## 3 Proposed Algorithm
### Modifications of the CIM Computation
#### 3.4.2 Clustering-Based Approach  
In this approach, for every \(\lambda\) data points, the clustering algorithm presented in [48] is applied to the attribute values. Each attribute value of \(\lambda\) data points is regarded as a one-dimensional vector and used as an input to the clustering algorithm. As a result, attributes with similar value ranges are grouped together. By using this grouping information, the similarity is calculated for each attribute group, and their average is defined as the similarity  
Figure 4: Visualization of a self-organizing result by CAEA. The dataset consists of six distributions, and each distribution has 15,000 data points [16]. (**a**) Dataset. (**b**) Generated Network. In (**a**), a black dot represents a data point. In (**b**), a circle represents a node, and a black line between nodes represents an edge. If the nodes have the same color, it means that they belong to the same cluster.  
between a data point \(\mathbf{x}_{n}\) and a node \(\mathbf{y}_{k}\). Specifically, by using the grouping information, the data point \(\mathbf{x}_{n}=(x_{n1},x_{n2},\ldots,x_{nd})\) is transformed into \(\mathbf{x}_{n}^{\mathrm{C}}=\left(\mathbf{u}_{n1},\mathbf{u}_{n2},\ldots, \mathbf{u}_{nl}\right)\)\((J\leq d)\) by the clustering algorithm, where \(\mathbf{u}_{j}\) represents the \(j\)th attribute group. Similarly, the node \(\mathbf{y}_{k}\) is transformed into \(\mathbf{y}_{k}^{\mathrm{C}}=\left(\mathbf{v}_{k1},\mathbf{v}_{k2},\ldots, \mathbf{v}_{kj}\right)\)\((J\leq d)\), where \(\mathbf{v}_{j}\) represents the \(j\)th attribute group. The dimensionality of each attribute group is represented as \(d^{\mathrm{C}}=\left\{d_{1},d_{2},\ldots,d_{J}\right\}\), where \(d_{j}\) is the dimensionality of the \(j\)th attribute group (i.e., the number of attributes in the \(j\)th attribute group).  
Here, the similarity between the data point \(\mathbf{x}_{n}^{\mathrm{C}}\) and the node \(\mathbf{y}_{k}\) is defined by the \(\mathrm{CIM}^{\mathrm{C}}\) as follows:  
\[\mathrm{CIM}^{\mathrm{C}}\left(\mathbf{x}_{n}^{\mathrm{C}},\mathbf{y}_{k}^{ \mathrm{C}},\mathbf{\omega}_{k}^{\mathrm{C}}\right) =\frac{1}{J}\sum_{j=1}^{J}\left[\kappa_{\rho_{j}}(0)-\hat{\mathbf{ C}}_{\mathrm{E}}^{\mathrm{C}}(\mathbf{u}_{j},\mathbf{v}_{j})\right]^{\frac{1}{2}}, \tag{25}\]  
\[\hat{\mathbf{C}}_{\mathrm{E}}^{\mathrm{E}}(\mathbf{u}_{j},\mathbf{v}_{j}) =\frac{1}{d_{j}}\sum_{i=1}^{d_{j}}\kappa_{\rho_{j}}(u_{i}-v_{i}), \tag{26}\]  
where \(\mathbf{y}_{k}^{\mathrm{C}}=\left(\mathbf{v}_{1},\mathbf{v}_{2},\ldots, \mathbf{v}_{J}\right)\) is a node \(\mathbf{y}_{k}\), but its attributes are grouped based on the attribute grouping of \(\mathbf{x}_{n}^{\mathrm{C}}\). A bandwidth \(\sigma_{j}\) is defined as follows:  
\[\sigma_{j}=\frac{1}{d_{j}}\sum_{i=1}^{d_{j}}\left[\left(\frac{4}{2+d_{j}} \right)^{\frac{1}{2+d_{j}}}\Gamma_{i}\lambda^{-\frac{1}{4+d_{j}}}\right], \tag{27}\]  
where \(\Gamma_{i}\) denotes a rescale operator which is defined by the standard deviation of the \(i\)th attribute value in the \(j\)th attribute group among the \(\lambda\) data points.  
In this paper, CAEAC with the clustering-based approach is called CAEAC-Clustering (CAEAC-C).  
The differences in attribute processing among the general approach, the individual-based approach (CAEAC-I), and the clustering-based approach (CAEAC-C) are shown in Figure 5. The source codes of CAEAC, CAEAC-I, and CAEAC-C are available on GitHub ([[https://github.com/Masuyama-lab/CAEAC](https://github.com/Masuyama-lab/CAEAC)]([https://github.com/Masuyama-lab/CAEAC](https://github.com/Masuyama-lab/CAEAC)), accessed on 21 September 2023).