Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 94

Fig. 3.10 Composite
histogram of copper values
from Muskox intrusion.
Original values x ik from
different rock types i ) were
made comparable with one
another by replacing them
by x 0 ik ¼ x ik /ave( x i ) where
ave( x i ) represents rock type
mean; horizontal line is for
log e (1+ x 0 ). Best-￿tting
curve is truncated normal
representing maximum-
entropy solution showing
truncated lognormality of
original data (Source:
Agterberg 1974 , Fig. 30)

Þ ˃

ðÞ X

˃ ð log e X ðÞ

ðÞ ¼ ʳ ʼ X

The coef￿cient of variation, therefore, is approximately equal to the standard
deviation of the logarithmically transformed data. This approximation is good if
ʳ < or its estimate from a sample can

be used as a criterion to distinguish between normality and lognormality. Hald
( 1952 ) has pointed out that if

0.5. From these considerations it follows that ʳ

1/3, a lognormal curve cannot be distinguished
from a normal curve. In general, a sample for which

ʳ <

ʳ > 0.5 is distinctly non-normal

but not necessarily lognormal. Because ʳ 0.8 for the rock types in Fig. 3.9 ,

non-normality is indicated.
In total, 622 copper values were used to estimate the means and standard devia-
tions plotted in Fig. 3.9 . Suppose that the copper values x ik for rock type i are divided
by their rock type mean. This transformation yields 622 new copper values x 0 with
overall average value equal to 1. The frequency distribution of these transformed
copper values was studied by Agterberg ( 1965 ). It is neither normal nor lognormal
but can be ￿tted by a truncated lognormal distribution for y 0 ¼ log e (1 + x 0 ), as shown

graphically in Fig. 3.10 . This result can be interpreted as an example of validity of the
maximum entropy criterion.
The concept of entropy is used in thermodynamics for evaluating the amount of
order or disorder in spatial con￿gurations of attributes. For example, the molecules of
an ideal gas can occur anywhere within a con￿ned space and their spatial con￿gu-
ration is completely random at any time. The entropy of the system then is at a
maximum (complete disorder). Shannon ( 1948 ) applied the concept of maximum
entropy to ordinary frequency distributions with n classes i ). If p i represents relative
frequency of occurrence in the i -th class, then a ￿rst constraint is

ʣ p i ¼ 1. Suppose

2 of the system is predetermined as a second constraint. The
entropy statistic S satis￿es:

that the variance ˃