Multiscale Fuzzy Temporal Pattern Mining: A Block-Decomposition Algorithm for Partial Periodic Associations in Event Data

Abstract

This paper introduces a dual-strategy model based on temporal transformation and fuzzy theory, and designs a partitioned mining algorithm for periodic frequent patterns in large-scale event data (3P-TFT). The model reconstructs original event data through temporal reorganization and attribute fuzzification, preserving data continuity distribution characteristics while enabling efficient processing of multidimensional attributes within a multi-temporal granularity calendar framework. The 3P-TFT algorithm employs temporal interval and object attribute partitioning strategies to achieve distributed mining of large-scale data. Experimental results demonstrate that this method effectively reveals hidden periodic patterns in stock trading events at specific temporal granularities, with volume–price association rules providing significant predictive and decision-making value. Furthermore, comparative algorithm experiments confirm that the 3P-TFT algorithm exhibits exceptional stability and adaptability across event databases with various cycle lengths, offering a novel theoretical tool for complex event data mining.

1. Introduction

In large-scale event data, periodic patterns refer to the recurrence of events following a fixed sequence within specific temporal intervals. Traditional periodic patterns require events to occur consistently at each observation point or demonstrate statistical significance, whereas partial periodic patterns accommodate scenarios where regularity only manifests during certain intervals [1]. This characteristic is more prevalent in real-world applications including: specific trading patterns in financial markets emerge predominantly during periods of high volatility; e-commerce user purchasing behaviors concentrate around promotional seasons; and disease transmission in healthcare may exhibit cyclical outbreaks under particular conditions. The identification and extraction of such partial periodic patterns hold significant value for enhancing decision-making efficiency, though their irregularity poses substantial challenges to existing analytical methodologies.

Research on partial periodic pattern mining primarily revolves around two core directions. First, in terms of innovative constraint formulation, scholars have introduced metric-based and temporal constraints to uncover periodic correlations among rare items. Kiran et al. [2] proposed replacing the conventional single minimum support threshold with multiple support thresholds to diversify the screening of frequent itemsets. Datta et al. [3] introduced the concept of rhythmic periods (RP) to address the omission of latent patterns caused by rigid periodic thresholds in existing periodic frequent pattern mining (PFPM) methods. Upadhya et al. [4] significantly improved rare pattern identification by incorporating temporal occurrence frequency parameters. To tackle sparse item challenges, Kiran et al. [5] devised a null-invariant measure based on relative periodic support, effectively resolving this issue. Rage et al. [6] advanced the field by proposing periodic confidence as a novel metric, coupled with an efficient algorithm, thereby addressing both sparse-item identification and computational inefficiency while enhancing overall mining performance. In temporal constraint research, Ale et al. [7] focused on extracting cyclical patterns confined to the lifecycle of target itemsets rather than the entire dataset, improving mining precision. Mahanta et al. [8] investigated time-interval lists to identify frequent periodic events exclusive to specific temporal windows. Kiran et al. [9] introduced a flexible local periodic constraint model incorporating maximum arrival time, minimum periodic support, and maximum distance thresholds, offering a robust solution for complex scenarios.

Second, in algorithmic efficiency and scalability optimization, researchers have concentrated on refining the intrinsic processing logic of mining algorithms. Pamalla et al. [10] proposed the 3P-ECLAT algorithm, which leverages compressed data structures and optimized search spaces to significantly enhance pattern recognition efficiency. Likhitha et al. [11,12] developed the k-PFPMiner algorithm, employing depth-first search to efficiently discover top-k periodic frequent patterns. Kiran et al. [13] integrated maximum scalar cardinality pruning to introduce the fuzzy periodic frequent pattern (FPFP) model, reducing computational overhead. Saideep et al. [14] adopted distributed parallel mining to substantially improve large-scale database processing.

Despite progress in redefining screening criteria (frequency) or narrowing mining scopes (temporal segments), fundamental challenges persist in both temporal and attribute dimensions when extracting partial periodic patterns from complex events [15]. These limitations hinder the accurate capture of real-world cyclical behaviors, particularly those influenced by multi-layered temporal structures and multi-dimensional attributes. This research gap not only restricts the applicability of partial periodic mining in high-dimensional data but also impedes the extraction of actionable insights from rich temporal datasets.

In the temporal dimension, existing methods often rely on predefined sliding windows to segment event data, transforming it into linear time-series representations. However, this approach suffers from two critical shortcomings: (1) event relationships are oversimplified as unidirectional or concurrent, disregarding true temporal dependencies and failing to capture nested multi-scale patterns [16]. For instance, stock price fluctuations exhibit intraday oscillations, cyclical adjustments, and quarterly trends—a complexity that existing methods cannot comprehensively model. (2) Temporal data mining techniques based on time-based enumeration (TBE) or segment-based enumeration (SBE) employ fixed intervals (e.g., hours, days, weeks), risking information loss or noise amplification due to inappropriate granularity selection [17]. For example, daily aggregation obscures micro-scale patterns like 20-min cardiac rhythm stability in habitual nappers. These constraints undermine the reliability of temporal decision support systems in complex scenarios.

In the attribute dimension, multi-faceted event characteristics introduce two key challenges: (1) Current methods discretize continuous attributes via rigid binning (e.g., low/medium/high price brackets), eroding distributional continuity and introducing boundary artifacts that obscure sensitivity gradients. (2) The combinatorial explosion of multi-dimensional attributes exponentially increases the difficulty of identifying latent correlations. For example, analyzing user behavior across age, income, and consumption habits simultaneously creates a high-dimensional search space where partial periodic patterns contingent on specific attribute combinations remain undetected, limiting practical utility.

To address these challenges, we propose a novel partial periodic pattern mining framework integrating temporal typing [18] and fuzzy attribute [19] transformation strategies. Our approach leverages temporal-type structures to resolve nested pattern detection and granularity selection issues, while fuzzy quantification softens attribute boundaries via label transformation [20,21,22], mitigating discretization artifacts and high-dimensional complexity. Building on this foundation, we construct a multi-granularity calendar pattern framework with temporal constraints to efficiently mine fuzzy temporal association rules and partial periodic patterns.

The paper is organized as follows: Section 2 formalizes the foundational concepts of fuzzy temporal event association with partial periodicity. Section 3 presents a dual-transformation block decomposition algorithm for mining partially periodic frequent patterns from fuzzy temporal association rules. Section 4 validates the framework through real-world dataset experiments and comparative analysis. Section 5 concludes with a discussion of contributions and future directions.

2. Partial Periodic Patterns in Fuzzy Temporal Events

To capture the multi-scale temporal characteristics of events and finer-grained events relationships, researchers have refined time into multi-level granularity units. Conducting periodic mining of temporal data at different levels of temporal granularity has become a significant research direction [18,23]. For instance, calendar patterns based on hierarchical temporal granularity structures are widely applied in periodicity mining. Within this framework, recurring association rules in calendar patterns are treated as partial periodic association rules, each of which must satisfy regularity conditions within specific time intervals in their corresponding calendar patterns [24].

In early studies, Li et al. pioneered the exploration of multi-granularity temporal event pattern discovery, based on the calendar framework [25]. Mahanta et al. further observed that the timestamps in events (e.g., calendar dates) inherently possess a hierarchical structure and can be treated as hierarchical data. Based on this, they proposed a method for detecting multi-type periodic patterns using timestamps [26,27]. Lee et al. demonstrated that defining periodicity through calendar-based frameworks significantly enhances the efficiency of discovering potential periodic association rules [28]. Hong et al., based on the lifecycle of events, divided the original time into hierarchical units of granularity and progressively extracted association rules and periodic events [23]. Additionally, Zhuo et al. extended calendar patterns by introducing temporal granularity precision parameters, proposing a fuzzy calendar time association rule mining algorithm that effectively improves prediction accuracy and reduces prediction intervals [29]. In summary, the hierarchical multi-granularity processing of time before mining association rules is a core strategy to efficiently explore periodic patterns.

Temporal types are also a fundamental unit of measurement for dividing the time axis as they can be applied to multi-granularity hierarchical processing of time. If v is the base temporal type of u, denoted as $B(v,u)$, then u is a coarser-grained temporal type relative to v, and Re $\mathrm{Re}len{\left(u\left(t\right)\right)}^v=l$ indicates that u and v differ by l relative time lengths. According to the definition of base temporal types, they form a partial-order relationship. For instance, finer-grained temporal types (such as seconds or minutes) can serve as base temporal types for coarser-grained temporal types (such as hours, days, weeks, or months), as any specific second clearly belongs to a particular minute, hour, and so on. However, it is important to note that the weekly temporal type is not a base temporal type for monthly or yearly temporal types.

Definition 1.

For any temporal types v and u satisfying relation $B(v,u)$, if there exists a set of base temporal factors $v_u^k$ between them such that $\mathrm{Re}len{\left(u\right)}^{v_u^k}\in \mathbb{N}$ and $\mathrm{Re}len{\left(v_u^k\right)}^v\in \mathbb{N}$, then $(u,v_u^k,v)$ constitutes a multi-granularity temporal format. Here, $\mathrm{Re}len{\left(u\right)}^{v_u^k}$ denotes the quantity of $v_u^k$ units in temporal type u, while $\mathrm{Re}len{\left(v_u^k\right)}^v$ represents the quantity of v units in $v_u^k$. According to the partial order relationship, $v_u^k$ and u satisfy the hierarchical subordination requirements, thus forming a multi-temporal calendar pattern $(u,v_u^k,v)$.

For instance, the most common multi-temporal calendar pattern $(u,v_u^k,v)$ = (Year, Month, Day) satisfies our definition since $\mathrm{Re}len{\left(Year\right)}^{Month}=12$ and $\mathrm{Re}len{\left(Month\right)}^{Day}$ ranges from 28 to 31, both being integers. This pattern can be specialized by restricting $v_u^k$ to a specific month, forming (*Year, December, *Day), which represents every December across multiple years. Similarly, we can create (*Year, *Month, 15th) to denote the middle day of each month, or (2023, *Month, *Day) to represent all days in 2023. These specialized patterns maintain the integer ratio relationship while providing nested temporal patterns. In practical applications, such restrictions enable precise identification of seasonal patterns like end-of-year financial activities or monthly reporting cycles.

In addition to temporal granularity, the granularity of event attributes is also a critical focus in periodic event mining. During the process of attribute value normalization, significant, detailed information is often lost. Consequently, achieving a balance between feature extraction and information preservation has emerged as a key challenge in ensuring the accuracy and reliability of event mining results. To address this problem, researchers have introduced the fuzzy set theory, which optimizes attribute processing by softening the hard boundaries of attributes, thus better supporting subsequent classification and association rule mining [20,21,22].

By referencing such operations, fuzzy temporal events can also be constructed [30]. Let us define a temporal type v over a time interval $\left[T,T^{\prime }\right]$, combined with an object set A and an attribute set S containing fuzzy states. We then define $\left(A,S,\left[T,T^{\prime }\right],v\right)$ as the fuzzy temporal event space of objects over the time interval $\left[T,T^{\prime }\right]$ with respect to the temporal type v. Within this space, any fuzzy temporal event is represented as $\left(A_l,s_{ij},d_i\left(t_k\right),v\left(t_k\right)\right)$, which can be described as state $s_{ij}$ for attribute $S_i$ of object $A_l$ occurring when at the temporal factor $v\left(t_k\right)$, taking the value $d_i\left(t_k\right)$. The state of the event is determined by the fuzzy membership degree ${\theta }_{ij}\left(d_i\left(t_k\right)\right)$ of the attribute value $d_i\left(t_k\right)$ to the attribute state $s_{ij}$, i.e., it is determined by the event function $E\left(A_l,s_{ij},d_i\left(t_k\right),v\left(t_k\right)\right)$.

To illustrate, consider a stock whose opening price increased by $3\%$ on 11 November 2021. To determine whether this qualifies as a “minor increase” state, we calculate the event function E(stock, minor price increase, $3\%$, 2021-11-11) = ${\theta }_{\mathrm{minor}\mathrm{price}\mathrm{increase}}\left(+3\%\right)$. If ${\theta }_{\mathrm{minor}\mathrm{price}\mathrm{increase}}\left(+3\%\right)$ exceeds the fuzzy threshold ${\alpha }_i\left({\alpha }_i\in \left[0,1\right]\right)$, then E(stock, minor price increase, +$3\%$, 2021-11-11) = 1, confirming the occurrence of the “minor price increase of the stock on 2021-11-11” event.

Within the interval $\left[T,T^{\prime }\right]$, the time interval during which the event first and last satisfies the membership threshold ${\alpha }_i$ is termed as the event lifespan, denoted as $\varphi \left(A_l,s_{ij}\left({\alpha }_i\right),v\right)$. Its length is represented as $len\left(\varphi \left(A_l,s_{ij}\left({\alpha }_i\right),v\right)\right)$, measured in units of the temporal type v.

Definition 2.

Let $r,h,o,z\in \left[0,\left.\infty \right)\right.$ be given integers, and let $\left[T,T^{\prime }\right]={\bigcup }_{k=1}^{n}v\left(t_k\right)$, where there are z adjacent temporal factors that follow an increasing trend over time $v\left(t_o\right)<v\left(t_{o+1}\right)<\cdots <v\left(t_z\right)$. The sequence of continuous events occurring at these temporal factors, denoted as: $\left(A_l,s_{ij}\left({\alpha }_i\right),d_i\left(t_o\right),v\left(t_o\right)\right),\left(A_l,s_{ij}\left({\alpha }_i\right),d_i\left(t_{o+1}\right),v\left(t_{o+1}\right)\right),\cdots ,\left(A_l,s_{ij}\left({\alpha }_i\right),d_i\left(t_z\right),v\left(t_z\right)\right)$, which is abbreviated as ${\left(A_l,s_{ij}\left({\alpha }_i\right)\right)}_o^z$. If the events within the sequence are combined solely using the logical OR operator ∨, it is simplified as ${\left(A_l,s_{ij}\left({\alpha }_i\right),\vee \right)}_o^z$; if the events are combined solely using the logical AND operator ∧, it is denoted as: ${\left(A_l,s_{ij}\left({\alpha }_i\right),\wedge \right)}_o^z$. If the events are combined using a mixture of logical ∨ and ∧ operators, it is represented as ${\left(A_l,s_{ij}\left({\alpha }_i\right),\vee \wedge \right)}_o^z$. If the internal combination pattern of the events is uncertain, it is abbreviated as ${\left(A_l,s_{ij}\left({\alpha }_i\right)\right)}_o^z$.

Similarly, after a gap of r temporal factors from $v\left(t_z\right)$, another sequence of h consecutive events may occur, which can be denoted as ${\left(A_l,s_{ij}\left({\alpha }_i\right),\square \right)}_{z+r}^h$, where □ represents the logical combination pattern (∨, ∧, $\vee \wedge$, or unspecified).

Note: For any temporal factor $v\left(t_k\right)$, if there is a lack of observation data, a special state $s_{\mathrm{missing}}$ is introduced so that: $d_i\left(t_k\right)=\mathrm{null}$ indicates missing observation. Define the missing value fuzzy membership function: ${\mu }_{\mathrm{missing}}\left(t_k\right)=[0,1]$, so that it is compatible with other states. The proportion of missing data is represented by $\rho ={\displaystyle \frac{|t_k|d_i\left(t_k\right)=\mathrm{null}|}{\left|\right[T,T^{\prime }\left]\right|}}$.

For example, ${\left(AAPL,Pric{e}_{sharp-rise},\wedge \right)}_5^1$ represents Apple stock prices rising sharply for five consecutive trading days (days 1–5), while ${\left(AAPL,Volum{e}_{high},\vee \right)}_6^3$ indicates the stock experiencing abnormally high trading volume on at least one day between days 6 and 8. If we observe ${\left(AAPL,RS{I}_{increasing},\wedge \right)}_{10}^2$ after a gap of $r=1$ day, this pattern represents the stock’s RSI increasing for two consecutive days (days 10–11), occurring after a one-day interval following the previous sequence. Such notation efficiently captures complex market patterns like “five days of sustained price momentum, followed by elevated trading activity, then, a brief pause before technical indicators signal potential overbought conditions”, which could indicate an impending reversal pattern in technical analysis.

Let the fuzzy event association rule with temporal constraints be of the form $X\stackrel{*}{\to }Y$, where the antecedent fuzzy temporal event (or group of events) $X=\left(\left(A_l,s_{ij}\left({\alpha }_i\right)\right),\left\{\wedge ,\vee \right\}\right)$ is formed by connecting one or more fuzzy temporal events using logical operators ∧ and ∨. Similarly, Y represents the subsequent fuzzy temporal event (or group of events) formed by connecting events in the same manner. Here, ∗ denotes the constraint condition under which the rule holds.

Definition 3.

Define an event group operation: If $E\left(X\stackrel{*}{\to }Y\right)=1$, then the rule $X\stackrel{*}{\to }Y$ is fulfilled.

If there exist two groups of events ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^{z}and{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h$ that satisfy $E\left({\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\wedge {\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h\right)=1$, it indicates that, after the occurrence of the group of events ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right),d_i\left(t_{k_i}\right),v\left(t_{k_i}\right)\right)}_o^z$, followed by an interval of r temporal factors, at least one fuzzy temporal event would occur in the group of events $\left(A_n,s_{p{q}_p}\left({\alpha }_p\right),d_i\left(t_{z+r}\right),v\left(t_{z+r}\right)\right),\cdots ,\left(A_n,s_{p{q}_p}\left({\alpha }_p\right),d_i\left(t_{z+r+h}\right),v\left(t_{z+r+h}\right)\right)$. In this case, the fuzzy temporal event association rule ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{{\alpha }_i|{\alpha }_p}{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h$ holds, where the fuzzy constraint $\ast ={\alpha }_i|{\alpha }_p$.

Note: When there are missing values in the sequence, conditional confidence is introduced: $\gamma ={\gamma }_0·(1-{\rho }_{\mathrm{seq}})$, where ${\gamma }_0$ is the confidence under complete data, and ${\rho }_{\mathrm{seq}}$ is the missing rate in the sequence.

The phenomenon of association among multiple events is more loosely defined and more commonly observed compared to the conditions for association rules between single events. For example, a certain stock may experience a price increase of more than $2\%$ within 3 min after the market opens, and approximately 2 min after this price increase, there would be an increase by more than $50\%$ in the trading volume within the subsequent 10-min time window. Such price–volume association patterns are highly prevalent in daily stock trading.

Property 1.

The temporal association interval r is not unique in the fuzzy temporal event association rule ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{{\alpha }_i|{\alpha }_p}{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h$. There may exist $r_1<r_2$, such that both rules ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{{\alpha }_i|{\alpha }_p}{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r_1}^h$ and ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{{\alpha }_i|{\alpha }_p}{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r_2}^h$ hold.

In the aforementioned case, where a price increase is followed by an approximately 2 min interval, considering that there is a 10-min time window buffer subsequently, the intermediate interval could also range from 1 to 3 min. Under such conditions, the association rule would still remain valid.

Definition 4.

The repeated occurrence of a fuzzy temporal event association rule over a time interval H under the temporal type v is denoted as: $N_H^v\left(X\stackrel{*}{\to }Y\right)=\sum E\left(X\stackrel{*}{\to }Y\right)$.

The number of repeated occurrences of the rule ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{{\alpha }_i|{\alpha }_p}{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h$ over the interval $\left[T,T^{\prime }\right]$ is defined as:

$$\begin{array}{c}{N}_{\left[T,T^{\prime }\right]}^v\left({\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{{\alpha }_i|{\alpha }_p}{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h\right)=\hfill \\ \sum _{k=1}^{n}E\left({\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{{\alpha }_i|{\alpha }_p}{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h\right)\hfill \end{array}$$

Note missing data compensation and introduce the weighted counting mechanism: $N_{[T,T^{\prime }]}^v$$\left(\mathrm{rule}\right)={\sum }_{k=1}^n{w}_k·E\left(\mathrm{rule}\right)$, where $w_k=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{complete}\mathrm{observation}\hfill \\ 1-{\rho }_k,\hfill & \mathrm{if}\mathrm{partly}\mathrm{missing}\hfill \end{array}\right.$, ${\rho }_k$ is the missing rate in the k-th rule instance.

Property 2.

When $N_{\left[T,T^{\prime }\right]}^v\left({\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{{\alpha }_i|{\alpha }_p}{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h\right)⩾2$, if the two event groups are identical and $r>0$, it describes a partially periodic phenomenon where a fuzzy temporal event (or group of events) repeats at regular intervals (r temporal factors) within a specific time period.

The scenario in which volume increases with price is often observed repeatedly during upward trends. However, if only a single event is involved, for example, a minor opening price increase in a stock followed by another minor opening price increase two days later, this indicates a periodic occurrence of the event over a partial time period, with a period of 2 days.

Definition 5.

If $MGF$ is in a multi-granularity format and $MGF$ = $\left(u,v_u^k,v\right)$ forms a multi-temporal calendar pattern, then the event groups X and Y constructed from the fuzzy multi-granularity temporal events $\left(A_l,S_i,d_{ij}\left(t_k\right),MGF\right)$, satisfy $X\stackrel{*}{\to }Y$ only in the set of specific temporal factors $v_u^k$. Such an event relationship is referred to as a fuzzy temporal (event) partial periodic association rule with temporal constraints, denoted as ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{\varphi }{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h$. Here, the fuzzy temporal constraint $\ast =\left({\alpha }_i|{\alpha }_p,v_u^k\right)=\varphi$, the periodic temporal type of the rule is u, the base temporal granularity is v, the temporal association interval is r, and the specific time period of occurrence is $v_u^k$.

Note that when facing sparse data, interval notation is allowed, and the fuzzy constraint is expanded to: $\ast =\left({\alpha }_i\right|{\alpha }_p,v_u^k,{\rho }_{\max })=\varphi$, where ${\rho }_{\max }$ is the maximum missing rate threshold for the rule to still be considered valid If the event $\left(A_l,S_{\mathrm{Highest}\mathrm{Price}},+10\%,\mathrm{2020.12.31}\right)$ occurs immediately after $\left(A_l,S_{\mathrm{Highest}\mathrm{Price}},+10\%,\mathrm{2020.12.31}\right)$, and this pattern repeats under the temporal constraint (*year, December, *day), then it indicates that the event follows an association rule with a daily base temporal granularity and an annual periodicity. For example, Stock $A_l$ exhibits an annual cycle where, every December, it experiences an opening price limit-up followed immediately by a highest price limit-up, resulting in a one-word limit-up scenario.

Property 3.

In the constraints for the establishment of fuzzy temporal periodic association rules, the larger the relative time length l between the temporal granularities u and v, the more values of the association interval r satisfy the rule, where $r\in [0,l]$.

Definition 6.

The strength of the fuzzy temporal partial periodic association rule ${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{\varphi }{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h$ occurring periodically under the multi-temporal calendar pattern $\left(v_{k-1},v_k,v_{k+1}\right)$ is denoted as $Str\left({\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{\varphi }{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h\right)$, which consists of the following two components:

(1) 

Periodic Cycle Frequency (PCF): Characterizes the occurrence frequency of the rule throughout its entire lifecycle, calculated as:

$$PCF={\displaystyle \frac{N_{v_{k-1}}^{v_k}\left({\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{\varphi }{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h\right)}{\mathrm{Re}len{\left(v_{k-1}\right)}^{\left[T,T\prime \right]}}},$$

(1)

where $v_{k-1}$ is the periodic temporal type, and $\mathrm{Relen}{\left(v_{k-1}\right)}^{[T,T^{\prime }]}$ represents the maximum number of repetitions with $v_{k-1}$ as the period within the time interval $[T,T^{\prime }]$.

(2) 

Rule Frequency (RF): Reflects the repetition frequency of the rule within specific periods, calculated as:

$$RF={\displaystyle \frac{N_{v_k}^{v_{k+1}}\left({\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{\varphi }{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h\right)}{len\left(\varphi \left({\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\right)\right)}},$$

(2)

where $v_{k+1}$ is the base temporal type.

The comprehensive calculation formula for periodic strength is:

$$\begin{array}{cc}\hfill Str\left(Rule\right)& ={\displaystyle \frac{DCF}{\mathrm{Relen}{\left(v_{k-1}\right)}^{[T,T^{\prime }]}}}\times \&\left[\lambda N_{v_{k-1}}^{v_k}\left(Rule\right)+(1-\lambda )\sum {\displaystyle \frac{N_{v_{k+1}}^{v_k}\left(Rule\right)}{len\left(\varphi \left({(A_m,s_{i{q}_i}\left({\alpha }_i\right))}_o^z\right)\right)}}\right],\hfill \end{array}$$

(3)

where: $DCF=1-\overline{\rho }$ is the data completeness factor, with $\overline{\rho }$ being the average missing rate. $\lambda \in [0,1]$ is the weighting coefficient between periodic cycle frequency and rule frequency.

3. Mining Algorithm for Periodic Patterns of Fuzzy Temporal Events

3.1. Algorithm Design Approach

In the mining of periodic event patterns, the core challenge lies in effectively managing both the temporal and attribute dimensions of events. This paper proposes a dual-dimensional block decomposition algorithm (3P-TFT) for mining periodic frequent patterns. 3P-TFT performs event reconstruction and feature extraction through temporal granularity transformation and fuzzy attribute processing, enabling comprehensive analysis of complex temporal event data.

In the temporal dimension, the concept of temporal types is introduced, and a multi-granularity, multi-level temporal calendar model is constructed to enable multi-dimensional representation of event sequence information. The proposed multi-level calendar model comprises three key layers: the base time unit layer, the occurrence period layer, and the combined period layer.

Base Time Unit Layer: this layer corresponds to the base temporal granularity v, transforming events into fuzzy temporal events with the smallest granularity in dual dimensions, such as seconds, minutes, or finer attribute state patterns. Occurrence Period Layer: this layer corresponds to the occurrence time interval $v_u^k$, where temporal units can be autonomously or adaptively set, such as days, weeks, or months as natural time cycle units, or domain-specific business cycles like trading days.

Combined Period Layer: this layer corresponds to the periodic temporal type u, forming $B(v_u^k,u)$ and $B(v,v_u^k)$, and combining the first two layers to construct nested periodic combinations, capturing composite periodic patterns. This hierarchical structure allows the algorithm to discover patterns at different temporal scales.

In the attribute dimension, fuzzy set theory is employed to reduce the dimensionality of event attribute values. By constructing membership functions and setting appropriate membership thresholds, numerical attributes are mapped into a finite set of attribute states. This approach effectively compresses the attribute state space while preserving the characteristics of numerical attributes, enabling efficient extraction and generalized description of numerical information.

In terms of key innovations, this algorithm contributes the following major advancements: (1) Proposes a dual-dimensional transformation framework, achieving unified processing of temporal information and attribute information; (2) Designs a multi-temporal calendar pattern that reveals complex partial periodic relationships across different scales; (3) Introduces fuzzy set theory for attribute value processing, effectively minimizing information loss.

3.2. Core Algorithm Description

3.2.1. Fuzzy Temporal Event Preprocessing at the Base Time Unit Layer

First, let us construct the basic elements of the algorithm. Let A denote a finite set of objects, and S denote a set of attributes, where $S=\left\{S_1,S_2,\cdots S_p\right\}$. For each attribute $S_i$, define its corresponding state set $\left\{s_{ij}\right\}\left(i\in \left[1,p\right],j\in \left[1,q\right]\right)$ and its fuzzy interval value range $[{\underline{d}}_{ij},{\overline{d}}_{ij}]$. To ensure the rationality of the fuzzy state partition, it is required that adjacent states’ fuzzy intervals have a certain degree of overlap.

Next, select the temporal type v to partition the time intervals $\left[T,T^{\prime }\right]$, generating a sequence of temporal factors $v\left(t_k\right)$, $k=1,2,\dots ,n$. Based on this, construct the fuzzy temporal event $\left(A,s_{ij},d_i\left(t_k\right),v\left(t_k\right)\right)$, which describes the occurrence of state $s_{ij}$ for attribute $S_i$ of object $A_l$ at the temporal factor $v\left(t_k\right)$, with the value $d_i\left(t_k\right)$.

Then, introduce the membership function ${\theta }_{s_{ij}}:R\to [0,1]$ to transform the fuzzy attribute states. For the observed value $d_i\left(t_k\right)$, calculate its membership degree to each state $s_{ij}$. Set a membership threshold ${\alpha }_i\in \left[0,1\right]$, and if ${\theta }_{s_{ij}}\left(d_i\left(t_k\right)\right)>{\alpha }_i$, the event is determined to have occurred. Through the above preprocessing steps, the fuzzy temporal event matrix can be obtained:  

$$F_{A{S}_{\theta }}\left(t\right)=\left[\begin{array}{c}E\left(A_l,{\overline{S}}_i,v\left(t_1\right)\right)\\ E\left(A_l,{\overline{S}}_i,v\left(t_2\right)\right)\\ ⋮\\ E\left(A_l,{\overline{S}}_i,v\left(t_n\right)\right)\end{array}\right]={\left[\begin{array}{cccc}{\theta }_{i1}\left(d_i\left(t_1\right)\right)& {\theta }_{i2}\left(d_i\left(t_1\right)\right)& \cdots & {\theta }_{i{q}_i}\left(d_i\left(t_1\right)\right)\\ {\theta }_{i1}\left(d_i\left(t_2\right)\right)& {\theta }_{i2}\left(d_i\left(t_2\right)\right)& \cdots & {\theta }_{i{q}_i}\left(d_i\left(t_2\right)\right)\\ ⋮& ⋮& \cdots & ⋮\\ {\theta }_{i1}\left(d_i\left(t_n\right)\right)& {\theta }_{i2}\left(d_i\left(t_n\right)\right)& \cdots & {\theta }_{i{q}_i}\left(d_i\left(t_n\right)\right)\end{array}\right]}_{n\times q_i}$$

(4)

Through the aforementioned processing, raw events are transformed into a structured fuzzy temporal event matrix representation, providing foundational data support for subsequent periodic pattern mining.

3.2.2. Candidate Periodic Association Rules at the Occurrence Period Layer

With selected temporal unit of the occurrence layer, we can construct a multi-temporal calendar pattern in the form of (u,$v_u^k$,v). Temporal unit of occurrence period layer partitions the fuzzy temporal event data into time intervals and decomposed into several independently processable subsets $\left\{D1,D2,\dots ,Dn\right\}$, providing foundational support for subsequent distributed computation.

Based on the block-stored fuzzy temporal event matrix, an improved Apriori algorithm is executed block by block. Using a layer-by-layer expansion strategy, the algorithm starts with single-item frequent sets and gradually constructs multi-item frequent sets. For each block data subset $Di$, frequent itemsets are independently computed to ensure that the subset meets the requirements of support and confidence. The frequent patterns mined from each block are then merged into global frequent patterns, generating candidate periodic association rules that satisfy the support and confidence thresholds. The formula for generating candidate rules is as follows:

$$\begin{array}{cc}\hfill & \mathrm{support}\left(\left(A_l,s_{ij}\left({\alpha }_i\right)\right)\stackrel{[T,T^{\prime }]v}{\to }{\left(A_l,s_{ij}\left({\alpha }_i\right)\right)}_r^h\right)\hfill \\ \hfill & ={\displaystyle \frac{N_{[T,T^{\prime }]}^v\left(\left(A_l,s_{ij}\left({\alpha }_i\right),d_i\left(t_k\right),v\left(t_k\right)\right)\wedge {\left(A_l,s_{ij}\left({\alpha }_i\right),d_i\left(t_k\right),v\left(t_k\right)\right)}_r^h\right)}{\mathrm{len}\left(\varphi \left(A_l,s_{ij}\left({\alpha }_i\right),v\right)\right)}}.\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \mathrm{confidence}\left(\left(A_l,s_{ij}\left({\alpha }_i\right)\right)\stackrel{[T,T^{\prime }]v}{\to }{\left(A_l,s_{ij}\left({\alpha }_i\right)\right)}_r^h\right)\hfill \\ \hfill & ={\displaystyle \frac{N_{[T,T^{\prime }]}^v\left(\left(A_l,s_{ij}\left({\alpha }_i\right),d_i\left(t_k\right),v\left(t_k\right)\right)\wedge {\left(A_l,s_{ij}\left({\alpha }_i\right),d_i\left(t_k\right),v\left(t_k\right)\right)}_r^h\right)}{N_{[T,T^{\prime }]}^v\left(A_l,s_{ij}\left({\alpha }_i\right),d_i\left(t_k\right)\right)}}.\hfill \end{array}$$

(6)

3.2.3. Partial Periodic Pattern Validation at the Combined Period Layer

By comprehensively evaluating the periodic occurrence strength and rule frequency strength, the rule frequency strength $RF\left(R\right)$ is first calculated, requiring $RF\left(R\right)\ge \mathrm{minsup}$, which measures the frequency of the rule across all block data. Next, the periodic strength threshold ${\tau }_{\min }$ is computed, and when $PCF\left(R\right)\ge {\tau }_{\min }$, the rule is considered to form a valid partial period. Finally, $Str\left(R\right)$ is calculated to measure the periodic occurrence strength of the association rule R, with the formula as follows:

$$\mathrm{Str}\left(R\right)={\displaystyle \frac{\mathrm{DCF}}{n}}\times \left[\lambda n\mathrm{PCF}\left(R\right)+(1-\lambda )\mathrm{RF}\left(R\right)\right],$$

(7)

where n represents the maximum number of repetitions within the mining time range.

A strength threshold $\eta$ is set, and when $Str\left(R\right)⩾\eta$, the association rule satisfying the condition is output as a partial periodic pattern. The complete logical framework of the fuzzy temporal Partial periodic pattern block decomposition mining algorithm is presented in Algorithm 1.

Algorithm 1 Main Algorithm: Fuzzy Temporal Partial Periodic Mining Algorithm (3P-TFT)

Require: Fuzzy temporal event list $case\_data$, parameters r, $minsup$, $minconf$, $minstr$ Ensure: Fuzzy periodic rule list $Rules$ 1: Initialize $u,v,w$ 2: $D\leftarrow \mathrm{Combine}(case\_data,v)$              ▹ Generate combined set using v as the temporal granularity factor 3: Split $D\leftarrow \left[u\left(t_1\right),u\left(t_2\right),u\left(t_3\right),\cdots ,u\left(t_K\right)\right]$ 4: and $l\leftarrow \mathrm{len}\left(u_w^j\right)=\mathrm{Abslen}\left(w\left(t_j\right)\right)$ ▹ Split the data and define the length of the signal segments and $l=len\left(u_w^j\right)=Abslen\left(w\left(t_j\right)\right)$ 5: $rules=\oslash$                                                                                                                          ▹ Initialize the rule set 6: for $i\leftarrow 1$ to K step 1 do                                                                                         ▹ Perform mining in blocks 7:     $D^{*}\leftarrow Fuzzy\left(u\left(t_i\right)\right)$               ▹ Transform the original temporal event set into fuzzy temporal events 8:     $bflist\leftarrow \mathrm{Unique}\left(D^{*},{\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\right)$     ▹$bflist$: Temporal constraint set for rule antecedents 9:     $aflist\leftarrow \mathrm{Unique}\left(D^{*},{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h\right)$ ▹$aflist$: Temporal constraint set for rule consequents 10:     $BF\leftarrow \mathrm{Frequent}\_\mathrm{Gen}(D^{*},bflist,minsup)$ 11:     $AF\leftarrow \mathrm{Frequent}\_\mathrm{Gen}(D^{*},aflist,minsup)$ 12:     $Rulelist\leftarrow \mathrm{Tomix}(BF,AF)$                ▹ Combine frequent sets through permutation 13:     for each R in $Rulelist$ do        ▹ Find association rules that meet the requirements 14:         if $\mathrm{conf}\left(R\right)\ge minconf$ then 15:            if R in $Rules$ then 16:                $Rules.R.strong\leftarrow Rules.R.strong+1$ 17:            else 18:                Append R to $Rules$ 19:            end if 20:         end if 21:     end for 22: end for 23: for $j\leftarrow 1$ to l step 1 do            ▹ Find partial periodic association rules that meet the requirements 24:     for each $rule$ in $Rules$ do 25:           $Rulestr\leftarrow \mathrm{Str}(rule,u)$▹ Calculate the periodic strength of the rule over the time period 26:           if $Rulestr\le minstr$ then 27:                  Delete $rule$ from $Rules$ 28:           end if 29:     end for 30: end for 31: return  $Rules$

4. Algorithm Testing and Result Analysis

To verify the effectiveness of the algorithm and model in mining daily stock trading data, we focus on examining their ability to identify partial periodic association rules that traditional models fail to detect. By analyzing characteristics such as periodic fluctuations in the stock market, this study aims to provide new perspectives and methods for stock market research and investment decision-making.

4.1. Data Preparation

Yanghe Co., Ltd. (Suqian City, China) (002304) was randomly selected as the experimental subject, and its trading data from Friday afternoons between 2010 and 2019 was extracted. The minimum temporal granularities were set to 1-min, 3-min, 5-min, 10-min, and 15-min. The periodic temporal type was defined as “year”, with the specific periodic time interval being “month”. The temporal calendar framework template was structured as (year, month, n min temporal type), and the rule template was denoted as:

$${\left(A_m,s_{i{q}_i}\left({\alpha }_i\right)\right)}_o^z\stackrel{\varphi }{\to }{\left(A_n,s_{p{q}_p}\left({\alpha }_p\right)\right)}_{z+r}^h$$

(8)

For the selected stock price and volume attributes, the attribute state values were obtained by calculating the difference between the current value and the corresponding value from the previous temporal interval, divided by the previous temporal value. These state values effectively describe the trend states of the opening price(OP), highest price(HP), lowest price(LP), closing price(CP), and trading volume(TV). After data preprocessing, the optimal number of classifications was determined to be five, using the Goodness of Variance Fit (GVF) method. Based on this, the price and volume attribute states were encoded as “Large Decrease (↓L)”, “Small Decrease (↓S)”, “Neutral (M)”, “Small Increase (↑S)”, or “Large Increase (↑L)”.

Considering the practical characteristics of the stock data and referencing common scenarios for fuzzy membership function adaptation, the trapezoidal function was selected as the membership function, as illustrated in Figure 1. The upper base of the trapezoid was set to the cluster center value expanded by $20\%$, while the lower base was defined as the midpoint between adjacent cluster centers.

The K-means algorithm was applied to cluster the attribute data, and based on practical considerations, five cluster centers were determined: for the price component, the centers were [−0.0051, −0.0011, 0.0011, 0.0051], and for the trading volume component, the centers were [−0.5297, −0.1782, 0.1782, 0.5297]. A fuzzy membership threshold of 0.5 was set to calculate the membership intervals for each attribute state and perform state encoding. Subsequently, the transformed fuzzy temporal event data were input into the fuzzy temporal event periodic mining algorithm. In the algorithm, the minimum support (minsup) was set to 0.2, the minimum confidence (minconf) to 0.6, and the minimum periodic strength (minstr) to 0.5, with the interval temporal factor, $r=1$, aiming to mine fuzzy partial periodic association rules between adjacent temporal factors.

4.2. Results Analysis

By applying 3P-TFT to explore adjacent partial periodic association rules among different attributes of the same object, the following conclusions were drawn.

(1) Discoverability of Fuzzy Temporal Partial Periodic Association Rules

In comparative experiments between fuzzy and conventional non-fuzzy data, the data were input into the model with a periodic strength threshold set to 0.5. The number of association rules obtained from both types of experiments is presented in

Table 1. Quantity of Temporal Association Rules.

	1-min	3-min	5-min	10-min	15-min
Conventional	13	27	24	14	9
Fuzzy	20	69	100	147	225

.

The experimental results demonstrate that 3P-TFT’s event processing method extracts a significantly higher number of periodic association rules compared to traditional rigid marking methods. Traditional methods produced a limited number of rules (only 13 at 1-min granularity) and primarily captured minor fluctuations around zero points (such as variations within $0.5\%$ of the opening price). While these findings align with stock volatility characteristics, they offer limited practical application value.

Comparative experiments revealed two key phenomena. First, traditional methods exhibit a unimodal characteristic: after reaching a peak at 3-min granularity (27 rules), the rule quantity decreases as granularity increases (only nine rules at 15-min intervals), reflecting their adaptability limitations for larger time windows. Second, 3P-TFT demonstrates significant scalability and robustness: from 1-min (20 rules) to 15-min (225 rules), showing super-linear growth (${\mathrm{R}}^2$ = 0.97), with a growth rate of $53\%$ in the 10–15 min interval. This indicates that 3P-TFT can more effectively capture periodic patterns in stock trading data at larger time granularities.

Performance differences intensify as the time granularity increases. At a 1-min granularity, the number of fuzzy temporal rules is 1.54 times that of traditional methods, while at a 15-min granularity, this ratio expands to 25 times. This phenomenon is also verified in the state distribution analysis (Figure 2). Under small granularity temporal settings (such as 1-min), small fluctuations in stock prices account for more than $60\%$ of all price states, exhibiting a spike distribution. As the temporal granularity increases (such as 10-min and 15-min), the distribution of various states tends to become more uniform. This divergence can be attributed to two main factors: on one hand, traditional methods are constrained by rigid thresholds, making it difficult to filter compound noise at larger granularities; on the other hand, 3P-TFT, through temporal recombination and attribute fuzzification, demonstrate significant advantages in processing time series data, effectively maintaining the stability of pattern recognition.

Experiments demonstrate that in practical applications, especially in scenarios such as financial analysis and trend prediction over larger time spans, 3P-TFT can extract richer and more valuable association rules, thereby supporting more comprehensive decision analysis.

(2) The Impact of Time Granularity on Stock Trading Trends and Patterns

The experimental results indicate that stock trading events exhibit different trends under various temporal granularities. As shown in Figure 2, in the 1-min temporal granularity, the price attributes of Yanghe Co., Ltd. demonstrated a slight upward trend for most of the past decade. However, this trend does not hold in other temporal granularities, such as the 10-min and 15-min scales, where the stock prices instead exhibited significant fluctuations. This demonstrates that the trading trends of the same stock vary significantly across different temporal granularities. Such differences highlight the necessity of conducting multi-temporal granularity analysis on stock trading. Therefore, for different temporal calendar models, the trading trends of the same stock would show notable variations, and it is necessary to employ multi-temporal granularity in stock trading analysis.

The differences in trading trends across temporal granularities are also reflected in the temporal distribution of effective partial periodic association rules. Figure 3 reveals that Yanghe Co., Ltd. exhibited distinct periodic patterns only on Friday afternoons in November under the 1-min temporal type, with weaker regularity observed in other months. In other temporal types, effective periodic patterns were distributed across different months: primarily in February for the 3-min type, in January, February, August, and December for the 5-min type, in August for the 10-min type, and in December for the 15-min type. This suggests that by analyzing trading events through multi-temporal transformation, hidden periodic patterns, such as seasonal and cyclical characteristics, can be discovered, thereby providing valuable insights for investment decision-making.

At the same time, certain consistencies can also be observed across different temporal granularities. Taking Yanghe Co., Ltd. as an example, trading data from the past decade shows that certain months consistently exhibit distinctive characteristics. For instance, a higher number of effective periodic rules can be identified every December. In contrast, trading events in June and September display a completely random pattern.

The above experimental results confirm that 3P-TFT can effectively uncover valuable association rules and periodic patterns across various temporal granularities.

(3) Constructing and Utilizing Trading Strategies Based on Periodic Rules

Based on the analysis of Yanghe Co., Ltd.’s 10-year trading data, with a minimum temporal granularity of 15-min intervals, the study focused on the trading activities during the last half-hour before market close, uncovering several effective price–volume periodic rules. These rules are valid only in specific months with periodic strength exceeding 0.65 and are not applicable in normal databases or other temporal granularity databases within the same time interval.

For instance, in

Table 2. Effective Periodic Association Rules for the 15-min Temporal Pattern.

Index	Rule Antecedent	Rule Consequent	Period (M)	Periodic Strength	Establishment Time
1	[CP↑S & CP↑L]	OP↑L	1	0.7121	[‘2010’, ‘2011’, ‘2012’,
					‘2013’, ‘2014’, ‘2015’, ‘2018’]
2	[CP↑S & CP↑L]	LP↑S	1	0.6524	[‘2010’, ‘2013’, ‘2014’,
					‘2015’, ‘2016’, ‘2018’]
3	[CP↑S & CP↑L]	LP↑S	5	0.7114	[‘2011’, ‘2012’, ‘2013’,
					‘2014’, ‘2016’, ‘2018’, ‘2019’]
4	[CP↑S & CP↑L]	OP↑S	2	0.6478	[‘2012’, ‘2014’, ‘2016’,
					‘2017’, ‘2018’, ‘2019’]
5	[CP↑S & TV↓S]	OP↑S	6	0.6638	[‘2010’, ‘2011’, ‘2012’,
					‘2014’, ‘2017’, ‘2018’]
6	OP↑S	OP↓S	3	0.6828	[‘2010’, ‘2011’, ‘2012’,
					‘2013’, ‘2015’, ‘2017’, ‘2018’]
7	[CP↓L & CP↓S]	[OP↓S & HP↓S]	7	0.7050	[‘2011’, ‘2012’, ‘2013’,
					‘2014’, ‘2016’, ‘2017’, ’2018’]
8	CP↓L	[OP↓L & HP↓S]	7	0.6982	[‘2012’, ‘2013’, ‘2014’,
					‘2016’, ‘2017’, ‘2018’, ‘2019’]
9	CP↓L	[OP↓L & HP↓L]	7	0.7056	[‘2012’, ‘2013’, ‘2014’,
					‘2015’, ‘2016’, ‘2018’, ‘2019’]
10	CP↑L	TV↓S	11	0.6855	[‘2010’, ‘2011’, ‘2012’,
					‘2013’, ‘2014’, ‘2015’, ‘2018’]
11	[HP↓S & CP↓L]	[OP↓L & OP↓S]	12	0.7104	[‘2011’, ‘2012’, ‘2013’,
					‘2015’, ‘2016’, ‘2017’, ‘2019’]

, Rule 6 indicates that on Friday afternoons in March each year, if the opening price shows a slight increase, the opening price in the subsequent 15 min will experience a slight decrease. Similarly, Rules 7, 10, and 11 also demonstrate periodic associations within specific months. Furthermore, Rules 2 and 3 reveal that on Friday afternoons in January and May each year, if the closing price shows a slight or significant increase, the lowest price in the following 15 min will also rise slightly. Rules 4 and 5 uncover a pattern on Friday afternoons in February and June: when both the closing price and trading volume experience a slight decrease, the opening price in the subsequent 15 min will show a slight increase.

The periodic rules listed in Table 1 encompass both individual events and combinations of multiple events, demonstrating predictive utility for price trends. Specifically, the first five rules forecast upward price movements, while the latter six rules predict declines in both price and trading volume. By leveraging these periodic rules, investors can capitalize on high-frequency trading within 30 min intervals during specific months to generate profits. For instance, according to Rules 1 and 2, in January on Friday afternoons, if Yanghe Co., Ltd. exhibits a slight increase in closing price or a concurrent decline in trading volume, a buy operation can be executed, followed by selling at the anticipated higher opening price to secure gains. Similarly, Rule 6 suggests that in March, when a modest rise in the opening price is observed, holdings should be sold to mitigate the risk of subsequent declines in the opening price.

4.3. Algorithm Evaluation

Current partial periodic pattern (3P) mining algorithms primarily focus on algorithmic refinements to enhance computational efficiency and scalability. State-of-the-Art approaches leveraging pruning techniques and depth-first search, such as the Partial Periodic Pattern Growth algorithm (3P-growth) [1,31] and its enhanced variant 3P-ECLAT [10], exhibit superior execution speeds. Although the proposed 3P-TFT algorithm employs fuzzy temporal event preprocessing followed by segmented window-based mining of frequent partial periodic patterns, both methodologies share identical evaluation metrics for periodic pattern mining. Specifically, the two primary constraints in 3P-growth and 3P-ECLAT—user-specified periodicity (per) and minimum periodic support (minPS)—correspond to the multi-temporal calendar pattern $\left(v_{k-1},v_k,v_{k+1}\right)$, where $v_{k-1}$ denotes periodicity and $Str$ represents periodic strength in 3P-TFT. To assess the computational efficiency of 3P-TFT, a comparative performance analysis was conducted against these benchmarks using the publicly available Japanese Air Pollution Dataset.

The experimental dataset comprises three months of PM 2.5 pollutant measurements collected by all sensors in Japan’s Ministry of the Environment SORAMAME network. The resulting database (hereafter referred to as the “Pollution Dataset”) is characterized by high density and dimensionality, consisting of numerous prolonged air pollution events. Key attributes of the data are summarized in

Table 3. Data Characteristics of the Air Pollution Database.

Dataset	Nature	Temper Type	lransaction Length (in Count)			Database Size (in Count)
			Min.	Avg.	Max.
Pollution	Dense	1 h	11	460	971	720

.

Experimental parameters included period lengths (corresponding to per in 3P-growth/3P-ECLAT and $v_k$ in 3P-TFT) of 1 h, 2 h, and 3 h, with periodic strength ($Str$) thresholds ranging from 0.5 to 0.55. Comparative results, illustrated in Figure 4, demonstrate that 3P-TFT exhibits progressively reduced execution times with increasing period lengths, showing minimal sensitivity to temporal granularity variations—a consequence of its unique segmented parallel mining architecture and efficient fuzzy preprocessing. While 3P-ECLAT outperforms 3P-growth, both algorithms experience rapid performance degradation with elevated periodicity thresholds. Regarding threshold robustness, 3P-TFT maintains stable execution times irrespective of minPS variations, whereas 3P-growth and 3P-ECLAT show significant latency increases under reduced thresholds due to expanded candidate pattern processing. Applicability analysis reveals 3P-TFT’s superior adaptability across diverse period lengths, particularly in parameter-tuning scenarios. Although 3P-growth demonstrates marginal efficacy in short-period, low-event contexts and 3P-ECLAT excels in long-period sparse datasets, 3P-TFT achieves optimal performance in most scenarios through exceptional stability and adaptive capability.

5. Conclusions

This study investigates effective methodologies for mining fuzzy temporal partial periodic association rules within timestamped event datasets. By processing the fuzzy states of attributes, the proposed approach achieves flexible dimensionality reduction and feature extraction while retaining critical information. The incorporation of temporal and multi-granular calendar patterns facilitates the identification and analysis of internal event relationships across diverse granularities and hierarchical levels. In contrast to traditional periodic mining models, the fuzzy temporal partial periodic model relaxes constraints on event transitions and period establishment conditions, thereby broadening its applicability. The proposed 3P-TFT partitions the problem into independently solvable sub-problems, resulting in a clear and concise program structure. Additionally, it leverages optimized data structures and parallel algorithms to enhance computational efficiency, making it particularly suitable for large-scale periodic mining tasks.

Experimental results indicate that applying 3P-TFT to Yanghe Share trading data uncovers more valuable periodic association rules than traditional methods. This algorithm allows traders to identify periodic association patterns at various time granularities, enabling the development of trading strategies that include market trend detection, cyclical fluctuation utilization, and risk control optimization, thereby enhancing investment returns.

Although this study is limited to mining partial periodic association rules between single events within temporal calendar frameworks in event databases, the proposed method exhibits significant extensibility. The rule template incorporates multiple adjustable parameters, enabling its extension to multi-level, multi-dimensional partial periodicity. Additionally, 3P-TFT is well-suited for handling periodicity with perturbations and evolution. Future research could explore mining techniques for diverse object event sets, event attributes, and approximate time intervals to uncover hidden structural patterns, thereby revealing order amidst disorder.