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We present several applications of non-linear data modeling, using principalmanifolds and principal graphs constructed using the metaphor of elasticity(elastic principal graph approach). These approaches are generalizations of theKohonen's self-organizing maps, a class of artificial neural networks. Onseveral examples we show advantages of using non-linear objects for dataapproximation in comparison to the linear ones. We propose four numericalcriteria for comparing linear and non-linear mappings of datasets into thespaces of lower dimension. The examples are taken from comparative politicalscience, from analysis of high-throughput data in molecular biology, fromanalysis of dynamical systems.

Principal manifolds and graphs in practice: from molecular biology to dynamical systems