STATISTICS FOR MATHEMATICAL PROPERTIES OF MAPS BETWEEN TIME-SERIES EMBEDDINGS

We develop a set of statistics which are intended to characterize in terms of probabilities or confidence levels whether two data sets are related by a mapping with certain mathematical properties. Given these statistics we can ask how confident we can be that the mapping is continuous, injective, differentiable, or has a differentiable inverse. The intended use is for experimental or numerical situations in which multiple time series are generated and one wants to know what relation exists among them, but the mapping between them is unknown or intractable. Examples of applications are testing filtered chaotic data for continuity and differentiability, testing two data sets for synchronization (in the most general sense), testing one data set for determinism forward and backward in time, and determining when transformations on two- or three-dimensional images are well behaved (diffeomorphisms). We test the statistics on several of these cases and show that they are useful for characterizing relations between data sets and for shedding light on phenomena which occur when data are transformed, for example, a dimension increase on filtering a chaotic data set.