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7 Figures# Application of a Modular Feedforward Neural Network for Grade Estimation

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

Figures - uploaded by Pejman Tahmasebi

Author content

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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Tahmasebi and Hezarkhani

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