Geoscience Reference
In-Depth Information
Fig. 2 .17
PDFs and CDFs for
categorical variables
Conditional expectations are linear functions of the data. All
linear combinations of Gaussian variables are also Gaussian,
and in particular, averages are Gaussian. Also, conditional
variances are data-values-independent, a property called
ho-
moscedasticity
.
In geostatistics, it is common to assume that the normal
scores of grade variables are multivariate Gaussian within
geologically defined domains. This is done for convenience
since the simple (co)kriging method provides exactly the
mean and variance of all conditional distributions, as de-
scribed in Chaps. 8-10.
Performing a univariate normal score transformation
guarantees a univariate Gaussian distribution, but there is no
guarantee of a multivariate Gaussian distribution. The trans-
formation does not remove nonlinearity or other constraints.
The proportional effect and heteroscedasticity is largely re-
moved by the transformation, but then it is reintroduced by
the back transformation. Transforming a multivariate distri-
bution is rarely done in mineral resource estimation because
of the complexity and requirement for many data.
Categorical Variables
The probability distribution of a dis-
crete or categorical variable is defined by the probability or
proportion of each category, that is,
p
k
, k
=
1, …, K
, where
there are
K
categories. The probabilities must be non-negative
and sum to 1.0. A table of the
p
k
values completely describes
the data distribution. Sometimes, however, it is convenient
to consider a histogram and cumulative histogram as shown
below (Fig.
2.17
):
The cumulative histogram is a series of step functions
for an arbitrary ordering of the discrete categories. Such a
cumulative histogram is not useful for descriptive purposes
but is needed for Monte Carlo simulation and data transfor-
mation. In general, but not always, the ordering does not
matter. The cases where the ordering affects the results will
be discussed later in the topic.
Consider
K
mutually exclusive categories
s
k
, k
=
1, …, K
.
This list is also exhaustive; that is, any location
u
belongs
to one and only one of these
K
categories. Let
i
(
u
; s
k
) be the
indicator variable corresponding to category s
k
, set to 1 if
location
u
in
s
k
, zero otherwise, that is:
