New Trivial Principal Component Method: System Modeling

Principal Component Analysis (PCA) is a powerful statistical technique used for understanding variation in the observed data and decomposing variation along eigenvectors, known as Principal Components (PCs), by considering variance-covariance structure of the data. Traditionally, eigenvectors that contain most of the variation or information are selected to reduce variables in data reduction. Eigenvalues of low magnitude are considered to be noise and often, not included in the dataset to accomplish dimensional reduction. Analogously, in Principal Component Regression (PCR), PCs with large eigenvalues are selected without considering correlation between the source variables and the dependent response. This inherent deficiency may lead to inferior regression modelling. While addressing this issue, an alternative to PCR is developed and proposed in this paper. In this method, a principal component associated with zero eigenvalue is termed Trivial Principal Component (TPC). This novel method involves the formulation of the TPC by including output response in the covariance matrix and then, extracting the Eigen-pairs. The TPC contains the relationship between the dependent response and the source variables and is used for extracting linear coefficients. In other words, the TPC is formulated to determine sensitivities taking into account correlation relationship between the output response and source variables. Example problems are presented to illustrate methodology and accuracy of the TPC method. Results of this method are applied on a practical production problem to make manufacturing changes for improved quality.