Geoscience Reference
In-Depth Information
14.7.8
Modeling 100 Association Matrix
Variables
A variogram is required for each of the 90 principal com-
ponents (10 sets/rows with 9 principal components in each).
As with the head grade variables these variograms were fit
with automatic variogram fitting software and visually in-
spected for inconsistencies.
Modeling the association matrix utilizes a combination of
the techniques previously discussed. The matrix is a 10 × 11
matrix where each row sums to 1.0 (or 100 %). Consider one
particular sample:
%UDQQHULWH&RIILQLWH 8UDQLQLWH3\ULWH&KDOFRS\ULWH%RUQLWH &KDOFRFLWH 2WKHU6XOSKLGHV $FLG6ROXEOH*DQJXH$FLG,QVROXEOH*DQJXH)UHH6XUIDFH
%UDQQHULWH
&RIILQLWH
8UDQLQLWH
3\ULWH
&KDOFRS\ULWH
%RUQLWH
&KDOFRFLWH
2WKHU6XOSKLGHV
$FLG6ROXEOH*DQJXH
$FLG,QVROXEOH*DQJXH
14.7.9
Special Considerations
for the Association Data
Each element in the matrix represents the % surface area
of interaction between minerals determined from mineral
liberation analysis. Each row sums to 1.0; however, each
column does not sum to a constant value as the values are
standardized by the proportions. There are a total of 100
elements in the matrix to be modeled, ignoring the diago-
nals. An assumption that the rows are independent is made
to reduce the problem to simulating 10 independent sets
(rows) of 10 dependent variables (columns). To maintain
the constant sum constraint the logarithmic transformation
is applied to each row resulting in the need to model 9
logarithmic variables. The PCA transformation is applied
to reproduce the correlation between variables in each row.
The principal components of each row are normal score
transformed and then simulated with SGS. There are a total
of 490 data available for simulation of association vari-
ables.
As with the grain size variables, the head grade simu-
lations provide a super secondary variable to use in col-
located SGS. There are a total of 23 (normal score PCA)
head grade simulations to be combined into a single super
secondary variable for each of the 100 elements in the as-
sociation matrix. The PCA transform is done in such a way
that the amount of data explained by each principal com-
ponent can be measured. Some components 'contain' more
information than others. In this case the first five compo-
nents of the head grade realizations contain over 75 % of
the information in the original head grades. Only the first 5
principal components generated in the head grade modeling
are combined into the super secondary variable to reduce the
computational requirements of the methodology. Moreover,
the super secondary variable is only used for the first 4 of the
9 principal components of the association variables. Because
there are 100 association variables to model, CPU time be-
comes an issue.
Missing or “null” values always pose a problem in composi-
tional data modeling. In this instance there are some entries
that are missing because a particular mineral does not appear
in a given sample. For rows that have some missing values
but still sum to 1.0, the missing values are reset to 0.0001 or
0.01 %. In some cases there are entire rows that are missing.
This is because the mineral does not appear at that location;
however, in these cases all values cannot be set to a small
value as they would not sum to 1.0. The solution undertaken
in this study was to remove the samples where the entire row
was missing.
When performing SGS at this location the values in that
particular row are simulated as if the data did not exist (in
fact this data does exist and has a value of zero). The mis
match between the missing values at this location and the
simulated values given the surrounding data can be fixed by
assigning a 0.0 proportion to the missing minerals, and the
mismatched association values can be ignored.
14.7.10
Histogram/Variogram Reproduction
There are 135 variables modeled in total. The histograms and
variograms reproduction for the first 3 realizations have been
analyzed. The following discussion compares the input his-
tograms and variograms to the realization outputs.
14.7.10.1 Head Grade Variables
The head grade variables reproduce the histogram quite
well (Fig.
14.69
) because of post-processing. Variogram re-
production is checked in normal score units of the principal
components.
