Methodological summary of flood frequency analysis

1
Methodological summary of flood frequency
analysis

• A.Zempléni
• (Eötvös Loránd University, Budapest)
• 13.04.2004

2
Analysis of extreme values

• Classical methods based on annual maxima
• Peaks-over-threshold methods utilize all floods
  higher than a given (high) threshold.
• Multivariate modelling
• Bayesian approach (dependence among parameters)
• Joint behaviour of extremes

3
Extreme-value distributions

• Let be independent, identically distributed
  random variables. If we can find norming
  constants an, bn such that
• has a nondegenerate limit, then this limit is
  necessarily a max-stable or so-called extreme
  value distribution.
• The conditions are related to the smoothness of
  the density of the sample elements, are fulfilled
  by all of the important parametric families.

X1, X2,,Xn
max(X1, X2,, Xn)-an/ bn
4
Characterisation of extreme-value distributions

• Limit distributions of normalised maxima
• Frechet (xgt0)
• is a positive parameter.
• Weibull (xlt0)
• Gumbel
• (Location and scale parameters can be
  incorporated.)

5
Another parametrisation
The distribution function of the generalised
extreme-value (GEV) distribution
if
? location, ?? scale, ? shape parameters ?gt0
corresponds to Frechet, ?0 to Gumbel ?lt0 to
Weibull distribution
6
Examples for GEV- densities
7
Check the conditions

• Are the observations (annual maxima)
• independent? It can be accepted for most of the
  stations.
• identically distributed? Check by
• comparing different parts of the sample. For
  details, see the next talk.
• fitting models, where time is a covariate.
• follow the GEV distribution?

8
Tests for GEV distributions

• Motivation limit distribution of the maximum of
  normalised iid random variables is GEV, but
• the conditions are not always fulfilled
• in our finite world the asymptotics is not always
  realistic
• Usual goodness-of-fit tests
• Kolmogorov-Smirnov
• ?2
• Not sensitive for the tails

9
Alternatives

• Anderson-Darling test
• Computation
• where ziF(Xi). Sensitive in both tails.
• Modification
• (for maximum upper tails). Its computation

10
Further alternatives

• Another test can be based on the stability
  property of the GEV distributions for any m ?N
  there exist am, bm such that F(x)Fm(amxbm) (x
  ?R)
• The test statistics
• Alternatives for estimation
• To find a,b which minimize h(a,b)
  (computer-intensive algorithm needed).
• To estimate the GEV parameters by maximum
  likelihood and plug these in to the stability
  property.

11
Limit distributions

• Distribution-free for the case of known
  parameters. For example
• where B denotes the Brownian Bridge over 0,1.
• As the limits are functionals of the normal
  distribution, the effect of parameter estimation
  by maximum likelihood can be taken into account
  by transforming the covariance structure.
• In practice simulated critical values can also
  be used (advantage small-sample cases).

12
Power studies

• For typical alternatives, the test A-D seems to
  outperform B. The power of h very much depends on
  the shape of the underlying distribution.
• The probability of correct decision (p0.05)

n Test 100 200 400 100 200 200 400
Distr. NB exp Normal
B 0.02 0.27 0.49 0.17 0.58 0.05 0.08
A-D 0.31 0.62 0.96 0.72 0.97 0.21 0.34
h 0.67 0.87 0.99 0.75 0.91 0.10 0.14
13
Applications

• For specific cases, where the upper tails play
  the important role (e.g. modified maximal values
  of real flood data), B is the most sensitive.
• When applying the above tests for the flood data
  (annual maxima windows of size 50), there were
  only a couple of cases when the GEV hypothesis
  had to be rejected at the level of 95.
• Possible reasons changes in river bed properties
  (shape, vegetation etc).

14
An example for rejection Szolnok water level,
1931-80
15
Estimation methods

• Maximum likelihood, based on the unified
  parametrisation (GEV) is the most widely used,
  with optimal asymptotic properties, if ?gt-0.5 (it
  is superefficient for -0.5gt?gt-1). We have applied
  it, with good results.
• Probability-weighted moments (PWM)
• Method of L-moments

16
Robustness of maximum likelihood estimators

• The effect of small observations is limited in
  our case (negative shape parameters) halving the
  smallest 3 values, the difference in return level
  estimators was not more than 5-8.
• However, for positive shape parameters the
  effect of smaller values seem to be larger.

17
Further investigations

• Confidence bounds should be calculated, possible
  methods
• based on asymptotic properties of maximum
  likelihood estimator
• profile likelihood
• resampling methods (bootstrap, jackknife)
• Bayesian approach
• Estimates for return levels, including confidence
  bounds

18
Confidence intervals

• For maximum likelihood
• By asymptotic normality of the estimator
• where is the (i,i)th element of the inverse
  of the information matrix
• By profile likelihood
• For other nonparametric methods by bootstrap.

19
Profile likelihood

• One part of the parameter vector is fixed, the
  maximization is with respect the other
  components
• l(?) is the log-likelihood function ?(?i , ?-i
  )
• Let X1,,Xn be iid observations. Under the
  regularity conditions for the maximum likelihood
  estimator, asymptotically
• (a chi-squared distribution with k degrees of
  freedom, if ?i is a k-dimensional vector).

20
Use of the profile likelihood

• Confidence interval construction for a parameter
  of interest
• where c? is the 1-? quantile of the ?12
  distribution.
• Testing nested models
• M1(?) vs. M0 (the first k components of ? 0).
• l1( M1 ), l0 (M0 ) are the maximized
  log-likelihood functions and D2l1( M1 )- l0
  (M0 ).
• M0 is rejected in favor of M1 if Dgtc?
• (c? is the 1-? quantile of the ?k2
  distribution).

21
Return levels

• zp return level, associated with the return
  period 1/p (the expected time for a level higher
  than zp to appear is 1/p)
• The quantiles of the GEV
• where
• Remark the probability that it actually appears
  before time 1/p is more than 0.5 (approx. 0.63 if
  p is small)

if ? ? 0
if ? 0
22
Return level plots
Continuous ? 0.2 broken ? -0.2

• on a logarithmic scale
• Linear if ? 0
• Convex, with a limit
• if ? lt 0
• Concave, if if ? gt 0.
• It can be used for diagnostics,
• if the observed data points
• are also plotted.

23
Example profile likelihood for 100-year return
level (Vásárosnamény)
Profile likelihood can be calculated (the return
level is considered as one of the parameters)
24
Investigation of the estimators

• Backtest estimators based on data from a shorter
  window. Quite often too many floods are observed
  above the estimated level - simulation studies
  may confirm if this is a significant deviation
  from the iid case (for details see a later talk
  about resampling techniques).
• Alternative model linear trend in the location
  parameter (the other parameters are supposed to
  be constant).
• Centred time-scale is used t(t-50.5)

25
Some results with time-varying location parameter
Location, type Linear estimator for ? Increment of loglikelihood
Tivadar,h 509.30.761t 0.75
Tivadar,q 1225 1.774t 0.1
Namény, h 616.01.205t 4.18
Szolnok, h 644.41.321t 5.08
Polgár, h 520.8 1.190t 5.70
Polgár, q 1709 0.455t 0.01
26
Peaks over threshold methods
If the conditions of the theorem about the
GEV-limit of the normalised maxima hold, the
conditional probability of X-u, under the
condition that Xgtu, can be given as

• if ygt0 and ,
  where
• H(y) is the so called generalized Pareto
  distribution
• (GPD).
• is the same as the shape parameter of the
  corresponding GEV distribution.

27
Densities of GPD with ?1 solid ?0.5, dotted
?-0.1, dots-and-lines ?-0.7, broken ?-1.3
28
Peaks over threshold methods

• Advantages
• More data can be used
• Estimators are not affected by the small floods
• Disadvantages
• Dependence on threshold choice
• Original daily observations are dependent
  declustering not always obvious (see
  Ferro-Segers, 2003 for a recent method).

29
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 or formal tests (similar to those presented
  earlier)
• Return levels/upper bounds can be estimated
• Our results for the flood data sometimes
  slightly lower return level estimators (reasons
  have to be analyzed) .

30
GPD fit Vásárosnamény, water level
shape-0.51, estimated upper endpoint940
cm the upper endpoint of its 95 conf. int.
1085 cm
31
Return level estimators by parts of the dataset
Vásárosnamény
32
Future

• Our plans to incorporate
• most recent data into
• the analyzis
• Plans for the future
• (engineers)
• to build temporal
• reservoirs
• to utilise our results in
• levy construction
• So we may hope to
• prevent such events
• to happen again.

33
Some references

• Ferro, T. A.- Segers, J. (2003) Inference for
  clusters of extreme values. Journal of Royal
  Statistical Soc. Ser. B. 65, p. 545-556.
• Kotz, S. Nadarajah, S. (2000) Extreme Value
  Distributions. Imperial College Press.
• Zempléni, A. (1996) Inference for Generalized
  Extreme Value Distributions Journal of Applied
  Statistical Science 4, p. 107-122.
• Zempléni, A. Goodness-of-fit tests in extreme
  value theory. (In preparation.)