# High-Dimensional Data Analysis Using Parameter Free Algorithm Data Point Positioning Analysis
## 3 Overview of Clustering Algorithms
### Affinity Propagation
In cases where the shadow for the kernel \(K\) is unknown or does not exist,the mean shift method can still be useful for estimating alternative modes, such as cluster centers, for a dataset with an unknown density function. #### 3.3.2 Some Special Kernels and Their Shadows

In this section, the main focus will be on investigating special kernels that possess shadows. More specifically, we delve into Gaussian kernels \(G^{p}(x)\), which are the most frequently utilized kernels that possess their own shadows [46]. \[G^{p}(x)=[g(\left\|x-x_{j}\right\|^{2})]^{p}=[exp\Big{\{}-\left\|x-x_{j} \right\|^{2}/\beta\Big{\}}]^{p} \tag{8}\]

with their shadows \(SG^{p}\) defined as

\[a(i,k)=\min 0,r(k,k)+\sum_{i^{\prime}\
eq i,i^{\prime}\
eq k}\max 0,r(i^{\prime},k)\]0

The process of mean shift, where \(x\) is reassigned as \(m_{G^{p}}(x_{j})\), is utilized to identify the modes of the density estimate \(\hat{f}SG^{p}(x)\). [PERSON] conducted a study on the behavior of mean shift in cluster analysis by employing a Gaussian kernel. The maximum entropy clustering algorithm is a specialized weight function-based Gaussian kernel mean shift. [PERSON] and [PERSON] employed a Gaussian kernel-induced distance measure to perform robust image segmentation using spatially constrained fuzzy c-means (FCM). [PERSON] and [PERSON] utilized \(\hat{f}SG^{p}(x)\) as an objective function for total similarity and derived a similarity-based clustering method (SCM) that could autonomously organize the number and size of clusters based on the data structure [46]. \[a(i,k)=\min 0,r(k,k)+\sum_{i^{\prime}\
eq i,i^{\prime}\
eq k}\max 0,r(i^{\prime},k)\]1

The Cauchy kernels, which are obtained from the Cauchy probability density function \(f(x)=(1/\pi)(1+x^{2})^{-1}\), where \(-\infty<x<\infty\), are noteworthy kernels. Their shadows are defined in the following manner [46]:

\[a(i,k)=\min 0,r(k,k)+\sum_{i^{\prime}\
eq i,i^{\prime}\
eq k}\max 0,r(i^{\prime},k)\]2

The process of mean shift using \(x\leftarrow m_{G^{p}}(x_{j})\) is used to locate the modes of the density estimate \(\hat{f}_{SCF}(x)\). The use of Cauchy kernels is not as prevalent as other types of kernels. To address the limitations of FCM in noisy environments, [PERSON] and [PERSON] proposed the possibilistic c-means (PCM) clustering algorithm by relaxing the constraint of the fuzzy c-partition's summation to 1. The possibilistic membership functions were subsequently utilized as Cauchy kernels. This is the only instance of Cauchy kernels being utilized in clustering that we are aware of. The flat kernel is the most basic kernel and is defined as follows [46]:

\[F(x)= \begin{cases}1&if\left\|x-x_{j}\right\|^{2}\leqslant 1,\\ 0&if\left\|x-x_{j}\right\|^{2}>1\end{cases}  \tag{12}\]

with the Epanechnikov kernel \(E(x)\), as its shadows

\[E(x)= \begin{cases}1-\left\|x-x_{j}\right\|^{2}&if\left\|x-x_{j} \right\|^{2}\leqslant 1,\\ 0&if\left\|x-x_{j}\right\|^{2}>1\end{cases}  \tag{13}\]

In addition, the Epanechnikov kernel's shadow is the biweight kernel \(B(x)\). \[B(x)= \begin{cases}(1-\left\|x-x_{j}\right\|^{2})^{2}&if\left\|x-x_{j}\right\| ^{2}\leqslant 1,\\ 0&if\left\|x-x_{j}\right\|^{2}>1\end{cases}  \tag{14}\]

To provide a more general explanation, we generalize the Epanechnikov kernel \(E(x)\) into the form of generalized Epanechnikov kernels \(K_{E}^{p}(x)\) with the parameter \(p\) defined as follows [46]:\[K_{E}^{p}(x)=[k_{E}(\left\|x-x_{j}\right\|^{2})]^{p}= \begin{cases}(1-\left\|x-x_{j} \right\|^{2}/\beta)^{p}&if\left\|x-x_{j}\right\|^{2}\leqslant\beta,\\ 0&if\left\|x-x_{j}\right\|^{2}>\beta\end{cases}  \tag{15}\]

As a result, the generalized Epanechnikov kernels \(K_{E}^{p}(x)\) have corresponding shadows \(SK_{E}^{p}(x)\), defined as [46]:

\[SK_{E}^{p}(x)=K_{E}^{p+1}(x),\ p>0 \tag{16}\]

When \(\beta=1\), functions \(K_{E}^{0}(x)\), \(K_{E}^{1}(x)\), and \(K_{E}^{2}(x)\) become \(F(x)\), \(E(x)\), and \(B(x)\), respectively.