## 3 Mixed-Attribute Fusion-Based Naive Bayesian Classifier  
As stated in the introduction, continuous attribute discretization and the attribute independence assumption limit the generalization performance of the NBC for the continuous attribute classification problem. To cope with these issues, recent studies either introduced complex structures to represent attribute dependencies or applied discretization techniques to transform mixed-attribute values into discrete attribute values [18; 19; 20]. This section presents a novel solution to the NBC-based mixed-attribute classification problem by considering attribute dependence and mixed-attribute transformation simultaneously. A mixed-attribute fusion strategy is designed to construct an NBC that can be trained based on the transformed continuous attributes with the minimum attribute dependence. For discrete attributes of an original mixed-attribute (OMA) data set \(\mathbb{D}\), the one-hot encoding technique [21] is applied to transform them into 0-1 numerical attributes; i.e., \(a_{ni}^{(m)}\), \(i=1,2,\cdots,\mathcal{D}_{1}\) is encoded as \(\left(e_{n1}^{(mi)},\cdots,e_{n\mathcal{C}_{i}}^{(mi)}\right)\) for the discrete attribute value of the \(n\)-th sample \(\mathbf{x}_{n}^{(mi)}\), \(n=1,2,\cdots,\mathcal{N}_{m}\), where the following is the case:  
\[e_{nk}^{(m)}=\left\{ \begin{array}{ll}1,&\text{if }a_{pi}^{(m)}=A_{k}^{(i)}\\ 0,&\text{if }a_{ni}^{(m)}\
eq A_{k}^{(i)}\end{array} \right. \tag{9}\]  
and \(\left\{A_{1}^{(i)},A_{2}^{(i)},\cdots,A_{\mathcal{K}_{i}}^{(i)}\right\}\) is \(\mathcal{K}_{i}\) categorical values corresponding to the \(i\)-th discrete attribute. Then, the one-hot encoded form of the original sample \(\mathbf{x}_{n}^{(m)}\) can be expressed as follows. \[\overline{\mathbf{x}}_{n}^{(m)}=\left(e_{n1}^{(m1)},\cdots,e_{n\mathcal{K}_{1} }^{(m1)},\cdots,e_{n1}^{(m\mathcal{D}_{1})},\cdots,e_{n\mathcal{K}_{D_{1}}}^{ (m\mathcal{D}_{1})},b_{n1}^{(m)},\cdots,b_{n\mathcal{D}_{2}}^{(m)}\right)\quad. \tag{10}\]In fact, the one-hot encoding technique causes attribute redundancy when extending a discrete attribute into multiple 0-1 numerical attributes. For example, the essence of the following transformation  
\[\left[ \begin{array}{c}A_{1}\\ A_{1}\\ A_{2}\end{array} \right]\rightarrow\left[ \begin{array}{cc}1&0\\ 1&0\\ 0&1\end{array} \right] \tag{11}\]  
is to represent a discrete attribute using two \"repetitive\" numerical attributes. The main role of one-hot encoding is to transform the discrete attribute values into numbers. The attribute independence assumption is not alleviated when constructing an NBC based on a one-hot encoded attribute (OHEA) data set. \[\overline{\mathrm{X}}=\left[ \begin{array}{cccccccc}\pi^{(1)}_{1}&\cdots& \pi^{(1)}_{\mathcal{N}_{1}}&\pi^{(2)}_{1}&\cdots&\pi^{(2)}_{\mathcal{N}_{2}}& \cdots&\pi^{(\mathcal{M})}_{\mathcal{N}_{\mathcal{M}}}&\cdots&\pi^{(\mathcal{ M})}_{\mathcal{N}_{\mathcal{M}}}\end{array} \right]^{\mathrm{T}}\enspace. \tag{12}\]  
Thus, an autoencoder neural network (ANN) is meticulously designed to solve the problems of attribute redundancy and attribute dependence mentioned above. An ANN is a special single hidden-layer feed-forward neural network that has the same input matrix and output matrix. The main purpose of training an ANN is to find the optimal input layer weight matrix \(\alpha=(\alpha_{rl})_{\mathcal{R}\times\mathcal{L}}=\left[ \begin{array}{cccc} \alpha_{1}&a_{2}&\cdots&\alpha_{\mathcal{L}}\end{array} \right]\) and output layer matrix \(\beta=(\beta_{lr})_{\mathcal{L}\times\mathcal{R}}\) so that the practical output matrix \(\overline{\mathrm{X}}^{\prime}\) approximates the true output matrix \(\overline{\mathrm{X}}\) as closely as possible, where \(\mathcal{R}=\sum\limits_{i=1}^{\mathcal{D}_{1}}\mathcal{K}_{i}+\mathcal{D}_{2}\) is the number of input layer nodes of the ANN and \(\mathcal{L}\) is the number of hidden layer nodes of the ANN.