27702788

International Journal of Mathematical and Computer Modelling, ISSN: 2051-4271, Vol.19, Issue.1 1129
© RECENT SCIENCE PUBLICATIONS ARCHIVES |September2014|$25.00 | 27702788|
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Fuzzy Mathematical Model for Detection of Lung Cancer Using a Multi-Nfclass with Confusion Fuzzy Matrix for Accuracy
Rana Waleed Hndoosh
Dept. of Software Engineering,
College of Computers Sciences and Mathematics, Mosul University, Iraq.
Email: rm.mr2100@yahoo.com
M. S. Saroa
Dept. of Mathematics, Maharishi Mar-kandeshawar University, Mullana, 133207, India
Email: mssaroa@yahoo.com.
Sanjeev Kumar
Dept. of Mathematics, Dr. B. R. Ambedkar University, Khandari Campus, Agra-282002, India
Email:sanjeevibs@yahoo.co.in.
ABSTRACT
In this paper, a multi neuron-fuzzy system is used for the classification and detection of lung cancer data. This model depends on a generic model of a fuzzy perceptron, which can be used to derive a neural fuzzy system for specific domains. The multi neuron-fuzzy classification (Multi-NFClass) model proposed that uses input, hidden layers, output, and subclasses that have a multitude in each class. This model derives fuzzy rules to classify patterns into a number of crisp classes. Firstly, an attempt is made to describe fuzzy if–then rules, and construction of the fuzzy if–then rule, that are determined by the simple steps when its antecedent fuzzy sets are specified by genetic operations, which results in a Multi-NFClass model. Six different membership functions (MF) are used for fuzzy classification, and after that, a confusion fuzzy matrix (CFM) technique to represent misclassifications is used. The diagonal in CFM shows the sample elements that have been classified correctly. Results are obtained by using MATLAB 2012b, to six programs, each program depends on a different type of MF. Therefore, it is applied for two times, the first performance with 100 rules that its results better than the second performance, which contains 80 rules. We also have proposed the neural network model that is dependent on the concept of the Fuzzy Bayesian classification (NN-FBC) model, and demonstrates its performance on Lung Cancer data. We have made a comparison between the previous results and the results obtained by NN-FBC, and have obtained improvement.
Keywords- Neuro-fuzzy system; Fuzzy classifier; Fuzzy inference for classification; Fuzzy perceptron; Nfclass; Confusion fuzzy matrix; Bayesian classification; ROC technique; Lung cancer diagnosis.
1. INTRODUCTION
Several Artificial Intelligence techniques including fuzzy systems and neural networks are successfully applied to a wide variety of decision-making problems in the area of medical detection, [Gliwa and Byrski, (2011)], [Kannan and Ramat (2008)]. We have extended the NFClass modelto a Multi Neuro-Fuzzy Classification (Multi- NFClass) model that is used to detect and handle the different types, stages, and treatments of Lung Cancer. Each case of patients has three basic classes as multi-
output variables that can be represented with a concept of fuzzy set as:
= 0.22 / SCLC (small cell lung cancer) + 0.78 / NSCLC (not small cell lung cancer);
= 0.12 / LD (limited disease) + 0.17 / SI + 0.19 / SII + 0.10 / SIIIA + 0.09 / SIIIB + 0.23 / SIV +0.10 / XD (extension disease);
= 0.30 / Ch (chemotherapy) + 0.39 / Ra (radiation) + 0.07 / Ch&Ra (chemotherapy and radiation) + 0.03 / Ch&Sur (chemotherapy before surgery) + 0.21 / Sur&Ch/Ra (surgery and chemotherapy or radiation).
A Multi-NFClass model can be represented as a particular type of fuzzy perceptron, because it allows for the representation of fuzzy classifiers from multi-inputs to multi-layers, and to classify real inputs to multi-outputs. Each one of the outputs can be formed as a separate class and each class has multi-subclasses, [Burak, et al. (2004)], [Detlef, (2003)], [Gliwa et al. (2011)]. In Section 2, an attempt is made to clarify the general concept of a Multi- NFClass with the definition of a multi-fuzzy system, a fuzzy classifier, and multi-layers of fuzzy perceptron with extension of this concept from three layers to L layers. [Amo et al. (2004)], [Asif and Choi (2001)], [Hua, Tzengb and Johannes et al. (2003)], [Ishibuchi et al. (1999)], [Ishibuchi et al. (2005)] and [Zhang (2000)]. Section 3 shows the structure of a Multi-NFClass model and a network of a Multi-NFClass processing structure, [Detlef (2003)], [Gliwa and Byrski (2011)] and [Svozil et al. (1997)]. In Section 4, we describe the concept of classical classifiers using a neural network model that depends on a fuzzy Bayesian classification, [Stephen et al. (2007)] and [Zhang (2000)]. The fuzzy inference model for classify- cation is explained in Section 5, [Guan et al. (2008)], [Inyaem et al. (2010)] and [Nakashimaa et al. (2007)]. In Section 6, we have described some concepts to represent the results of classification by confusion fuzzy matrices model that are used to find the values of sensitivity and specificity. Depending on it, we can represent the receiver operating characteristics (ROC) curve, [Cinthia et al., (2007)], [Dwivedi et al. (2012)], [Hassan (2010)], and [Plamen and Zhou (2008)]. Section 7 applies all previous models on real data of lung cancer, and we have obtained very good results with smaller values of the misclassification rate MR. This misclassification measurement is a good tool for determining the quality of

a classifier and it is sometimes called "accuracy”, [Binaghi, et al. (1999)], [Jain and Abraham (2003)]. Some concluding remarks are given through the Section 8.
2. MULTI NEURO FUZZY CLASSIFICAT- IONMODEL
Fuzzy systems are be used for classification models that can be explained as a particular case of approximation function. To define it, let us consider data points characterized by , where * +, which corresponds to outputs { }. The aim is to find an unknown function ( ) . In a classification model the pattern; have to be specified to one of various classes, [Amo et al. (2004)], [Asif and Choi, (2001)]. The model of classification can be easily converted into a model of approximation function by assigning a set with patterns ( ), [Ishibuchi et al. (1999)]. In this paper, the concept of multi-layers has been extended, [Burak et al. (2004)]. Some mathematical concepts for fuzzy system are as follows:
2.1. Multi-Fuzzy System
Let represent a multi fuzzy system mapping defined as:
, (1)
where is called an input space, is called an output space, and ( ) and
( ) represents an input, and output vectors, respectively.
Suppose is a fuzzy rule base that determines the fuzzy system structure, [Hua and Tzengb (2003)]:
( ). (2)
Each rule is (n, m)-tuple of fuzzy sets:
( ), (3)
where , - is a fuzzy set of input variables and , - are a fuzzy set of output variables . We can define:
( ) ( ) ( )( ), (4)
where are a number of outputs, because there are multi-outputs.
( ) . ( { ̅ })/ ( ) (5)
which is ̅ , - ( ) and the function max is a s-norm, with
( ) ( { * ( ) ( )+ }) ( ) (6)
The function min is a t-norm, and the function defuzz is a defuzzification model that is used to convert an output fuzzy set ̅ into a crisp output value, [Bansal, (2011)].
2.2. Fuzzy Classifier
Fuzzy classifier is a fuzzy system and that is:
, - , (7)
Its rule-base consists of particular types of fuzzy rules of the form ( ), where * + is a class label. We define:
( ) ( ) with
( ) ( ) { * ( ) ( )+} ( ) (8)
where max is a s-norm, and ( ) is the consequent of rules , [Svozil et al. (1997)]. The output of a fuzzy classifier is a vector whose components denote the membership degree of a processed pattern to the available classes.
2.3. Multi-Layer of Fuzzy Perceptron Model
Let, a fuzzy system is represented by . A multi-layer fuzzy perceptron is a network representation of a in the form of a neural network( ), with the following specification:
1) , (9)
Set is a non-empty set of neurons and is the index set of , where with holds. * + is called an input layer, hidden layers and * + output layer, [Gliwa and Byrski (2011)].
2) The network structure (connections) is defined as:
( ), (10)
where ( ) is the set of all fuzzy subset of , and there are only connections ( ) with ( * +).
3) The function defines an activation function in order to compute the activation :
i. For input and hidden layers :
( ) , (11)
ii. For output layer :
( ) ( ), ( ) (12)
4) The function defines an output function to compute the output :
i. For input and hidden layers :
( ) , (13)
ii. For output layer :
( ) , ( ) ( ) (14)
5) The function defines a propagation function for each to compute the net input :

i. For input layer :
, (15)
ii. For hidden layers :
( ( )) , -,
̅ * ( ̅ )( ̅)+ (16)
where min is a t-norm.
iii. For output layer :
(, - ( ))( ) ( ), and , -
( ) ̅ ( )* ( ̅ ( ̅ )( ))+, (17)
where max is a s-norm
If the fuzzy sets ( ̅ ) ̅ ( ) and are monotonic on their support, and ( ̅ )( ) such that ( ̅ )( ) then propagation function of an output layer can alternatively be defined as:
( ) { Σ ̅ ( ̅ )( ̅) ̅ ( ) Σ ̅ ̅ ( ) (18)
To compute the output in this case with ( )
6) The external input function ( ) defines as:
( ) { (19)
3. STRUCTURE of an MULTI-NFCLASS MODEL
A Multi-NFClass represents a fuzzy classifier with a set of class labels * +. Elements of the model are treated as neurons: input, hidden, and output neurons. A network representation of a Multi-NFClass model is a fuzzy perceptron model with the following specifications:
1) The input layer processes input data. The network structure is a mapping, ( ) such as (10), that is given by, [Nakashimaa et al. (2007)] and [Svozil et al. (1997)]:
( ) { ( ) ( ) ( ) . ( )/ (20)
where , - , - , -.
2) The neurons of the hidden layer represent membership functions, fuzzy rules and weighted rules, [Hndoosh et al. (2012)], [Hua and Tzengb (2003)]. The antecedents of fuzzy rules become weights for the rule neurons in the hidden layer. The consequence of a weighted rule is connecting with the output layer, [Detlef and Johannes et al. (2003)]. An activation function in order to compute the activation that is used to make connections between:
i. Inputs neuron and MFs .
ii. MFs and rules neuron by use t-norm.
iii. Rules neuron and weighted rules.
Next, the last layer of hidden layers has multi independent fuzzy output that connects weighted rules with subclasses in order to obtain multi class with different subclass as an output layer. See Fig. 1.
3) The consequents of an output layer calculated by the activation value of a given classes. The consequents of output-layer depending on the maximum function. After the calculation of an activation of output neurons, the neuron with the highest activation is chosen as result of classification that depending on equation (14) as following:
( ) ( ) ( ) (21)
where, we defuzzify the output fuzzy set by a defuzzification function that calculates the height of the output fuzzy set, [Asif and Choi (2001)] and [Hassan (2010)].
Improving rules set is often understood as an increase of classification accuracy, reducing the model complexity or a combination of these two approaches. A large number of parameters are often needed to obtain high accuracy, but it sometimes causes a difficulty in understanding. Reducing the number of parameters sometimes increases classification accuracy, because the model with a large number of parameters can over fit the learning data and lose the generalization variable.
4. NEURAL NETWORK with FUZZY BAYESIAN CLASSIFICATION
Let denote a class characteristic with a finite domain ( ) classes, and a number characteristic with a finite domains ( ) ( ).
A pattern is described by its characteristic values ( ) ( ). Bayes classifiers perform a likelihood idea of classification where they compute the class of a new pattern by evaluating each class from ( ). Formally, ( ) they evaluate the likelihood as:
( ), (22)
That a pattern with characteristic values as the given new pattern has the class , to improve readability, we may use ( ) like a limitation for ( ), also ( ) like a reduction for ( ) Moreover, the number of training patterns is often too small to give a good evaluation for all these values.

Now, the idea of Bayes classification is starting with the Bayes rule, [Bansal (2011)] and [Zhang (2000)]:
( ) ( ) ( ) ( ) , with
( | ) ( | ) ( ) ( ) (23)
The second step, suppose all characteristic values are independent, given us the likelihood evaluate
( | ) . ( | ) ( | )/ ( ) ( ) ( ) (24)
Bayes classification of patterns ( | ) is computed ( ) to find the maximum value. Moreover, the evaluation of the likelihood values given as:
( ) | | ( ) , ( ) ( ) and
( ) | | ( ) (25)
Further Bayes classification for discrete characteristic values is implemented for a pattern by foretelling its class to the value ( ) with maximum likelihood ( | ) computed as Bayes using evaluations for the needed likelihoods.
Now, pattern does not have exactly one value ( ) for each characteristic , but has each value ( ) to a degree , -. This matches to the concept of a linguistic variable in fuzzy logic, where the characteristic names a linguistic variable and each of the values of this characteristic matches to a linguistic term. We have classified to a degree , - for each class ( ). We suppose these degrees are normalized for all patterns:
Σ ( )Σ ( ) (26)
Consider the degree as the likelihood ( ) a pattern has a characteristic . This is not the full meaning of a fuzzy truth degree, but we may see it, as it allows us to extend the Fuzzy Bayes classification.
( ) ( ) (27)
We suppose all characteristic values and the values of class are independent in this artificial likelihood distribution for . Then, we can compute the likelihood ( ) a pattern has class . Firstly, supposing the class depends only on these values. ( ) Σ ( ) ( )
Then, we make the assumption that the characteristic values of pattern are independent of each other. Note that these likelihoods are subject of our explanation of the fuzzy reality degrees. We obtain as: ( ) Σ ( ) ( ) ( )
( ) Σ ( ) ( ), (28)
Application of the Fuzzy Bayes rule to ( ) yields:
Figure 1. A Multi-NFClass processing structure.
__________________________________________________________________________________________________
:
:
:
:
Input layer
Hidden Layers
Output layer
(Types of LC)
(Stages of LC)
(Treatments of LC)

( ) Σ ( ) ( ) ( ) ( )
Next, we apply the same independent assumption as in the classical case: ( ) Σ ( ) ( ) ( ) ( ) ( ) ( )
Now, we move constant factors in front of the sum. Finally, we find using distributive:
( ) ( )(.Σ ( ) ( ) / .Σ ( ) ( ) /) (29)
To apply the Fuzzy Bayes to our classification task, we must evaluate the likelihoods used in (29). To find the size of a set, we compute the sum of the degrees and the patterns belonging to that set. So, we find:
( ) .Σ / ( ) , ( ) (Σ ) ( ) and
( ) .Σ / (Σ ) ( ) (30)
5. FUZZY INFERENCE MODEL for CLASSIFICATION
The task of fuzzy classification is to generate an appropriate fuzzy partition of the variable space. In this context, the word appropriate means that the number of misclassified patterns is very small or zero, [Guan et al. (2008)].
Let an n-dimensional pattern of space. Let us suppose that our pattern classification model is a subclass in the m-dimensional pattern of space , - for each class, being . We also suppose that real vectors ( ) are given as training patterns from the subclass. We can define the subclass of each class in the following way:
, (31)
where { * ⏟ ⏟ + ( ) }
The equivalence relation that causes the function class is unknown, but a training set is given by:
{. ( )/ } (32)
The classification model is the task of finding the mapping from to the set of m classes, using only the information brought by the training set, [Ishibuchi et al. (2005)], [Jain and Abraham (2003)].
Consider a FIS that contains multiple-input and multiple- output with a TS model, in which the antecedents of every rule define the membership of a pattern to a fuzzy relation, and the consequents are the membership values of the pattern to each class. We can represent a general form of the fuzzy rules as, [Burak et al. (2004)]: ( ) ( ) ( )
( ) ( ( ) (33)
where label of the fuzzy if–then rule,( ), antecedent fuzzy sets on the interval [0, 1], the consequent subclass (i.e., one of the given subclass), and are the certainty degree of the fuzzy if– then rule .
In our fuzzy classification model, the following simple steps determined by the consequent subclass and the certainty degree of each fuzzy if–then rule when its antecedent fuzzy sets specified by genetic operations:
Step (1): For each training pattern and the fuzzy if–then rule , we computed the compatibility degree by the product operation:
( ) ( ) ( ) ( ) (34)
where ( ) , - is a membership function of , ( ) is a number of inputs, ( ) is a number of rules, and * is a product operator.
Step (2): For each subclass, we compute the sum of the compatibility degrees of the training patterns with the fuzzy if–then rules .
( ) Σ ( ) (35)
where ( ) is the sum of the compatibility degrees of the training patterns in ( ) is a number of classes and ( ) is a number of subclasses.
Step (3): We find ̅ that has the maximum value of ( )
̅ ( ) { ( ) ( )} (36)
If more than one subclass takes the maximum value, then the consequent of the fuzzy if–then rule cannot be determined uniquely. In this case, let be .
Step (4): If the consequent is empty, then the certainty degrees of the fuzzy if–then rule is , else the certainty degrees is computed as follows:
{ ( ) Σ( ( ) ( ) ) ̅ Σ ( ) (37)
We must repeat these steps for each class. The computation of the certainty degree by (37) seems to be a bit complicated at a glance. When the antecedent fuzzy sets of each fuzzy if–then rule are given, we can compute the consequent subclass for each class and the certainty degree by the rule of generation steps. We can represent the last results for all classes by using the confusion fuzzy matrix model that is explained in the next section.

6. CONFUSION FUZZY MATRIX with CLASSIFICATION MODEL
The main problem in classification model has been introduced with uncertainty by the error matrix, where the classification, gradual membership in several classes is allowed for each element of sample data. We let be the set of classification data assigned to , and the set of reference data assigned to , with ( ) ( ) where is a number of classes.
The fuzzy classifications context and may be considered fuzzy sets having the MFs, [Cinthia et al., (2007)] and [Dwivedi et al. (2012)]:
( ) , -, and ( ) , - (38)
where ( ) ( ) represent the gradual membership of the sample element in and as indicated in the classification and reference data, respectively.
We use fuzzy set operators within the confusion matrix building procedure to provide a confusion fuzzy matrix. The task to the element involves the calculation the membership degree in the fuzzy intersection set ( ) as the following:
( ) . / (39)
Further, the cardinality of the fuzzy set intersection provides the global value of the generic element in row n and column m calculated on the overall sample data set, [Binaghi et al. (1999)], [Kannan and Ramat (2008)]:
| | Σ ( ) (40)
So, depending on the (40), we can represent the confusion fuzzy matrix by the matrix:
[ ] (41)
In case of multi-membership, the generic element represents the cardinality of the intersection set between classification data and reference data calculated according to (40), [Hassan (2010)]. The element in the confusion fuzzy matrix denotes a fuzzy number belong to [0, 1], due to the fact that in (40) scalar cardinality is applied to the fuzzy set . As in the conventional case, ( Σ ) ( Σ ) represent the total membership degrees specified to the class for classification and reference data, respectively. The simplest index is fuzzy overall accuracy FOA. This is calculated by dividing the total of the main diagonal by the total number of membership degrees found in reference data explaining FOA such a measure of the total match between classification and reference data are as follows, [Plamen and Zhou (2008)]:
Σ where Σ Σ (42)
From (42), we obtain the total classification accuracy. For each class, we can obtain the fuzzy producer’s accuracy FPA related to omission errors, together with the fuzzy user’s accuracy FUA related to commission errors. All these measures, FOA, FPA, and FUA, are limited to the range [0, 1]. We can define the fuzzy user’s accuracy, the fuzzy producer’s accuracy, and the average of fuzzy user's and producer's accuracy, respectively as:
( ) ⁄, ( ) ⁄and
( ) . (43)
The elements of the main diagonal in the CFM represent the number of sample elements that have been classified correctly, while the elements of off the diagonal represented the misclassifications rate (MR), [Binaghi et al. (1999)] and [Cinthia et al. (2007)]. The MR is also called simply "accuracy”, which can be computed, depending on the confusion fuzzy matrix (41) as:
ΣΣ (44)
The sums of the confusion fuzzy matrix elements over row and column are noted , and , respectively. When considering one in particular, one may distinguish four types of instances: true positives (TP) and false positives (FP) are instances of classification and misclassification as , whereas true negatives (TN) and false negatives (FN) are instances of classification and misclassification as , respectively. The corresponding counts are defined as
where is the total number of classified instances that is defined by (42), [Dwivedi et al. (2012)]. The true positives rate (TPR) and true negatives rate (TNR) are both reference-oriented, (i.e. they consider the confusion fuzzy matrix columns (reference classes)). The former is also called sensitivity, but the latter is alternatively called specificity and both measures range from zero to one.
( ) , ( ) (45)
Next, we can use the technique of the receiver operating characteristics (ROC) curve that is focused on sensitivity and specificity. They were originally designed as a tool in communication theory to visually determine optimal operating points for signal recognizer. We can represent the space of ROC graph by 2-dimentional in which the sensitivity ( ) is plotted on thevertical axis and the specificity ( ) is plotted on the horizontal axis [Plamen and Zhou (2008)].
7. APPLICATION
The building of a Multi-NFClass model depends on a fuzzy inference model for classification. Firstly, we must create antecedent fuzzy sets for each observation that depend on input variables. The numeric variables must be processed by using fuzzy values. We can determine the

structure of fuzzy input variables and the outputs of classes as shown in Table 1. The fuzzy sets are not parameterized, but they store the membership degrees for each variable value. The rule-base should be built by an expert, so by which it causes few errors as possible. See Fig. 2. The expert should be determining the number of rules, so that all input vectors are covered by the rules. We select the suitable consequents that are related with antecedents for each observation. This model is applied on actual data of lung cancer.
7.1.Lung Cancer (LC) Dataset Lung cancer LC is a disease characterized by uncontrolled cell growth in biology of the lung. It is the most common of all cancer deaths in both men and women. The main types of LC are small-cell lung cancer (SCLC), and non-small-cell lung cancer (NSCLC).The most common symptoms are coughing, weight loss, shortness of breath, and chest pains etc, and the common cause of LC is long-term exposure to tobacco smoke, family history, radon gas, and asbestos. LC may be seen on chest radiograph and CT-scan. The diagnosis is confirmed with a biopsy. The treatment depends on the type of cancer, the stage (degree of spread), and the person's overall health. Common treatments include surgery, chemotherapy, and radiotherapy. NSCLC is sometimes treated with surgery, whereas SCLC usually responds better to chemotherapy and radiotherapy. The Jordan Hospital Club Cancer Center, Specialized Hospital Hazem Al Hafez/ Iraq (Oncology and Nuclear) and King Hussein Cancer Center/ Jordan provided dataset of LC. The data set contains 250 patients. Each case represented by sixteen multi-input variables corresponding to three main classes as multi-output variables. See Table 1. Each output (class) has multi-subclasses as following: The first output (Class1) is a type of LC classifier with two subclasses:
Table 1: The structure of input fuzzy variables and the output classes labels Inputs of lung cancer Outputs of lung cancer Input variables Fuzzy variable Fuzzy value Left bound Mean bound Right bound Classes Code subclass Fuzzy variable Code Fuzzy value Gender Low (Female) -0.4 0 0.7 (class 1) Types of LC SCLC 1 High (Male) 0.3 1 1.4 Age Less than 30 -0.4 0 0.4 NSCLC 2 Average 0.25 0.5 0.75 Greater than 70 0.6 1 1.4 (class 2) Stages of LC LD 1 Smokes Less (No Smokes) -0.4 0 0.4 More (Smokes) 0.2 1 1.4 SI 2 Tobacco& Alcohol Less (No T and Co) -0.4 0 0.4 More (T and Co) 0.2 1 1.4 SII 3 Family history Low (No Genetic) -0.4 0 0.6 High (Genetic) 0.3 1 1.4 SIIIA 4 Asbeston Less (No Asbeston) -0.4 0 0.75 More (Asbeston) 0.25 1 1.4 SIIIB 5 Radon Low (No Radon) -0.4 0 0.8 High (Radon) 0.2 1 1.4 SIV 6 Weight loss Low (Weight loss) -0.4 0 0.4 Medium (Weight loss) 0.2 0.5 0.8 XD 7 High (Weight loss) 0.6 1 1.4 Coughing Increasing (Cough) 0.5 0.9 1.4 (class 3) Stages of LC Ch 1 Sputum with blood Increasing (SpB) 0.35 0.95 1.4 Chest pain Increasing (ChP) 0.55 0.85 1.4 Ra 2 Voice change Increasing (Vch) 0.25 0.75 1.4 RP and CLC Increasing (RPn) 0.4 0.85 1.4 Ch&Ra 3 X-ray and CT-scan Low (X-ray) -0.4 0 0.5 High (X-ray&CT scan) 0 1 1.4 Ch&Sur 4 Sputum cytology Low (No Sputum cyt) -0.4 0 0.6 High(Test Sputum cyt) 0.4 1 1.4 Sur&Ch/Ra 5 Biopsy Low (No Biopsy) -0.4 0 0.7 High (Test Biopsy) 0.3 1 1.4

Yes
Yes
Create new fuzzy sets from all missing input and add them to
If antecedent list
If ( )
Add to antecedent list
Initialize
Create new antecedent( )
Create rule (( ) ) and add it to rule list
Create new empty antecedent
( ) ( ) * +{ ( )( )}
Start
Add ( ) to antecedent
If input is not missing
Create new empty antecedent
Yes
No
No
Yes
Yes
If not all class C
Yes
No
If antecedent list
No
If class(c)=cons( )
If is missing
No
D
Yes
No
, - , -
If all symbolic feature
D
No
If rule base list
Yes
Yes
If ( )
No
No
Normalize ( )
Calculate performance c
If rule list
No
Yes
Calculate cons ( ) & resolve conflicts
If select rule
If select rule each class
Yes
No
Yes
Select rule algorithm
Select rule each class algorithm
Stop
Figure 2. Flowchart of a Multi-NFClass model.

=0.22/ SCLC+0.78/ NSCLC;
The second output (Class2) is a stage of LC classifier with seven subclasses:
=0.12/ LD+0.17/ SI+0.19/ SII+0.10/ SIIIA+0.09/ SIIIB+0.23/ SIV+0.10/ XD;
The third output (Class3) is a treatment of LC that classifier with five subclasses as:
=0.30/ Ch + 0.39/ Ra +0.07/ Ch&Ra + 0.03/ Ch&Sur +0.21/ Sur&Ch/Ra;
7.2. Discussion and Results
A Multi-NFClass represents a fuzzy classifier with a set of classes* +. We can examine the performance of our Multi-NFClass model with multi- inputs for LC data that consists of 250 cases (patients) as training data. There are sixteen continuous inputs ( ).Each input has been represented by linguistic terms, such as an input fuzzy variable (weight loss) has antecedent fuzzy set that represented by three linguistic terms {low, medium, high} as fuzzy values. There are five inputs ( ) have one continuous linguistic term, but only two inputs ( ) have three linguistic terms, while nine inputs ( ) have two linguistic terms as antecedent fuzzy sets of each fuzzy if–then rule. The maximum number of possible fuzzy if–then rules calculated ( rules) that have general form such as (33). The number of rules is considered for all possible cases. However, these cases are not possible with our application that dealing with illness cases that is related to the lives of patients. We have to be very carefully to detect the type of LC and to determine the stage of LC that has spread, as well as determine the treatments required for each case. There are many cases, in which it is impossible to make a relationship between the inputs and outputs. For e.g. when the type of LC belongs to the SCLC with the stage (XD), and the cancer cells have spread to outside the lung as far as the brain, then it is impossible to remove the tumor from the brain and only chemotherapy and radiation can be used as treatment. Depending on many of the conditions, we could control and build the rule-base set of a Multi-NFClass model. Consequently, we have specified an initial number of rules (100 from 4608 rules) at a first performance of Multi- NFClass model. For examining the validity of the proposed number of rules, we can suggest other number of rules in which less than 100 rules (such as 80 rules) at a second performance. We have created a basic program depending on steps of a Multi-NFClass and fuzzy inference for classification models, which have been, explained in Section 2, 3, and 5. We have used software of MATLAB in order to build the program. The basic program applied and performed 6th times, each time with different type of MF. These procedures performed two times: the first time with 100 rules and the second time with 80 rules. In order to make a comparison between them, the outputs of program Table 2: Comparison between the difference types of MF and its fuzzy misclassifications. Type of MF Number of misclassification Misclassification of classes Type of LC Stage of LC Treatment of LC Triangular MF 11 0.04 0.03 0.04 Trapezoidal MF 10 0.03 0.03 0.04 Gaussian MF 12 0.03 0.06 0.03 Bell-Shape MF 16 0.00 0.09 0.07 Product of 2 Sigmoid MF 13 0.03 0.06 0.04 Pi-Shape MF 11 0.02 0.04 0.05
have three main classes in which each class obtained only one subclass depending on the case of the patient. The best performance of a classification model, when it has a maximum number of the classification, and when the misclassification and the rules have a minimum number. We have obtained spatial classification of a type, stage, and treatment of LC for each patient. Consequently, we tried to compare between actual (reference) and classification classes in order to obtain the misclassification for each patient. Depending on a misclassification rate, we have made comparison between the results of the six programs that have different types of MF. See Table 2. Table 2 contains the results of the misclassifications that performed a Multi-NFClass model with different MF for each class. The minimum number of the misclassification was with a Trapezoidal MF, which there are three misclassifications for LC type and stage, while four misclassifications for LC treatment. The largest number of misclassification was with Bell-Shape MF.
The best performance of a Multi-NFClass model was with Trapezoidal MF that represented by using confusion fuzzy matrix (CFM). From (38)-(41), we could compute the matrices CFM between the reference and classification data. We have repeated these procedures two times with 100 and 80 rules for each class as shown in Tables 3-5. The CFM has been shown in the classifications and misclassifications values. The classifications are the green squares on the matrices diagonal, while the misclassifications formed the values on red parts from CFM. From the Tables 3-5, we noted the misclassifications of the LC type, stage, and treatment were 0.03, 0.02, 0.03, respectively with 100 rules, while with 80 rules they were 0.13, 0.17, and 0.11, respectively. Consequently, we proved the results with 100 rules being better than it being with 80 rules is.
From (42) and (43), we can compute the fuzzy producer’s accuracy (FPA) thus the omission error (OE) (OE=100- FPA) and the fuzzy user’s accuracy (FUA) thus the commission error (CE) (CE=100- FUA) for each class.

Table 3A: Confusion fuzzy matrix of the LC types with 80 rules 80 rules Reference data Row total SCLC NSCLC Classification data SCLC 0.21 0.12 0.33 NSCLC 0.01 0.66 0.67 Column total 0.22 0.78 0.87/0.13 Table 3B: Confusion fuzzy matrix of the LC types with 100 rules 100 rules Reference data Row total SCLC NSCLC Classification data SCLC 0.22 0.02 0.24 NSCLC 0.01 0.75 0.76 Column total 0.23 0.77 0.97/ 0.03 Table 4A: Confusion fuzzy matrix of the LC stages with 80 rules 80 rules Reference data LD SI SII SIIIA SIIIB SIV XD Row total Classification data LD 0.12 0.03 0.02 0.02 0.01 0.03 0.01 0.24 SI 0.00 0.13 0.01 0.00 0.00 0.00 0.00 0.14 SII 0.00 0.00 0.15 0.00 0.00 0.00 0.00 0.15 SIIIA 0.00 0.00 0.00 0.08 0.01 0.00 0.00 0.09 SIIIB 0.00 0.00 0.01 0.00 0.07 0.01 0.00 0.09 SIV 0.00 0.00 0.00 0.00 0.00 0.19 0.00 0.19 XD 0.00 0.01 0.00 0.00 0.00 0.00 0.09 0.10 Column total 0.12 0.17 0.19 0.10 0.09 0.23 0.10 0.83 0.17 Table 4B: Confusion fuzzy matrix of the LC stages with 100 rules 100 rules Reference data LD SI SII SIIIA SIIIB SIV XD Row total Classification data LD 0.12 0.00 0.00 0.00 0.00 0.00 0.0 0.12 SI 0.00 0.16 0.00 0.00 0.00 0.00 0.0 0.16 SII 0.00 0.00 0.18 0.00 0.00 0.00 0.0 0.18 SIIIA 0.00 0.00 0.00 0.01 0.01 0.00 0.0 0.11 SIIIB 0.00 0.00 0.01 0.00 0.08 0.01 0.0 0.10 SIV 0.00 0.00 0.00 0.00 0.00 0.22 0.0 0.22 XD 0.00 0.01 0.00 0.00 0.00 0.00 0.1 0.11 Column total 0.12 0.17 0.19 0.10 0.09 0.23 0.1 0.96 0.04 Table 5A: Confusion fuzzy matrix of the LC treatments with 80 rules 80 rules Reference data Ch Ra Ch& Ra Ch& Sur Sur& Ch/Ra Row total Classification data Ch 0.28 0.03 0.00 0.00 0.03 0.34 Ra 0.01 0.35 0.00 0.00 0.00 0.36 Ch&Ra 0.00 0.00 0.05 0.00 0.00 0.05 Ch&Sur 0.00 0.00 0.00 0.03 0.00 0.03 Sur&Ch/Ra 0.01 0.01 0.02 0.00 0.18 0.22 Column total 0.30 0.39 0.07 0.03 0.21 0.89 0.11 Table 5B: Confusion fuzzy matrix of the LC treatments with 100 rules 100 rules Reference data Ch Ra Ch& Ra Ch& Sur Sur& Ch/Ra Row total Classification data Ch 0.27 0.00 0.00 0.00 0.00 0.27 Ra 0.01 0.39 0.00 0.00 0.00 0.40 Ch&Ra 0.00 0.00 0.07 0.00 0.00 0.07 Ch&Sur 0.00 0.00 0.00 0.03 0.00 0.03 Sur&Ch/Ra 0.02 0.00 0.00 0.00 0.21 0.23 Column total 0.30 0.39 0.07 0.03 0.21 0.97 0.03 Table 6: Average of omission and commission errors of the subclasses classification with 100 rules Class Subclass FPA OE FUA CE Class1 SCLC NSCLC 88% 12% 95% 5% 99% 1% 96% 4% AFPA of class 1 =6.5% AFUA of class 1 =4.5% Class 2 LD 100% 0% 100% 0% SI 94% 6% 100% 0% SII 95% 5% 100% 0% SIIIA 100% 0% 91% 9% SIIIB 89% 11% 80% 20% SIV 96% 4% 100% 0% XD 100% 0% 91% 9% AFPA of class 2 =3.7% AFUA of class 2 =5.4% Class 3 Ch 90% 10% 100% 0% Ra 100% 0% 98% 2% Ch&Ra 100% 0% 100% 0% Ch&Sur 100% 0% 100% 0% Sur&Ch/ Ra 100% 0% 91% 9% AFPA of class 3 =2% AFUA of class 3 =2.2%

Depending on the results of OE and CE, we can be find average of fuzzy producer's accuracy (AFPA) and Average of fuzzy producer's accuracy (AFUA), respectively. See Table 6. The AFPA of the LC type, stage, and treatment were 6.5%, 3.7%, 2% respectively, while The AFUA of the LC type, stage, and treatment were 4.5%, 5.4%, and 2.2%, respectively. Depending on (42) and (44), we have calculated FOA and MR with 100 and 80 rules. The MR of the LC type, stage, and treatment were 0.03, 0.041, and 0.03 with 100 rules respectively, while with 80 rules they were 0.149, 0.205, and 0.123, respectively. See Table 7. This classification covers all data, and there are no unclassified cases. The classification performance is very good and the classifier is very compact.
Second part, depending on the concept of neural network with fuzzy Bayesian classification (NN-FBC) that has explained through (22)-(30), we have built a program of a NN-FBC model in order to detect LC. The results of a NN-FBC model can be represented by the confusion matrices (CM) for each class. The FOA of the LC type, stage, and treatment by using a NN-FBC model were 0.98, 0.94, and 0.91, respectively. See Table 9. We have made comparisons between a Multi-NFClass model and NN- FBC model. The MR of the LC type, stage, and treatment by using a NN-FBC model were 0.02, 0.064, and 0.099, respectively. See Table 8. Then, we have proved the performance of the proposed model (Multi-NFClass) very well and it worked properly when it was compared with another model. From (45), we tried to compute the sensitivity and the specificity in order to represent the results of a Multi-NFClass model by using the concept of the receiver-operating characteristic (ROC). The best classifiers will have a line going as far as possible towards the top left corner of the quadrant. The ROC curves have represented all the subclasses of the LC with different colors. See Fig. 3.
8. CONCLUSION
In this paper, we have developed a Multi-NFClass as a natural extension of an NFClass and have focused on the fuzzy inference for classification model that have been extended to multi-antecedents and multi-consequents. We have proposed a Multi-NFClass and NN-FBC models and demonstrated its performance on LC data collected from Jordan Hospital Club Cancer Center, Specialized Hospital Hazem Al Hafez, and King Hussein Cancer Center. Construction of the fuzzy if–then rules that are determined by the simple steps when its antecedent fuzzy sets are specified by genetic operations, which is performed a Multi-NFClass model. The minimum number of the misclassification was with a Trapezoidal MF, in which there are three misclassifications for LC type and stage, while four misclassifications for LC treatment. The MR of the LC type, stage, and treatment were 0.03, 0.041, and 0.03 with 100 rules respectively, while the results were 0.149, 0.205, and 0.123 with 80 rules, respectively. Moreover, by using a NN-FBC model it is 0.02, 0.064, and 0.099, respectively. We have proved the performance of a Multi-NFClass very well and it worked properly when it was compared with NN-FBC model. The concept of neural network we applied in this paper uses a fuzzy Bayesian classification in order to built NN-FBC model. Applications of the proposed models remain to be studied. Since lung cancer is a special case of cancer, it would be interesting to see how proposed models can be extended to detect and handle other types of cancer. This is one of our future research directions.
ACKNOWLEDGEMENTS
The author wishes to thank the reviewers for their excellent suggestions that have been incorporated into this paper. I would like to thank and acknowledge Professor Deepak Gupta, Head Department of Mathematics, M.M. University, Mullana, Ambala, India, who advised me to improve my work.
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Table 7: The fuzzy overall accuracy FOA and misclassification MR with different number of rules Rule number Fuzzy class FOA MR AFUPA 80 Type of LC 0.87 0.149 0.1425 Stage of LC 0.83 0.205 0.153 Treatment of LC 0.89 0.123 0.108 100 Type of LC 0.97 0.030 0.055 Stage of LC 0.96 0.041 0.045 Treatment of LC 0.97 0.030 0.021 Table 8: Comparison between a Multi-NFClass and a NN-FBC models Fuzzy classes Classification model FOA MR Type of LC Multi-NFClass model 0.97 0.030 NN-FBC model 0.98 0.020 Stage of LC Multi-NFClass model 0.96 0.041 NN-FBC model 0.94 0.064 Treatment of LC Multi-NFClass model 0.97 0.030 NN-FBC model 0.91 0.099

(A) ROC of the LC types
(B) ROC of the LC stages
(C) ROC of the LC treatments
Figure 3: The receiver operating characteristics ROC of the classes.
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