## 2 Methods  
Nonnegative matrix factorization (NMF) has now become a popular technique for the analysis of high-dimensional data, as it extracts sparse and significant characteristics with its weights from a set of nonnegative data vectors [28]. NMF is widely used in various fields such as artificial intelligence (AI), bioinformatics, computational biology, and demography [29; 30; 31; 32]. NMF is a kind of technique of an unsupervised cluster analysis, which provides a simplified decomposition of the data matrix by the nonnegativity constraint on its outputs [33]. NMF can give the features of data that are easier to integrate then with other analytical methods. The primary approach of NMF is to estimate matrices \(W\) and \(H\) as a local minimum of the optimization problem [34], and it was popularized by [PERSON] and [PERSON] in 1999 [35]. The NMF general equation is as follows:  
\[X\approx W\times\ H \tag{1}\]  
\(X\) is set up for \(n\) data points with \(m\) dimensions \(m\times n\), i.e., \(X\in\mathbb{R}^{m\ n}\) (linear combination of \(W\) and \(H\)). By the estimation of NMF, the \(m\) original dimensions are reduced to \(k\), which is the low rank approximation of \(X\) (\(k\leq\) min (\(m\), \(n\))) and creates \(W\in\mathbb{R}^{m\ k}\) and \(H\in\mathbb{R}^{k\ n}\). Each row of \(W\) gives weights of the basis that commonly appears in some rows of \(X\), while each column of \(H\) gives the basis made up of the set of a column's attributes in \(X\). In other words, each row vector in \(X\) is approximated by a linear combination of the \(k\)th column of \(H\), weighted by the corresponding row vector of \(W\). A basic NMF is described in Equation (1), where the NMF solutions are not normalized. Given a nonnegative matrix \(X\) and a desired factorization rank \(k\), the objective of the NMF algorithm is to find an approximation of \(X\)-\(WH\). After initializing nonnegative matrices \(W\) and \(H\) randomly, they are iteratively updated to minimize the loss function \(D(X\), \(WH)\), shown in Equation (2), specified as a generalized Frobenius distance in Equation (3). \[\underset{W,H\geq 0}{\text{Min}}[D(X,WH)+R(W,H)] \tag{2}\]  
where \(D\) is the function in approximation based on the Frobenius distance, a loss function; and \(R\) is an optional regularization function, which is smoothness or sparsity of the matrices \(W\) and \(H\). \[D(X,WH)=\frac{1}{2}\sum_{i=1}^{m}\sum_{j=1}^{n}\left[X_{ij}-(WH)_{ij}\right]^{2} \tag{3}\]  
Equation (2) can be optimized by the alternated estimation of \(W\) and \(H\) as follows: In each iteration, \(W\) and \(H\) are updated by using the coupled multiplicative update rules as shown in Equations (4) and (5):  
\[W_{ik}\gets W_{ik}\frac{(\sum_{l}\frac{H_{lj}X_{il}}{(WH)_{ij}})}{\sum_{l }H_{kl}} \tag{4}\]  
\[H_{kj}\gets H_{kj}\frac{(\sum_{l}\frac{W_{lk}X_{lj}}{(WH)_{ij}})}{\sum_{l }W_{lk}} \tag{5}\]  
In practice, the optimization of Equations (4) and (5) is calculated from an arbitrary matrix with randomized elements, so then the estimated \(W\) and \(H\) are not unique but robust under an appropriate \(k\). If \(k\) is not adequate, however, \(W\) and \(H\) become unstable, which makes the interpretation of the estimated basis difficult. To alleviate this problem, some existing methods enforce a simplicity of the basis by adding regularization terms to the NMF objective function [28]. Furthermore, in this study, the \(W\) matrix is standardized as \(W^{\prime}\) where rows of \(W\) are simply summed up, and each component (\(w_{mk}\)) is divided by its row sum. For the \(H\) matrix, standardization is performed by summing up the entries in each column, and each component (\(l_{kn}\)) is divided by its column sum to form \(H^{\prime}\) (influence factors of the corresponding samples). The standardization by Equations (6) and (7) makes it easier to compare the cluster membership and the interpretation of the basis.