A Generic Scheme for the Design of Efficient On-Line Algorithms for Lattices

A major issue with large dynamic datasets is the processing of small changes in the input through correspondingly small rearrangements of the output. This was the motivation behind the design of incremental or on-line algorithms for lattice maintenance, whose work amounts to a gradual construction of the final lattice by repeatedly adding rows/columns to the data table. As an attempt to put the incremental trend on strong theoretical grounds, we present a generic algorithmic scheme that is based on a detailed analysis of the lattice transformation triggered by a row/column addition and of the underlying sub-structure. For each task from the scheme we suggest an efficient implementation strategy and put a lower bound on its worst-case complexity. Moreover, an..

A Formal Concept Analysis Approach to Association Rule Mining: The QuICL Algorithms

Association rule mining (ARM) is the task of identifying meaningful implication rules exhibited in a data set. Most research has focused on extracting frequent item (FI) sets and thus fallen short of the overall ARM objective. The FI miners fail to identify the upper covers that are needed to generate a set of association rules whose size can be exploited by an end user. An alternative to FI mining can be found in formal concept analysis (FCA), a branch of applied mathematics. FCA derives a concept lattice whose concepts identify closed FI sets and connections identify the upper covers. However, most FCA algorithms construct a complete lattice and therefore include item sets that are not frequent. An iceberg lattice, on the other hand, is a concept lattice whose concepts contain only FI sets. Only three algorithms to construct an iceberg lattice were found in literature. Given that an iceberg concept lattice provides an analysis tool to succinctly identify association rules, this study investigated additional algorithms to construct an iceberg concept lattice. This report presents the development and analysis of the Quick Iceberg Concept Lattice (QuICL) algorithms. These algorithms provide incremental construction of an iceberg lattice. QuICL uses recursion instead of iteration to navigate the lattice and establish connections, thereby eliminating costly processing incurred by past algorithms. The QuICL algorithms were evaluated against leading FI miners and FCA construction algorithms using benchmarks cited in literature. Results demonstrate that QuICL provides performance on the order of FI miners yet additionally derive the upper covers. QuICL, when combined with known algorithms to extract a basis of association rules from a lattice, offer a best known ARM solution. Beyond this, the QuICL algorithms have proved to be very efficient, providing an order of magnitude gains over other incremental lattice construction algorithms. For example, on the Mushroom data set, QuICL completes in less than 3 seconds. Past algorithms exceed 200 seconds. On T10I4D100k, QuICL completes in less than 120 seconds. Past algorithms approach 10,000 seconds. QuICL is proved to be the best known all around incremental lattice construction algorithm. Runtime complexity is shown to be O(l d i) where l is the cardinality of the lattice, d is the average degree of the lattice, and i is a mean function on the frequent item extents