Abstract
Since the beginning of the eighties there has been an expanding activity in analyzing time series in terms of (reconstructed) attractors and their dimensions. For a surveys and further references, see [ABST, 1993], [C.,1991], and [GSS,1991j. The notion of dimension, which was based on considerations concerning deterministic time series, describes roughly how many parameters one needs to specify the possible states on the attractor of the underlying dynamical system. Another approach, proposed by Cheng and Tong, is to analyze a time series primarily in terms of the order, which can be heuristically described as the number of successive elements of a time series which determine the state of the underlying system [CT, 1992]; this notion was introduced in the context of non-deterministic systems, the state should be interpreted as an (abstract) notion which summerizes the information from the past, as far as it is of influence on the future; this notion of state makes sense for both deterministic and stochastic systems. This same idea of order was also studied by Savit and Green [SG, 1991]. Their approach is closer to the above mentioned considerations concerning deterministic systems. The notions of order and dimension are different, but there are clear relations, e.g. the order should at least be equal to the dimension. The main purpose of this paper is to give a survey of these different approaches and to discuss in this context some new numerical examples.
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Takens, F. (1996). Estimation of dimension and order of time series. In: Broer, H.W., van Gils, S.A., Hoveijn, I., Takens, F. (eds) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol 19. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7518-9_19
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DOI: https://doi.org/10.1007/978-3-0348-7518-9_19
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