On Wed, Sep 25, 2013 at 9:36 AM, Neal Becker <ndbeck...@gmail.com> wrote:
> David Goldsmith wrote:
>
> > Is this a valid algorithm for generating a 3D Wiener process? (When I
> > graph the results, they certainly look like potential Brownian motion
> > tracks.)
> >
> > def Wiener3D(incr, N):
> > r = incr*(R.randint(3, size=(N,))-1)
> > r[0] = 0
> > r = r.cumsum()
> > t = 2*np.pi*incr*(R.randint(3, size=(N,))-1)
> > t[0] = 0
> > t = t.cumsum()
> > p = np.pi*incr*(R.randint(3, size=(N,))-1)
> > p[0] = 0
> > p = p.cumsum()
> > x = r*np.cos(t)*np.sin(p)
> > y = r*np.sin(t)*np.sin(p)
> > z = r*np.cos(p)
> > return np.array((x,y,z)).T
> >
> > Thanks!
> >
> > DG
>
> Not the kind of Wiener process I learned of. This would be the integral of
> white noise. Here you have used:
>
> 1. discrete increments
> 2. spherical coordinates
>
>
I agree with Neal: that is not a Wiener process. In your process, the
*angles* that describe the position undergo a random walk, so the particle
tends to move in the same direction over short intervals.
To simulate a Wiener process (i.e. Brownian motion) in 3D, you can simply
evolve each coordinate independently as a 1D process.
Here's a simple function to generate a sample from a Wiener process. The
dimension is determined by the shape of the starting point x0.
import numpy as np
def wiener(x0, n, dt, delta):
"""Generate an n-dimensional random walk.
The array of values generated by this function simulate a Wiener
process.
Arguments
---------
x0 : float or array
The starting point of the random walk.
n : int
The number of steps to take.
dt : float
The time step.
delta : float
delta determines the "speed" of the random walk. The random
variable
of the position at time t, X(t), has a normal distribution whose
mean
is the position at time t=0 and whose variance is delta**2*t.
Returns
-------
x : numpy array
The shape of `x` is (n+1,) + x0.shape.
The first element in the array is x0.
"""
x0 = np.asfarray(x0)
shp = (n+1,) + x0.shape
# Generate a sample numbers from a normal distribution.
r = np.random.normal(size=shp, scale=delta*np.sqrt(dt))
# Replace the first element with 0.0, so that x0 + r.cumsum() results
# in the first element being x0.
r[0] = 0.0
# This computes the random walk by forming the cumulative sum of
# the random sample.
x = r.cumsum(axis=0)
x += x0
return x
Warren
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