J. Intelligent Learning Systems & Applications, 2010, 2: 86-96
doi:10.4236/jilsa.2010.22012 Published Online May 2010 (http://www.SciRP.org/journal/jilsa)
Copyright © 2010 SciRes. JILSA
Machine Learning Algorithms and Their
Application to Ore Reserve Estimation of Sparse
and Imprecise Data
Sridhar Dutta1, Sukumar Bandopadhyay2, Rajive Ganguli3, Debasmita Misra4
1Mining Engineer and MineSight Specialist, Mintec, Inc. & MineSight® Applications, Tucson, USA; 2Mining Engineering, Fair-
banks, USA; 3Mining Engineering, University of Alaska Fairbanks, Fairbanks, USA; 4Geological Engineering, University of Alaska
Fairbanks, Fairbanks, USA.
Email: sridhar.dutta@mintec.com, {sbandopadhyay, rganguli}@alaska.edu, ffdm1@uaf.edu
Received November 2nd, 2009; revised January 6th, 2010; accepted January 15th, 2010.
Traditional geostatistical estimation techniques have been used predominantly by the mining industry for ore reserve
estimation. Determination of mineral reserve has posed considerable challenge to mining engineers due to the geo-
logical complexities of ore body formation. Extensive research over the years has resulted in the development of several
state-of-the-art methods for predictive spatial mapping, which could be used for ore reserve estimation; and recent ad-
vances in the use of machine learning algorithms (MLA) have provided a new approach for solving the problem of ore
reserve estimation. The focus of the present study was on the use of two MLA for estimating ore reserve: namely, neural
networks (NN) and support vector machines (SVM). Application of MLA and the various issues involved with using
them for reserve estimation have been elaborated with the help of a complex drill-hole dataset that exhibits the typical
properties of sparseness and impreciseness that might be associated with a mining dataset. To investigate the accuracy
and applicability of MLA for ore reserve estimation, the generalization ability of NN and SVM was compared with the
geostatistical ordinary kriging (OK) method.
Keywords: Machine Learning Algorithms, Neural Networks, Support Vector Machine, Genetic Algorithms, Supervised
1. Introduction
Estimation of ore reserve is essentially one of the most
important platforms upon which a successful mining op-
eration is planned and designed. Reserve estimation is a
statistical problem and involves determination of the
value (or quantity) of the ore in unsampled areas from a
set of sample data (usually drill-hole samples) X1, X2,
X3, …. Xn collected at specific locations within a deposit.
During this process, it is assumed that the samples used
to infer the unknown population or underlying function
responsible for the data are random and independent of
each other. Since the accuracy of grade estimation is one
of the key factors for effective mine planning, design,
and grade control, estimation methodologies have un-
dergone a great deal of improvement, keeping pace with
the advancement of technology. There are a number of
methodologies [1-6] that can be used for ore reserve estima-
tion. The merits and demerits associated with these
methodologies determine their application for a particu-
lar scenario. The most common and widely used methods
are the traditional geostatistical estimation techniques of
kriging. Typically, the previously mentioned criteria of
randomness and independence among the samples are
rarely observed. The samples are correlated spatially, and
this spatial relationship is incorporated in the traditional
geostatistical estimation procedure. The resulting infor-
mation is contained in a tool known as the “variogram
function,” which describes both graphically and numeri-
cally the continuity of mineralization within a deposit.
The information can also be used to study the anisot-
ropies, zones of influence, and variability of ore grade
values in the deposit. Although kriging estimators find a
wide range of application in several fields, their estima-
tion ability depends largely on the quality of usable data.
Usable data applies to the presence of good and sufficient
data to map the spatial correlation structure. Their per-
formance is also appreciably better when a linear rela-
tionship exists between the input and output patterns. In
real life, however, this is extremely unlikely. Even though
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data87
there are a number of kriging versions, such as log-nor-
mal kriging and indicator kriging that apply certain spe-
cific transformations to capture nonlinear relationships,
they may not be efficient enough to capture the broad
nature of spatial nonlinearity.
Modernization and recent developments in computing
technologies have produced several machine learning
algorithms (MLA), for example, neural networks (NN)
and support vector machines (SVM), that operate non-
linearly. These artificial MLA learn the underlying func-
tional relationship inherently present in the data from the
samples that are made available to them. The attractive-
ness of these nonlinear estimators lies in their ability to
work in a black-box manner. Given sufficient data and
appropriate training, they can learn the relationship be-
tween input patterns (such as coordinates) and output
patterns (such as ore grades) in order to generalize and
interpolate ore grades for areas between drill holes. With
this approach, no assumptions must be made about fac-
tors or relationships of ore grade spatial variations, such
as linearity, between boreholes.
This study investigated ore-reserve estimation capa-
bilities of NN and SVM in the Nome gold deposit under
data-sparse conditions. The performance of these MLA is
validated by comparing results with the traditional ordi-
nary kriging (OK) technique. Several issues pertaining
to model development are also addressed. Various es-
timation errors, namely, root mean square error (RMSE),
mean absolute error (MAE), bias or mean error (ME),
and Pearson’s correlation coefficient, were used as
mea-sures to assess the relative performance of all the
2. Nome Gold Deposit and Data Sparseness
The Nome district is located on the south shore of the
Seward Peninsula roughly at latitude 64°30’N and lon-
gitude 165°30’W. It is 840 km west of Fairbanks and
860 km northwest of Anchorage (Figure 1). Placer gold
at Nome was discovered in 1898. Gold and antimony
have been produced from lode deposits in this district,
and tungsten concentrates have been produced from re-
sidual material above the scheelite-bearing lodes near
Nome. Other valuable metals, including iron, copper,
bismuth, molybdenum, lead, and zinc, are also reported
in the Nome district.
[7] and [8] studied the Nome ore deposit and presented
an excellent summary regarding its origins by chroni-
cling their exploration and speculating on the chronology
of events in the complex regional glacial history that al-
lowed the formation and preservation of the deposit.
Apart from the research just mentioned, several inde-
pendent agencies have carried out exploration work in
this area over the last few decades. Figure 2 shows the
composition of the offshore placer gold deposit. Alto-
gether, around 3500 exploration drill holes have been
made available by the various sampling explorations in
the 22,000-acre Nome district. The lease boundary is
arbitrarily divided into nine blocks named Coho, Halibut,
Herring, Humpy, King, Pink, Red, Silver, and Tomcod.
These blocks represent a significant gold resource in the
Nome area that could be mined economically.
The present study was conducted in the Red block of
the Nome deposit. Four hundred ninety-seven drill-hole
samples form the data used for the investigation. Al-
though the length of each segment of core sample col-
lected from bottom sediment of the sea floor varied con-
siderably, the cores were sampled at roughly 1 m inter-
vals. On average, each hole was drilled to a depth of 30 m
underneath the sea floor. Even though a database com-
piled from the core samples of each drill hole was made
available, an earlier study by [9] revealed that most of the
gold is concentrated within the top 5 m of bottom sedi-
ment of the sea floor. As a result, raw drill-hole samples
were composited of the first 5 m of sea floor bottom
sediment. These drill-hole composites were eventually
used for ore grade modeling using NN, SVM, and Geo-
Preliminary statistical analysis conducted on drill-hole
composites from the Red block displayed a significantly
large grade variation, with a mean and standard deviation
of 440.17 mg/m3 and 650.58 mg/m3, respectively. The
coefficient of variation is greater than one, which indi-
cates the presence of extreme values in the dataset. Spa-
tial variability of the dataset was studied and character-
ized through a variography study. Figure 3 presents a
spatial plot showing an omni-directional variogram for
gold concentration in the data set. From the variogram
plot, it can be observed that there is a small amount of
the regional component. Large proportions of spatial
variability occur from the nugget effect, indicating the
presence of a poor spatial correlation structure in the de-
posit over the study area. Poor spatial correlation, in
general, tends to suggest that prediction accuracy for this
deposit might not be reliable. Hence, borehole data are
sparse for reserve estimation, considering the high spatial
variation of ore grade that is commonly associated with
placer gold deposit. A histogram plot of the gold data is
presented in Figure 4. The histogram plot illustrates that
the gold values are positively skewed. A log-normal dis-
tribution may be a suitable fit to the data. Visual por-
trayal of the histogram plot also reveals that the gold
datasets are composed of a large proportion of low values
and a small proportion of extremely high values. Closer
inspection of the spatial distribution of high and low
gold-grade values portrays a distinct spatial characteristic
of the deposit. For example, the high values do not ex-
ibit any regular trend. Instead, one or two extremely h
Copyright © 2010 SciRes. JILSA
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data
Figure 1. Location plot of the studied area
high values occasionally occur in a mix of low values.
This may pose a particular difficulty in ore grade model-
ing, since the pattern of occurrence of extremely high
values is somewhat unpredictable.
As it is discernable that the available gold data are
sparse and exhibit a low level of spatial correlation, spa-
tial modeling of these datasets is complex. Prediction
accuracy may be further reduced if the problem of sparse
data is not addressed. Prediction accuracy not only de-
pends on the type of estimation method chosen but also,
largely, on the model data subsets on which the model is
built. Since learning models are built by exploring and
capturing similar properties of the various data subsets,
these data subsets should be statistically similar to each
Copyright © 2010 SciRes. JILSA
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data 89
Figure 2. Offshore placer gold deposit
Figure 3. Omni-directional variogram plot
other and should reflect the statistical properties of the
entire dataset. Statistical similarity ensures that the com-
parisons made for the model built on the training dataset
and tested on the prediction dataset are logical [10,11].
Traditionally used practices of random division of data
might fail to achieve the desired statistical properties
when data are sparse and heterogeneous. Due to sparse-
ness, limited data points categorized into data subsets by
random division might result in dissimilarity of the data
Figure 4. Histogram plot Red block
subsets [12].Consequently, overall model performance
will be decreased. In order to demonstrate the severity of
data sparseness in random data division, a simulation
study was conducted using the Nome datasets. One hun-
dred random data divisions were generated, in which
sample members for training, calibration, and validation
subsets were chosen randomly. The reason for the choice
of three data subsets is presented in Section 3.0.1. Statis-
Copyright © 2010 SciRes. JILSA
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data
tical similarity tests of the three data subsets, using
analysis of variance (ANOVA) and Wald tests were
conducted. Data division was based on a consideration of
all the attributes associated with the deposit, namely, the
x-coordinate, y-coordinate, water-table depth, and gold
concentration. The results of the simulation study made
obvious the fact that almost one-quarter of the data divi-
sions are bad during random division of data due to the
existing sparseness. This figure can be regarded as quite
significant. The unreliability of random data division is
further explored through inspection of a bad data division.
A statistical summary for one of the arbitrarily selected
random data divisions for the dataset is presented in Ta-
ble 1. From the table, it is seen that both the mean and
standard deviation values are significantly different.
Therefore, careful subdivision of data during model de-
velopment is essential. Various methodologies were in-
vestigated in this regard for proper data subdivision un-
der such a modeling framework, including the applica-
tion of genetic algorithms (GA) [3,5,13,14] and Koho-
nen networks [11,14]. A detailed description of the the-
ory and working principle of these methodologies can be
found in any NN literature [15,16].
3. Nome Gold Reserve Estimation
When estimating ore grade, northing, easting, and wa-
ter-table depth were considered as input variables, and
gold grade associated with drill-hole composites up to a
depth of 5 m of sea floor was considered an output vari-
able. The next few sections describe the application of
NN and SVM to ore reserve estimation, along with vari-
ous issues that arose while using NN and SVM for ore
reserve modeling.
3.1 NN for Grade Estimation
Neural networks form a computational structure inspired
by the study of biological neural processing. This struc-
ture exhibits certain brain-like capabilities, including per-
ception, pattern recognition, and pattern prediction in a
variety of situations. As with the brain, information pro-
cessing is done in parallel using a network of “neurons.”
As a result, NN have capabilities that go beyond algo-
rithmic programming and work exceptionally well for
nonlinear input-output mapping. It is this property of
nonlinear mapping that makes NN appealing for ore
grade estimation.
There is a fundamental difference in the principles of
OK and NN. While OK utilizes information from local
samples only, NN utilize information from all of the
samples. Ordinary Kriging is regarded as a local estima-
tion technique, whereas NN are global estimation tech-
niques. If any nonlinear spatial trend is present in a de-
posit, it is expected that the NN will capture it reasonably
well. The basic mechanisms of NN have been discussed
at length in the literature [15,17]. A brief discussion of
NN theory is presented below to provide an overview of
the topic.
In NN, information is processed through several in-
terconnected layers, where each layer is simply repre-
sented by a group of elements designated as neurons.
Basic NN architecture is made of an input layer consist-
ing of inputs, one or more hidden layers consisting of a
number of neurons, and the output layer consisting of
outputs. Typical network architecture, having three lay-
ers, is presented in Figure 5. Note that while the input
layer and the output layer have a fixed number of ele-
ments for a given problem, the number of elements in the
hidden layer is arbitrary. The basic functioning of NN
involves a manipulation of the elements in the input layer
and the hidden layer by a weighing function to generate
network output. The goodness of the resulting output
(how realistic it is) depends upon how each element in
the layers is weighted to capture the underlying phe-
nomenon. As it is apparent that the weights associated
with the interconnections largely decide output accuracy,
they must be determined in such a way as to result in
minimal error. The process of determination of weights is
called learning or training during which, depending upon
the output, NN adjust weights iteratively based on their
contribution to the error. This process of propagating the
effect of the error onto all the weights is called back-
propagation . It is during the process of learning that NN
map the patterns pre-existing in the data by reflecting the
changes in data fluctuations in a spatial coordinate. The
sample dataset for a given deposit is used for this pur-
pose. Therefore, given the spatial coordinates and other
relevant attributes as input and the grade attribute as
output, NN will be able to generate a mapping function
through a set of connection weights between the input
and output. Hence, output, O, of a neural network can be
regarded as a function of inputs, X, and connection
weights, W: O = (X), where is a mapping function.
Training of NN involves finding a good mapping func-
tion that maps the input-output patterns correctly. This is
done, as previously described, by adjusting connection
weights between the neurons of a network, using a suit-
able learning algorithm while simultaneously fixing the
network architecture and activation function.
An additional criterion for optimization of the NN ar-
chitecture is to choose the network with minimal gener-
alization error. The main goal of NN modeling is not to
generate an exact fit to the training data, but to generalize
a model that will represent the underlying characteristics
of a process. A simple model may result in poor gener-
alization, since it can not take into account all the intrica-
ies present in the data. On the other hand, a too-complex c
Copyright © 2010 SciRes. JILSA
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data 91
Table 1. Statistical summary of one of the random divisions for the Red dataset
Mean Standard Deviation
Overall Training Calibration Validation Overall Training Calibration Validation
X 3941.8 3947.7 3838.2 4032.6 456.54 436.89 505.50 425.5
Y 10198 10174 10350 10097 469.75 483.19 487.22 384.06
Gold 440.17 297.75 781.77 385.00 650.58 353.31 10340 475.12
WTD 8.4845 8.94 6.6242 9.414 5.2063 5.1679 5.3559 4.69
Figure 5. A typical neural network architecture
model is flexible enough to fit data with anomalies or
noise. Therefore, complexity of a model should be well
matched to improve generalization properties of the data.
Past research has been devoted to improving the gener-
alization of NN models, including techniques such as
regularization, quick-stop training, and smoothing, and
combining several learning models using various ensem-
ble techniques like bragging and boosting [1,18]. In the
present study, a quick-stop training method is employed
to improve the NN model generalization. Quick-stop
training is based on the notion that generalization per-
formance varies over time as the network adapts during
training [15]. Using this method, the dataset is split into
three subsets: training, calibration, and validation. The
network actually undergoes training on the training set.
However, the decision to stop the training is made on the
network’s performance in the calibration set. The error
for the training set decreases monotonically with an in-
creasing number of iterations; however, the error for the
calibration set decreases monotonically to a minimum,
and then starts to increase as the training continues. A
typical profile of the training error and the calibration
error of a NN model is presented in Figure 6. This ob-
served behavior occurs because, unlike the training data,
the calibration data are not used to train the network. The
calibration data are simply used as an independent meas-
ure of model performance. Thus, it is possible to stop
over-training or under-training by monitoring the per-
formance of the network on the calibration subset, and
then stop the training when the calibration error starts
increasing. In order to make a valid model-performance
measurement, the training, calibration, and validation sub-
sets should have similar statistical properties. Thus, the
members of the data in the training, calibration, and vali-
dation subsets should be selected in such a way that the
three data subsets acquire similar statistical properties.
Once the data subsets are obtained, a NN model is de-
veloped based on the NN architecture and learning rule
to generate model outputs.
3.2 NN Grade Estimation Results
The Levenberg-Marquardt backpropagation (LMBP)
learning algorithm was used in conjunction with slab
architecture, as shown in Figure 7, for NN modeling.
The hidden layer consisted of 12 neurons. The number of
hidden neurons chosen was based on the minimum gen-
eralization errors of NN models while experimenting
with a different number of hidden nodes in the hidden
slabs. A MATLAB code was developed for conducting
all the studies associated with NN modeling. The model
datasets were obtained by following an integrated ap-
proach using data segmentation and GA. Data segmenta-
tion involved the division of data into three prime seg-
ments of high-, medium-, and low-grade gold concentra-
tions. This division was based on a visual inspection of
the histogram plot. Figure 8 presents a schematic dia-
gram of data segmentation and the GA approach. After
data segmentation, GA was applied in each of the seg-
ments: segment 1, segment 2, and segment 3. The mem-
bers of the training, calibration, and the validation data-
sets were selected using GA from each of the segments.
Once the members for the training, calibration, and vali-
dation data were chosen, the selected members from the
segments were appended together to form respective
subsets. Table 2 presents the summary statistics of the
mean and standard deviation values for all variables of
the three data subsets and the entire dataset. Observe that
the mean and standard deviation values are similar for all
the data subsets. The histogram plots of the three subsets
Copyright © 2010 SciRes. JILSA
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data
Figure 6. A typical profile of training and calibration error
of a NN model [16]
Figure 7. Slab architecture for NN modeling
Figure 8. Data segmentation and genetic algorithms for
data divisions
and the entirety of the nine datasets are presented in
Figure 9. From the figure, it can be seen that all the data
subsets assume an almost identical shape to that of the
overall dataset, and that the skewness of the data in the
three subsets is preserved. Table 3 presents summary
statistics of the generalization performance of the NN
model for the Red dataset, while Figure 10 presents a
scatterplot of the actual values versus predicted values of
the validation data subset for the Red block.
3.3 SVM for Ore Grade Estimation
The SVM method is based on statistical learning theory
(SLT) and performing structural risk minimization (SRM).
Popularly known as support vector regression (SVR) for
its regression abilities, the SVR not only has a solid
mathematical background but also is robust to noise in
measurements [19-21]. Support vector regression ac-
quires knowledge from the training data by building a
model, during which the expected risk, R, is approxi-
mated and minimized by the empirical risk, Remp. This
process always involves a generalization error bound and
is given by
R(h) Remp (h) + (h) (1)
where R is the bound on the testing error, Remp is the em-
pirical risk on the training data, and is the confidence
term that depends on the complexity of the modeling
function. Though a brief explanation of how the SVR
approach works is described below, interested readers are
referred to [20-22]. Given the training dataset {(xi, yi), i =
1, 2,….L}, where xi is the input variable and yi is the
output variable, the idea of SVR is to develop a linear
regression hyperplane expressed in Equation (2), which
allows, at most, ε deviation for the true values, yi, in the
training data (see Figure 11) and at the same time
searches for a solution that is as flat as possible [21].
() ()
xW xb
where Wo is the optimum weight vector, b is the bias, and
(x) is a mapping function used to transform the input
variable in the input space to a higher dimensional fea-
ture space. This transformation allows the handling of
any nonlinearity that might exist in the data. The desired
flatness is obtained by seeking a small w [21]. In reality,
however, a function that approximates all the (xi, yi) pairs
with ε precision may not be feasible. Slack variables εi,
* [23] are introduced in such cases that allow the incor-
poration of some amount of error (see Figure 11). The
problem of obtaining a small w and at the same time re-
stricting the errors to, at most, ε deviation after introduc-
ing the slack variables can be obtained by solving the
following convex quadratic optimization problem:
minimize 2*
subject to *
wx byi
In Equation (3), both the empirical risk, realized by the
training error Σ (εi + εi
*), as well as the confidence term,
Copyright © 2010 SciRes. JILSA
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data
Copyright © 2010 SciRes. JILSA
Table 2. Statistical summary of data division using GA (Red)
Mean Standard Deviation
Overall Training Calibration Validation Overall Training Calibration Validation
X 3941.8 3950 3935.6 3931.6 456.54 456.6 457.6 458.6
Y 10198 10194 10218 10186 469.75 471.2 467.2 472.4
Gold 440.17 461.99 418.46 418.59 650.58 673.97 627.89 628.8
WTD 8.4845 8.38 8.63 8.54 5.2063 5.23 5.19 5.19
Figure 11. The soft margin loss settings for linear SVM [21]
realized by the ||w||2 term (expressed by Equation (1)) are
minimized. An optimum hyperplane is obtained by solv-
ing the above minimization problem employing the Kha-
rush-Kuhn-Tucker (KKT) conditions [24] which results
in minimum generalization error. The final formulation
to obtain the SVR model predictions is given by
Figure 9. Histogram plot of Red dataset
()()( )()
xx b
 
 
Table 3. Generalization performance of the models for the
Red block
Statistics SVM NN OK
Mean Error -8.1 -1.30 33.54
Mean Absolute
Error 341.2 351.50 353.02
Root Mean
Squared Error 563.13 564.34 565.23
R2 0.234 0.191 0.193
* are the weights corresponding to individual
input patterns, K(xi, x) is a user-defined kernel function,
and b is the bias parameter. Figure 12 presents a list of
commonly used kernels. The most commonly used ker-
nel function is an isotropic Gaussian RBF defined by
Kx xe
where σ is the kernel bandwidth. The solution of this
optimization problem might result in zero weight for
some input patterns and non-zero weight for the rest. The
patterns with zero weight are redundant and are insig-
nificant to the model structure. On the other hand, input
patterns with non-zero weights are termed support vec-
tors (SV), and they are vital to obtaining model predic-
tions. As the number of support vectors increases, so
does model complexity. The main parameters that influ-
ence SVR model performance are the C, σ, and ε. Pa-
rameter C is a trade-off between empirical risk and the
weight vector norm ||w||. It decreases empirical risk but,
at the same time, increases model complexity, which
Figure 10. Actual vs. predicted for the validation data of
Red block
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data
Figure 12. Commonly used kemels in SVM
deteriorates model generalization. Parameter ε defines
the width of the insensitivity zone, and controls the
number of SV. The effect of increase in ε is a decrease in
the number of SV, which results in smoothing of the final
solution during modeling of noisy data. Similarly, note
from Equation (5) that a higher value of kernel width, σ,
has a smoothing effect on the solution. Optimal values of
these parameters can be obtained by a grid-search pro-
3.4 SVM Grade Estimation Results
Out of the numerous options available for the choice of
kernel function, a RBF kernel was selected, and the
recommendations of [20] and [25] were considered care-
fully in the development of the SVM-based model. As
per recommendations, the input data were first scaled
assuming a uniform distribution. In other words, the
scaled value of an attribute was calculated using the
maximum and minimum values of the attribute. The drill
holes were used to estimate SVM parameters, the cost
function (C), the radial basis kernel parameter (σ), and
the error margin (ε). Optimal SVM parameters were de-
termined based on a K-fold cross-validation technique
applied to the training dataset. The K-fold cross-valida-
tion approach splits the available data into more or less K
equal parts. Of the K parts of the data, only K-1 parts of
the data were used to find the SVM estimate and calcu-
late the error of the fitted model, and for predicting the
kth part of the data as part of the validation process. The
procedure was then repeated for k = 1, 2, . . ., K, and the
selection of parameters was based on the minimum pre-
diction error estimates over all K parts. The value of K is
based on the shape of a “learning curve” [26], which is a
plot of the training error versus the training size. For
given SVM parameters, the training errors are calculated
by progressively estimating the SVM model for in-
creased training data size, thereby constituting the learn-
ing curve. From the learning curve, an optimum training
size (or in other words, the number of folds, K) can be
obtained where the error is minimal. In this study, the
optimum value of K was found to be 10. Once the value
of K is obtained, the SVM model is trained using K-fold
cross validation. Training and testing involves a thorough
grid search for optimal C and σ values. Thus, unlike NN,
where training involves passing a dataset through hidden
layers to optimize the weights, optimal training of SVM
involves estimation of parameters C and σ through a grid
search such that the error is minimized. The optimum
values for C and σ for the Red block was found to be
0.53 and 9.5. These values are depicted by the troughs
and flat regions of the error surface in Figure 13. Once
the optimal values for the SVM model parameters were
determined, the model was tested for its generalization
ability on validation datasets for the Red block. Figure
14 shows the performance of the SVM model in predict-
ing gold grade for the Red block, while performance sta-
tistics are presented in Table 3.
4. Summary and Conclusions
Nome gold reserve estimation is challenging because of
the geologic complexity associated with placer gold de-
posits and because of sparse drill holes. Each drill hole
contains information on northing (Y-coordinate), easting
(X-coordinate), water-table depth, and gold grade in
mg/ m3, as well as other relevant information. For grade
estimation, northing, easting, and water-table depth were
considered input variables, and gold grade was consid-
ered an output variable. Gold grade up to a 5 m sea floor
depth, were considered. The gold grade associated with
the Nome deposit Red block was estimated using two
MLA—the NN method and the SVM method—and their
performance were compared using the traditional geosta-
tistical OK technique. Various issues involved in the use
of these techniques for grade estimation were discussed.
Based on the results from this study, the SVM-based
model produced better estimates as compared with the
other two methods. However, the improvement was only
marginal, which may be due to the presence of extreme
data values. The various criteria used to compare model
performance were the mean error (ME), the mean abso-
lute error (MAE), the root mean squared error (RMSE),
and the coefficient of determination (R2). Generally, a
model with less error and high R2 is considered a better
fit. Since the improvements were only marginal, a sum-
mary statistic was developed to compare the three mod-
els. This summary statistic, termed the skill value, is an
entirely subjective measurement, expressed by Equation
(6) [14,27-29]. Numerous skill measures can be devised;
however, the one proposed in this study considers the
ME, MAE, and RMSE equally, and applies scaling to the
R2 so that it is of the same order of magnitude as the oth-
ers. Note that the lower the skill value, the better the
method is. In this way, various methods can be ranked
Copyright © 2010 SciRes. JILSA
Machine Learning Algorithms and Their Application to Ore Reserve Estimation of Sparse and Imprecise Data95
Figure 13. Effect of cost and kernel width variation on error (Red)
Figure 14. Scatterplot for actual vs. predicted grade (Red)
Table 4. Model performances based on skill values
Statistics SVM NN OK
Skill Value 989.03 998.03 1032.49
Rank 01 02 03
based on their skill values, that is, their overall perform-
ance on the prediction dataset.
‘skill value’ = abs (ME) + MAE + RMSE + 100 × (1 – R2)
Table 4 presents skill values and ranks for the various
methods that were used on the prediction dataset. It can
be seen from the table that the MLA performed signifi-
cantly better than the traditional kriging method. The
difference in the skill values is mainly due to the high
variation in the R2 (Table 3).
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