Weighted principal component analysis: a weighted covariance eigendecomposition approach

TL;DR

This paper introduces a new PCA method that uses weighted covariance eigendecomposition via spectral methods, effectively handling weighted and missing data to identify significant data patterns.

Contribution

The paper proposes a novel PCA approach based on spectral decomposition of weighted covariance matrices, improving pattern detection with weighted and missing data.

Findings

Method effectively identifies significant data patterns.

Demonstrates high accuracy on real and simulated data.

Fast and flexible implementation available.

Abstract

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.