## 4 Properties of the Algorithm and Its Extensions  
For the derivation of Algorithm 1, the constraint \(u_{j}\in[0,1]\) was not incorporated in the derivation, and according to Equation (13), it is also possible that negative values or values larger than \(1\) are possible for \(u_{j}\) if \(\alpha\) is not chosen sufficiently large. For illustration purposes, we consider the simple dataset, \(\{1,2,3,\dots,10\}\). Figure 1 shows the objective function (left) and its derivative (right) for this dataset with the parameter settings \(\delta=10\) and \(\alpha=30\). It should be noted that \(\delta\) must be interpreted as the squared noise distance. One can easily identify the two local maxima and the local minimum in between the objective function and the corresponding roots of the derivative. Due to the construction of the dataset, the local minimum of the objective function--the cluster centre--is at 5.5 and the curve is symmetric with respect to this point. In this case, \(\alpha\) is sufficiently large, resulting in only positive membership degrees. For Figure 2, \(\alpha\) was changed to the smaller, value \(\alpha=10\). In this case, \(\alpha\) is too small, and the local minimum of the objective function vanishes while the two local maxima are joined together. The derivative has only one real root. Computing the membership degrees with the cluster centre at 5.5, Equation (13) yields partly negative values. In this case, Algorithm 1 would fail because it relies on three different real roots of the derivative of the objective function. To amend this problem, we modify Algorithm 1 by checking whether the derivative in Equation (15) has three different real roots. If not, \(\alpha\) is increased step by step by a constant factor until it is large enough to produce three different real roots for Equation (15). Algorithm 2 describes this modified version in detail. Figure 1: Objective function according to Equation (4) (**left**) and its derivative according to Equation (15) (**right**) for the dataset \(\{1,2,3,\ldots,10\}\) with \(\delta=10\) and \(\alpha=30\). Figure 2: Objective function according to Equation (4) (**left**) and its derivative according to Equation (15) (**right**) for the dataset \(\{1,2,3,\ldots,10\}\) with \(\delta=10\) and \(\alpha=10\). Instead of using the iterative procedure for the adjustment of the parameter \(\alpha\), one could also use a \"worst case scenario\" for \(\alpha\) to guarantee that membership degrees are between \(0\) and \(1\). For a given value of \(\delta\), Equation (13) satisfies the constraint \(0\leq u_{j}\leq 1\) if and only if  
\[\Big{|}\frac{\delta-d_{j}}{2\alpha}\Big{|}\ \leq\ \frac{1}{2} \tag{16}\]  
holds for all \(j\in\{1,\ldots,n\}\) or, equivalently  
\[\alpha\ \geq\ \max_{j\in\{1,\ldots,n\}}\big{|}\delta-d_{j}\big{|}. \tag{17}\]  
Because \(d_{j}\) is the squared distance to the cluster centre, which must not lie between the smallest and largest data point, Equation (17) is satisfied if  
\[\alpha\ \geq\ \max\Bigl{\{}\delta,\left(\max\{x_{1},\ldots,x_{n}\}-\min\{x_{1}, \ldots,x_{n}\}\right)^{2}-\delta\Bigr{\}}. \tag{18}\]  
Equation (18) is very conservative because it considers the extreme cases where the cluster centre coincides with one of the data points--\(d_{j}=0\)--or it is one of the extreme points--the smallest or largest value--in the dataset--\(d_{j}=(\max\{x_{1},\ldots,x_{n}\}-\min\{x_{1},\ldots,x_{n}\})^{2}\). Choosing \(\alpha\) according to Equation (18) can lead to a very large value for \(\alpha\). It is obvious from Equation (13) that \(\alpha\to\infty\) implies \(u_{j}\to\frac{1}{2}\). If there are outliers and one extreme outlier \(x_{o}\) in the dataset, Equation (18) would still lead to a small membership degree for the extreme outlier because \(d_{o}\approx\left(\max\{x_{1},\ldots,x_{n}\}-\min\{x_{1},\ldots,x_{n}\}\right)^ {2}\) would hold for the extreme outlier.