Principal component analysis (PCA) has been a widely used technique for dimension reduction while retaining essential information. However, the ordinary PCA lacks interpretability, especially when dealing with large scale data. To address this limitation, sparse PCA (SPCA) has emerged as an interpretable variant of ordinary PCA. However, the ordinary SPCA relies on solving a challenging non-convex discrete optimization problem, which maximizes explained variance while constraining the number of non-zero elements in each principal component. In this paper, we propose an innovative least angle SPCA technique to address the computational complexity associated with SPCA, particularly in ultrahigh dimensional data, by sequentially identifying sparse principal components with minimal angles to their corresponding components extracted through ordinary PCA. This sequential identification enables solving the optimization problem in polynomial time, significantly reducing computational challenges. Despite its efficiency gains, our proposed method also preserves the main attributes of SPCA. Through comprehensive experimental results, we demonstrate advantages of our approach as a viable alternative for dealing with the computational difficulties inherent in ordinary SPCA. Notably, our method emerges as an efficient and effective solution for conducting ultrahigh dimensional data analysis, enabling researchers to extract meaningful insights and streamline data interpretation.

In the era of Industry 4.0, the complexity of semiconductor production is growing very fast, raising the possibility of unnoticed defective wafers and subsequent wasteful use of resources. One of the key advantages of Industry 4.0 is the accessibility to big data, which can be obtained from a number of sensors, including multiple sensor data and extensive data repositories. Recently, engineers have developed data fusion strategies for virtual metrology (VM) prediction models to effectively handle data from multiple sources. This research explores a novel approach for data-driven VM prediction model for multi-source data, namely multi-source ensemble method with random source selection. By utilizing the bagging principle for multi-source data and tree-based prediction paradigms, the proposed approach randomly selects subsets of data sources to construct each tree learner, thus reducing interdependence among the trees and minimizing the risk of overfitting, which can be a challenge faced by existing tree-based prediction models. To validate and illustrate the practical applicability of our proposed method, we use real-world data from the plasma etching process, aiming to provide potential benefits and effectiveness of our methodology.