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Association rules algorithm

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Program Information

Name: Association rules algorithm
Domain: Algorithm
Functionality: Association rules algorithm in data mining
Input: We use $O$ to indicate the original test case,and $F$ indicate the follow-up test case. We add an attribution $A_{fp}$ on the original test case and the additional attribution values are all the same, namely, the attribution is no relations with the original test case ones. For the source test case, add a column attribution(transaction item) $E_{rp}$ that is the same distribution with one attribution $E_{op}$ of the original test case to grain the follow-up test case. 
Output: $O^{(k)}(k=1,2,\cdots ,m)$ denotes the number of the $k$-frequent itemset of the original test cases; $\textrm{OC}^{(k)}(k=1,2,\cdots ,m)$ denotes $k$-frequent itemset of the original test cases; $F^{(k)}(k=1,2,\cdots ,m,m+1)$ is the number of the $k$-frequent itemset of the follow-up cases; $\textrm{FC}^{(k)}(k=1,2,\cdots ,m,m+1)$ is the k-frequent itemset of the follow-up cases. The number of itemsets including the value of $E_{op}$ denotes $|E_{op}^{(k)}|$ in the $\textrm{OC}^{(k)}$.The frequent itemset including the addition attribution $E_{op}$ that is the same as $E_{fp}$ denotes $\textrm{OC}_{op}^{(k)}$ 

Reference

 AN EVALUATION APPROACH FOR THE PROGRAM OF ASSOCIATION RULES ALGORITHM BASED ON METAMORPHIC RELATIONS
https://doi.org/10.1007/s11767-012-0743-9 

MR Information

MR1------Affine transformation

Description:
Property: $F^{(k)}=O^{(k)},\textrm{FC}^{(k)}=\textrm{OC}^{(k)},k=1,2,\cdots ,m$
Source input: $O$
Source output: $O^{(k)},\textrm{OC}^{(k)}$
Follow-up input: $F$
Follow-up output: $F^{(k)},\textrm{FC}^{(k)}$
Input relation:
Output relation: $F_{(k)}=O_{(k)},\textrm{FC}^{(k)}=\textrm{OC}^{(k)},k=1,2,\cdots ,m$
Pattern:

MR2------Row transformation 

Description:
Property: $F^{(k)}=O^{(k)},\textrm{FC}^{(k)}=\textrm{OC}^{(k)},k=1,2,\cdots ,m$ 
Source input:  $O$
Source output: $O^{(k)},\textrm{OC}^{(k)}$ 
Follow-up input: $F$ 
Follow-up output: $F^{(k)},\textrm{FC}^{(k)}$ 
Input relation:  
Output relation: $F^{(k)}=O^{(k)},\textrm{FC}^{(k)}=\textrm{OC}^{(k)},k=1,2,\cdots ,m$ 
Pattern:

MR3------Column transformation 

Description:
Property: $F^{(k)}=O^{(k)},\textrm{FC}^{(k)}=\textrm{OC}^{(k)},k=1,2,\cdots ,m$ 
Source input: $O$ 
Source output: $O^{(k)},\textrm{OC}^{(k)}$ 
Follow-up input: $F$ 
Follow-up output: $F^{(k)},\textrm{FC}^{(k)}$ 
Input relation:  
Output relation: $F^{(k)}=O^{(k)},\textrm{FC}^{(k)}=\textrm{OC}^{(k)},k=1,2,\cdots ,m$ 
Pattern:

MR4------Addition of uninformative attribution 

Description:
Property: $F^{(k)}=O^{(k-1)}+O^{(k)}$ and $O^{(0)}=1,O^{(m+1)}=0 \Rightarrow \textrm{FC}^{(k)}=\{\textrm{OC}^{(k-1)}+A_{fp} \cup \textrm{OC}^{(k)}\}$ and $\textrm{OC}^{(0)}=\varnothing,\textrm{OC}^{(m+1)}=\varnothing$ 
Source input: $O$ 
Source output:  $O^{(k)},O^{(k-1)},\textrm{OC}^{(k)},\textrm{OC}^{(k-1)}$
Follow-up input: $F$ 
Follow-up output: $F^{(k)},\textrm{FC}^{(k)}$ 
Input relation:  
Output relation: $F^{(k)}=O^{(k-1)}+O^{(k)}$ and $O^{(0)}=1,O^{(m+1)}=0 \Rightarrow \textrm{FC}^{(k)}=\{\textrm{OC}^{(k-1)}+A_{fp} \cup \textrm{OC}^{(k)}\}$ and $\textrm{OC}^{(0)}=\varnothing,\textrm{OC}^{(m+1)}=\varnothing$ 
Pattern:

MR5------Addition of informative attribution 

Description:
Property: $F^{(k)}=O^{(k-1)}+|E_{op}^{(k-1)}| + |E_{op}^{(k)}|$ and $|E_{op}^{(0)}=0|,O^{(m+1)}=0,|E_{op}^{m+1}=0|$ $\Rightarrow$ $\textrm{FC}^{(k)}=\textrm{OC}^{(k)} \cup \{\textrm{OC}^{(k-1)}_{op}+E_{fp}\} \cup \textrm{OC}_{op}^{(k)}-E_{op}+E_{fp}$ and  $\textrm{OC}^{(0)}=\varnothing,\textrm{OC}^{(m+1)}=\varnothing$ 
Source input: $O$ 
Source output: $O^{(k)},O^{(k-1)},\textrm{OC}^{(k)},\textrm{OC}^{(k-1)}$ 
Follow-up input: $F$ 
Follow-up output: $F^{(k)},\textrm{FC}^{(k)}$ 
Input relation:  
Output relation: $F^{(k)}=O^{(k-1)}+|E_{op}^{(k-1)}| + |E_{op}^{(k)}|$  $|E_{op}^{(0)}=0|,O^{(m+1)}=0,|E_{op}^{m+1}=0|$ $\Rightarrow$ $\textrm{FC}^{(k)}=\textrm{OC}^{(k)} \cup \{\textrm{OC}^{(k-1)}_{op}+E_{fp}\} \cup \textrm{OC}_{op}^{(k)}-E_{op}+E_{fp}$ and  $\textrm{OC}^{(0)}=\varnothing,\textrm{OC}^{(m+1)}=\varnothing$ 
Pattern:

MR6------Copy the train set 

Description:
Property: $F^{(k)}=O^{(k)},\textrm{FC}^{(k)}=\textrm{OC}^{(k)},k=1,2,\cdots ,m$ 
Source input:  $O$
Source output: $O^{(k)},\textrm{OC}^{(k)}$ 
Follow-up input:  $F$
Follow-up output:  $F^{(k)},\textrm{FC}^{(k)}$
Input relation:  
Output relation: $F^{(k)}=O^{(k)},\textrm{FC}^{(k)}=\textrm{OC}^{(k)},k=1,2,\cdots ,m$ 
Pattern:

MR7------Remove an attribution 

Description:
Property: $F^{(k)}=O^{(k)}-|E_{op}^{(k)}| \Rightarrow \textrm{FC}^{(k)}=\textrm{OC}^{(k)}-\textrm{OC}^{(k-1)}_{op}$ 
Source input: $O$ 
Source output:  $O^{(k)},\textrm{OC}^{(k)}$
Follow-up input: $F$ 
Follow-up output: $F^{(k)},\textrm{FC}^{(k)}$ 
Input relation:  
Output relation: $F^{(k)}=O^{(k)}-|E_{op}^{(k)}| \Rightarrow \textrm{FC}^{(k)}=\textrm{OC}^{(k)}-\textrm{OC}^{(k-1)}_{op}$ 
Pattern:
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