# Mining Temporal Association Rules with Temporal Soft Sets

Authors: Xiaoyan Liu; Feng Feng; Qian Wang; Ronald R. Yager; Hamido Fujita; José Carlos R. Alcantud
Journal: Journal of Mathematics
 (2021)

---

## Abstract

Traditional association rule extraction may run into some difficulties due to ignoring the temporal aspect of the collected data. Particularly, it happens in many cases that some item sets are frequent during specific time periods, although they are not frequent in the whole data set. In this study, we make an effort to enhance conventional rule mining by introducing temporal soft sets. We define temporal granulation mappings to induce granular structures for temporal transaction data. Using this notion, we define temporal soft sets and theirQ-clip soft sets to establish a novel framework for mining temporal association rules. A number of useful characterizations and results are obtained, including a necessary and sufficient condition for fast identification of strong temporal association rules. By combining temporal soft sets with NegNodeset-based frequent item set mining techniques, we develop the negFIN-based soft temporal association rule mining (negFIN-STARM) method to extract strong temporal association rules. Numerical experiments are conducted on commonly used data sets to show the feasibility of our approach. Moreover, comparative analysis demonstrates that the newly proposed method achieves higher execution efficiency than three well-known approaches in the literature.


---

## Body

## 1. Introduction

In modern society, vast amounts of data are produced and collected daily by all walks of life. With an increasing amount of data, there has been an urgent need for developing powerful models, methods and apparatuses to facilitate data analysis. In response to this demand, data mining has emerged and become a fast-growing research field with various fascinating topics and practical applications. Data mining is a multidisciplinary field, which involves applied mathematics, computer science, information science, statistics, and other disciplines. In the process of knowledge discovery in databases (KDD), data mining is viewed as the most essential step in which sophisticated methods are applied to extract knowledge or patterns from data. As shown in Figure1, six fundamental tasks in data mining are association rule mining, clustering, classification, regression, summarization, and sequence analysis. In a more general perspective, some researchers treat data mining as a synonym for KDD. Data mining has proven to be useful in a myriad of areas including biological statistics [1], case-based reasoning [2], factor analysis of heart disease [3], pattern classification [4], and group role assignment [5].Figure 1
The generic framework of KDD.Association rule mining, such as association analysis and association rule learning, is of great importance in the realm of knowledge discovery and data mining. It was originally proposed in [6] with the aim to find frequent patterns in transactional databases and potential association rules between different item sets. Most rule extraction algorithms belong to one of the following two categories. The first one is known as the class of “candidate generation” methods with the Apriori [7] algorithm as its typical representative. The main drawback of these methods is that all of them require multiple database scans. The second category consists of “pattern growth” methods such as the FP-growth algorithm [8], which relies on the tree-based data structure (like FP-trees) to store basic information about frequent item sets. More specifically, it does not generate candidate sets of items, and it does not require multiple scans of the database by saving basic information about frequent sets of items into a custom-built data structure. In addition, Zaki [9] proposed another lower I/O costs vertical mining algorithm, called the equivalence class transformation (ECLAT). However, the performance of ECLAT can be affected in dense databases. By using the bitmap representation of sets, Aryabarzan et al. [10] presented a crucial data structure named NegNodeset and developed the NegNodeset-based Frequent Itemset Mining (negFIN) algorithm. The prominent features of the negFIN algorithm are three-fold. Firstly, it makes use of bitwise operators in order to extract NegNodesets of item sets. Secondly, it significantly reduces the complexity of computing supports. Lastly, it generates frequent item sets by using the structure called set-enumeration tree, and meanwhile, it efficiently prunes the search space with the promotion method. Djenouri et al. [11] developed an efficient parallel genetic algorithm for extracting diversified association rules in big data sets. To further improve pattern mining in big data, Luna et al. [12] designed several sophisticated algorithms which rely on a novel paradigm called MapReduce and related implementation named Hadoop. Nevertheless, it should be noticed that the abovementioned rule extraction methods may sometimes produce redundant or incoherent association rules. In view of this, Feldman et al. [13] proposed the maximal association rule, which is a novel complementary apparatus to extract interesting association rules that are frequently lost when using regular association rules. Amir et al. [14] contributed to additional developments regarding exact conceptualization and efficient identification of maximal association rules. In addition to objective measures such as support, confidence, and correlation, some researchers have been interested in considering subjective measures such as risk, interest, and utility to discover useful item sets and association rules. In particular, a new research direction named utility pattern mining [15–17] has received considerable attention in recent years.Temporal association rule mining (TARM) is one of the most fascinating topics in the field of association rule mining. It has been successfully applied to a wide range of domains such as cancer treatment [18], gene analysis [19], and web mining [20]. Depending on whether the time variable is considered as an implied or integral component, Segura-Delgado et al. [21] systematically classified the existing TARM approaches into two main categories. Agrawal and Srikant [22] coined the terminology of sequential pattern to facilitate the analysis of a transaction database. Inspired by this seminal idea, many scholars have conducted in-depth research with regard to sequential rule mining. Zhai et al. [23] designed a time constraint-based rule mining algorithm, called T-Apriori, to analyse the sequence of ecological events. Gan et al. [24] presented a projection-based utility mining method which is useful for mining high-utility sequential patterns from sequence data. Hong et al. [25] constructed a hierarchical granular framework to enhance TARM by considering different levels of time granules. Song et al. [26] detected changes of customer behavior by using temporal association rules mining from customer profiles and sales data at different time snapshots. Yun et al. [27] designed an efficient algorithm to discover high-utility patterns from incremental databases by constructing a global data structure through a single scan.Molodtsov’s soft set theory [28] provides a formal framework for coping with uncertainty. Its basic principle relies on the perspective of parameterization, suggesting that one should recognize uncertainly defined objects from various facets, and every solo feature yields an approximate description of this object. Maji et al. [29] soon presented several operations of soft sets to complement [28]. Ali et al. [30] introduced several new operations to consolidate the basis of soft set theory. Babitha and Sunil [31] extended the ideas of functions and relations by virtue of soft set theory. Feng and Li [32] clarified the relations among several kinds of soft subsets and discovered that soft sets satisfy new algebraic properties. By the combination of soft sets and fuzzy sets, Maji et al. [33] proposed a hybrid concept named as fuzzy soft sets. Later on, several more complicated extensions of soft sets have been developed and investigated [34–38]. Ali and Shabir [39] developed some logic connectives in (fuzzy) soft set theory. In [40], a distance-based algorithm was designed for fuzzy soft set parameter reduction. Several works pointed out that rough sets, soft sets, and fuzzy sets are closely connected models [41–43]. They model uncertainty from independent perspectives, namely, gradualness, granularity, and parameterization. Feng et al. initiated several hybrid structures combining rough sets, soft sets, and fuzzy sets [44]. Taking a soft set as the underlying granulation structure, Feng et al. [45] proposed soft rough sets. Soft sets and related extensions have been widely used in many distinct domains, such as decision-making [46–51], valuation of assets [52], clustering [53], medical diagnosis [54], parameter reduction [55], feature selection [56], data analysis [57], BCK/BCI-algebras [58–60], graph theory [61], and computational biology [62]. The reader is referred to John’s latest monograph [63] for more details regarding soft set theory and its applications.With the assistance of soft set theory, Herawan and Deris [64] made an innovative proposal of identifying association rules from transaction data sets. Their pioneering work opened up a new research direction, aiming at developing soft set-based approach to rule extraction. Some concepts were first introduced in [65] to study the approximate reasoning theory based on soft sets, inclusive of logical formulas over soft sets, and basic soft truth degree of formulas. Feng et al. [66] revisited Herawan and Deris’s initial idea and refined several important notions to promote (maximal) association rule mining by virtue of soft set theory. Two important observations motivate us to continue this line of exploration:(1)
The ignorance of the temporal aspect of data in the abovementioned association rule extraction approaches [64, 66] may cause some limitations. For instance, some item sets are indeed frequent within certain time periods, even if they are not frequent in the whole data set and the entire time-span. Nonetheless, it is meaningful to discover such item sets since a commodity may sell exceptionally well in a specific season but not during the rest of the year.(2)
The identification of temporal frequent item sets, as an essential step in TARM process, can be facilitated by integrating time as a new component into soft set theory. In fact, the BitMap Coding (BMC) tree [10] must be built to generate node sets corresponding to frequent 1-item sets in the NegNodeset-based frequent item set mining process. The bit value at the index of each temporal frequent 1-item set can be combined to form the bitmap code of a temporal frequent item set. This indicates that temporal soft sets and Q-clip soft sets to be introduced in current work will provide a helpful apparatus for the construction of BMC trees.To address these issues, the current study focuses on enhancing association rule extraction with the aid of temporal soft sets. The main contributions of this study are summarized as follows:(1)
We define some new concepts such as temporal granulation mappings, temporal soft sets andQ-clip soft sets in order to establish a conceptual framework for extracting temporal association rules(2)
We present a number of useful characterizations and results within the established framework, including a necessary and sufficient condition for fast identification of strong temporal association rules(3)
We develop an effective approach, called negFIN-STARM, to extract strong temporal association rules by virtue of temporal soft sets and NegNodeset-based frequent item set miningThe rest of this paper is arranged in the following way: Section2 provides the rudiments with regard to TARM. Section 3 proposes several fundamental notions such as temporal soft sets and Q-clip soft sets. Section 4 focuses on soft temporal association rule mining to develop the negFIN-STARM approach. Section 5 is devoted to numerical experiments and comparative analysis of four different methods for extracting temporal association rules. Section 6 concludes this research and points out future research directions.


## 2. Temporal Association Rules

This section focuses on temporal association rules. First, we quote some fundamental definitions from [19].Definition 1 (see [19]).
An item endowed with a time-stamp is called a temporal item. A temporal item set means a nonempty setI˙ of temporal items.Definition 2 (see [19]).
Assume thatT is a set of transactions on a temporal item set I˙ and the positive integer α is the selected support threshold. Then, we say that I˙ is a temporal frequent item set with respect to T and α, when supportTI˙≥α.Definition 3 (see [19]).
A pair of disjoint temporal item sets is called a temporal association rule (TAR). LetRHS and LHS, respectively, represent the right and left temporal item sets. Then, we denote a TAR by LHS>ΔRHS, where the time-stamp of every temporal item in LHS precedes that, of any temporal item in RHS, Δ is the interval of two different time-stamps.Since temporal items in a transaction are associated with respective time-stamps, TARs can be generated by finding temporal frequent item sets in the temporal transaction set with intervalΔ. TARs defined in [19] are therefore useful for capturing temporal dependence among items within different time spans.Nevertheless, it is also interesting to see that some item sets are indeed frequent within certain period of time, even if they are not frequent in the whole data set during the entire time-span. In order to better describe such cases, we revisit some basic concepts in TARM and refine them in what follows.Suppose thatI=i1,…,iI is an item domain. Any subset t of I is a transaction, and a transaction data set consists of a set D=t1,…,tD formed by all transactions under inspection. Each transaction in D has a unique transaction identifier (TID).In classical association rule extraction, an item setX is a subset of I. When it is formed by k distinct items, we call it a k-item set. To simplify notation, the item set is is denoted by is. An item set X appears in t (alternatively, t supports X, when X⊆t).Now, letP=p1,p2,…,pP be a collection of pairwise disjoint periods of time. If t∈D is related to a unique period pt=τt∈P (indicating that t occurs during the period pt), then pt is called the period marker of t. In fact, this defines a mapping τ from D to P such that τt=pt. In what follows, T=D,τ,P is called a temporal transaction data set.Definition 4.
Assume thatT=D,τ,P is a temporal transaction data set, t∈D, Q⊆P, and X is an item set. Then, t supports X in T during a period in Q if X⊆t and τt∈Q.Definition 5.
Assume thatT=D,τ,P is a temporal transaction data set, Q⊆P and Z is an item set. The set(1)ΔTQZ=t∈D:τt∈Q∧Z⊆t,is the temporal realization of Z in T during a period in Q.The setΔTQZ consists of all the transactions in D which contain all the items in Z and occur during a period in Q. The cardinality of this set is written as STQZ, called the temporal support of Z in T during a period in Q. For simplicity, ΔTpkZ and STpkZ are written as ΔTpkZ and STpkZ, respectively.Definition 6.
LetT=D,τ,P be a temporal transaction data set and Q⊆P. Given two disjoint nonempty item sets G,Z⊆I, an expression G⟹QZ is called a temporal association rule (TAR).We refer toZ and G as the consequent and antecedent of the rule G⟹QZ. The rule G⟹pkZ is simply written as G⟹pkZ.Definition 7.
LetT=D,τ,P be a temporal transaction data set, Q⊆P and G⟹QZ be a TAR. Then, the temporal realization of G⟹QZ in T is given by(2)ΔTQG⟹QZ=ΔTQG∪Z.The cardinality of the setΔTQG∪Z, denoted by STQG⟹QZ, is called the temporal support of G⟹QZ.Definition 8.
The temporal confidence of a TARG⟹QZ is given by(3)CTQG⟹QZ=STQG⟹QZSTQG=STQG∪ZSTQG.
In particular,CTQG⟹QZ=0 if STQG=0.The temporal confidence serves as an essential measure in the evaluation of temporal association rules. It reflects the strength of the association between antecedent and the consequent of a TAR during concerned periods.For simplicity,ΔTpkG⟹pkZ, STpkG⟹pkZ, and CTpkG⟹pkZ are written as ΔTpkG⟹pkZ, STpkG⟹pkZ, and CTpkG⟹pkZ, respectively. Let ℕ∗ stand for the set of all positive integers. To find significant and interesting TARs from a temporal transaction data set T=D,τ,P, the users or experts should specify the minimum temporal support (min-TS) αQ∈ℕ∗ and the minimum temporal confidence (min-TC) βQ∈0,1 for a given subset Q of P. An item set Z is temporal frequent during a period in Q if STQZ≥αQ. A TAR G⟹QZ is frequent during a period in Q if STQG⟹QZ≥αQ. If CTQG⟹QZ≥βQ, G⟹QZ is a confident TAR during a period in Q. A TAR G⟹QZ is strong during a period in Q if it is both frequent and confident.The next example illustrates some concepts mentioned above.Example 1.
Consider a temporal transaction data set adapted from [25]. Let us assume that T=D,τ,P be a sample temporal transaction data set, where D=t1,t2,…,t16 consisting of all the transactions. Assume that every t∈D is related to a unique period pt=τt∈P, where P=p1,p2,p3,p4. From Table 1, it can be seen that T is divided into four parts by P. For example, the item set γ appears in the transaction t2 and t3 during the period p1.
Now, let us consider the subsetQ=p1 of P. By Definition 5, we have ΔTp1γ=t2,t3 and STp1γ=2. In a similar fashion, ΔTp1δ=t1,t2,t4 and STp1δ=3. In addition, the 2-item set γ,δ appears in the transaction t2, and transaction t2 occurs during the period p1. Thus by Definition 4, we can say that t2 supports γ,δ in T during the period p1. Also, it is clear that ΔTp1γ,δ=t2 and STp1γ,δ=1.
Next, we consider the TARγ⟹p1δ. By Definition 7, its temporal realization in T is(4)ΔTp1γ⟹p1δ=ΔTp1γ∪δ=t2,and the temporal support of this rule is(5)STp1γ⟹p1δ=STp1γ∪δ=1.
By Definition8, the temporal confidence of this rule is(6)CTp1γ⟹p1δ=STp1γ⟹p1δSTp1γ=STp1γ∪δSTp1γ=50%.
Finally, assume thatαp1=1 and βp1=35%. We conclude that γ⟹p1δ is a strong TAR during the period p1.Table 1
An example of a temporal transaction data set.
PeriodTIDItemsp1t1δp1t2γ,δp1t3γp1t4δp2t5α,γ,δp2t6α,β,γ,δp2t7β,γ,δp2t8α,δp3t9βp3t10α,γp3t11α,β,γp3t12β,γp4t13β,δp4t14β,γ,δp4t15βp4t16β,γ,δ


## 3. Temporal Soft Sets

In this section, we define some new concepts such as temporal granulation mappings, temporal soft sets, andQ-clip soft sets which will play a role of fundamental importance in this study. In the following, ℧ represents a universal set of objects and E stands for the parameter space consisting of all parameters associated with objects in ℧. The power set of ℧ is written as 2℧.Definition 9 (see [28]).
A soft setG=F^,K over ℧ is an ordered pair, in which K⊆E and F^:K⟶2℧ is called the approximate function of G.Definition 10 (see [67]).
Assume that℧ and K are nonempty finite sets of alternatives and attributes, respectively. The pair ℐ=℧,K is called an information system (IS), when every attribute k∈K can be identified with an information function k:℧⟶Vk and Vk is the value set of k.WhenG=F^,K is a soft set over ℧, it naturally induces an IS ℐG=℧,K in the following fashion. Given every k∈K and v∈℧, associate the corresponding information function k:℧⟶Vk=0,1 as follows:(7)kv=0,whenv∉F^k,1,otherwise.Definition 11.
LetP be a set of pairwise disjoint periods of time. Then, τ:℧⟶P is called a temporal granulation mapping.Definition 12.
A temporal soft set (TSS) over℧ is a quadruple T=F^,K,τ,P such that(1)
F^,K is a soft set over ℧(2)
τ:℧⟶P is a temporal granulation mappingThe soft setF^,K is said to be the underlying soft set (USS) of the TSS T. We also refer to ℘=P,τ as a temporal granulation of ℧. The TSS T, as an abstract representation of data, can additionally capture temporal information, which is unable to be expressed by its underlying soft set.Definition 13.
Assume thatT=F^,K,τ,P is a TSS over ℧ and Q⊆P. Then the Q-clip of T is a soft set TQ=G^,K over(8)τ−1Q=u∈℧|τu∈Q,where G^k=F^k∩τ−1Q for all k∈K.Note thatp-clip soft set is simply called p-clip soft set. Next, we consider an example that illustrates the abovementioned notions.Example 2.
The Nobel Prizes are awarded annually to individuals and organizations in recognition of outstanding contributions in several categories: literature, chemistry, physics, physiology or medicine, and peace. In the following, we focus on three types of prizes, which are the Nobel Prizes in Physics (NPP), Physiology or Medicine (NPPM), and Chemistry (NPC).
We consider(9)℧=v1,v2,…,v9,as a universal set that consists of all Nobel Prizes in scientific categories, namely, NPP, NPPM, and NPC awarded between 1901 and 1903. Detailed information regarding these prizes can be found in Table 2. Suppose that C=c1,c2,…,c6 is a set of parameters, containing all the affiliation countries associated with the prizes in ℧. More specifically, let cii=1,2,…,6 stand for “Denmark,” “France,” “Germany,” “The Netherlands,” “Sweden,” and “United Kingdom,” respectively. Based on the information in Table 2, we can construct a soft set F^,C over ℧, with its approximate function defined as F^c1=v9, F^c2=v8, F^c3=v1,v2,v3,v4, F^c4=v5, F^c5=v7, and F^c6=v6.
In addition, a temporal granulation℘=Y,τ of ℧ can be derived from Table 2 in a natural way. In fact, let Y=y1,y2,y3 with yi=1900+i for i=1,2,3. Then, the temporal granulation mapping τ:℧⟶Y is given by(10)τv3=τv2=τv1=y1=1901,(11)τv6=τv5=τv4y2=1902,(12)τv9=τv8=τv7=y3=1903.
The intuitive meaning ofτ is apparent. For instance, equation (10) says that the prizes v1, v2, and v3 were bestowed in 1901. With this mapping, we can construct a TSS T=F^,C,τ,Y over ℧, as shown in Table 3. As seen from the equations (10)–(12), the temporal granulation mapping τ:℧⟶Y induces a partition of ℧ as follows:(13)τ−1yi|yi∈Y=v4,v5,v6,v7,v8,v9,v1,v2,v3.
Finally, by Definition13, the yi-clip soft set Tyi=G^i,C of the TSS T for i=1,2,3 are as follows:(1)
They1-clip of T is a soft set Ty1=G^1,C over τ−1y1, where G^1c3=τ−1y1=v1,v2,v3 and G^1cj=∅  for all cj∈C with j≠3(2)
They2-clip of T is a soft set Ty2=G^2,C over τ−1y2, where G^2c3=v4, G^2c4=v5, G^2c6=v6, and G^2cj=∅  for all cj∈C with j≠3,4,6(3)
They3-clip of T is a soft set Ty3=G^3,C over τ−1y3, where G^3c1=v9, G^3c2=v8, G^3c5=v7, and G^3cj=∅  for all cj∈C with j≠1,2,5Table 2
Information on NPCs, NPPs, and NPPMs (1901–1903).
AwardYear of awardPrize categoryAffiliation countryv11901NPCGermanyv21901NPPGermanyv31901NPPMGermanyv41902NPCGermanyv51902NPPThe Netherlandsv61902NPPMUnited Kingdomv71903NPCSwedenv81903NPPFrancev91903NPPMDenmarkTable 3
Tabular representation of the TSST=F^,C,τ,Y.
℧c1c2c3c4c5c6τ⋅v1001000y1v2001000y1v3001000y1v4001000y2v5000100y2v6000001y2v7000010y3v8010000y3v9100000y3


## 4. Soft Temporal Association Rule Mining

This section aims to establish a formal framework for mining TARs by means of TSSs. LetP be a set of pairwise disjoint periods of time and Q⊆P throughout this section.Definition 14.
(see [66]). Assume that G=F^,K is a soft set over ℧ and v∈℧. Then, we call(14)CoGv=k∈K:v∈F^k,as the parameter coset of the alternative v in G.It can be seen thatCoGv contains all the parameters that the alternative v meets, according to the information contained in G.Definition 15.
Assume thatT=F^,K,τ,P is a TSS over ℧ with its USS G=F^,K. For any ∅≠B⊆K,(15)ΔTQB=v∈τ−1Q:B⊆CoGv,is called the Q-realization of B in the TSS T.Whenv0∈ΔTQB, it is said that B is Q-supported by the alternative v0∈℧. The Q-support of B in T is the cardinality of ΔTQB, represented by suppTQB. Note that ΔTpB and suppTpB are respectively written as ΔTpB and suppTpB.Definition 16.
Assume thatT=F^,K,τ,P is a TSS over ℧ and G,Z are two disjoint non-empty subsets of K. We call the expression G⟹QZ as a temporal association rule (TAR) in the TSS T. The non-empty parameter sets Z and G are respectively called consequent and antecedent of the TAR G⟹QZ.Definition 17.
Suppose thatT=F^,K,τ,P is a TSS over ℧ and G⟹QZ is a TAR in T. We refer to(16)ΔTQG⟹QZ=ΔTQG∪Z,as the Q-realization of G⟹QZ in the TSS T.TheQ-support of G⟹QZ, written as suppTQG⟹QZ, is the cardinality of ΔTQG⟹QZ. For convenience, G⟹pZ, ΔTpG⟹pZ and suppTpG⟹pZ are simply written as G⟹pZ, ΔTpG⟹pZ and suppTpG⟹pZ, respectively.Proposition 1.
Assume thatT=F^,K,τ,P is a TSS over ℧ and fix ∅≠B⊆K. Then(17)ΔTQB=∩k∈BF^k∩τ−1Q.Proof.
We denote byG the USS of the TSS T=F^,K,τ,P. Let x∈ΔTQB. Equation (15) assures B⊆CoGx and x∈τ−1Q. By Definition 14, x∈F^k when k∈B. Thus we have(18)x∈∩k∈BF^k∩τ−1Q.
This proves(19)ΔTQB⊆∩k∈BF^k∩τ−1Q.
Now, suppose thaty∈∩k∈BF^k∩τ−1Q. Then y∈τ−1Q and y∈F^k for any k∈B. From the definition of the parameter coset CoGy, it follows that B⊆CoGy and y∈τ−1Q. Hence y∈ΔTQB, which also shows that(20)∩k∈BF^k∩τ−1Q⊆ΔTQB.
Therefore we derive that(21)ΔTQB=∩k∈BF^k∩τ−1Q.
This ends the proof.
By Proposition1, the following results can be deduced.Corollary 1.
Assume thatT=F^,K,τ,P is a TSS over ℧ and TQ=G^,K is its Q-clip soft set. Then we have(22)ΔTQB=∩k∈BG^k,for all non-empty subset B of K.Corollary 2.
Assume thatT=F^,K,τ,P is a TSS over ℧ and K1,K2 are subsets of K. Then we have(23)K1⊆K2⟹ΔTQK2⊆ΔTQK1.Proposition 2.
Assume thatT=F^,K,τ,P is a TSS over ℧ and G⟹QZ is a TAR in T. Then,(24)ΔTQG⟹QZ=ΔTQG∩ΔTQZ=∩k∈G∪ZF^k∩τ−1Q.Proof: .
For simplicity, letJ1 and J2 stand for ∩k∈GF^k and ∩k∈ZF^k, respectively. According to Definition 17,(25)ΔTQG⟹QZ=ΔTQG∪Z.
By Proposition1, we have(26)ΔTQG⟹QZ=∩k∈G∪ZF^k∩τ−1Q=J1∩J2∩τ−1Q=J1∩τ−1Q∩J2∩τ−1Q=ΔTQG∩ΔTQZ.
This ends the proof.Remark 1.
The above assertion reveals that theQ-realization of a TAR G⟹QZ in a TSS coincides with the intersection of the Q-realizations of the consequent and antecedent of G⟹QZ.By Proposition2, the following results can be deduced.Corollary 3.
Assume thatT=F^,K,τ,P is a TSS over ℧ and TQ=G^,K is its Q-clip soft set. Then,(27)ΔTQG⟹QZ=∩k∈G∪ZG^k,where G⟹QZ is a TAR in T.Corollary 4.
Assume thatT=F^,K,τ,P is a TSS over ℧ and G⟹QZ is a TAR in T. Then,(28)minsuppTQG,suppTQZ≥suppTQG⟹QZ.Definition 18.
Assume thatT=F^,K,τ,P is a TSS over ℧ and G⟹QZ is a TAR in T. The Q-confidence of G⟹QZ is given by(29)confTQG⟹QZ=suppTQG⟹QZsuppTQG,ifsuppTQG≠0,0,ifsuppTQG=0.For convenience,confTpG⟹pZ is simply written as confTpG⟹pZ.Theorem 1.
Assume thatT=F^,K,τ,P is a TSS over ℧ and G⟹QZ is a TAR in T. Then, G⟹QZ is strong during a period in Q if and only if(30)suppTQG⟹QZ≥maxαQ,βQ⋅suppTQG,where αQ∈ℕ∗ is the min-TS and βQ∈0,1 is the min-TC.Proof.
Suppose thatG⟹QZ is strong in T during a period in Q. Then, we have(31)suppTQG⟹QZ≥αQ,confTQG⟹QZ=suppTQG⟹QZsuppTQG≥βQ.
It follows that(32)suppTQG⟹QZ≥βQ⋅suppTQG.
Thus, we have(33)suppTQG⟹QZ≥maxαQ,βQ⋅suppTQG.
Conversely, letG⟹QZ be a temporal association rule in T such that(34)suppTQG⟹QZ≥maxαQ,βQ⋅suppTQG.
It follows that(35)suppTQG⟹QZ≥αQ,suppTQG⟹QZ≥βQ⋅suppTQG.
Hence, we deduce that(36)suppTQG⟹QZsuppTQG=confTQG⟹QZ≥βQ.
Thus,G⟹QZ is strong in T during a period in Q, completing the proof.
Using the aforementioned concepts and results, we can obtain the following result.Proposition 3.
Suppose thatT=F^,K,τ,P is a TSS over ℧ and G⟹QZ is a TAR in T with suppTQG⟹QZ=suppTQG. Then, the following are equivalent:(1)
G is temporal frequent during a period in Q(2)
G⟹QZ is frequent during a period in Q(3)
G⟹QZ is confident during a period in Q(4)
G⟹QZ is strong during a period in QTo illustrate the new notions above, we consider the following example, which is a continuation of Example2.Example 3.
Assume thatA=a1,a2,a3 is a set of parameters, consisting of the three types of prizes under consideration, i.e., a1 is NPC, a2 is NPP, and a3 is NPPM. Before using the proposed concepts regarding soft temporal association rule mining for mathematical modeling and analysis, we now first establish another TSS based on the information in Table 2. The TSS T=F^,B,τ,Y over ℧ is shown in Table 4, where the parameter set B=C∪A and the temporal granulation mapping τ:℧⟶Y is identical with what is defined in Example 2. In what follows, let us consider three different cases in which Q1=y1,y2, Q2=y2, and Q3=Y=y1,y2,y3, respectively. Suppose that the min-TS αQ1=αQ2=1 and αQ3=2. The min-TC βQi=75% for i=1,2,3.
Let us first focus on the case whenQ1=y1,y2. Recall first that τ−1Q1=v1,v2,…,v6. By Definition 13, the Q1-clip of the TSS T is a soft set TQ1=G1^,B over τ−1Q1, where G1^c3=v1,v2,v3,v4, G1^c4=v5, G1^c6=v6, G1^a1=v1,v4, G1^a2=v2,v5, G1^a3=v3,v6 and G1^cj=∅  for all cj∈C with j=1,2,5. By Proposition 1 and Corollary 1, we can easily get(37)ΔTQ1a1=F^a1∩τ−1Q1=G1^a1=v1,v4,ΔTQ1c3=F^c3∩τ−1Q1=G1^c3=v1,v2,v3,v4.
Next, we consider the TARa1⟹Q1c3. By Proposition 2 and Corollary 3, its Q1-realization in T can be calculated as follows:(38)ΔTQ1a1⟹Q1c3=ΔTQ1a1∩ΔTQ1c3=F^a1∩F^c3∩τ−1Q1=G1^a1∩G1^c3=v1,v4.
In fact, as indicated by Corollary3, the Q1-realization of the TAR a1⟹Q1c3 in the TSS T is completely determined by the approximate function of the corresponding Q1-clip TQ1=G1^,B. It is clear that the Q1-support of this rule is(39)suppTQ1a1⟹Q1c3=2.
By Definition18, the Q1-confidence of this rule is(40)confTQ1a1⟹Q1c3=suppTQ1a1⟹Q1c3suppTQ1a1=100%.
Hence, by definition,a1⟹Q1c3 is strong during a period in Q1. On the other hand, we can draw the same conclusion from Theorem 1 since(41)suppTQ1a1⟹Q1c3>maxαQ1,βQ1⋅suppTQ1a1=max1,1.5=1.5.
Note also that(42)suppTQ1a1⟹Q1c3=suppTQ1a1=2,suppTQ1a1>αQ1=1.
Thus, we can also conclude thata1⟹Q1c3 is strong during a period in Q1 by Proposition 3. This rule indicates that “From 1901 to 1902, all the Nobel Prizes in Chemistry were awarded to Germany.” Conversely, we can consider the TAR c3⟹Q1a1. Its Q1-support is(43)suppTQ1c3⟹Q1a1=2>αQ1,but its Q1-confidence is(44)confTQ1c3⟹Q1a1=50%<βQ1=75%.
Hence, this rule is frequent but not confident during a period inQ1. It reveals that “From 1901 to 1902, 50% of the Nobel Prizes awarded to Germany pertain to the category of chemistry.”
Now, let us consider the second case whenQ2=y2. Similarly, we can get(45)suppTy2c3⟹y2a1=suppTy2c3=αy2=1,confTy2c3⟹y2a1=100%>βy2=75%.
Hence, we conclude thatc3⟹y2a1 is strong during the period y2. This rule says that “In 1902, Germany was only awarded the NPC, instead of the NPP or NPPM.”
Finally, we consider the third case whenQ3=Y=y1,y2,y3. Clearly, τ−1Q3=v1,v2,…,v9=℧ in this case. It follows that the Q3-clip of the TSS T is a soft set TQ3=G3^,B over ℧, which coincide with the USS F^,B of the TSS T. That is, G3^e=F^e for all e∈B. By Proposition 1 and Corollary 1, we have(46)ΔTQ3a3=F^a3∩℧=G3^a3=v3,v6,v9,ΔTQ3c6=F^c6∩℧=G3^c6=v6.
Next, let us consider the TARa3⟹Q3c6, which can also be seen as an association rule a3⟹c6 in conventional sense. By Proposition 2 and Corollary 3, its Q3-realization in T can be calculated as follows:(47)ΔTQ3a3⟹Q3c6=ΔTQ3a3∩ΔTQ3c6=F^a3∩F^c6∩℧=G3^a3∩G3^c6=v6.
Obviously,suppTQ3a3⟹Q3c6=1. By Definition 18, we also have(48)confTQ3a3⟹Q3c6=suppTQ3a3⟹Q3c6suppTQ3a3=13.
It is clear that(49)suppTQ3a3⟹Q3c6=1<αQ3=2,confTQ3a3⟹Q3c6=13<βQ3=0.75.
Thus, we deduce thata3⟹Q3c6 is neither frequent nor confident during a period in Q3. This rule reveals that “From 1901 to 1903, only one Nobel Prize in Physiology or Medicine was awarded to the United Kingdom.” In addition, it can be seen that the rule c3⟹Q3a3 is neither frequent nor confident during a period in Q3 since(50)suppTQ3c3⟹Q3a3=1<αQ3=2,confTQ3c3⟹Q3a3=0.25<βQ3=0.75.
This rule says that “From 1901 to 1903, only a quarter of the Nobel Prizes awarded to Germany pertain to the category of physiology or medicine.”
Compared with the case ofQ3 consisting of all time periods, we see that some rules such as a1⟹Q1c3 and c3⟹y2a1 can only be identified as strong TARs when we restrict to the cases of Q1 or Q2 consisting of fewer time periods. This is mainly due to the fact that some item sets can be frequent during certain time periods rather than all of them. In a nutshell, we conclude that the TARM based on TSSs can help find some strong TARs which might be ignored in conventional rule extraction process.
Based on the results obtained in this section and the concepts such as TSSs andQ-clip soft sets proposed in Section 3, we present a novel TARM method by combining NegNodeset-based frequent item set mining with TSS-based rule mining. Our method will be abbreviated as negFIN-STARM in the sequel. The pseudocode description of the negFIN-STARM method is given in Algorithm 1. This algorithm takes a temporal transaction data set T, a set Q⊆P, the min-TS αQ, and the min-TC βQ as the input. The output of Algorithm 1 is the class SRTART,Q,αQ,βQ, which contains all strong TARs during a period in Q. The main procedure of the negFIN-STARM method can be divided into three stages:(1)
In the first stage, we construct a TSST=F^,I,τ,P over D from the provided temporal transaction data set T. Then, according to Definition 13, we determine τ−1Q and construct the Q-clip soft set TQ=H^,I of the TSS T. Next, we derive the IS ℐTQ=τ−1Q,I from the Q-clip soft set TQ=H^,I of T.(2)
In the second stage, NegNodeset-based frequent item set mining technique and temporal soft sets are combined for generating all temporal frequent item sets. More specifically, we first employ the information function ofℐTQ=τ−1Q,I to construct the BMC tree. Then, the Nodesets of all frequent 1-item sets are generated by traversing the BMC tree. Furthermore, we identify the NegNodesets of all frequent k-item sets k≥2. Eventually, the set-enumeration tree is built to generate the class TFISTQ,αQ, which consists of all temporal frequent item sets. These item sets will function as potential consequents and antecedents for finding strong TARs. Here, we would like to emphasize a crucial issue. To apply the NegNodeset-based frequent item set mining, the BMC tree must be built to generate the node set related to every frequent 1-item set. Each frequent item set is represented by a bitmap code, and every frequent 1-item set is mapped to one of its bits. In other words, the bit value at the corresponding index of each temporal frequent 1-item set can be combined to form the bitmap code of the temporal frequent item set. It is worth noting that the use of TSSs and Q-clip soft sets can facilitate the calculation of bitmap codes and the construction of BMC trees in this important stage.(3)
In the last stage, by Corollary3, we can calculate the Q-realization of G⟹QZ using the Q-clip soft set TQ for all G,Z∈TFISTQ,αQ which are disjoint. Next, by Theorem 1, it is easy to check whether or not the G⟹QZ is strong during a period in Q. If this is true, then we put G⟹QZ into the class SRTART,Q,αQ,βQ.Table 4
Tabular representation of the TSST=F^,B,τ,Y.
℧c1c2c3c4c5c6a1a2a3τ⋅v1001000100y1v2001000010y1v3001000001y1v4001000100y2v5000100010y2v6000001001y2v7000010100y3v8010000010y3v9100000001y3Algorithm 1: The negFIN-STARM method.
Input: a temporal transaction data set T=D,τ,P, a set Q⊆P, the min-TS αQ∈ℕ∗, and the min-TC βQ∈0,1.Output: the class SRTART,Q,αQ,βQ that contains all strong TARs during a period in Q.(1)
Construct a TSST=F^,I,τ,P over D from the temporal transaction data set T with the item domain I(2)
Calculateτ−1Q and construct the Q-clip soft set TQ=H^,I of T(3)
Construct the ISℐTQ=τ−1Q,I induced by TQ=H^,I(4)
Construct the BMC tree byℐTQ=τ−1Q,I(5)
Traverse the BMC tree to get the Nodesets of all frequent 1-item sets(6)
Identify the NegNodesets of all frequentk-item sets k≥2(7)
Build the set-enumeration tree to generate the classTFISTQ,αQ, which consists of all temporal frequent item sets(8)
forG∈TFISTQ,αQdo(9)
forZ∈TFISTQ,αQdo(10)
ifG∩Z=∅   then(11)
CalculateΔTQG⟹QZ=∩a∈G∪ZH^a;(12)
end if(13)
ifsuppTQG⟹QZ=ΔTQG⟹QZ≥maxαQ,βQ⋅suppTQG  then(14)
PutG⟹QZ into SRTART,Q,αQ,βQ;(15)
end if(16)
end for(17)
end for(18)
returnSRTART,Q,αQ,βQ;


## 5. Numerical Experiments

In this section, we conduct numerical experiments on the commonly used chess and mushroom data sets to compare the performance of our newly presented negFIN-STARM approach with three well-known approaches in the literature, namely, the T-Apriori [23], T-FPGrowth [8], and T-ECLAT methods [9]. Hereinafter, the abbreviations such as T-Apriori, T-FPGrowth, and T-ECLAT stand for Temporal Apriori, Temporal FPGrowth, and Temporal Eclat, respectively.


### 5.1. Running Environment

The numerical experiment was conducted on a laptop computer equipped with a 2.00 GHz Intel Core i7 processor and 8 GB of RAM running the 64-bit Microsoft Windows 10 operating system. The algorithms are coded in Java 13.0.1 using IntelliJ IDEA 2019.2.2. The performance of the selected methods is evaluated by the runtime over the aforementioned data sets. For higher accuracy, the codes corresponding to the four methods were executed 5 times under the same conditions. The comparison is made in terms of the average values of the runtime.


### 5.2. Description of Data Sets

Two commonly used data sets are employed for comparing our method with existing methods mentioned above. These data sets are available from the open-source data mining library SPMF (The SPMF library athttp://www.philippe-fournier-viger.com/spmf/.) founded by Philippe Fournier-Viger. The first data set is the chess data set adapted based on the UCI chess data set. The second one is the mushroom data drawn from The Audubon Society Field Guide to North American Mushrooms. Table 5 gives a basic description of these data sets.Table 5
Basic description of the data sets.
Data setTypeItemsTransactionsPeriodsChessReal7531965MushroomReal119812417


### 5.3. Results and Comparative Analysis

At first, we conduct numerical experiments and comparative analysis of four different methods using the chess data set. This data set contains 3196 transactions, each uniquely related to a period inP=p1,p2,p3,p4,p5. There are 75 different items in the item domain of this data set. We consider the cases when Q=p1 and Q=p5. The min-TS and min-TC are simply denoted by αpi and βpii=1,5, respectively. The runtime comparison based on the chess data set of four methods under different thresholds is shown in Figure 2. More details regarding the average runtime (in milliseconds) of four methods on the chess data set are listed in Tables 6 and 7.Figure 2
The runtime comparison of four methods based on the chess data set. (a)Q=p1 and αp1=500. (b) Q=p1 and βp1=85%. (c) Q=p5 and αp5=500. (d) Q=p5 and βp5=85%.
(a)(b)(c)(d)Table 6
Execution time (ms) of four methods on the chess data set whenQ=p1.
Methodαp1=500βp1=85%βp1=75%βp1=85%βp1=95%αp1=400αp1=450αp1=500negFIN-STARM3684.7863342.8241960.21434 331.8579424.5313342.824T-FPGrowth3867.4533535.8032164.63634 692.8639634.3653535.803T-ECLAT4266.3503924.4202547.60137 159.24710 479.5333924.420T-apriori5020.1864732.6393408.66753 857.78512 905.5584732.639Table 7
Execution time (ms) of four methods on the chess data set whenQ=p5.
Methodαp5=500βp5=85%βp5=75%βp5=85%βp5=95%αp5=400αp5=450αp5=500negFIN-STARM1861.9701819.1801566.78211 266.1453901.7631819.180T-FPGrowth1943.7521905.2751646.24311 556.5464049.5391905.275T-ECLAT2100.1312094.2291822.63712 908.2274460.3292094.229T-apriori2861.2862844.4422582.78718 667.5177362.8132844.442As shown in Figure2(a), the negFIN-STARM method is faster than the T-Apriori, T-ECLAT, and T-FPGrowth methods when Q=p1, the min-TS αp1=500, and the min-TC βp1 is 75%, 85%, and 95%, respectively. In addition, Figure 2(b) illustrates that the negFIN-STARM method runs faster than the other methods when βp1=85% and αp1 is designated as 400, 450, and 500, respectively.From Figure2(c), we see that the negFIN-STARM method is faster than the T-Apriori, T-ECLAT, and T-FPGrowth methods when Q=p5, αp5=500, and βp5 is designated as 75%, 85%, and 95%, respectively. In addition, Figure 2(d) illustrates that our new method performs better than those existing methods when βp5=85% and αp5 is set as 400, 450, and 500, respectively.Furthermore, the quantity comparison of the obtained TARs based on the chess data set under different thresholds is demonstrated in Figure3. In brief, we can find that the rule number decreases when the threshold increases.Figure 3
The quantity comparison of TARs based on the chess data set. (a)Q=p1 and αp1=500. (b) Q=p1 and βp1=85%. (c) Q=p5 and αp5=500. (d) Q=p5 and βp5=85%.
(a)(b)(c)(d)More specifically, Figure3(a) shows that if Q=p1, the min-TS αp1=500, and the min-TC βp1 is specified as 75%, 85%, and 95%, the number of TARs is 1481 178, 1290 849, and 348 100, respectively. In addition, as illustrated in Figure 3(b), when the min-TC βp1=85%, and αp1 is specified as 400, 450, and 500, the rule number is 26 632 597, 6750 662, and 1290 849, respectively.Figure3(c) shows that when Q=p5, αp5=500, and βp5 is specified as 75%, 85%, and 95%, the rule number is 312 092, 282 769, and 76 564, respectively. In addition, as shown in Figure 3(d), when βp5=85% and αp5 is specified as 400, 450, and 500, the rule number is 9024 350, 1926 626, and 282 769, respectively.Similarly, we also conduct numerical experiments and comparative analysis of four methods using the mushroom data set. This data set contains 8124 transactions, each uniquely related to a period inP=p1,p2,…,p17. There are 119 different items in the item domain of this data set. We consider the cases when Q=p1 and Q=p10. The min-TS and min-TC are simply denoted by αpi and βpii=1,10, respectively. The runtime comparison with respect to the mushroom data set of four methods under different thresholds is shown in Figure 4. The quantity comparison of the obtained TARs based on the mushroom data set under different thresholds is illustrated in Figure 5. The average runtime (in milliseconds) of four methods on the mushroom data set is listed in Tables 8 and 9.Figure 4
The runtime comparison of four methods based on the mushroom data set. (a)Q=p1 and αp1=400. (b) Q=p1 and βp1=70%. (c) Q=p10 and αp10=300. (d) Q=p10 and βp10=70%.
(a)(b)(c)(d)Figure 5
The quantity comparison of TARs based on the mushroom data set. (a)Q=p1 and αp1=400. (b) Q=p1 and βp1=70%. (c) Q=p10 and αp10=300. (d) Q=p10 and βp10=70%.
(a)(b)(c)(d)Table 8
Execution time (ms) of four methods on the mushroom data set whenQ=p1.
Methodαp1=400βp1=70%βp1=50%βp1=70%βp1=90%αp1=300αp1=400αp1=500negFIN-STARM23 301.31416 805.88210 229.61920 402.43316 805.8824147.764T-FPGrowth23 471.04316 936.19110 358.53420 530.65816 936.1914228.420T-ECLAT24 027.20117 564.73010 956.09121 303.87917 564.7304496.445T-apriori24 912.99618 563.46511 896.17523 838.72518 563.4654757.468Table 9
Execution time (ms) of four methods on the mushroom data set whenQ=p10.
Methodαp10=300βp10=70%βp10=50%βp10=70%βp10=90%αp10=250αp10=300αp10=350negFIN-STARM24 591.73723 351.53622 622.41144 517.33723 351.53622 317.219T-FPGrowth24 691.05323 444.93422 721.84644 671.87323 444.93422 453.963T-ECLAT25 243.84523 984.70723 251.53745 474.46723 984.70722 990.058T-apriori26 033.49024 824.82824 074.11047 534.27324 824.82823 866.514The above numerical experiments demonstrate that our newly proposed method is a helpful apparatus for mining TARs. The comparative analysis illustrates that the negFIN-STARM method performs better than three well-known existing methods, which are the T-Apriori, T-FPGrowth, and T-ECLAT methods.


## 5.1. Running Environment

The numerical experiment was conducted on a laptop computer equipped with a 2.00 GHz Intel Core i7 processor and 8 GB of RAM running the 64-bit Microsoft Windows 10 operating system. The algorithms are coded in Java 13.0.1 using IntelliJ IDEA 2019.2.2. The performance of the selected methods is evaluated by the runtime over the aforementioned data sets. For higher accuracy, the codes corresponding to the four methods were executed 5 times under the same conditions. The comparison is made in terms of the average values of the runtime.


## 5.2. Description of Data Sets

Two commonly used data sets are employed for comparing our method with existing methods mentioned above. These data sets are available from the open-source data mining library SPMF (The SPMF library athttp://www.philippe-fournier-viger.com/spmf/.) founded by Philippe Fournier-Viger. The first data set is the chess data set adapted based on the UCI chess data set. The second one is the mushroom data drawn from The Audubon Society Field Guide to North American Mushrooms. Table 5 gives a basic description of these data sets.Table 5
Basic description of the data sets.
Data setTypeItemsTransactionsPeriodsChessReal7531965MushroomReal119812417


## 5.3. Results and Comparative Analysis

At first, we conduct numerical experiments and comparative analysis of four different methods using the chess data set. This data set contains 3196 transactions, each uniquely related to a period inP=p1,p2,p3,p4,p5. There are 75 different items in the item domain of this data set. We consider the cases when Q=p1 and Q=p5. The min-TS and min-TC are simply denoted by αpi and βpii=1,5, respectively. The runtime comparison based on the chess data set of four methods under different thresholds is shown in Figure 2. More details regarding the average runtime (in milliseconds) of four methods on the chess data set are listed in Tables 6 and 7.Figure 2
The runtime comparison of four methods based on the chess data set. (a)Q=p1 and αp1=500. (b) Q=p1 and βp1=85%. (c) Q=p5 and αp5=500. (d) Q=p5 and βp5=85%.
(a)(b)(c)(d)Table 6
Execution time (ms) of four methods on the chess data set whenQ=p1.
Methodαp1=500βp1=85%βp1=75%βp1=85%βp1=95%αp1=400αp1=450αp1=500negFIN-STARM3684.7863342.8241960.21434 331.8579424.5313342.824T-FPGrowth3867.4533535.8032164.63634 692.8639634.3653535.803T-ECLAT4266.3503924.4202547.60137 159.24710 479.5333924.420T-apriori5020.1864732.6393408.66753 857.78512 905.5584732.639Table 7
Execution time (ms) of four methods on the chess data set whenQ=p5.
Methodαp5=500βp5=85%βp5=75%βp5=85%βp5=95%αp5=400αp5=450αp5=500negFIN-STARM1861.9701819.1801566.78211 266.1453901.7631819.180T-FPGrowth1943.7521905.2751646.24311 556.5464049.5391905.275T-ECLAT2100.1312094.2291822.63712 908.2274460.3292094.229T-apriori2861.2862844.4422582.78718 667.5177362.8132844.442As shown in Figure2(a), the negFIN-STARM method is faster than the T-Apriori, T-ECLAT, and T-FPGrowth methods when Q=p1, the min-TS αp1=500, and the min-TC βp1 is 75%, 85%, and 95%, respectively. In addition, Figure 2(b) illustrates that the negFIN-STARM method runs faster than the other methods when βp1=85% and αp1 is designated as 400, 450, and 500, respectively.From Figure2(c), we see that the negFIN-STARM method is faster than the T-Apriori, T-ECLAT, and T-FPGrowth methods when Q=p5, αp5=500, and βp5 is designated as 75%, 85%, and 95%, respectively. In addition, Figure 2(d) illustrates that our new method performs better than those existing methods when βp5=85% and αp5 is set as 400, 450, and 500, respectively.Furthermore, the quantity comparison of the obtained TARs based on the chess data set under different thresholds is demonstrated in Figure3. In brief, we can find that the rule number decreases when the threshold increases.Figure 3
The quantity comparison of TARs based on the chess data set. (a)Q=p1 and αp1=500. (b) Q=p1 and βp1=85%. (c) Q=p5 and αp5=500. (d) Q=p5 and βp5=85%.
(a)(b)(c)(d)More specifically, Figure3(a) shows that if Q=p1, the min-TS αp1=500, and the min-TC βp1 is specified as 75%, 85%, and 95%, the number of TARs is 1481 178, 1290 849, and 348 100, respectively. In addition, as illustrated in Figure 3(b), when the min-TC βp1=85%, and αp1 is specified as 400, 450, and 500, the rule number is 26 632 597, 6750 662, and 1290 849, respectively.Figure3(c) shows that when Q=p5, αp5=500, and βp5 is specified as 75%, 85%, and 95%, the rule number is 312 092, 282 769, and 76 564, respectively. In addition, as shown in Figure 3(d), when βp5=85% and αp5 is specified as 400, 450, and 500, the rule number is 9024 350, 1926 626, and 282 769, respectively.Similarly, we also conduct numerical experiments and comparative analysis of four methods using the mushroom data set. This data set contains 8124 transactions, each uniquely related to a period inP=p1,p2,…,p17. There are 119 different items in the item domain of this data set. We consider the cases when Q=p1 and Q=p10. The min-TS and min-TC are simply denoted by αpi and βpii=1,10, respectively. The runtime comparison with respect to the mushroom data set of four methods under different thresholds is shown in Figure 4. The quantity comparison of the obtained TARs based on the mushroom data set under different thresholds is illustrated in Figure 5. The average runtime (in milliseconds) of four methods on the mushroom data set is listed in Tables 8 and 9.Figure 4
The runtime comparison of four methods based on the mushroom data set. (a)Q=p1 and αp1=400. (b) Q=p1 and βp1=70%. (c) Q=p10 and αp10=300. (d) Q=p10 and βp10=70%.
(a)(b)(c)(d)Figure 5
The quantity comparison of TARs based on the mushroom data set. (a)Q=p1 and αp1=400. (b) Q=p1 and βp1=70%. (c) Q=p10 and αp10=300. (d) Q=p10 and βp10=70%.
(a)(b)(c)(d)Table 8
Execution time (ms) of four methods on the mushroom data set whenQ=p1.
Methodαp1=400βp1=70%βp1=50%βp1=70%βp1=90%αp1=300αp1=400αp1=500negFIN-STARM23 301.31416 805.88210 229.61920 402.43316 805.8824147.764T-FPGrowth23 471.04316 936.19110 358.53420 530.65816 936.1914228.420T-ECLAT24 027.20117 564.73010 956.09121 303.87917 564.7304496.445T-apriori24 912.99618 563.46511 896.17523 838.72518 563.4654757.468Table 9
Execution time (ms) of four methods on the mushroom data set whenQ=p10.
Methodαp10=300βp10=70%βp10=50%βp10=70%βp10=90%αp10=250αp10=300αp10=350negFIN-STARM24 591.73723 351.53622 622.41144 517.33723 351.53622 317.219T-FPGrowth24 691.05323 444.93422 721.84644 671.87323 444.93422 453.963T-ECLAT25 243.84523 984.70723 251.53745 474.46723 984.70722 990.058T-apriori26 033.49024 824.82824 074.11047 534.27324 824.82823 866.514The above numerical experiments demonstrate that our newly proposed method is a helpful apparatus for mining TARs. The comparative analysis illustrates that the negFIN-STARM method performs better than three well-known existing methods, which are the T-Apriori, T-FPGrowth, and T-ECLAT methods.


## 6. Conclusions

This paper is devoted to enhancing association rule mining by virtue of temporal soft sets. The notion of temporal granulation mappings was defined to induce the granular structure of a given temporal transaction data set. With the help of temporal granulation mappings, we introduced temporal soft sets and theirQ-clip soft sets, which enable us to establish a conceptual framework for extracting TARs. Specially, we presented a number of useful characterizations and related results within this framework, including a necessary and sufficient condition for fast identification of strong TARs. An illustrative example regarding the Nobel Prizes was presented to show how these concepts and results can help facilitate TARM. We also developed a novel method, named negFIN-STARM, for extracting strong TARs by taking advantage of both temporal soft sets and NegNodeset-based frequent item set mining techniques. In addition, two commonly used data sets were employed to verify the feasibility of the negFIN-STARM method. Numerical results have shown that the negFIN-STARM method has better performance than existing approaches such as T-Apriori, T-ECLAT, and T-FPGrowth. It is robust with respect to the selection of different min-TS and min-TC thresholds as well. In future, it will be interesting to investigate the mining of maximal TARs using TSSs and consider its potential applications to dynamic detection, fault diagnosis, and optimal control in industrial processes.



---

*Source: 7303720-2021-11-29.xml*