scipy.stats.rvs_ratio_uniforms#
- scipy.stats.rvs_ratio_uniforms(pdf, umax, vmin, vmax, size=1, c=0, random_state=None)[source]#
Generate random samples from a probability density function using the ratio-of-uniforms method.
- Parameters:
- pdfcallable
A function with signature pdf(x) that is proportional to the probability density function of the distribution.
- umaxfloat
The upper bound of the bounding rectangle in the u-direction.
- vminfloat
The lower bound of the bounding rectangle in the v-direction.
- vmaxfloat
The upper bound of the bounding rectangle in the v-direction.
- sizeint or tuple of ints, optional
Defining number of random variates (default is 1).
- cfloat, optional.
Shift parameter of ratio-of-uniforms method, see Notes. Default is 0.
- random_state{None, int,
numpy.random.Generator
, numpy.random.RandomState
}, optionalIf seed is None (or np.random), the
numpy.random.RandomState
singleton is used. If seed is an int, a newRandomState
instance is used, seeded with seed. If seed is already aGenerator
orRandomState
instance then that instance is used.
- Returns:
- rvsndarray
The random variates distributed according to the probability distribution defined by the pdf.
Notes
Given a univariate probability density function pdf and a constant c, define the set
A = {(u, v) : 0 < u <= sqrt(pdf(v/u + c))}
. If (U, V) is a random vector uniformly distributed over A, then V/U + c follows a distribution according to pdf.The above result (see [1], [2]) can be used to sample random variables using only the pdf, i.e. no inversion of the cdf is required. Typical choices of c are zero or the mode of pdf. The set A is a subset of the rectangle
R = [0, umax] x [vmin, vmax]
whereumax = sup sqrt(pdf(x))
vmin = inf (x - c) sqrt(pdf(x))
vmax = sup (x - c) sqrt(pdf(x))
In particular, these values are finite if pdf is bounded and
x**2 * pdf(x)
is bounded (i.e. subquadratic tails). One can generate (U, V) uniformly on R and return V/U + c if (U, V) are also in A which can be directly verified.The algorithm is not changed if one replaces pdf by k * pdf for any constant k > 0. Thus, it is often convenient to work with a function that is proportional to the probability density function by dropping unneccessary normalization factors.
Intuitively, the method works well if A fills up most of the enclosing rectangle such that the probability is high that (U, V) lies in A whenever it lies in R as the number of required iterations becomes too large otherwise. To be more precise, note that the expected number of iterations to draw (U, V) uniformly distributed on R such that (U, V) is also in A is given by the ratio
area(R) / area(A) = 2 * umax * (vmax - vmin) / area(pdf)
, where area(pdf) is the integral of pdf (which is equal to one if the probability density function is used but can take on other values if a function proportional to the density is used). The equality holds since the area of A is equal to 0.5 * area(pdf) (Theorem 7.1 in [1]). If the sampling fails to generate a single random variate after 50000 iterations (i.e. not a single draw is in A), an exception is raised.If the bounding rectangle is not correctly specified (i.e. if it does not contain A), the algorithm samples from a distribution different from the one given by pdf. It is therefore recommended to perform a test such as
kstest
as a check.References
[2]W. Hoermann and J. Leydold, “Generating generalized inverse Gaussian random variates”, Statistics and Computing, 24(4), p. 547–557, 2014.
[3]A.J. Kinderman and J.F. Monahan, “Computer Generation of Random Variables Using the Ratio of Uniform Deviates”, ACM Transactions on Mathematical Software, 3(3), p. 257–260, 1977.
Examples
>>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng()
Simulate normally distributed random variables. It is easy to compute the bounding rectangle explicitly in that case. For simplicity, we drop the normalization factor of the density.
>>> f = lambda x: np.exp(-x**2 / 2) >>> v_bound = np.sqrt(f(np.sqrt(2))) * np.sqrt(2) >>> umax, vmin, vmax = np.sqrt(f(0)), -v_bound, v_bound >>> rvs = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=2500, ... random_state=rng)
The K-S test confirms that the random variates are indeed normally distributed (normality is not rejected at 5% significance level):
>>> stats.kstest(rvs, 'norm')[1] 0.250634764150542
The exponential distribution provides another example where the bounding rectangle can be determined explicitly.
>>> rvs = stats.rvs_ratio_uniforms(lambda x: np.exp(-x), umax=1, ... vmin=0, vmax=2*np.exp(-1), size=1000, ... random_state=rng) >>> stats.kstest(rvs, 'expon')[1] 0.21121052054580314