Periodic pattern mining aims to identify patterns that recur at regular or near-regular intervals in temporal data. Early work introduced the concept of periodic frequent patterns and proposed pruning-based strategies to efficiently discover such patterns in transactional databases [14]. Subsequent studies demonstrated the usefulness of periodic pattern mining in real-world temporal applications, including public transportation analysis and urban mobility modeling [11,15].

While these approaches effectively capture temporal regularity, they are largely designed for single-sequence settings. In most cases, periodicity is evaluated within one long sequence, and the question of whether a pattern exhibits consistent periodic behavior across multiple sequences is not explicitly addressed. Moreover, periodic pattern mining is commonly based on binary item occurrence, without considering quantitative information associated with items. Even studies that extend periodic analysis to multiple sequences [16] typically focus on temporal aspects alone and do not incorporate quantitative constraints. This limits their ability to support applications where both periodicity and quantitative significance are important.

Another relevant line of research focuses on mining patterns based on quantitative or utility-based criteria. Early work on quantitative association rules established mechanisms for evaluating item importance using numerical values rather than frequency alone [18]. This direction later evolved into weighted and high-utility pattern mining, enabling the discovery of patterns that reflect contribution or utility instead of pure occurrence counts [19]. More recently, high-utility sequential pattern mining has been explored in uncertain or dynamic environments, illustrating the flexibility of quantitative evaluation in sequential data analysis [13].

Despite their effectiveness, quantitative and high-utility approaches typically do not consider temporal periodicity. Patterns are evaluated mainly from the perspective of contribution or importance, and sequences are often processed independently. As a result, these methods cannot ensure that a pattern is both quantitatively significant and periodically recurrent across multiple sequences, which is a key requirement in many real-world scenarios.

Pattern discovery across multiple sequences has also received increasing attention. Methods in this category aim to identify patterns that occur in a sufficient number of sequences, thereby capturing population-level or system-level behaviors. Representative examples include studies on correlated periodic patterns in multiple sequences [16] and approaches that combine periodicity with utility measures in multi-sequence settings [17]. These works move beyond single-sequence analysis and highlight the importance of cross-sequence consistency.

Despite these advances, the proposed MHQPFPS algorithm differs from the two most closely related methods, MRCPPS [16] and MHUPFPS [17], in several important aspects. MRCPPS focuses on correlated periodic pattern mining over multiple sequences, but it does not take into account the quantitative contribution of patterns, since pattern significance is evaluated mainly from occurrence information and correlation. MHUPFPS extends periodic pattern mining to the high-utility setting, but it uses absolute utility values rather than a normalized measure. Therefore, pattern significance may be difficult to compare directly across sequences with different lengths or quantitative scales. In contrast, MHQPFPS evaluates each pattern by a normalized quantitative ratio bounded in [0,1], which provides a more comparable measure across heterogeneous sequences. A second difference lies in the treatment of periodicity regularity. While MRCPPS considers a standard-deviation-related constraint in the rare correlated periodic pattern setting, and MHUPFPS uses maximum periodicity bounds, neither method combines period stability control with normalized quantitative evaluation in the multi-sequence frequent pattern setting considered here. MHQPFPS introduces the period standard deviation threshold maxStd to further distinguish patterns with relatively stable periodic behavior from those that only satisfy the maximum period requirement. A third difference is related to the pruning strategy. MHQPFPS derives a sequence-level upper bound for the cross-sequence quantitative ratio of pattern extensions. This upper bound supports safe pruning during the depth-first search process and helps improve mining efficiency. While MRCPPS also employs a sequence-level upper bound (termed upBondRa) based on support and periodicity conditions, its upper bound additionally incorporates the bond correlation constraint and does not extend to quantitative measures, since MRCPPS does not incorporate quantitative evaluation. MHUPFPS does not report a comparable upper-bound pruning strategy.

Overall, these differences indicate that existing multi-sequence studies only partially cover the problem addressed in this work, which motivates the summary and gap analysis presented next.

Table 1 summarizes the main distinctions between MHQPFPS and the most closely related methods from five aspects: support for multiple sequences, periodicity consideration, quantitative measure, period stability constraint, and upper-bound-based pruning.

As shown in Table 1, existing studies cover only part of the problem addressed in this work. Some methods model periodicity, whereas others emphasize quantitative importance or cross-sequence behavior. However, these aspects are usually considered separately rather than within a unified mining framework. As a result, prior studies do not jointly address periodic recurrence within each sequence, quantitative contribution in a normalized form, and consistency across multiple sequences in the frequent-pattern setting considered here.

These observations motivate the present study. Accordingly, this work formulates the problem of discovering High-Quantitative Periodic Frequent Patterns across multiple sequences, where periodicity, quantitative contribution, and cross-sequence consistency are considered jointly. To solve this problem efficiently, we further develop an algorithmic framework based on HQPFPS-list and upper-bound-guided pruning.

MHQPFPS follows a depth-first pattern-growth strategy, similar to classical prefix-based mining frameworks. The mining process starts from all single-item patterns and recursively extends a current prefix pattern P by appending items with lexicographically larger identifiers. For a pattern P={i1,i2,…,ik} with items ordered lexicographically such that i1<i2<⋯<ik, we denote by last(P) the largest item ik. The extension of P therefore considers only items i satisfying i>last(P), ensuring that each itemset is enumerated exactly once. For each extended pattern P∪{i}, an HQPFPS-list is constructed to efficiently compute all required statistics of the pattern in each sequence of the database.

A key observation underlying the pruning strategy is the anti-monotonicity of the support ratio defined in Section 3. Specifically, for any two patterns X and Y such that Y⊇X, and for any sequence S, the following property holds:supRa(Y,S)≤supRa(X,S).

This property implies that, as a pattern is extended, its ability to satisfy support-related and periodicity-related constraints cannot improve. Based on this observation, an upper bound on the achievable cross-sequence ratio of any pattern extension can be derived, which enables the algorithm to prune unpromising branches early and significantly reduce the search space.

Definition 10 (Candidate Sequence Set): For a pattern X, its candidate-supporting sequence set is defined as candSeq(X)={S∈D∣supRa(X,S)≥minSupRa ∧ maxPer(X,S)≤maxPr}.

Remark: Only supRa(X,S) and maxPer(X,S) are included in candSeq(X), since both measures satisfy anti-monotonicity with respect to pattern extension. In contrast, quRa(X,S) and stanDev(X,S) are not anti-monotonic and therefore cannot be used to derive valid upper bounds for pruning. This will be described later in the pruning procedure

Definition 11 (Upper-Bound Sequence Ratio): The upper bound on the high-quantitative periodic sequence ratio achievable by a pattern X is defined as upRa(X)=|candSeq(X)||D|.

Lemma 1 (Anti-Monotonicity Properties): For any two patterns X⊆Y and any sequence S, the following properties hold: 1.

supRa(Y,S)≤supRa(X,S);

Proof: (1) Since Y⊇X, every occurrence of Y necessarily contains X, which implies TR(Y,S)⊆TR(X,S). Therefore, supRa(Y,S)=|TR(Y,S)||S|≤|TR(X,S)||S|=supRa(X,S).

(2) As TR(Y,S) is a subsequence of TR(X,S), the distances between consecutive occurrences of Y cannot be smaller than those of X. Consequently, maxPer(Y,S)≥maxPer(X,S).◻

Theorem 1 (Safe Upper Bound): For any pattern extension Y⊇X, the following inequality holds:hqSeqRa(Y)≤upRa(X),where hqSeqRa(⋅) is defined in Section 3.

Proof: By Lemma 1, for any sequence S∈D, supRa(Y,S)≤supRa(X,S)andmaxPer(Y,S)≥maxPer(X,S).

Hence, any sequence that can satisfy both supRa(Y,S)≥minSupRa and maxPer(Y,S)≤maxPr must already belong to candSeq(X). It follows that hqPrSeq(Y)⊆candSeq(X).

Dividing both sides by |D| yields hqSeqRa(Y)=|hqPrSeq(Y)||D|≤|candSeq(X)||D|=upRa(X).◻

Corollary 1 (Pruning Condition): If upRa(X)<minSeqRa,then for any extension Y⊇X, hqSeqRa(Y)<minSeqRa.

In this case, the entire search branch rooted at pattern X can be safely pruned (Algorithm 1, Lines 5–6.)

To efficiently evaluate candidate patterns across multiple sequences, MHQPFPS employs a vertical representation structure referred to as the HQPFPS-list. This structure is designed to compactly maintain all information required for computing support, periodicity, and quantitative measures, thereby avoiding repeated scans of the original database.

For a pattern X and a sequence S in which X occurs, the HQPFPS-list stores the following information:

sid: the identifier of sequence S;

For a pattern X, its HQPFPS-list is denoted byPFL(X)={(sid,TR,q-list,supRa,maxPer,stanDev,quRa)}.

All entries in PFL(X) are computed in accordance with the formal definitions given in Section 3. When a pattern is extended during the mining process, the corresponding HQPFPS-list can be constructed and updated directly from existing lists, without rescanning the original database. This property is essential for ensuring the efficiency of the proposed depth-first search framework.

Based on the pruning strategy and the HQPFPS-list structure introduced above, the MHQPFPS algorithm enumerates all High-Quantitative Periodic Frequent Patterns from a multi-sequence database. The overall procedure is outlined in Algorithm 3.

Given a database D and user-specified thresholds (minSupRa,maxPr,maxStd,minHqRa,minSeqRa), the algorithm begins with a single scan of D to construct the HQPFPS-list PFL({i}) for each 1-itemset {i}. For every item i∈I and sequence S∈D, the statistics supRa({i},S), maxPer({i},S), stanDev({i},S), and quRa({i},S) are computed directly from PFL({i}). These values are then used to derive the candidate sequence set candSeq({i}) and the corresponding upper bound upRa({i})=|candSeq({i})|/|D|.

A 1-itemset {i} is considered promising only if upRa({i})≥minSeqRa (Algorithm 3, Line 5). For such patterns, the procedure HQPFP-CHECK (Algorithm 4 Lines 1–12) is invoked to verify whether {i} satisfies all single-sequence constraints (Definition 8) as well as the cross-sequence constraint (Definition 9). If {i} is an HQPFPS, it is added to the global result set. The algorithm then calls the recursive procedure DFS-SEARCH (Algorithm 1) to further extend {i}.

The DFS-SEARCH procedure follows a depth-first pattern-growth strategy. Given a current prefix pattern P, it generates extensions by appending items that are lexicographically larger than the last item in P. For each extension Y=P∪{i}, a new HQPFPS-list PFL(Y) is constructed by combining PFL(P) and PFL({i}) using Algorithm 2. Since both lists record transaction identifiers, PFL(Y) can be constructed efficiently through intersection operations, without scanning the original database.

From PFL(Y), the algorithm computes the candidate sequence set candSeq(Y) and the corresponding upper bound upRa(Y). If upRa(Y)<minSeqRa, Corollary 1 guarantees that no extension of Y can satisfy the HQPFPS condition, and the entire branch rooted at Y is pruned (Algorithm 1, Lines 5–6). Otherwise, HQPFP-CHECK (Algorithm 4) is applied to Y. If Y satisfies all constraints, it is reported as an HQPFPS, and DFS-SEARCH is recursively invoked on Y.

Algorithm 2 details the construction of PFL(Y) from PFL(P). For each record (sid,TR,q-list,supRa,maxPer,stanDev,quRa) in PFL(P), the transaction list TR(Y,S) is obtained by intersecting TR(P,S) and TR({i},S) (Algorithm 2, Line 4). If TR(Y,S) is non-empty, the sequence-level statistics supRa(Y,S), maxPer(Y,S), stanDev(Y,S), and quRa(Y,S) are recomputed, a new q-list is generated, and the corresponding record is inserted into PFL(Y).

Finally, the procedure HQPFP-Check (Algorithm 4) counts the number of sequences in which a pattern satisfies all single-sequence constraints using PFL(X). Dividing this count by |D| yields hqSeqRa(X), which determines whether X is an HQPFPS.

All experiments were carried out in a Python 3 environment using PyCharm on a MacBook Air running macOS, equipped with an Apple M4 processor and 16 GB of unified memory. The MHQPFPS algorithm was implemented in a single-threaded manner to provide a fair and controlled evaluation of its computational characteristics.

Four datasets were used in the experimental study, including three real-world datasets—FIFA, Bike, and Leviathan—and one synthetic dataset, T23L68KD15K. All datasets were obtained from the SPMF Library. The real-world datasets represent typical application scenarios of sequential and periodic pattern mining, while the synthetic dataset allows controlled adjustment of data characteristics, facilitating a systematic evaluation of algorithm behavior under different conditions.

The primary goal of the experiments is to investigate how the parameters minSupRa and minSeqRa influence both the efficiency and the output of MHQPFPS. We denote by MHQPFPS(x,y,z) an instance of the algorithm configured with minSeqRa=x, minHqRa=y, and maxPr=z. Unless otherwise specified, the parameter maxStd is fixed throughout the experiments, and minSupRa is varied as reported in the corresponding figures.

To provide a reference point for comparison, a baseline configuration is defined using relatively loose thresholds, allowing a large number of high-quantitative periodic patterns to be generated: maxPr = 20, maxStd = 10, minHqRa = 0.01, minSeqRa = 0. All other experimental configurations are evaluated relative to this baseline in terms of runtime, number of discovered patterns, and memory consumption.

Runtime: Fig. 2 reports the runtime of MHQPFPS under different combinations of minSupRa and minSeqRa across the four datasets. Overall, runtime decreases steadily as minSupRa increases, because a higher support-ratio threshold reduces the number of candidate patterns and correspondingly shrinks the effective search space.

This behavior is most pronounced on the Bike and T23L68KD15K datasets, where the runtime curves decrease monotonically with minSupRa under configurations such as MHQPFPS (0.001,0.15,10). The parameter minSeqRa also plays a critical role in runtime performance. When minSeqRa is set to a small positive value (e.g., 0.0001 or 0.001), the upper-bound pruning mechanism eliminates unpromising candidates at an earlier stage of the search, thereby reducing the number of HQPFPS-lists that must be constructed and evaluated.

For example, on the FIFA dataset, setting minSeqRa=0.0001 reduces runtime by nearly an order of magnitude compared with the baseline configuration. This result indicates that incorporating cross-sequence constraints into the upper-bound pruning strategy can substantially reduce runtime, particularly when the candidate search space is large.

Number of patterns: Fig. 3 illustrates how the number of discovered HQPFPS varies with minSupRa and minSeqRa. Increasing either threshold leads to a smaller set of discovered patterns, which is consistent with the stricter constraints imposed on periodicity, support ratio, and cross-sequence occurrence. On the FIFA dataset, the baseline configuration consistently produces a large number of patterns (approximately 3000). However, when minSeqRa is increased to 0.0001, the number of discovered patterns decreases to fewer than 100, representing a reduction of more than 96% relative to the baseline.

A similar trend is observed on the Bike dataset: raising minSupRa from 0.01 to 0.7 reduces the number of output patterns from 67 to 48. These results indicate that a considerable proportion of high-quantitative periodic patterns occur in only a limited number of sequences. The cross-sequence constraint enforced by minSeqRa is therefore essential for filtering out such sequence-specific patterns and retaining only those that are more representative at the database level.

Memory usage: Fig. 4 shows the memory consumption of MHQPFPS under different parameter configurations. On the FIFA dataset, memory usage under the baseline setting MHQPFPS(0,0.01,20) decreases from approximately 366 to 164 MB as minSupRa increases from 0.01 to 0.5. This behavior is consistent with the reduction in the number of maintained patterns, since higher support thresholds limit the number of HQPFPS-list that must be stored simultaneously.

For the synthetic T23L68KD15K dataset, the influence of minSeqRa on memory consumption is marginal, and the corresponding curves remain nearly flat. A plausible explanation is that this dataset contains relatively few sequences and transactions. Consequently, variations in minSeqRa primarily affect the depth of the search process rather than the total number of pattern lists retained in memory.

Overall, these results indicate that minSupRa is the dominant factor governing memory consumption, whereas minSeqRa mainly contributes to runtime reduction by effectively shrinking the search space through early pruning.

To isolate the effect of the main components of MHQPFPS, we compare the full method with two reduced variants: MHQPFPS-noUB, which disables the upper-bound pruning condition while keeping all other components unchanged, and MHQPFPS-noQ, which removes the quantitative ratio constraint quRa(X,S)≥minHqRa. All three variants are evaluated on the same four datasets under identical parameter settings.

Fig. 5 reports the runtime comparison. MHQPFPS consistently runs faster than MHQPFPS-noUB across all datasets. The speedup is most pronounced on Leviathan, where MHQPFPS-noUB maintains a near-constant runtime of approximately 7.3 s regardless of minSupRa, while MHQPFPS decreases to approximately 0.3 s at minSupRa =0.1, corresponding to a speedup of about 20×. On T23L68KD15K, MHQPFPS-noUB maintains a runtime of approximately 0.5 s, compared with 0.05 s for MHQPFPS, corresponding to a reduction of about 10×. MHQPFPS-noQ closely follows the full MHQPFPS in all runtime curves, indicating that the efficiency gain mainly comes from the upper-bound pruning mechanism.

Fig. 6 shows the number of discovered patterns. Across all datasets and parameter settings, MHQPFPS and MHQPFPS-noUB discover exactly the same pattern set, which empirically supports the correctness of the pruning strategy. MHQPFPS-noQ consistently discovers more patterns than the full MHQPFPS, since it retains periodically frequent patterns regardless of quantitative contribution. On FIFA, MHQPFPS-noQ returns up to 375 patterns at minSupRa =0.01, while MHQPFPS discovers none under the same setting. On Leviathan, MHQPFPS-noQ discovers up to 250 patterns, compared with approximately 10 for MHQPFPS. The gap between the two variants corresponds to patterns that satisfy periodicity and support constraints but do not satisfy the quantitative ratio requirement.

Fig. 7 shows memory consumption. MHQPFPS-noUB generally uses more memory than MHQPFPS on most datasets, since the absence of pruning forces the algorithm to maintain a larger number of active HQPFPS-list simultaneously. An exception is observed on FIFA at minSupRa =0.01, where MHQPFPS exhibits a transient memory peak before the effect of pruning becomes apparent; as minSupRa increases, this peak quickly disappears and MHQPFPS remains below MHQPFPS-noUB. Overall, the results show that the upper-bound pruning mechanism reduces runtime and memory consumption without changing the discovered pattern set.

To examine the influence of the two threshold parameters maxStd and minHqRa, Figs. 8 and 9 report the sensitivity of MHQPFPS to these two parameters, respectively, with all other parameters fixed at minSupRa =0.1, maxPr =10, and minSeqRa =0.001.

Runtime. Across both parameters and all four datasets, runtime shows no clear monotonic trend and varies only within a narrow range. This behavior is consistent with the algorithm design: the upper-bound pruning condition depends only on supRa and maxPer, which are anti-monotonic, whereas maxStd and minHqRa affect only the final HQPFP verification step and do not alter the candidate search space explored during DFS.

Pattern count. For maxStd (Fig. 8), FIFA and Leviathan exhibit a clear increase in pattern count followed by saturation. On FIFA, the count rises from 1 at maxStd=1 to 6 at maxStd=5 and remains stable thereafter, indicating that no patterns in this dataset have a period standard deviation exceeding 5. On Leviathan, saturation occurs at maxStd=3 with a maximum of 10 patterns.

For minHqRa (Fig. 9), FIFA and Leviathan exhibit a sharp decline in pattern count. On FIFA, the pattern count drops from 126 at minHqRa=0.10 to fewer than 10 at minHqRa=0.15, a reduction of over 92% within a single threshold step. On Leviathan, the count decreases from 71 to approximately 10 over the same interval. This suggests that a large fraction of periodically frequent patterns in these datasets carry quantitative ratios concentrated in the interval [0.10,0.15].

Bike remains insensitive to both parameters across all tested values, reflecting that its patterns inherently exhibit low period variance and high quantitative contribution. T23L68KD15K yields no patterns under the experimental configuration, consistent with the results reported in Figs. 3 and 6.

Overall, these results show that maxStd and minHqRa mainly act as quality-filtering thresholds whose influence on the output is data-dependent, while their effect on runtime remains limited.

To examine the sensitivity of MHQPFPS to quantitative values, we conduct a controlled experiment on T23L68KD15K by replacing all item quantities with values drawn from three distributions while preserving the original sequence structure: (A) Uniform 𝒰(1,10), in which all values are evenly distributed; (B) Normal 𝒩(5,1.5), truncated to [1,10], in which values concentrate around the mean; and (C) Long-tail (Pareto with α=1.5), in which the majority of values are small while a minority are disproportionately large. All other parameters are fixed at minSupRa=0.05, maxPr=10, maxStd=10, minSeqRa=0.001, and minHqRa is varied across {0.01,0.05,0.10,0.15,0.20,0.25,0.30}.

Fig. 10 shows the number of discovered patterns under each distribution. Under the long-tail distribution, the pattern count drops sharply at low minHqRa values, since a small number of high-quantity transactions dominate the sequence-level total q(S), causing most patterns to exhibit low quRa values. Even a modest increase in minHqRa therefore eliminates a large fraction of candidates. Under the uniform distribution, the decline is more gradual, as quantitative contributions are spread more evenly across transactions and no single transaction disproportionately inflates q(S). The normal distribution exhibits intermediate sensitivity, consistent with its moderate degree of concentration around the mean.

Fig. 11 shows that runtime remains relatively stable across all three distributions and all tested minHqRa values. This suggests that the pruning behavior is not strongly affected by the quantitative value distribution under the tested settings, although the number of discovered patterns is clearly influenced by how quantitative values are distributed.

These results indicate that MHQPFPS is sensitive to the distribution of quantitative values mainly through its effect on quRa and the resulting number of discovered patterns. Accordingly, the practical choice of minHqRa should take into account the quantitative characteristics of the target dataset.