Weighted principal component analysis: a weighted covariance eigendecomposition approach

We present a new straightforward principal component analysis (PCA) method based on the diagonalization of the weighted variance–covariance matrix through two spectral decomposition methods: power iteration and Rayleigh quotient iteration. This method allows one to retrieve a given number of orthogonal principal components amongst the most meaningful ones for the case of problems with weighted and/or missing data. Principal coefficients are then retrieved by fitting principal components to the data while providing the final decomposition. Tests performed on real and simulated cases show that our method is optimal in the identification of the most significant patterns within data sets. We illustrate the usefulness of this method by assessing its quality on the extrapolation of Sloan Digital Sky Survey quasar spectra from measured wavelengths to shorter and longer wavelengths. Our new algorithm also benefits from a fast and flexible implementation.