Discovering non-compliant window co-occurrence patterns

Abstract

Given a set of trajectories annotated with measurements of physical variables, the problem of Non-compliant Window Co-occurrence (NWC) pattern discovery aims to determine temporal signatures in the explanatory variables which are highly associated with windows of undesirable behavior in a target variable. NWC discovery is important for societal applications such as eco-friendly transportation (e.g. identifying engine signatures leading to high greenhouse gas emissions). Challenges of designing a scalable algorithm for NWC discovery include the non-monotonicity of popular spatio-temporal statistical interest measures of association such as the cross-K function which renders the anti-monotone pruning based algorithms (e.g. Apriori) inapplicable for such interest measures. In our preliminary work, we proposed two upper bounds for the cross-K function and a top-down multi-parent tracking approach that uses these bounds for filtering out uninteresting candidate patterns and then applies a minimum support (i.e. frequency) threshold as a post-processing step to filter out chance patterns. In this paper, we propose a novel bi-directional pruning approach (BDNMiner) that combines top-down pruning based on the cross-K function threshold with bottom-up pruning based on the minimum support threshold to efficiently mine NWC patterns. Case studies with real world engine data demonstrates the ability of the proposed approach to discover patterns which are interesting to engine scientists. Experimental evaluation on real-world data show that the proposed approach yields substantial computational savings compared to prior work.

Appendix A: Proofs of preliminary results section

Lemma 1

Given an NWC pattern C and a time lag δ, $Upper_{Loc}(|C \overset {\delta }{\bowtie } W_{N}|)$ is an upper bound of $|C \overset {\delta }{\bowtie } W_{N}|$

Proof

For every NWC pattern {S _i } consisting of a single event-sequence where {S _i }⊆ C, 1≤i≤D i m(C), we have |{S _i }|≥|C|, where |{S _i }| and |C| are the cardinality of the patterns {S _i } and C in the time series, respectively. Since {S _i }⊆ C, we also have $|\{S_{i}\} \overset {\delta }{\bowtie } W_{N}| \geq |C \overset {\delta }{\bowtie } W_{N}|$, ∀ 1 ≤i≤D i m(C). Then, $Upper_{Loc}(|C \overset {\delta }{\bowtie } W_{N}|)$ = $\min \limits _{\{S_{i}\} \in C, 1\leq i \leq Dim(C)}(|\{S_{i}\} \overset {\delta }{\bowtie } W_{N}|) \geq |C \overset {\delta }{\bowtie } W_{N}|$. □

Lemma 2

Given an NWC pattern C and a time lag δ, Lower(|C|) is a lower bound of |C|.

Proof

Any superset pattern of C has a cardinality smaller than or equal to C. Therefore, Lower(|C|) = |superset(C)|≤|C|. □

Theorem 1

Given an NWC pattern C and a time lag δ, $UB_{local}(\hat {K}_{C,W_{N}}(\delta ))$ is an upper bound of $\hat {K}_{C,W_{N}}(\delta )$.

Proof

Using Lemmas 1 and 2, we have $\hat {K}_{C,W_{N}}(\delta ) = \frac {T}{|W_{N}|} \times \frac {|C \overset {\delta }{\bowtie } W_{N}|}{|C|} \leq \frac {T}{|W_{N}|} \times \frac {Upper_{Loc}(|C \overset {\delta }{\bowtie } W_{N}|)}{Lower|C|} = UB_{local}(\hat {K}_{C,W_{N}}(\delta ))$ □

Theorem 2

Given an NWC pattern C and a time lag δ, $UB_{lattice}(\hat {K}_{C,W_{N}}(\delta ))$ is an upper bound of $\hat {K}_{C,W_{N}}(\delta )$.

Proof

Since Definition 8 differs from Definition 7 only in the min term being replaced by a max term, the proof of Theorem 2 is straightforward from Theorem 1. □

Lemma 3

Given an NWC pattern C and a time lag δ, $UB_{lattice}(\hat {K}_{C,W_{N}}(\delta ))$ is monotonically decreasing with decreasing Dim(C) if Lower(|C|) is kept monotonically increasing. In other words, given two NWC patterns C and C’ where C’ ⊂ C, then if Lower(|C ^′ |)≥Lower(|C|), then $UB_{lattice}(\hat {K}_{C^{\prime },W_{N}}(\delta )) \leq UB_{lattice}(\hat {K}_{C,W_{N}}(\delta ))$.

Proof

Let C ^′⊂ C where C and C ^′ are two NWC patterns. Then ∀S _i (v _i )∈ C ^′, where 1≤i≤ Dim(C ^′), we have S _i (v _i )∈ C. Therefore, $Upper_{Lat}(|C^{\prime } \overset {\delta }{\bowtie } W_{N}|) = \max \limits _{\{S_{i}\} \in C^{\prime }, 1\leq i \leq Dim(C^{\prime })}(|{S_{i}} \overset {\delta }{\bowtie } W_{N}|) \leq \max \limits _{\{S_{i}\} \in C, 1\leq i \leq Dim(C)}(|{S_{i}} \overset {\delta }{\bowtie } W_{N}|) = Upper_{Lat}(|C \overset {\delta }{\bowtie } W_{N}|)$ (A). Also, since L o w e r(|C|) is kept monotonically increasing as Dim(C) decreases, then L o w e r(|C ^′|)≥L o w e r(|C|) (B). From (A) and (B), we have $UB_{lattice}(\hat {K}_{C,W_{N}}(\delta )) = \frac {T}{|W_{N}|} \times \frac {Upper_{Lat}(|C \overset {\delta }{\bowtie } W_{N}|)}{Lower(|C|)} \geq \frac {T}{|W_{N}|} \times \frac {Upper_{Lat}(|C^{\prime } \overset {\delta }{\bowtie } W_{N}|)}{Lower(|C^{\prime }|)}$ = $UB_{lattice}(\hat {K}_{C^{\prime },W_{N}}(\delta ))$ □