**1. Introduction**

62 Fuzzy Inference System – Theory and Applications

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A fuzzy inference system employing fuzzy if then rules able to model the qualitative aspects of human expertise and reasoning processes without employing precise quantitative analyses. This is due to the fact that the problem in acquiring knowledge from human experts is that much of the information is uncertain, inconsistent, vague and incomplete (Khoo and Zhai, 2001; Tsaganou et al., 2002; San Pedro and Burstein, 2003; Yang et al., 2005). The drawbacks of FIS are that a lot of trial and error effort need to be taken into account in order to define the best fitted membership functions (Taylan and Karagözoğlu, 2009) and no standard methods exist for transforming human knowledge or experience into the rule base (Jang, 1993).

Evaluation and reasoning of student's learning achievement is the process of determining the performance levels of individual students in relation to educational objectives (Saleh and Kim, 2009). Although Fuzzy inference system is a potential technique to reason the student's performance, as well as to present his/her knowledge status (Nedic et al., 2002; Xu et al., 2002; Kosba et al. 2003), it is a challenge when more than one factor involve in determining the student's performance or knowledge status (Yusof et. al, 2009). Hence, the reasoning of the student's performance for multiple factors is difficult. This issue is critical considering that the human experts' knowledge is insufficient to analyze all possible conditions as the information gained is always incomplete, inconsistent, and vague.

Addressing these matters, this work proposes a Neuro-Fuzzy Inference System (ANFIS), which combines fuzzy inference system and neural network, in order to produce a complete fuzzy rule base system (Jang, 1993). The fuzzy system represents knowledge in an interpretable manner, while the neural networks have the learning ability platform to optimize its parameters. Hence, ANFIS has the capability to perform parameter-learning rather than structural learning (Lin and Lu, 1996). ANFIS is expected to recognize other decisions that are previously not complete, in both the antecedents and consequent parts of the fuzzy rules. Unfortunately, too many fuzzy rules will result in a large computation time and space (Jamshidi, 2001). Therefore, reduction of knowledge is possible to be applied to determine the selection of important attributes that can be used to represent the decision system (Chen, 1999) based on the theory of rough sets. Fig. 1 shows the proposed fuzzy inference system.

A Concise Fuzzy Rule Base to Reason Student Performance Based on Rough-Fuzzy Approach 65

b. Time (T) is the average duration, *x2*, taken by a student to answer the each question of a learning unit and with three term sets: *fast* (*T1*), *average* (*T2*), and *slow* (*T3*). The average of time (*x2*) is obtained by dividing the total time to answer a set of given questions by

> �� <sup>=</sup> <sup>∑</sup> �� � ��� �

Measurement of time can be done by using the distribution method. Fig. 2 shows the T-score

The time taken to answer each question (��) can be calculated by using the equation (3).

10(�� � ���) ��

100

The numbered "10" is distance value of standard deviation from mean, while numbered "50" is value of mean. T-score is divided by 100 so that able to get the value in the range 0 to 1. c. Attempt (A) is the average number of tries , *x3*, for a given learning unit, in which it is counted after student give a wrong answer for the first attempt and the question will repeat again for student to answer until correct. The term sets involve: *a few* (*A1*), *average* (*A2*), and *many* (*A3*). The average of attempt (*x3*) is calculated as equation (4). Dividing the total number of tries to answer a set of given questions by the total number of

+ 50

distribution, in which the mean is 50 and the standard deviation is 10.

Fig. 2. T-score distribution for time taken to answer question

�� is the time spent to answer the *i-*th question

��� is mean score for the time spent distribution �� is the standard deviation for the *i*-th question

�� is the time spent by the student

questions in the set.

�� =

(2)

(3)

Where :

Where :

Where :

*mi* is marks from each question

� is total number of questions

�� is the time spent to answer the *i-*th question

*Q* is total number the question in the set

the total number of questions, see equation (2).

Fig. 1. The proposed Fuzzy Inference System

This chapter is divided into six sections. Section 1 is the introduction and the problem statements. Section 2 discusses about the student modeling and learning criteria. Section 3 presents the Human Expert Fuzzy Inference System model that defines the data representation and the rule base acquired from the human experts. Section 4 describes the ANFIS approach to form a complete fuzzy rule base to solve the problem of incomplete and vague decisions made by human. Section 5 presents the proposed Rough-Fuzzy approach to determine important attributes and refine the fuzzy rule base into a concise fuzzy rule base. Finally, section 6 presents the conclusions of the work.

#### **2. Student modeling and the learning criteria**

Student model represents the knowledge about the student's behavior and learning performance. In this work, student's performance are classified into three categories, named as Has Mastered (HM), Moderately Mastered (MM), and Not Mastered (NM). The conditions that determine the decision made about the student's performance is also depend on the criteria set by the human expert. There are four input conditions namely, the *score* (*S*), *time* (*T*), *attempts* (*A*), and *helps* (*H*) in which each of the input condition is represented by three term sets with values (Norazah, 2005).

a. Score (S) is the average scoring, *x1*, which gains from each question of a learning unit and the term sets is represented by *low* (*S1*), *moderate* (*S2*), and *high* (*S3*). It can be found by dividing the total marks for a set of given questions by the total number of questions (*Q*) in the set, as shown in equation (1).

$$\omega\_1 = \frac{\sum\_{l=1}^{Q} m\_l}{Q} \tag{1}$$

Where :

64 Fuzzy Inference System – Theory and Applications

This chapter is divided into six sections. Section 1 is the introduction and the problem statements. Section 2 discusses about the student modeling and learning criteria. Section 3 presents the Human Expert Fuzzy Inference System model that defines the data representation and the rule base acquired from the human experts. Section 4 describes the ANFIS approach to form a complete fuzzy rule base to solve the problem of incomplete and vague decisions made by human. Section 5 presents the proposed Rough-Fuzzy approach to determine important attributes and refine the fuzzy rule base into a concise fuzzy rule base.

Student model represents the knowledge about the student's behavior and learning performance. In this work, student's performance are classified into three categories, named as Has Mastered (HM), Moderately Mastered (MM), and Not Mastered (NM). The conditions that determine the decision made about the student's performance is also depend on the criteria set by the human expert. There are four input conditions namely, the *score* (*S*), *time* (*T*), *attempts* (*A*), and *helps* (*H*) in which each of the input condition is represented by

a. Score (S) is the average scoring, *x1*, which gains from each question of a learning unit and the term sets is represented by *low* (*S1*), *moderate* (*S2*), and *high* (*S3*). It can be found by dividing the total marks for a set of given questions by the total number of questions

> �� <sup>=</sup> <sup>∑</sup> �� � ��� �

Fig. 1. The proposed Fuzzy Inference System

Finally, section 6 presents the conclusions of the work.

**2. Student modeling and the learning criteria** 

three term sets with values (Norazah, 2005).

(*Q*) in the set, as shown in equation (1).

*mi* is marks from each question *Q* is total number the question in the set

b. Time (T) is the average duration, *x2*, taken by a student to answer the each question of a learning unit and with three term sets: *fast* (*T1*), *average* (*T2*), and *slow* (*T3*). The average of time (*x2*) is obtained by dividing the total time to answer a set of given questions by the total number of questions, see equation (2).

$$\mathbf{x}\_2 = \frac{\sum\_{l=1}^{Q} T\_l}{Q} \tag{2}$$

Where :

� is total number of questions

�� is the time spent to answer the *i-*th question

Measurement of time can be done by using the distribution method. Fig. 2 shows the T-score distribution, in which the mean is 50 and the standard deviation is 10.

Fig. 2. T-score distribution for time taken to answer question

The time taken to answer each question (��) can be calculated by using the equation (3).

$$T\_l = \frac{\frac{10(X\_l - \overline{X}\_l)}{\sigma\_l} + 50}{100} \tag{3}$$

Where :

(1)

�� is the time spent to answer the *i-*th question

�� is the time spent by the student

��� is mean score for the time spent distribution

�� is the standard deviation for the *i*-th question

The numbered "10" is distance value of standard deviation from mean, while numbered "50" is value of mean. T-score is divided by 100 so that able to get the value in the range 0 to 1.

c. Attempt (A) is the average number of tries , *x3*, for a given learning unit, in which it is counted after student give a wrong answer for the first attempt and the question will repeat again for student to answer until correct. The term sets involve: *a few* (*A1*), *average* (*A2*), and *many* (*A3*). The average of attempt (*x3*) is calculated as equation (4). Dividing the total number of tries to answer a set of given questions by the total number of questions in the set.

A Concise Fuzzy Rule Base to Reason Student Performance Based on Rough-Fuzzy Approach 67

possible students learning performance. Bases on a survey done by Norazah (2005), there are only 18 decisions about the student's behavior are formed with certainty from seven subject matter experts; and these decisions are considered as the acceptable rules. All other

Scores (*S*) *x1* ≥ 75% High 75% ≥ *x1* ≥ 35% Md *x1* < 35% Low Time (*T*) *x2* < 40 Fast 60 ≥ *x2* ≥ 40 Avg *x2* > 60 Slow Attempt (*A*) *x3* < 25% A Few 75% ≥ *x3* ≥ 25% Avg *x3* > 75% Many Help (*H*) *x4* < 25% Little 75% ≥ *x4* ≥ 25% Avg *x4* > 75% Needed

Human expert's FIS uses a collection of fuzzy membership functions and rules to reason about student's performance. FIS consists of a fuzzification interface, a rule base, a database,

To compute the output of this fuzzy inference system given the inputs, four steps has to be

a. Compare the input variables with the membership functions on the antecedent part to obtain the membership values of each linguistic label. This step is called fuzzification. b. Combine the membership values on the premise part to get firing strength of each rule. c. Generate the qualified consequents or each rule depending on the firing strength. d. Aggregate the qualified consequents to produce a crisp output. This step is called

In the fuzzification stage, the input and output of the fuzzy inference system are determined. Table 2 and Table 3 exhibit examples of the four input and one output

**Fuzzy input variable Fuzzy linguistic terms Numerical range (normalized)** 

Moderate, High}

Average, Slow}

Average, Many}

Average, Needed}

**Moderately Mastered 75 ≥ y ≥ 25**

**Value Label Value Label Value Label** 

**Not Mastered y < 25** 

[0.14, 0.0] [0.12, 0.55] [0.14, 1.0]

[0.15, 0.0] [0.08, 0.5] [0.15, 1.0]

[0.12, 0.0] [0.12, 0.5] [0.12, 1.0]

[0.12, 0.0] [0.12, 0.5] [0.12, 1.0]

decisions that are not certain and have conflicts are being discarded from the rules.

**Criteria item Has Mastered**

followed (Norazah, 2005):

defuzzification.

**3.1 Fuzzification** 

**y >75**

Table 1. The criteria for the student's performance

**3. Human expert Fuzzy Inference System** 

Score (S) {Low,

Time (T) {Fast,

Attempt (A) {A few,

Help (H) {Little,

Table 2. The input variables of the Fuzzy Inference System

a decision-making unit, and finally a defuzzification interface.

$$\mathbf{x}\_3 = \frac{\sum\_{l=1}^{Q} \mathbf{t}\_l}{Q} \tag{4}$$

Where :

� is total number of questions

The number of attempt (��) is determined by calculating the number of attempts made (��) to answer a given question and dividing it by the maximum number of attempts (��) allowed for the question.

$$t\_l = \frac{a\_l}{P\_l} \tag{5}$$

Where :

*ai* is the number of attempts made to answer a given question

*Pi* is the maximum number of attempts allowed for the question

d. Help (H) is the average amount of help, *x4*, of a learning unit where it able to help student by giving some hints or notes to answer the question. The term sets involve: *little* (*H1*), *average* (*H2*),and *needed* (*H3*).

The average amount of help (*x4*) is calculated as equation (6), by dividing the total amount of help accessed by a student in answering a set of given questions by the total number of questions in the set.

$$\mathbf{x}\_4 = \frac{\sum\_{l=1}^{Q} h\_l}{Q} \tag{6}$$

Where :

� is the total number of questions

*hi* is the total amount of help accessed by a student

The amount of help (ℎ�) is found by calculating the number of help (��) links that a student accessed while answering a given question and dividing it by the maximum number of help links (����) provided for a given question.

$$h\_l = \frac{l\_l}{H\_{max}}\tag{7}$$

The output consequent of the student model is the student's performance and can be represented as *has mastered* (*P1*), *moderately mastered* (*P2*) and *not mastered* (*P3*) for the output. A student is classified as *has mastered* in a particular learning unit, when the student earns high scores (i.e. greater than 75%) with below 40% of time spent, not exceeding 25% of number of tries needed and number of helps. Besides that, a student is classified as *moderately mastered* when the student earns a moderate score, with moderate time spent, tries more than once, and number of help needed. For example, a moderate score would be rated in between 35% and 75%, time spent between 40% and 60%, tries between 25% and 75%, and help between 25% and 75%. Furthermore, a student is classified as *not mastered*  when the student has a low score with a lot of time, many tries and many help needed. However, in acquiring knowledge from the human experts is that, they cannot decide on all possible students learning performance. Bases on a survey done by Norazah (2005), there are only 18 decisions about the student's behavior are formed with certainty from seven subject matter experts; and these decisions are considered as the acceptable rules. All other decisions that are not certain and have conflicts are being discarded from the rules.


Table 1. The criteria for the student's performance
