**1. Introduction**

A concept lattice [1] is a conceptual hierarchy constructed according to the binary relationship between objects and attributes in data sets. As an effective tool for knowledge representation, a concept lattice is widely used in knowledge discovery, rule mining, information retrieval, and other fields because of its accuracy and completeness [2–4].

Concept lattice theory mainly focuses on the following aspects: A concept lattice extent model [5–7], concept lattice construction and rule extraction [8–10], concept lattice merging [11–15], concept lattice reduction [16,17], concept lattice modification [18], etc.

In a classic concept lattice, the concept extents have all the attributes or only one attribute, sometimes. Hence, the support and confidence degree of the extracted association rules would be reduced greatly. To solve this problem, the authors have put forward a new concept lattice structure: An interval concept lattice, and the construction methods, compression, and maintenance of lattice structure were studied [19–22].

From the perspective of a concept lattice, the relationship between intents is association rules, while the relationship between extents is its embodiment. A concept lattice is the unity of intent and extent, and the relationship between its nodes also reflects the relationship between the generalization and specialization of concepts. Therefore, a concept lattice is suitable for the application of a basic data structure in association rules mining. Many scholars have conducted in-depth research on rule mining based on a concept lattice [23–28].

Previously, we studied the structure characteristics of an interval concept lattice and gave two measurement standards of the uncertainty rule—precision and uncertainty. Then, a mining model of interval association rule with parameters was constructed [29]. The algorithm can mine and optimize rules according to the adjustment of parameters, which is of grea<sup>t</sup> significance to the mining of rules with uncertainties. Next, the complex relationship between interval parameters and association rules and the optimization algorithm of the rule base were given [30]. By adjusting the parameters, the purpose of controlling and optimizing rules was achieved.

In the era of data explosion, people's demand for data processing is getting higher and higher. The real-time updating of data requires the e fficient processing of dynamic data. For example, in the supermarket shopping system, the massive transaction information generated every day can only mine local association rules, but cannot provide a timely and accurate decision-making plan for decision-makers as a whole. However, the time and space complexity of the process will increase rapidly with the increase of the amount of data, and the mining association rules will be missing. Therefore, it is necessary to study the dynamic mining of interval association rules in order to grasp uncertain rules in real time.

The authors have studied the consistency of interval concept lattices, discussed the decision theorem of the concept of consistent intent, and designed a vertical union algorithm of interval concept lattices based on the breadth-first principle [31]. Furthermore, the sequential traversal method was used to scan the lattice structure, and a union algorithm of interval concept lattices was proposed from a transverse point of view [32]. Based on the research results of the union algorithm of interval concept lattice, this paper carries out the dynamic mining of interval association rules.

## **2. Concepts and Methods**

## *2.1. Interval Concept Lattice*

**Definition 1 ([33]).** *For the formal context* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*)*, where U* = {*<sup>x</sup>*1, *x*2, ··· , *<sup>x</sup>*3} *is the object sets and each xi*(*<sup>i</sup>* ≤ *n*) *denotes an object; A* = {*<sup>a</sup>*1, *a*2, ··· , *am*} *is the attribute set, and each aj*(*j* ≤ *m*) *denotes an attribute; R is the binary relationship between U and A*. *R* ⊆ *U* × *a. If* (*x*, *a*) ∈ *R, then we record that x has the attribute a, and write as xRa*.

**Definition 2 ([33]).** *For the formal context* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*)*, operators f*, *g are defined as follows:*

∀*x* ∈ *U*, *f*(*x*) = - *y* ∀*<sup>y</sup>* ∈ *A*, *xRy, i.e., f is the mapping between x and its attributes;* ∀*y* ∈ *a*, *g*(*y*) = - *x* ∀*<sup>x</sup>* ∈ *U*, *xRy, i.e., g is the mapping between y and its objects.*

**Definition 3 ([33]).** *For the formal context* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*)*, if f*(*X*) = *Y*, *g*(*Y*) = *X for X* ⊆ *U*,*Y* ⊆ *A, then the sequence* < *X*,*Y* > *is called a formal concept, or concept for short. X is the extent and Y is the intent.*

Rough concept lattice *RL*(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) based on rough set theory was studied in Reference [7], where the upper approximation extent and lower approximation extent refer to the maximal concept set and the minimal concept set respectively which have all the attributes in *Y* ⊆ *A*.

**Definition 4 ([33]).** *For the formal context* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *and its rough concept lattice RL*(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*)*,* (*<sup>M</sup>*, *<sup>N</sup>*,*<sup>Y</sup>*) *is the rough concept. Set an interval* [α, β] (0 ≤ α ≤ β ≤ <sup>1</sup>)*, then* α *upper bound extent M*α *and* β *lower bound extent M*β *are:*

$$M^{\alpha} = \left\{ \mathbf{x} \middle| \mathbf{x} \in M\_{\prime} \middle| f(\mathbf{x}) \cap Y \middle| \, |\, |Y| \ge a, 0 \le \alpha \le 1 \right\} \tag{1}$$

$$M^{\otimes} = \left\{ \mathbf{x} \middle| \mathbf{x} \in M, \left| f(\mathbf{x}) \cap Y \right| / |Y| \ge \beta, 0 \le \alpha \le \beta \le 1 \right\} \tag{2}$$

*Y is the concept intent and* |*Y*| *is the number of elements in Y, that is base number*. *M*α *refers to the objects which may be covered by* α × |*Y*| *attributes or more in Y*. *M*β *refers to the objects which may be covered by* β × |*Y*| *attributes or more in Y*.

**Definition 5 ([33]).** *Let* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *be a formal context and* (*M*<sup>α</sup>, *<sup>M</sup>*β, *Y*) *be an interval concept. Then, Y is the intent; M*α *is the* α *upper bound extent and M*β *is the* β *lower bound extent.*

**Definition 6.** *Suppose that* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *has two interval concepts,* (*M*<sup>α</sup>1 , *<sup>M</sup>*β1,*Y*1) *and* (*M*<sup>α</sup>2 , *<sup>M</sup>*β2,*Y*2) *. If the two meet Y*1 ⊆ *Y*2*,* |*Y*2| − |*Y*1| = 1*, <sup>M</sup>*<sup>α</sup>1= *<sup>M</sup>*<sup>α</sup>2*and M*β1= *<sup>M</sup>*β2*, then* (*M*<sup>α</sup>1, *<sup>M</sup>*β1,*Y*1) *is called the redundant concept.*

**Definition 7.** *Suppose that* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *has an interval concept,* (*M*<sup>α</sup>, *<sup>M</sup>*β,*<sup>Y</sup>*) *. If it meets M*α = *M*β = ∅*, then* (*M*<sup>α</sup>, *<sup>M</sup>*β,*<sup>Y</sup>*) *is called the empty concept.*

**Definition 8.** *Suppose* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *has an interval concept, C* = (*M*<sup>α</sup>, *<sup>M</sup>*β,*<sup>Y</sup>*) *. If C is neither the redundant concept nor the empty concept, then C is called the existence concept. <sup>L</sup>*βα(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *is a collection of all the existence concepts.*

**Definition 9.** *<sup>L</sup>*βα(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *refers to all the* [α, β] *interval concepts, which include: Existence concepts, redundant concepts, and empty concepts, that is:*

$$(M\_1^{\alpha}, M\_1^{\beta}, \mathcal{Y}\_1) \le (M\_2^{\alpha}, M\_2^{\beta}, \mathcal{Y}\_2) \Leftrightarrow \mathcal{Y}\_1 \supseteq \mathcal{Y}\_2. \tag{3}$$

*Then "*≤*"is called the partial order relationship of <sup>L</sup>*βα(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*).

**Definition 10.** *If all the concepts in <sup>L</sup>*βα(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *meet "*≤*", then <sup>L</sup>*βα(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) *is called interval concept lattice on the formal context* (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*).

**Definition 11.** *In the interval concept lattice <sup>L</sup>*βα(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*)*, if C* = (*M*<sup>α</sup>, *<sup>M</sup>*β,*<sup>Y</sup>*) ∈ *<sup>L</sup>*βα(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*)*, then the layer of the Lattice Structure is* |*A*| + 1 *and node C is at Layer* |Y| *. In particular, when Y* = ∅*, C was recorded on the zeroth layer.*

## *2.2. Interval Association Rules*

Formal context (*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) can describe a database, where *U* represents an object set; *A* represents an attribute set. For *x* ∈ *U*, *a* ∈ *A*, *xRa* represents the item-set where *a* belongs to *x*.

**Definition 12.** *Given the minimal support threshold* θ*, for any interval concept node C, if the number of objects in the upper bound extent is not less than* |*U*| × θ*, then C is called the* α*-upper bound frequent node, and the corresponding Y is called the* α*-upper bound frequent item-set; if the number of objects in the lower bound extent is not less than* |*U*| × θ*, then C is called the* β*-lower bound frequent node, the corresponding Y is called the* β*-lower bound frequent item-set.*

The father and son concept in the interval concept lattice does not have a specific relationship in frequency, which is different from the classical concept lattice.

If the association rule *A* ⇒ *B* corresponds to the interval concept node (*C*1, *C*2)(*C*1 = (*M*<sup>α</sup>1 , *<sup>M</sup>*β1, *<sup>Y</sup>*1), *C*2 = (*M*<sup>α</sup>2, *<sup>M</sup>*β2,*Y*2)) and *C*1 ≥ *C*2, then rule *A* ⇒ *B* is generated by node binary (*<sup>C</sup>*1,*C*2).

The α-upper bound association rules and the β-lower bound association rules can be extracted by two lower bound extents of the interval concept. The calculation methods of confidence and support are as follows:

The α-upper bound rule *A* ⇒ *B*:

$$Conf(A \Rightarrow B) = \left| M\_2^a \right| / \left| M\_1^a \right| \tag{4}$$

$$Support(A \Rightarrow B) = \left| M\_2^a \right| / |\mathcal{U}| \tag{5}$$

The β-lower bound *A* ⇒ *B*:

$$Conf(A \Rightarrow B) = \left| M\_2^{\mathcal{S}} \right| / \left| M\_1^{\mathcal{S}} \right| \tag{6}$$

$$Support(A \implies B) = \left| M\_2^{\mathcal{C}} \right| / |\mathcal{U}| \tag{7}$$

**Definition 13.** *Given the minimal support threshold* θ *and the minimal confidence threshold c*. *Node binary* (*C*1,*C*2) *is called* α*-upper bound candidate binary which consists of two frequent concept nodes C*1 = (*M*<sup>α</sup>1 , *<sup>M</sup>*β1,*Y*1) *and C*2 = (*M*<sup>α</sup>2 , *<sup>M</sup>*β2,*Y*2)*, where <sup>M</sup>*<sup>α</sup>2 ⊆ *<sup>M</sup>*<sup>α</sup>1 *and M*<sup>α</sup>2 /*M*<sup>α</sup>1 ≥ *c. When* (*C*1,*C*2) *meets M*β2 ⊆ *M*β1 *and M*β2/*M*β1 ≥ *c, it is called* β*-lower bound candidate binary*.

**Definition 14.** *For interval association rule A* ⇒ *B, if A* ∪ *B is frequent item-sets, Support*(*A* ⇒ *B*) ≥ θ *and Con f*(*A* ⇒ *B*) ≥ *c, i.e., P*(*A* ∪ *B*)/*P*(*A*) ≥ *c, then it is called the strong association rule.*

**Definition 15.** *If A* ⇒ *B is the strong association rule, then C* ⇒ *D must be the strong association rule, then we say "A* ⇒ *B can derive C* ⇒ *D*".

**Theorem 1.** *If C* ⊂ *D, then A* ⇒ *B can derive A* ⇒ *C*.

**Proof.** *C* ⊂ *B* ⇒ |*C*| < |*B*| ⇒ |*A* ∪ *C*| < |*A* ∪ *B*| ⇒ *Support*(*A* ⇒ *C*) = |*C*| < |*U*| < *Support*(*A* ⇒ *B*) = |*B*|/|*U*| and *Con f*(*A* ⇒ *C*) = |*A* ∪ *C*|/|*A*| < *Con f*(*A* ⇒ *B*) = |*A* ∪ *B*|/|*A*|. -

**Theorem 2.** *In the interval concept lattice, if* (*C*1,*C*2) *and* (*C*1,*C*3) *are candidate binaries and C*3 > *C*2*, then all the rules in Rules*(*C*1, *C*3) *can be derived from Rules*(*C*1, *<sup>C</sup>*2).

**Definition 16.** *Suppose A* ⇒ *B is* α*-upper bound association rule derived from* (*<sup>C</sup>*1,*C*2)*. The upper bound extent of C*1 *is <sup>M</sup>*<sup>α</sup>1 = {*<sup>x</sup>*1, *x*2, ... , *xm*, }*. The intent of C*1 *is Y*1.*The upper bound extent of C*2 *is <sup>M</sup>*<sup>α</sup>2= {*<sup>o</sup>*1, *o*2, ... , *om*, }*. The intent of C*2 *is Y*2*. The precision of A* ⇒ *B is*

$$PD\_{A \Rightarrow B} = \min\left\{ \min\_{i=1}^{m} \frac{|\mathbf{x}\_i \mathbf{Y} \cap \mathbf{Y}\_1|}{|\mathbf{Y}\_1|}, \min\_{i=1}^{n} \frac{|o\_i \mathbf{Y} \cap \mathbf{Y}\_2|}{|\mathbf{Y}\_2|} \right\} \tag{8}$$

*The uncertainty of A* ⇒ *B is UDA*⇒*<sup>B</sup>* = 1 − *PDA*⇒*B*.

**Definition 17.** *Let* Ω = {*Rule*1,*Rule*2, ··· ,*Rulek*} *denotes to* α *-rules set, the uncertainty of Rulei is UD*<sup>α</sup>−*Ri, then the uncertainty of* α *-rules sets is*

$$\text{LID}\_{\alpha-\text{Rhasset}} = \max\_{i=1}^{k} (\text{LID}\_{\alpha-\text{Ri}}) \tag{9}$$

*Let* Ω = {*Rule*1,*Rule*2, ··· ,*Rulem*} *denote to* β *-rules set, the uncertainty of Rulej is UD*β−*Ri, then the uncertainty of* β *-rules sets is*

$$\text{LID}\_{\beta-\text{Rluesset}} = \max\_{j=1}^{m} (\text{LID}\_{\beta-\text{Ri}}) \tag{10}$$

*and the uncertainty of interval association rules is UD* = max(*UD*<sup>α</sup>−*Rluesset*, *UD*β−*Rluesset*). **Theorem 3.** *Suppose the minimal support threshold* θ *and the minimal confidence threshold c . In the same context, when one of the interval parameters* α *and* β *is unchanged and the other is larger, the process of extracting association rules will change as follows:*


## **3. Algorithm and Results**

In the mining method of interval association rules, the lattice structure is united with the vertical union algorithm first, and then the association rules are extracted by using the parametric association rules mining algorithm. Because the structure of the interval concept lattice is complex, the memory space required is large, and the time complexity of mining interval association rules after vertical merging is high, this method does not meet the requirements of the current era of large data. In this section, the principle of a vertical union algorithm of interval association rules is proposed, and on this basis, a dynamic mining algorithm of interval association rules is designed.

#### *3.1. Vertical Union Principle of Interval Association Rules*

If the interval concept lattices *L* β α(*<sup>U</sup>*1, *<sup>A</sup>*1,*R*1) and *L* β α(*<sup>U</sup>*2, *<sup>A</sup>*2,*R*2) are consistent and *A*1 = *A*2 = *A*. *U*1 ∩ *U*2 = φ, then *L* β α(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*) can be gotten through the union of the two. Suppose the minimal support threshold θ and the minimal confidence threshold *c*, we can extract the association rules of *L* β α(*<sup>U</sup>*1, *<sup>A</sup>*,*R*1) and *L* β α(*<sup>U</sup>*2, *<sup>A</sup>*,*R*2). Here, α − *Rluesset*∗ is the upper bound association rules set derived from *L* β α(*<sup>U</sup>*1, *<sup>A</sup>*,*R*1). α − *Rluesset*∗∗ is the upper bound association rules set derived from *L* β α(*<sup>U</sup>*2, *<sup>A</sup>*,*R*2); α − *Rluesset* is the upper bound association rules set derived from *L* β α(*<sup>U</sup>*, *<sup>A</sup>*,*<sup>R</sup>*).

$$\mathsf{C}^{\*}\_{1} = (M^{a\*}\_{1}, M^{\beta\*}\_{1}, Y\_{1}), \mathsf{C}^{\*}\_{2} = (M^{a\*}\_{2}, M^{\beta\*}\_{2}, Y\_{2}) \text{ and } \mathsf{C}^{\*}\_{1}, \mathsf{C}^{\*}\_{2} \in \overline{L^{\beta}\_{\alpha}(L\_{1}, A, R\_{1})};$$

$$\mathsf{C}^{\*}\_{1} = (M^{a\*}\_{1}, M^{\beta\*}\_{1}, Y\_{1}), \mathsf{C}^{\*}\_{2} = (M^{a\*}\_{2}, M^{\beta\*}\_{2}, Y\_{2}) \text{ and } \mathsf{C}^{\*}\_{1}, \mathsf{C}^{\*}\_{2} \in \overline{L^{\beta}\_{\alpha}(L\_{2}, A, R\_{2})};$$

$$\mathsf{C}\_{1} = (M^{a}\_{1}, M^{\beta}\_{1}, Y\_{1}), \mathsf{C}\_{2} = (M^{\alpha}\_{2}, M^{\beta}\_{2}, Y\_{2}) \text{ and } \mathsf{C}\_{1}, \mathsf{C}\_{2} \in \overline{L^{\beta}\_{\alpha}(L\_{1}, A, R)}.$$

Now, taking the upper bound extent as an example (the union principle of lower bound extent and upper bound extent), the vertical union of interval association rules α − *Rluesset*∗ and α − *Rluesset*∗∗ is carried out as follows:

**Theorem 4.** *If C*∗ 1, *C*∗ 2 *constitutes the association rule, and Rules*(*C*∗ 1, *C*∗ 2)<sup>∈</sup><sup>α</sup> − *Rluesset*∗ *; C*∗∗ 1 , *C*∗∗ 2 *the constitutes association rule, and Rules*(*C*∗∗ 1,*C*∗∗ 2 )<sup>∈</sup>α − *Rluesset*∗∗*, then Rules*(*<sup>C</sup>*1,*C*2)<sup>∈</sup><sup>α</sup> − *Rluesset*.

.

$$\begin{array}{l} \textbf{Proof.} \;Rides(\textbf{C}\_{1},\textbf{C}\_{2}^{\*}) \in \alpha-\text{Rluess} \textbf{et}^{\*} \text{ and } \textsc{Ruless} \left(\textbf{C}\_{1}^{\*},\textbf{C}\_{2}^{\*}\right) \in \alpha-\text{Rluess} \textbf{et}^{\*\ast}, \text{ so:}\\ \textbf{R relation 1:} \;M\_{2}^{a\ast} \leq M\_{1}^{a\ast} \text{ and } M\_{2}^{a\ast} \leq M\_{1}^{a\ast}, \text{ then } M\_{2}^{a\ast} \cup M\_{2}^{a\ast\ast} \leq M\_{1}^{a\ast} \cup M\_{1}^{a\ast\ast}, \text{ i.e., } M\_{2}^{a\ast} \leq M\_{1}^{a\ast} \text{ and } M\_{2}^{a\ast} \leq M\_{1}^{a\ast}.\\ \text{R relation 2:} \;\frac{|M\_{2}^{a\ast}|}{|M\_{1}^{a\ast}|} \geq c \text{ and } \frac{|M\_{2}^{a\ast}|}{|M\_{1}^{a\ast\ast}|} \geq c, \text{ then} \\ \\ |M\_{2}^{a\ast}| + |M\_{2}^{a\ast}| \geq c \text{( $M\_{1}^{a\ast}| + |M\_{1}^{a\ast\ast}|)$ } \cdot \frac{|M\_{2}^{a\ast}|}{|M\_{1}^{a\ast}| + |M\_{2}^{a\ast\ast}|} \geq c, \text{i.e., } \frac{|M\_{2}^{a\ast}|}{|M\_{1}^{a\ast}|} \geq c. \end{array}$$

Then *Rules*(*<sup>C</sup>*1,*C*2)<sup>∈</sup><sup>α</sup> − *Rluesset* can be derived from two relations. -

1

1

1

**Theorem 5.** *If at least one of the two association rules Rules*(*<sup>C</sup>*<sup>∗</sup>1, *<sup>C</sup>*<sup>∗</sup>2) *and Rules*(*<sup>C</sup>*∗∗1 , *<sup>C</sup>*∗∗2 ) *in <sup>L</sup>*βα(*<sup>U</sup>*1, *<sup>A</sup>*,*R*1) *and <sup>L</sup>*βα(*<sup>U</sup>*2, *<sup>A</sup>*,*R*2) *does not exist, then there must be no Rules*(*<sup>C</sup>*1,*C*2) *in* α − *Rluesset*.

Because the vertical union of interval association rules is based on the objects set with certain proportion attributes, it is necessary to introduce support, confidence, accuracy, and uncertainty to measure accurately.

The upper bound frequencies of *<sup>C</sup>*<sup>∗</sup>1,*C*<sup>∗</sup>2are θ∗1and <sup>θ</sup><sup>∗</sup>2, respectively.

The upper bound frequencies of *<sup>C</sup>*∗∗1 ,*C*∗∗2are θ∗∗1and <sup>θ</sup>∗∗2, respectively.

The upper bound frequencies of *<sup>C</sup>*1,*C*2are θ1and <sup>θ</sup>2, respectively.

 The support degree, confidence degree, accuracy, and uncertainty of *Rules*(*<sup>C</sup>*1,*C*2) obtained by vertical union of association rules *Rules*(*<sup>C</sup>*<sup>∗</sup>1, *<sup>C</sup>*<sup>∗</sup>2) and *Rules*(*<sup>C</sup>*∗∗1 , *<sup>C</sup>*∗∗2 ) can be deduced from frequency to its solution formula, as follows.

**Theorem 6.** *The support degree of Rules*(*<sup>C</sup>*1,*C*2) *is:*

$$Support(C\_1 \Rightarrow C\_2) = \frac{|\mathcal{U}\_1|\partial\_2^\* + |\mathcal{U}\_2|\partial\_2^{\*\*}}{|\mathcal{U}\_1| + |\mathcal{U}\_2|}\tag{11}$$

**Theorem 7.** *The confidence degree of Rules*(*<sup>C</sup>*1,*C*2) *is*

$$\text{Conf}(\mathbb{C}\_1 \Rightarrow \mathbb{C}\_2) = \frac{|\mathcal{U}\_1|\mathcal{O}\_2^\* + |\mathcal{U}\_2|\mathcal{O}\_2^{\*\*}}{|\mathcal{U}\_1|\mathcal{O}\_1^\* + |\mathcal{U}\_2|\mathcal{O}\_1^{\*\*}}\tag{12}$$

**Theorem 8.** *Rules*(*<sup>C</sup>*1,*C*2) = *MinRules*(*<sup>C</sup>*<sup>∗</sup>1,*C*<sup>∗</sup>2),*Rules*(*<sup>C</sup>*∗∗1 ,*C*∗∗2 ).

*3.2. Dynamic Mining Algorithms for Interval Association Rules*

## 3.2.1. Algorithm Design

According to Theorems 4 and 5, the result of the vertical union of interval association rules only occurs in the data with the same interval association rules and adjacent data. In order to ensure that the mined interval association rules are not lost, the algorithm retains the non-united rules on the basis of a vertical union of the same rules, which facilitates the mining of interval association rules later, and measures the occurrence frequency and frequency of the same rules with frequency. The basic idea of the algorithm is that whenever an interval association rule is generated, it is transformed into an array representation, and it is mined with the set of interval rules that have been united and retain the non-united rules, so that the interval rules can be aggregated again and again.

In order to distinguish different rules, association rules mined from interval concept lattices are stored in the form of arrays *RS*[*i*] = *Rule*, *FN*1, *FN*2, *U*, *Support*,*Con f*, *PD*, *UD*, *Flag*, *Num*.

*RS*[*i*] represents the *i*th association rule in the interval association rules set *RS*;

*Rule* represents the interval association rules *Rule*(*Cx*,*Cy*);

*FN*1, *FN*2 represents the frequency degree of frequency nodes *Cx* and *Cy* in *Rule*(*Cx*,*Cy*);

*U* represents the number of objects in the context;

*Support*, *Con f*, *PD* and *UD* represents the support, confidence, accuracy, and uncertainty degree of *Rule*(*Cx*,*Cy*);

*Flag* marked according to whether rules have been merged or not. *Flag* = 0 denotes that rules are not united vertically; *Flag* = 1 indicates that the rules have been vertically united; *Num* denotes the number of occurrences of rule *Rule*(*Cx*,*Cy*) in previously united rule sets.

Based on the above algorithm principle and analysis, we design a dynamic extraction and union algorithm of interval association rules, DMA (Dynamic Mining Algorithm). See Algorithm 1.

**Algorithm 1**. DMA (Dynamic Mining Algorithm)

**Input: Association rule sets** *RS*1,*RS*2, ... *RSk*

**Output: Association rule set** *RS*

**Step1** *RS* = *RS*1;

Rule Vertical Union (*RS*,*RSk*)

**Step2** The interval association rules in the rule set *RSk* is stored in the form of arrays. Set *RS*∗ and initialize. Comparing the *Rule* of rule set *RS* and *RSk*, we unit interval association rules vertically according to Theorems 4 and 5. For the rules that have been united in *RS* and *RSk*, let *Flag* = 1 and put the united rule number in *RS*<sup>∗</sup>. According to Theorems 6–8, calculate the frequency,*Support*, *Con f*, *PD*, and *UD* of *Cx* and *Cy* in *RS*<sup>∗</sup>. *Flag*=1, *Num* = *Num* + 1. Delete the rules of *Flag* = 1 in *RS* and *RSk*, and renumbering. Let rules number in *RS*∗ add the renumbering number in *RS*. Putting the remain rules of *RS* into *RS*<sup>∗</sup>; then renumbering in *RSk*, and putting the remain rules of *RSk* into *RS*<sup>∗</sup>.

 ...

```
1 {
2 g = 0;
3 RS[g]∗ = Rule, FN1, FN2, U, Support,Con f, PD, UD, Flag, Num
4 { Rule = ∅;
5 FN1 = FN2 = 0;
6 U = RS[1].U + RSk[1].U;
7 Support = Con f = PD = UD = Flag = 0;
8 Num = 1;
9 }
10 For ( i=1;|RS|; i++)
11 {For (j=1;|RSk|; j++)
12 If(RS[i].Flag = RSk[j].Flag = 0|| RS[i].Rule = RSk[j].Rule)
13 { g + 1;
14 Num + 1;
15 RS[i].Flag = 1;
16 RSk[j].Flag = 1;
17 RS[g]∗.FN1 =RS[i].U∗RS[i].FN1+RSk [j].U∗RSk [j].FN1 RS[i].U+RSk [j].U
18 RS[g]∗.FN2 =RS[i].U∗RS[i].FN2+RSk [j].U∗RSk [j].FN2 RS[i].U+RSk [j].U
19 RS[g]∗.Support = RS[g]∗.FN2
20 RS[g]∗.Con f =RS[i].U∗RS[i].FN2+RSk [j].U∗RSk [j].FN2 RS[i].U∗RS[i].FN1+RSk [j].U∗RSk [j].FN1
21 RS[g]∗.PD =min-
                               RS[i].PD,RSk[j].PD;
22 RS[g]∗.UD =1 − RS[g]∗.PD;
23 }
24 }
25 For each RS[i] in RS;
26 {If RS[i].Flag = 0;
27 g + 1;
28 RS∗[g] = RS[i];}
29 For each RSk[j] in RSk;
30 {If RSk[j].Flag = 0;
31 g + 1;
32 RS∗[g] = RSk[j];}
33 }
  Step3 RS = RS∗
```
## 3.2.2. Algorithm Analysis

The main content of DMA is embodied in the function Rule Vertical Union (*RS*, *RSk*). Lines 1 to 9 implement the initialization of parallel rule set *RS*<sup>∗</sup>. Lines 10–12 nested for loop implements searching for interval association rules in *RS* and *RSk* that correspond to the same rules and are not united. Lines 13 to 16 number the interval association rules found in *RS*<sup>∗</sup>. Num+1 is used to count the rule, and the corresponding rule *Flag* in united *RS*, *RSk* is recorded as 1. Lines 17 to 25 implement the assignment of interval association rule *RS*∗ [*g*]. In lines 26–34, the unconsolidated rules in *RS*, *RSk* are put into *RS*∗ to maintain the diversity of the united rule set.

Compared with Step 2, the algorithm achieves the step requirements, and the assignment of Step 1 and Step 3 make the DMA algorithm dynamic, so that the algorithm has integrity and correctness. Compared with the general method, the algorithm realizes the merging of rules to rules, eliminating the process of vertical merging of interval concept lattices, thus greatly reducing the time complexity and space complexity of the implementation process.

If the number of association rules in two sub lattices is *n* and *m* respectively, the time complexity of the algorithm is less than *<sup>O</sup>*(*n* × *m*) + *<sup>O</sup>*(*n*) + *<sup>O</sup>*(*m*). Compared with the interval association rule mining algorithm with parameters, the algorithm is more e fficient.

## *3.3. Example Study*

Set the formal contexts as shown in Tables 1 and 2:

**Table 1.** Formal context *FC*1.


**Table 2.** Formal context *FC*2.


Set α = 0.6, β = 0.7, θ = 50%, *c* = 60%.

The DMA algorithm is used to mine interval association rules in parallel. Take the upper bound interval association rules as an example (the mining of lower bound association rules is similar).

Among them, the upper bound frequent nodes from *FC*1 and *FC*2 are shown in Tables 3 and 4 respectively, and the corresponding upper bound association rules are shown in Tables 5 and 6 respectively.




**Table 4.** The frequent nodes of *FC*2.



The DMA algorithm is used to mine the previous association rules of *FC*1 and *FC*2 in parallel. The parallel results of the di fferent association rules are the same part of the combined association rules deleted in Tables 5 and 6, respectively.

The parallel results of the same association rules are shown in Table 7.


**Table 6.** The previous rules and its measurement results on *FC*2.

**Table 7.** The previous rules set and its measurement results after vertical union.


## **4. Discussion**

Considering the characteristics of interval concept lattices, the vertical union method of interval concept lattices and the principle of mining association rules with parameters were studied in this paper. Considering the intrinsic relationship between interval association rules, this paper presents some metrics, such as the confidence and credibility of united rules, and proposes a dynamic mining algorithm for interval association rules, which realizes rule aggregation and keeps the diversity of interval association rules. The rationality and efficiency of the algorithm are proved by algorithm analysis and case study, which provides timely, effective and abundant decision information for decision makers in data analysis. The next step is to optimize the algorithm for incomplete information systems in the context of large data. In the mining process, the stability and timing choice of the rule union is another problem to be studied in depth.

**Author Contributions:** Y.Y. contributed to the main body of the paper. The main models and algorithms were constructed; R.Z. was mainly responsible for the writing of the paper, colleagues performed data processing; B.L. guided the overall process of the paper.

**Funding:** This research was funded by the National Natural Science Foundation of China (61370168, 61472340), Natural Science Foundation of Hebei Province (F2016209344).

**Acknowledgments:** The authors thank the support of Hebei Key Laboratory for Data Science and Application and the technical assistance of Chunying Zhang and Lihong Li from North China University of Science and Technology.

**Conflicts of Interest:** The authors declare no conflicts of interest.
