# Fuzzy Clustering Approaches Based on Numerical Optimizations of Modified Objective Functions
## 4 Experimental Studies  
A series of experimental procedures are carried out to analyze the modified fuzzy clustering algorithms. These algorithms were applied on several slightly distorted versions of benchmark datasets of the UCI Machine Learning Repository (where some artificially noise data are added) in order for their performance to be evaluated. The employed datasets in these studies are Banknotes, Liver, Yeast, Dermatology, Vehicle Silhouettes, and LANDSAT Satellite with an additional quantity of 2-3% artificial noise points, which are randomly positioned at a distance from the cluster centers being 5-15% larger than the distance of the farthest genuine point from the respective cluster center [31]. In the original datasets, these artificially added noise points are labeled according to the cluster the random generation relied on, which is expected to be the closest cluster, too, though not mathematically guaranteed. As these points typically lie near cluster boundaries, they are expected to exhibit partial memberships across clusters. The Banknotes dataset contains four numerical features derived from wavelet-transformed images of a group of Swiss banknotes, along with the class labels indicating for each banknote whether it is genuine or counterfeit. The Liver dataset (Indian Liver Patient Dataset) contains two demographic features (age and sex) and eight biochemical test results related to liver health, along with the class label indicating whether the patient is healthy or has a liver disorder. The Yeast dataset contains eight numerical features representing various physicochemical properties of proteins, such as sequence length and localization signals, as well as the class, which is the sub-cellular localization of proteins across ten possible classes. The Dermatology dataset contains patient data about some skin diseases manifested with redness and scaling. The dataset comprises 34 clinical attributes, including histological, pathological, and subjective features, used to classify six types of skin diseases. The Vehicle Silhouettes dataset contains 18 numerical features extracted from the silhouettes of vehicle images. It comprises instances of four classes (Double Decker Bus, Opel Manta 400, Saab 9000, and Chevrolet Van). The LANDSAT Satellite dataset contains multi-spectral values of pixels from satellite images. It comprises 36 numerical features derived from 3 \(\times\) 3 pixel neighborhoods in satellite images and the class representing the land cover type (e.g., red soil, grey soil, etc.). The details of the original datasets, including the number of features, number of instances, and number of clusters, are summarized in Table 1.  
Although the categorization into classes for the instances of these datasets is well known, this information will not be provided to the clustering algorithms, so they will operate in an unsupervised way. The process starts with the generation of the clustering results, which are then compared with the known natural classes of the dataset to evaluate the performance of the algorithms. The number of clusters hyper-parameter is set to the respective natural number of clusters of each dataset. The regularization hyper-parameter \(\lambda\) is tuned via grid search among the candidate values [0.1, 0.25, 0.5, 1, 2, 4]. The learning rate is set to 0.01, but it is progressively decreased by 10% in case of divergence. Beyond accuracy assessment, the algorithms will also be examined for their sensitivity to noise, allowing for a more thorough evaluation of their performance. Two criteria are employed for the performance evaluation, which are the fuzzy adjusted Rand index (Fuzzy ARI) and fuzzy normalized mutual information (Fuzzy NMI). The fuzzy adjusted Rand index is a generalization of the Rand index, adjusting this metric to operate with soft (partial) memberships of the instances in the clusters.