Geoscience Reference
In-Depth Information
terpreted as a separate population altogether (see for example
Parker
1991
). There are a number of methods to deal with out-
liers at the time of variography and resource estimation.
In general, outliers or extreme values are considered on
a case-by-case basis with sensitivity studies and considering
their impact on local and global resource estimates.
2.2.6
Multiple Variable Distributions
Mineral resource estimation commonly considers multiple
variables. The multiple variables could be geometric attri-
butes of the deposit or grades such as thickness, gold, silver,
or copper grades. They could be the same grade sampled at
different locations. Bivariate and multivariate statistics are
used in these cases. There are many references to multivari-
ate statistics, such as Dillon and Goldstein (
1984
).
The cumulative distribution function and probability den-
sity function can be extended to the bivariate case. Let
X
and
Y
be two different RVs. The bivariate cdf of
X
and
Y
,
F
XY
(
x, y
) and the pdf of
X
and
Y
f
XY
(
x, y
) are defined as
Fig. 2.8
Scatterplot of Bitumen vs. Fines Gaussian variables
example in Fig.
2.8
has a negative covariance because the
relationship is dominated by an inverse relationship.
The correlation coefficient between random variables
X
and
Y
is defined as the covariance between
X
and
Y
divided
by the standard deviations of the
X
and
Y
variables:
F
( ,
x y
)
=
Prob X
{
≤
x
, and
Y
≤
y
}
XY
and
=
Co
v
{
X
,
Y
}
2
∂
F
(, )
xy
ρ
XY
XY
f
(, )
xy
=
σ
X
σ
Y
XY
∂∂
xy
We could also define a bivariate histogram, that is, divide the
range of the
X
and
Y
variables into bins and plot bivariate fre-
quencies. It is more common to simply plot a scatterplot of
paired samples on arithmetic or logarithmic scale. Figure
2.8
shows an example from the oil sands in Northern Alberta,
Canada, after transformation to a Gaussian variable.
The means and variances of each variable are used as
summary statistics. The covariance is used to characterize
bivariate distributions:
The correlation coefficient is a dimensionless measure
between − 1 (a perfect inverse linear relationship) and + 1
(a perfect direct linear relationship). Independence between
the two variables means that the correlation coefficient is
zero, but the reverse is not necessarily true. A covariance or
correlation coefficient of zero means there is no dominant di-
rect or inverse relationship, but the variables may be related
in a nonlinear manner.
Second order moments like the variance and covariance
are significantly affected by outlier data. Some outlier pairs
can destroy an otherwise good correlation or enhance an oth-
erwise poor correlation, see Fig.
2.10
. The sketch on the left
illustrates a case where some outliers would make an oth-
erwise good correlation appear low; the sketch on the right
shows a case where a few outliers make an otherwise poor
correlation appear high.
The
rank
correlation is more robust with respect to outli-
ers, and is obtained by calculating the correlation coefficient
on the rank order of the data. Each data variable is replaced
by its rank position in the dataset, and then the correlation
coefficient is calculated using the rank positions.
It is common for both correlation coefficients to be shown
on experimental cross plots as in Fig.
2.8
where a direct
comparison of the two correlation coefficients can be made.
Their difference highlights whether there are data features,
Cov X Y
{
,
}
=
E
{[
X
−
m
][
Y
−
m
]}
=
E XY
{
}
−
m
m
X
Y
XY
+∞
+∞
∫∫
=
dx
(
x
−
m
)(
y
−
m
)
f
( ,
x y dy
)
X
Y
XY
−∞
−∞
The unit of the covariance is the product of the units of the
two variables, for example, g/t Au multiplied by thickness in
meters. Since these units are hard to understand or interpret,
it is common for the covariance to be standardized.
The covariance describes whether the bivariate relation-
ship is dominated by a direct or an inverse relationship,
see Fig.
2.9
. The product of [
X
−
m
X
][
Y
−
m
Y
] is positive in
quadrants II and IV; it is negative in quadrants I and III. The
expected value is the average of the product over all pairs.
The example of Fig.
2.9
has a positive covariance, while the
