# An Optimal and Stable Algorithm for Clustering Numerical Data
## 3 Proposed Clustering Algorithm with a Constant Seeding Selection
### Proposed Algorithm  
We call the new algorithm with zero-point seeding the Zk-AMH algorithm. The main difference between the Zk-AMH algorithm and the k-AMH numeric algorithm is the replacement of random seeding with zero-point multidimensional seeding. In addition, from an implementation perspective, each k-object will be compared to \(n\) objects rather than \((n-k)\) objects, as imposed by the original k-AMH numeric algorithm. This difference exists because the seeding objects are not taken from the \(n\) objects. Thus, the Zk-AMH algorithm employs the procedure and steps as described for the k-AMH numeric algorithm above exactly, except in terms of the seeding selection (Step 2). From the perspective of implementation, the zero-seeding assigns zero values of multidimensional spaces (attributes). Therefore, let \(O=O_{1},O_{2},\ldots,O_{k}\) be a set of origins and \(\hat{Z}_{l,j}\in O_{m}\), where \(1\leq l\leq k\), and \(1\leq j\leq m\) is the zero-seeding assigned to each \(\hat{Z}\) with \(O\). For example, if \(k=3\) and \(m=5\), the initial seeding may resemble \(\hat{Z}_{1}=\{0,0,0,0,0\}\), \(\hat{Z}_{2}=\{0,0,0,0,0,0\}\), and \(\hat{Z}_{3}=\{0,0,0,0,0,0\}\). Therefore, the initial objective function as Equation (8) is first obtained by calculating the distance \(X\) and \(\hat{Z}\) as Equation (1), which is based on the zero-seeding. This initial objective function must be maximal according to our theorem below. **Theorem 1**. : _Let \(\hat{Z}_{l}\in O\) be the zero-seeding for \(1\leq l\leq k\). \(\hat{Z}_{l}\) is replaced with \(X_{i}\) for \(1\leq i\leq n\) to serve as the medoid if and only if_  
\[F(W,D)^{i}<F(W,D)^{\beta};\forall i,1\leq i\leq n\text{ where }\beta\text{ is the zero- seeding}\]  
Proof. : Let \(X=\{X_{1},X_{2},\ldots,X_{n}\}\) be a set of \(n\) objects, \(O=\{O_{1},O_{2},\ldots,O_{k}\}\)be a set of zero-point seeding, and \(\hat{Z}=\{\hat{Z}_{1},\hat{Z}_{2},\ldots,\hat{Z}_{k}\}\) be a set of medoids (centers of clusters) for \(k\) clusters. Let \(d_{euc}(X_{i},O_{l})\) be the distance between object \(X_{i}\) and zero-seeding \(O_{l}\), and let \(d_{euc}(X_{i},\hat{Z}_{l})\) be the distance between object \(X_{i}\) and medoid \(\hat{Z}_{l}\). Thus, based on Equation (1), \(d_{euc}(X_{i},O_{l})\) is given by  
Figure 2: Proposed seeding approach with zero multidimensional spaces seeding regarding k-AMH numeric procedure. (**a**) Three clusters (\(C_{1},C_{2}\), and \(C_{3}\)) initially refer to the same point: the origin. (**b**) Two clusters (\(C_{2}\) and \(C_{3}\)) initially refer to the origin, whereas \(C_{1}\) has obtained the final object using k-AMH numeric procedure. (**c**) Remaining cluster \(C_{3}\) initially refers to the origin, whereas \(C_{2}\) has obtained the final object. (**d**) Final k-combination objects. **Note:** This concept will be implemented in the k-AMH framework where the process of finding the final medoids (Cluster center) involves checking for each object \(X_{i}\) and replacing it, if the cost function is minimized. For the case above, the final cluster centers for \(C_{1}\), will be obtained first, followed by \(C_{2}\), and finally \(C_{3}\). The final cluster centers denoted as \(C_{1},C_{2}\), and \(C_{3}\) are actually, and eventually, represented by the arbitrary objects X owing to the k-AMH procedure. \[d_{euc}(X,O) =\sqrt{\sum_{j=1}^{m}|x_{j}-o_{j}|^{2}},\] \[=\sqrt{\sum_{j=1}^{m}|x|^{2}}\]  
and \(d_{euc}(X_{i},\hat{Z}_{l})\) is given by  
\[d_{euc}(X,\hat{Z})=\sqrt{\sum_{j=1}^{m}|x_{j}-z_{j}|^{2}}\]  
Therefore, \(d_{euc}(X,O)\) must be greater than \(d_{euc}(X,\hat{Z})\). We write \(F(W,D)^{i}\), where \(\hat{Z}_{l}\in X\), and \(F(W,D)^{\beta}\), where \(\hat{Z}=O\), as  
\[F(W,D)=\sum_{l=1}^{k}\sum_{i=1}^{n}w_{li}^{\psi}d_{euc}(X_{i},\hat{Z}_{l})\]  
As \(d_{euc}(X,O)\) is greater than \(d_{euc}(X,\hat{Z})\) and \(w_{li}^{\psi}\)and \(d_{euc}(X_{i},\hat{Z}_{l})\) are non-negative, the product \((w_{li}^{\psi}d_{euc}(X_{i},\hat{Z}_{l}))^{i};\forall i,1\leq i\) must be less than \((w_{li}^{\psi}d_{euc}(X_{i},\hat{Z}_{l}))^{\beta}\).