abdel1

1

Abdel H. El-ShaarawiNational Water Research Institute and

Department of Mathematics and Statistics, McMaster University

[email protected]

Data-driven and Physically-based Models for Characterization of Processes in Hydrology,

Hydraulics, Oceanography and Climate Change

January 6-28, 2008IMS, 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 Distributions with Applications to Rainfall Modeling. Environmetrics

Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applications. 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

Examples of Extreme Events Data3. 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.

10

Examples of Extreme Events Data6. 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.

11

Nile meter

12

U.S. Bureau of the census, Watson and Pauly (2002)

Living resources: food security

13

Niagara River Fraser River

14

Upstream-Downstream Water Quality MonitoringHuman 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 DayDaily Flow

(m3/s)

Nitrogen Total

Dissolved (mg/L)

Phosphorus Total (mg/L)

Daily Flow (m3/s)

Nitrogen Total

Dissolved (mg/L)


Daily Flow (m3/s)

Nitrogen Total

Dissolved (mg/L)


Daily Flow (m3/s)

Nitrogen Total

Dissolved (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/A3/2/1912 62 #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A 538 #N/A #N/A3/3/1912 63 #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/A12/28/2003 362 8.07 #N/A #N/A 108 #N/A #N/A 429 #N/A #N/A 730 #N/A #N/A12/29/2003 363 7.79 #N/A #N/A 104 #N/A #N/A 423 #N/A #N/A 720 #N/A #N/A12/30/2003 364 7.61 #N/A #N/A 101 #N/A #N/A 417 #N/A #N/A 720 #N/A #N/A12/31/2003 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 km2Fraser River at Red Pass Fraser River at Hansard Fraser River at Marguerite Fraser River at Hope

15

Time Plots: Fraser River @ Hope

Time

log(

Flow

)

1950 2000 2050 2100

67

89

Year

log

(Flo

w)

1920 1940 1960 1980 2000

67

89

Julian

log

(Flo

w)

0 100 200 300

67

89

16

Evolution of the Flow along the Fraser River

0 100 200 300

0.0

00

.05

0.1

00

.15

day

Est

ima

ted

Co

nce

ntr

atio

n o

f TP

(mg

/L)

Hope

Red PassHansardMargueriteHope

Hansard/Red Pass

0 100 200 300

51

01

52

02

53

03

5

day

Ra

tio

of th

e T

P C

on

ce

ntr

atio

n

0 100 200 300

1.0

1.5

2.0

2.5

3.0

day

Ra

tio o

f th

e T

P C

on

cen

tra

tion

17

Max of log (Flow) at Hope

Years

Lo

g(m

ax.

flow

)

1920 1940 1960 1980 2000

8.6

8.8

9.0

9.2

9.4

9.6

Log(max.flow(i))

Lo

g(m

ax.

flow

)(i+

1)

8.6 8.8 9.0 9.2 9.4 9.6

8.6

8.8

9.0

9.2

9.4

9.6

Gumble.Q

Ord

er.

log

flo

w

-1 0 1 2 3 4

8.6

8.8

9.0

9.2

9.4

9.6

18

Some Results for Max (Hope)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Probability Plot

Empirical

Mod

el

6000 8000 10000 12000

6000

1000

014

000

Quantile Plot

Model

Em

piri

cal

6000

1000

014

000

Return Period

Ret

urn

Leve

l

0.1 1 10 100 1000

Return Level Plot Density Plot

z

f(z)

6000 10000 14000

0.00

000

0.00

010

0.00

020

19

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 10.9011.10 11.30 11.30 11.55 11.75 12.85 12.90 13.40

1750 1800 1850 1900 1950 2000

Year

-10

010

20

Tem

p

Maximum and Minimum Tempertue for Basel 1755-1991

Two More Example: wave-height & Temperature (Basel)

20

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

23

Seoul Rainfall Data

24

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)

25

Approximate expression for probability of compliance with the regulations

Let

b and

a

)(

)(

x

xg

where b is the geometric mean ; a is single sample maximum )(x is the pdf of standard normal distribution

)(x is the CDF of standard normal distribution

)())((1Pr ngnob

)()1

)((2Pr

2

n

gg

ganob

26

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

27

Ratio of single sample rejection probability to that of the mean rule (n = 5,10 and 20)

nagprob

bXprob nn

1

)(1log

)(1

)(1log

)(

28

14 16 18 20 22 24

0.0

0.1

0.2

0.3

Teperature

Pro

bab

ilit

y

Non- parametricGumbel

16 18 20 22

Temperature

0.0

0.2

0.4

0.6

0.8

Gu

mb

el

de

nsi

ty 1775- 18541775- 19911855- 1991

Figure 2. Fit ted Gumbel density for the full data and its two subsect ions

The Temperature Data: Change-Point

29

Relative Likelihood Function for the Change Point

Year

Re

lativ

e L

ike

liho

od

1750 1800 1850 1900 1950

0.0

0.2

0.4

0.6

0.8

1.0

30

Relative Likelihood function for the Change Point (Temp. Data)

x

f(x)

10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

0.5

Segment1Segment2

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

31

Q-Q plots for the two segements

Theoritical Quantile

Obs

erve

d Q

uant

ile

-1 0 1 2 3 4

1819

20

Theoritical Quantile

Ob

se

rve

d Q

ua

ntile

0 2 4

16

18

20

22

32

Return Levels

Return Level

Re

turn

Pe

rio

d

10 15 20 25 30

02

00

04

00

06

00

08

00

0

mu1,sigma1mu2,sigma2

33

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; ), that fits the data reasonably well

2. Estimate the parameters, and quantiles, X(p), of such a model

35


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.

40

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; ), depending on unknown vector-valued parameter .

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, and the maximum, Xn:n order

statistics.

44

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: nxFxF )()(min 11

nxFxF )()(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; ), 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).

47Generalized 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.

48Generalized 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

x

x

xH

50

Minimal Generalized Extreme Value

The cumulative distribution function

(cdf) of the minimal GEVm distribution

is:

.,)(

expexp

,,)(

exp

)(

/

0

0

1

111

x

x

xL

51

Relationship Between GEVM and GEVm

Theorem:

If the cdf of X is L(, , ), then the cdf

of Y = X is H(, , ).

Implication:

One form of the cdf can be obtained

from the other.

52

Maximal Generalized Extreme Value

The GEVM family has three-

parameters:

• is a location parameter

• is a scale parameter ( > 0)

• is a shape parameterThe parameter is the most important

of the three. The pth quantile is (0 < p

< 1): )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 > 0. Its cdf is:

.,

,,,exp)(

otherwise

xxxH

1

0

54


2. The Maximal Gumbel distribution is

obtained when = 0. Its cdf is:

;)(

expexp)(

xxH

,x

55


3. The Maximal Frechet distribution is

obtained when < 0. Its cdf is:

.,,exp

,,,)( 0

00

xx

xxH

56

Weibull, Gumbel, and Frechet

Weibull and Frechet converge to

Gumbel

0

0

57

Summary

The GEV family can be used when:

1. The cdf of the parent population, F(x; ), 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:

1. Complete Data: All n observations are available.

2. Maxima/Minima: Only maxima or minima are available

3. Exceedances over/under a threshold: 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.

61Generalized 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(, ), with

cdf

.,

,,/)(

/

/

0

0

1

11 1

xMe

xxF

62Generalized Maximal Pareto Distribution

The GPDM family has a two-

parameters:

• is a scale parameter ( > 0)

• is a shape parameter

The pth quantile is (0 < p < 1):

/)()( ppx 11

Note that when .)(Var,/ X21

63

Special Cases of the Maximal GPD

The GPDM has three special cases:

1. When = 0, the GPDM reduces to

the Exponential distribution with

mean .

2. When = 1, the GPDM reduces to

the Uniform U(0, ).

3. When < 0, the GPDM becomes the

Pareto distribution.

64Generalized 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 < + / , for > 0

x > + / , for > 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 > 1, the MLE do not exist (the

likelihood can be made infinite).

70


• When < 1, the mean and higher

moments do not exist. So, MOM and

PWM do not exist when < 1.

• The PWM estimators are good for

cases where –0.5 < < 0.5.

• Outside this range of , the PWM

estimates may not exist, and if they do exist their performance worsens

as increases.

71Recently 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 :: ),,;(

njnj pxF :: ),,;(

nrnr pxF :: ),,;(

73


2. Substitute the cdf of the GEVM, we

obtain: )log()/( :: nini px 1

)log()/( :: njnj px 1

)log()/( :: nrnr px 1

These are three equations in three unknowns: , , and .

74


To solve these equations, we eliminate and , and obtain:

,)( 011

jr

jrijr A

ADg

wherenr

niir p

pA

:

:

loglog

nrni

nrnjijr xx

xxD

::

::

Solving this equation for by the bisection method, we obtain an initial estimate .ˆirj

75


Substituting in two of the above equations and solve for and :

irjirj

ninr

nrniirjijr

pp

xx

ˆ

:ˆ

:

::

loglog

ˆˆ

irj

irj

irjniirj

ijr

p

ˆ

ˆ:logˆ

ˆ

1

76


Theorem: The initial estimates are asymptotically normal and consistent.

Final estimates of , , and are obtained by combining all possible triplets

,ˆ,ˆ,ˆ irjirjirj

and obtain efficient estimates using a suitable function such as the trimmed mean.

,ˆ,ˆ,ˆ irjirjirj

77The 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

i

n

ixf

);(log :1

ni

n

ixf


2. The modified likelihood:

A Taylor series expansion of

around

gives

);(log : nii xf )( :: nini XE

250

1niiniii

n

ixcxba :.:)(


3. Let )()( :: gXE nini

),;( :: nini pFX 1

where are plotting positions.

)/()(: bnaip ni

4. Substitute these in the modified likelihood and solve for .


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:1. Complete Data: All n observations

are available.

2. Maxima/Minima: Only maxima or minima are available

3. Exceedances over/under a threshold: Only observations exceeding a given threshold are available

Use GPD.Use GPD.

Use GEV.Use GEV.



abdel1

Engineering

extreme value tributions

data analysis

statistics of extremes

statistical applications

science applications

areas of applications

average rainfall

references castillo