## 3 Proposed Method  
We propose a novel classifier-based elementary cellular automaton that uses the variance of the initial rule matrices. In this section, several preliminaries are presented, and some essential equations are simplified. Additionally, we present the motivation for the model based on the limitations of the traditional classifier. In Table 1, we first introduce some notations that are frequently used in this study. First, let us provide an example of the equations for finding the rule matrices according to Equations (10) and (11) before analyzing the problem. \[[R^{+}]_{ij}= \begin{cases}1,&[U_{p}]_{ij}\geq\omega_{p}\\ 0,&\text{otherwise}\end{cases}  \tag{10}\]  
\[[R^{-}]_{ij}= \begin{cases}1,&[U_{n}]_{ij}\geq\omega_{n}\\ 0,&\text{otherwise}\end{cases}  \tag{11}\]  
where \(R^{+}\) and \(R^{-}\) are the rule matrices of the positive and negative classes, respectively; \(\omega_{p}\) and \(\omega_{n}\) are the cutting-point decisions used to eliminate some values from the initial rule matrices \(U_{p}\) and \(U_{n}\) in the \(i\)th column and \(j\)th row, respectively, where \(i,j=1,2,3\ldots,n\) in order to generate rule matrices with the best classification accuracy. **Definition 1** (conflict value). : _A conflict value is the difference between the same position element of the initial rule matrices \([U_{p}]ij\) and \([U_{n}]ij\), arising from the creation processes of the rule matrices \([R^{+}]ij\) and \([R^{-}]ij\). The conflict value can be represented by Equation (12)._  
\[\mathit{Conflict\ value}= \begin{cases}|[U_{p}]_{ij}-[U_{n}]_{ij}|,&[U_{p}]_{ij} \text{ and }[U_{n}]_{ij}\geq 1\\ 0,&[U_{p}]_{ij}\text{ or }[U_{n}]_{ij}=0\end{cases}  \tag{12}\]  
where \(\mathit{Conflict\ value}\in I^{+}\)  
**Lemma 1**. : _The higher the conflict value the lower the classification accuracy._  
Proof of Lemma 1.: Equations (10) and (11) show that all elements of \([U_{p}]_{ij}\) and \([U_{n}]_{ij}\) would convert to 1 if their value is greater than 0. Assume that \(U_{p}\gg U_{n}\) when \(\lim\limits_{x\rightarrow\infty}[U_{p}]_{ij}=\infty\) and \(\lim\limits_{x\rightarrow\infty}[U_{n}]_{ij}=1\) then  
\[ \begin{split}\mathit{Conflict\ value}&=|[U_{p}]_{ij} -[U_{n}]_{ij}|\\ &=\infty-1\\ &=\infty\end{split}  \tag{13}\]  
From the above Equation (13), the conflict value \([U_{p}]ij\) is much greater than \([U_{n}]ij\). The conflict value is equivalent to the maximum value. However, from (10) and (11), it can be observed that small values are equalized with large values. Unfair behavior leads to an unreasonable increase in the amount of data. This increase in the number affects the prediction of the test data because both \([U_{p}]ij\) and \([U_{n}]ij\) are converted to 1 in the rule ordering process. **Definition 2** (percentage of variation coefficient). : _The coefficient of variation (\(CV\)) is defined as the ratio of the standard deviation (\(S\)) in Equation (14) and the sample mean (\(\bar{x}\)) in Equation (15)._  
\[\bar{x}=\frac{\sum\limits_{k=1}^{n}x_{k}}{N} \tag{14}\]  
where \(\bar{x}\) represents the sample mean of the elements at the same position in the initial rule matrices \([U_{p}]\) and \([U_{n}]\). \(x_{k}\) is the observed value, where \(k=(1,2,\ldots,n)\), and \(N\) represents