Fuzzification, Rule Propagation and Defuzzification

Initially, in the fuzzification process (Hopgood 2000), the crisp inputs for the variables SMS and D are converted to the corresponding fuzzy values, using the membership functions (Figure 6.2). If the fuzzy values are readily available (for example, from the previous calculations) the fuzzification process will not be necessary. Thereafter, in the rule propagation process, the relevant fuzzy rules are identified from Table 6.2, using the fuzzy values of SMS and D. These are then systematically applied to obtain fuzzy values for the output variable PAS. Finally, in the defuzzification process, the appropriate PAS level (or the crisp value for PAS) can be determined.

For example, a learner with SMS=65 failed a question with D=35.

In the fuzzification process, according to the fuzzy membership function given in Figure 6.2, the relevant degree of membership can be determined:

For Mental State (SMS):

SMS = 65 Æ

µ

Strong (65) =0.25,

µ

Medium (65) =0.75; and, for Difficulty (D):

The relevant fuzzy rules are given in the rows of Table 6.2. Since the answer is incorrect, it is only necessary to consider column P = Incorrect. For example, the first row represents the following rule:

IF Strength of Mental State (SMS) is Strong AND Difficulty Level (D) of the test is Low AND Answer is Incorrect

THEN

Learner needs a PASwL1 type pedagogical action – they might have made a careless mistake (slip), and therefore, they may be given a chance to try again.

Similarly, further rules can be formulated from the other rows of the Table 6.2 It can be noted, the fuzzy AND rule is:

µ

X AND Y (c) = min (

µ

X (c),

µ

Y (c)). and the fuzzy OR rule is:

µ

X OR Y (c) = max (

µ

X (c),

µ

Y (c)).

For a complex if-then rule, using AND in the condition part, the confidence of the necessary part will be given by the minimum value of the confidence in the individual statements that make up the condition: similarly for OR rules, the maximum value will be given.

In the rule propagation, the following rules give the intermediate values for the degree of membership for PASw:

SMS(Strong) & D(Low) Æ PASw(PASwL1)

That is,

µ

PASwL1 ( C ) = min (0.25, 0.25) =0.25,

SMS(Strong) & D(Moderate) Æ PASw(PASwL2) That is,

µ

PASwL2 ( C ) = min (0.25, 0.75) =0.25,

SMS(Medium) & D(Low) Æ PASw(PASwL2)

That is,

µ

PASwL2 ( C ) = min (0.75, 0.25) =0.25,

SMS(Medium) & D(Moderate)Æ PASw(PASwL3)

That is,

µ

PASwL3 ( C ) = min (0.75, 0.75) =0.75,

µ

PASwL1 ( C ) = 0.25

µ

PASwL2 ( C ) = max (0.25, 0.25) =0.25

µ

PASwL3 ( C ) = 0.75

Finally, in the defuzzification process, the crisp value C of PASw can be determined, using the fuzzy membership function given in Figure 6.3. However, at this juncture, the Learner model only needs to suggest the appropriate level of pedagogical actions for a particular learner and, therefore, the model only needs to select a PASw level between the five options. In other words, the Learner model needs to select a range, rather than a single numerical value for PAS. Therefore, in defuzzification, it is sufficient to select the PASw level (the linguistic value) with the highest degree of membership. If two PAS levels have an equal degree of membership, the highest PAS level will be selected.

This results in:

max (

µ

PASwL1 ( C ),

µ

PASwL2 ( C ),

µ

PASwL3 ( C ) ) = max (0.25, 0.25, 0.75) = 0.75 The highest degree of membership is 0.75, and the corresponding fuzzy value is PASwL3

Therefore, the required PASw level is: PASwL3

If a learner with SMS=65 fails a question with D=35, the system proposes a third level pedagogical action for the incorrect answers (PASwL3). This means the feedback should include the explanation for each option, such as why it is correct or incorrect and later another MC test, at the same scaffolding stage, will be offered to the learner.

Alternatively, a complex defuzzification process may be used to determine the crisp value of PASw. Being the single numerical representation for the required pedagogical action for a learner in a particular MC test, this value may be stored in the Learner model for later usage. An illustration of the defuzzification process is included here for completeness. The Larsen’s Product Operation Rule (Hopgood 2000) combined with

mirror rule at extremes is used for the defuzzification process in this illustration. Initially

in the defuzzification process, the fuzzy membership function (Figure 6.3) will be adjusted, based on the current possibilities (the calculated degree of membership). In Larsen’s Rule, the membership functions are multiplied by their corresponding possibilities.

the mirror rule at the extremes. At the boundaries (here at 0), the fuzzy set for PASwL1 is reflected on an imaginary mirror. At this time, the centroid method can be used to find the

crisp value. In this example, the balance point along the fuzzy variable axis, for the composite shape of the three shaded triangles in Figure 6.5, will give the defuzzified (crisp) value.

In general, the centroid C along the X axis will be given by the following formula:

=

∑_∑

i i i

A

A

x

C

Where, Ai is the area of a triangle and xi is the distance of its centroid from the origin.

Since the triangles are equilateral, and the bases of the triangles are equal, the crisp value may be calculated by the following formula:

=

∑_∑

i i i

x

PASw

μ

μ

Where,

µ

i denotes the calculated possibilities (degrees of memberships)

Therefore, the crisp value for PASw will be:

= 35 PASw = 0.75 +0.25 + 0.25 0.75*50 +0.25*25 + 0.25*0 12. PasL5 PasL4 PasL3 PasL2 PasL1 0 10 20 30 40 50 60 70 80 90 100 25 75 1 0.25 0.75

Figure 6.5 Defuzzification, using Larsen’s Product Rule

| | | | | | | | | | |

PAS Degree

of Membership

In document Designing CBL systems for complex domains using problem transformation and fuzzy logic : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science at Massey University, Palmerston North, New Z (Page 154-158)