Discrete Guide Table (DGT)#
Required: probability vector (PV) or the PMF along with a finite domain
Speed:
Set-up: slow (linear with the vector-length)
Sampling: very fast
DGT samples from arbitrary but finite probability vectors. Random numbers are generated by the inversion method, i.e.
Generate a random number U ~ U(0,1).
Find smallest integer I such that F(I) = P(X<=I) >= U.
Step (2) is the crucial step. Using sequential search requires O(E(X)) comparisons, where E(X) is the expectation of the distribution. Indexed search, however, uses a guide table to jump to some I’ <= I near I to find X in constant time. Indeed the expected number of comparisons is reduced to 2, when the guide table has the same size as the probability vector (this is the default). For larger guide tables this number becomes smaller (but is always larger than 1), for smaller tables it becomes larger.
On the other hand the setup time for guide table is O(N), where N denotes the length of the probability vector (for size 1 no preprocessing is required). Moreover, for very large guide tables memory effects might even reduce the speed of the algorithm. So we do not recommend to use guide tables that are more than three times larger than the given probability vector. If only a few random numbers have to be generated, (much) smaller table sizes are better. The size of the guide table relative to the length of the given probability vector can be set by the guide_factor parameter:
>>> import numpy as np
>>> from scipy.stats.sampling import DiscreteGuideTable
>>>
>>> pv = [0.18, 0.02, 0.8]
>>> urng = np.random.default_rng()
>>> rng = DiscreteGuideTable(pv, random_state=urng)
>>> rng.rvs()
2
By default, the probability vector is indexed starting at 0. However, this
can be changed by passing a domain parameter. When domain is given
in combination with the PV, it has the effect of relocating the
distribution from (0, len(pv)) to (domain[0], domain[0] + len(pv)).
domain[1] is ignored in this case.
>>> rng = DiscreteGuideTable(pv, random_state=urng, domain=(10, 13))
>>> rng.rvs()
10
The method also works when no probability vector but a PMF is given.
In that case, a bounded (finite) domain must also be given either by
passing the domain parameter explicitly or by providing a support
method in the distribution object:
>>> class Distribution:
... def __init__(self, c):
... self.c = c
... def pmf(self, x):
... return x ** self.c
... def support(self):
... return 0, 10
...
>>> dist = Distribution(2)
>>> rng = DiscreteGuideTable(dist, random_state=urng)
>>> rng.rvs()
9
Note
As DiscreteGuideTable expects PMF with signature
def pmf(self, x: float) -> float, it first vectorizes the
PMF using np.vectorize and then evaluates it over all the
points in the domain. But if the PMF is already vectorized,
it is much faster to just evaluate it at each point in the domain
and pass the obtained PV instead along with the domain.
For example, pmf methods of SciPy’s discrete distributions
are vectorized and a PV can be obtained by doing:
>>> from scipy.stats import binom
>>> from scipy.stats.sampling import DiscreteGuideTable
>>> dist = binom(10, 0.2) # distribution object
>>> domain = dist.support() # the domain of your distribution
>>> x = np.arange(domain[0], domain[1] + 1)
>>> pv = dist.pmf(x)
>>> rng = DiscreteGuideTable(pv, domain=domain)
Domain is required here to relocate the distribution
The size of the guide table relative to the probability vector may be set using
the guide_factor parameter. Larger guide tables result in faster generation
time but require a more expensive setup.
>>> guide_factor = 2
>>> rng = DiscreteGuideTable(pv, random_state=urng, guide_factor=guide_factor)
>>> rng.rvs()
2
Unfortunately, the PPF is rarely available in closed form or too slow when
available. The user only has to provide the probability vector and the
PPF (inverse CDF) can be evaluated using ppf method. This
method calculates the (exact) PPF of the given distribution.
For example to calculate the PPF of a binomial distribution with \(n=4\) and \(p=0.1\): we can set up a guide table as follows:
>>> n, p = 4, 0.1
>>> dist = stats.binom(n, p)
>>> rng = DiscreteGuideTable(dist, random_state=42)
>>> rng.ppf(0.5)
0.0
Please see 1 and 2 for more details on this method.
References#
- 1
UNU.RAN reference manual, Section 5.8.4, “DGT - (Discrete) Guide Table method (indexed search)” https://statmath.wu.ac.at/unuran/doc/unuran.html#DGT
- 2
H.C. Chen and Y. Asau (1974). On generating random variates from an empirical distribution, AIIE Trans. 6, pp. 163-166.