# Dynamic Parallel Mining Algorithm of Association Rules Based on Interval Concept Lattice

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## Abstract

**:**

## 1. Introduction

## 2. Concepts and Methods

#### 2.1. Interval Concept Lattice

**Definition**

**1**

**([33]).**For the formal context$(U,A,R)$, where$U=\{{x}_{1},{x}_{2},\cdots ,{x}_{3}\}$is the object sets and each${x}_{i}(i\le n)$denotes an object;$A=\{{a}_{1},{a}_{2},\cdots ,{a}_{m}\}$is the attribute set, and each${a}_{j}(j\le m)$denotes an attribute;$R$is the binary relationship between$U$and$A$. $R\subseteq U\times a$. If$(x,a)\in R$, then we record that$x$has the attribute$a$, and write as$xRa$.

**Definition**

**2**

- $\forall x\in U,f(x)=\left\{y\right|\forall y\in A,xRy\}$, i.e.,$f$is the mapping between$x$and its attributes;
- $\forall y\in a,g(y)=\left\{x\right|\forall x\in U,xRy\}$, i.e.,$g$is the mapping between$y$and its objects.

**Definition**

**3**

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

**Definition**

**4**

**([33]).**For the formal context$(U,A,R)$and its rough concept lattice$RL(U,A,R)$,$(M,N,Y)$is the rough concept. Set an interval$[\alpha ,\beta ]\text{\hspace{0.05em}}(0\le \alpha \le \beta \le 1)$, then$\alpha $upper bound extent${M}^{\alpha}$and$\beta $lower bound extent${M}^{\beta}$are:

**Definition**

**5**

**([33]).**Let$(U,A,R)$be a formal context and$({M}^{\alpha},{M}^{\beta},Y)$be an interval concept. Then,$Y$is the intent;${M}^{\alpha}$is the$\alpha $upper bound extent and${M}^{\beta}$is the$\beta $lower bound extent.

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

#### 2.2. Interval Association Rules

**Definition**

**12.**

**Definition**

**13.**

**Definition**

**14.**

**Definition**

**15.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Definition**

**16.**

**Definition**

**17.**

**Theorem**

**3.**

- (1)
- The number of frequent nodes generated does not increase;
- (2)
- The node with the largest intent in frequent nodes does not increase in intent cardinality.
- (3)
- The number of candidate binary arrays generated does not increase;
- (4)
- The number of generated association rules does not increase.

## 3. Algorithm and Results

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

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

#### 3.2. Dynamic Mining Algorithms for Interval Association Rules

#### 3.2.1. Algorithm Design

Algorithm 1. DMA (Dynamic Mining Algorithm) |

Input: Association rule sets $R{S}_{1},R{S}_{2},\dots R{S}_{k}\dots $Output: Association rule set $RS$Step1 $RS=R{S}_{1}$;Step2 The interval association rules in the rule set $R{S}_{k}$ is stored in the form of arrays. Set $R{S}^{*}$ and initialize. Comparing the $Rule$ of rule set $RS$ and $R{S}_{k}$, we unit interval association rules vertically according to Theorems 4 and 5. For the rules that have been united in $RS$ and $R{S}_{k}$, let $Flag=1$ and put the united rule number in $R{S}^{*}$. According to Theorems 6–8, calculate the frequency,$Support$, $Conf$, $PD$, and $UD$ of ${C}_{x}$ and ${C}_{y}$ in $R{S}^{*}$. $Flag$=1, $Num$ = $Num$ + 1. Delete the rules of $Flag=1$ in $RS$ and $R{S}_{k}$, and renumbering. Let rules number in $R{S}^{*}$ add the renumbering number in $RS$. Putting the remain rules of $RS$ into $R{S}^{*}$; then renumbering in $R{S}_{k}$, and putting the remain rules of $R{S}_{k}$ into $R{S}^{*}$.Rule Vertical Union ($RS,R{S}_{k}$) - 1
- {
- 2
- $g=0$;
- 3
- $RS{[g]}^{*}=\{Rule,F{N}_{1}^{},F{N}_{2}^{},U,Support,Conf,$$PD,UD,Flag,Num\}$
- 4
- { $Rule=\varnothing $;
- 5
- $F{N}_{1}^{}=F{N}_{2}^{}=0$;
- 6
- $U=RS[1].U+R{S}_{k}[1].U$;
- 7
- $Support=Conf=PD=UD=Flag=0$;
- 8
- $Num=1$;
- 9
- }
- 10
- For ( i=1;$\left|RS\right|$; i++)
- 11
- {For (j=1;$\left|R{S}_{k}\right|$; j++)
- 12
- If($RS[i].Flag=R{S}_{k}[j].Flag=0$||$RS[i].Rule=R{S}_{k}[j].Rule$)
- 13
- { $g+1$;
- 14
- $Num+1$;
- 15
- $RS[i].Flag=1;$
- 16
- $R{S}_{k}[j].Flag=1;$
- 17
- $RS{[g]}^{*}.F{N}_{1}=$$\frac{RS[i].U*RS[i].F{N}_{1}+R{S}_{k}[j].U*R{S}_{k}[j].F{N}_{1}}{RS[i].U+R{S}_{k}[j].U}$
- 18
- $RS{[g]}^{*}.F{N}_{2}=$$\frac{RS[i].U*RS[i].F{N}_{2}+R{S}_{k}[j].U*R{S}_{k}[j].F{N}_{2}}{RS[i].U+R{S}_{k}[j].U}$
- 19
- $RS{[g]}^{*}.Support=RS{[g]}^{*}.F{N}_{2}$
- 20
- $RS{[g]}^{*}.Conf=$$\frac{RS[i].U*RS[i].F{N}_{2}+R{S}_{k}[j].U*R{S}_{k}[j].F{N}_{2}}{RS[i].U*RS[i].F{N}_{1}+R{S}_{k}[j].U*R{S}_{k}[j].F{N}_{1}}$
- 21
- $RS{[g]}^{*}.PD=$$\mathrm{min}\{RS[i].PD,R{S}_{k}[j].PD\}$;
- 22
- $RS{[g]}^{*}.UD=$$1-RS{[g]}^{*}.PD$;
- 23
- }
- 24
- }
- 25
- For each $RS[i]$ in $RS$;
- 26
- {If $RS[i].Flag=0$;
- 27
- $g+1$;
- 28
- $R{S}^{*}[g]=RS[i]$;}
- 29
- For each $R{S}_{k}[j]$ in $R{S}_{k}$;
- 30
- {If $R{S}_{k}[j].Flag=0$;
- 31
- $g+1$;
- 32
- $R{S}^{*}[g]=R{S}_{k}[j]$;}
- 33
- }
Step3 $RS=R{S}^{*}$ |

#### 3.2.2. Algorithm Analysis

#### 3.3. Example Study

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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a | b | c | d | e | |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 0 | 0 |

2 | 0 | 0 | 0 | 1 | 0 |

3 | 1 | 1 | 0 | 1 | 0 |

4 | 1 | 0 | 1 | 0 | 1 |

a | b | c | d | e | |
---|---|---|---|---|---|

(1) | 1 | 0 | 1 | 1 | 0 |

(2) | 1 | 1 | 0 | 1 | 0 |

(3) | 0 | 1 | 0 | 1 | 1 |

Frequent Node | Frequent Degree | Frequent Node | Frequent Degree |
---|---|---|---|

a | 75% | acd | 100% |

b | 50% | ace | 50% |

c | 75% | ade | 50% |

d | 50% | bcd | 75% |

ab | 50% | bce | 50% |

ac | 50% | cde | 50% |

abc | 75% | abcd | 50% |

abd | 50% | abce | 50% |

abe | 75% | abcde | 75% |

Frequent Node | Frequent Degree | Frequent Node | Frequent Degree |
---|---|---|---|

a | 67% | acd | 67% |

b | 67% | ade | 100% |

d | 100% | bcd | 100% |

ad | 67% | bde | 67% |

bd | 67% | cde | 67% |

abc | 67% | abcd | 67% |

abd | 100% | abde | 67% |

abe | 67% | abcde | 100% |

Rule | Support | Confidence | Accuracy | Uncertainty | Frequent Number |
---|---|---|---|---|---|

$a\Rightarrow bcde$ | 75% | 100% | 60% | 40% | 1 |

$abc\Rightarrow de$ | 75% | 100% | 60% | 40% | 1 |

$abe\Rightarrow cd$ | 75% | 100% | 60% | 40% | 1 |

$acd\Rightarrow be$ | 75% | 75% | 60% | 40% | 1 |

$a\Rightarrow bce$ | 50% | 67% | 75% | 25% | 1 |

$c\Rightarrow abe$ | 50% | 67% | 75% | 25% | 1 |

$ac\Rightarrow be$ | 50% | 100% | 75% | 25% | 1 |

$abc\Rightarrow e$ | 50% | 67% | 67% | 33% | 1 |

$abe\Rightarrow c$ | 50% | 67% | 67% | 33% | 1 |

$ace\Rightarrow b$ | 50% | 100% | 67% | 33% | 1 |

$a\Rightarrow bcd$ | 50% | 67% | 75% | 25% | 1 |

$b\Rightarrow acd$ | 50% | 100% | 75% | 25% | 1 |

$ab\Rightarrow cd$ | 50% | 100% | 75% | 25% | 1 |

$abc\Rightarrow d$ | 50% | 67% | 67% | 33% | 1 |

$abd\Rightarrow c$ | 50% | 100% | 67% | 33% | 1 |

$bcd\Rightarrow a$ | 50% | 67% | 67% | 33% | 1 |

$c\Rightarrow de$ | 50% | 67% | 67% | 33% | 1 |

$c\Rightarrow be$ | 50% | 67% | 67% | 33% | 1 |

$a\Rightarrow de$ | 50% | 67% | 67% | 33% | 1 |

$a\Rightarrow ce$ | 50% | 67% | 67% | 33% | 1 |

$c\Rightarrow ae$ | 50% | 67% | 67% | 33% | 1 |

$ac\Rightarrow e$ | 50% | 100% | 67% | 33% | 1 |

$a\Rightarrow be$ | 75% | 100% | 67% | 33% | 1 |

$a\Rightarrow bd$ | 50% | 67% | 67% | 33% | 1 |

$b\Rightarrow ae$ | 50% | 100% | 67% | 33% | 1 |

$ab\Rightarrow e$ | 50% | 100% | 67% | 33% | 1 |

$a\Rightarrow bc$ | 75% | 100% | 67% | 33% | 1 |

$a\Rightarrow c$ | 50% | 67% | 100% | 0% | 1 |

$c\Rightarrow a$ | 50% | 67% | 100% | 0% | 1 |

$a\Rightarrow b$ | 50% | 67% | 100% | 0% | 1 |

$b\Rightarrow a$ | 50% | 100% | 100% | 0% | 1 |

Rule | Support | Confidence | Accuracy | Uncertainty | Frequent Number |
---|---|---|---|---|---|

$d\Rightarrow abce$ | 100% | 100% | 60% | 40% | 1 |

$abd\Rightarrow ce$ | 100% | 100% | 60% | 40% | 1 |

$ade\Rightarrow bc$ | 100% | 100% | 60% | 40% | 1 |

$bcd\Rightarrow ae$ | 100% | 100% | 60% | 40% | 1 |

$b\Rightarrow ade$ | 67% | 100% | 75% | 25% | 1 |

$d\Rightarrow abe$ | 67% | 67% | 75% | 25% | 1 |

$bd\Rightarrow ae$ | 67% | 100% | 75% | 25% | 1 |

$abd\Rightarrow e$ | 67% | 67% | 67% | 33% | 1 |

$abe\Rightarrow d$ | 67% | 100% | 67% | 33% | 1 |

$ade\Rightarrow b$ | 67% | 67% | 67% | 33% | 1 |

$bde\Rightarrow a$ | 67% | 100% | 67% | 33% | 1 |

$a\Rightarrow bcd$ | 67% | 100% | 75% | 25% | 1 |

$d\Rightarrow abc$ | 67% | 67% | 75% | 25% | 1 |

$ad\Rightarrow bc$ | 67% | 100% | 75% | 25% | 1 |

$abc\Rightarrow d$ | 67% | 100% | 67% | 33% | 1 |

$abd\Rightarrow c$ | 67% | 67% | 67% | 33% | 1 |

$acd\Rightarrow b$ | 67% | 100% | 67% | 33% | 1 |

$bcd\Rightarrow a$ | 67% | 67% | 67% | 33% | 1 |

$d\Rightarrow ce$ | 50% | 67% | 67% | 33% | 1 |

$b\Rightarrow de$ | 67% | 100% | 67% | 33% | 1 |

$d\Rightarrow be$ | 67% | 67% | 67% | 33% | 1 |

$bd\Rightarrow e$ | 67% | 100% | 67% | 33% | 1 |

$d\Rightarrow bc$ | 100% | 100% | 67% | 33% | 1 |

$d\Rightarrow ae$ | 100% | 100% | 67% | 33% | 1 |

$a\Rightarrow cd$ | 67% | 100% | 67% | 33% | 1 |

$d\Rightarrow ac$ | 67% | 67% | 67% | 33% | 1 |

$ad\Rightarrow c$ | 67% | 100% | 67% | 33% | 1 |

$b\Rightarrow ae$ | 67% | 100% | 67% | 33% | 1 |

$d\Rightarrow ab$ | 100% | 100% | 67% | 33% | 1 |

$a\Rightarrow bc$ | 67% | 100% | 67% | 33% | 1 |

Rule | Support | Confidence | Accuracy | Uncertainty | Frequent Number |
---|---|---|---|---|---|

$a\Rightarrow bcd$ | 57% | 80% | 75% | 25% | 2 |

$abc\Rightarrow d$ | 57% | 80% | 67% | 33% | 2 |

$abd\Rightarrow c$ | 57% | 80% | 67% | 33% | 2 |

$bcd\Rightarrow a$ | 57% | 67% | 67% | 33% | 2 |

$a\Rightarrow bc$ | 71% | 100% | 67% | 33% | 2 |

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## Share and Cite

**MDPI and ACS Style**

Yang, Y.; Zhang, R.; Liu, B.
Dynamic Parallel Mining Algorithm of Association Rules Based on Interval Concept Lattice. *Mathematics* **2019**, *7*, 647.
https://doi.org/10.3390/math7070647

**AMA Style**

Yang Y, Zhang R, Liu B.
Dynamic Parallel Mining Algorithm of Association Rules Based on Interval Concept Lattice. *Mathematics*. 2019; 7(7):647.
https://doi.org/10.3390/math7070647

**Chicago/Turabian Style**

Yang, Yafeng, Ru Zhang, and Baoxiang Liu.
2019. "Dynamic Parallel Mining Algorithm of Association Rules Based on Interval Concept Lattice" *Mathematics* 7, no. 7: 647.
https://doi.org/10.3390/math7070647