and X j is given by

σi j = E (xi ’ µi )(x j ’ µ j )

Cov Xi , X j

ρi j = . (2.33)

= (xi ’ µi )(x j ’ µ j ) f xi x j (xi , x j ) d xi d x j

Var(Xi )Var X j

R2

The correlation coef¬cient always takes values

between the ith and jth elements of X. Note that in the interval [’1, 1]. The absolute value of the

the covariance matrix is symmetric: the covariance coef¬cient is exactly 1 when Xi is linearly related

between Xi and X j is the same as that between to X j , that is, when constants a and b exist so

X j and Xi . The diagonal elements of are the that Xi = a + bX j . Here the correlation is +1

variances of the individual random variables that if b is positive and ’1 if b is negative. Values

form the random vector X. That is, σii = Var(Xi ).

2

of the correlation coef¬cient between ’1 and +1

The covariance matrix is positive-de¬nite. are an indication of the extent to which there

Covariances describe the tendency of jointly is a linear relationship between the two random

distributed random variables to vary in concert. If variables. In fact, ρi2j can be interpreted usefully

the deviations of Xi and X j from their respective as the proportion of the variance of one of the

means tend to be of the same sign, the covariance variables that can be represented linearly by the

between Xi and X j will be positive, and if other (see also [18.2.7] and Section 8.2).

the deviations tend to have opposite signs, the As an example, consider the 1933“84 segment

covariance will be negative. of the Southern Oscillation Index (SOI) (Fig-

As in the discrete variable case, the covariance ure 1.4). Superimposed on the graph is Wright™s

is zero if Xi and X j are independent. This SST index of the SO [426]. The SST index carries

occurs because the expectation of a product roughly the same information about the Southern

of independent random variables factors into a Oscillation on time scales of a year or more. The

product of expectations. Note, however, that the estimated correlation between the monthly mean

reverse need not be true (see the example in values of these indices is 0.67. We will examine

[2.8.14]). this example in more detail in Section 8.2.

The effect of scaling on covariance is similar to A word of caution about the correlation

that which occurs in the scalar case (see (2.6)). If coef¬cient: it is not always a measure of the

A is a k — m matrix with k ¤ m, then extent to which there is a deterministic relationship

between two random variables. In fact, two

Σ A X ,A X = AΣAT .

random variables may well be related through a

deterministic, nonlinear function and yet have a

14 Nearby values in all atmospheric and oceanic ¬elds

correlation of zero.

are related to one another. In fact, without this property

initialization of numerical weather prediction models would

require a much denser observing network than exists today.

2.8.8 Mapping the Correlation Matrix: Tele-

Objective analysis and data assimilation techniques, which

are used to initialize forecast models, make extensive use of connection Patterns. The various combinations

the covariance structure of the atmosphere. Climate forecast

of correlations between the ith and jth compo-

systems based on coupled ocean/atmosphere models also make

nents of a random vector X form the correlation

extensive use of such techniques to initialize the oceanic

matrix. If X represents a ¬eld, a (possibly gridded)

components of these models.

2.8: Random Vectors 41

upper right half shows spatial correlations of the

low-frequency variations while the lower left half

shows the longitudinal correlations at the synoptic

time scale. The diagrams are read as follows. If we

read across from 0—¦ on the vertical scale and up

from 40—¦ E on the horizontal scale we see that the

(simultaneous) correlation between low-frequency

time scale variations at 0—¦ and 40—¦ E is about ’0.3.

The banded structure in the lower left re¬‚ects

the midlatitude stormtracks. The strongest (neg-

ative) correlations are found in a band that is

about 30—¦ off the diagonal. When there is a deep

low at a given longitude, it is likely that there

will be a high 30—¦ to the east or west, and vice

versa. The organization of the correlation minima

in bands indicates that the disturbances propagate

(the direction of this propagation cannot be read

from this diagram).

The correlation structure is no longer banded

at time scales of 10 or more days. On these time

Figure 2.8: Correlation matrices for the simulta- scales, height anomalies east of the dateline are

neous variations of 500 hPa height along 50—¦ N strongly connected with anomalies of opposite

in the synoptic time scale (lower left) and the sign over North America (this re¬‚ects the PNA-

low-frequency transpose (upper right) band. Only pattern, [3.1.7]); other links appear over Europe

negative correlations are shown. From Fraedrich and Asia, and over the East Atlantic and Europe.

et al. [127].

2.8.9 Multivariate Normal Distribution. The

m-dimensional random vector X has a multivariate

set of observations in space, then the jth row

normal distribution with mean µ and covariance

(or column) of the correlation matrix contains the

matrix Σ if its joint probability density function is

correlations between the ¬eld at the jth location

given by

and all other locations. When this row is mapped

we obtain a spatial pattern of correlations that T ’1

1

e’(x’µ) Σ (x’µ) . (2.34)

f X (x) =

climatologists call a teleconnection pattern or (2π|Σ|)1/2

teleconnection map. A map is considered ˜inter-

A bivariate normal density function is shown

esting™ if it exhibits large correlations at some

distance from the ˜base point™ j, and if it suggests in Figure 2.9. Like its univariate counterpart, the

physically plausible mechanisms (such as wave distribution is symmetric across all planes which

pass through the mean. The spread, or dispersion,

propagation).

Such maps often unveil large-scale ˜teleconnec- of the distribution is determined by the covariance

tions™ between a ¬xed base point and distant areas. matrix Σ.

An important property of the multivariate

We deal with these teleconnection maps in some

normal distribution is that linear combinations of

detail in 17.4.

The entire correlation matrix may be plotted normal random variables are again distributed as

when the ¬eld is one-dimensional. For example, normal random variables. In particular, let A be a

Fraedrich, Lutz, and Spekat [127] analysed daily full rank m — m matrix of constants with m ¤

500 hPa geopotential height along 50—¦ N. The m. Then Y = AX de¬nes an m -dimensional

annual cycle was removed from the data, and then random vector which is distributed multivariate

two different time ¬lters (see 17.5) were applied to normal with mean vector Aµ and covariance

separate the synoptic disturbances (2.5 to 6 days) matrix AΣAT (Graybill [147]).

from low-frequency atmospheric variability (time An immediate consequence of this result is

scales greater than 10 days). that all marginal distributions of the multivariate

The correlation matrices (only negative corre- normal distribution are also normal. That is,

lations are shown) of these two one-dimensional individual elements of a normal random vector X

random vectors are shown in Figure 2.8. The are normally distributed and subsets of elements of

2: Probability Theory

42

This can be rewritten as

2

(xi ’µi )2

m m

’

1 2σi 2

f X (x) = = f X i (xi ).

√ e

2π σi

1

i=1 i=1

x2

0

Thus, if the non-diagonal elements Σ are