🚀 go-pugleaf

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
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