Mining Temporal Association Rules with Temporal Soft Sets

Department of Applied Mathematics, School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China School of Economics and Management, Beijing University of Posts and Telecommunications, Beijing 100 876, China Machine Intelligence Institute, Iona College, New Rochelle, NY 10 801, USA Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI), University of Granada, Granada, Spain I-SOMET Incorporated Association, Morioka, Iwate, Japan BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, E37007 Salamanca, Spain


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 Figure 1, 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].
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. e first one is known as the class of "candidate generation" methods with the Apriori [7] algorithm as its typical representative. e main drawback of these methods is that all of them require multiple database scans. e second category consists of "pattern growth" methods such as the FP-growth algorithm [8], which relies on the tree-based data structure (like FPtrees) 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. e 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][16][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 projectionbased utility mining method which is useful for mining highutility 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][35][36][37][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][42][43]. ey 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][47][48][49][50][51], valuation of assets [52], clustering [53], medical diagnosis [54], parameter reduction [55], feature selection [56], data analysis [57], BCK/BCI-algebras [58][59][60], graph theory [61], and computational biology [62]. e 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. eir 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) e 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) e 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. e 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. is 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.
e main contributions of this study are summarized as follows: (1) We define some new concepts such as temporal granulation mappings, temporal soft sets and Q-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 NegNodesetbased frequent item set mining e rest of this paper is arranged in the following way: Section 2 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 neg-FIN-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.

Temporal Association Rules
is 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 set _ I of temporal items.
Definition 2 (see [19]). Assume that T is a set of transactions on a temporal item set _ I and the positive integer α is the selected support threshold. en, we say that _ I is a temporal frequent item set with respect to T and α, when support T( _ I) ≥ α.
Definition 3 (see [19]). A pair of disjoint temporal item sets is called a temporal association rule (TAR). Let RHS and LHS, respectively, represent the right and left temporal item sets. en, we denote a TAR by LHS > (Δ)RHS, 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 interval Δ. TARs defined in [19] are therefore useful for capturing temporal dependence among items within different time spans.

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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 that I � i 1 , . . . , i |I| is an item domain. Any subset t of I is a transaction, and a transaction data set consists of a set D � t 1 , . . . , t |D| formed by all transactions under inspection. Each transaction in D has a unique transaction identifier (TID).
In classical association rule extraction, an item set X 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 i s is denoted by i s . An item set X appears in t (alternatively, t supports X, when X ⊆ t).
Now, let P � p 1 , p 2 , . . . , p |P| be a collection of pairwise disjoint periods of time. If t ∈ D is related to a unique period p t � τ(t) ∈ P (indicating that t occurs during the period p t ), then p t is called the period marker of t. In fact, this defines a mapping τ from D to P such that τ(t) � p t . In what follows, T � (D, τ, P) is called a temporal transaction data set.
Definition 5. Assume that T � (D, τ, P) is a temporal transaction data set, Q ⊆ P and Z is an item set. e set is the temporal realization of Z in T during a period in Q.
e set Δ Q T (Z) consists of all the transactions in D which contain all the items in Z and occur during a period in Q. e cardinality of this set is written as S Q T (Z), called the temporal support of Z in T during a period in Q. For Definition 6. Let T � (D, τ, P) be a temporal transaction data set and Q ⊆ P. Given two disjoint nonempty item sets G, Z ⊆ I, an expression G ⟹ Q Z is called a temporal association rule (TAR).
We refer to Z and G as the consequent and antecedent of Definition 7. Let T � (D, τ, P) be a temporal transaction data set, Q ⊆ P and G ⟹ Q Z be a TAR. en, the temporal realization of G ⟹ Q Z in T is given by Definition 8. e temporal confidence of a TAR G ⟹ Q Z is given by In particular, e 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.
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) α Q ∈ N * and the minimum temporal confidence (min-TC) β Q ∈ (0, 1] for a given subset e 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 � t 1 , t 2 , . . . , t 16 consisting of all the transactions. Assume that every t ∈ D is related to a unique period p t � τ(t) ∈ P, where P � p 1 , p 2 , p 3 , p 4 . From Table 1, it can be seen that T is divided into four parts by P. For example, the item set c appears in the transaction t 2 and t 3 during the period p 1 . Now, let us consider the subset Q � p 1 of P. By Definition 5, we have Δ In addition, the 2-item set c, δ appears in the transaction t 2 , and transaction t 2 occurs during the period p 1 .
us by Definition 4, we can say that t 2 supports c, δ in T during the period p 1 . Also, it is clear that Δ and the temporal support of this rule is

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By Definition 8, the temporal confidence of this rule is Finally, assume that α p 1 � 1 and β p 1 � 35%. We conclude that c ⟹ p 1 δ { } is a strong TAR during the period p 1 .

Temporal Soft Sets
In this section, we define some new concepts such as temporal granulation mappings, temporal soft sets, and Q-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 [. e power set of [ is written as 2 [ .
Definition 10 (see [67]). Assume that [ and K are nonempty finite sets of alternatives and attributes, respectively. e pair I � ([, K) is called an information system (IS), when every attribute k ∈ K can be identified with an information function k: [ ⟶ V k and V k is the value set of k.
When G � (F, K) is a soft set over [, it naturally induces an IS I G � ([, K) in the following fashion. Given every k ∈ K and v ∈ [, associate the corresponding information function k: { } as follows: Definition 11. Let P be a set of pairwise disjoint periods of time. en, τ: [ ⟶ P is called a temporal granulation mapping.
is a temporal granulation mapping e soft set (F, 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 [. e TSS T, as an abstract representation of data, can additionally capture temporal information, which is unable to be expressed by its underlying soft set. where Note that p -clip soft set is simply called p-clip soft set. Next, we consider an example that illustrates the abovementioned notions.

Example 2.
e 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 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 � c 1 , c 2 , . . . , c 6 is a set of parameters, containing all the affiliation countries associated with the prizes in [. More specifically, let c i (i � 1, 2, . . . , 6) stand for "Denmark," "France," "Germany," " e 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 In addition, a temporal granulation ℘ � (Y, τ) of [ can be derived from Table 2 in a natural way. In fact, let Y � y 1 , y 2 , y 3 with y i � 1900 + i for i � 1, 2, 3. en, the temporal granulation mapping τ: [ ⟶ Y is given by e intuitive meaning of τ is apparent. For instance, equation (10) says that the prizes v 1 , v 2 , and v 3 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: Finally, by Definition 13, the y i -clip soft set T y i � (G i , C) of the TSS T for i � 1, 2, 3 are as follows:

Soft Temporal Association Rule Mining
is section aims to establish a formal framework for mining TARs by means of TSSs. Let P 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 ∈ [. en, we call as the parameter coset of the alternative v in G.
It can be seen that Co G (v) contains all the parameters that the alternative v meets, according to the information contained in G.
is called the Q-realization of B in the TSS T. Definition 16. Assume that T � (F, K, τ, P) is a TSS over [ and G, Z are two disjoint non-empty subsets of K. We call the expression G ⟹ Q Z as a temporal association rule (TAR) in the TSS T. e non-empty parameter sets Z and G are respectively called consequent and antecedent of the TAR G ⟹ Q Z.
Definition 17. Suppose that T � (F, K, τ, P) is a TSS over [ and G ⟹ Q Z is a TAR in T. We refer to Proof. We denote by G the USS of the TSS T � (F, K, τ, P). Let x ∈ Δ Q T (B). Equation (15) assures B ⊆ Co G (x) and x ∈ τ − 1 (Q). By Definition 14, x ∈ F(k) when k ∈ B. us we have is proves  Table 3: Tabular representation of the TSS T � (F, C, τ, Y).
Journal of Mathematics Now, suppose that y ∈ ( ∩ k∈B F(k)) ∩ (τ − 1 (Q)). en y ∈ τ − 1 (Q) and y ∈ F(k) for any k ∈ B. From the definition of the parameter coset Co G (y), it follows that B ⊆ Co G (y) and y ∈ τ − 1 (Q). Hence y ∈ Δ Q T (B), which also shows that erefore we derive that is ends the proof. By Proposition 1, the following results can be deduced.

Corollary 1. Assume that T � (F, K, τ, P) is a TSS over [ and T Q � (G, K) is its Q-clip soft set. en we have
for all non-empty subset B of K.

Corollary 2. Assume that T � (F, K, τ, P) is a TSS over [
and K 1 , K 2 are subsets of K. en we have

Proposition 2. Assume that T � (F, K, τ, P) is a TSS over [ and G ⟹ Q Z is a TAR in T. en,
Proof: . For simplicity, let J 1 and J 2 stand for ∩ k∈G F(k) and ∩ k∈Z F(k), respectively. According to Definition 17, By Proposition 1, we have is ends the proof. □ Remark 1. e above assertion reveals that the Q-realization of a TAR G ⟹ Q Z in a TSS coincides with the intersection of the Q-realizations of the consequent and antecedent of G ⟹ Q Z.
By Proposition 2, the following results can be deduced.

Corollary 3. Assume that T � (F, K, τ, P) is a TSS over [ and T Q � (G, K) is its Q-clip soft set. en,
where G ⟹ Q Z is a TAR in T.

Corollary 4.
Assume that T � (F, K, τ, P) is a TSS over [ and G ⟹ Q Z is a TAR in T. en, For convenience, conf

Theorem 1. Assume that T � (F, K, τ, P) is a TSS over [ and G ⟹ Q Z is a TAR in T. en, G ⟹ Q Z is strong during a period in Q if and only if
where α Q ∈ N * is the min-TS and β Q ∈ (0, 1] is the min-TC.
Proof. Suppose that G ⟹ Q Z is strong in T during a period in Q. en, we have It follows that us, we have Conversely, let G ⟹ Q Z be a temporal association rule in T such that It follows that Hence, we deduce that Journal of Mathematics 7 us, G ⟹ Q Z is strong in T during a period in Q, completing the proof.
Using the aforementioned concepts and results, we can obtain the following result.
en, the following are equivalent:

strong during a period in Q
To illustrate the new notions above, we consider the following example, which is a continuation of Example 2.
Example 3. Assume that A � a 1 , a 2 , a 3 is a set of parameters, consisting of the three types of prizes under consideration, i.e., a 1 is NPC, a 2 is NPP, and a 3 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. e 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 Q 1 � y 1 , y 2 , Q 2 � y 2 , and Q 3 � Y � y 1 , y 2 , y 3 , respectively. Suppose that the min-TS α Q 1 � α Q 2 � 1 and α Q 3 � 2.
Let us first focus on the case when Q 1 � y 1 , y 2 . Recall first that τ − 1 (Q 1 ) � v 1 , v 2 , . . . , v 6 . By Definition 13, the Q 1 -clip of the TSS T is a soft set 6 and G 1 (c j ) � ∅ for all c j ∈ C with j � 1, 2, 5. By Proposition 1 and Corollary 1, we can easily get Next, we consider the TAR a 1 ⟹ Q 1 c 3 . By Proposition 2 and Corollary 3, its Q 1 -realization in T can be calculated as follows: In fact, as indicated by Corollary 3, the Q 1 -realization of the TAR a 1 ⟹ Q 1 c 3 in the TSS T is completely determined by the approximate function of the corresponding Q 1 -clip T Q 1 � (G 1 , B). It is clear that the Q 1 -support of this rule is By Definition 18, the Q 1 -confidence of this rule is Hence, by definition, a 1 ⟹ Q 1 c 3 is strong during a period in Q 1 . On the other hand, we can draw the same conclusion from eorem 1 since Note also that us, we can also conclude that a 1 ⟹ Q 1 c 3 is strong during a period in Q 1 by Proposition 3. is rule indicates that "From 1901 to 1902, all the Nobel Prizes in Chemistry were awarded to Germany." Conversely, we can consider the TAR c 3 ⟹ Q 1 a 1 . Its Q 1 -support is but its Q 1 -confidence is Hence, this rule is frequent but not confident during a period in Q 1 . 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 when Q 2 � y 2 . Similarly, we can get Hence, we conclude that c 3 ⟹ y 2 a 1 is strong during the period y 2 . is rule says that "In 1902, Germany was only awarded the NPC, instead of the NPP or NPPM." Finally, we consider the third case when Q 3 � Y � y 1 , y 2 , y 3 . Clearly, τ − 1 (Q 3 ) � v 1 , v 2 , . . . , v 9 � [ in this case. It follows that the Q 3 -clip of the TSS T is a soft set T Q 3 � (G 3 , B) over [, which coincide with the USS (F, B) of the TSS T. at is, G 3 (e) � F(e) for all e ∈ B. By Proposition 1 and Corollary 1, we have 8 Journal of Mathematics Next, let us consider the TAR a 3 ⟹ Q 3 c 6 , which can also be seen as an association rule a 3 ⟹ c 6 in conventional sense. By Proposition 2 and Corollary 3, its Q 3 -realization in T can be calculated as follows: Obviously, supp It is clear that us, we deduce that a 3 ⟹ Q 3 c 6 is neither frequent nor confident during a period in Q 3 . is 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 c 3 ⟹ Q 3 a 3 is neither frequent nor confident during a period in Q 3 since is 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 of Q 3 consisting of all time periods, we see that some rules such as a 1 ⟹ Q 1 c 3 and c 3 ⟹ y 2 a 1 can only be identified as strong TARs when we restrict to the cases of Q 1 or Q 2 consisting of fewer time periods. is 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 and Q-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. e pseudocode description of the negFIN-STARM method is given in Algorithm 1. is algorithm takes a temporal transaction data set T, a set Q ⊆ P, the min-TS α Q , and the min-TC β Q as the input. e output of Algorithm 1 is the class SRTAR(T, Q, α Q , β Q ), which contains all strong TARs during a period in Q. e main procedure of the negFIN-STARM method can be divided into three stages: (1) In the first stage, we construct a TSS T � (F, I, τ, P) over D from the provided temporal transaction data set T. en, according to Definition 13, we determine τ − 1 (Q) and construct the Q-clip soft set T Q � (H, I) of the TSS T. Next, we derive the IS I T Q � (τ − 1 (Q), I) from the Q-clip soft set T Q � (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 of I T Q � (τ − 1 (Q), I) to construct the BMC tree. en, the Nodesets of all frequent 1item sets are generated by traversing the BMC tree. Furthermore, we identify the NegNodesets of all frequent k-item sets (k ≥ 2). Eventually, the setenumeration tree is built to generate the class TFIS(T Q , α Q ), which consists of all temporal frequent item sets. ese 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 Table 4: Tabular representation of the TSS T � (F, B, τ, Y).
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 Corollary 3, we can calculate the Q-realization of G ⟹ Q Z using the Q-clip soft set T Q for all G, Z ∈ TFIS(T Q , α Q ) which are disjoint. Next, by eorem 1, it is easy to check whether or not the G ⟹ Q Z is strong during a period in Q. If this is true, then we put G ⟹ Q Z into the class SRTAR(T, Q, α Q , β Q ).

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.

Running Environment.
e 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. e algorithms are coded in Java 13.0.1 using IntelliJ IDEA 2019.2.2. e 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. e comparison is made in terms of the average values of the runtime.

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

Results and Comparative
Analysis. At first, we conduct numerical experiments and comparative analysis of four different methods using the chess data set. is data set contains 3196 transactions, each uniquely related to a period in P � p 1 , p 2 , p 3 , p 4 , p 5 .
ere are 75 different items in the item domain of this data set. We consider the cases when Q � p 1 and Q � p 5 . e min-TS and min-TC are simply denoted by α p i and β p i (i � 1, 5), respectively. e 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.
(1) Construct a TSS T � (F, I, τ, P) over D from the temporal transaction data set T with the item domain I (2) Calculate τ − 1 (Q) and construct the Q-clip soft set T Q � (H, I) of T (3) Construct the IS I T Q � (τ − 1 (Q), I) induced by T Q � (H, I) (4) Construct the BMC tree by I T Q � (τ − 1 (Q), I) (5) Traverse the BMC tree to get the Nodesets of all frequent 1-item sets (6) Identify the NegNodesets of all frequent k-item sets (k ≥ 2) (7) Build the set-enumeration tree to generate the class TFIS(T Q , α Q ), which consists of all temporal frequent item sets (8) for G ∈ TFIS(T Q , α Q ) do (9) for Z ∈ TFIS(T Q , α Q ) do (10) if G ∩ Z � ∅ then (11) Calculate end if (16) end for (17) end for (18) return SRTAR(T, Q, α Q , β Q ); ALGORITHM 1: e negFIN-STARM method. 10 Journal of Mathematics From Figure 2(c), we see that the negFIN-STARM method is faster than the T-Apriori, T-ECLAT, and T-FPGrowth methods when Q � p 5 , α p 5 � 500, and β p 5 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 β p 5 � 85% and α p 5 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 Figure 3. In brief, we can find that the rule number decreases when the threshold increases.
Similarly, we also conduct numerical experiments and comparative analysis of four methods using the mushroom data set.
is data set contains 8124 transactions, each uniquely related to a period in P � p 1 , p 2 , . . . , p 17 . ere are 119 different items in the item domain of this data set. We consider the cases when Q � p 1 and Q � p 10 . e min-TS and min-TC are simply denoted by α p i and β p i (i � 1, 10), respectively. e runtime comparison with respect to the mushroom data set of four methods under different thresholds is shown in Figure 4. e quantity comparison of the obtained TARs based on the mushroom data set under different thresholds is illustrated in Figure 5. e average runtime (in milliseconds) of four methods on the mushroom data set is listed in Tables 8 and 9.
e above numerical experiments demonstrate that our newly proposed method is a helpful apparatus for mining TARs. e comparative analysis illustrates that the negFIN-STARM method performs better than three well-known Temporal confidence (%)   Quantity of rules Temporal support (c) Q � p 10 and α p 10 � 300. (d) Q � p 10 and β p 10 � 70%. Table 9: Execution time (ms) of four methods on the mushroom data set when Q � p 10 .

Conclusions
is paper is devoted to enhancing association rule mining by virtue of temporal soft sets. e 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 their Q-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.

Data Availability
e data used to support the findings of this study are available from the SPMF library at http://www.philippefournier-viger.com/spmf/ founded by Dr. Philippe Fournier-Viger.

Conflicts of Interest
e authors declare that there are no conflicts of interest.