.. _sampling-srou:

Simple Ratio-of-Uniforms (SROU)
===============================

.. currentmodule:: scipy.stats.sampling

* Required: PDF, area under PDF if different than 1
* Optional: mode, CDF at mode
* Speed:

  * Set-up: fast
  * Sampling: slow

SROU is based on the ratio-of-uniforms method that uses universal inequalities for constructing
a (universal) bounding rectangle. It works for T-concave distributions with T(x) = -1/sqrt(x).

    >>> import numpy as np
    >>> from scipy.stats.sampling import SimpleRatioUniforms

Suppose we have the normal distribution:

    >>> class StdNorm:
    ...     def pdf(self, x):
    ...         return np.exp(-0.5 * x**2)

Notice that the PDF doesn't integrate to 1. We can either pass the exact
area under the PDF during initialization of the generator or an upper
bound to the exact area under the PDF. Also, it is recommended to pass
the mode of the distribution to speed up the setup:

    >>> urng = np.random.default_rng()
    >>> dist = StdNorm()
    >>> rng = SimpleRatioUniforms(dist, mode=0,
    ...                           pdf_area=np.sqrt(2*np.pi),
    ...                           random_state=urng)

Now, we can use the `rvs` method to generate samples from the distribution:

    >>> rvs = rng.rvs(10)

If the CDF at mode is available, it can be set to improve the performance of `rvs`:

    >>> from scipy.stats import norm
    >>> rng = SimpleRatioUniforms(dist, mode=0,
    ...                           pdf_area=np.sqrt(2*np.pi),
    ...                           cdf_at_mode=norm.cdf(0),
    ...                           random_state=urng)
    >>> rvs = rng.rvs(1000)

We can check that the samples are from the given distribution by visualizing
its histogram:

.. plot::
    :alt: " "

    >>> from scipy.stats.sampling import SimpleRatioUniforms
    >>> from scipy.stats import norm
    >>> import matplotlib.pyplot as plt
    >>> class StdNorm:
    ...     def pdf(self, x):
    ...         return np.exp(-0.5 * x**2)
    ... 
    >>> urng = np.random.default_rng()
    >>> dist = StdNorm()
    >>> rng = SimpleRatioUniforms(dist, mode=0,
    ...                           pdf_area=np.sqrt(2*np.pi),
    ...                           cdf_at_mode=norm.cdf(0),
    ...                           random_state=urng)
    >>> rvs = rng.rvs(1000)
    >>> x = np.linspace(rvs.min()-0.1, rvs.max()+0.1, 1000)
    >>> fx = 1/np.sqrt(2*np.pi) * dist.pdf(x)
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, fx, 'r-', lw=2, label='true distribution')
    >>> ax.hist(rvs, bins=10, density=True, alpha=0.8, label='random variates')
    >>> ax.set_xlabel('x')
    >>> ax.set_ylabel('PDF(x)')
    >>> ax.set_title('Simple Ratio-of-Uniforms Samples')
    >>> ax.legend()
    >>> plt.show()

The main advantage of the method is a fast setup. This can be beneficial if one
repeatedly needs to generate small to moderate samples of a distribution with
different shape parameters. In such a situation, the setup step of
`sampling.NumericalInverseHermite` or `sampling.NumericalInversePolynomial` will
lead to poor performance. As an example, assume we are interested to generate
100 samples for the Gamma distribution with 1000 different shape parameters
given by `np.arange(1.5, 5, 1000)`.

    >>> import math
    >>> class GammaDist:
    ...     def __init__(self, p):
    ...         self.p = p
    ...     def pdf(self, x):
    ...         return x**(self.p-1) * np.exp(-x)
    ...
    >>> urng = np.random.default_rng()
    >>> p = np.arange(1.5, 5, 1000)
    >>> res = np.empty((1000, 100))
    >>> for i in range(1000):
    ...     dist = GammaDist(p[i])
    ...     rng = SimpleRatioUniforms(dist, mode=p[i]-1,
    ...                               pdf_area=math.gamma(p[i]),
    ...                               random_state=urng)
    ...     with np.testing.suppress_warnings() as sup:
    ...         sup.filter(RuntimeWarning, "invalid value encountered in double_scalars")
    ...         sup.filter(RuntimeWarning, "overflow encountered in exp")
    ...         res[i] = rng.rvs(100)

See [1]_, [2]_, and [3]_ for more details.

References
----------
.. [1] UNU.RAN reference manual, Section 5.3.16,
       "SROU - Simple Ratio-of-Uniforms method",
       http://statmath.wu.ac.at/software/unuran/doc/unuran.html#SROU
.. [2] Leydold, Josef. "A simple universal generator for continuous and
       discrete univariate T-concave distributions." ACM Transactions on
       Mathematical Software (TOMS) 27.1 (2001): 66-82
.. [3] Leydold, Josef. "Short universal generators via generalized ratio-of-uniforms
       method." Mathematics of Computation 72.243 (2003): 1453-1471
