Nonlinear manifold representation in natural systems: The SOMersault

Natural systems often contain rhythmically fluctuating individual components which, when combined, can result in nonlinear patterns such as cycles, helixes, and parabolas. The self-organizing map (SOM) is a widely used artificial neural network for exploratory data analysis of high dimensional, multivariate data sets, however it encounters limitations when dealing with such highly nonlinear patterns. The SOMersault method is an expansion of the SOM, effective for gaining an understanding of patterns and clusters in natural data sets containing a low dimensional nonlinear manifold set amongst complex high dimensional data measurements. Data clusters become ordered with respect to the nonlinear degrees of freedom in the data, and patterns extracted are closely related to the data they represent. Results are shown on synthetic and real world data, involving a global set of river basins, with clustering and pattern extraction improvements displayed visually and quantified through a new set of geodesic error measures.