An efficient algorithm for weighted PCA (Q1965988)

A three-way array \(X\) that contains \(p\) frontal slices \(X_1,\dots,X_p\) all of the same order can be analysed by the TUCKALS3 generalization of principal component analysis (PCA). The replicated PCA algorithm, being the variant of TUCKALS3, minimizes \(\sum_{k=1}^p \|X_k-h_kEF'\|^2,\) where \(E\) is a column-wise orthonormal matrix, \(F\) is a matrix and \(h\) is a \(p\)-vector. The authors propose a new algorithm, called the weighted PCA algorithm, that uses less computational time that the replicated PCA algorithm. It is shown that both algorithms produce identical convergent sequences of functional values but the number of floating points per full iterative cycle used by weighted PCA is less than that of replicated PCA.