PENELOPE 2003 - OECD Nuclear Energy Agency

1.2. Random sampling methods 13
can be generated from a single random number. Continuing with our graphical picture,
assume that the N bars in the square are aligned consecutively to form a segment of
length N (bottom of fig. 1.3). To sample x, we can generate a single random value
ξN, which is uniformly distributed in (0,N) and determines one of the segment pieces.
The result of the sampling is the label of the selected piece. Explicitly, the sampling
algorithm proceeds as follows:
(i) Generate a random number ξ and set R = ξN + 1.
(ii) Set i = [R] and r = R − i.
(iii) If r > F i , deliver x = K i .
(iv) Deliver x = i.
We see that the sampling of x involves only the generation of a random number and
one comparison (irrespective of the number N of possible outcomes). The price we
pay for this simplification reduces to doubling the number of memory locations that
are needed: the two arrays K i and F i are used instead of the single array p i (or P i ).
Unfortunately, the calculation of alias and cutoff values is fairly involved and this limits
the applicability of Walker’s algorithm to distributions that remain constant during the
course of the simulation.
1.2.4 Rejection methods
The inverse transform method for random sampling is based on a one-to-one correspondence
between x and ξ values, which is expressed in terms of a single-valued function.
There is another kind of sampling method, due to von Neumann, that consists of sampling
a random variable from a certain distribution [different from p(x)] and subjecting
it to a random test to determine whether it will be accepted for use or rejected. These
rejection methods lead to very general techniques for sampling from any PDF.
The rejection algorithms can be understood in terms of simple graphical arguments
(fig. 1.4). Consider that, by means of the inverse transform method or any other available
sampling method, random values of x are generated from a PDF π(x). For each sampled
value of x we sample a random value y uniformly distributed in the interval (0, Cπ(x)),
where C is a positive constant. Evidently, the points (x, y), generated in this way, are
uniformly distributed in the region A of the plane limited by the x-axis (y = 0) and
the curve y = Cπ(x). Conversely, if (by some means) we generate random points (x, y)
uniformly distributed in A, their x-coordinate is a random variable distributed according
to π(x) (irrespective of the value of C). Now, consider that the distribution π(x) is such
that Cπ(x) ≥ p(x) for some C > 0 and that we generate random points (x, y) uniformly
distributed in the region A as described above. If we reject the points with y > p(x),
the accepted ones (with y ≤ p(x)) are uniformly distributed in the region between the
x-axis and the curve y = p(x) and hence, their x-coordinate is distributed according to
p(x).
A rejection method is thus completely specified by representing the PDF p(x) as
p(x) = Cπ(x)r(x), (1.47)