VM-NSP: Vertical Negative Sequential Pattern Mining with Loose Negative Element Constraints

Negative sequential patterns (NSPs) capture more informative and actionable knowledge than classic positive sequential patterns (PSPs) due to the involvement of both occurring and nonoccurring behaviors and events, which can contribute to many relevant applications. However, NSP mining is nontrivial, as it involves fundamental challenges requiring distinct theoretical foundations and is not directly addressable by PSP mining. In the very limited research reported on NSP mining, a negative element constraint (NEC) is incorporated to only consider the NSPs composed of specific forms of elements (containing either positive or negative items), which results in many valuable NSPs being missed. Here, we loosen the NEC (called loose negative element constraint (LNEC)) to include partial negative elements containing both positive and negative items, which enables the discovery of more flexible patterns but incorporates significant new learning challenges, such as representing and mining complete NSPs. Accordingly, we formalize the LNEC-based NSP mining problem and propose a novel vertical NSP mining framework, VM-NSP, to efficiently mine the complete set of NSPs by a vertical representation (VR) of each sequence. An efficient bitmap-based vertical NSP mining algorithm, bM-NSP, introduces a bitmap hash table--based VR and a prefix-based negative sequential candidate generation strategy to optimize the discovery performance. VM-NSP and its implementation bM-NSP form the first VR-based approach for complete NSP mining with LNEC. Theoretical analyses and experiments confirm the performance superiority of bM-NSP on synthetic and real-life datasets w.r.t. diverse data factors, which substantially expands existing NSP mining methods toward flexible NSP discovery.