A unified approach to attractor reconstruction

In the analysis of complex, nonlinear time series, scientists in a variety ofdisciplines have relied on a time delayed embedding of their data, i.e.attractor reconstruction. The process has focused primarily on heuristic andempirical arguments for selection of the key embedding parameters, delay andembedding dimension. This approach has left several long-standing, but commonproblems unresolved in which the standard approaches produce inferior resultsor give no guidance at all. We view the current reconstruction process asunnecessarily broken into separate problems. We propose an alternative approachthat views the problem of choosing all embedding parameters as being one andthe same problem addressable using a single statistical test formulateddirectly from the reconstruction theorems. This allows for varying time delaysappropriate to the data and simultaneously helps decide on embedding dimension.A second new statistic, undersampling, acts as a check against overly long timedelays and overly large embedding dimension. Our approach is more flexible thanthose currently used, but is more directly connected with the mathematicalrequirements of embedding. In addition, the statistics developed guide the userby allowing optimization and warning when embedding parameters are chosenbeyond what the data can support. We demonstrate our approach on uni- andmultivariate data, data possessing multiple time scales, and chaotic data. Thisunified approach resolves all the main issues in attractor reconstruction.Comment: 22 pages, revised version as submitted to CHAOS. Manuscript is currently under review. 4 Figures, 31 reference