# Incremental Algorithm for Association Rule Mining under Dynamic Threshold
## 2 Association Rule Terms  
This section introduces and defines the important key terms used in ARM domain and the approach proposed in this study. The market basket was applied to clarify the terms and to illustrate the meaning. All the terms employed in this study are defined as follows:  
**Definition 1**. : _The'set of items' was given in the following:_  
\[I=\{i_{1},i_{2},i_{3},...,i_{m}\}\]  
_where \(i\) refers to the attribute in the database. In market basket analysis, it denotes the product. It is a chance for the set of items to function as the range pool of attributes value in the database transaction. Based on the case study of ARM, the set of items refers to product found in the supermarket. Grounded on Definition 1, the term 'transaction' is defined as follows in the next definition:_  
**Definition 2**. : _The transaction, \(T\): It is known as the main element of the database with a pair structure: \(T=(tid,I^{\prime})\), Where tid is the transaction identifier, while \(I^{\prime}\) is a subset of 1. The purchase of a set product by a customer is stored on a record or transaction market database. This operation is composed of two essential aspects; the 1D for this record, and the items bought (\(I^{\prime}\))._  
From Definition 2, the 'transaction database' is defined as follows:  
**Definition 3**. : _Transaction database (\(DB\)) refers to a collection of transactions \(\{T\}\) over 1. \(DB\) is the source database that is scanned for the first time to generate new knowledge. From Definition 3, a special definition is derived for the modified database._  
**Definition 4**. : _The modified transaction database (\(db+\)) is a collection of transactions \(\{T\}\) over 1, whereby \(\{T\}\) is manipulated after mining \(DB\). Simply put, the new transactions are transactions that have been either updated or deleted. Based on Definitions 1-4, the term 'itemset' is given in the following:_  
**Definition 5**. : _Itemset In the core point of ARM, the itemset is characterized as \(x=\{i_{1},i_{2},i_{3},...,i_{m}\}\), where \(x\) refers to a set of items that is supposed to befall frequently in a database as a pattern. The itemset has several features with'support' being the most important one, defined as follows:_  
**Definition 6** (itemset support). : _The itemset \(x\) support denotes the counting of \(T\) in \(D\) that contains the itemset \(x\), such as:_  
\[Support(x,D)=|tid|(tid,i^{\prime})D,X\subseteq I^{\prime},I\subseteq I^{\prime}|\]Further clarification regarding the term 'itemset support' is related in the following example:  
**Example 1**. : _If \(x=i_{1},i_{2},i_{3}\) is an itemset in a \(D\) database that has \(10\) rows and \(x\) can be found in four transactions, then we have the following support of \(x\):_  
\[Support(x)=4/10=0.4\]  
Based on Definitions 5 and 6,'support threshold' is described as given in the following:  
**Definition 7**. : _Support threshold (\(MinSupp\)) is the number or the ratio of itemset occurrence. If the itemset set has occurrence count exceeding \(\alpha\), it is considered as a frequent itemset. Nonetheless, the value of \(\alpha\) is \(0\leq\alpha(x)\leq|D|\), with \(|D|\) reflecting the database size. \(\alpha(x)\) refers to the count of \(T\) that contains \(x\) itemset in the item set._  
**Definition 8**. : \(X\Rightarrow Y\) _represents the association rule form, where \(X,Y\) and \(X\cap Y=\emptyset\). \(X\) is the body or the antecedent, whereas \(Y\) is the head or the consequence of the rule._  
**Definition 9**. : _The confidence of an association rule \(X\Rightarrow Y\) is the conditional probability of having \(Y\) embedded in a transaction, given that \(X\) is contained in that transaction._  
\[Confidence(X\Rightarrow Y,D)=Supprt(X\cup Y,D)/Support(X,D)\]  
_The concept of association rule confidence is drawn in the following example._  
**Example 2**. : _Let \(D=T\) be a database of transactions.