# High-Dimensional Data Analysis Using Parameter Free Algorithm Data Point Positioning Analysis
## 2 Related Work  
Numerous clustering algorithms have been introduced in past years, as evidenced by the literature [20,21]. For instance, in Reference [22], a method utilizing association rules is proposed to cluster customer transactions within a market database. The study conducted in [23] focused on exploring algorithms for clustering categorical databases using non-linear dynamical systems. According to [PERSON] [24], the process of clustering splits data patterns into subsets in such a manner that patterns with comparable characteristics are grouped in the same cluster. Therefore, the patterns are organized into a well-formed assessment, which identifies the population from which the sample was selected. For clustering, the data, K-means algorithm was implemented in [25,26]. [PERSON] et al. [27] presented a global k-means algorithm, an incremental concept to clustering that dynamically adds one cluster center at a time using a deterministic global search technique, consisting of the sizes of the dataset executions of the k-means algorithm from suitable initial positions. The authors of [28] presented an affinity propagation clustering technique for pattern recognition. In their approach, each data point exchanges signals (called responsibility and availability) in an iterative manner to identify acceptable examples. To ascertain the affinities of the exemplars in the network, certain preferences for criteria must be established. The success of this algorithm relies on well-defined criteria to attain its ideal aim, even if there are no beginning parameters that must be set. Its inability to handle groups of arbitrary forms is another issue. The DNo, DNg, and DNs indices have been recognized as simple internal validity measures for clustering [29, 30, 31]. These indices assess the relationship between cluster size and inter-cluster distance. Specifically, they calculate the ratio of the minimum distance between two clusters to the size of the largest cluster, aiming to maximize the index values. The DB index, proposed by [PERSON] and [PERSON] [32], quantifies the average similarity between each cluster and its most similar one. It aims to maximize the distances between clusters while minimizing the distance between the cluster centroid and other data objects. The silhouette value, introduced by [PERSON] [33], measures the similarity of an object to its own cluster (cohesion) compared to other clusters (separation). Ranging from \(-1\) to +1, a higher silhouette value indicates a better fit to the object's own cluster and a poorer fit to neighboring clusters. Positive and negative large silhouette widths (SWs) signify proper and improper clustering, respectively. Objects with a SW value close to zero indicate a lack of clear discrimination between clusters. Another cluster validity measure is the gap statistic, which is based on a statistical hypothesis test [34]. It assesses the change in within-cluster dispersion relative to the expected change under an appropriate reference null distribution. In Reference [35], a comparative study was conducted to evaluate five different clustering algorithms using gene expression time series datasets of the Saccharomyces cerevisiae yeast. The experiments employed a k-fold cross-validation procedure to compare the performance of the algorithms. The results showed that k-means, dynamical clustering, and SOM consistently achieved high accuracy across all experiments. In Reference [36], the performances of k-means, single linkage, and simulated annealing (SA) were assessed using various partitions obtained through validation indexes. They introduced a novel validation index called the I index, which measured separation based on the maximum distance between clusters and compactness based on the sum of distances between objects and their respective centroids. The study concluded that the I index demonstrated the highest reliability among the considered indices, attaining its maximum value when the number of clusters was appropriately chosen.