Widely linear complex partial least squares for latent subspace regression

The method of partial least squares (PLS) has become a preferred tool for ill-posed linear estimation problems in the real domain, both in the regression and correlation analysis context. However, many modern applications involve complex-valued data (e.g. smart grid, sensor networks) and would benefit from corresponding well-posed latent variable regression analyses. To this end, we propose a PLS algorithm for physically meaningful latent subspace regression with complex-valued data. For rigour, this is achieved by taking into account full complex second-order augmented statistics to produce a robust widely linear estimator for general improper complex-valued data which may be highly correlated or colinear. The so-derived widely linear complex PLS (WL-CPLS) is shown to allow for effective joint latent variable decomposition of complex-valued data, while accounting for computational intractabilities in the calculation of a generalised inverse. This makes it possible to also determine the joint-subspace identified within the proposed algorithm, when applied to univariate outputs. The analysis is supported through both simulations on synthetic data and a real-world application of frequency estimation in unbalanced power grids. Finally, the ability of WL-CPLS to identify physically meaningful components is demonstrated through simultaneous complex covariance matrix diagonalisation.