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being studied. The cell can be square, rectangular or have another shape. One
property of this model is that the complement (1
X
) also is the quantile of a
normal random variable with different mean but the same variance. The fact that
percentage values for a rock type and its complement can both have a frequency
distribution which satisfies the same equation may be of interest in the study of
closed number systems. If the values for a set of random variables sum to one at all
observation points, these variables cannot have the same type of distribution if one
or more of them has a normal, lognormal, or gamma distribution. On the other hand,
each variable in a set of random variables summing to one can have a frequency
ˁZʲ
p
1
ˁ
distribution function given by
X
¼
ʦ
:
An example of a complement
2
(1
X
) will be provided in Fig. 12.53. The close resemblance of probits to logits
was discussed in Sect.
12.2.1
(Fig.
12.8
). The logit of (1
X
) is - logit (
X
).
If this model is valid, observed cell values should have a frequency distribution
that plots as a straight line on “prob-prob” paper with standard normal quantile
scales along both
X
- and
Y
-axis. Then the parameters
can be estimated from
the intercept and slope of this straight line. In this respect the approach seems to
form a natural extension of two
Q
-
Q
straight line fitting techniques that are widely
applied: use of log-log paper in fractal-multifractal analysis, and log-prob paper in
fitting a lognormal frequency distribution to data.
In the next section, two previously used data sets (felsic metavolcanics in the
Bathurst and Abitibi areas) will be re-analyzed. The multifractal method of
moments also will be applied to the larger of these two data sets (Abitibi felsic
metavolcanics).
Matheron's (
1974
,
1976
) discrete Gaussian model can be applied to cell values
(
cf
. Agterberg
1981
,
1984
) as follows: Let
X
1
with density function
f
(
x
1
) represent a
random variable for average concentration values of small cells with size
S
1
and
X
1
,
with
f
(
x
2
), that of large cells with size
S
2
. Suppose that
X
1
can be transformed into
Z
1
by
X
1
¼
ˈ
1
(
Z
1
) and
X
2
into
Z
2
by
X
2
¼
ˈ
2
(
Z
2
) so that the random variable
Z
1
, with
marginal density function
ʲ
and
ˁ
ˆ
(
z
1
), and
Z
2
with
ˆ
(
z
2
), together satisfy the bivariate
standard normal density function
=
1
z
1
z
2
2
ˆ
ð
z
1
;
z
2
Þ
¼
p
1
exp
2
ˁ
z
1
z
2
þ
21
ˁ
2
ˀ
ˁ
2
where
represents the product-moment correlation coefficient of
Z
1
and
Z
2
.In
general, if the regression of
X
1
on
X
2
satisfies
E
(
X
1
|
X
2
)
ˁ
X
2
,
f
(
x
2
) can be derived
from
f
(
x
1
). The interpretation for a binary pattern is as follows. Suppose that
X
2
represents the average value for amount of pattern in a large cell superimposed on
the map, while
X
1
is this value for a small cell sampled at random from within the
large cell. If the small cell is made infinitely small,
X
1
becomes a binary random
variable that can be written as
X
0
for presence or absence of the pattern at a point.
Consequently,
¼
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