**5. Rough-fuzzy approach**

ANFIS approach described in Section 4 has successfully formed a complete fuzzy rule that able to solve the problem of incomplete and vague decisions made by human. However, not all rules generated are significant and thus it is important to extract only the most significant rules in order to improve the classification accuracy. In this work, we propose Rough-Fuzzy approach to refine the fuzzy rule base into a concise fuzzy rule base (refer Fig. 11).

Fig. 11. The rough–fuzzy approach to constructing concise fuzzy rules

#### **5.1 Rough fuzzy phases**

76 Fuzzy Inference System – Theory and Applications

because the 18 rules in this rule base were carefully selected to give full certainty for decisions. However, we found that not all situations covered by this 18 fuzzy rules and still have some rules are not stated. On the contrary, the complete fuzzy rule base in ANFIS is complete but still got some rules are inconsistent and the decision output is not logically. Although all situations for all four attributes are covered by the set of 81 rules, some of the rules have been found to have unnecessary conditions. Thus, the increment of the training data need to done, so that the ANFIS based on 69 training datasets able to eliminate the unnecessary conditions and the illogical decisions. Finally, the ANFIS model is consistent and complete; all situations for all four attributes are covered by the set of 69 training data,

**Fuzzy Rule Base Input data patterns** 

Human Experts 62.00% 66.00% ANFIS (44) 69.14% 89.60% ANFIS (54) 85.19% 99.47% ANFIS (64) 96.30% 99.73% ANFIS (69) 100.00% 100.00%

ANFIS approach described in Section 4 has successfully formed a complete fuzzy rule that able to solve the problem of incomplete and vague decisions made by human. However, not all rules generated are significant and thus it is important to extract only the most significant rules in order to improve the classification accuracy. In this work, we propose Rough-Fuzzy

approach to refine the fuzzy rule base into a concise fuzzy rule base (refer Fig. 11).

Fig. 11. The rough–fuzzy approach to constructing concise fuzzy rules

**81 1500** 

and there are no missing rules.

**5. Rough-fuzzy approach** 

Table 5. Percentage of successful classifications correctly

The three main phases in the rough-fuzzy approach are data pre-processing, reduct computation and data post-processing as shown in Fig. 11 and described as follows:

**Phase 1.** Data pre-processing.

In this phase, the complete fuzzy rules are converts from linguistic terms into numeric values that correspond to the rough set format.

**Phase 2.** Reduct computation.

The fuzzy rules are mapped into a decision system format, discretisation of data, computation of reducts from data and derivation of rules from reducts.


The reduct computation stage determines the selection of an important attribute that can be used to represent the decision system (Carlin et al., 1998). It is used to reduce the decision system, thus generating more concise rules. The rough set approach employs two important concepts related to reduction: one is related to reduction of rows, and the other one is related to reduction of columns (Chen, 1999). With the notion of an indiscernibility class, the rows with certain properties are grouped together, while with the notion of dispensable attributes, the columns with less important attributes are removed. Another essential concept in reduct computation is the lower and upper approximations, in which the computation involved in the lower approximation produces rules that are certain, while the computation involved in the upper approximation produces possible rules (Øhrn, 2001).

d. Rule Generation. A reduct is converted into a rule by binding the condition attribute values of the object class from which the reduct is originated to the corresponding attribute.

#### **Phase 3.** Data post-processing

The rules in rough set format are converted into linguistic terms of the concise fuzzy rule base.

#### **5.2 Rough fuzzy experiment**

In Section 4, there are 81 datasets that represent every possible value of the fuzzy rules with full certainty. This dataset is used for the development of the ANFIS model. Using Rosetta as rough set tool, the genetic algorithm with object reduct is the method used for computing reducts (Øhrn, 2001). This method implements a genetic algorithm for computing minimal hitting sets as described by Vinterbo and Øhrn (2000). Using rough set, we trained the fuzzy

A Concise Fuzzy Rule Base to Reason Student Performance Based on Rough-Fuzzy Approach 79

Rules generated from reduct are representative rules extracted from the data set. Since a reduct is not unique, rule sets generated from different reducts contain different sets of rules

*R3* Score = moderate AND Attempt = a few => Performance = moderately mastered

*R5* Score= high AND Attempt = average => Performance = moderately mastered *R6* Score= high AND Attempt = a few AND Help = little => Performance = has

*R7* Score= high AND Attempt = a few AND Help = average => Performance =

*R8* Score= high AND Attempt = a few AND Help = needed => Performance =

For example, the given reduct from Table 8 i.e. reduct {Score, Attempt}, is presented by three

*R3* : IF Score = moderate AND Attempt = a few THEN Performance = moderately mastered *R4* : IF Score= moderate AND Attempt = average THEN Performance = moderately

*R5* : IF Score= high AND Attempt = average THEN Performance = moderately mastered

A unique feature of the rough set method is its generation of rules that played an important role in predicting the output. Table 10 listed the rule generation analysis by Rosetta and provides some statistics for the rules which are support, accuracy, coverage and length. The rule coverage and accuracy are measured to determine the reliability of the rules. Below is

a. The rule support is defined as the number of records in the training data that fully

b. The rule accuracy is defined as the number of RHS support divided by the number of

c. The conditional coverage is the fraction of the records that satisfied the IF conditions of the rule. It is obtained by dividing the support of the rule by the total number of records

d. The decision coverage is the fraction of the training records that satisfied the THEN conditions. It is obtained by dividing the support of the rule by the number of records

e. The rule length is defined as the number of conditional elements in the IF part.

*R4* Score= moderate AND Attempt = average => Performance = moderately

as shown in Table 9.

mastered

mastered

Table 9. Rule Generation

LHS support.

in the training sample.

mastered

moderately mastered

moderately mastered

rules as shown in Table 9 namely *R3*, *R4*, and *R5*.

the definition of the rule statistics (Bose, 2006).

exhibit property described by the IF condition.

in the training that satisfied the THEN condition.

*R1* Score = low => Performance = not mastered *R2* Attempt = many => Performance = not mastered

**Rule set Rules** 

rules incrementally with different training data set that consist of 44, 54, 64 and 69 input data patterns as described in Section 4. The purpose of the iteration with different input patterns of ANFIS is to ensure that the decision is agreed by human expert.

Table 6 shows the number of reducts, the number of rules and the rule percentage of rough set experiment with different input patterns. The result shows that ANFIS with 69 input patterns generates more concise rule with less number of reducts and less number of rules extracted compared to ANFIS with other pattern.


Table 6. Number of reducts and rules based on different input patterns

To determine whether the performance of the concise fuzzy rule base is consistent with the performance of the complete fuzzy rule base, each rule bases of input patterns is compared.

Table 7 shows that the decision output given by both the rule bases of each input patterns has very small differences (in terms of its mean square error). This result confirms that the concise fuzzy rule base does not degrade the performance of the complete fuzzy rule base.

It can be seen from Table 7 that ANFIS with 69 input patterns matched exactly as predicted by experts with MSE value equal to zero. The reducts and rules generated by rough set for ANFIS with 69 input patterns is chosen for further discussion.


Table 7. MSE result of Complete vs Concise Rule Base

Furthermore, Table 8 shows four object-related reduct generated by Rosetta for ANFIS with 69 input patterns. All reducts has 100% support, which mean that all objects are mapped deterministically into a decision class. In other words, the support for the decision rule is the probability of an object to be covered by the description that belongs to the class (Grzymala-Busse, 1991).


Table 8. Object-related reduct based on ANFIS 69 model


Rules generated from reduct are representative rules extracted from the data set. Since a reduct is not unique, rule sets generated from different reducts contain different sets of rules as shown in Table 9.

Table 9. Rule Generation

78 Fuzzy Inference System – Theory and Applications

rules incrementally with different training data set that consist of 44, 54, 64 and 69 input data patterns as described in Section 4. The purpose of the iteration with different input

Table 6 shows the number of reducts, the number of rules and the rule percentage of rough set experiment with different input patterns. The result shows that ANFIS with 69 input patterns generates more concise rule with less number of reducts and less number of rules

**Model No of Reducts No of Rules Percentage of Rules** 

To determine whether the performance of the concise fuzzy rule base is consistent with the performance of the complete fuzzy rule base, each rule bases of input patterns is compared. Table 7 shows that the decision output given by both the rule bases of each input patterns has very small differences (in terms of its mean square error). This result confirms that the concise fuzzy rule base does not degrade the performance of the complete fuzzy rule base. It can be seen from Table 7 that ANFIS with 69 input patterns matched exactly as predicted by experts with MSE value equal to zero. The reducts and rules generated by rough set for

> **Complete Rule Base (81 Rules) Concise Rule Base MSE**  ANFIS with 44 input patterns 23 Rules 4.76E07 ANFIS with 54 input patterns 16 Rules 1.02E07 ANFIS with 64 input patterns 13 Rules 3.70E10 ANFIS with 69 input patterns 8 Rules 0.00

Furthermore, Table 8 shows four object-related reduct generated by Rosetta for ANFIS with 69 input patterns. All reducts has 100% support, which mean that all objects are mapped deterministically into a decision class. In other words, the support for the decision rule is the probability of an object to be covered by the description that belongs to the class (Grzymala-

> **Class Reduct Support**  *C1* {Score} 100 *C2* {Attempt} 100 *C3* {Score , Attempt} 100 *C4* {Score , Attempt, Help} 100

patterns of ANFIS is to ensure that the decision is agreed by human expert.

1. Human expert 6 13 16% 2. ANFIS with 44 input patterns 11 23 28% 3. ANFIS with 54 input patterns 9 16 20% 4. ANFIS with 64 input patterns 7 13 16% 5. ANFIS with 69 input patterns 4 8 10% Table 6. Number of reducts and rules based on different input patterns

ANFIS with 69 input patterns is chosen for further discussion.

Table 7. MSE result of Complete vs Concise Rule Base

Table 8. Object-related reduct based on ANFIS 69 model

Busse, 1991).

extracted compared to ANFIS with other pattern.

For example, the given reduct from Table 8 i.e. reduct {Score, Attempt}, is presented by three rules as shown in Table 9 namely *R3*, *R4*, and *R5*.

*R3* : IF Score = moderate AND Attempt = a few THEN Performance = moderately mastered

*R4* : IF Score= moderate AND Attempt = average THEN Performance = moderately mastered

*R5* : IF Score= high AND Attempt = average THEN Performance = moderately mastered

A unique feature of the rough set method is its generation of rules that played an important role in predicting the output. Table 10 listed the rule generation analysis by Rosetta and provides some statistics for the rules which are support, accuracy, coverage and length. The rule coverage and accuracy are measured to determine the reliability of the rules. Below is the definition of the rule statistics (Bose, 2006).


A Concise Fuzzy Rule Base to Reason Student Performance Based on Rough-Fuzzy Approach 81

significant. Therefore, this work proposes the Rough-Fuzzy approach that able to reduce the complete fuzzy rule base into a concise fuzzy rule base. This approach able to determine the selection of important attributes that can be used to represent the fuzzy rule base system. Therefore, the condition space is reduced by taking only a few conditions to achieve a reasonable size of the condition subspace. Moreover, the proposed concise fuzzy rule base is

The authors are especially grateful to the members of the Soft Computing Research Group (SCRG), Faculty of Computer Science and Information Systems, University of Technology Malaysia, for their encouraging support to this work. The authors would also like to thank Universiti Teknologi Malaysia (UTM) for their financial support under Research University Grant Vot. No. Q.J130000.7128.01H82 and Q.J130000.7128.02J57 as well as the FRGS Grant -

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**7. Acknowledgment** 

Vot No. 4F026(NT:2000957).

II, pp. 222–229.

**8. References** 


*Legend:* 

RS – Rule Sets, RSupp – Rule Support, RA – Rule Accuracy, CA – Conditional Coverage, DC – Decision Coverage, RL – Rule Length

Table 10. Rule Generation Analysis

Coverage gives a measure of how well the objects describe the decision class. The conditional coverage is measured by the ratio of the number of rules that fulfil the conditional part of the rules to the overall number of rules in the sample. Meanwhile, the decision coverage is measured by the ratio of the number of rules that give decision rules to the overall number of rules in the sample. Accuracy gives a measure of how trustworthy the rule is in the condition. It is the probability that an arbitrary object belonging to Class *C* is covered by the description of the reduct (Grzymala-Busse, 1991). According to Pawlak (1998), an accuracy value of 1 indicates that the classes have been classified into decision classes with full certainty and consistency.

For example, there are 27 objects that fulfil the conditional part of the rule *R1*, compared with the overall 81 rules in the sample. Therefore, the conditional coverage of this rule is about 0.3333. In addition, the decision for the performance and learning efficiency with the value of not mastered is used once in the fuzzy rule base and it is only given to rule *R1*. Therefore, the decision coverage for this rule is 1. Finally, the accuracy value of this rule is 1, which means that this rule belongs to Class *C1* and is covered. Thus, it is said to have full certainty and is consistent. In conclusion, because all of the rules in Table 10 have accuracy values of 1, the concise fuzzy rules are reliable because they are covered, have full certainty, and are consistent.
