Simple methods for the optimization of complex-valued kurtosis as a projection index

Simple and rapidly executed iterative algorithms for the optimization of the complex-valued kurtosis as a projection index are developed, which have the following advantages: maximization and minimization in response to requirements, no requirements for sphering and strong-uncorrelation transformation, no assumptions for circular or non-circular source signals, and the option to choose uncorrelated scores or orthogonal projection vectors. With the advancement of modern techniques, complex-valued data have become more important in chemistry and many other areas. The data collected are often multi-dimensional. This imposes an increasing demand on the tools used for the analysis of complex-valued data. In multivariate data analysis, projection pursuit is a useful and important technique that in many cases gives better results than principal component analysis. One important projection pursuit variant uses the real-valued kurtosis as its projection index and has been shown to be a powerful approach to address different problems. However, using the complex-valued kurtosis as a projection index to deal with complex-valued data is rare. This is, to a great extent, due to the lack of simple and fast optimization algorithms. In this work, simple and rapidly executed optimization algorithms for the complex-valued kurtosis used as a projection index are proposed. The developed algorithms have a variety of advantages: no requirement for sphering or strong-uncorrelation transformation of the data in advance, no assumption for the latent components (source signals) to be circular or non-circular, search for maxima or minima on users' requirements, and users having the option to choose uncorrelated scores or orthogonal projection vectors. The mathematical development of the algorithms is described and simulated and real experimental data are employed to demonstrate the utility of the proposed algorithms. Copyright (c) 2015 John Wiley & Sons, Ltd.