scipy.stats.crystalball — SciPy v1.15.2 Manual (original) (raw)
scipy.stats.crystalball = <scipy.stats._continuous_distns.crystalball_gen object>[source]#
Crystalball distribution
As an instance of the rv_continuous class, crystalball object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.
Notes
The probability density function for crystalball is:
\[\begin{split}f(x, \beta, m) = \begin{cases} N \exp(-x^2 / 2), &\text{for } x > -\beta\\ N A (B - x)^{-m} &\text{for } x \le -\beta \end{cases}\end{split}\]
where \(A = (m / |\beta|)^m \exp(-\beta^2 / 2)\),\(B = m/|\beta| - |\beta|\) and \(N\) is a normalisation constant.
crystalball takes \(\beta > 0\) 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 loc and scale parameters. Specifically, crystalball.pdf(x, beta, m, loc, scale) is identically equivalent to crystalball.pdf(y, beta, m) / scale withy = (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
Examples
import numpy as np from scipy.stats import crystalball import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
beta, m = 2, 3 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 cdf and 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()
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. |