Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 25

1.3.3.3 Rank correlation method
The Spearman rank correlation between X 1 and X 2 , denoted by ρ 12 , is the Pearson cor-
relation between the ranks of X 1 and X 2 . Namely, each X samples are converted into its
ranks. For instance, if there are five (X 1 , X 2 ) samples (random sequence initiated using
randn['state', 13]):

=

() 1 () 2 () 3 () 4 () 5

XXXXX
XXXXX

1 1 1 1 1

X T

() () 1 () 2 () 3 () 4 5

2 2 2 2 2

(1.43)

120067 . . 174047 . − . 168 .

=

− 128094 . − . 027143 . − . − 051 .

Then the ranks are

32514
23514

X r T = (1.44)

The Spearman rank correlation ρ 12 is simply the Pearson correlation between the two
rows of ranks in Equation 1.44 . The Spearman rank correlation ρ 12 quantifies how well the
relationship between two variables can be described as a monotonic function. For ρ 12 = 1,
the relationship between the two variables is perfectly positively monotonic (not necessarily
linear), and for ρ 12 = −1, the relationship is perfectly negatively monotonic. In MATLAB, ρ 12
can be estimated using the function corr( X 1 , X 2 ,'type', 'Spearman').
In general, the Spearman correlation ρ 12 is not the same as the Pearson correlation δ 12 .
Table 1.6 s hows an example where X and Y are perfectly monotonically correlated. In fact,
Y = X 3 . The Pearson correlation δ = 0.906 is computed by corr( X , y , 'type', 'Pearson'),
whereas the Spearman correlation ρ = 1 is computed by the MATLAB function corr( X , y ,
'type', 'Spearman'). They are not equal. The Pearson correlation δ is not exactly 1 because
it quantifies “linear correlation,” and in this case, the relationship between X and Y is
nonlinear. The Spearman correlation ρ is exactly 1 because the ranks of X and Y are iden-
tical (see the fourth and fifth columns of Table 1.6 ) . As a result, the Spearman correlation
ρ quantifies how well the relationship between (X, Y) can be described as a monotonic
function.
It is well known that the Pearson product-moment correlation δ 12 between the bivariate
normal random variables (X 1 , X 2 ) is approximately equal to its Spearman rank correlation
ρ 12 . It is therefore possible to estimate δ 12 using the rank correlation ρ 12 .

Table 1.6 Values of (X, Y) data points
Index k X Y X rank Y rank

1 0 0 1 1

2 1 1 2 2

3 2 8 3 3

4 3 27 4 4

5 4 64 5 5

6 5 125 6 6