**3. Human expert Fuzzy Inference System**

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

To compute the output of this fuzzy inference system given the inputs, four steps has to be followed (Norazah, 2005):


#### **3.1 Fuzzification**

66 Fuzzy Inference System – Theory and Applications

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

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

> �� <sup>=</sup> �� ��

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:

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

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

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

> ℎ� <sup>=</sup> �� ����

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

*ai* is the number of attempts made to answer a given question *Pi* is the maximum number of attempts allowed for the question

*little* (*H1*), *average* (*H2*),and *needed* (*H3*).

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

links (����) provided for a given question.

Where :

Where :

Where :

for the question.

questions in the set.

� is the total number of questions

� is total number of questions

(4)

(5)

(6)

(7)

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


Table 2. The input variables of the Fuzzy Inference System

A Concise Fuzzy Rule Base to Reason Student Performance Based on Rough-Fuzzy Approach 69

Fuzzy rules are a collection of linguistic statements that describe how the fuzzy inference system should make a decision regarding classifying an input or controlling an output. Fig. 5 presents the four inputs and one output reasoning of the student's performance procedure

Fig. 5. Fuzzy reasoning procedures for Human Expert FIS model of Student's Performance

The rule *Ri* is the *i-*th rule in the fuzzy rule base system, the *µi* is the membership function of the antecedent part of the *i*-th rule for each input variable and *wi* is the weight of the consequent of each rule. For example, for input1 is score and the membership function can classified as low, moderate or high. If *score* is *high* and *time* is *fast* and *attempt* is *a few* and *help* is *little* then *student performance* is *has mastered*. This process of taking input such as *score* and processing it through membership functions to determine the "high" score is called fuzzification. Based on the human experts' experience and knowledge about the students' performance, 18 initial rules that are certain have been constructed as shown in Table 4.

The outputs of all of the fuzzy rules must now be combined to obtain one fuzzy output distribution. The output membership functions on the right-hand side of the figure are combined using the fuzzy operator AND to obtain the output distribution shown on the lower right corner of the Fig. 5. For a zero-order Sugeno model, the output level *z* is a constant. The output level *zi* of each rule is weighted by the firing strength *wi* of the rule (Lin and Lu, 1996). For example, for an ∩ rule with input 1 = *x* and input 2 = *y*, the firing strength

 *wi* = *F1*(*x*) ∩ *F2*(*y*) (9)

*R*i have four input variables and one output variable as shown below:

*Ri*: IF *S* is *µi1* AND *T* is *µi2* AND *A* is *µi3* AND *H* is *µi4* THEN *P* is *wi*

**3.3 Combining outputs into an output distribution** 

is as shown in equation (9).

for zero order Sugeno fuzzy model. Each input has its own membership function.

**3.2 Creating fuzzy rules** 

variables, in which each of the variables consists of three term values and labels as discussed in Section 2. The fuzzy output follows the zero-order Sugeno style inference, in which the output value of each fuzzy rule is a constant (Sivanandam et al., 2007). Fig. 3 shows the four inputs and one single output for the Human Expert FIS.


Table 3. The output variables of the Fuzzy Inference System

Fig. 3. Four inputs and single output for the Human Expert FIS

The membership function of the input is expressed by a Gaussian function specified by two parameters {�� �}, and the membership value is derived by the formula in Fig. 4.

Fig. 4. Gaussian shape function ��������(�� ����)

$$\text{Gaussian}(\mathbf{x}; \ \sigma, \mathbf{c}) = \exp\left(-\left[\frac{\mathbf{x} - \mathbf{c}}{2\sigma}\right]^2\right) \tag{8}$$

Where :

*c* represents the membership function's center

*σ* determines the membership function's width

#### **3.2 Creating fuzzy rules**

68 Fuzzy Inference System – Theory and Applications

variables, in which each of the variables consists of three term values and labels as discussed in Section 2. The fuzzy output follows the zero-order Sugeno style inference, in which the output value of each fuzzy rule is a constant (Sivanandam et al., 2007). Fig. 3 shows the four

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

The membership function of the input is expressed by a Gaussian function specified by two

��������(�� �� �) � ��� �� ����

�� � �

� (8)

parameters {�� �}, and the membership value is derived by the formula in Fig. 4.

0.0 0.5 1.0

Moderately Mastered, Has Mastered}

inputs and one single output for the Human Expert FIS.

Performance (P) {Not Mastered,

Table 3. The output variables of the Fuzzy Inference System

Fig. 3. Four inputs and single output for the Human Expert FIS

Fig. 4. Gaussian shape function ��������(�� ����)

*c* represents the membership function's center *σ* determines the membership function's width

Where :

Fuzzy rules are a collection of linguistic statements that describe how the fuzzy inference system should make a decision regarding classifying an input or controlling an output. Fig. 5 presents the four inputs and one output reasoning of the student's performance procedure for zero order Sugeno fuzzy model. Each input has its own membership function.

Fig. 5. Fuzzy reasoning procedures for Human Expert FIS model of Student's Performance

*R*i have four input variables and one output variable as shown below:

*Ri*: IF *S* is *µi1* AND *T* is *µi2* AND *A* is *µi3* AND *H* is *µi4* THEN *P* is *wi*

The rule *Ri* is the *i-*th rule in the fuzzy rule base system, the *µi* is the membership function of the antecedent part of the *i*-th rule for each input variable and *wi* is the weight of the consequent of each rule. For example, for input1 is score and the membership function can classified as low, moderate or high. If *score* is *high* and *time* is *fast* and *attempt* is *a few* and *help* is *little* then *student performance* is *has mastered*. This process of taking input such as *score* and processing it through membership functions to determine the "high" score is called fuzzification. Based on the human experts' experience and knowledge about the students' performance, 18 initial rules that are certain have been constructed as shown in Table 4.

#### **3.3 Combining outputs into an output distribution**

The outputs of all of the fuzzy rules must now be combined to obtain one fuzzy output distribution. The output membership functions on the right-hand side of the figure are combined using the fuzzy operator AND to obtain the output distribution shown on the lower right corner of the Fig. 5. For a zero-order Sugeno model, the output level *z* is a constant. The output level *zi* of each rule is weighted by the firing strength *wi* of the rule (Lin and Lu, 1996). For example, for an ∩ rule with input 1 = *x* and input 2 = *y*, the firing strength is as shown in equation (9).

$$w\_i = F\_1(\mathbf{x}) \cap F\_2(\mathbf{y})\tag{9}$$

A Concise Fuzzy Rule Base to Reason Student Performance Based on Rough-Fuzzy Approach 71

rules and known facts. The ANFIS model is proposed to form a complete fuzzy rule bases so

It is necessary to take into consideration the scarcity of data and the style of input space partitions. For example, for a single input problem, usually 10 data points are necessary to come up with a good model (Jang et al., 1997). Details on ANFIS model structure will be

The ANFIS model structure consists of four nodes for input layer, the nodes of hidden layer and one node for output layer as presented in Fig. 6. The input layer represents the antecedent part of the fuzzy rule, which is the student's learning behavior such as the scores (*S*) earned, the time (*T*) spent, the attempts (*A*), and helps (*H*); the output layer represents the consequent part of the rule, i.e. the student's performance (*P*). The size of the hidden

In this work, the ANFIS model is trained with 18 fuzzy rules obtained from the human expert. These rules are considered as the rules that are certain. After that, 81 potential fuzzy rules are used for testing the network that represent the 3 3 3 3 rule antecedents.

From the Fig. 6, every nodes of the same layer have similar functions. Layer 1 is the input

layer and the neurons in this layer simply pass external crisp signals to Layer 2.

that all possible input conditions of the fuzzy rules are being generated.

described in section 4.1.

**4.1 ANFIS model structure** 

layer is determined experimentally.

Fig. 6. ANFIS model structure


Where:

*F1* and *F2* are the membership functions for input 1 and 2, respectively

Table 4. Initial fuzzy rules determine by human experts

#### **3.4 Defuzzification of output distribution**

The input for the defuzzification process is a fuzzy set and the output is a single number crispness recovered from fuzziness. Given a fuzzy set that encompasses a range of output values, we need to return one number, thereby moving from a fuzzy set to a crisp output. The final output of the system is the weighted average of all rule outputs, computed as in equation (10).

$$Final\ Output = \frac{\sum\_{l=1}^{N} w\_{l}z\_{l}}{\sum\_{l=1}^{N} w\_{l}}\tag{10}$$

Finally, all the outputs of datasets for reasoning of the student's performance in the human expert FIS have been recorded.

Next section describes the ANFIS approach to form a complete fuzzy rule base to solve the problem of incomplete and vague decisions made by human.
