A systematic approach to the assessment of fuzzy association rules

Abstract

In order to allow for the analysis of data sets including numerical attributes, several generalizations of association rule mining based on fuzzy sets have been proposed in the literature. While the formal specification of fuzzy associations is more or less straightforward, the assessment of such rules by means of appropriate quality measures is less obvious. Particularly, it assumes an understanding of the semantic meaning of a fuzzy rule. This aspect has been ignored by most existing proposals, which must therefore be considered as ad-hoc to some extent. In this paper, we develop a systematic approach to the assessment of fuzzy association rules. To this end, we proceed from the idea of partitioning the data stored in a database into examples of a given rule, counterexamples, and irrelevant data. Evaluation measures are then derived from the cardinalities of the corresponding subsets. The problem of finding a proper partition has a rather obvious solution for standard association rules but becomes less trivial in the fuzzy case. Our results not only provide a sound justification for commonly used measures but also suggest a means for constructing meaningful alternatives.

Appendices

Appendix A: Additive generator of a t-norm

An additive generator of a t-norm ⊗ is a mapping ${f}:{[0,1]}\rightarrow{[0,\infty]}$ such that

• f is continuous and monotone decreasing,

• $f(1)= 0$,

• $\alpha \otimes \beta = f^{(-1)}(f(\alpha) + f(\beta))$ for all $0 \leq \alpha,\beta \leq 1$,

where $f^{(-1)}(\cdot)$ denotes the pseudo-inverse of $f(\cdot)$, i.e.,

$$f^{(-1)}(x) = \left\{\begin{array}{c@{\quad}l} y & \text{ if } 0 \leq x \leq f(0) \text{ and } y = f(x)\\ 0 & \text{ if } f(0) < x \leq \infty\end{array}\right.$$

for all $x \geq 0$. It can be shown that an additive generator does exist for each Archimedian t-norm, and that this generator is unique up to a positive multiplicative constant (Ling, 1965). (A continuous t-norm ⊗ is Archimedian if $\alpha \otimes \alpha < \alpha$ for all $0 < \alpha < 1$.)

Appendix B: Ordinal sums

Suppose n t-norms $\otimes^1 \cdots \otimes^n$ to be given. Moreover, let $u_1 \ldots u_n$ and $v_1 \ldots v_n$ be numbers such that $0 \leq u_1 < v_1 \leq u_2 < v_2 \leq \dots \leq u_n < v_n \leq 1$. The ordinal sum ⊗ of $\otimes^1 \cdots \otimes^n$ is given by

$$\alpha \otimes \beta \,\,{\stackrel{\rm df}{=}} \left\{\begin{array}{l@{\quad}l} u_i + (v_i-u_i) \left(\displaystyle \frac{\alpha-u_i}{v_i-u_i} \otimes^i \frac{\beta-u_i}{v_i-u_i} \right) & \text{ if } \alpha, \beta \in [u_i,v_i] \\ [0.2cm] \min(\alpha, \beta) & \text{ otherwise}\end{array} \right. ,$$

for all $0 \leq \alpha , \beta \leq 1$.

As an example, the t-norm (15) is obtained for $n=1$, $\otimes^1 =$ product, $u_1=0, v_1=\gamma$.