(untitled) (Contemporary Mathematics, 2025)
The complex structures of vast amounts of data provide a considerable challenge to researchers. Dimensional reduction methods are reliable and transform high-dimensional data into lower-dimensional representations while preserving most original information. Principal Component Analysis (PCA) is a commonly used approach for dimensionality reduction that transforms data into a lower-dimensional space while preserving important information. The variability within the sample may affect the stability and reliability of PCA results. The constraint of existing approaches compromises the accuracy of PCA stability assessments in practical data scenarios. These methodologies frequently depend on linear assumptions and encounter difficulties when addressing high-dimensional data. This study used the bootstrap method to assess the stability of PCA by assessing the variability of eigenvalues and principal components over several bootstrap iterations. We evaluate how stability metrics, particularly confidence intervals for eigenvalues and the proportion of variance clarified, can assist in determining the optimal number of principal components. The results indicate that the bootstrap provides a helpful framework for evaluating the robustness of PCA and guiding informed decisions on dimensionality reduction in many applications, including data compression, visualization, and classification. Moreover, the results illustrate the efficacy of this method in enhancing the reliability and interpretability of PCA findings among distinct data-driven research endeavors. This study enhances understanding of how principal component analysis (PCA) tackles data unpredictability while delivering valuable insights for professionals in several disciplines.