Encyclopedia of Environmental Science and Engineering, Volume I and II

STATISTICAL METHODS FOR ENVIRONMENTAL SCIENCE 1131

Another way of looking at correlation is by consider-

ing the regression of one variable on another. Figure 7

shows the relation between two variables, for two sets of

bivariate data, one with a 0.0 correlation, the other with a

correlation of 0.75. Obviously, estimates of type value of one

variable based on values of the other are better in the case of

the higher correlation. The formula for the regression of y on

x is given by

yrxy

y

xy x

x



ˆ 

ˆ

( ˆ )

(ˆ )

.

m

s

m

s

(25)

A similar equation exists for the regression of x on y.

A number of other correlation measures are available.

For ranked data, the Spearman correlation coefficient, or

Kendall’s tau, are often used. Measures of correlation appro-

priate for frequency data also exist. See Siegel.

MULTIVARIATE ANALYSIS

Measurements may be available on more than two variables

for each experiment. The environmental field is one which

offers great potential for multivariate measurement. In areas of

environmental concern such as water quality, population stud-

ies, or the study of the effects of pollutants on organisms, to

name only a few, there are often several variables which are of

interest. The prediction of phenomena of environmental inter-

est, or such as rainfall, or floods, typically involves the consid-

eration of many variables. This section will be concerned with

some problems in the analysis of multivariate data.

Multivariate Distributions

In considering multivariate distributions, it is useful to define

the n -dimensional random variable X as the vector

XXXX′
[, 12 ,Κ, ].n

(26)

The elements of this vector will be assumed to be con-

tinuous unidimensional random variables, with density

functions f 1 (x 1 ), Ff 2 (x 2 )K,fn(xn) and distribution functions

F 1 (x 1 ),F 2 (x 2 )K,Fn(xn) Such a vector also has a joint distribu-

tion function.

Fx x(, ,, ) (^12 ΚΚxnn=PX^1 x^1 ,,X xn)

(27)

where P refers to the probability of all the stated conditions

occurring simultaneously.

The concepts considered previously in regard to univari-

ate distribution may be generalized to multivariate distri-

butions. Thus, the expected value of the random vector, X,

analogous to the mean of the univariate distribution, is

EX( ′)
[ (EX EX^12 ), ( ),KEX( n)],

(28)

where the E ( X i ) are the expected values, or means, for the

univariate distributions.

Generalization of the concept of variance is more com-

plicated. Let us start by considering the covariance between

two variables,

(^) sij
EX EX X[ i ( i)][ jEX( j)]. (29)
The covariances between each of the elements of the vector
X can be computed; the covariances of the i th and j th ele-
ments will be designed as sij If i = j the covariance is the
r = 0.0
r = 0.75
X
X
FIGURE 7
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