## 3 Background  
This section provides detailed preliminary background information to understand the frequent regular identical itemsets and top-K frequent regular identical itemsets which are used and presented in the proposed approach in the next section. Several definitions are presented along with their mathematical equations to provide the background knowledge of the area and proposed method. The notions used in the equations are presented and explained in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**S. No** & **Symbol/Term** & **Explanation** \\ \hline
1 & \(tr\) & Incoming transaction \\ \hline
2 & \(t_{b}/T_{b}\) & Transaction buffer \\ \hline
3 & \(\theta_{r}\) & Regularity threshold \\ \hline
4 & \(\theta_{s}\) & Support threshold \\ \hline
5 & \(1 SI-Tables\) & Item support index table \\ \hline
6 & \(P\) & Possible set of candidate itemsets \\ \hline
7 & \(F_{l}\) & Frequent regular patterns or patterns \\  & & generated at the parent node \\ \hline
8 & \(C_{l}\) & Candidate itemsets generated or \\  & & available at the parent node \\ \hline
9 & \(F_{l+1}\) & New generated frequent pattern \\ \hline
10 & \(C_{l+1}\) & New set of candidate itemsets/patterns \\ \hline
11 & \(l\) & Set of all items in a dataset \\ \hline
12 & X & Itemset or pattern \\ \hline
13 & \(t_{j}\) & Individual transaction \\ \hline
14 & \(I_{FR}\) & Set of frequent regular patterns \\ \hline \hline \end{tabular}
\end{table}
Table 2: Introduction and explanation of the notions used in the mathematic equations.  
Let \(\text{I}=\{i_{1},i_{2},\ldots\ \ldots\.\ i_{n}\}\) present the set of all items. A set \(\text{X}=\{i_{j},\ldots\ \ldots\.\ i_{k}\}\subseteq I\)\(\text{j}\leq k,j,k\ \in[1,n]\) is called an itemset or pattern. A transaction dataset \(\text{T}=\{t_{1},t_{2},\ldots,t_{m}\}\) over dataset D contain \(m\) number of total transactions. Every transaction \(t_{i}\in\text{T}\) is 2-tuple (i.e., \(t_{i}=\text{(TID, Y)}\) where \(\text{TID}\in[1,m]\) and Y is a pattern). If a pattern \(\text{X}\subseteq Y\) is found in transaction \(t_{i}\in T\), then the transaction ID (i.e., TID) of the corresponding transaction is written as \(t_{j}^{x},\ j\ \in[1,m]\). The support count of pattern \(X\) is the set of all corresponding TIDs where transactions contained X as patterns (i.e., \(T^{x}=\left\{t_{j}^{x},\ldots\ldots,t_{k}^{x}\right\}\), \(j,k\ \in[1,m]\)\(j\leq k\)). The period of pattern \(X\) is the number of transactions where \(X\) as a pattern does not appear between any two consecutive transactions \(t_{j},\ t_{j+1}\ \in T,\ J\in[1,m]\). The regularity of pattern \(X\) is the maximum number of transactions between two consecutive transactions \(t_{j},t_{j+1}\ \in T^{x}\ j\in[1,m]\) where \(X\) does not occur in the transaction dataset (i.e., \(P^{x}=max\left(P_{1}^{x},P_{2}^{x},\ldots,P_{|T^{x}|-1}^{x}\,P_{|T^{x}|}^{x}\right)\)). On a user-supplied support threshold and regularity threshold, \(R\) is the set of patterns qualifying aforementioned constraint (i.e., \(R=\{\forall i=1\cdots n\ X_{i}|\forall i=1\cdots n\ support(X_{i})\geq\theta_{s}\wedge\ \forall i=1\cdots n\ Reg(X_{i})\leq\ \theta_{r}\}\)). The support count of all the patterns in \(R\) can be collected in a set of distinct support counts \(S=\{\forall i=1,\ldots,m\ S_{i}=\text{max\_supp}_{1}(R)\}\). The patterns in \(R\) can be classified into identical frequent regular (IFR) patterns and top-K identical frequent regular patterns.