1048 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
an overall framework to model the recovery potential of a
tailings storage facility.
The PSM is a least absolute shrinkage and selection
operator (LASSO)-regularized logistic regression, fed with
individual particle data obtained with automated mineral-
ogy: size (equivalent circle diameter, ECD), shape (aspect
ratio and solidity), modal composition, and surface com-
position. Data transformation consists of:
• Log transformation of size and shape variables,
• The square of the log-transformed ECD is added as
a new variable,
• Modal and surface composition vectors are individu-
ally closed to sum to 100% (van den Boogaart and
Tolosana-Delgado, 2013),
• Adding a categorical variable indicating the main
mineral composing the particle (in mass) to the data-
set—namely mainmineral.
Particle data, compiled in a training dataset that contains
the same number of particles for each process stream, is
used to train the PSM. From all particle data, 70% is used
for model training and the remaining 30% for validation.
The response variable of the logistic regression is particle
class (i.e., concentrate or tailings). The model allows for
interaction between size and shape variables and main-
mineral (as in an analysis of covariance framwork, a.k.a
ANCOVA) is used in the model to capture a singular rela-
tion between recovery and particle size and shape for each
mineral. After training the logistic regression, model inter-
cepts are adjusted to reflect the actual particle balance of the
process (i.e., prior adjustment).
Model validation is done by comparing the predicted
mass and composition of each processing product to that
observed in the test. For that, a bootstrap approach is used:
• Resample the validation particle dataset with the
same number of particles and replacement,
• Predict the probability of recovery of individual par-
ticles with the trained model,
• Assign particles to each processing stream based on
the computed probabilities,
• Compute the bulk mass and composition of each
processing stream,
• Compare to observed values.
This process is repeated 1000 times to capture model and
process uncertainties. Uncertainties of observed stream
composition values are captured by resampling their par-
ticle dataset, as proposed by Blannin et al. (2021). In this
study, all four concentrates collected in the flotation experi-
ment are represented as a single flotation product in the
PSM to simplify the analysis of results. The model is vali-
dated with each of the five composite samples.
Recovery Interpolation
With the obtained PSM global model, the MLA samples
from Blannin et al. (2022) are processed to obtain forecasts
of the mass pulls of each mineral to concentrates and tail-
ings. The prior adjustment applied for each sample follows
the particle balance observed in the flotation experiments
of its corresponding composite family. Given the goal to
recover sulfides, we pooled the mass of all minerals in two
groups: sulfide minerals on the one hand (group “sulfide”),
and the remaining minerals on the other (group “gangue”).
This gives, for each sample, predictions of four numbers:
• Msc: proportion of the feed mass corresponding to
sulfides reporting to the flotation concentrate,
• Mst: proportion of the feed mass corresponding to
sulfides reporting to the flotation tailings,
• Mgc: proportion of the feed mass corresponding to
gangue reporting to the flotation concentrate,
• Mgt: proportion of the feed mass corresponding to
gangue reporting to the flotation tailings.
With these four numbers, grades and recoveries are readily
available as
R M Mst
M
R M Mst
M
Gsc M M
M
Gst M M
M
sc sc
sc
st sc
st
sc gc
sc
st gt
st
=+=+
=+=+
(1)
where Rij and Gij respectively represent the recovery and
the grade of component i in product j. The four numbers
(Msc to Mgt) are almost always strictly positive and their
sum is necessarily one, as the sum of the masses of all pos-
sible components reporting to all possible output streams
must be equal to the feed mass. This allows us to treat them
geostatistically with the methods of compositional data
analysis (Tolosana-Delgado et al., 2019), which propose a
preliminary log-ratio transformation
ln M
Msc
M
Mst
M
M
gt gt gt
gc
1 g g2 g3 ===(2)
prior to the calculation of an empirical variogram.
Variograms describe the spatial dependence structure
exhibited by the target variables at any two locations
within the study area: direct variograms measure the loss
of dependence of the observations of a variable measured
at two locations as these two locations are taken farther
and farther apart cross variograms do the same concerning
the co-dependence between two variables observed at two
an overall framework to model the recovery potential of a
tailings storage facility.
The PSM is a least absolute shrinkage and selection
operator (LASSO)-regularized logistic regression, fed with
individual particle data obtained with automated mineral-
ogy: size (equivalent circle diameter, ECD), shape (aspect
ratio and solidity), modal composition, and surface com-
position. Data transformation consists of:
• Log transformation of size and shape variables,
• The square of the log-transformed ECD is added as
a new variable,
• Modal and surface composition vectors are individu-
ally closed to sum to 100% (van den Boogaart and
Tolosana-Delgado, 2013),
• Adding a categorical variable indicating the main
mineral composing the particle (in mass) to the data-
set—namely mainmineral.
Particle data, compiled in a training dataset that contains
the same number of particles for each process stream, is
used to train the PSM. From all particle data, 70% is used
for model training and the remaining 30% for validation.
The response variable of the logistic regression is particle
class (i.e., concentrate or tailings). The model allows for
interaction between size and shape variables and main-
mineral (as in an analysis of covariance framwork, a.k.a
ANCOVA) is used in the model to capture a singular rela-
tion between recovery and particle size and shape for each
mineral. After training the logistic regression, model inter-
cepts are adjusted to reflect the actual particle balance of the
process (i.e., prior adjustment).
Model validation is done by comparing the predicted
mass and composition of each processing product to that
observed in the test. For that, a bootstrap approach is used:
• Resample the validation particle dataset with the
same number of particles and replacement,
• Predict the probability of recovery of individual par-
ticles with the trained model,
• Assign particles to each processing stream based on
the computed probabilities,
• Compute the bulk mass and composition of each
processing stream,
• Compare to observed values.
This process is repeated 1000 times to capture model and
process uncertainties. Uncertainties of observed stream
composition values are captured by resampling their par-
ticle dataset, as proposed by Blannin et al. (2021). In this
study, all four concentrates collected in the flotation experi-
ment are represented as a single flotation product in the
PSM to simplify the analysis of results. The model is vali-
dated with each of the five composite samples.
Recovery Interpolation
With the obtained PSM global model, the MLA samples
from Blannin et al. (2022) are processed to obtain forecasts
of the mass pulls of each mineral to concentrates and tail-
ings. The prior adjustment applied for each sample follows
the particle balance observed in the flotation experiments
of its corresponding composite family. Given the goal to
recover sulfides, we pooled the mass of all minerals in two
groups: sulfide minerals on the one hand (group “sulfide”),
and the remaining minerals on the other (group “gangue”).
This gives, for each sample, predictions of four numbers:
• Msc: proportion of the feed mass corresponding to
sulfides reporting to the flotation concentrate,
• Mst: proportion of the feed mass corresponding to
sulfides reporting to the flotation tailings,
• Mgc: proportion of the feed mass corresponding to
gangue reporting to the flotation concentrate,
• Mgt: proportion of the feed mass corresponding to
gangue reporting to the flotation tailings.
With these four numbers, grades and recoveries are readily
available as
R M Mst
M
R M Mst
M
Gsc M M
M
Gst M M
M
sc sc
sc
st sc
st
sc gc
sc
st gt
st
=+=+
=+=+
(1)
where Rij and Gij respectively represent the recovery and
the grade of component i in product j. The four numbers
(Msc to Mgt) are almost always strictly positive and their
sum is necessarily one, as the sum of the masses of all pos-
sible components reporting to all possible output streams
must be equal to the feed mass. This allows us to treat them
geostatistically with the methods of compositional data
analysis (Tolosana-Delgado et al., 2019), which propose a
preliminary log-ratio transformation
ln M
Msc
M
Mst
M
M
gt gt gt
gc
1 g g2 g3 ===(2)
prior to the calculation of an empirical variogram.
Variograms describe the spatial dependence structure
exhibited by the target variables at any two locations
within the study area: direct variograms measure the loss
of dependence of the observations of a variable measured
at two locations as these two locations are taken farther
and farther apart cross variograms do the same concerning
the co-dependence between two variables observed at two