_hypergeom.py

# MIT License
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# Copyright (c) 2020 Yuxin Wang
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"""This module implements the sf (survival function) of hypergeometric distribution. Note that sf(x) = 1 - cdf(x)
where cdf is the cumulative density function, but implementation-wise it gives a better precision than (1 - cdf).
We try to mimic the interfaces of the counterparts in scipy (scipy.stats.hypergeom.sf and scipy.stats.hypergeom.pmf).
However, our implementation is much more efficient thanks to the JIT compiler numba, without loss of too
much precision. (difference with scipy is < 10^-9, see tests/test_hypergeom.py)

References:
https://en.wikipedia.org/wiki/Hypergeometric_distribution
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.hypergeom.html
https://github.com/distributions-io/hypergeometric-cdf/blob/bf59133188731fe0631f1f03a8ac641ad35470bf/lib/number.js
https://stackoverflow.com/a/47725299/5148356
Wu, Trong. "An accurate computation of the hypergeometric distribution function."
ACM Transactions on Mathematical Software (TOMS) 19.1 (1993): 33-43.
"""

import math
from logging import getLogger
import numba
import numpy as np

logger = getLogger(__name__)


@numba.njit(numba.float64(numba.int_, numba.int_))
def _ln_binomial(n, k):
    """log of binomial coefficient function (n k), i.e., n choose k"""
    if k > n:
        raise ValueError
    if k == n or k == 0:
        return 0
    if k * 2 > n:
        k = n - k
    return math.lgamma(n + 1) - math.lgamma(k + 1) - math.lgamma(n - k + 1)


@numba.njit(numba.float64(numba.int_, numba.int_, numba.int_, numba.int_))
def pmf(k, M, n, N):
    """returns the pmf of hypergeometric distribution for given parameters. This interface mimics scipy's hypergeom.pmf
    :param k: input value
    :param M: the total number of objects
    :param n: the total number of Type 1 objects
    :param N: the number of draws
    :return: the probability mass function for given parameter
    """
    if N > M:
        raise ValueError
    if k > n or k > N:
        return 0
    elif N > M - n and k + M - n < N:
        return 0
    return math.exp(_ln_binomial(n, k) + _ln_binomial(M - n, N - k) - _ln_binomial(M, N))


@numba.njit(numba.float64(numba.int_, numba.int_, numba.int_, numba.int_))
def sf(k, M, n, N):
    """returns the survival function of hypergeometric distribution for given parameters. This equals (1 - cdf) but we
    try to be more precise than (1 - cdf). This interface mimics scipy.stats.hypergeom.sf.
    :param k: input value
    :param M: the total number of objects
    :param n: the total number of Type 1 objects
    :param N: the number of draws
    :return: the cumulative density for given parameter
    """
    if N > M:
        raise ValueError(
            'The number of draws (N) is larger than the total number of objects (M)')
    if k >= min(n, N):
        return 0
    elif k < 0:
        return 1
    # calculating the pmf is expensive, use the following recursive definition for performance:
    # P(X=i) = (i / (n - i + 1)) * ((M - n + i - N) / (N - i + 1)) * P(X=i+1)
    # P(X=i) = ((n - i) / (i + 1)) * ((N - i) / (M - n + i + 1 - N)) * P(X=i-1)

    # the hypergeometric distribution is centered around N * (n / M),
    # i.e. pmf has the largest value when k = N * (n / M)
    # therefore, for fewer iterations, we use forward recursive definition to calculate P(X > k) for k > N * (n / M)
    # otherwise we use backward recursive definition to calculate P(X <= k) and return 1 - P(x <= k)
    # this also gives use more precise result when pmf(k) ~= 0 since the error will be significant and propagated
    # through the recursion
    if k > N * n / M:
        pmf_i = pmf(k + 1, M, n, N)
        result = pmf_i
        for i in range(k + 1, N):
            pmf_i *= ((n - i) / (i + 1)) * ((N - i) / (M - n + i + 1 - N))
            result += pmf_i

        # temp = np.arange(k+1, N)
        # pmf_i = pmf_i * (((n-temp)/(temp+1))@((N-temp)/(M-n+temp+1-N)))
        # result +=pmf_i

        return result
    else:
        pmf_i = pmf(k, M, n, N)
        result = pmf_i
        for i in range(k, 0, -1):
            pmf_i *= (i / (n - i + 1)) * ((M - n + i - N) / (N - i + 1))
            result += pmf_i

        # temp = np.arange(k, 0, -1)
        # pmf_i = pmf_i * ((temp/(n-temp+1))@((M-n+temp-N)/(N-temp+1)))
        # result+= pmf_i
        
        return 1-result