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- From: mbk@gibbs.ucsd.edu (Matt Kennel)
- Newsgroups: sci.fractals
- Subject: Re: Interpolation & Chaotic TS
- Date: 6 Nov 1992 06:40:38 GMT
- Organization: Institute For Nonlinear Science, UCSD
- Lines: 35
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- sassoj@rs6416.ecs.rpi.edu (John J. Sasso Jr.) writes:
- : Given a chaotic time series with a fractal dimension, does anyone know of
- : any theory or whatever which relates splines to the interpolation of the
- : chaotic time series corresponding to the given fractal dimension? For
- : example, if I have a signal that can be best interpolated by cubic splines,
- : then that is what I would use (using a linear or quadratic spline would not
- : do so well). Now, given a chaotic time series (perhaps one similar to that
- : of Brownian motion), it would seem that you cannot interpolate it with any
- : spline of some order, or you need a very high order spline in order to
- : interpolate it accurately. This may sound crazy, but would it be that you
- : would need a spline of some fractional order (in relation to the fractal
- : dimension of the time series) in order to do an accurate interpolation? Or,
- : would the spline needed have to have fractal properties itself (a fractal
- : interpolating a fractal, so to speak.
-
- What do you mean by "interpolating a fractal?"
-
- A fractal attractor certainly has a fractal measure and support--but this
- is a set, not a function.
-
- If you mean predicting some value, say the future point of a time series,
- given the past and a data set of obseved dynamics, the fractal structure
- of the attractor does not really matter--the dynamics (i.e. evolution
- function) is supposed to be a smooth function of phase space and thus
- easily approximated by splines, even though you observe data only on
- some fractal set.
-
- : John
-
-
- --
- -Matt Kennel mbk@inls1.ucsd.edu
- -Institute for Nonlinear Science, University of California, San Diego
- -*** AD: Archive for nonlinear dynamics papers & programs: FTP to
- -*** lyapunov.ucsd.edu, username "anonymous".
-