## 2 Method to Extract the Rule-Based Knowledge
### Notations and Definitions  
According to the application of this study, cerebrovascular examination data records of patients \(p=\{p1,\ p2,\ \ldots\}\) were collected and denoted as set \(D\). A set of \(D=\left\{\left(a_{i},\ sbp_{i},\ bmi_{i},\ fg_{i},\ tc_{i}\right),\ mri_{i} \right\}\)\(\left|1\leq i\leq|P|\right\}\) consists of six features: age \((a_{i}\in A)\), systolic blood pressure \(\left(sbp_{i}\in SBP\right)\), body-mass index \((bmi_{i}\in BMI)\), fasting glucose \(\left(fg_{i}\in FG\right)\), and total cholesterol \((tc_{i}\in TC)\), which correspond to MRI results as class labels \(M=\{miri_{1},miri_{2},\ldots\}\). Furthermore, there is a collection of distinct items from each feature, denoted \(I=\{i_{1},i_{2},\ldots\}\), thus \(A\), \(SBP\), \(BMI\), \(FG\), \(TC\subseteq I\). Set \(D\) can be written as \(D=\{(d_{i},\ mri_{i})\big{|}1\leq i\leq|P|\}\), where \(d_{i}=\left(a_{i},\ sbp_{i},\ bmi_{i},\ fg_{i},\ tc_{i}\right)\) for simplicity. Furthermore, one notes that there is no patient with a \(BMI\) value in both the obese and underweight ranges at the same time in medical case. Table 1 gives an illustration of database \(D\). To discover useful information from the database \(D\), conventional ARM is a popular data-mining technique to assist experts--i.e., medical practitioners, in this case--in decision making. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**PatientID** & **Age** & **SBP** & **MRI\_Brain** \\ \hline p1 & Adult & Normal & Normal \\ p2 & Elderly & Hypertension I & Abnormal \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ pn & Adolescent & Normal & Normal \\ \hline \hline \end{tabular}
\end{table} 
Table 1: An example of a cerebrovascular examination dataset in Taiwan. Useful information can be represented as a rule \(\mathcal{R}:(X\to Y)\), where \(X,Y\subseteq 1\). Hence, \(X\) and \(Y\) can also be said to be item sets. To determine an association rule \(\mathcal{R}\), two measurements need to be defined, e.g., support and confidence values (\(sup(X)\) and \(conf(X\to Y)\), respectively), as follows:  
\[sup(X)=P(X)=\frac{|d\in D;\ x\subseteq d|}{|D|} \tag{1}\]  
\[conf(X\to Y)=P(Y|X)=\frac{sup(X\to Y)}{sup(X)}=\frac{|d\in D;\ X\subseteq d,\ Y\subseteq d|}{sup(X)} \tag{2}\]  
Based on [23], \(X\) or \(Y\) can be considered a frequent itemset if their support values satisfy a predefined minimum support threshold, \(\sigma\), as defined below in Definition 1. **Definition 1**. : _(Frequent itemset)_  
Given a minimum support threshold \(\sigma\in[0,\ 1]\), then an itemset \(X\) is a frequent itemset if \(sup(X)\geq\sigma\). In conventional ARM, a generate-and-test approach is employed to collect frequent itemsets and subsequently build ARs based on two predefined threshold values. Formally, conventional ARM is defined as follows. **Definition 2**. : _(Conventional ARM)_  
Given minimum support and confidence thresholds \(\sigma\) and \(\delta\in[0,\ 1]\), and two frequent itemsets \(X\) and \(Y\), we extract a set of association rules \(\mathcal{R}=\{(Xi\to Yi)\big{|}Xi\in X,\ Yi\in Y\}\) that fulfill the following conditions:  
1. \(sup(Xi)\geq\sigma,\ \forall Xi\in X\); and
2. \(conf(Xi\to Yi)\geq\delta,\ \forall Yi\in Y\). However, conventional ARM is computationally expensive in terms of run-time and memory usage. To overcome this drawback, this study focused on ARM using a hybrid approach [19]--with a vertical database format (the detailed illustration of a vertical format on \(D\) is given in Table 3 in Section 2.3.1)--and presents NRARs. Furthermore, there are five types of non-redundant frequent item sets discussed in this study: FCI, generator of FCI (GFCI), minimal GFCI (mGFCI), non-subsumption FCI (nSFCI), and non-circular FCI (nCFCI). Henceforth, this study defines all non-redundant frequent item sets and subsequently generates a set of concise rules, so-called non-anomalous association rules (NAARs), based on the aforementioned types of non-redundant frequent item sets.