Abdel1

1
Abdel H. El-Shaarawi
National Water Research Institute and
Department of Mathematics and Statistics,
McMaster University
Abdel.el-shaarawi@ec.gc.ca
Data-driven and Physically-based Models for
Characterization of Processes in Hydrology,
Hydraulics, Oceanography and Climate Change
January 6-28, 2008
IMS, Singapore
Modeling Extreme Events Data

2
Outline
• Some references
• Examples of extreme events data
• Types of extreme events data
• Commonly used models for extremes:
• Distributions of order statistics
• Generalized extreme value distributions
• Generalized Pareto distributions
• Parameter and quantile estimation of extremes
• Summary and concluding remarks

3
References
Beirlant Jan, Yuri Goegebeur, Johan Segers and Jozef Teugels (2004),
Statistics of Extremes: Theory and Applications, NewYork: John Wiley
& Sons.
Castillo, E. and Hadi, A. S. (1994), Parameter and Quantile Estimation
for the Generalized Extreme-Value Distribution, Environmetrics, 5, 417–
432.
Castillo, E. and Hadi, A. S. (1995), A Method for Estimating Parameters
and Quantiles of Continuous Distributions of Random Variables,
Computational Statistics and Data Analysis, 20, 421–439.

4
References
Castillo, E., Hadi, A. S., Balakrishnan, N., and Sarabia, J. M. (2006), Extreme Value
and Related Models in Engineering and Science Applications,
New York: John Wiley & Sons.
Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values.
Springer-Verlag, London, England.
El-Shaarawi, A. H., and Hadi, A. S.,Modified Likelihood Function for Parameter
and Quantile Estimation, Work in progress.
Nadarajah, S. and El-Shaarawi, A. H. (2006). On the Ratios for Extreme Value
tributions with Applications to Rainfall Modeling. Environmetrics
Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applica
London: Imperial College Press.

5
Software: S-plus & R
• Stuart Coles S-plus package available at
URL:http://www.math.lancs.ac.uk./~coless
• extRemes R package available at
http://www.isse.ucar.edu/extremevalues

6
Examples of Extreme Events Data
In many statistical applications, the interest
is centered on estimating some population
characteristics based on random samples
taken from a population under study.
For example, we wish to estimate:
• the average rainfall,
• the average temperature,
• the median income,
• … etc.

7
In other areas of applications, we are not
interested in estimating the average but
rather in estimating the maximum or the
minimum.
1. Ocean Engineering: In the design of
offshore platforms, breakwaters, dikes
and other harbor works, engineers rely
upon the knowledge of the probability
distribution of the maximum, not the
average wave height.
Some Examples:

8
2. Structural Engineering: Modern building
codes and standards require:
• Estimation of extreme wind speeds and
their recurrence intervals during the
lifetime of the building.
• Knowledge of the largest loads acting
on the structure during its lifetime.
• Seismic incidence: the maximum
earthquake intensity during the lifetime
of the building.

9
3. Designing Dams: Engineers would not be
interested in the probability distribution of
the average flood, but in the maximum
floods.
4. Agriculture: Farmers would be interested
in both the minimum and maximum rain
fall (drought versus flooding).
5. Insurance companies would be
interested in the maximum insurance
claims.

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6. Pollution Control: The pollution of air and
water has become a common problem in
many countries due to large concentrations
of people, traffic, and industries (producing
smoke, human, chemical, nuclear wastes,
etc.). Government regulations, require
pollution indices to remain below a given
critical level. Thus, the regulations are
satisfied if, and only if, the largest pollution
concentration during the period of interest
is less than the critical level.

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U.S. Bureau of the census, Watson and Pauly (2002)
Living resources: food security

14
Upstream-Downstream Water Quality Monitoring
Human and Ecosystem Health: Regulations and Control
S0 ⇒ S1 ⇒ S2 ⇒ . . . Sk-1 ⇒ Sk
Niagara River
• Overview of U-D M: Purpose, Design and Examples
• Univariate Series and Ratio (Trend & Seasonality)
• Bivariate Series
Fraser River Several Stations
Date Julian Day
Daily Flow
(m3/s)
Nitrogen
Total
Dissolved
(mg/L)
Phosphor
us Total
(mg/L)
Daily Flow
(m3/s)
Nitrogen
Total
Dissolved
(mg/L)
Phosphor
us Total
(mg/L)
Daily Flow
(m3/s)
Nitrogen
Total
Dissolved
(mg/L)
Phosphor
us Total
(mg/L)
Daily Flow
(m3/s)
Nitrogen
Total
Dissolved
(mg/L)
Phosphor
us Total
(mg/L)
3/1/1912 61 #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A 538 #N/A #N/A
12/27/2003 361 8.4 #N/A #N/A 113 #N/A #N/A 430 #N/A #N/A 740 #N/A #N/A
12/31/2003 365 7.49 #N/A #N/A 96.7 #N/A #N/A 416 #N/A #N/A 720 #N/A #N/A
1700 km2 18000 km2 114000 km2 217000 km2
Fraser River at Red Pass Fraser River at Hansard Fraser River at Marguerite Fraser River at Hope

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Time Plots: Fraser River @ Hope
Time
log(Flow)
1950 2000 2050 2100
6789
Year
log(Flow)
1920 1940 1960 1980 2000
6789
Julian
log(Flow)
0 100 200 300
6789

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Evolution of the Flow along the Fraser River
0 100 200 300
0.000.050.100.15
day
EstimatedConcentrationofTP(mg/L)
Hope
Red Pass
Hansard
Marguerite
Hope
Hansard/Red Pass
0 100 200 300
5101520253035
day
RatiooftheTPConcentration
0 100 200 300
1.01.52.02.53.0
day
RatiooftheTPConcentration

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Max of log (Flow) at Hope
Years
Log(max.flow)
1920 1940 1960 1980 2000
8.68.89.09.29.49.6
Log(max.flow(i))
Log(max.flow)(i+1)
8.6 8.8 9.0 9.2 9.4 9.6
8.68.89.09.29.49.6
Gumble.Q
Order.logflow
-1 0 1 2 3 4
8.68.89.09.29.49.6

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Some Results for Max (Hope)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Probability Plot
Empirical
Model
6000 8000 10000 12000
60001000014000
Quantile Plot
Model
Empirical
60001000014000
ReturnLevel
0.1 1 10 100 1000
Return Level Plot Density Plot
f(z)
6000 10000 14000
0.000000.000100.00020

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Yearly maximum significant wave-height data 1949-1976
5.60 6.55 6.65 7.35 7.80 7.90 8.00 8.50
9.05 9.15 9.40 9.60 9.80 9.90 10.85 10.90
11.10 11.30 11.30 11.55 11.75 12.85 12.90 13.40
1750 1800 1850 1900 1950 2000
Year
-1001020
Temp
Maximum and Minimum Tempertue for Basel 1755-1991
Two More Example: wave-height & Temperature (Basel)

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Two Stations: Ratio of GEV Distributions W=X/(X+Y)

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Current U.S. Environmental Protection Agency (USEPA) guidelines for:
a) designated beaches specify a 30-day geometric mean and a single-
sample sample maximum corresponding to the 75th
percentile based
on that 30-day mean [USEPA, 1986].
b) drinking water specify the arithmetic mean coliform density of all
standard samples examined per month shall not exceed one per
100ml.
EPA recent workshop to establish Recreational Water Quality Criteria,
Chapel Hill, North Carolina last February:
Objective was not only to determine compliance but also to relate
waterborne illness to bacteriological indicator’s density
1. Estimation of Chemical Concentrations and Loadings (Ecosystem
Health)
Microbiological Regulations (Human health)

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Approximate expression for probability of compliance with the regulations
Let and
where b is the geometric mean ;
a is single sample maximum
is the pdf of standard normal distribution
is the CDF of standard normal distribution

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Sample size n=5 and 10 # of simulations =10000

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Ratio of single sample rejection probability to that of the mean rule (n = 5,10 and 20)
{ }
( )η
ξ
ρ
nagprob
bXprob n
n
Φ−
Φ−
=
<−
<−
=
1
)(1
log
)(1
)(1
log
)(

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14 16 18 20 22 24
0.00.10.20.3
Teperature
Probability
Non- parametric
Gumbel
16 18 20 22
Temperature
0.00.20.40.60.8
Gumbeldensity
1775- 1854
1775- 1991
1855- 1991
Figure 2. Fitted Gumbel density for the full data and its two subsections
The Temperature Data: Change-Point

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Relative Likelihood Function for the Change Point
Year
RelativeLikelihood
1750 1800 1850 1900 1950
0.00.20.40.60.81.0

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Relative Likelihood function for the Change Point
(Temp. Data)
Mu Sigma
First segment 18.40425 0.7638125
Second segment 18.11343 1.301378
E(X) Var(X)
First segment 18.84512 0.9596703
Second segment 18.86459 2.785836

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Q-Q plots for the two segements
Theoritical Quantile
ObservedQuantile
-1 0 1 2 3 4
181920
Theoritical Quantile
ObservedQuantile
0 2 4
16182022

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Return Levels
Return Level
ReturnPeriod
10 15 20 25 30
02000400060008000
mu1,sigma1
mu2,sigma2

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Outline
• Some references



34
Types of Extreme Events Data
The choice of model and estimation
methods depends on the type of available
data.
Data, x1, x2, …, xn, drawn from a possibly
unknown population, are available.
We wish to:
1. Find an appropriate parametric model,
F(x; θ), that fits the data reasonably well
2. Estimate the parameters, θ, and
quantiles, X(p), of such a model

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Examples:
1. Complete Data: All n observations are
available.
Daily/Monthly energy consumption
• Daily/Monthly rain fall, stream discharge
or flood flow

36
Examples:
2. Maxima/Minima: Only maxima or minima
are available.
• Maximum/minimum daily/monthly
temperatures
• Maximum daily/monthly wave heights
• Maximum daily/monthly wind speeds,
pollution concentrations, etc.

37
3. Exceedances over/under a threshold:
When using yearly maxima (minima),
then an important part of the information
large (small) values (other than the two
extremes occurring the same year) is
lost. The alternative is to use the
exceedances over (under) a given
threshold.

38
Exceedances Over/Under a Threshold
We are interested in events that cause
failure such as exceedances of a random
variable over a threshold value.
For example, waves can destroy a
breakwater when their heights exceed a
given value, say 9 meters. Then it does not
matter whether the height of a wave is 9.5,
10 or 12 meters because the consequences
of these events are similar.

39
So, only failure causing observations
exceeding a given threshold are available.
Definition:
Let X be a random variable and u be a given
threshold value. The event {X = x} is said to
be an exceedance at the level u if X > u.

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Summary: Types of Data
Extreme events data come in one of three
types:
1. Complete observations,
2. Maxima/Minima, or
3. Exceedances over/under a threshold value

41
Outline
• Some references




42
Commonly Used Models for Extremes
The choice of model depends on the type of
available data:
• Distributions of Order Statistics (DOS):
Used when we have complete data
• Generalized Extreme Value (GEV)
Distribution (AKA: Von Mises Family):
Used for maxima/minima type of data
• Generalized Pareto Distribution (GPD):
Used for exceedances over/under
threshold type of data

43
Distributions of Order Statistics
Let X1, X2, …, Xn be a sample of size n from a
possibly unknown cdf F(x; θ), depending on
unknown vector-valued parameter θ.
Let X1:n < X2:n < … < Xn:n be the corresponding
order statistics.
Xi:n is called the ith order statistic.
Of particular interest is the minimum, X1:n,

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Distributions of Order Statistics
The distributions of the the order statistics
are well know. For example:
• The cdf of the maximum order statistics is:
• The cdf of the minimum order statistics is:
[ ]n
xFxF )()(min −−= 11
[ ]n
xFxF )()(max =

45
Problems with Distributions of OS
The distributions of the order statistics have
the following practical problems:
1. The cdf of the parent population, F(x; θ),
is usually unknown
2. When the data consist only of maxima or
minima, the sample sizes are usually
unknown

46
Non-Degenerate Limiting Distributions
The answer to the above problem is:
Theorem:
1. The only non-degenerate cdf family
satisfying (1) is the Maximal Generalized
Extreme Value Distribution (GEVM).
2. The only non-degenerate cdf family
satisfying (2) is the Minimal Generalized
Extreme Value Distribution (GEVm).

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Generalized Extreme Value Distributions
Thus, there are two GEV distributions, one
maximal, GEVM, and one minimal, GEVm.
The GEV (AKA, Von Mises) distributions
were introduced by Jenkinson (1955).
They are used when we have a large
sample or the observations themselves are
either minima or maxima.
Their cdf are given later.

48
Generalized Extreme Value Distributions
The GEV distributions are now widely used
to model extremes of natural and
environmental data. Examples are found in:
• Flood Studies Report of the USA’s Natural
Environment Research Council (1975)
• Several articles in Tiago de Oliveira (1984)
• Hosking, Wallis, and Wood (1985)
• Castillo et al. (2006)

49
Maximal Generalized Extreme Value
The cumulative distribution function (cdf) of
the maximal GEVM distribution is:







=









 −
−−
≠










 −
−−
=
.,
)(
expexp
,,
)(
exp
)(
/
0
0
1
1
κ
δ
λ
κ
δ
λκ
κ
x
x
xH

50
Minimal Generalized Extreme Value
The cumulative distribution function (cdf) of
the minimal GEVm distribution is:







=









 −
−−−
≠










 −
+−−
=
.,
)(
expexp
,,
)(
exp
)(
/
0
0
1
11
1
κ
δ
λ
κ
δ
λκ
κ
x
x
xL

51
Relationship Between GEVM and GEVm
Theorem:
If the cdf of X is L(λ, δ, κ), then the cdf of
Y = − X is H(−λ, δ, κ).
Implication:
One form of the cdf can be obtained from
the other.

52
Maximal Generalized Extreme Value
The GEVM family has three-parameters:
• λ is a location parameter
• δ is a scale parameter (δ > 0)
• κ is a shape parameter
The parameter κ is the most important of
the three. The pth quantile is (0 < p < 1):
[ ]( )κ
κδλ )log()/()( ppx −−+= 1

53
Special Cases of the Maximal GEV
The family of GEVM has three special cases:
1. The Maximal Weibull distribution is
obtained when κ > 0. Its cdf is:
( )




>≤


 −−
=
.,
,,,exp
)(
otherwise
xx
xH
1
0βλ
δ
λ β

54
2. The Maximal Gumbel distribution is
obtained when κ = 0. Its cdf is:
;
)(
expexp)(









 −
−−=
δ
λx
xH
∞<<−∞≥ λλ,x

55
3. The Maximal Frechet distribution is
obtained when κ < 0. Its cdf is:
( )



>≥



−
−
><
=
.,,exp
,,,
)(
0
00
βλ
λ
δ
βλ
β
x
x
x
xH

56
Weibull, Gumbel, and Frechet
Weibull and Frechet converge to Gumbel
0<κ
0>κ

57
Summary
The GEV family can be used when:
1. The cdf of the parent population, F(x; θ),
is unknown
2. The sample size is very large (no
degeneracy problems)
3. The data consist only of maxima or
minima (we do not need to know the
sample sizes)

58
Outline
• Some references






59
Recall the three types of extreme
events data:
available.
2. Maxima/Minima: Only maxima or
minima are available
Only observations exceeding a given
threshold are available
Use distributions of order statistics if we know
F(x) and n is not too large; else, use GEV.
Use GPD.Use GPD.
Use GEV.Use GEV.

60
As mentioned earlier, we are interested in
events that cause failure such as
exceedances of a random variable over a
threshold value.
The differences between the actual values
and the threshold value are called
exceedances over/under the threshold.

61
Generalized Maximal Pareto Distributions
Pickands (1975) demonstrates that when
the threshold tends to the upper end of the
random variable, the exceedances follow a
generalized Pareto distribution, GPDM(α, κ),
with cdf
( )



=−
≠−−
= −
.,
,,/
)( /
/
0
0
1
11
1
κ
κακ
α
κ
xM
e
x
xF

62
Generalized Maximal Pareto Distribution
The GPDM family has a two-parameters:
• α is a scale parameter (α > 0)
• κ is a shape parameter
The pth quantile is (0 < p < 1):
( ) κα κ
/)()( ppx −−= 11
Note that when .)(Var,/ ∞=−≤ X21κ

63
Special Cases of the Maximal GPD
The GPDM has three special cases:
1. When κ = 0, the GPDM reduces to the
Exponential distribution with mean α.
2. When κ = 1, the GPDM reduces to the
Uniform U(0, α).
3. When κ < 0, the GPDM becomes the
Pareto distribution.

64
Generalized Minimal Pareto Distribution
A similar family exists for the case of
exceedances under a threshold. These are
called the the Generalized Minimal Pareto
distributions or the Reversed Generalized
Pareto distributions.

65
Outline
• Some references








66
Parameter and Quantile Estimation
Available estimation methods include:
1. The maximum likelihood (MLE):
Jenkinson (1969)
Prescott and Walden (1980, 1983)
Smith (1984, 1985)
2. The method of moments (MOM)

67
Parameter and Quantile Estimation
3. The probability weighted moments (PWM):
Greenwood et al. (1979), Hosking et al. (1985)
4.The Elemental Percentile method (EPM): Castillo
and Hadi (1995)
5.Order Statistics (Least Squares): El-Shaarawi
5. Modified Likelihood Function (MLF): El-
Shaarawi and Hadi (work in progress).

68
Problems With Traditional Estimators
Traditional methods of estimation (MLE and
the moments-based methods) have
problems because:
• The range of the distribution depends on
the parameters:
x < λ + δ / κ, for κ > 0
x > λ + δ / κ, for κ > 0
So, MLE do not have the usual asymptotic
properties.

69
• The MLE requires numerical solutions.
• For some samples, the likelihood may not
have a local maximum.
• For κ > 1, the MLE do not exist (the
likelihood can be made infinite).

70
• When κ < −1, the mean and higher
moments do not exist. So, MOM and PWM
do not exist when κ < −1.
• The PWM estimators are good for cases
where –0.5 < κ < 0.5.
• Outside this range of κ, the PWM
estimates may not exist, and if they do
exist their performance worsens as κ
increases.

71
Recently Proposed Estimation Methods
4. The Elemental Percentile method (EPM):
Castillo and Hadi (1995)
5. Modified Likelihood Function (MLF):
El-Shaarawi and Hadi (work in progress).
This leaves us with two recently proposed
methods for estimating the parameters and
quantiles of the extreme models:

72
Elemental Percentile method (EPM)
1. Initial estimates are obtained by equating
three distinct order statistics to their
corresponding percentiles:
nini pxF :: ),,;( =κδλ
njnj pxF :: ),,;( =κδλ
nrnr pxF :: ),,;( =κδλ

73
2. Substitute the cdf of the GEVM, we
obtain:
( )κ
κδλ )log()/( :: nini px −−+= 1
( )κ
κδλ )log()/( :: njnj px −−+= 1
( )κ
κδλ )log()/( :: nrnr px −−+= 1
These are three equations in three
unknowns: λ, δ, and κ.

74
To solve these equations, we eliminate λ
and δ, and obtain:
,)( 0
1
1
=
−
−
−= κ
κ
κ
jr
jr
ijr
A
A
Dg
where
nr
ni
ir
p
p
A
:
:
log
log
=
nrni
nrnj
ijr
xx
xx
D
::
::
−
−
=
Solving this equation for κ by the bisection
method, we obtain an initial estimate .ˆirjκ

75
Substituting in two of the above
equations and solve for λ and δ:
( )
( ) ( ) irjirj
ninr
nrniirj
ijr
pp
xx
κκ
κ
δ ˆ
:
ˆ
:
::
loglog
ˆˆ
−−−
−
=
irjκˆ
( )( )
irj
irj
niirj
ijr
p
κ
κ
δ
λ ˆ
ˆ
:logˆ
ˆ −−
=
1

76
Theorem: The initial estimates
are asymptotically normal and consistent.
Final estimates of λ, δ, and κ are obtained
by combining all possible triplets
,ˆ,ˆ,ˆ irjirjirj λδκ
and obtain efficient estimates
using a suitable function such as the
trimmed mean.
,ˆ,ˆ,ˆ irjirjirj λδκ

77
The Modified Likelihood Function (MLF)
The MLF method can be thought of as a
marriage between the maximum likelihood
method and the method of moments. The
ideas behind the method are:
1. The log likelihood function is:
);(log)(
1
θθ i
n
i
xf∑=
=

);(log :
1
θni
n
i
xf∑=
=

78
2. The modified likelihood:
A Taylor series expansion of
around
gives
);(log : θnii xf= )( :: nini XE=α
( )2
50
1
niiniii
n
i
xcxba :.:)( −+∑≅
=
θ

79
3. Let )()( :: θα gXE nini ==
↓
),;( :: θnini pFX 1−
=
↓
where are plotting
positions.
)/()(: bnaip ni +−=
4. Substitute these in the modified
likelihood and solve for θ.

80
We think this will be a happy marriage, but
to be sure we are:
Investigating (analytically and using
simulation) the properties of the proposed
estimators and their dependence on the
choice of the plotting positions pi:n.
This is still work in progress.

81
Outline
• Some references









82
Summary
The choice of models for extremes
depends on the type of data available:
available.
2. Maxima/Minima: Only maxima or
minima are available
Only observations exceeding a given
threshold are available Use GPD.Use GPD.
Use GEV.Use GEV.

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