scipy.stats.invgauss#
- scipy.stats.invgauss = <scipy.stats._continuous_distns.invgauss_gen object>[source]#
- An inverse Gaussian continuous random variable. - As an instance of the - rv_continuousclass,- invgaussobject inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.- Methods - rvs(mu, loc=0, scale=1, size=1, random_state=None) - Random variates. - pdf(x, mu, loc=0, scale=1) - Probability density function. - logpdf(x, mu, loc=0, scale=1) - Log of the probability density function. - cdf(x, mu, loc=0, scale=1) - Cumulative distribution function. - logcdf(x, mu, loc=0, scale=1) - Log of the cumulative distribution function. - sf(x, mu, loc=0, scale=1) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(x, mu, loc=0, scale=1) - Log of the survival function. - ppf(q, mu, loc=0, scale=1) - Percent point function (inverse of - cdf— percentiles).- isf(q, mu, loc=0, scale=1) - Inverse survival function (inverse of - sf).- moment(order, mu, loc=0, scale=1) - Non-central moment of the specified order. - stats(mu, loc=0, scale=1, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(mu, loc=0, scale=1) - (Differential) entropy of the RV. - fit(data) - Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments. - expect(func, args=(mu,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) - Expected value of a function (of one argument) with respect to the distribution. - median(mu, loc=0, scale=1) - Median of the distribution. - mean(mu, loc=0, scale=1) - Mean of the distribution. - var(mu, loc=0, scale=1) - Variance of the distribution. - std(mu, loc=0, scale=1) - Standard deviation of the distribution. - interval(confidence, mu, loc=0, scale=1) - Confidence interval with equal areas around the median. - Notes - The probability density function for - invgaussis:\[f(x; \mu) = \frac{1}{\sqrt{2 \pi x^3}} \exp\left(-\frac{(x-\mu)^2}{2 \mu^2 x}\right)\]- for \(x \ge 0\) and \(\mu > 0\). - invgausstakes- muas a shape parameter for \(\mu\).- The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- invgauss.pdf(x, mu, loc, scale)is identically equivalent to- invgauss.pdf(y, mu) / scalewith- y = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.- A common shape-scale parameterization of the inverse Gaussian distribution has density \[f(x; \nu, \lambda) = \sqrt{\frac{\lambda}{2 \pi x^3}} \exp\left( -\frac{\lambda(x-\nu)^2}{2 \nu^2 x}\right)\]- Using - nufor \(\nu\) and- lamfor \(\lambda\), this parameterization is equivalent to the one above with- mu = nu/lam,- loc = 0, and- scale = lam.- This distribution uses routines from the Boost Math C++ library for the computation of the - ppfand- isfmethods. [1]- References [1]- The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/. - Examples - >>> import numpy as np >>> from scipy.stats import invgauss >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Get the support: - >>> mu = 0.145 >>> lb, ub = invgauss.support(mu) - Calculate the first four moments: - >>> mean, var, skew, kurt = invgauss.stats(mu, moments='mvsk') - Display the probability density function ( - pdf):- >>> x = np.linspace(invgauss.ppf(0.01, mu), ... invgauss.ppf(0.99, mu), 100) >>> ax.plot(x, invgauss.pdf(x, mu), ... 'r-', lw=5, alpha=0.6, label='invgauss pdf') - Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed. - Freeze the distribution and display the frozen - pdf:- >>> rv = invgauss(mu) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') - Check accuracy of - cdfand- ppf:- >>> vals = invgauss.ppf([0.001, 0.5, 0.999], mu) >>> np.allclose([0.001, 0.5, 0.999], invgauss.cdf(vals, mu)) True - Generate random numbers: - >>> r = invgauss.rvs(mu, size=1000) - And compare the histogram: - >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> ax.legend(loc='best', frameon=False) >>> plt.show() 