# Toward Automatic Cardiomyocyte Clustering and Counting through Hesitant Fuzzy Sets
## 3 Methods and Algorithms
### Hesitant Fuzzy Sets for Clustering  
We will focus on the relevant contents of hesitation fuzzy sets (HFSs) [19] and correlation coefficients of HFSs [44]. **Definition 1**. : _[_19_]_ _let \(X\) be a reference set, a hesitant fuzzy set (HFS) \(A\) on \(X\) is defined in terms of a function \(h_{A}(x)\) when applied to \(X\) returns a finite subset of [0,1], i.e.,_  
\[A=\{<x,h_{A}(x)>\big{|}x\in X\} \tag{1}\]  
where \(h_{A}(x)\) is a set of some different values in [0,1], representing the possible membership degrees of the element \(x\in X\) to \(A\). For convenience, we call \(h_{A}(x)\) a hesitant fuzzy element (HFE). Let \(X=\{x_{1},x_{2},\cdots,x_{n}\}\) be a discrete universe of discourse, \(A\) and \(B\) be two HFSs on \(X\) denoted as \(A=\{<x_{i},h_{A}(x_{i})>\big{|}x_{i}\in X,i=1,2,\cdots,n\}\) and \(B=\{<x_{i},h_{B}(x_{i})>\big{|}x_{i}\in X,i=1,2,\cdots,n\}\), respectively. It is noted that the number of values in different HFEs may be different. To compute the correlation coefficients between two HFSs, let \(l_{i}=\max\{l\ (h_{A}(x_{i})),l\ (h_{B}(x_{i}))\}\) for each \(x_{i}\) in \(X\), where \(1\ (h_{A}(x_{i}))\) and \(l\ (h_{B}(x_{i}))\) represent the number of values in \(h_{A}(x_{i})\) and \(h_{B}(x_{i})\), respectively. When \(1\ (h_{A}(x_{i}))\
eq 1\ (h_{B}(x_{i}))\), one can make them to have the same number of elements by adding some elements to the HFE, which has less number of elements. In terms of the pessimistic principle, the smallest element will be added, while in the opposite case, the optimistic principle may be adopted. In the present work, we use the former case. Especially, if \(1\ (h_{A}(x_{i}))<1\ (h_{B}(x_{i}))\), then \(h_{A}(x_{i})\) should be extended by adding the minimum value in it until it has the same length as \(h_{B}(x_{i})\). This idea has been successfully applied to distance and similarity measures for HFSs. According to [19], a hesitant fuzzy set (HFS) \(A\) on a reference set \(X\) is defined in terms of a function \(h_{A}(x)\), which takes \(X\) as input and returns a finite subset of [0,1]. The correlation [44] between two HFS's \(A\) and \(B\) are defined as:  
\[C_{HFS}(A,B)=\sum_{i=1}^{n}\ (\frac{1}{l_{i}}\sum_{j=1}^{l_{i}}h_{A_{\omega(j)}}(x_ {i})\cdot h_{B_{\omega(j)}}(x_{i})) \tag{2}\]  
where \(A_{\delta(i)}\) indicates the \(j\)-th largest value in \(A\). In practical applications, the elements \(X_{i}\)'s \((i=1,2,\cdots,n\ \text{in}\ X)\) all have different weights. Let \(w=(w_{1},w_{2},\cdots,w_{n})^{T}\) be the weight vector of \(x_{i}\) with \(w_{i}\geq 0\), and \(\sum\limits_{i=1}^{n}w_{i}=1\). In our proposed algorithm, we normalize the correlation coefficient as follows:  
\[\rho_{HFS}(\mathrm{A},\mathrm{B})= \tag{3}\]  
**Definition 2**. : _[_44_]_ _Let \(A_{j}(j=1,2,\cdots,m)\) be \(m\) HFSs, and \(C=\left(\rho_{ij}\right)_{m\times m}\) be a correlation matrix, where \(\rho_{ij}=\rho\left(A_{i},A_{j}\right)\) denotes the correlation coefficient of two HFSs \(A_{i}\) and \(A_{j}\) and satisfies: 1. \(0\leq\rho_{ij}\leq 1\), \(i,j=1,2,\cdots,m\); 2. \(\rho_{ii}=1\), \(i=1,2,\cdots,m\); 3. \(\rho_{ij}=\rho_{ji}\), \(i,j=1,2,\cdots,m\)._  
We will introduce the related properties of the correlation coefficient of the hesitant fuzzy set in the literature [45] to explain, as follows. **Definition 3**. : _Let \(C=\left(\rho_{ij}\right)_{m\times m}\) be a correlation matrix, if \(C^{2}=C^{\circ}C=\left(\overline{\rho}_{ij}\right)_{m\times m}\), then \(C^{2}\) is called a composition matrix of \(C\), where_  
\[\overline{\rho}_{ij}=\max_{k}\left\{\min\left(\rho_{ik},\rho_{kj}\right)\right\},\ i,j=1,2,\cdots,m. \tag{4}\]  
**Theorem 1**. : _Let \(C=\left(\rho_{ij}\right)_{m\times m}\) be a correlation matrix. Then, the composition matrix \(C^{2}=C^{\circ}C=\left(\overline{\rho}_{ij}\right)_{m\times m}\) is also a correlation matrix._  
**Theorem 2**. : _Let \(C\) be a correlation matrix.