Generation of uniform random points on Sαn .
José C. S. de Miranda
Instituto de Matemática e Estatı́stica, USP,∗
05508-090 , São Paulo, SP
E-mail: [email protected] .
1
Introduction
The generation of random variates is a central
issue when performing simulations of random
systems. The use of random variates is also
a very important building block in any Monte
Carlo method of integration. Specially on multidimensional domains, the need to efficiently
generate random points becomes apparent.
Methods for generation of uniformly distributed random points on the n-dimensional
sphere have been developed among which we
cite:
• Acceptance-rejection followed by projection,
2
Main Result
The contribution of this article is theorem 2.1.
This provides a generalization of Box-Muller
method.
The generation of uniform random points on
n
Sα can be performed once we know the following:
Theorem 2.1 Let X1 , ..., Xn+1 be independent and identically distributed random variables on the real line with density function
given by:
fX (x) =
1
exp(−|x|α )
Aα
where α ∈ R∗+ and
• Polar coordinates generation,
Z
Aα =
• The Box-Muller method.
exp(−|x|α )dx,
R
Then the random point
Acceptance-rejection plus projection is not a
convenient method for high dimensions. This
is a consequence of the fact that the volume of
the n + 1 dimensional unit radius ball tends to
zero as n goes to infinity.
Polar coordinates generation presents the
problem
of having to solve equations of the
R
form 0φk sink (y)dy = uk for all 1 ≤ k ≤ n − 1,
where uk are i.i.d. uniform random variables on
[0, 1]. An alternative way to generate the angles
φk is to use an acceptance rejection technique
which by its turn will also have a high rejection
rate for high dimension.
The Box-Muller method has the great advantage of no rejection whatever the dimension is.
In this article we will present a method for
generation of uniform random points on Sαn =
P
α
{x ∈ R : n+1
i=1 |xi | = 1}.
∗
This work is partially supported by FAPESP grant
03/10105-2. The author thanks O.L.S.Jesus Christ.
(X1 , ..., Xn+1 )
X= P
1
α α
( n+1
i=1 |Xi | )
is uniformly distributed on Sαn
The random variables Xi can be obtained
by numerical cumulative-distribution-inversion
applied to uniform random variables on [0, 1]
in case of general α.
References
[1] X.R. Li, Generation of random points uniformly distributed in hyperellipsoids, in
First IEEE Conference on Control Applications,ieeexplore.ieee.org, 1992.
[2] N.Madras, “Lectures on Monte Carlo
Methods”, Fields Institute Monographs,
AMS Providence, Rhode Island, 2002.