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}