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MathGroup Archive 2007

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Re: Plot a simple function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74876] Re: Plot a simple function
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Mon, 9 Apr 2007 06:16:56 -0400 (EDT)
  • References: <evab3s$d6b$1@smc.vnet.net>

Paul K. wrote:

>Hi, a simple query. Do you know how could I plot the entropy function
>in Mathematica? I wish to produce an image similar to this one:
>
>http://upload.wikimedia.org/wikipedia/commons/c/c9/Binary_entropy_plot.p=
ng
>
>Do you know how could I plot mutual information between two random
>variables X and Y in a 3D surface in Mathematica?
>
>Thanks!
>
>Paul.
>
>
> 
>
I think you may be looking at multifractal entropy?
The first curve can be done two ways:
1) The logistic way:
y[x_]=-4*x*(1-x)
Plot[y[x],{x,0,1},PlotRange->{{0,1},{0,1}}]
The "And" like plot:
x0 = t;
y0 = p;
z0 = y[t]*y[p];
ParametricPlot3D[{x0, y0, z0}, {t, 0, 1}, {p, 0, 1}]
2) The sine like way:
f[x_]=Sin[Pi*t]
Plot[Sin[Pi*t],{t,0,1},PlotRange->{{0,1},{0,1}}]
x1 = t;
y1 = p;
z1 = f[t]*f[p];
ParametricPlot3D[{x1, y1, z1}, {t, 0, 1}, {p, 0, 1}]

They aren't the same result:

Plot[y[t]-f[t],{t,0,1},PlotRange->{{0,1},{0,1}}]

x2=t;
y2=p;
z2=y[t]*y[p]-f[t]*f[p];
ParametricPlot3D[{x2,y2,z2},{t,0,1},{p,0,1}]

I think the Logistic one is the better of the two theoretically.
Using an Hurst like exponent H they can be made to be the same:
H[h_] = h /. Solve[Sin[Pi*t]^h - y[t] == 0, h]
Plot[H[t], {t, 0, 1}, PlotRange -> {{0, 1}, {0, 1}}]
g[t_]=FullSimplify[f[t]^H[t]]

The H function is a very slippery one.
Although the derivitive existes at t=1/2
you have to use a limit to get the value:
D[Log[-4(-1 + t) t]/Log[Sin[=CF=80 t]], t]
Limit[H[t], t -> 1/2]
{8/Pi^2}
N[%]
{0.810569}


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