Separation of deterministic signals using independent component analysis (ICA)

Independent Component Analysis (ICA) represents a higher-order statistical technique that is often used to separate mixtures of stochastic random signals into statistically independent sources. Its benefit is that it only relies on the information contained in the observations, i.e. no parametric a-priori models are prescribed to extract the source signals. The mathematical foundation of ICA, however, is rooted in the theory of random signals. This has led to questions whether the application of ICA to deterministic signals can be justified at all? In this context, the possibility of using ICA to separate deterministic signals such as complex sinusoidal cycles has been subjected to previous studies. In many geophysical and geodetic applications, however, understanding long-term trend in the presence of periodical components of an observed phenomenon is desirable. In this study, therefore, we extend the previous studies with mathematically proving that the ICA algorithm with diagonalizing the 4th order cumulant tensor, through the rotation of experimental orthogonal functions, will indeed perfectly separate an unknown mixture of trend and sinusoidal signals in the data, provided that the length of the data set is infinite. In other words, we justify the application of ICA to those deterministic signals that are most relevant in geodetic and geophysical applications.