import numpy as np
from scipy._lib._util import check_random_state


def 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, `~np.random.RandomState`, `~np.random.Generator`}, optional
        If `random_state` is `None` the `~np.random.RandomState` singleton is
        used.
        If `random_state` is an int, a new ``RandomState`` instance is used,
        seeded with random_state.
        If `random_state` is already a ``RandomState`` or ``Generator``
        instance, then that object is used.
        Default is None.

    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
    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 `~scipy.stats.kstest` as a check.

    References
    ----------
    .. [1] 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
    --------
    >>> from scipy import stats

    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
    >>> np.random.seed(12345)
    >>> rvs = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=2500)

    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]
    0.33783681428365553

    The exponential distribution provides another example where the bounding
    rectangle can be determined explicitly.

    >>> np.random.seed(12345)
    >>> rvs = stats.rvs_ratio_uniforms(lambda x: np.exp(-x), umax=1,
    ...                                vmin=0, vmax=2*np.exp(-1), size=1000)
    >>> stats.kstest(rvs, 'expon')[1]
    0.928454552559516

    """

    if vmin >= vmax:
        raise ValueError("vmin must be smaller than vmax.")

    if umax <= 0:
        raise ValueError("umax must be positive.")

    size1d = tuple(np.atleast_1d(size))
    N = np.prod(size1d)  # number of rvs needed, reshape upon return

    # start sampling using ratio of uniforms method
    rng = check_random_state(random_state)
    x = np.zeros(N)
    simulated, i = 0, 1

    # loop until N rvs have been generated: expected runtime is finite.
    # to avoid infinite loop, raise exception if not a single rv has been
    # generated after 50000 tries. even if the expected numer of iterations
    # is 1000, the probability of this event is (1-1/1000)**50000
    # which is of order 10e-22
    while simulated < N:
        k = N - simulated
        # simulate uniform rvs on [0, umax] and [vmin, vmax]
        u1 = umax * rng.uniform(size=k)
        v1 = rng.uniform(vmin, vmax, size=k)
        # apply rejection method
        rvs = v1 / u1 + c
        accept = (u1**2 <= pdf(rvs))
        num_accept = np.sum(accept)
        if num_accept > 0:
            x[simulated:(simulated + num_accept)] = rvs[accept]
            simulated += num_accept

        if (simulated == 0) and (i*N >= 50000):
            msg = ("Not a single random variate could be generated in {} "
                   "attempts. The ratio of uniforms method does not appear "
                   "to work for the provided parameters. Please check the "
                   "pdf and the bounds.".format(i*N))
            raise RuntimeError(msg)
        i += 1

    return np.reshape(x, size1d)
