Transcript Árvíz-modellezési kérdések

New approaches in extreme-value
modeling
A.Zempléni, A. Beke, V. Csiszár
(Eötvös Loránd University, Budapest)
Flood Risk Workshop,
08.07.2002
Analysis of extreme values
• Probably the most important part of the project (its
aim: to estimate return levels – values which are
supposed to be observed once in a given period,
100 years for example).
• Classical methods: based on annual maxima (other
values are not used).
• Peaks-over-threshold methods: utilize all values
higher than a given (high) threshold.
Extreme-value distributions (for
modeling annual maxima)
Let X1, X2,…,Xn
be independent, identically
distributed random variables. If we can find
norming constants an, bn such that
[max(X1, X2,…, Xn)-an]/ bn
has a nondegenerate limit, then this limit is
necessarily a max-stable or so-called extreme
value distribution.
Characterisation of extreme-value
distributions
• Limit distributions of normalised maxima:

F ( x)  exp(  x )
Frechet:
(x>0)
 is a positive parameter.

F
(
x
)

exp(

(

x
)
) (x<0)
Weibull:

F ( x)  exp(  exp(  x))
Gumbel:
(Location and scale parameters can be
incorporated.)
Estimation methods
• Maximum likelihood, based on the unified
parametrisation:
x   1/ 
F ( x)  exp{ [1  
] }
x

0
• if 1  

• the most widely used, with optimal asymptotic
properties, if ξ>-0.5
• Probability-weighted moments (PWM)
• Method of L-moments
Probability-weighted moments
• Analogous to the method of moments
• It puts more weight to the high values
r ( )  E(ZF r (Z ))
• The estimators are got by equating the
empirical and the theoretical weighted
moments and solving the equations for the
parameter vector.

 r ( )   zFr ( z)dFn ( z)

Method of L-moments
• The basic characteristics (mean, variance,
skewness and kurtosis) of the observed
distribution are equated to their respective
theoretical values.
• These values can be estimated by the help
of the probability weighted moments.
Comparison
• Maximum likelihood is preferable, since
– asymptotic properties are known, allowing the
construction of confidence intervals
– covariates can be incorporated into the model
• For the other methods, there is no firm
theory.
Further investigations
• Estimates for return levels
• Confidence bounds should be calculated,
possible methods
– based on asymptotic properties of maximum
likelihood estimator
– profile likelihood
– resampling methods (bootstrap, jackknife)
– Bayesian approach
Confidence intervals
• For maximum likelihood:
– By asymptotic normality of the estimator:
 ~ N ( , i ,i )
where  i,i is the (i,i)th element of the inverse of
the information matrix
– By profile likelihood
• For other nonparametric methods by
bootstrap.
Profile likelihood
• One coordinate of the parameter vector is fixed,
the maximization is with respect the other
components:
 p (i )  max (ii )
i
• Its main advantages:
– The uncertainty can be visualized
– More exact (asymmetric) confidence bounds
– Model selection for nested models is possible by the
likelihood ratio test
Model diagnostics
• Probability plot (P-P plot), the points:
i
(n)
ˆ
{( F ( xi ),
)}
n 1
• Quantile plot (Q-Q plot), the points:
i
( n ) ˆ 1
{( xi , F (
))}
n 1
Both diagram should be close to the unit diagonal if
the fit is good.
An example: a simulated 100element sample of unit exponentials
Peaks over threshold methods
• Those events are considered extreme, which
exceed a given (high) threshold
• Advantages:
– More data can be used
– Estimators are not affected by the small “floods”
• Disadvantages:
– Dependence on threshold choice
– Declustering not always obvious
Theoretical foundations
Let X1, X2,…,Xn be independent, identically
distributed random variables. If the normalised
maximum of this sequence converges to an
extreme value distribution (with parameters μ,σ,ξ),
then
y 1/ 
P( X  u  y | X  u )  1  (1  ~ )

if y>0 and 1  y / ~  0 where ~     (   u )
The asymptotics holds if n and u increases.
Inference
• Similar to the annual maxima method:
– Maximum likelihood is to be preferred
– Confidence bounds can be based on profile
likelihood
– Model fit can be analyzed by P-P plots and Q-Q
plots
– Return levels/upper bounds can be estimated
Threshold selection
Mean excess plot:
For any u (threshold), plot the mean of X-u (for
those observations for which X>u) against u.
If the Pareto model is true, this plot should be nearly
linear.
The interpretation is made difficult by the great
variability near the upper endpoint of the
observations.
Another, very recent method:
maximum cross-entropy
• Kullback introduced the concept of probabilistic
distance (cross-entropy) of a posterior distribution
h(t) from a prior f(t):

 h( x ) 
dx
I (h, f )   h( x) ln 
 f ( x) 

• The method (Pandey, 2002) minimizes the crossentropy of x(t) (the observed quantile function of
exceedances) with respect to its prior estimate y(t),
which is chosen as the exponential (motivated by
its central role within the GPD family).
Moment conditions
PWM constraints:
1
k
x
(
u
)
u
du  bk (k  0,1,, N )

0
(usually N=3 is used), where
 i  1

 X i

1 i 1  k 
bk 
n  n  1


 k 
n
is unbiased estimator for the kth PWM, based on the
ordered sample of size n
Some results for Vásárosnamény
• Six threshold values were used: 440, 480, 520,
560, 600 and 640 centimetres.
• As constraints, we considered the first four
PMWs, that is, N=3.
• The estimated 100-, 500-, and 1000-year return
levels
threshold
100-year
500-year
1000-year
440cm
1099
1279
1356
480cm
1035
1190
1257
520cm
1002
1141
1202
560cm
985
1112
1168
600cm
982
1102
1155
640cm
965
1072
1119
Comments
• The results were not as stable as it was
claimed in the original paper.
• We intended to add bootstrap confidence
bounds to the estimates, but this was too
time consuming in its original version and
the used simplifications have not proven to
be realistic.
Stationary sequences
• If the independence does not hold (as it is the case
for the original daily observations), the limit of the
normalized maxima is still a GEV distribution, if
the dependence among far away observations tend
to diminish. (See the talk of S. Gáspár.) So the
GEV model for annual maxima has sound
theoretical background.
• For POT models, the maximum of the clusters of
exceedances may be used. (Clusters need to be
defined).
To cope with nonstationarity
• Linear regression-type models can be
incorporated into the maximum likelihood
framework
• Profile likelihood, likelihood-ratio tests can
be performed for nested models
Water-level data example:
Vásárosnamény
• At least two observations per day for each station
(there are approx. 50 of them) for 100 years.
• Reduction: one observation per day.
Annual maxima
The fitted model (shape=-0.4459,
right-endpoint: 778 cm)
Another station (estimated max: 945 cm)