Re: A functional measure of roughness
From: Roger L. Bagula (rlbtftn_at_netscape.net)
Date: 08/22/04
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Date: Sun, 22 Aug 2004 14:27:29 GMT
The measure of roughness based on the Bezier
which is like a Lyapunov exponent average, but more sensative:
It turns out to be like a second derivative:
Measure[n]=Sum[Log[1+Abs[f''(x(i))]/4],{i,1,n}]/n
Limit[Measure[n],n-> Infinity]=rho
The roughness measure is relate to the Lyapunov exponent average by
k=Sum[Log[f'(x(i))],{i,1,n}]/n/Sum[Log[1+Abs[f''(x(i))]/4],{i,1,n}]/n
k is close to 2 for the primes.
So I have actually found two measures.
Roger Bagula wrote:
> 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/
>
>
> ------------------------------------------------------------------------
>
-- 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/
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