XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 1449
inputs allow ore processing responses to be incorporated
into mine planning, stockpile management and production
scheduling for better control of feed to the process plant.
The importance of testing sufficient samples in this
context has been captured by Garrido (2019) who states:
“In general, highly variable domains (high variance)
are “difficult” GMU*s to model or process and require
more analysis than homogeneous domains. If a data-
base has a poor sampling density, then the short-term
models will be smoothed and will not represent the
real variability of the mineral feed, losing the short-
term predictability and, therefore, decreasing the
metallurgist’s ability to react preventively before the
change of ore type occurs.”
It is therefore the accuracy of the models built to pre-
dict short-term fluctuations in processing response that is
important. This observation points to an alternative route
to selection of the minimum number of samples required
for metallurgical testing, based on the measured errors in
the predictive models.
How Many Samples Do We Need?—Let The Data
Speak
The authors propose that in establishing the minimum
number of samples required for evaluation of a domain or
ore type, the principle should be to let the data speak. That
is, once the available testwork results produce robust pre-
dictive models that do not vary significantly as additional
testwork results are added, there is little benefit from testing
more samples.
Error or uncertainty in a predictive model will, at
some point, cease to be reduced by the addition of more
samples and testwork data. This is because the accuracy of
the model will be limited by the accumulated errors asso-
ciated with geological sample heterogeneity, subsampling,
measurements of masses of samples and testwork products,
geochemical analysis, and errors introduced by operator
behaviour or sub-optimal function of the equipment during
the metallurgical test. In many deposits, important charac-
teristics that impact processing response, such as grain size
and texture, are not measured or captured in the geological
database and their absence contributes to the total error of
the prediction.
Therefore, the aim of the method proposed in this
paper is to measure the errors in the predictions as the
number of samples and test results increases and identify
the point at which the addition of further data does not
*GMU: Geo Metallurgical Unit (sic)
significantly improve the model. If the samples tested cover
the full extent of multivariate ore characteristics and the
predictive model is applied to the entire relevant drill hole
database and/or the resource block model, it can be said
that planning and evaluation will not be improved by fur-
ther testing.
The challenge then is to identify useful measures of
robustness, apply them to the predictive models and estab-
lish the criteria for the stage of study by which a model is
accepted as being sufficiently robust.
Possible candidates include:
• Standard error of model must be similar to that
of replicate testing. This is not a practical solution
because there is rarely sufficient replicate testing data
(if any at all).
• Adjusted R-squared metric. Whilst this is a com-
monly used measure of the performance of predictive
models, it is not necessarily robust, and it does not
provide a measure of error.
• Confidence limits based on the assumption that
errors are normally distributed.
For this study, we chose the root mean squared error
(RMSE) as our metric to compare predictive models.
RMSE is a measure of the average error and by assuming
that the errors are normally distributed, confidence inter-
vals, with which the mining industry is familiar, may be
estimated.
We tested the impact of increasing numbers of test-
work samples using a set of 41 Drop Weight index (DWi)
test results from an iron oxide copper gold project. This is
a data set that was considered possibly too small to dem-
onstrate that the limit of useful metallurgical testing has
been reached. For the test, we derived a much larger syn-
thetic data set by using a simulation method known as the
Probabilistic Graphical Model (PGM). The synthetic data
set consists of 10,000 samples with the same multivariate
statistical characteristics as the real set of 41 samples. For the
purpose of this analysis, we nominated DWi as the target
variable for prediction and nominated the following seven
numerical features and one categorical feature to set up a
predictive model: gold, silver, sulphur, copper, iron, and
magnetic iron grades, density, and lithology (categorical).
We considered a simple linear model (linear regression
(LR)) and a Cubist model. In linear regression, the goal is
to find the parameter values, or coefficients, that minimize
the quadratic sum of the differences between the actual
values and the estimated values. The Cubist algorithm is a
machine learning model that incorporates decision trees and
multivariate linear regression along with ensemble learning.
inputs allow ore processing responses to be incorporated
into mine planning, stockpile management and production
scheduling for better control of feed to the process plant.
The importance of testing sufficient samples in this
context has been captured by Garrido (2019) who states:
“In general, highly variable domains (high variance)
are “difficult” GMU*s to model or process and require
more analysis than homogeneous domains. If a data-
base has a poor sampling density, then the short-term
models will be smoothed and will not represent the
real variability of the mineral feed, losing the short-
term predictability and, therefore, decreasing the
metallurgist’s ability to react preventively before the
change of ore type occurs.”
It is therefore the accuracy of the models built to pre-
dict short-term fluctuations in processing response that is
important. This observation points to an alternative route
to selection of the minimum number of samples required
for metallurgical testing, based on the measured errors in
the predictive models.
How Many Samples Do We Need?—Let The Data
Speak
The authors propose that in establishing the minimum
number of samples required for evaluation of a domain or
ore type, the principle should be to let the data speak. That
is, once the available testwork results produce robust pre-
dictive models that do not vary significantly as additional
testwork results are added, there is little benefit from testing
more samples.
Error or uncertainty in a predictive model will, at
some point, cease to be reduced by the addition of more
samples and testwork data. This is because the accuracy of
the model will be limited by the accumulated errors asso-
ciated with geological sample heterogeneity, subsampling,
measurements of masses of samples and testwork products,
geochemical analysis, and errors introduced by operator
behaviour or sub-optimal function of the equipment during
the metallurgical test. In many deposits, important charac-
teristics that impact processing response, such as grain size
and texture, are not measured or captured in the geological
database and their absence contributes to the total error of
the prediction.
Therefore, the aim of the method proposed in this
paper is to measure the errors in the predictions as the
number of samples and test results increases and identify
the point at which the addition of further data does not
*GMU: Geo Metallurgical Unit (sic)
significantly improve the model. If the samples tested cover
the full extent of multivariate ore characteristics and the
predictive model is applied to the entire relevant drill hole
database and/or the resource block model, it can be said
that planning and evaluation will not be improved by fur-
ther testing.
The challenge then is to identify useful measures of
robustness, apply them to the predictive models and estab-
lish the criteria for the stage of study by which a model is
accepted as being sufficiently robust.
Possible candidates include:
• Standard error of model must be similar to that
of replicate testing. This is not a practical solution
because there is rarely sufficient replicate testing data
(if any at all).
• Adjusted R-squared metric. Whilst this is a com-
monly used measure of the performance of predictive
models, it is not necessarily robust, and it does not
provide a measure of error.
• Confidence limits based on the assumption that
errors are normally distributed.
For this study, we chose the root mean squared error
(RMSE) as our metric to compare predictive models.
RMSE is a measure of the average error and by assuming
that the errors are normally distributed, confidence inter-
vals, with which the mining industry is familiar, may be
estimated.
We tested the impact of increasing numbers of test-
work samples using a set of 41 Drop Weight index (DWi)
test results from an iron oxide copper gold project. This is
a data set that was considered possibly too small to dem-
onstrate that the limit of useful metallurgical testing has
been reached. For the test, we derived a much larger syn-
thetic data set by using a simulation method known as the
Probabilistic Graphical Model (PGM). The synthetic data
set consists of 10,000 samples with the same multivariate
statistical characteristics as the real set of 41 samples. For the
purpose of this analysis, we nominated DWi as the target
variable for prediction and nominated the following seven
numerical features and one categorical feature to set up a
predictive model: gold, silver, sulphur, copper, iron, and
magnetic iron grades, density, and lithology (categorical).
We considered a simple linear model (linear regression
(LR)) and a Cubist model. In linear regression, the goal is
to find the parameter values, or coefficients, that minimize
the quadratic sum of the differences between the actual
values and the estimated values. The Cubist algorithm is a
machine learning model that incorporates decision trees and
multivariate linear regression along with ensemble learning.