Linguistic frequent pattern mining using a compressed structure

Abstract

Traditional association-rule mining (ARM) considers only the frequency of items in a binary database, which provides insufficient knowledge for making efficient decisions and strategies. The mining of useful information from quantitative databases is not a trivial task compared to conventional algorithms in ARM. Fuzzy-set theory was invented to represent a more valuable form of knowledge for human reasoning, which can also be applied and utilized for quantitative databases. Many approaches have adopted fuzzy-set theory to transform the quantitative value into linguistic terms with its corresponding degree based on defined membership functions for the discovery of FFIs, also known as fuzzy frequent itemsets. Only linguistic terms with maximal scalar cardinality are considered in traditional fuzzy frequent itemset mining, but the uncertainty factor is not involved in past approaches. In this paper, an efficient fuzzy mining (EFM) algorithm is presented to quickly discover multiple FFIs from quantitative databases under type-2 fuzzy-set theory. A compressed fuzzy-list (CFL)-structure is developed to maintain complete information for rule generation. Two pruning techniques are developed for reducing the search space and speeding up the mining process. Several experiments are carried out to verify the efficiency and effectiveness of the designed approach in terms of runtime, the number of examined nodes, memory usage, and scalability under different minimum support thresholds and different linguistic terms used in the membership functions.

Appendix

Lemma 1

For an termset X, if Sup(X) or rSup(X) is less than the minimum support threshold, then any supersets (extension) of X is not multiple fuzzy frequent pattern and should be pruned.

Proof

∀ transaction $T\supseteq X^{\prime }$,

• $\because$

• $X^{\prime }$ is an extension of X, $(X^{\prime } - X) = (X^{\prime }/X)$, we can obtain that $X\subseteq X^{\prime }\subseteq T\Rightarrow (X^{\prime }/X)\subseteq (T/X)$,

• $\therefore$

• $fv(X^{\prime }, T) = fv(X, T)\cup fv((X^{\prime } - X), T) = min(fv(X, T), fv(X^{\prime }/X, T))\leq fv(X, T)$ and $min(fv(X, T), fv(X^{\prime }/X, T))\leq fv(X^{\prime }/X, T) = rmrfv(X, T)$.

Suppose that X.tids denotes the set of tids of X,

• $\because$

• $X\subseteq X^{\prime }\Rightarrow X^{\prime }.tids\subseteq X.tids$,

• $\therefore$

• $\frac {{\sum }_{id(T)\in X^{\prime }.tids}fv(X^{\prime }, T)}{N}\leq \frac {{\sum }_{id(T)\in X.tids}fv(X, T)}{N}\Rightarrow Sup(X) < minSup$.

Furthermore, we can obtain that $\frac {{\sum }_{id(T)\in X^{\prime }.tids}rmrfv(X^{\prime }, T)}{N}\leq \frac {{\sum }_{id(T)\in X.tids}rmrfv(X, T)}{N}\Rightarrow rSup(X) < minSup$. □

Lemma 2

For a termset X, if Sup(X) or relative remaining support rSup(X) is less than the minimum support threshold, then any supersets (extension) of X is not a MFFP and should be discarded.

Proof

• $\because$

• $X\subseteq X^{\prime }\Rightarrow X^{\prime }.tids\subseteq X.tids$,

• $\therefore$

• $Sup(X^{\prime }) = \frac {{\sum }_{id(T)\in X.tids}fv(X^{\prime }, T)}{N} = \frac {{\sum }_{id(T)\in X^{\prime }.tids}min(fv(X, T), fv(X^{\prime }/X, T)}{N}\\ \leq \frac {{\sum }_{id(T)\in X^{\prime }.tids}min(fv(X, T), rmrfv(X, T)}{N} = \frac {{\sum }_{id(T)\in Q^{\prime }}fv(X, T) +{\sum }_{id(T)\in Q^{\prime \prime }}rmrfv(X, T)}{N}= rSup(X)\leq minSup$.

Note that suppose $Q^{\prime }\cup Q^{\prime \prime } = X^{\prime }.tids$ and $Q^{\prime }\cap Q^{\prime \prime } = \null$, $T\in Q^{\prime }, fv(X, T) < rmrfv(X, T)$, and $T\in Q^{\prime }, fv(X, T)\geq rmrfv(X, T)$. □