use of extreme value theory in engineering...

1

ESREL 2008 Valencia September 22-25 1

Use Use ofof Extreme Extreme ValueValue TheoryTheory in in EngineeringEngineering DesignDesign..

Enrique Castillo, Carmen Castillo and Roberto Mínguez

University of

Castilla-La Mancha

University of

Cantabria

2


Outline of the Talk

Outline of the talk

Motivation

Order statistics

ExceedancesExample

Conclusions

Limit distributions

3


MOTIVATION

4


Motivation

It is clear that extreme values appear in natural calamities and disasters .

5


FloodsQuantifying uncertainty in flood magnitude estimators is an important

problem in floodplain development, including risk assessment forfloodplain management, risk-based design of hydraulic structures

and estimation of expected annual flood damages.

An aerial view shows the flood-affected banks of the Koshi river in the border area between Supaul and Saharsa in the easternIndian state of Bihar.

6


Volcanoes

Volcanoes can cause a lot of trouble to

sourrounding areas

7


Tsunamis

8


Tsunamis

Tsunamis are related to

earthquakes but with a good information

system they can be predicted some

time ahead and some damages

can be disminished

9


Tsunami

10


Tsunami

Tsunami in Banda Aceh, Indonesia, January 2005

11


Tornados

Tornadoescause a lot of

damage and theirintensities

must be predicted with precision

12


Motivation

The questions are: Should all engineering works be designed to survive natural disasters?

or

should we wait for them to occur and then proceed torepair damages?

Is this justified from an economical point of view?

13


Motivation

Apart from natural calamities and disasters whereextremes become clearly apparent, extremes (maxima

and minima) appear in engineering normal life

Engineering desing is always based on maxima (loads, winds, earthquakes, snowfall, pollution levels,

radioactivity levels, etc.) and minima (strengths, droughts , lack of water supply, etc.), because they are

the main cause of failures.

14


Ocean engineering

The design of breakwaters, dikes, and other harbor works rely upon the knowledge of the probability distribution of the highest waves.

Storm frequency and its incidence on the location of interest is of crucial interest in the design of safe engineering works.

15


Ocean engineering

Wave action can cause important losses of lifes and a lot of damage.

Engineering design requires the knowledge of statistical behavior of waves.

16


Structural engineering

A correct structure design requires a precise estimation of the probabilities of occurrence of extreme winds, loads or earthquakes, and also of the strength of materials involved in engineering works.

Mississippi River after a bridge collapse in Minneapolis, Aug. 1, 2007.

17


Structural engineering.

Minneapolis Bridge Collapse

Failures are normally due to extreme values of loads (maxima) orstrengths (minima), and also to human errors.

Most of the times they are due to a combination of them.

18


LandslidesA correct design requires a precise estimation of the probabilities of

occurrence of landslides, which requires determination of the weakest sliding lines.

In this case a calculus of variations or an optimization problem must be solved to obtain the critical sliding line (safety factor), or the probability of failure.

19


Offshore platforms

Quantifying uncertainty in wave magnitude and period is an important problem in offshore platform design.

20


Motivation

The field of extremes, maxima and minima of random variables, has attracted the attention of engineers, scientists and statisticians for many years, because engineering works need to be designed for extreme conditions and this forces people to pay special attention to singular (extreme) values more than to regular (or mean) values.

The statistical theory for dealing with mean values is very different from the statistical theory required for extremes, so that one cannot solve the above indicated problems without a specialized knowledge on the statistical theory of extremes.

In many statistical applications, the interest is centered on estimating some population central characteristics (e.g., the average rainfall, the average temperature, the median income, etc.). However, in some other areas of applications, engineers are interested in estimating the maximum, the minimum or high or low order statistics.

21


Statistical orientation

To illustrate, assume that instead of being required to read the words(the usual case):

Blue Red Yellow Blue BlackGrey Red Blue Grey Yellow

An extreme value statistician has a different point of view and usually thinksof maximum or minimum values.

A common statistician tends to think mainly of mean values.

It is not easy to change the mind to become a extreme value statistician. In some cases it is better to start from scratch.

you are asked to read the color of the following words:

Blue Red Yellow Blue BlackGrey Red Blue Grey Yellow

22


Order statistics

Ordered sample :

Statistic of order r:

Density function :

Cdf of maximum :

Cdf of minimum :

Sample :

23


Joint density of several order statistics

Using a multinomial experiment, illustrated in the figure, one can easily obtain the joint density of any susbset of

order statistics.

24


Distribution of order statistics (1 to 5)

Uniform

Normal

25


Distribution of the first and last five (Normal n=20)Distribution of the last five

Distribution of the first five

26


Distribution of maxima of samples of sizes 2, 10, 100, 1000 and 10000

Normal

Gumbel

MinimalGumbel

27


Limit distributions (independence case)

28


Limit distributions of maxima and minima

Since we observe a displacement to the right and a change of slopewith increasing n, we use location and scale constants to get a stable distribution H(x).

The role of b is to change the slope of the cdf.n

The role of a is to produce a displacement of the cdf.nA similar strategy is done for minima.

29


Limit distributions

30


Limit distributions for maxima (independence case)

31


Limit distributions for minima (independence case)

32


Limit distributions (independence case)

33


Weibull and reverse Weibull distributions

34


Gumbel and reverse Gumbel distributions

35


Frechet and reverse Frechet distributions

36


Determining the domain of attraction for maxima

A practical and very important conclusion of theprevious results is the following:

The maximal Weibull, Gumbel and Frechet families can be used to approximate the true distribution of maximum

coming from any distribution F(x).

Similarly, the minimal Weibull, Gumbel and Frechetfamilies can be used to approximate the true distribution

of minimum coming from any distribution F(x).

37


Limit distributions

38


Determining the domain of attraction for maxima

An interesting practical problem is the following:

Given the cdf F(x) of a random variable, what is themaximal H(x) limit distribution?

The following theorem solves this problem.

In other words,

Which of the limit distributions Weibull, Gumbel andFrechet is the adequate family to fit maxima data coming

from a given F(x)?

39


Normalizing constants for maxima

What are the sequences of constantsto be used in the expression

to obtain the limit H(x)?

The following theorem solves this problem.

40


Determining the domain of attraction for minima

41


Normalizing constants for minima

42


Limit distributions

Using these twotheorems we get the

following tablewhich includes the

most commondistributions

43


Domain of attraction for maxima

Unfortunately, in practical cases we do not know F(x), and these methods are useless.

However, we normally have a sample from F(x).

What to do to determine the domain of attraction?

Maximal probability papersenlarge the right tails of distributions

44


Domain of attraction for minima

Minimal probability papersenlarge the left tails of distributions

45


Practical recommendations

The Weibull distribution is not possible if the random variable is unlimited in the tail of interest.

The Frechet distribution is not possible if the random variable is limited in the tail of interest.

Consequently, Weibull or Frechet distributions can be discarded as candidates to approximate maxima or minima because of physical reasons.

The Gumbel distribution can be approximated as much as possible by Weibull or Frechet distributions.

Consequently, Gumbel distribution can be avoided as candidates to approximate maxima or minima.

Finally, Weibull or Frechet can be selected for any case, depending on the limited or unlimited character of the variable in the tail of interest.

46


Guidelines for selecting the domain of attraction

1.- Use physical considerations to eliminate some domains of attractions. If the random variable is limited in the tail of interest, eliminate the Fréchetdomain of attraction; otherwise, eliminate the Weibull domain of attraction.

6.- In case of doubt between Gumbel and Weibull, choose the Weibull model.In case of doubt between Gumbel and Fréchet, choose the Fréchet model.

2.- If the data is maxima (minima), draw the data on the Maximal (Minimal) Gumbel Probability Paper.

3.- If the tail of interest (the right for maxima and left for minima) shows a linear trend, the domain of attraction is the Maximal (Minimal) Gumbel family.

4.- If the tail of interest has a vertical asymptote, then the domain of attraction is the Maximal (Minimal) Weibull.

5.- If the tail of interest has a horizontal asymptote, then the domain of attraction is the Maximal (Minimal) Fréchet.

47


Limit distributions of order statistics

48


Limit distributions of central order statistics

In particular if r(n)/n = p

49


Limit distributions of high order statistics

50


Limit distributions of low order statistics

51


Limit distributions of low and high order statistics

52


Limit distributions for the case of dependence

Questions similar to those formulated for the case of independentobservations also arise in the case of dependent observations.

In particular, we address the following questions:

1. Is the GEVD family of distributions the only limit family?

2. Under what conditions do the limit distributions for the independentcase remain valid for dependent observations?

53



As would be expected, the dependent observations case is more complicatedthan the independent case.

However, only partial knowledge is required.

One of the main reasons for this is that, while the independent case can be formulated in terms of the marginal distributions, the dependence case requires more information about the joint distribution of the random variables involved.

This implies that different joint distributions (different dependence conditions) can lead to the same limit distributions (this happens when the partial informationrequired coincide for both cases).

In the case of dependence any limit cdf is possible.

54



55



56



57


Limit distributions for ARMA models

58


Exceedances

Future number of exceedances of the m-th largest

Mean:

Variance:

Observation of the past Prediction of future

Can the number of exceedancesof the m-th largest observation be predicted?:

59


Limit distributions of exceedances

Generalized Pareto distribution

60


Limit distributions of exceedances

61


Example: Design of a breakwater

62


Example

63


Example

64


Example

65


Example

66


Example

67


Example

68


Example

69


Example

70


Conclusions

The case of extreme values is completely diferent from the case of mean values and requires especial statistical tools. Fail to recognize this feature leads to fatal errors.

In the case of independence, the limit distributions belong to the Von Mises family (Weibull, Gumbel and Frechet). Some of them can be excluded from physical or engineering considerations.

In the dependence case any limit distribution is possible, but for weak dependence, the same distribution arise.

The case of exceedances over a certain threshold can be dealt with the generalized Pareto distribution, which permits using more information than the maxima and minima.

Engineering practice is full of examples in which extreme value distributions must be used.

71


Some references of our work

Castillo, E., A. S. Hadi, N. Balakrishnan, and J. M. Sarabia (2004). Extreme Value and Related Models with Applications in Engineering and Science. New York: Wiley.

Coles, S. (2001). An introduction to statistical modeling of extreme values.Springer Series in Statistics. London: Springer-Verlag London Ltd.

Books

Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics(Second ed.). Malabar, Florida: Robert E. Krieger.

Leadbetter, M. R., G. Lindgren, and H. Rootzén (1983). Extremes and related properties of random sequences and processes. Springer Series in Statistics. New York: Springer-Verlag.

Arnold, B. C., N. Balakrishnan, and H. N. Nagaraja (1992). A first course in order statistics. New York: John Wiley and Sons Inc.

Castillo, E. (1988). Extreme Value Theory in Engineering. Academic Pres. New York.

72


Castillo, E., Losada, M., Mínguez, R., Castillo, C., and Baquerizo, A. (2004a). An optimal engineering design method that combines safety factors and failure probabilities: Application to rubble-mound breakwaters. Journal of Waterways, Ports, Coastal and Ocean Engineering, ASCE 130(2), 77–88.

Castillo, E., Conejo, A., Mínguez, R., and Castillo, C. (2003). An alternative approach for addressing the failure probability-safety factor method with sensitivity analysis. Reliability Engineering and System Safety82(2), 207–216.

Castillo, E., Mínguez, R., Ruíz-Terán, A., and Fernández-Canteli, A. (2004b). Design and sensitivity analysis using the probability safety-factor method. An application to retaining walls. Structural Safety 26, 159–179.

Castillo, E., Mínguez, R., Ruíz-Terán, A., and Fernández-Canteli, A. (2005). Design of a composite beam using the probability-safety factor method. International Journal for Numerical Methods in Engineering 62, 1148–1182.


Articles

73


Mínguez, R., Castillo, E., and Hadi, A. S. (2005). Solving the inverse reliability problem using decomposition techniques. Structural Safety 27, 1–23.

Castillo, C., Mínguez, R., Castillo, E., and Losada, M. (2006). An optimal engine-ering design method with failure rate constraints and sensitivity analysis. Application to composite breakwaters. Coastal Engineering 53, 1–25.

Mínguez, R., Castillo, E., Castillo, C., and Losada, M. (2006). Optimal cost design with sensitivity analysis using decomposition techniques. Application to composite breakwaters. Structural Safety 28, 321–340.


Mínguez, R., Castillo, E., and Hadi, A. S. (2005). Solving the inverse reliability problem using decomposition techniques. Structural Safety 27, 1–23.

74


THIS IS THE END

THANKS FOR

YOUR ATTENTION

75


Outline of the Talk

1. Motivation with some examples

2. Order statistics.2.1 Distribution of order statistics2.2 Some examples

3. Limit distributions.3.1 The case of maxima and minima. Domains of attraction3.2 The case of order statistics3.3 The case of dependence.3.4 Exceedances. Generalized Pareto distribution.

4. Example of a breakwater4.1 Joint distributions.

76



77


Structural engineering

A correct structure design requires a precise estimation of the modes of vibration and minimum tensile and compression strengths.

Tacoma Bridge Collapse

78


Pollution studies

With the existence of large concentrations of people (producing smoke, human wastes, etc.) or the appearance of new industries (chemical, nuclear, etc.), the pollution of air, rivers, and coasts has become a common problem for many countries. The pollutant concentration, expressed as the amount of pollutant per unit volume (of air or water), is forced, by government regulations, to remain below a given critical level.

Aznalcollar disasterin Seville (FeS2)

79


Lightning

80


Return periods

A return period also known as a recurrence interval is an estimate of the interval of time between events like an earthquake, flood or river discharge flow of a certain intensity or size.

It is a statistical measurement denoting the average recurrence interval over an extended period of time of a certain event.

It is usually required for risk analysis (i.e. whether a project should be allowed to go forward in a zone of a certain risk) and also to dimension structures so that they are capable of withstanding an event of a certain return period

81


Return periods

In mean, the event will occur once every years

82


Return periods

Return periods can be calculated from yearly data directly, or from other period data after correction.

From yearly data, they are obtained as follows:

83



84



85


Motivation

Extreme values appear in natural calamities of great magnitude, as for example:

1. the extraordinary dry spell in the western regions of the UnitedStates and Canada during the summer of 2003,

2. the devastating earthquake that destroyed almost the entire historic Iranian city of Bam in 2003,

3. the massive snowfall in the eastern regions of the United Statesand Canada during February 2004,

4. the destructive hurricanes and devastating floods that affect many parts of the world.

use of extreme value theory in engineering...

Documents