Concept-drift Patterns for Fuzzy Association Rules

In this part, , we generalize the original concept-drift patterns in [24] to quantitative transactions. The following different concept drift patterns of fuzzy

sj

association rules are considered. The first one is the fuzzy emergent patterns in which both the conditional and the consequent parts between two fuzzy association rules from two different databases are the same but the fuzzy support values of the conditional or consequent parts are different. The second one is the unexpected change for fuzzy association rules. It considers two rules in different databases with similar change of the condition parts, but their consequent parts are quite different. The last one also considers unexpected change for fuzzy association rules, but it considers the two rules with similar consequent parts and quite different conditional parts. They are described below. The added and perished concept drift patterns are not considered in the paper.

(A) The fuzzy emerging change

In fuzzy emerging patterns, both the conditional and the consequent parts between two fuzzy association rules from two different databases are the same but the fuzzy support values of the conditional or consequent parts are different. There are three kinds of fuzzy support change for fuzzy emerging patterns. The first case is that the fuzzy support values of the conditional terms between two fuzzy association rules are similar but the fuzzy support values of the consequent terms are different. The second case is that the fuzzy support values of the conditional terms are different but the fuzzy support values of the consequent terms are similar. The third case is that the fuzzy support values of the conditional terms and the consequent terms are both different. Two rules

with the similar support values in both the consequent and conditional parts are not considered since they do not change significantly and are thus not emerging patterns.

In order to calculate fuzzy concept-drift patterns, the following formula modified from [24] is adopted to estimate the similarity ps of the premise (conditional) part in two fuzzy association rules:

𝑐𝑐𝑠𝑠 = �

ℓ_{𝑖𝑖𝑗𝑗}× ∑_{𝑗𝑗∈𝐴𝐴𝑖𝑖𝑖𝑖}𝑥𝑥_{𝑖𝑖𝑗𝑗𝑗𝑗}

�𝐴𝐴𝑖𝑖𝑗𝑗� , 𝑖𝑖𝑖𝑖�𝐴𝐴𝑖𝑖𝑗𝑗� ≠ 0 0, 𝑖𝑖𝑖𝑖�𝐴𝐴𝑖𝑖𝑗𝑗� = 0

(4-7)

The notation in this formula is briefly explained as follows:

𝑐𝑐𝑠𝑠_{𝑖𝑖𝑗𝑗}: The degree of premise similarity between two rules rit and rjt+k ,0 ≤ 𝑠𝑠𝑖𝑖𝑗𝑗 ≤ 1,

ℓ_{𝑖𝑖𝑗𝑗}: The degree of attribute match of the premise part between two rules rit

and rjt+k,

�𝐴𝐴_{𝑖𝑖𝑗𝑗}�: The number of attributes common to both conditional parts of r^it and rjt+k,

𝑥𝑥_{𝑖𝑖𝑗𝑗𝑗𝑗}: The degree of attribute value (linguistic term) match of the k-th matching

attribute in 𝐴𝐴𝑖𝑖𝑗𝑗.

In the above formula, ℓ_{𝑖𝑖𝑗𝑗} can be defined by the following formula [24]:

ℓ𝑖𝑖𝑗𝑗 = �𝐴𝐴𝑖𝑖𝑗𝑗�

max��𝑋𝑋_𝑖𝑖^𝑡𝑡�, �𝑋𝑋_𝑗𝑗^{𝑡𝑡+𝑗𝑗}�� , (4-8) where |𝑋𝑋_𝑖𝑖^𝑡𝑡| and �𝑋𝑋_𝑗𝑗^{𝑡𝑡+𝑗𝑗}� are the numbers of attributes in the premise parts of r^it and rjt+k, respectively. 𝑥𝑥_{𝑖𝑖𝑗𝑗𝑗𝑗} represents the degree of attribute value (linguistic term) match

of the k-th matching attribute in 𝐴𝐴_{𝑖𝑖𝑗𝑗}. It can be defined as follows by considering fuzzy

match:

𝑥𝑥𝑖𝑖𝑗𝑗𝑗𝑗 = 1 − �𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖_𝑐𝑐𝑖𝑖𝑠𝑠𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑖𝑖𝑖𝑖𝑗𝑗

𝑖𝑖_𝑗𝑗− 1 �

𝛼𝛼

, (4-9)

where nk is the number of membership functions the k-th attribute, 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖_𝑐𝑐𝑖𝑖𝑠𝑠𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑖𝑖 is the number of intervals between the linguistic values of the two

same attributes in the two rules rit and rjt+k, and α is a parameter controlling the effect of different linguistic values. For example, if an attribute has only three linguistic terms:

high, middle, low. Then the value of interval_distance between high and middle is 1, and between high and low is 2.

After the premise similarity of fuzzy rules is defined, the similarity cs of the

consequent parts in two fuzzy association rules is designed as follows:

𝑐𝑐𝑠𝑠𝑖𝑖𝑗𝑗 = 𝑐𝑐𝑖𝑖𝑗𝑗 × �1 − �𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖_𝑐𝑐𝑖𝑖𝑠𝑠𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑖𝑖𝑖𝑖𝑗𝑗

𝑖𝑖_𝑗𝑗− 1 �

𝛼𝛼

�, (4-10)

where cij = 1 if the consequent attributes (not including values) in the two rules rit

and rjt+k are the same, and cij = 0 otherwise.

If both the psij and sij values are equal to or larger than a predefined threshold value T, then the supports of the two rules are then checked according to the three cases

mentioned above.

If the rule similarity measure is less than the threshold, it means the conditional

terms are similar in these two fuzzy association rules and if the rule similarity measure is large than the threshold, it means the conditional terms are quite different.

Below is an example to show the concept above. Assume the membership functions for the purchased amount of apples are the same as those in Figure 4.1. Also assume there are two fuzzy association rules from two different databases are shown in Table 4.3.

Table 4.3: The first case of an emerging pattern for two fuzzy association rules Database Fuzzy Association Rules

D^t (Apple.Low,0.5), (Mike.High,0.6) → (Bread.High,0.7) D^t+k (Apple.Low,0.5), (Mike.High,0.6) → (Bread.High,0.5)

In Table 4.3, both the premise similarity and the conditional similarity between the two association rules are very similar (actually the same), thus we judge their fuzzy support change. The fuzzy support values at the premise parts of the two rules are the same, but the value at the consequent parts of the two rules are different with 0.2. If 0.2 is larger than the threshold, then it is the first case of an emerging pattern.

The two fuzzy association rules in Table 4.4 are another example. Both the premise similarity and the conditional similarity between the two association rules are very similar (actually the same), thus we judge their fuzzy support change. The fuzzy support values at the premise parts of the two rules are different, but the values at the consequent parts of the two rules are the same. It is the second case of an emerging pattern.

Table 4.4: The second case of an emerging pattern for two fuzzy association rules

Database Fuzzy Association Rules

D^t (Apple.Low,0.5), (Mike.High,0.9) → (Bread.High,0.7) D^t+k (Apple.Low,0.5), (Mike.High,07) → (Bread.High,0.7)

At last, Table 4.5 shows an example for an emerging pattern for two fuzzy association rules. Both the premise similarity and the conditional similarity between the two association rules are the same. The fuzzy support values of the premise and consequent terms are different. It is thus the third case of an emerging pattern.

Table 4.5: The third case of an emerging pattern for two fuzzy association rules Database Fuzzy Association Rules

D^t (Apple.Low,0.4), (Mike.High,1) → (Bread.High,0.7) D^t+k (Apple.Low,0.4), (Mike.High,0.6) → (Bread.High,0.9) (B) The fuzzy unexpected change

There are two kinds of fuzzy concept drift patterns for unexpected change. The first one is that the premise similarity between the two association rules is very high, but the consequent similarity of the two rules is not high. The second one is the contrary.

That is, the consequent similarity between the two association rules is very high, but the premise similarity of the two rules is not high. It can be judged by a threshold in a way similar to that for emerging patterns.

Below is an example to show the first case. In Table 4.6, there are two fuzzy association rules. The premise similarity of the two rules is high, which is 1. If there are only three membership functions for Bread, then the consequent similarity of the two rules is 0.5. If the threshold is set at 0.6, then they are thought of as quite different.

It is then the first case of the unexpected change.

Table 4.6: The first case of the unexpected change for two fuzzy association rules

Database Fuzzy Association Rules

D^t (Apple.Low,0.4), (Mike.High,0.9) → (Bread.Middle,0.4) D^t+k (Apple.Low,0.4), (Mike.High,0.9) → (Bread.Low,0.2)

Finally, an example is given to show the concept drift which has different premise terms but similar consequent terms. Table 4.7 shows an example for this case. In this case, the consequent similarity of the two rules is 1, but the premise similarity of the two rules is 0.5. If the threshold is set at 0.6, then they are thought of as quite different.

It is then the second case of the unexpected change

Table 4.7: The second case of the unexpected change for two fuzzy association rules

Database Fuzzy Association Rules

D^t (Apple.Middle,0.6), (Mike.High,0.9) → (Bread.High,0.7) D^t+k (Apple.High,0.8), (Mike.Low,0.3) → (Bread.High,0.7)