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Accepted
Mathematical Geosciences manuscript No.
(will be inserted by the editor)
Modeling Geospatial Uncertainty of Geometallurgical
Variables with Bayesian Models and Hilbert–Kriging
J´
ulio Hoffimann ·Jos´
e Augusto ·Lucas
Resende ·Marlon Mathias ·Douglas
Mazzinghy ·Matheus Bianchetti ·Mˆ
onica
Mendes ·Thiago Souza ·Vitor Andrade ·
Tarc´
ısio Domingues ·Wesley Silva ·Ruberlan
Silva ·Danielly Couto ·Elisabeth Fonseca ·
Keila Gonc¸alves
Received: date / Accepted: date
Abstract In mine planning, geospatial estimates of variables such as comminution
indexes and metallurgical recovery are extremely important to locate blocks for which
the energy consumption at the plant is minimized and for which the recovery of
minerals is maximized. Unlike ore grades, these variables cannot be modeled with
traditional geostatistical methods, which rely on the availability of large number of
samples for variogram estimation, on the additivity of variables for change of support,
among other issues. Past attempts to build geospatial models of geometallurgical vari-
ables have failed to address some of these issues, and most importantly, did not con-
sider adequate mathematical models for uncertainty quantification. In this work, we
propose a new methodology that combines Bayesian predictive models with Kriging
in Hilbert spaces to quantify the geospatial uncertainty of such variables in realis-
tic industrial settings. The results we obtained with data from a real deposit indicate
that the proposed approach may become an interesting alternative to geostatistical
simulation.
Keywords bayesian modeling ·kriging ·hilbert spaces ·drop weight test ·bond
work index ·metallurgical recovery ·geostatistics ·geometallurgy
J´
ulio Hoffimann (
), Jos´
e Augusto, Lucas Resende, Marlon Mathias
Instituto de Matem´
atica Pura e Aplicada
E-mail: julio.hoffimann@impa.br
Douglas Mazzinghy
Universidade Federal de Minas Gerais
Matheus Bianchetti, Mˆ
onica Mendes, Thiago Souza, Vitor Andrade, Tarc´
ısio Domingues, Wesley Silva,
Ruberlan Silva, Danielly Couto, Elisabeth Fonseca, Keila Gonc¸ alves
Vale S.A.
Accepted
2 J´
ulio Hoffimann et al.
1 Introduction
In the mining industry, the exploitation of mineral resources is planned on the basis of
geospatial models of ore grades, and more recently, geometallurgical variables such
as comminution indexes (e.g., DWT, BWI) and metallurgical recovery. Laboratory
experiments to estimate geometallurgical variables are time-consuming compared to
the chemical analysis that is performed meter-by-meter along drill holes, and con-
sequently, only a few (<100) core samples of greater support (e.g., 10 meters) are
subject to comminution and flotation tests. Moreover, the distributions of these vari-
ables cannot be Gaussian (e.g., bounded support). In particular, it is well-known that
comminution indexes and metallurgical recovery are not additive—the component-
weighted average of two sample values is not a good estimator of the corresponding
value in the blend (Tavares and Kallemback 2013 Carrasco et al. 2008)—and that this
property hinders the use of linear geostatistical models. Ignoring these issues can lead
to bias in geospatial estimates and suboptimal mine planning (Campos et al. 2021).
Although geostatistical methods such as Kriging and Gaussian simulation have
been extensively proposed for ore grades (Journel 2003), these same methods cannot
be directly applied to geometallurgical variables for various reasons, which include
reduced number of samples for variogram estimation, non-additive variables, non-
Gaussian distributions, among others. Past attempts to build geospatial models of
geometallurgical variables did not address these issues or considered modeling as-
sumptions that are not valid in industrial settings.
A common strategy to circumvent the modeling challenges in geometallurgy con-
sists of (1) applying nonlinear transformations to the data, (2) modeling the data as
if it was Gaussian, and (3) undoing the transformations (Boisvert et al. 2013 Deutsch
et al. 2015). In spite of its appeal, this strategy cannot handle physical constraints
that are crucial in downstream applications (e.g., material balance, positivity of hard-
ness) nor it can quantify uncertainty adequately. Consequently, models created with
this strategy usually require detailed human intervention and ad-hoc modifications to
work in practice. They are not robust to small variations in the data, cannot be applied
online, nor can be easily transferred across different types of deposits.
For specific deposits, it is sometimes possible to exploit relationships between
a set of auxiliary variables to design geospatial models of primary geometallurgical
variables. As an example, consider the metallurgical recovery of copper, defined as
the mass ratio of copper in the concentrate by copper in the feed in a flotation process.
If the numerator and denominator are linearly related, then it is possible to use CoK-
riging to estimate both (additive) variables simultaneously (assuming enough samples
are available for variogram estimation), and consequently estimate the ratio at each
mining block (Adeli et al. 2021). Although clever, this model is not general enough
for wider use by the industry, which is usually concerned with nonlinear recoveries
from locked cycle tests.
Finally, and most importantly, the misspecification of Gaussian distributions for
geometallurgical (or transformed) variables may lead to poor uncertainty estimates.
Together with the reduced number of samples that is common in geometallurgical
modeling, this is extremely undesirable.
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Title Suppressed Due to Excessive Length 3
In this work, we propose a new methodology that combines Bayesian predictive
models (Gelman 2014) with Kriging in Hilbert spaces (Hilbert-Kriging) (Menafoglio
et al. 2013) to quantify the geospatial uncertainty of geometallurgical variables. Bayesian
models allow for the incorporation of domain expertise in the form of prior distri-
butions and physical relations, which counterweights the reduced number of sam-
ples. Hilbert-Kriging enables the interpolation of non-Gaussian distributions from
drill holes to mining blocks. As a result, the proposed methodology provides quick
(≈6 minutes) probabilistic estimates of non-Gaussian variables across space without
compute-intensive geostatistical simulation.
The paper is organized as follows. In section 2, we define Bayesian models for
different geometallurgical variables and explain how the output of these models is
used in Hilbert–Kriging. All the steps of the proposed methodology are illustrated
using data from a real copper deposit. In section 3, we discuss the results and possible
technical challenges associated with the application of the methodology in practice.
In section 4, we conclude the work and point to future research directions.
2 Methodology
2.1 Overview
The flowchart in Figure 1 illustrates the steps of the proposed methodology, which
starts with three tables: comminution,flotation, and drillholes. The comminution
and flotation tables contain data for a reduced number of rock samples—cylinders—
that are subject to comminution and flotation tests. The content of these tables will
be described in more detail in the following sections. The drillholes table is the
standard table in the mining industry that contains chemical analysis data for all rock
samples along the drill holes with their respective X, Y, Z coordinates.
In the first step, the comminution and flotation tables are used to infer the
posterior distribution of geometallurgical variables according to a set of Bayesian
models. These models take chemical (or mineralogy) data as input and output full
probability distributions for Drop Weight Test (DWT), Bond Work Index (BWI) and
metalurgical recovery of copper in a Locked Cycle Test (LCT). By assuming that the
samples in the drillholes table are composited to a length that is similar to the
length of the samples in the other two tables, the Bayesian models can be directly
applied to the rock samples in the drill holes. The result of this second step consists
of a new predictions table that contains probabilistic predictions for all rock sam-
ples along the drill holes. In the final step, the probability distributions of DWT, BWI
and LCT are interpolated for all blocks with Hilbert–Kriging, which accounts for the
change of support. The final result is a block model where each block has a full prob-
ability distribution for each geometallurgical variable. No geoestatistical simulation
is involved.
In Figure 2, we illustrate the geospatial configuration of the samples from the real
copper deposit investigated in this work. The nature and number of comminution
and flotation samples invalidate the use of traditional geostatistical methods.
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4 J´
ulio Hoffimann et al.
Fig. 1: Flowchart of the proposed methodology
2.2 Bayesian models
Hereafter, we assume that the reader is familiar with Bayesian statistics. The excellent
introductory book “Bayesian Methods for Hackers: Probabilistic Programming and
Bayesian Inference” by Cameron Davidson-Pilon (Davidson-Pilon 2015) is freely
available online for those who are not familiar with basic concepts such as Bayes
rule, prior distributions, etc.
The following sections describe three specific Bayesian models for DWT, BWI
and LCT, but other Bayesian models could have been adopted instead. Each model
takes a vector x
x
x∈Rpof explanatory variables as input and outputs probabilistic pre-
dictions of the response geometallurgical variable. In this work, the explanatory vari-
ables are chemical compositions, which are also available in the drillholes table.
In order to facilitate the specification of prior distributions and to reduce the effects
of abundant elements, the explanatory variables are transformed with a centred-log-
ratio (CLR) (Aitchison 2003) followed by a normal-quantile (Barnett and Deutsch
2012) (a.k.a. normal-score) transform.
Accepted
Title Suppressed Due to Excessive Length 5
Fig. 2: Geospatial configuration of comminution,flotation and drillholes
samples in a real copper deposit
2.2.1 Drop Weight Test
In this comminution test, a standard weight is dropped from three heights h1,h2,h3on
nrock samples and particle size distributions are recorded after each impact φ(i)
hj,i=
1,2,...,n,j=1,2,3. For each height hjthere is a corresponding potential energy
Ehj, which is considered an input to the test. The value t(i)
10 (hj) = φ(i)
hj(l/10)is defined
as the fraction of particles with size smaller or equal to l/10 where lis the original
size of each and every sample (i.e., l(1)=l(2)=·· · =l(n)=l).
The standardized formula
t10(h) = A1−e−bEh(1)
relates the potential energy to the fraction of fine material (Napier-Munn et al. 1999).
The parameters A>0 and b>0 are fitted for each sample with least squares, consid-
ering the three energy levels. Let Ji(A,b) = ∑3
j=1t(i)
10 (hj)−A1−e−bEhj2
be the
sum of squares for the i-th sample. It can be written in matrix form as
Ji(A,b) = ∥t
t
t(i)−A1
1
1−e−bE
E
E∥2
2(2)
with E
E
E= (Eh1,Eh2,Eh3)and t
t
t(i)= (t(i)
10 (h1),t(i)
10 (h2),t(i)
10 (h3)). The energy levels, which
are also standardized in the industry, are shared across all samples. The optimal solu-
tion is obtained via minimization:
A(i),b(i)=argmin
A,b>0
Ji(A,b)(3)
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6 J´
ulio Hoffimann et al.
Finally, the drop weight test index is defined as DWT(i)=A(i)×b(i). It summa-
rizes the fragility of the sample in the following sense: the higher is the index, the
more fragile is the sample.
In Appendix A, we demonstrate that the DWT index cannot be a linear function of
any given set of explanatory variables. In particular, it cannot be modeled with linear
geostatistical models. Motivated by this fact, and aware of the variables involved in
the computation of the index, we propose the following Bayesian hierarchical model:
t10(hj) = t10 (hj−1) + ∆(hj)(4)
∆(hj)∼LogitNormal(µj,σ2
j)(5)
µj=αj+⟨β
β
βj,x
x
x⟩(6)
αj∼Normal(δj,σ2
j)(7)
β
β
βj∼Normal(0
0
0,τ2
jI
I
I)(8)
The model defines random variables for the fractions of fine material in terms
of increments ∆(hj) = t10(hj)−t10 (hj−1). These increments are random variables
themselves with a LogitNormal(µj,σ2
j)distribution. The mean parameter µj=αj+
⟨β
β
βj,x
x
x⟩of this distribution is assumed to be an affine combination of explanatory
variables with normally-distributed coefficients β
β
βjand intercept αj. By fixing the
hyperparameters of the model δj,σj,τj, we can sample the prior distribution of frac-
tions, and consequently, the prior distribution of DWT as illustrated in Figure 3.
Fig. 3: Prior distribution of t10(h1),t10(h2),t10 (h3), and consequently of DWT. Data
from the comminution table illustrated with vertical lines
Accepted
Title Suppressed Due to Excessive Length 7
The hyperparameters δjlocate the centers of the three density plots. They are
initialized with empirical averages from the comminution table:
δ1=1
n
n
∑
i=1
t(i)
10 (h1)(9)
δ2=1
n
n
∑
i=1
t(i)
10 (h2)−t(i)
10 (h1)(10)
δ3=1
n
n
∑
i=1
t(i)
10 (h3)−t(i)
10 (h2)(11)
The hyperparameters σjand τjspecify the prior uncertainty around these centers.
They are initialized with fixed large values to encompass all possible physical values.
All hyperparameters can be adjusted by the domain expert if he/she believes that the
default prior distribution for the deposit is different.
2.2.2 Bond Work Index
The Bond Work Index (BWI) in kW h/tis an index that quantifies the resistance of a
sample to grinding according to a test carried out in a laboratory mill. The laboratory
mill has 305mm as internal diameter and 305mm in length. The mill has a smooth
lining with rounded corners and no lifters. The ball media charge has approximately
20.1kg distributed in different sizes starting with 38mm (Bond 1961). Ultimately, the
BWI index is estimated using the formula
BWI =49.0
A0.23M0.82 10.0
√P80 −10.0
√F80 ,(12)
where F80 and P80 are the apertures in µmfor which 80% of the material in the feed
and in the product passes through; Mis the moability index in g/rev; and Ais the
size of the screen used in µm.Equation 12 was derived by Bond decades ago. Even
though its exponents were calibrated around that time with a specific data set, the
formula is still widely used in the industry for a variety of mineral deposits, possibly
adopting correction factors for specific cases (Rowland 1975). In this work, we adopt
the formula as is, but emphasize the need for a full data-driven approach without
preset constants.
To improve the convergence of Bayesian inference in subsection 2.3 and to facili-
tate the specification of prior distributions, we normalize the variables as follows. Let
mF=1
n∑n
i=1F80(i)be the empirical average of F80 in the comminution table, and
let G=P80/F80 be the ratio of apertures. For a fixed screen size A, we propose the
Accepted
8 J´
ulio Hoffimann et al.
following Bayesian hierarchical model:
F80
mF∼Normal(µF,σ2
F)(13)
G∼LogitNormal(µG,σ2
G)(14)
P80
mF
=GF80
mF
(15)
M∼Normal(µM,σ2
M)(16)
µF=αF+⟨β
β
βF,x
x
x⟩(17)
µG=αG+⟨β
β
βG,x
x
x⟩(18)
µM=αM+⟨β
β
βM,x
x
x⟩(19)
αF∼Normal(1,σ2
F)(20)
αG∼Normal(mG,σ2
G)(21)
αM∼Normal(mM,σ2
M)(22)
β
β
βF∼Normal(0
0
0,τ2
FI
I
I)(23)
β
β
βG∼Normal(0
0
0,τ2
GI
I
I)(24)
β
β
βM∼Normal(0
0
0,τ2
MI
I
I)(25)
Due to the normalization by mF, the distribution of αFis centered at 1. The hy-
perparameters mGand mMlocate the center of the other two density plots in Figure 4.
They are initialized with empirical averages from the comminution table:
mG=1
n
n
∑
i=1
P80(i)
F80(i)(26)
mM=1
n
n
∑
i=1
M(i)(27)
Similar to the DWT model, the hyperparameters σF,σG,σM, and τF,τG,τM
specify the prior uncertainty around these centers. They are initialized with fixed
large values. By fixing the hyperparameters of the model, we can sample the prior
distribution of F80, G,P80, M, and consequently of BWI using Equation 12.
2.2.3 Locked Cycle Test
A Locked Cycle Test (LCT) is a low-cost test to estimate the metallurgical recovery
of a given ore in an industrial-scale flotation circuit (Agar 2000). The LCT carried out
in this project considered an arrangement of cells that are known as “Rougher” and
“Cleaner” to separate the mass of copper sulphides from the gangue minerals that are
present in the rock sample, see Figure 5.
In a first stage, a sample of mass mis fed into the Rougher cell with a known
chemical composition x
x
x= (xCu,xAu,...,xFe ). A fraction frof this mass is recovered
in the Rougher concentrate with an enriched grade of copper xCu
r≥xCu. In a second
Accepted
Title Suppressed Due to Excessive Length 9
Fig. 4: Prior distribution of F80, G,M, and consequently of BWI. Data from the
comminution table illustrated with vertical lines
Fig. 5: Schematic illustration of “Rougher-Cleaner” cells inside a locked cycle test.
Data from flotation table shown in blue color.
stage, the mass in the Rougher concentrate mris fed into the Cleaner cell (or a se-
quence of such cells) and a new fraction fcis recovered with an even richer grade
xCu
c≥xCu
r. The metallurgical recovery of copper at the end of the second stage is
defined as the mass ratio of copper in the Cleaner concentrate by copper in the feed:
RCu
rc =mCu
c
mCu =mCu
c
mcmc
mrmr
m m
mCu =xCu
cfcfr
1
xCu (28)
Equation 28 is only valid for an open circuit. In reality, the metallurgical recovery
is the result of many cycles (e.g., 9) of flotation stages. Because the Rougher stage is
more widely available, we define the intermediate recovery
RCu
r=xCu
rfr
1
xCu (29)
Accepted
10 J´
ulio Hoffimann et al.
and map it to the final recovery with an affine transformation α+βRCu
r. These mod-
eling steps lead to the following Bayesian hierarchical model:
LCT ∼LogitNormal(µ,σ2)(30)
µ=α+βRCu
r(31)
α∼Normal(a,σ2)(32)
β∼Normal(0,τ2)(33)
RCu
r=xCu
rfr
1
xCu (34)
xCu ∼LogitNormal(µx,σ2
x)(35)
fr∼LogitNormal(µf,σ2
f)(36)
xCu
r∼LogitNormal(µr,σ2
r)(37)
µx=αx+⟨β
β
βx,x
x
x⟩(38)
µf=αf+⟨β
β
βf,x
x
x⟩(39)
µr=αr+⟨β
β
βr,x
x
x⟩(40)
αx∼Normal(rx,σ2
x)(41)
αf∼Normal(rf,σ2
f)(42)
αr∼Normal(rr,σ2
r)(43)
β
β
βx∼Normal(0
0
0,τ2
xI
I
I)(44)
β
β
βf∼Normal(0
0
0,τ2
fI
I
I)(45)
β
β
βr∼Normal(0
0
0,τ2
rI
I
I)(46)
The hyperparameters rx,rf,rrand alocate the center of the density plots in
Figure 6. They are initialized with empirical averages from the flotation table:
rx=1
n
n
∑
i=1
xCu(i)(47)
rf=1
n
n
∑
i=1
fr(i)(48)
rr=1
n
n
∑
i=1
xCu
r
(i)(49)
a=1
n
n
∑
i=1
LCT(i)(50)
The hyperparameters σx,σf,σr,σand τx,τf,τr,τspecify the prior uncertainty
around these centers. They are initialized with fixed large values. By fixing the hy-
perparameters of the model, we can sample the prior distribution of xCu,fr,xCu
r, and
consequently of LCT as illustrated in Figure 6.
Accepted
Title Suppressed Due to Excessive Length 11
Fig. 6: Prior distribution of xCu,fr,xCu
r, and consequently of LCT. Data from the
flotation table illustrated with vertical lines
2.3 Inference and predictions
Having specified the three Bayesian models in subsection 2.2, we can now proceed
with Bayesian inference to obtain the posterior distribution of all the modeled vari-
ables given the data from the comminution and flotation tables. This posterior
distribution can then be used to make probabilistic predictions on unseen samples
from the drillholes table.
In this section, we describe the content of these three tables in more detail, and
illustrate predictions of DWT, BWI and LCT for arbitrarily chosen samples to solidify
the concepts presented so far.
2.3.1 Input tables
The comminution table contains data that are used to calibrate the DWT and BWI
models. We recall that the DWT index is the product of two parameters that are
obtained via least squares and three fractions of fine material known as t10(h1),t10(h2)
and t10(h3). Similarly, the BWI index is obtained with a standardized formula in terms
of F80, P80, Mand A(see Equation 12). Therefore, the table must contain all these
7 columns that are directly available from the corresponding laboratory experiments.
Besides these columns, the table must also contain the columns with explanatory
variables x
x
x.
The flotation table contains data that are used to calibrate the LCT model. We
recall that the final metallurgical recovery of copper in the locked cycle test is ob-
tained from the intermediate recovery in the Rougher stage, which is in turn obtained
from the grade of copper and mass fraction in the Rougher concentrate, xCu
rand fr,
and from the grade of copper in the feed xCu. Therefore, the table must contain these
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12 J´
ulio Hoffimann et al.
3 variables. Unlike the other two models, this model also requires a column with the
target LCT variable. Besides these 4 columns, the table must also contain the columns
with explanatory variables x
x
x.
Finally, the drillholes table contains the X, Y and Z columns with the coordi-
nates of samples along the drill holes and the columns with explanatory variables x
x
x.
See the data availability section for more details.
2.3.2 Bayesian inference
Given the models in subsection 2.2 and the data from the comminution and flotation
tables, we can proceed and perform Bayesian inference with an extension of Hamilto-
nian Monte Carlo known as the No-U-Turn Sampler (NUTS) (Hoffman and Gelman
2014). The result is a collection of samples from the joint posterior distribution of
all variables in the proposed models. In Figure 7, we illustrate the prior and poste-
rior distributions of DWT, BWI and LCT. As expected, probabilities get relocated
from regions where there are few or zero samples to regions where there is a greater
number of samples.
Fig. 7: Prior and posterior distributions of DWT, BWI and LCT illustrating the relo-
cation of probability to regions where there is a greater number of samples
We can also visualize the joint posterior distribution of any subset of latent vari-
ables, and check that all samples lie inside high density regions. This is illustrated in
Figure 8 for variables t10(h1),t10 (h2), and t10(h3).
Most importantly, it is during Bayesian inference that we learn the joint posterior
distribution of coefficients for all explanatory variables. This distribution is used to
make predictions on unseen samples as explained next.
Accepted
Title Suppressed Due to Excessive Length 13
Fig. 8: Joint posterior distribution of t10(h1),t10(h2), and t10(h3). Samples (white
dots) in high density regions (red areas)
2.3.3 Predictions on drill holes
For any rock sample with a volume (i.e., support) similar to the volume of the sam-
ples in the comminution and flotation tables, we can use the joint posterior dis-
tribution of coefficients to make probabilistic predictions of all other variables in the
Bayesian models. We simply evaluate the models forward using the explanatory vari-
ables x
x
x(e.g., chemical analysis, mineralogy data) of the rock sample and thousands
of likely values of coefficients obtained with Bayesian inference. In particular, this
procedure generates thousands of likely values of DWT, BWI and LCT for the rock
sample of interest as illustrated in Figure 9.
We observe that the posterior mean is not a good estimator of these three ge-
ometallurgical variables due to the asymmetry of their distributions, and that a naive
approach with Gaussian distributions would have failed to capture this property. We
also notice how the specification of the prior distribution is crucial to constrain the
values of the variables to physical ranges. Even with a reduced number of samples
(≈50), we can still obtain useful uncertainty intervals for DWT, BWI and LCT.
In order to avoid storing thousands of likely values of DWT, BWI and LCT for
each rock sample in the drillholes table, we integrate the probability density func-
tions (pdfs) in Figure 9 into probability mass functions (pmfs) in Figure 10. This inte-
gration consists of counting how many values lie on predefined bins (i.e., histogram).
In the case of DWT, we use bins from an industry standard (Chieregati and Delboni Jr
2001): ETA (0,10), EXA (10,20), MTA (20,30), ALT (30,40), MDA (40,50), MED
(50,60), MDB (60,70), BAI (70,90), MTB (90,110), ETB (110,120).
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14 J´
ulio Hoffimann et al.
Fig. 9: Probabilistic predictions of DWT, BWI and LCT for arbitrarily chosen sam-
ples from the comminution and flotation tables. Values observed in the tables lie
within the density plots. Posterior mean is not a good estimator of observed values
Fig. 10: Corresponding probability mass functions for arbitrarily chosen samples in
Figure 9
The predicted pmfs of DWT, BWI and LCT for all rock samples in the drillholes
table are stored in a new predictions table along with the X, Y and Z coordinates
of the samples. This table is the input to the next step of the methodology.
Accepted
Title Suppressed Due to Excessive Length 15
2.4 Hilbert–Kriging
Here we propose the use of Hilbert–Kriging (Menafoglio et al. 2013) for direct in-
terpolation of pdfs (or pmfs) at mining blocks given the predictions table. We
briefly review the main concepts behind the framework and explain how an appropri-
ate choice of a Hilbert space can address the issue of non-additivity with geometal-
lurgical variables.
2.4.1 Main concepts
Consider a collection of kgeoreferenced objects (e.g., numbers, functions) in a Hilbert
space (i.e., vector space with well-defined inner product) H:
z(s
s
s1),z(s
s
s2),...,z(s
s
sk)∈H(51)
where s
s
s1,s
s
s2,...,s
s
sk∈D⊂R3are the locations of the objects. For example, con-
sider vectors with dcomponents (i.e., H=Rd) and the usual inner product ⟨x,y⟩=
∑d
i=1xiyi, or square-integrable functions (i.e., H=L2) with the inner product ⟨f,g⟩=
Rf(t)g(t)dt (Giraldo et al. 2010). The goal of the Hilbert-Kriging framework is to de-
fine an estimator of the object z(s
s
s)at a new location s
s
s∈Das a weighted combination
of the available objects:
ˆz(s
s
s) = λ1·z(s
s
s1) + λ2·z(s
s
s2) + ···+λk·z(s
s
sk)(52)
The estimator in Equation 52 is defined in terms of the scalar multiplication (·)
and the vector addition (+) in H. Since the choice of the inner product (⟨,⟩)induces
a norm, a distance, and consequently a notion of variance in H, the Universal Krig-
ing system of equations for optimal weights can be generalized to Hilbert spaces via
constrained minimization of estimation variance:
minimize
λ1,λ2,...,λk
Var (ˆz(s
s
s)−z(s
s
s)) (53)
subject to E[ˆz(s
s
s)] = m(s
s
s)(54)
where m(s
s
s) = ∑L
l=0alfl(s
s
s)is the drift, flare pre-specified monomials in s
s
s, and alare
coefficient objects in the Hilbert space.
In practice, the generalization of a modern Kriging implementation to a Hilbert–
Kriging implementation consists of two main modifications. First, the coefficient ob-
jects must be estimated with an implementation of generalized least-squares that sup-
ports objects in Hilbert spaces:
ˆ
a
a
a= (F
F
F⊤Σ
Σ
Σ−1F
F
F)−1F
F
F⊤Σ
Σ
Σ−1z
z
z(55)
where z
z
z= (z(s
s
s1),z(s
s
s2),...,z(s
s
sk)) ∈Hkis the vector with all available objects, Σ
Σ
Σis
the covariance matrix between all objects, F
F
Fis the monomial matrix for all locations
s
s
s1,s
s
s2,...,s
s
skand ˆ
a
a
a= ( ˆa0,ˆa1,..., ˆaL)∈HL+1is the vector of estimated coefficients
objects (see section 2.4 of Hilbert-Kriging paper).
Accepted
16 J´
ulio Hoffimann et al.
Second, the empirical variogram (assuming an intrinsic stationary model) must
be estimated in terms of the induced norm (∥·∥) in H:
ˆ
γ(h
h
h) = 1
2|N(h
h
h)|∑
(i,j)∈N(h
h
h)∥z(s
s
si)−z(s
s
sj)∥2(56)
where N(h
h
h) = (i,j):s
s
si−s
s
sj≈h
h
his the usual set of pairs of locations aligned with
the lag vector h
h
h∈R3.
Very informally, we say that the choice of scalar multiplication and vector addi-
tion determine the geometry of weighted combinations (Equation 55), and that the
choice of inner product determines the structure of geospatial dependence in Equa-
tion 56 (Menafoglio and Petris 2016). Given that the objects of interest in this work
are pdfs (or pmfs) predicted with Bayesian models at drill hole samples, we propose
the use of a specific Hilbert space known as the Aitchison space for Hilbert-Kriging
interpolation (Menafoglio et al. 2014).
2.4.2 Aitchison space
In 1986, the statistician J. Aitchison introduced the branch of statistics known today
as compositional data analysis to cope with nonlinear constraints on the entries of
vector variables (Aitchison 2003). He developed a vector space Awhere vectors
p
p
p= (p1,p2,...,pm)of real entries satisfy
p1,p2,...,pm≥0 (57)
m
∑
i=1
pi=P(58)
The first constraint, known as the non-negativity constraint, reflects the fact that
sometimes entries in a vector only contain relative information (a.k.a. proportions).
The second constraint, known as the fixed-sum constraint, exists to guarantee that the
initial amount of quantify Pis preserved after vector operations. The specific value P
is not relevant and is often replaced by P=1 after careful re-normalization.
We note that pmfs satisfy both constraints, and therefore we can leverage the
operations of the space Ato interpolate these objects without ever producing invalid
probability distributions. The scalar multiplication and vector addition are defined as
λ·p
p
p=C(pλ
1,pλ
2,...,pλ
m)(59)
p
p
p+q
q
q=C(p1q1,p2q2,...,pmqm)(60)
with λ∈Ra scalar, p
p
p,q
q
q∈A, pmfs with mbins, and C(p
p
p) = p
p
p
∑m
i=1pithe closure (or
re-normalization) operation. As already discussed in the previous section, these oper-
ations determine the geometry of weighted combinations. In Figure 11, we illustrate
how two pmfs morph into one another according to these definitions.
The inner product defined as
⟨p
p
p,q
q
q⟩=∑
i<j
log pi
pj
log qi
qj
(61)
Accepted
Title Suppressed Due to Excessive Length 17
Fig. 11: Scalar multiplication and vector addition in Aitchison space determine how
pmfs p
p
p,q
q
q∈A(top-left and bottom-right plots) morph into one another as we vary
λ∈Rin the weighted combination (1−λ)p
p
p+λq
q
q
considers ratios (relative information) of entries as opposed to absolute values. It
induces a norm ∥p
p
p∥=p⟨p
p
p,p
p
p⟩and a distance d(p
p
p,q
q
q) = ∥p
p
p−q
q
q∥that can be written
as
d(p
p
p,q
q
q) = v
u
u
t∑
i<jlog pi
pj−log qi
qj2
(62)
The distance in Equation 62 determines the structure of geospatial dependence. It
is used for empirical variogram estimation in Equation 56.InFigure 12, we illustrate
how this distance increases as we relocate probability mass across different bins in
Figure 11. It is important to note that these definitions require pmfs with non-zero
entries. This is a well-known limitation of the Aitchison space that we overcome
with the addition of a very small value to all bins (a.k.a. Laplace smoothing).
Finally, we emphasize that Hilbert–Kriging of pmfs in the Aitchison space en-
joys all the features of traditional Kriging, including change of support. In addition,
Hilbert–Kriging of pmfs is less prone to non-additivity issues. The scalar multipli-
cation and vector addition in the Aitchison space assure that the asymmetries of the
pmfs are preserved and that linear combinations of pmfs are good estimators of the
resulting shape of the distribution.
2.4.3 Variography
Before we can perform Hilbert–Kriging of pmfs, we first need to model the variogram
of these objects. Our goal in this section is to illustrate the shape of such variograms
for DWT, BWI and LCT. Since our intuition for geospatial dependence of pmfs is
Accepted
18 J´
ulio Hoffimann et al.
Fig. 12: Inner product in Aitchison space determines distances between pmfs as prob-
ability mass is relocated across different bins in Figure 11
limited, we do not attempt to interpret these shapes as we usually do with variograms
of scalar variables (e.g., ore grades).
Figure 13 illustrates the empirical variograms of the three residual pmfs along
the downhole direction after subtracting a drift pmf of zero degree (a.k.a. Ordinary
Hilbert–Kriging). Because the distance in Equation 62 is very sensitive to relocation
of probability mass (see Figure 12), these variograms show high nugget effect. In the
same visualization, we plot the corresponding theoretical models fitted with weighted
least-squares using weights that are proportional to the bin counts.
From the fitted theoretical models in Figure 13, we observe that the pmfs of DWT
and BWI display a visible correlation length (a.k.a. range) for the support of the
samples in the predictions table (10m), whereas the pmf of LCT does not. For
simplicity, and because we do not have enough evidence to support an anisotropic
theoretical model, we assume that actual variograms of pmfs are not a function of
direction, and use these theoretical models fitted along the downhole direction as our
omnidirectional models in Hilbert–Kriging.
2.4.4 Geostatistical estimation
Having modeled the variograms of the three residual pmfs, we can proceed and per-
form Hilbert-Kriging to estimate the pmfs at all mining blocks of size 30m×30m×
15m. The change of support is performed as usual by integration of the specified var-
iogram within each block. Let p
p
p= (p1,p2,..., pm)be an estimated pmf at a given
block. We adopt the mode η(p
p
p) = argmaxipias our final estimate of the correspond-
ing (block-support) geometallurgical variable and the entropy H(p
p
p) = −∑ipilog pi
as our measure of uncertainty.
Accepted
Title Suppressed Due to Excessive Length 19
Fig. 13: Empirical variograms of residual pmfs of DWT, BWI and LCT along drill-
hole direction (orange). Theoretical models (green) fitted with weighted least-squares
using weights that are proportional to the bin counts (gray)
First, we consider the entropy maps of DWT, BWI and LCT in Figure 14,Fig-
ure 15 and Figure 16. We note that these maps cannot be easily obtained from the
geospatial configuration of samples alone in Figure 2. Unlike the Kriging variance,
which is only a function of the geospatial configuration and specified variogram,
these maps also encode non-linear relations built into the Bayesian models of sub-
section 2.2. In particular, the entropy maps of the two comminution variables—DWT
and BWI—are similar, indicating that they may indeed reflect uncertainty about the
mechanical competence of the mining block to be extracted.
After considering the geospatial uncertainty, we proceed and look at the mode
maps of the three geometallurgical variables. In this specific deposit, the mode maps
of DWT and LCT are uninteresting because most predictions of these two variables
lie in the same bin interval during the integration of pdfs into pmfs in subsubsec-
tion 2.3.3. These homogeneous maps are not an issue, since they are just a conse-
quence of the binning choices adopted by the industry. The mode map of BWI is the
only heterogeneous map for the specified number of bins. It is shown in Figure 17.
The block model with estimated pmfs is the final result of the proposed methodol-
ogy. In Table 1, we summarize the wall time of each major step in our implementation
of the methodology, which was executed with 8 parallel threads in an Intel®Xeon®
Platinum 8354H CPU @ 3.10GHz CPU.
3 Discussion
As with any other academic work, there are technical challenges that need to be
discussed before any serious adoption of the technology by the industry.
Accepted
20 J´
ulio Hoffimann et al.
Fig. 14: Geospatial uncertainty of DWT measured by entropy of estimated pmfs
Fig. 15: Geospatial uncertainty of BWI measured by entropy of estimated pmfs
In previous sections, we highlighted the major strengths of the proposed method-
ology, which are:
1. Robust predictions of (non-Gaussian) geometallurgical variables based on do-
main expertise (e.g., nonlinear relations, prior distributions).
2. Probabilistic predictions from which one can easily obtain uncertainty estimates
(e.g., entropy) to assess risk and make strategic decisions.
3. Online predictions that get updated during the development of the mine as soon
as new data becomes available.
Accepted
Title Suppressed Due to Excessive Length 21
Fig. 16: Geospatial uncertainty of LCT measured by entropy of estimated pmfs
Fig. 17: Most likely interval of BWI measured by mode of estimated pmfs
And we also emphasized that it can generate quick, geospatial, probabilistic esti-
mates of geometallurgical variables without compute-intensive geostatistical simula-
tion (e.g., Gaussian simulation).
In this section, we would like to highlight the weaknesses of the adopted math-
ematical models and discuss important implementation details that are necessary to
make the solution work well in practice. We organize the discussion as a series of
issues below.
Accepted
22 J´
ulio Hoffimann et al.
Table 1: Wall times of different steps of the methodology. Inference performed with
≈60 samples from the comminution and flotation tables, prediction performed
with 2000 samples from the drillholes table and Hilbert-Kriging performed with
≈3500 blocks. Wall time per geometallurgical variable is ≈6 minutes
Inference Prediction Hilbert-Kriging Total time
DWT 344 sec 177 sec 3 sec 8.7 min
BWI 83 sec 177 sec 3 sec 4.3 min
LCT 67 sec 253 sec 3 sec 5.3 min
3.1 Choice of explanatory variables
Bayesian models are generally very good at constraining their predictions to accept-
able ranges even when the explanatory variables x
x
xare not well selected. In this work,
we did experiments with transformed chemical compositions (see subsection 2.2) as
our explanatory variables given their widespread availability. However, after a se-
ries of experiments with subcompositions, we concluded that these variables may not
necessarily be the best predictors of DWT, BWI nor LCT. For instance, we noticed
that Bayesian metrics such as the Kullback-Leibler score did not vary much with the
addition of new chemical elements to the list of variables. Similar experiments with
mineralogical compositions led to the same conclusions.
We believe that the predictive performance of the proposed Bayesian models can
be considerably improved with a more thoughtful selection of explanatory variables.
In particular, we believe that the resistance of a rock sample to grinding is associ-
ated with textural features more than it is associated with chemical or mineralogical
compositions. Additionally, extrinsic factors such as operating speeds, circulating
volumes, etc. could be taken into account for the prediction of both comminution and
flotation variables.
3.2 Covariate shift
Although the specification of prior distributions give domain experts control of the
likely ranges of all latent variables in Bayesian models, covariate shift of the explana-
tory variables may still be an issue (Hoffimann et al. 2021). In particular, when the
comminution and flotation tables are too localized in the deposit, it is wise to
consider transfer learning methods before making predictions with the drillholes
table (Pan and Yang 2010 Weiss et al. 2016).
3.3 Compositing restrictions
The proposed methodology assumes that the samples in the drillholes table are
composited to lengths that are comparable to the lengths of samples in the comminution
and flotation tables. If for some reason the comminution and flotation tests are per-
formed with samples of radically varying lengths, it may not be easy to leverage all
the data in Bayesian inference.
Accepted
Title Suppressed Due to Excessive Length 23
In practice, laboratory tests are standardized and performed with samples of simi-
lar support. Hence, we understand that it is always possible to composite chemical or
mineralogical compositions in drill holes to standardized supports in geometallurgy.
3.4 Oversimplification of physics
To quickly test the methodology with real data from a copper deposit, we opted for
oversimplified models of BWI and LCT. The physical processes that are involved
in these two tests require dynamical modeling, which was out of the scope of this
particular project.
In the case of the BWI index, we adopted the Bond formula in Equation 12, which
relies on a series of empirical observations and constants derived decades ago using
a specific data set. If the actual deposit shows different behavior, then this behavior
can only be incorporated in the model via manual modification of the constants.
Regarding the LCT model, we replaced important aspects of the flotation process
by a simple affine map from the Rougher stage to the final metallurgical recovery in
the locked cycle test. This oversimplification certainly compromises the predictive
performance of the model, and requires the measurement of the target LCT variable
in the flotation table.
3.5 Variable scaling
Bayesian inference algorithms are sensitive to variable ranges. It is important to scale
all variables in a Bayesian model or introduce auxiliary variables to improve the
convergence of the associated Markov chains. Fortunately, it is usually possible to
scale a variable by a non-zero mean value, or shrink the variable with a nonlinear
transformation in the design of new models.
As an example, we introduced the variable G=P80/F80 ∈[0,1]in the BWI
model to improve the convergence of the NUTS algorithm with default hyperpa-
rameters. Our attempts to work directly with F80 and P80 values in µmled to low
effective sample size and poor mixing of chains.
3.6 Binning choices
In subsubsection 2.3.3, bin intervals were chosen to convert posterior samples into
probability mass functions (pmfs). In the case of DWT, these intervals were chosen
from an industry standard, whereas in the case of BWI and LCT, a number of bins
was chosen to reflect a given level of detail.
It is clear that the choice of bins affects the predicted pmfs. The number of bins
must be large enough to approximate well the underlying probability density func-
tions (pdfs) and small enough to make Hilbert-Kriging feasible in conventional com-
puters.
Accepted
24 J´
ulio Hoffimann et al.
3.7 Laplace smoothing
The definition of inner product in Equation 61 assumes non-zero entries in the pmfs,
and this is a well-known issue of the Aitchison space. In order to perform variography
and Hilbert-Kriging, one must re-normalize the result with the addition of a small
threshold value to all entries. The addition of this threshold value can be formulated
as Laplace smoothing, which in turn can be interpreted as a form of prior knowledge.
Nevertheless, the threshold value is ad-hoc, and care must be taken to preserve the
original shape of the pmf as much as possible.
3.8 Variogram interpretation
Unlike variograms of scalar regionalized variables, the variograms used in this method-
ology are variograms of pmf objects. In this case, the notion of variance relies on the
(Fr´
echet) distance induced by the inner product between pmfs, and consequently it
becomes more difficult to associate ranges and sills in Figure 13 with physical equiv-
alents.
4 Conclusions
In this paper, we addressed the problem of geostatistical interpolation of geometallur-
gical variables with a novel combination of Bayesian modeling and Hilbert-Kriging.
The proposed methodology produces quick (≈6 minutes), geospatial, probabilistic
estimates of non-Gaussian variables without ad-hoc transformations and compute-
intensive geostatistical simulation.
We applied the methodology to a real copper deposit to illustrate that it can work
in practice at industrial scale. By considering three geometallurgical variables from
comminution and flotation tests, we showed that the proposed approach honors the
shape of the posterior (non-Gaussian) distribution at each mining block, including
their characteristic asymmetry. Additionally, all these probabilistic predictions pre-
sented satisfactory geospatial continuity according to simple visualizations of mode
and entropy maps.
One of the main practical challenges encountered during the application of the
methodology was the interpretation of variograms of probability mass functions. In
this work, we assumed that the empirical variograms along the downhole direction
were a good approximation of an ominidirectional model. More experiments with
different data sets are needed to assess the extent to which this assumption is valid,
and to get more intuition about this notion of geospatial continuity.
Finally, we believe that this work proves the concept that more sophisticated
mathematical models are well-suited to address major challenges in geometallurgi-
cal modeling. Future work could consider new Bayesian models inspired by physics,
new data sets from different types of deposits, and more quantitative assessments
designed in partnership with the industry.
Accepted
Title Suppressed Due to Excessive Length 25
Acknowledgements
The authors acknowledge the leadership of the IMPA–Vale partnership, a research
collaboration between the Instituto de Matem´
atica Pura e Aplicada and Vale S.A.
Software citations
This project was developed with state-of-the-art open-source libraries and the Ju-
lia programming language (Bezanson et al. 2017). The proposed Bayesian models
were written with Turing.jl, a general-purpose probabilistic programming language
for robust, efficient Bayesian inference and decision making. (Ge et al. 2018). The
Hilbert-Kriging method is available in GeoStats.jl, an extensible framework for high-
performance geostatistics (Hoffimann 2018).
Data availability
Obfuscated versions of the comminution,flotation and drillholes tables are
publicly available at https://doi.org/10.5281/zenodo.6336137 (Hoffimann et al. 2022).
The data obfuscation procedure does not affect any of the presented results.
Contributorship statement
We adopt the CRediT taxonomy. J´
ulio Hoffimann: Conceptualization, Methodol-
ogy, Software, Validation, Formal analysis, Investigation, Resources, Data Curation,
Writing - Original Draft, Visualization, Supervision, Project administration. Jos´
e Au-
gusto: Methodology, Software, Formal analysis, Investigation, Data Curation. Lu-
cas Resende: Methodology, Software, Formal analysis, Investigation, Data Curation.
Marlon Mathias: Software, Formal analysis, Data Curation. Douglas Mazzinghy:
Conceptualization, Validation, Resources, Writing - Review & Editing. Matheus
Bianchetti, Mˆ
onica Mendes, Thiago Souza, Vitor Andrade, Tarc´
ısio Domingues:
Validation, Data Curation. Wesley Silva, Ruberlan Silva, Danielly Couto: Data Cu-
ration. Elisabeth Fonseca: Conceptualization, Validation, Resources, Data Curation,
Project administration. Keila Gonc¸alves: Funding acquisition.
Declaration of interests
The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.
Accepted
26 J´
ulio Hoffimann et al.
Appendix
A DWT is not linear
Given the objective function Ji(A,b) = ∥t
t
t(i)−A1
1
1−e−bE
E
E∥2
2we show that the optimal parameters A(i),b(i)=
argminA,b>0Ji(A,b)cannot be both linear functions of a common set of explanatory variables x
x
x.
First, we note that the derivatives must vanish at the optimal solution:
d
dA Ji=−2t
t
t(i)−A1
1
1−e−bE
E
E⊤1
1
1−e−bE
E
E=0 (63)
d
db Ji=−2At
t
t(i)−A1
1
1−e−bE
E
E⊤E
E
E◦e−bE
E
E=0 (64)
where ◦is the Hadamard (or entrywise) product. From the first equation, we obtain the following expres-
sion for A(i)in terms of b(i):
A(i)=⟨t
t
t(i),1
1
1−e−b(i)E
E
E⟩
||1
1
1−e−b(i)E
E
E||2
2
(65)
From the second equation, we obtain the following implicit relation for b(i)after substituting the
expression obtained above for A(i):
*t
t
t(i)−⟨t
t
t(i),1
1
1−e−b(i)E
E
E⟩
||1
1
1−e−b(i)E
E
E||2
2
(1
1
1−e−b(i)E
E
E),E
E
E◦e−b(i)E
E
E+=0 (66)
Denoting by
proj1
1
1−e−b(i)E
E
Et
t
t(i)=⟨t
t
t(i),1
1
1−e−b(i)E
E
E⟩
||1
1
1−e−b(i)E
E
E||2
2
(1
1
1−e−b(i)E
E
E),(67)
we can rewrite the implicit relation as
t
t
t(i)−proj1
1
1−e−b(i)E
E
Et
t
t(i),E
E
E◦e−b(i)E
E
E=0.(68)
We now know that the pair (A(i),b(i))of optimal parameters must satisfy Equation 65 and Equation 68.
It is clear that if b(i)is a linear function of x
x
x, then A(i)cannot be a linear function of x
x
x, and vice-versa. For
similar reason, the product DWT(i)=A(i)×b(i)cannot be a linear function of x
x
x.
References
Adeli A, Dowd P, Emery X, Xu C (2021) Using cokriging to predict metal recovery accounting for non-
additivity and preferential sampling designs. Minerals Engineering 170:106923
Agar G (2000) Calculation of locked cycle flotation test results. Minerals Engineering 13:1533–1542
Aitchison J (2003) The statistical analysis of compositional data. Caldwell, N.J: Blackburn Press, ISBN
978-1930665781
Barnett R M, Deutsch C V (2012) Practical Implementation of Non-linear Transforms for Modeling Ge-
ometallurgical Variables. Dordrecht: Springer Netherlands, ISBN 978-94-007-4153-9, 409–422
Bezanson J, Edelman A, Karpinski S, Shah V B (2017) Julia: A fresh approach to numerical computing.
SIAM review 59(1):65–98
Boisvert J, Rossi M, Ehrig K, Deutsch C (2013) Geometallurgical modeling at olympic dam mine, south
australia. Mathematical Geosciences 45
Bond F (1961) Crushing and grinding calculations parts 1 and 2. British Chemical Engineering 6:378–385,
543–548
Campos P H A, Costa J F C L, Koppe V C, Bassani M A A (2021) Geometallurgy-oriented mine scheduling
considering volume support and non-additivity. Mining Technology 0(0):1–11
Carrasco P, Chil`
es J P, S´
eguret S (2008) Additivity, metallurgical recovery, and grade. VIII International
Geostatistics Congress, GEOSTATS 2008 1188
Accepted
Title Suppressed Due to Excessive Length 27
Chieregati A C, Delboni Jr H (2001) Nova metodologia de caracterizac¸ ˜
ao de min´
erios aplicada a projetos
de moinhos ag/sag. In VI SHMMT/XVIII ENTMME
Davidson-Pilon C (2015) Bayesian Methods for Hackers: Probabilistic Programming and Bayesian Infer-
ence. Addison-Wesley Professional, 1st edition, ISBN 0133902838
Deutsch J L, Palmer K, Deutsch C V, Szymanski J, Etsell T H (2015) Spatial modeling of geometallurgical
properties: Techniques and a case study. Natural Resources Research 25(2):161–181
Ge H, Xu K, Ghahramani Z (2018) Turing: a language for flexible probabilistic inference. In International
Conference on Artificial Intelligence and Statistics, AISTATS 2018, 9-11 April 2018, Playa Blanca,
Lanzarote, Canary Islands, Spain, 1682–1690
Gelman A (2014) Bayesian data analysis. Boca Raton: CRC Press, ISBN 978-1439840955
Giraldo R, Delicado P, Mateu J (2010) Ordinary kriging for function-valued spatial data. Environmental
and Ecological Statistics 18(3):411–426
Hoffimann J (2018) Geostats.jl – high-performance geostatistics in julia. Journal of Open Source Software
3(24):692, ISSN 2475-9066
Hoffimann J, Augusto J, Resende L, Mathias M, Mazzinghy D, Bianchetti M, Mendes M, Souza T, An-
drade V, Domingues T, Silva W, Silva R, Couto D, Fonseca E, Gonc¸ alves K (2022) Geomet dataset
Hoffimann J, Zortea M, de Carvalho B, Zadrozny B (2021) Geostatistical learning: Challenges and oppor-
tunities. Frontiers in Applied Mathematics and Statistics 7, ISSN 2297-4687
Hoffman M D, Gelman A (2014) The no-u-turn sampler: Adaptively setting path lengths in hamiltonian
monte carlo. Journal of Machine Learning Research 15:1593–1623
Journel A G (2003) Mining geostatistics. Caldwell, N.J: Blackburn Press, ISBN 978-1930665910
Menafoglio A, Guadagnini A, Secchi P (2014) A kriging approach based on aitchison geometry for the
characterization of particle-size curves in heterogeneous aquifers. Stochastic Environmental Research
and Risk Assessment 28(7):1835–1851
Menafoglio A, Petris G (2016) Kriging for hilbert-space valued random fields: The operatorial point of
view. Journal of Multivariate Analysis 146:84–94
Menafoglio A, Secchi P, Rosa M D (2013) A Universal Kriging predictor for spatially dependent functional
data of a Hilbert Space. Electronic Journal of Statistics 7(none):2209 – 2240
Napier-Munn T, Morrell S, Morrison R, Kojovic T (1999) Mineral Comminution Circuits Their Operation
and Optimisation. JKMRC Monograph Series in Mining and Mineral Processing 2
Pan S J, Yang Q (2010) A survey on transfer learning. IEEE Transactions on Knowledge and Data Engi-
neering 22(10):1345–1359
Rowland J C A (1975) The tools of power: How to evaluate grinding mill performance using the bond
work index to measure grinding efficiency. In AIME Annual Meeting, Arizona
Tavares L M, Kallemback R D (2013) Grindability of binary ore blends in ball mills. Minerals Engineering
41:115–120, ISSN 0892-6875
Weiss K, Khoshgoftaar T M, Wang D (2016) A survey of transfer learning. Journal of Big Data 3(1)
