1 ’ |ρ| respectively. The mean of X is given by µ =

Thus, the closer |ρ| is to 1, the more concentrated (µ1 , µ2 ) where µ1 and µ2 are the means of X1

the variation about the principal axis will be. The and X2 respectively. The covariance matrix of X is

bottom of Figure 2.10 illustrates this with another given by

hypothetical vector wind distribution, in which

ρU V is increased to 0.8. 11 12

=

21 22

2.8.14 Example. Let us consider two univariate where 11 is the covariance matrix of X1 , 22

normal random variables, is the covariance matrix of X2 , 12 , which is

called the cross-covariance matrix, is the m 1 — m 2

U, V ∼ N (0, 1),

matrix of covariances of elements of X1 with X2 ,

related through U = AV, where A is a discrete and 21 = T

12 . The marginal distribution of

X1 is N (µ1 , 11 ), and that of X2 has a similar

random variable such that

form. From (2.31), we obtain that the conditional

P (A = 1) = P (A = ’1) = 1/2 .

distribution of X1 , given X2 = x2 , is also

Both random variables have a standard deviation multivariate normal with conditional mean

of 1, so by (2.33) the correlation ρU V between U ’1

µ1|2 = µ1 + 12 22 (x2 ’ µ2 ) (2.36)

and V is equal to the covariance

∞ and conditional covariance matrix

ρU V = uv fU V (u, v) dudv

’1

’∞ = ’ 12 .

T

(2.37)

11|2 11 12 22

∞

= u 2 fU (u) du a — P (A = a) The proof may be found in [281] or [147].

’∞

a

It is interesting to note that the conditional mean

of X1 depends upon X2 when 12 = 0 (i.e., when

1

= u 2 fU (u) du

a

X1 and X2 are dependent upon each other).

2

a=±1

= 0.

2.8.16 More on Conditional Distributions”

This should not, however, lead us to the conclusion Optional.15 The conditional mean (2.36) can be

that they are independent, since U2 = V2 . thought of as a linear speci¬cation of the value of

Examination of the probability density functions X1 that is based on X2 . The speci¬cation is linear

adds more satisfying evidence that these variables because the conditional mean is a vector of linear

are dependent: combinations of the elements of X2 .

The speci¬cation skill can be determined by

fU V (u, v)

fU |V =v (u) = computing the cross-covariances between the

f V (v)

vector of speci¬cation errors and random vectors

0 if u = ±v

= X1 and X2 . Useful speci¬cations will have errors

1/2 if u = ±v. with near zero covariance with X1 and exactly zero

That is, fU |V =v = fU . The variables are covariance with X2 . The interpretation in the ¬rst

case is that the speci¬cation accounts for almost

dependent, since the joint (bivariate) density

all of the variation in X1 because the errors have

function fU V cannot be represented as the product

little variation in common with X1 . In the second

of the two marginal distributions fU and f V (see

case, the interpretation is that all the information in

[2.8.5]).

Thus, we have found an example in which 15 Interested readers may want to return to this subsection

two dependent normal random variables have zero after reading Chapter 8. This material is presented here because

correlation. However, this does not contradict it ¬‚ows naturally from the previous subsection.

2.9: Extreme Value Distributions 45

the extremes of wind pressure loading which are

X2 about X1 that is obtainable by linear methods is

likely to occur during the life of the structures.

contained in the speci¬cation.

The roofs of houses built in high latitudes must

The speci¬cation errors are given by

be able to withstand extreme snow loads. Insurers

’1

X1|2 = X1 ’ µ1 + 22 (X2 ’ µ2 ) .

12 who underwrite the ¬nancial risk associated with

these natural risks must have good estimates of the

The covariance between the speci¬cation errors

size and impact of extreme events in order to set

and X2 is zero as required:

their premiums at a pro¬table level.

Cov X2 , X1|2 Extreme value analysis is the branch of

probability and statistics that is used to make

’1 T

= E X2 X1 ’ (µ1 + 22 (X2 ’ µ2 ))

12

inferences about the size and frequency of

= 0. extreme events. The basic paradigm used varies

with application but generally has the following

The covariance between the speci¬cation errors

components:

and X1 is

• data gathering;

Cov X1 , X1|2

= E X1 X1 ’ (µ1 + Σ12 Σ’1 (X2 ’ µ2 )) • identi¬cation of a suitable family of probabil-

T

22

ity distributions, one of which is to be used

= Σ11|2 .

to represent the distribution of the observed

extremes;

To determine from this whether the speci¬cation • estimation of the parameters of the selected

is skilful, one could compute the proportion of the model;

total variance of X1 that is explained by X2 . The

total variance of a random vector is simply the sum • estimation of return values for periods of

of the variances of the individual random variables ¬xed length. Return values are thresholds

that make up the vector. This is equal to the sum of which are exceeded, on average, once per

the diagonal elements (or trace) of the covariance return period.

matrix. Thus, a measure of the skill, s, is

We will discuss each of these items brie¬‚y in the

s = 1 ’ tr( 11|2 )/tr( 11 ) following subsections.

’1 T

22 12 )

tr( 12

= .

11 )

tr( 2.9.1 Data Gathering. Typically, the objects of

Note that s = 0 if 12 = 0 and that s = 1 study in extreme value analysis are collections of

1/2 1/2

when 12 = 11 22 . In fact, s cannot be greater annual maxima of parameters that are observed

daily, such as temperature, precipitation, wind

than 1.

speed and stream ¬‚ow. Thus, observations are

required on two time scales.

2.9 Extreme Value Distributions

• Observations are taken on short time scales

2.9.0 Introduction. Many practical problems

over a ¬xed time interval to obtain a single

encountered in climatology and hydrology require

extreme value. For example, they might

us to make inferences about the extremes of a

consist of daily precipitation accumulations

probability distribution. For example, the designs

for a year. The maximum of the 365

of emergency measures in river valleys, ¬‚oodways,

observations is retained as the extreme daily

hydro-electric reservoirs, and bridges are all

precipitation accumulation for the year, while

constrained in one way or another by the largest

the rest of the observations serve only to

stream ¬‚ow which is expected to occur over the

determine the extreme value.