A functional measure of rouchness

From: Roger Bagula (tftn_at_earthlink.net)
Date: 08/21/04

  • Next message: Roger Bagula: "IFS: Bezier and fractional partial angular derivative" 
    Date: Sat, 21 Aug 2004 16:52:20 GMT

    In thinking of a way to get a better than Lyapunov , Hausdorff or
    Kolmogorov
    measure of dimension , I thought of this:
    F(curve)=0 if smooth and continuous
    F(curve)<>0 if rough or discontinuous
    The best measure of dimensional roughness (Mandelbrot's way of
    expressing it) is the
    Lyapunov exponent (or maybe the Hurst exponent?).
    Box counting or capacity/ entropy dimension of the Kolmogorov type
    is too big most of the time
    while Hausdorff being very cut-off measure like
    is usually too small.
    The trouble with Lyapunov is that it depends on a derivative
    and unless you are talking about a fractional derivative,
    many fractal functions are of the Weierstrass fractal type
    where the classical derivative doesn't exist.

    I did some work on Bezier functions in IFS in the past
    and fractional partial derivatives of an angular sort as well.
    I came to realize that the three point Bezier function of an iterative
    sequence in n:
    Bezier[p,n]=p^2*f(n+2+2*p*(1-p)*f(n+1)+(1-p)^2*f(n)
    is such that if smooth and continuous:
    f(n+1)=Bezier[1/2,n]=f(n+2)/4+f(n+1)/2+f(n)/4
    So that the function :
     delta[n]=f(n+2)/4+f(n+1)/2+f(n)/4-f(n+1)
    is a measure of the roughness.
    Putting this measure in an Lyapunov average type function:
    Measure[n]=Sum[Log[1+delta[i]],{i,1,n}]/n
    I tried this out by comparing it to a known rough set, the primes
    and it's Lyapunov integer difference average.
    In this experiment the new Bezier roughness measure performs better than the
    Lyapunov equivalent over the same range in detecting roughness.
    Respectfully, Roger L. Bagula

    tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
    URL : http://home.earthlink.net/~tftn
    URL : http://victorian.fortunecity.com/carmelita/435/

    BEZIER_ROUGHNESS_PRIMES.gif
  • Next message: Roger Bagula: "IFS: Bezier and fractional partial angular derivative"

    Relevant Pages

    • Re: A functional measure of roughness
      ... The measure of roughness based on the Bezier ... The roughness measure is relate to the Lyapunov exponent average by ... > and fractional partial derivatives of an angular sort as well. ...
      (sci.fractals)
    • Re: A functional measure of roughness
      ... The measure of roughness based on the Bezier ... The roughness measure is relate to the Lyapunov exponent average by ... > and fractional partial derivatives of an angular sort as well. ...
      (sci.math)
    • Re: A functional measure of roughness
      ... The measure of roughness based on the Bezier ... The roughness measure is relate to the Lyapunov exponent average by ... > and fractional partial derivatives of an angular sort as well. ...
      (sci.nonlinear)
    • A functional measure of rouchness
      ... Lyapunov exponent. ... Box counting or capacity/ entropy dimension of the Kolmogorov type ... and fractional partial derivatives of an angular sort as well. ... is a measure of the roughness. ...
      (sci.math)
    • A functional measure of rouchness
      ... Lyapunov exponent. ... Box counting or capacity/ entropy dimension of the Kolmogorov type ... and fractional partial derivatives of an angular sort as well. ... is a measure of the roughness. ...
      (sci.nonlinear)
    • Quantcast