in¬‚uence the diameter, slope and other parameters of a city™s

the Generalized Extreme Value (GEV), Weibull, storm sewer system. An estimate of the 25-year event that is

Pareto, and Wakeby distributions are also used in too large will result in the building of a sewer system that has

extreme value analysis. larger capacity, and therefore higher cost, than necessary.

2: Probability Theory

48

model, will be exceeded on average once every

return period.

Return values are simply the upper quantiles of

the ¬tted extreme value distribution. For example,

suppose that the random variable Y represents

an annual extreme maximum and that Y has

probability density function f Y (y). The 10-year

return value for Y is the value Y(10) such that

∞

P Y > Y(10) = f Y (y) dy = 1/10.

y(10)

In general, the T -year return value for the annual

maximum, say Y(T ) , is the solution of

Figure 2.11: An example of the probability

density function f Y (y) and cumulative distribution ∞

f Y (y) dy = 1/T.

function FY (y) of an extreme value distribution

Y(T )

(for annual maxima). This particular distribution

is the Gumbel with parameters u = ln 6 and That is, the T -year return values are simply points

» = 1 (see [2.9.8]). The location of 2, 5, 10, 100, on the abscissa such that the area under the right

and 1000 year return values are indicated by the hand tail of the density function is 1/T . The

vertical bars. Note that the two-year return value concept is illustrated in Figure 2.11.

corresponds to FY (y) = 1/2, the 10-year value Return values for extreme minima are similarly

corresponds to FY (y) = 9/10, etc. Also note that computed using the tail areas under the left hand

the distribution has a much wider right hand tail tail of a suitable extreme value distribution.

than that of distributions we have become familiar

with.

2.9.7 Example: Daily Maximum Temperature.

As an example, consider the change in the

• the method of probability weighted moments, extremes of the daily maximum temperature at 2 m

and height that might occur as a result of a doubling

of the atmospheric CO2 concentration (see Zwiers

• the method of L-moments. and Kharin [448]). Zwiers and Kharin showed that

the annual extremes of temperature can be well

Optimality considerations in repeated sampling

represented by the EV-I distributions in both the

generally lead to the use of the method of

1—CO2 and 2—CO2 climates of the ˜CCC GCMII™

maximum likelihood (see [5.3.8]). On the other

General Circulation Model.22 Estimates of the 10-

hand, the method of L-moments is more robust.

year return values derived from the ˜control run™

This method is less affected by occasional

1—CO2 are displayed in Figure 2.12 (top). These

observational errors or data transcription errors

values verify reasonably well in general terms.

(such as a misplaced decimal point) than other

However, values at speci¬c locations should not

¬tting methods. The method of probability

be compared directly with return values estimated

weighted moments (see Hosking, Wallis, and

from station data because climate simulations can

Wood [184]) is closely related to the method of

not be considered reliable at length scales shorter

L-moments. The ordinary method of moments is

than a few grid lengths.

also frequently used because of simplicity and

Figure 2.12 (bottom) illustrates the change

convention considerations.21

induced in the 10-year return value by a doubling

of CO2 . The globally averaged increase is about

2.9.6 Return Values. The last step in an extreme

3.1 —¦ C. The corresponding value for the increase

value analysis is usually to compute ˜return values™

in the 10-year return value of the daily minimum

for preset periods (e.g., 10, 50, 100 years). These

temperature is 5.0 —¦ C, indicating that the shape

values are thresholds that, according to the ¬tted

of the temperature distribution might change

substantially with increasing CO2 concentrations.

21 The ordinary method of moments is similar to the method

of L-moments [2.6.7, 2.6.9]. Instead of matching population

22 The Canadian Climate Centre GCMII (McFarlane et al.

L-moments to estimated L-moments, ordinary population

moments (mean, variance, skewness, and kurtosis) are matched [270]). The CCC 2—CO2 experiment is described by Boer,

with corresponding estimates. McFarlane, and Lazare [52].

2.9: Extreme Value Distributions 49

In addition to the overall warming caused the domain of attraction of the EV-I distribution.

by the change in the radiative balance of the The essential element that controls convergence is

model climate under CO2 doubling, there are simply the point at which the right hand tail of the

also a variety of interesting physical effects that distribution generating the individual observations

contribute to the spatial structure of the changes begins to behave as the right hand tail of the ex-

in the return values. For example, daily maximum ponential distribution. Distributions for which the

temperatures are no longer constrained by the maximum of a sample of size n converges to the

effect of melting ice at the location of the 1—CO2 Gumbel slowly exhibit exponential behaviour only

sea ice margin. Also, the soil dries and the for observations that are many standard deviations

albedo of the land surface increases over the from the centre of the distribution.

Northern Hemisphere land masses, leading to a The EV-I distribution is a two-parameter

substantial increase in the extremes of modelled distribution with a location parameter u and a

scale parameter ». The density function of an EV-I

daily maximum temperature.

random variable Y is given by

2.9.8 Gumbel Distribution. To conclude this

f Y (y; u, ») = exp{’[(y ’ u)/» + e’(y’u)/» ]}.

section on extreme values, we provide a sim-

ple derivation of the Gumbel or EV-I distribu- The mean and the variance of Y are given by

tion [149]. Let X1 , . . . , Xn represent n indepen-

µY = u + γ »

dent, identically distributed, exponential random

variables observed on the short time scale. These Var(Y ) = »2 π 2 /6,

random variables might, for example, represent a

sample of n wind pressure measurements which, as where γ is Euler™s constant. The L-moments are:

we have noted previously, have a distribution that

(1)

is close to exponential.23 The distribution function » = u + γ »

»(2) = » ln 2

for any one of these random variables is

γ1L = 0.1699

F (x; ») = P (X < x) = 1 ’ e’x/» .

X

γ2L = 0.1504 .

Let Y be the maximum of {X1 , . . . , Xn }. Then

Y < y if and only if Xi < y for each i = 1, . . . , n. As noted above, return values are obtained

Using independence, we obtain that by inverting the distribution. For example, if the

Gumbel distribution were ¬tted to annual maxima,

n

then the T -year return value, say Y(T ) , is the

P (Y < y) = P (Xi < y)

solution of

i=1

= FX (y; »)n

1/T = P Y > Y(T )

= (1 ’ e’y/» )n

= 1 ’ FY (y(T ) ; u, »)

≈ exp{’ne’y/» }.

= 1 ’ exp{’e’(Y(T ) ’u)/» }. (2.38)

The quality of the approximation improves with

increasing n, that is, if each extreme is obtained Solving (2.38) yields

from a larger sample of observations collected on

the short time scale. After a bit more manipulation, y(T ) = u ’ » ln ’ ln(1 ’ 1/T ) .

we see that, as n increases inde¬nitely, the

2.9.9 Other Approaches. Another approach

distribution function of Y takes the form

to extreme value analysis that we have not

FY (y; u, ») = P (Y < y) = exp{’e’(y’u)/» }. discussed is the so-called peaks-over-threshold

approach. In contrast to analysing annual (or

This is the distribution function of the Gumbel

other period) maxima, the peaks-over-threshold

or EV-I distribution. Convergence to this distribu-