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Application of a Modular Feedforward Neural Network for Grade Estimation

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DOI: 10.1007/s11053-011-9135-3}
Abstract
This article presents new neural network (NN) architecture to improve its ability for grade estimation. The main aim of this study is to use a specific NN which has a simpler architecture and consequently achieve a better solution. Most of the commonly used NNs have a fully established connection among their nodes, which necessitates a multivariable objective function to be optimized. Therefore, the more the number of variables in the objective function, the more the complexity of the NN. This leads the NN to trap in local minima. In this study, a new NN, in which the connections based on the final performance are eliminated, is used. Toward this aim, several network architectures were tested, and finally a network which yielded the minimum error was selected. This selected network has low complexity and connection among nodes which help the learning algorithm to converge rapidly and more accurately. Furthermore, this network has this ability to deal with the small number of data sets. For testing and evaluating this new method, a case study of an iron deposit was performed. Also, to compare the obtained results, some common techniques for grade estimation, e.g., geostatistics and multilayer perceptron (MLP) were used. According to the obtained results, this new NN architecture shows a better performance for grade estimation. KeywordsNeural network–local minima–geostatistics–Gol-Gohar–modular
Application of a Modular Feedforward Neural Network
for Grade Estimation
Pejman Tahmasebi
1,2
and Ardeshir Hezarkhani
1
Received 29 April 2010; accepted 6 January 2011
This article presents new neural network (NN) architecture to improve its ability for grade
estimation. The main aim of this study is to use a specific NN which has a simpler archi-
tecture and consequently achieve a better solution. Most of the commonly used NNs have a
fully established connection among their nodes, which necessitates a multivariable objective
function to be optimized. Therefore, the more the number of variables in the objective
function, the more the complexity of the NN. This leads the NN to trap in local minima. In
this study, a new NN, in which the connections based on the final performance are elimi-
nated, is used. Toward this aim, several network architectures were tested, and finally a
network which yielded the minimum error was selected. This selected network has low
complexity and connection among nodes which help the learning algorithm to converge
rapidly and more accurately. Furthermore, this network has this ability to deal with the small
number of data sets. For testing and evaluating this new method, a case study of an iron
deposit was performed. Also, to compare the obtained results, some common techniques for
grade estimation, e.g., geostatistics and multilayer perceptron (MLP) were used. According
to the obtained results, this new NN architecture shows a better performance for grade
estimation.
KEY WORDS: Neural network, local minima, geostatistics, Gol-Gohar, modular.
INTRODUCTION
One of the most important aspects of mining
activities is grade estimation values of which are
very critical for both mining exploration and
exploitation. Therefore, several attempts have been
made in the past to increase the accuracy of grade
estimation. One of the most important methods is
geostatistical technique which is based on random
components in space and samples’ spatial relation-
ships. Furthermore, the application of geostatis-
tical methods needs a considerable knowledge of
mathematics, skills, and time to be applied so as to
get the preferred solution. Therefore, all of these
constraints led the researchers to find some alter-
native methods which could be applied easily.
In the last two decades, artificial neural net-
works (ANNs), as nonlinear alternative methods,
have been applied for various aspects of mining and
geology tasks. NN (neural network), by using a set
of connections (weights) and a global algorithm, is
expected to provide a much finer performance when
there is a complex relationship between the loca-
tions and grade distribution, in comparison with the
geostatistical methods which are mainly based on
local algorithms. Examples of these wide applica-
tions of ANNs could be found in the literature
(Yama and Lineberry, 1999; Koike, Matsuda, and
Gu, 2001; Ke, 2002; Matı
´
as and others, 2004; Porwal,
Carranza, and Hale, 2004; Weller and others, 2005;
1
Department of Mining, Metallurgy and Petroleum Engineering,
Amirkabir University of Technology (Tehran Polytechnic),
Hafez Ave. No. 424, Tehran, Iran.
2
To whom correspondence should be addressed; e-mail:
Pejman@aut.ac.ir.
2011 International Association for Mathematical Geology
Natural Resources Research ( 2011)
DOI: 10.1007/s11053-011-9135-3
Lacassie and others, 2006; Singer, 2006; Weller and
others, 2006; Weller, Harris, and Ware, 2007).
However, several problems concerning the
efficiency of NNs, e.g., weights, topology, training
parameters, etc., which impact on NNÕs performance
have been investigated widely. For example, in
the recent studies, Samanta, Bandopadhyay, and
Ganguli (2004) applied simulated annealing for NN
training in which no improvement by their proposed
method was achieved. Mahmoudabadi, Izadi, and
Menhaj (2009), in a case with a small data set,
optimized a NN with LM (Levenberg–Marquardt)
and genetic algorithms to improve its perfor-
mance for grade estimation. Also, Tahmasebi and
Hezarkhani (2010) used genetic algorithms for
optimizing the parameters and topology of NNs, and
their results showed improvement compared with
ordinary NN.
In the most of the NN applications, finding the
optimum topology and weights is one of the most
important challenges. Therefore, based on the above
mentioned studies, several researchers tried to
overcome these problems with alternative optimi-
zation methods. In this study, use of a new modular
NN is attempted, which because of its especial
structure and links, offers an alternative choice to
the modeler. Indeed, in this proposed method, it is
possible to subtract the networkÕs size and manipu-
late the internal processing elements of network
in such a way that the preferred responses are
obtained. For this purpose and for illustrating this
new approach, an iron deposit was used, and
the obtained results have been compared with the
ordinary NN (Multilayer perceptrons) and Kriging.
MODULAR FEEDFORWARD NETWORKS
(MFN)
It is clear that there are some problems related
to NN designing, which led the authors to investigate
some other methods that produce more reliable
results (Tahmasebi and Hezarkhani, 2010). In view
of these problems (e.g., learning weights and NNs
topology), the authors have to use some other
methods to make amends for the errors caused and
improve the network performance. More informa-
tion could be found in Tahmasebi and Hezarkhani
(2010). One of these methods is the application of
modular NN which will be explained below.
Actually, the principle of modular feedforward
networks (MFNs) is similar to that of the multilayer
perceptron (MLP). In other words, this kind of NN
is composed of several MLPs wherein the input will
be given to some parallel MLPs. Finally, the out-
comes will be combined yielding a result. This aim
would be achieved by setting several topologies and
structures within the network which by using the
supposed sub-module helps the network to work
faster.
One of the advantages of the MFNs over the
MPL network is the former’s network size which has
been dwindled (Fig. 1). Actually, since the connec-
tion would not be fully used in this network, the
network size and interconnections will be mini-
mized. Consequently, the more the minimizing of
the network size, the less the complexity that will be
achieved as a result of using a fewer weights in
network topology with lower weight. Obviously,
Figure 1. Comparison of MLP and MFN in aspect of connec-
tion paths and nodes (a: MLP and b: MFN).
Tahmasebi and Hezarkhani
subtraction of networkÕs interconnection leads the
network to work more rapidly as well as helps it to
use a smaller training samples, which is one of the
most important objective in mining and geology
activities—by using the minimum amount of avail-
able data, more accurate and reliable results could
be obtained (Ballard, 1987; Jacobs, Jordan, and
Barto, 1991; Jacobs and Jordan, 1993).
CASE STUDY
In this study, NN was applied for grade esti-
mation of an iron ore deposit. The Gol-Gohar iron
mine is located 60 km southwest of Sirjan city of
Kerman province which is located in the southeast-
ern part of Iran, which lies between 293¢¢ and 297¢¢
latitude and between 5515¢¢ and 5524¢¢ longitude
(Fig. 2) formed in six separated anomalies confined
within ca. 10 km length and 4 km width. Anomaly
No. 3 is the biggest anomaly at this mine. On the
basis of exploration work, the total weight of ore
reserves of anomaly No. 3 is calculated as 616
million tons, with an average grade of 54.3% Fe.
Subsidence in this region is not prevalent, and the
mineral occurrence is uniform. Data used in this
section included 65 exploratory boreholes in regular
grid in which 1222 m of cores has been obtained and
composited at 6 m length.
EXPERIMENTAL RESULTS
Data Processing
All of the available data were organized as
input (x, y, and z) and output (Fe grade). Then, this
data set was divided into three subsets consisting of
training (70%), validation (15%), and testing (15%).
In this study, a validation data set is required to
prevent from over fitting and for early stopping,
because no criteria were used to stop the training
process when the network performance for training
reduced. Then, the available data were broken down
randomly into mentioned different data sets. Also, it
is important to prepare the available data in a shape
which are suitable for NN training. Therefore, the
entire available data was normalized into [0 1]. Also,
it should be mentioned that, for the entire data
processing and NN modeling, MATLAB (R2009a)
was used.
Modeling
Kriging Modeling
It is clear that geostatistical method has been
widely applied in grade estimation, and by the recent
progress, its applications are increasing. The
important reasons which lead the geostatistical
methods to be more popular could be because of
their being unbiased and providing the minimum of
variance. Among all of the geostatistical methods, in
this study, Kriging (Ordinary Kriging, OK) method
was used to compare its results with other applied
methods.
For this case study (Gol-Gohar iron deposit),
fortunately several studies have been done.
According to results of application of Kriging to this
deposit, owing to the lack of the available sam-
ple and very complex structures, no directional
variogram could be found; therefore, an omnivario-
gram was used (Fig. 3). According to this modeling, a
spherical model with two structures was known as the
best fit omnidirectional variogram. The mean square
error of this modeling has an average of 0.172
(Mahmoudabadi, Izadi, and Menhaj, 2009).
The insufficiency of variogram (or two-point)-
based geostatistical methods in this case study was
confirmed. That is why several researchers have
suggested using the geostatistical simulation-based
method which tried to overcome some of the
Figure 2. Histogram of composited data for iron (%) samples.
Application of a Modular Feedforward Neural Network
variogramÕs shortcomings. The aim of geostatistical
simulation is to combine different realizations to
make an estimation which is more probable and with
a less uncertainty (Table 1).
MLP Neural Network
In MLP, the most common way to determine
the optimal structure is trial and error method. For
this case, by changing several network parameters,
such as the number of hidden layers and their neu-
rons, and substitution of the inputs, final structure of
NN will be obtained. In other words, in this case,
since the user has no tool to understand the most
appropriate values of parameters, the only solution
will be to run several networks for different condi-
tions and save the best obtained one. The results of
this section are shown in Table 2. It is important to
mention that we used Levenberg–Marquardt (LM)
and tansig for training algorithm and output function
of NN, respectively. With the use of sigmoidal out-
put functions, one can define popular continuous-
graded response neurons. For example, the logistic
output function is
rðI=sÞ¼
1
1 þ e
I=s
The constant s determines the slope of the
logistic function. There are many functions similar in
r-shape to the logistic function, forming a broad
class of sigmoidal functions.
Also, as mentioned in ‘‘Data Processing’’ sec-
tion, all of the available data were divided into three
sections (training, validation, and testing). We used
validation data set for testing the network along
training process, and if the error of training
increases, then the training process will be stopped.
After making sure that the network is well
trained, the testing phase, which is associated with
all of the optimal functions, for the number of neu-
rons in the hidden layers will be started. Also,
according to Table 2, the optimal structure of the
network was found to be 3-12-1. This structure has
the lowest MSE (Mean Square Error), and NMSE
(Normalized Mean Square Error) with the maxi-
mum R value (correlation coefficient) for the testing
data set. It is clear that the preferred values for the
mentioned parameters are 0, 0, and 1, respectively.
Furthermore, for more than a single layer, sev-
eral networks were used, but the results were
unsatisfactory.
One of the reasons which leads the MLP to be
more time consuming is its parameters which have a
direct effect on the networkÕs performance. For
example, using a lot of neurons in hidden layer could
decrease the error in training phase, while such a use
often causes to increase the error in testing phase.
Most of the NN’s problems are due to a lot of ele-
ments which make the NN to be more complex.
These complexities could be improved by using a
more heuristics optimization method or a method
Figure 3. Omnivariogram of Fe distribution in Gol-Gohar iron
deposit.
Table 1. Summary Statistics of the Data Sets
Data Mean (%) SD
X-Coordinate 0.0125 0.5131
Y-Coordinate 0.2763 0.3980
Z-Coordinate 0.0850 0.4207
Fe grade 0.3101 0.4589
Table 2. Obtained Results of MLP for Several Neurons in
Hidden Layer with Their Correspond MSE
Number of Neurons in Hidden Layer MSE
4 0.0378
6 0.0371
8 0.0379
12 0.0362
16 0.0372
20 0.0368
24 0.0371
28 0.0374
30 0.0372
Tahmasebi and Hezarkhani
which could decrease the number of connections or
element effectively.
Furthermore, the lack of data is a common
problem in all fields of geosciences. This issue will be
more important when the acquisition of the data
required spending too much time and money.
Therefore, the necessity for a method which could
deal with a small data set is obligatory. Preferably,
this method should be able to capture the complex
relationship among the data sets and be operative
under different conditions.
Modular Feedforward Network (MFN)
The modularity is a mimic of human brain
which, by subdividing the brain, is trying to make
acceleration and help one to improve human per-
formance. Thus, in complex and complicated con-
ditions, it will be useful to simplify the problem by
dividing and allocating each of its parts to a specific
section. In other words, in NN analogy it is impor-
tant to reduce the complexity and help the network
to increase its performance.
Based on above idea, in this section, we try to
use modularity properties in NN. For this aim, the
NN should be down sized in such a way that it helps
it to increase its performance. Actually, our aim is to
reduce the complexity of objective function which
should be optimized. In traditional NNs, the net-
work by its training algorithm tries to solve an
objective function parameters of which are the
weights and biases. Therefore, mostly NN will be
more similar to solving an optimization problem.
For this reason, some researchers tried to combine
certain optimization algorithm with NN (such as
simulated annealing, genetic algorithm). Accord-
ingly, similar to optimization problems in which one
tries to reduce the dimension of the problem under
study by some dimensional reduction techniques,
such as principle component analysis (PCA) or
Factor Analysis (FA), to solve the problem much
easier, in this study, the aim is to reduce the NNÕs
dimension. Another aim is to overcome the prob-
lems of a small data set which is also very important.
Several strategies could be applied for dimen-
sional reduction of objective function in NN. For
example, one could reduce the number of inputs
which provides an acceptable answer. However,
input reduction is a routine work which is being
widely used in NN. Another solution could be the
minimizing of the internal connections (such as
weight vectors) of NN. This method could reduce
the dimensionality of objective function, and sub-
sequently help the NN find the global minimum
faster and more accurately, because in traditional
NN, one of the reasons which prevents the training
algorithm to find the global minimum is the high
dimensionality of objective function. Therefore, it
leads the algorithm to trap in local minimum instead
of global minimum. Hence, variable reduction of
objective function could have a great beneficial effect
on both CPU time and the accuracy of solutions.
Also, this has another advantage. By reducing the
networkÕs size, it permits us to use a small data set.
In other words, prevalent NNs are extremely sensi-
tive to the number of data, and, in the conditions of
data deficiency, their training image will trap in local
minima. Because, there are a lot of variables in
objective function which consequently needs a large
data set to be able to capture the variabilities and
escape of local minima. Therefore, if we could make
the objective function smaller, then it will be possi-
ble for us to use a small data set, because we could
be sure that the network will not be trapped in local
minima, since the size of network, and consequently
the objective function, was decreased.
The criterion which was used for this aim was
based on making a structure that results in a mini-
mum error relative to MLP network. The structure
of this network is shown in Figure 4. In other words,
the methodology could be summarized that, in the
first try, all the connections same as those of MFN
are retained. Then, some of the weight vectors will
Figure 4. The modular NN structure for Fe grade estimation.
Application of a Modular Feedforward Neural Network
be omitted, and the network will be tested by this
new structure. This procedure (addition and elimi-
nation) will be continued until a situation in which
both CPU time and accuracy were decreased and
increased, respectively. It should be noted that in the
each iteration, some of the weights vectors will be
added or omitted and, in this section, three possible
outcomes could be encountered:
If no changes were observed in the perfor-
mance of NN, then that weight vector will be
omitted.
If by deleting or adding a weight vector an
improvement was observed, then that vector
will be added or omitted, respectively.
If adding or deleting a weight vector led to a
decline in the network performance, then, it
should not be deleted or added.
The above process will be continued until we
reach a stationary nature of performance wherein no
improvement could be observed. However, one of
the main problems in MFN’s application is finding
the best or the near-best structure which was done
by trial and error method in this study. Definitely,
the obtained structure is just applicable for this case
study, and for more and newer data sets, the above
process should be repeated. Therefore, in a new
application, the transfer functions or other parame-
ters will not be changed, and the only item which
should be changed is whether or not of some of
weight vectors (or connections) in NN. The results
of comparison of MLP and MFN networks are
shown in Figure 5.
There are several points which could be con-
cluded from Figure 5. It is obvious that by applying
this new method, in respect of both CPU time
and accuracy, a significant improvement will be
acquired. In other words, this method is able to
achieve a better solution in a less time. This claim is
clear form Figure 5 in which after less than 1000
epochs, MFN reaches a lower level of MSE. One of
the reasons that causes this successful implementa-
tion is the MFN-specific structure. This topology lets
the users to select the most appropriate structure
which enables the network to have a lower com-
plexity and connections. Another reason for this
difference is the higher sensitivity of MLP to the
lack of data or small data set. A point that should be
clarified here is about data set. The meaning of small
data set could be expressed by two definitions: the
first definition, which is the clearer one, meaning a
data set which has a few data; and the second one
meaning a data set which could not convey the
complexity or variability. Furthermore, in compari-
son with other relevant estimation methods, MFN
shows a better performance which will be discussed
in the following.
It is clear that most of the geostatistical methods
are based on spatial relationships and their depen-
dency. This concept is concealed in variogram. Also,
one of the problems of variogram is that it is based on
two points. Therefore, it dose not let the geostatis-
tical methods to overcome complex and problematic
tasks in mining, petroleum, and hydrogeology. This
problem shows itself in the current case study.
According to the obtained results of variogram
modeling, this method could not yield an acceptable
and reliable result. Also, to improve the problems of
application of geostatistics for such mentioned con-
ditions, new methods based on training image have
been extended in which more than two points could
be considered in geostatistical studies. More infor-
mation about these methods could be found in
Guardiano and Srivastava (1993), Strebelle (2000),
Zhaneg, Switzer, and Journel (2006), Arpat (2005),
Mariethoz, Renard, and Straubhaar (2010).
Another method which was used in this study
was NN, and its results are shown in Table 2. This
method partially achieves a better result than geo-
statistics. In general, one of the NNs’ advantages is
their ability to capture the complex spatial rela-
tionships of data. Also, this method is a dynamic
method which can justify itself with multimodal
data. Moreover, in contrast to other methods, this
method needs much less time and knowledge which
helps it to be used extensively. Like other methods,
this method has some problems too. These prob-
lems could be addressed in several categories. For
example, one of the important problems is its
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Epoch
MSE
MLP Network MFN Network
Figure 5. Comparison of MLP and MFN networkÕs results.
Tahmasebi and Hezarkhani
optimal topology which could have negative effects
on the results. Non-optimal parameters including
number of neurons in the first and hidden layer could
lead the network to trap in local minima. Another
problem which could be mentioned is those of applied
weights, learning, and momentum rate in finding the
global minima (Tahmasebi and Hezarkhani, 2010).
As mentioned, for dealing with these problems, the
suing of hybrid models is suggested (Tahmasebi and
Hezarkhani, 2010). The obtained results of these
hybrid methods indicate its ability to estimate the
grade finely. However, generally, one of the reasons
for the network having error is its complexity, and in
addition, its having several weights that make a
multivariable function’s optimization of CPU very
time consuming and produce a large amount of
uncertainty. Therefore, one strategy could be down
sizing the network structure in such a way that it dose
not choke its performance. These circumstances led
us to apply MFN.
In MFN, user can modify the network structure
in a way that enables it to reach a lower error level
so that it is possible to eliminate and change the
connections of network to decrease the NN con-
nections and its complexity. This connection reduc-
tion enables the network to have less computational
cycles and could thus help it to converge more
quickly. Indeed, in the usual MLP networks, since all
of the nodes are connected, it causes to increase
both networkÕs complexity and CPU time. There-
fore, this new approach (MFN) is very easy, and the
network could reach the global minima with a high
insurance rate and much more ease. Furthermore, in
this kind of network, since the units are not fully
connected, the objective function which should be
minimized by NN will be smaller, and it becomes
more generalized. Finally, MFN, due to its small size
and low dimensional space, could be useful and
applicable in several fields such as mining, petro-
leum, and other geosciences branches to have easier,
more accurate, and faster simulations, and especially
for a case which has a small data set. Also, future
researches might explore a solution by which it
should be possible to make the MFN automatically
save the time and eliminate the trial and error
method.
CONCLUSION
Complexity of NN structure is one of the prob-
lems in grade estimation. One of the reasons is the
lack of data as well as complicated spatial distribu-
tion of input–output and fully connected NN. Indeed,
in the prevalent NN, because of large connection
between the nodes, some problems like trapping in
local minima could happen. Also, these connections
could lead the network to make an objective function
based on their variables composed of weights, biases,
etc., and finally they cause the network to have
higher complexity. Therefore, a possible solution
could be the reduction in the network size according
to its performance. This strategy leads to test several
architectures based on their ability to reach the best
solution. Actually, this new network is a modular
type of the original network which is fully connected,
whereas in the former the nodes are partially con-
nected. These partial connections between nodes
have some advantages. One of the main advantages
of MFN is their special small and individual structure
which let the user to manage and control the NN
training and derive a good performance when
encountered with small data sets. For example, in
this small and customized NN, it is easy to train and,
because of simplification of objective function, its
efficiency will increase. The most obvious finding of
this study is the increase and decrease of the accuracy
of results and CPU time, respectively. According to
the obtained results which were compared with rel-
evant grade estimation (geostatistics and NN), this
method could demonstrate its significant improve-
ment in performance with regard to the estimation of
the iron grade with high efficiency and accuracy.
Therefore, it is suggested that, for escaping from
local minima, saving the time, reducing the network’s
complexity, and having more accurate results, the
modelers test (or may replace) this methodology for
their future studies.
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