# Distributed Action-Rule Discovery Based on Attribute Correlation and Vertical Data Partitioning
## 2 Background
### Action Rules  
Action rules (or actionable patterns), first introduced by [PERSON] and [PERSON] in [20], describe possible transitions of objects from one state to another with respect to a distinguished attribute called the decision attribute. Strategies for discovering them are divided into two types: rule-based and object-based. The rule-based approach consists of two main steps: (1) using a standard learning method to detect interesting patterns in the form of classification rules, association rules, or clusters, and (2) employing an automatic or semi-automatic strategy to inspect these results and derive possible action strategies. These strategies can provide insights into how the values of some attributes should be modified to move desirable objects into a desired group. The object-based approach assumes that actionable patterns are directly extracted from a database. Examples of rule-based strategies for extracting action rules are System DEAR (presented in [21]) and System DEAR2 (presented in [22]). [PERSON]'s group has implemented these systems as modules of his LISP Miner software package (see [23; 24; 25]). Another example of a rule-based approach to action rules discovery is a strategy based on tree classifiers (see [26]). It should be mentioned here that the quality and quantity of action rules discovered heavily depend on the classification rules used in that process. To our knowledge, there is only one method based on discernibility functions in rough set theory that can discover all classification rules from a given dataset (see [27]). The more classification rules we have, the more action rules can be discovered. Object-based strategies for action rule mining have the same limitations--they do not discover all action rules [28]. An interesting concept is an action rules schema, which can be constructed directly from so-called action reducts. Action-rules schemas cover all action rules. Can we propose a strategy for generating all action rules from action-rules schemas? Additional examples of strategies for action rules discovery can be found in publications by [PERSON] (see [29; 30]) and [PERSON] (see [31]). Action rules have been used in various domains, such as medical (see [5; 6; 32; 33; 34]) business (see [2; 35; 36]), music (see [7; 12]),and art (see [8; 9]). The rule-based approach has two main steps: (1) mining classification rules and (2) generating action rules from those classification rules. A classification rule \(r_{1}\) is defined as a term:  
\[r_{1}=[((A,a_{1})\wedge(B,b_{1})\wedge(C,c_{1})\wedge(E,e_{1}))\Rightarrow(D,d_{ 1})], \tag{1}\]  
where, in the rule's antecedent, \(a_{1}\), \(b_{1}\), \(c_{1}\), and \(e_{1}\) are attribute values associated with attributes \(A\), \(B\), \(C\), \(E\), correspondingly. Attribute value \(d_{1}\), associated with attribute \(D\), is a predicted target value called the consequent or the decision. The support and confidence are calculated for each classification rule as a performance measure. The support is measured by the number of objects (or rows in the dataset) that match both the antecedent and consequent of a given rule. The confidence is the \"ratio between\" a given rule's support and the total number of items that \"satisfy [just] the antecedent\" [37]. Action rules can be discovered directly from the dataset or constructed from classification rules. An action rule is defined as follows [20]:  
\[r=[[a_{1}\wedge g_{2}\wedge(B,b_{1}\to b_{2})\wedge(H,h_{2})]\Rightarrow(D,d_{ 1}\to d_{2})], \tag{2}\]  
In the above, \(a_{1}\) and \(g_{2}\) are stable attributes, meaning they are attributes that do not change over time (their values cannot change). The next parts \((B,b_{1}\to b_{2})\) and \((H,\to h_{2})\) show the flexible attributes. These are the attributes that can change over time. The final piece, \((D,d_{1}\to d_{2})\), is our consequent, the target value we wish to change.