Assessing different norms in nonlinear analysis of noisy time series

Abstract

Many methods dealing with the analysis of multivariate data involve computations of point interdistances in state space. In time series analysis, the points are reconstructed from scalar data, most commonly using the method of delay coordinates. While, theoretically, all norms are considered equivalent in measuring interdistances, they perform differently in the presence of noise. The statistical description of three of the most popular norms, L 1, L 2, and L ∞, revealed certain shortcomings of each, depending on the corresponding noise-free interdistances. For chaotic time series, the effect of noise on the different measures also varies with the reconstruction. Estimating the correlation dimension for simulated noisy data using the three norms confirmed the statistical analysis. Generally, the L 2 norm turns out to be the most robust over a range of time series types and reconstructions, whereas the L 1 and L ∞ norms seem to deteriorate markedly for specific types of systems or reconstructions. Furthermore, the investigation gave some more insight into the role of the reconstruction parameters, showing in particular, the importance of the time window length τ w.