Re: Problem with Fractional Brownian Motion
- To: mathgroup at smc.vnet.net
- Subject: [mg63693] Re: Problem with Fractional Brownian Motion
- From: Roger Bagula <rlbagulatftn at yahoo.com>
- Date: Tue, 10 Jan 2006 01:48:48 -0500 (EST)
- References: <dptbr2$t4t$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Paolo Tarpanelli wrote:
> To simulate a Fractional Brwonian Motion I have implemented this algorithm
>
> H := 0.9;
> n := 5;
> m = 100;
> dFBM = Table[n^(-H)/Gamma[H + 1/2]*((Sum[Xa?1 + n*(m + t)-i?*i^(H-1/2),{i,1,n*t}]))+((Sum[Ya?1+n*(m-1+t)-i*((n+1)^(H-1/2)- i^(H-1/2),{i,1,n(m-1)}])), {t, 1, 100}];
>
> where Xa and Ya are two random vector that I have created previously.
> The problem is that the process is negative(!) while it must to be positive!
>
> Paolo
> ***************
> From my blackberry
> ****************
>
Here's a method I came up with two years ago (about) and posted to my
chaos theory egroup:
Clear[x, a]
(* by Roger L. Bagula 7 July 2004©*)
(* projective line function: interpolates a line to a circle in angular
terms*)
x[a_] = (1 + Sqrt[1 - a^2])/2;
(* seed random at date/ time*)
SeedRandom[];
(* fractional Brownian dimension as H = 3 - a0*)
a0 = Log[4]/Log[3];
digits = 150
(* Fractional brownian distribution as defined by Kenneth Falconer page
250 Fractal Geometry 1990 John Wiley and sons publisher*)
noise = Table[Exp[-(x[Sin[2*Pi*Random[]]]^2 +
x[Sin[2*Pi*Random[]]]^2)*((((n - digits/2)/digits)^2 + ((m -
digits/2)/100)^2)^a0)/2]/Sqrt[2*Pi], {n, 1,
digits}, {m, 1, digits}];
ListDensityPlot[noise, PlotRange -> All]
g1 = ListPlot3D[noise, PlotRange -> All, ViewPoint -> {2.011,
-2.721,-0.076}]
Show[g1, ViewPoint -> {3.034, -1.123, 0.992}]