Estimating dynamical dimensions from noisy observations

Knowledge of the dynamical dimension mitigates the “curse of dimensionality” by permitting analysis in dimension lower than that of the original state vectors. The description length quantifies complexity and so allows us to use Occam's razor to estimate the dynamical dimension underlying noisily observed data. Applying our method, based on the description length, to a coarsely sampled scalar time series requires the choice of only one parameter; an embedding dimension. For the three systems considered in this study observed amid observational noise, a single choice of embedding dimension does provide reasonable estimates of the dynamical dimension. The spatial distribution of local estimates of dynamical dimension aids visualisation and provides extra insight into the geometric structure of many systems.