# scipy.stats.rvs_ratio_uniforms

scipy.stats.rvs_ratio_uniforms(pdf, umax, vmin, vmax, size=1, c=0, random_state=None)

  Generate random samples from a probability density function using the ratio-of-uniforms method.

  Parameters:
    pdf callable

      A function with signature pdf(x) that is proportional to the probability density function of the distribution.

    umax float

      The upper bound of the bounding rectangle in the u-direction.

    vmin float

      The lower bound of the bounding rectangle in the v-direction.

    vmax float

      The upper bound of the bounding rectangle in the v-direction.

    size int or tuple of ints, optional

      Defining number of random variates (default is 1).

    c float, optional.

      Shift parameter of ratio-of-uniforms method, see Notes. Default is 0.

    random_state{None, int, `numpy.random.Generator`,

      > `numpy.random.RandomState`}, optional

      If seed is None (or np.random), the `numpy.random.RandomState` singleton is used. If seed is an int, a new `RandomState` instance is used, seeded with seed. If seed is already a `Generator` or `RandomState` instance then that instance is used.

  Returns:
    rvs ndarray

      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]` where

  - `umax = 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 unnecessary 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

  [1] (1,2)

  L. Devroye, “Non-Uniform Random Variate Generation”, Springer-Verlag, 1986.

  [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