Kernel Density Estimation for Dynamical Systems

© 2018 Hanyuan Hang, Ingo Steinwart, Yunlong Feng and Johan A.K. Suykens. We study the density estimation problem with observations generated by certain dynamical systems that admit a unique underlying invariant Lebesgue density. Observations drawn from dynamical systems are not independent and moreover, usual mixing concepts may not be appropriate for measuring the dependence among these observations. By employing the C-mixing concept to measure the dependence, we conduct statistical analysis on the consistency and convergence of the kernel density estimator. Our main results are as follows: First, we show that with properly chosen bandwidth, the kernel density estimator is universally consistent under L1-norm; Second, we establish convergence rates for the estimator with respect to several classes of dynamical systems under L1-norm. In the analysis, the density function f is only assumed to be Holder continuous or pointwise Holder controllable which is a weak assumption in the literature of nonparametric density estimation and also more realistic in the dynamical system context. Last but not least, we prove that the same convergence rates of the estimator under L1-norm and L1-norm can be achieved when the density function is Holder continuous, compactly supported, and bounded. The bandwidth selection problem of the kernel density estimator for dynamical system is also discussed in our study via numerical simulations.

Publication year:2018

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BOF-publication weight:2

CSS-citation score:1

Authors:International

Authors from:Higher Education

Accessibility:Open