scipy.stats.nchypergeom_wallenius#
- scipy.stats.nchypergeom_wallenius = <scipy.stats._discrete_distns.nchypergeom_wallenius_gen object>[source]#
- A Wallenius’ noncentral hypergeometric discrete random variable. - Wallenius’ 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 draw a pre-determined N objects from a bin one by one. - As an instance of the - rv_discreteclass,- nchypergeom_walleniusobject 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 \[p(x; N, n, M) = \binom{n}{x} \binom{M - n}{N-x} \int_0^1 \left(1-t^{\omega/D}\right)^x\left(1-t^{1/D}\right)^{N-x} dt\]- for \(x \in [x_l, x_u]\), \(M \in {\mathbb N}\), \(n \in [0, M]\), \(N \in [0, M]\), \(\omega > 0\), where \(x_l = \max(0, N - (M - n))\), \(x_u = \min(N, n)\), \[D = \omega(n - x) + ((M - n)-(N-x)),\]- and the binomial coefficients are defined as \[\binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.\]- nchypergeom_walleniususes 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 Wallenius’ noncentral hypergeometric distribution is distinct from Fisher’s noncentral hypergeometric distribution, which models take a handful of objects from the bin at once, finding out afterwards that N objects were taken. 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_wallenius.pmf(k, M, n, N, odds, loc)is identically equivalent to- nchypergeom_wallenius.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]- “Wallenius’ noncentral hypergeometric distribution”, Wikipedia, https://en.wikipedia.org/wiki/Wallenius’_noncentral_hypergeometric_distribution - Examples - >>> import numpy as np >>> from scipy.stats import nchypergeom_wallenius >>> 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_wallenius.stats(M, n, N, odds, moments='mvsk') - Display the probability mass function ( - pmf):- >>> x = np.arange(nchypergeom_wallenius.ppf(0.01, M, n, N, odds), ... nchypergeom_wallenius.ppf(0.99, M, n, N, odds)) >>> ax.plot(x, nchypergeom_wallenius.pmf(x, M, n, N, odds), 'bo', ms=8, label='nchypergeom_wallenius pmf') >>> ax.vlines(x, 0, nchypergeom_wallenius.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_wallenius(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_wallenius.cdf(x, M, n, N, odds) >>> np.allclose(x, nchypergeom_wallenius.ppf(prob, M, n, N, odds)) True - Generate random numbers: - >>> r = nchypergeom_wallenius.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.