scipy.stats.crystalball#
- scipy.stats.crystalball = <scipy.stats._continuous_distns.crystalball_gen object>[source]#
- Crystalball distribution - As an instance of the - rv_continuousclass,- crystalballobject 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(beta, m, loc=0, scale=1, size=1, random_state=None) - Random variates. - pdf(x, beta, m, loc=0, scale=1) - Probability density function. - logpdf(x, beta, m, loc=0, scale=1) - Log of the probability density function. - cdf(x, beta, m, loc=0, scale=1) - Cumulative distribution function. - logcdf(x, beta, m, loc=0, scale=1) - Log of the cumulative distribution function. - sf(x, beta, m, loc=0, scale=1) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(x, beta, m, loc=0, scale=1) - Log of the survival function. - ppf(q, beta, m, loc=0, scale=1) - Percent point function (inverse of - cdf— percentiles).- isf(q, beta, m, loc=0, scale=1) - Inverse survival function (inverse of - sf).- moment(order, beta, m, loc=0, scale=1) - Non-central moment of the specified order. - stats(beta, m, loc=0, scale=1, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(beta, m, 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=(beta, m), 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(beta, m, loc=0, scale=1) - Median of the distribution. - mean(beta, m, loc=0, scale=1) - Mean of the distribution. - var(beta, m, loc=0, scale=1) - Variance of the distribution. - std(beta, m, loc=0, scale=1) - Standard deviation of the distribution. - interval(confidence, beta, m, loc=0, scale=1) - Confidence interval with equal areas around the median. - Notes - The probability density function for - crystalballis:\[\begin{split}f(x, \beta, m) = \begin{cases} N \exp(-x^2 / 2), &\text{for } x > -\beta\\ N A (B - x)^{-m} &\text{for } x \le -\beta \end{cases}\end{split}\]- where \(A = (m / |\beta|)^m \exp(-\beta^2 / 2)\), \(B = m/|\beta| - |\beta|\) and \(N\) is a normalisation constant. - crystalballtakes \(\beta > 0\) and \(m > 1\) as shape parameters. \(\beta\) defines the point where the pdf changes from a power-law to a Gaussian distribution. \(m\) is the power of the power-law tail.- The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- crystalball.pdf(x, beta, m, loc, scale)is identically equivalent to- crystalball.pdf(y, beta, m) / 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.- Added in version 0.19.0. - References [1]- “Crystal Ball Function”, https://en.wikipedia.org/wiki/Crystal_Ball_function - Examples - >>> import numpy as np >>> from scipy.stats import crystalball >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Get the support: - >>> beta, m = 2, 3 >>> lb, ub = crystalball.support(beta, m) - Calculate the first four moments: - >>> mean, var, skew, kurt = crystalball.stats(beta, m, moments='mvsk') - Display the probability density function ( - pdf):- >>> x = np.linspace(crystalball.ppf(0.01, beta, m), ... crystalball.ppf(0.99, beta, m), 100) >>> ax.plot(x, crystalball.pdf(x, beta, m), ... 'r-', lw=5, alpha=0.6, label='crystalball 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 = crystalball(beta, m) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') - Check accuracy of - cdfand- ppf:- >>> vals = crystalball.ppf([0.001, 0.5, 0.999], beta, m) >>> np.allclose([0.001, 0.5, 0.999], crystalball.cdf(vals, beta, m)) True - Generate random numbers: - >>> r = crystalball.rvs(beta, m, 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() 