# High-Dimensional Data Analysis Using Parameter Free Algorithm Data Point Positioning Analysis
## 3 Overview of Clustering Algorithms
### Affinity Propagation
To identify the modes of the estimated density function \(\hat{f}SKE^{p}(x)\), we employ the mean shift process using \(x\) replaced by \(m_{K_{E}^{p}}(x_{j})\). In total, we have three categories of kernels and their corresponding counterparts, namely Gaussian kernels \(G^{p}(x)\) and their shadows \(SG^{p}(x)=G^{p}(x)\), Cauchy kernels \(C^{p}(x)\) and their shadows \(SC^{p}(x)=C^{p-1}(x)\), and generalized Epanechnikov kernels \(K_{E}^{p}(x)\) and their shadows \(SK_{E}^{p}(x)=K_{E}^{p+1}(x)\). The corresponding density estimates can be found using the mean shift process with any of these three kernel categories. The performance of the mean shift procedure for kernel density estimation is significantly affected by the normalization and stabilization parameters, denoted by \(\beta\) and \(p\), respectively. We will explore this topic in the next section. ### K-Means

K-means clustering is a popular unsupervised machine learning algorithm used for clustering data points into groups based on their similarity [50; 51]. The algorithm aims to minimize the sum of squared distances between the data points and their assigned cluster centers. Here are the main steps of the algorithm:

1. Choose the number of clusters \(K\) that you want to identify and randomly initialize \(K\) cluster centers. 2. Assign each data point to its nearest cluster center. This can be done using Euclidean distance or other distance measures. 3. Calculate the mean of the data points in each cluster to obtain the new cluster centers. 4. Repeat steps 2 and 3 until the cluster centers converge, i.e., when the assignments of the data points to clusters no longer change or change minimally. Here are the equations for the steps:

1. Assign each data point \(x_{i}\) to its nearest cluster center \(c_{j}\): \[\underset{j}{argmin}||x_{i}-c_{j}||^{2}\] (17) where \(|.|\) denotes the Euclidean distance. 2. Calculate the mean of the data points in each cluster to obtain the new cluster centers: \[c_{j}=\frac{1}{\left|C_{j}\right|}\sum_{x_{i}\in C_{j}}x_{i}\] (18) where \(C_{j}\) is the set of data points assigned to cluster \(j\). 3. Repeat steps 2 and 3 until convergence. The algorithm may converge to a local optimum rather than a global one, so it is often run multiple times with different random initializations to find the best result. K-means clustering can be extended to handle different data types and distances, as well as more complex clustering problems, such as non-convex clusters or variable cluster sizes. However, the algorithm is sensitive to the choice of \(K\) and the initial cluster centers, and it may not work well with noisy or high-dimensional data. ## 4 Data Point Positioning Analysis Algorithm

The DPPA algorithm is required because clustering is an essential data analysis technique that is used in various fields, such as data mining, machine learning, and pattern recognition. Clustering helps to identify patterns, group similar data points, and extract useful information from large datasets. However, traditional clustering algorithms, like k-means, DBSCAN, Mean Shift, and affinity propagation, have some limitations that DPPA aims to address. The k-means algorithm requires the number of clusters to be specified beforehand, which may not be known in advance. The DBSCAN algorithm assumes that the clusters have similar densities, which may not be valid in all cases. The Mean Shift algorithm is sensitive to the choice of the bandwidth parameter and may lead to overfitting or underfitting. The affinity propagation algorithm requires the selection of a damping factor, which can significantly affect the clustering results. In contrast, the DPPA algorithm does not require any prior knowledge of the number of clusters or the density of the data. The algorithm works by calculating the distances between each pair of adjacent data points, and then using this information to identify neighborhoods of data points.