# A One-Phase Tree-Structure Method to Mine High Temporal Fuzzy Utility Itemsets
## 3 Problem Definition  
Assume \(TQD\) denote a temporal quantitative transaction database. It may have: \(TQD\) = {\(T_{1}\), \(T_{2}\),..., \(T_{z}\)}, where \(z\) is the number of transactions. Let \(I\) be the set of items in \(TQD\). Thus, \(I\) = {\(i_{1}\), \(i_{2}\), \(i_{3}\),..., \(i_{m}\)}, where \(m\) is the number of items. Each transaction \(T_{y}\) in \(TQD\) has its identifier denoted \(TID\). Each item \(i_{m}\) in a transaction \(T_{y}\) includes its quantity sold, denoted \(v_{ym}\), and unit profit called the external utility and denoted \(s(i_{m})\). There is a set of time periods in \(TQD\), denoted \(P\) = {\(p_{1}\), \(p_{2}\), \(p_{3}\),..., \(p_{r}\)}, where \(r\) is the number of time periods. Table 1 shows a database with five different items and three time periods. The unit profit values of the items are listed in Table 2.  
Given the temporal fuzzy utility threshold \(\lambda\) and the membership functions of each item, such as those in Figure 1, the problem is to find all the high temporal fuzzy utility \begin{table}
\begin{tabular}{c c c c} \hline \hline _Period_ & _TID_ & _Quantitative Items_ & _Transformed Fuzzy Terms_ \\ \hline \multirow{3}{*}{\(P_{1}\)} & \(T_{1}\) & {\(a\):6}, {\(c\):2} & (\(1/a.M\)), (0.67/\(c.L\)) \\  & \(T_{2}\) & {\(a\):4}, {\(b\):4} & (\(0.67/b.L+0.33/b.M\)) \\  & \(T_{3}\) & {\(a\):1}, {\(c\):1} & (\(0.33/c.L\)), (0.33/\(a.L\)) \\  & \(T_{4}\) & {\(a\):3}, {\(b\):3}, {\(c\):4}, {\(d\):4}, {\(e\):2} & (\(1/a.L\)), (\(1/b.L+0.67/e.L\)) \\  & \(T_{5}\) & {\(a\):3}, {\(b\):6}, {\(e\):2} & (\(1/a.L\)), (\(1/b.M\). ), (0.67/\(e.L\). ) \\  & \(T_{6}\) & {\(a\):3}, {\(d\):4}, {\(e\):2} & (\(1/a.L+0.67/e.L\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: An example of temporal quantitative transaction database. itensets from the temporal database. The following definitions [16] are first given to explain the mining problem. **Definition 1**. : _Let the h-th linguistic term of the item \(i_{m}\) be denoted \(R_{mh}\). The fuzzy utility \(fu_{ymh}\) of the linguistic term \(R_{mh}\) in a transaction \(T_{y}\) is defined as:_  
\[fu_{ymh}=\ast v_{ym}\ast s(i_{m}) \tag{1}\]  
_where \(v_{ym}\) is the quantitative value of the item \(i_{m}\) in the transaction \(T_{y}\), \(u_{ymh}\) is the fuzzy value of \(v_{ym}\) in \(R_{mh}\), and \(s(i_{m})\) is the unit profit value of \(i_{m}\)._  
Assume Figure 1 shows the three membership functions for item \(a\). Its quantity in \(T_{2}\) is 4, which is transformed into \((0.67/a.L+0.33/a.M)\). The fuzzy utility of \(R_{a.L}\) is thus \(0.67\times 4\times 2\), which is \(5.36\). **Definition 2**. : _The transactional fuzzy utility \(fty\) of a transaction \(T_{y}\) is the summation of the fuzzy utility values of all linguistic terms in \(T_{y}\)._  
**Definition 3**. : _A fuzzy itemset includes one or more linguistic terms with no terms derived from the same item._  
Take transaction \(T_{6}\) as an example. (\(a.L\), \(d.L\)) is a valid fuzzy itemset with two fuzzy items, but (\(d.L\), \(d.M\)) is not because it is derived from the same item \(d\). **Definition 4**. : _Let \(X\) be a fuzzy itemset. The fuzzy utility of \(X\) in \(T_{y}\), denoted \(fu_{yX}\), is:_  
\[=\ast\sum_{R_{mh}\subseteq X}v_{ym}\ast s(i_{m}) \tag{2}\]  
_where the \(\mu_{yX}\) is the minimum scalar cardinality in \(R_{mh}\) in \(X\)._  
Take the fuzzy itemset (\(a.L\), \(d.L\)) in \(T_{6}\) as an example. The fuzzy value of (\(a.L\), \(d.L\)) in \(T_{6}\) is \(0.67\) (=min(1,0.67)). Thus, its \(fu_{6_{6},\ (a.L\), \(d.L\)) value is calculated as \(0.67\times[(3\times 2)+(4\times 2)]=9.38\). **Definition 5**. : _The start transaction period (STP) of an item is its first appearing time period in TQD._  
**Definition 6**. : _The lasting transaction periods of an itemset \(X\), denoted as LTP\({}_{X}\), is the set of the periods from the latest STP among the items in \(X\) to the end of TQD._  
**Definition 7**.