## 3 Proposed Algorithm
### CIM-Based ART with Edge and Age
#### 3.3.1 Initialization Process for Nodes and a Bandwidth of a Kernel Function in the CIM  
In the case that CAEA does not have any nodes, i.e., a set of prototype node \(\mathcal{Y}=\varnothing\), the first \((\lambda/2)\)th training data points \(\mathcal{X}_{\text{init}}=\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots\mathbf{x}_{ \lambda/2}\}\) directly become prototype nodes, i.e., \(\mathcal{Y}_{\text{init}}=\{\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{ \lambda/2}\}\), where \(\mathbf{y}_{k}=\mathbf{x}_{k}\) (\(k=1,2,\ldots,\lambda/2\)) and \(\lambda\in\mathbb{Z}^{+}\) is a predefined parameter of CAEA. This parameter is also used for a node deletion process that is explained in Section 3.3.4. In an ART-based clustering algorithm, a vigilance parameter (i.e., a similarity threshold) plays an important role in a self-organizing process. Typically, the similarity threshold is data-dependent and specified by hand. On the other hand, CAEA uses the minimum pairwise CIM value between each pair of nodes in \(\mathcal{Y}_{\text{init}}=\{\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{ \lambda/2}\}\), and the average of pairwise CIM values is used as the similarity threshold \(V_{\text{threshold}}\), i.e.,  
\[V_{\text{threshold}}=\frac{1}{\lambda/2}\sum_{i=1}^{\lambda/2}\min_{j\
eq i} \left[\text{CIM}\big{(}\mathbf{y}_{i},\mathbf{y}_{j},\sigma\big{)}\right], \tag{5}\]  
where \(\sigma\) is a kernel bandwidth in the CIM. In general, the bandwidth of a kernel function can be estimated from \(\lambda\) instances belonging to a certain distribution [46], which is defined as follows:  
\[\mathbf{\Sigma}=U(F_{v})\mathbf{\Gamma}\lambda^{-\frac{1}{2\
u+d}}, \tag{6}\]  
\[U(F_{v})=\left(\frac{\pi d^{d/2}2^{d+\
u-1}(\
u!)^{2}R(F)^{d}}{\
u\kappa_{v}^{ 2}(F)[(2\
u)!!+(d-1)(\
u!!)^{2}]}\right)^{\frac{1}{2\
u+d}}, \tag{7}\]  
where \(\mathbf{\Gamma}\) denotes a rescale operator (\(d\)-dimensional vector), which is defined by a standard deviation of each of the \(d\) attributes among \(\lambda\) instances, \(\
u\) is the order of a kernel, the single factorial of \(\
u\) is calculated by the product of integer numbers from \(1\) to \(\
u\), the double  
Figure 3: Surface plot of CIM. factorial notation is defined as \((2\
u)! !=(2\
u-1)\cdot 5\cdot 3\cdot 1\) (commonly known as the odd factorial), \(R(F)\) is a roughness function, and \(\kappa_{\
u}(F)\) is the moment of a kernel. The details of the derivation of (6) and (7) can be found in [46]. In this paper, we use the Gaussian kernel for CIM. Therefore, \(\
u=2\), \(R(F)=(2\sqrt{\pi})^{-1}\), and \(\kappa_{\
u}^{2}(F)=1\). Then, (7) is rewritten as follows:  
\[\mathbf{H}=\left(\frac{4}{2+d}\right)^{\frac{1}{2+d}}\mathbf{\Gamma}\lambda^{- \frac{1}{4+d}}. \tag{8}\]  
Equation (8) is known as Silverman's rule [47]. Here, \(\mathbf{H}\) contains the bandwidth of a kernel function in CIM. In (8), \(\mathbf{\Sigma}\) contains the bandwidth of each attribute (i.e., \(\mathbf{\Sigma}\) is a \(d\)-dimensional vector). In this paper, the median of the \(d\) elements of \(\mathbf{\Sigma}\) is selected as a representative bandwidth of the Gaussian kernel in the CIM, i.e.,  
\[\sigma=\text{median}(\mathbf{\Sigma}). \tag{9}\]  
In CAEA, the initial prototype nodes \(\mathcal{Y}_{\text{init}}=\{\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_ {\lambda/2}\}\) have a common bandwidth of the Gaussian kernel in the CIM, i.e., \(\mathcal{S}_{\text{init}}=\{\sigma_{1},\sigma_{2},\ldots,\sigma_{\lambda/2}\}\) where \(\sigma_{1}=\sigma_{2}=\cdots=\sigma_{\lambda/2}\). When a new node \(\mathbf{y}_{K+1}\) is generated from \(\mathbf{x}_{n}\), a bandwidth \(\sigma_{K+1}\) is estimated from the past \(\lambda/2\) data points, i.e., \(\{\mathbf{x}_{n-\lambda/2},\ldots,\mathbf{x}_{n-2},\mathbf{x}_{n-1}\}\) by using (6) and (7). As a result, each new node has a different bandwidth \(\sigma\) depending on the distribution of training data points. In addition, a set of counters \(\mathcal{M}=\{M_{1},M_{2},\ldots,M_{\lambda/2}\}\), where \(M_{1}=M_{2}=\cdots=M_{\lambda/2}=1\), is defined.