Orthogonal projection, embedding dimension and sample size in chaotic time series from a statistical perspective

By studying systematically the orthogonal projections, in a particular sense associated with a (random) time series admitting a possibly chaotic skeleton and in a sequence of suitably defined L2-spaces, we describe a geometric characterisation of the notion of embedding dimension within a statistical framework. The question of sample size requirement in the statistical estimation of the said dimension is addressed heuristically, ending with a pleasant surprise: the curse of dimensionality may be lifted except in the excessively stringent cases.

[1]  H. Tong,et al.  On consistent nonparametric order determination and chaos , 1992 .

[2]  P. Robinson,et al.  Hypothesis Testing in Semiparametric and Nonparametric Models for Econometric Time Series , 1989 .

[3]  D. Ruelle,et al.  Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems , 1992 .

[4]  J. D. Farmer,et al.  Nonlinear modeling of chaotic time series: Theory and applications , 1990 .

[5]  P. Lewis,et al.  Nonlinear Modeling of Time Series Using Multivariate Adaptive Regression Splines (MARS) , 1991 .

[6]  D. Ruelle,et al.  The Claude Bernard Lecture, 1989 - Deterministic chaos: the science and the fiction , 1990, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  Ping Zhang.  Model Selection Via Multifold Cross Validation , 1993 .

[8]  Qiwei Yao,et al.  Quantifying the influence of initial values on nonlinear prediction , 1994 .

[9]  R. Bhansali,et al.  Some properties of the order of an autoregressive model selected by a generalization of Akaike∘s EPF criterion , 1977 .

[10]  H. Tong,et al.  On residual sums of squares in non-parametric autoregression , 1993 .

[11]  Howell Tong,et al.  Nonparametric function estimation in noisy chaos , 1993 .

[12]  H. Akaike.  A Bayesian extension of the minimum AIC procedure of autoregressive model fitting , 1979 .

[13]  Kung-Sik Chan,et al.  A Note on Noisy Chaos , 1994 .

[14]  M. Stone.  An Asymptotic Equivalence of Choice of Model by Cross‐Validation and Akaike's Criterion , 1977 .

[15]  Qiwei Yao,et al.  On subset selection in non-parametric stochastic regression , 1994 .

[16]  G. P. King,et al.  Extracting qualitative dynamics from experimental data , 1986 .

[17]  H. Akaike.  A new look at the statistical model identification , 1974 .

[18]  Jens Ledet Jensen,et al.  Chaotic dynamical systems with a View towards statistics: a review , 1993 .

[19]  George G. Szpiro.  Measuring dynamical noise in dynamical systems , 1993 .

[20]  R. Shibata.  Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process , 1980 .

[21]  Leonard A. Smith.  Intrinsic limits on dimension calculations , 1988 .

[22]  I. Johnstone,et al.  Empirical functionals and e cient smoothing parameter selection , 1992 .

[23]  P. G. Drazin,et al.  Nonlinear systems: Frontmatter , 1992 .

[24]  B. G. Quinn,et al.  The determination of the order of an autoregression , 1979 .

[25]  David Ruelle,et al.  Deterministic chaos: the science and the fiction , 1995 .

[26]  H. Tong.  Akaike’s Approach can Yield Consistent Order Determination , 1994 .

[27]  Royal-statistical-society meeting on chaos , 1992 .

[28]  G. P. King,et al.  Interpretation of Time Series from Nonlinear Systems , 1992 .

[29]  H. Tong.  Non-linear time series. A dynamical system approach , 1990 .