Abstract. Complex principal component analysis (CPCA) is a useful linear method for dimensionality reduction of data sets characterized by propagating patterns, where the CPCA modes are linear functions of the complex principal component (CPC), consisting of an amplitude and a phase. The use of non-linear methods, such as the neural-network based circular non-linear principal component analysis (NLPCA.cir) and the recently developed non-linear complex principal component analysis (NLCPCA), may provide a more accurate description of data in case the lower-dimensional structure is non-linear. NLPCA.cir extracts non-linear phase information without amplitude variability, while NLCPCA is capable of extracting both. NLCPCA can thus be viewed as a non-linear generalization of CPCA. In this article, NLCPCA is applied to bathymetry data from the sandy barred beaches at Egmond aan Zee (Netherlands), the Hasaki coast (Japan) and Duck (North Carolina, USA) to examine how effective this new method is in comparison to CPCA and NLPCA.cir in representing propagating phenomena. At Duck, the underlying low-dimensional data structure is found to have linear phase and amplitude variability only and, accordingly, CPCA performs as well as NLCPCA. At Egmond, the reduced data structure contains non-linear spatial patterns (asymmetric bar/trough shapes) without much temporal amplitude variability and, consequently, is about equally well modelled by NLCPCA and NLPCA.cir. Finally, at Hasaki, the data structure displays not only non-linear spatial variability but also considerably temporal amplitude variability, and NLCPCA outperforms both CPCA and NLPCA.cir. Because it is difficult to know the structure of data in advance as to which one of the three models should be used, the generalized NLCPCA model can be used in each situation.
Structure preserving integrators for solving (non-)linear quadratic optimal control problems with applications to describe the flight of a quadrotor