scipy.stats.nchypergeom_fisher#
- scipy.stats.nchypergeom_fisher = <scipy.stats._discrete_distns.nchypergeom_fisher_gen object>[source]#
- A Fisher’s noncentral hypergeometric discrete random variable. - Fisher’s noncentral hypergeometric distribution models drawing objects of two types from a bin. M is the total number of objects, n is the number of Type I objects, and odds is the odds ratio: the odds of selecting a Type I object rather than a Type II object when there is only one object of each type. The random variate represents the number of Type I objects drawn if we take a handful of objects from the bin at once and find out afterwards that we took N objects. - As an instance of the - rv_discreteclass,- nchypergeom_fisherobject inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.- See also - Notes - Let mathematical symbols \(N\), \(n\), and \(M\) correspond with parameters N, n, and M (respectively) as defined above. - The probability mass function is defined as \[p(x; M, n, N, \omega) = \frac{\binom{n}{x}\binom{M - n}{N-x}\omega^x}{P_0},\]- for \(x \in [x_l, x_u]\), \(M \in {\mathbb N}\), \(n \in [0, M]\), \(N \in [0, M]\), \(\omega > 0\), where \(x_l = \max(0, N - (M - n))\), \(x_u = \min(N, n)\), \[P_0 = \sum_{y=x_l}^{x_u} \binom{n}{y}\binom{M - n}{N-y}\omega^y,\]- and the binomial coefficients are defined as \[\binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.\]- nchypergeom_fisheruses the BiasedUrn package by Agner Fog with permission for it to be distributed under SciPy’s license.- The symbols used to denote the shape parameters (N, n, and M) are not universally accepted; they are chosen for consistency with - hypergeom.- Note that Fisher’s noncentral hypergeometric distribution is distinct from Wallenius’ noncentral hypergeometric distribution, which models drawing a pre-determined N objects from a bin one by one. When the odds ratio is unity, however, both distributions reduce to the ordinary hypergeometric distribution. - The probability mass function above is defined in the “standardized” form. To shift distribution use the - locparameter. Specifically,- nchypergeom_fisher.pmf(k, M, n, N, odds, loc)is identically equivalent to- nchypergeom_fisher.pmf(k - loc, M, n, N, odds).- References [1]- Agner Fog, “Biased Urn Theory”. https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf [2]- “Fisher’s noncentral hypergeometric distribution”, Wikipedia, https://en.wikipedia.org/wiki/Fisher’s_noncentral_hypergeometric_distribution - Examples - >>> import numpy as np >>> from scipy.stats import nchypergeom_fisher >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Calculate the first four moments: - >>> M, n, N, odds = 140, 80, 60, 0.5 >>> mean, var, skew, kurt = nchypergeom_fisher.stats(M, n, N, odds, moments='mvsk') - Display the probability mass function ( - pmf):- >>> x = np.arange(nchypergeom_fisher.ppf(0.01, M, n, N, odds), ... nchypergeom_fisher.ppf(0.99, M, n, N, odds)) >>> ax.plot(x, nchypergeom_fisher.pmf(x, M, n, N, odds), 'bo', ms=8, label='nchypergeom_fisher pmf') >>> ax.vlines(x, 0, nchypergeom_fisher.pmf(x, M, n, N, odds), colors='b', lw=5, alpha=0.5) - Alternatively, the distribution object can be called (as a function) to fix the shape and location. This returns a “frozen” RV object holding the given parameters fixed. - Freeze the distribution and display the frozen - pmf:- >>> rv = nchypergeom_fisher(M, n, N, odds) >>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1, ... label='frozen pmf') >>> ax.legend(loc='best', frameon=False) >>> plt.show()   - Check accuracy of - cdfand- ppf:- >>> prob = nchypergeom_fisher.cdf(x, M, n, N, odds) >>> np.allclose(x, nchypergeom_fisher.ppf(prob, M, n, N, odds)) True - Generate random numbers: - >>> r = nchypergeom_fisher.rvs(M, n, N, odds, size=1000) - Methods - rvs(M, n, N, odds, loc=0, size=1, random_state=None) - Random variates. - pmf(k, M, n, N, odds, loc=0) - Probability mass function. - logpmf(k, M, n, N, odds, loc=0) - Log of the probability mass function. - cdf(k, M, n, N, odds, loc=0) - Cumulative distribution function. - logcdf(k, M, n, N, odds, loc=0) - Log of the cumulative distribution function. - sf(k, M, n, N, odds, loc=0) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(k, M, n, N, odds, loc=0) - Log of the survival function. - ppf(q, M, n, N, odds, loc=0) - Percent point function (inverse of - cdf— percentiles).- isf(q, M, n, N, odds, loc=0) - Inverse survival function (inverse of - sf).- stats(M, n, N, odds, loc=0, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(M, n, N, odds, loc=0) - (Differential) entropy of the RV. - expect(func, args=(M, n, N, odds), loc=0, lb=None, ub=None, conditional=False) - Expected value of a function (of one argument) with respect to the distribution. - median(M, n, N, odds, loc=0) - Median of the distribution. - mean(M, n, N, odds, loc=0) - Mean of the distribution. - var(M, n, N, odds, loc=0) - Variance of the distribution. - std(M, n, N, odds, loc=0) - Standard deviation of the distribution. - interval(confidence, M, n, N, odds, loc=0) - Confidence interval with equal areas around the median.