marcell Dekker.
• Genest, C. Quessy, J.-F. and Rémillard, B.
  (2003). Goodnes-of-fit Procedures for Copula
  Models Based on the Integral Probability
  Transformation. GERAD.
• Leadbetter, M. R. - Lindgren, G. and Rootzen, H.
  (1983). Extremes and Related Properties of Random
  Sequences and Processes, Springer.
• Zempléni, A. (2004). Goodness-of-fit test in
  extreme value applications. Discussion paper No.
  383, SFB 386, Statistische Analyse Diskreter
  Strukturen, TU München.