Multivariate bounded support Kotz mixture model with semi-supervised projected model-based clustering

Data clustering is a crucial technique in data analysis, aimed at identifying and grouping similar data points to uncover underlying structures within a dataset. We propose a new unsupervised clustering approach using a multivariate bounded Kotz mixture model (BKMM) for data modeling when the data lie within a bounded support region. In many real applications, BKMM effectively handles observed data that fall within these limits, accurately modeling and clustering the observations. In BKMM, parameter estimation is performed by maximizing the log-likelihood using Expectation-Maximization (EM) algorithm and the Newton-Raphson method. Additionally, we explore the enhancements in clustering performance through semi-supervised learning by incorporating a small amount of labeled data to guide the clustering process. Thus, we propose a bounded Kotz mixture model using a semi-supervised projected model-based clustering method (BKMM-SeSProC) to obtain hidden cluster labels. Model selection in mixtures is essential for determining the optimal number of mixture components, and we introduce a minimum message length (MML) model selection criterion to find the best number of clusters in the BKMM-SeSProC approach. A greedy forward search is applied to estimate the optimal number of clusters. We use the same datasets to evaluate our proposed models, BKMM and BKMM-SeSProC, for data clustering. Additionally, we utilize MML model selection with BKMM-SeSProC to determine the number of components. Initially, we validate both proposed models and the model selection process in various medical applications. Furthermore, to assess their broader performance, we test the models on image datasets, including Alzheimer's disease, lung tissue, and gastrointestinal tract images for disease recognition, and the CIFAR-100 dataset for object categorization. BKMM is compared with the Kotz mixture model (KMM), Student's t mixture model (SMM), Laplace mixture model (LMM), bounded Gaussian mixture model (BGMM), and Gaussian mixture model (GMM) under similar experimental settings across all datasets. To evaluate the performance of BKMM and BKMM-SeSProC, several performance metrics are employed. To find the best number of clusters for BKMM-SeSProC, we examine the effectiveness of MML model selection against seven different criteria. The experimental results demonstrate that the proposed BKMM outperforms the compared models, KMM, SMM, LMM, BGMM, and GMM, in all applications. Additionally, the semi-supervised projected model-based clustering shows better performance across all evaluation metrics compared to unsupervised BKMM.