# -*- coding: utf-8 -*- # # Author: Travis Oliphant 2002-2011 with contributions from # SciPy Developers 2004-2011 # import warnings from collections.abc import Iterable import ctypes import numpy as np from scipy._lib.doccer import (extend_notes_in_docstring, replace_notes_in_docstring) from scipy._lib._ccallback import LowLevelCallable from scipy import optimize from scipy import integrate from scipy import interpolate import scipy.special as sc import scipy.special._ufuncs as scu from scipy._lib._util import _lazyselect, _lazywhere from . import _stats from ._rvs_sampling import rvs_ratio_uniforms from ._tukeylambda_stats import (tukeylambda_variance as _tlvar, tukeylambda_kurtosis as _tlkurt) from ._distn_infrastructure import (get_distribution_names, _kurtosis, _ncx2_cdf, _ncx2_log_pdf, _ncx2_pdf, rv_continuous, _skew, valarray, _get_fixed_fit_value, _check_shape) from ._ksstats import kolmogn, kolmognp, kolmogni from ._constants import (_XMIN, _EULER, _ZETA3, _XMAX, _LOGXMAX, _SQRT_2_OVER_PI, _LOG_SQRT_2_OVER_PI) # In numpy 1.12 and above, np.power refuses to raise integers to negative # powers, and `np.float_power` is a new replacement. try: float_power = np.float_power except AttributeError: float_power = np.power def _remove_optimizer_parameters(kwds): """ Remove the optimizer-related keyword arguments 'loc', 'scale' and 'optimizer' from `kwds`. Then check that `kwds` is empty, and raise `TypeError("Unknown arguments: %s." % kwds)` if it is not. This function is used in the fit method of distributions that override the default method and do not use the default optimization code. `kwds` is modified in-place. """ kwds.pop('loc', None) kwds.pop('scale', None) kwds.pop('optimizer', None) if kwds: raise TypeError("Unknown arguments: %s." % kwds) ## Kolmogorov-Smirnov one-sided and two-sided test statistics class ksone_gen(rv_continuous): r"""Kolmogorov-Smirnov one-sided test statistic distribution. This is the distribution of the one-sided Kolmogorov-Smirnov (KS) statistics :math:`D_n^+` and :math:`D_n^-` for a finite sample size ``n`` (the shape parameter). %(before_notes)s Notes ----- :math:`D_n^+` and :math:`D_n^-` are given by .. math:: D_n^+ &= \text{sup}_x (F_n(x) - F(x)),\\ D_n^- &= \text{sup}_x (F(x) - F_n(x)),\\ where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF. `ksone` describes the distribution under the null hypothesis of the KS test that the empirical CDF corresponds to :math:`n` i.i.d. random variates with CDF :math:`F`. %(after_notes)s See Also -------- kstwobign, kstwo, kstest References ---------- .. [1] Birnbaum, Z. W. and Tingey, F.H. "One-sided confidence contours for probability distribution functions", The Annals of Mathematical Statistics, 22(4), pp 592-596 (1951). %(example)s """ def _pdf(self, x, n): return -scu._smirnovp(n, x) def _cdf(self, x, n): return scu._smirnovc(n, x) def _sf(self, x, n): return sc.smirnov(n, x) def _ppf(self, q, n): return scu._smirnovci(n, q) def _isf(self, q, n): return sc.smirnovi(n, q) ksone = ksone_gen(a=0.0, b=1.0, name='ksone') class kstwo_gen(rv_continuous): r"""Kolmogorov-Smirnov two-sided test statistic distribution. This is the distribution of the two-sided Kolmogorov-Smirnov (KS) statistic :math:`D_n` for a finite sample size ``n`` (the shape parameter). %(before_notes)s Notes ----- :math:`D_n` is given by .. math:: D_n &= \text{sup}_x |F_n(x) - F(x)| where :math:`F` is a (continuous) CDF and :math:`F_n` is an empirical CDF. `kstwo` describes the distribution under the null hypothesis of the KS test that the empirical CDF corresponds to :math:`n` i.i.d. random variates with CDF :math:`F`. %(after_notes)s See Also -------- kstwobign, ksone, kstest References ---------- .. [1] Simard, R., L'Ecuyer, P. "Computing the Two-Sided Kolmogorov-Smirnov Distribution", Journal of Statistical Software, Vol 39, 11, 1-18 (2011). %(example)s """ def _get_support(self, n): return (0.5/(n if not isinstance(n, Iterable) else np.asanyarray(n)), 1.0) def _pdf(self, x, n): return kolmognp(n, x) def _cdf(self, x, n): return kolmogn(n, x) def _sf(self, x, n): return kolmogn(n, x, cdf=False) def _ppf(self, q, n): return kolmogni(n, q, cdf=True) def _isf(self, q, n): return kolmogni(n, q, cdf=False) # Use the pdf, (not the ppf) to compute moments kstwo = kstwo_gen(momtype=0, a=0.0, b=1.0, name='kstwo') class kstwobign_gen(rv_continuous): r"""Limiting distribution of scaled Kolmogorov-Smirnov two-sided test statistic. This is the asymptotic distribution of the two-sided Kolmogorov-Smirnov statistic :math:`\sqrt{n} D_n` that measures the maximum absolute distance of the theoretical (continuous) CDF from the empirical CDF. (see `kstest`). %(before_notes)s Notes ----- :math:`\sqrt{n} D_n` is given by .. math:: D_n = \text{sup}_x |F_n(x) - F(x)| where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF. `kstwobign` describes the asymptotic distribution (i.e. the limit of :math:`\sqrt{n} D_n`) under the null hypothesis of the KS test that the empirical CDF corresponds to i.i.d. random variates with CDF :math:`F`. %(after_notes)s See Also -------- ksone, kstwo, kstest References ---------- .. [1] Feller, W. "On the Kolmogorov-Smirnov Limit Theorems for Empirical Distributions", Ann. Math. Statist. Vol 19, 177-189 (1948). %(example)s """ def _pdf(self, x): return -scu._kolmogp(x) def _cdf(self, x): return scu._kolmogc(x) def _sf(self, x): return sc.kolmogorov(x) def _ppf(self, q): return scu._kolmogci(q) def _isf(self, q): return sc.kolmogi(q) kstwobign = kstwobign_gen(a=0.0, name='kstwobign') ## Normal distribution # loc = mu, scale = std # Keep these implementations out of the class definition so they can be reused # by other distributions. _norm_pdf_C = np.sqrt(2*np.pi) _norm_pdf_logC = np.log(_norm_pdf_C) def _norm_pdf(x): return np.exp(-x**2/2.0) / _norm_pdf_C def _norm_logpdf(x): return -x**2 / 2.0 - _norm_pdf_logC def _norm_cdf(x): return sc.ndtr(x) def _norm_logcdf(x): return sc.log_ndtr(x) def _norm_ppf(q): return sc.ndtri(q) def _norm_sf(x): return _norm_cdf(-x) def _norm_logsf(x): return _norm_logcdf(-x) def _norm_isf(q): return -_norm_ppf(q) class norm_gen(rv_continuous): r"""A normal continuous random variable. The location (``loc``) keyword specifies the mean. The scale (``scale``) keyword specifies the standard deviation. %(before_notes)s Notes ----- The probability density function for `norm` is: .. math:: f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}} for a real number :math:`x`. %(after_notes)s %(example)s """ def _rvs(self, size=None, random_state=None): return random_state.standard_normal(size) def _pdf(self, x): # norm.pdf(x) = exp(-x**2/2)/sqrt(2*pi) return _norm_pdf(x) def _logpdf(self, x): return _norm_logpdf(x) def _cdf(self, x): return _norm_cdf(x) def _logcdf(self, x): return _norm_logcdf(x) def _sf(self, x): return _norm_sf(x) def _logsf(self, x): return _norm_logsf(x) def _ppf(self, q): return _norm_ppf(q) def _isf(self, q): return _norm_isf(q) def _stats(self): return 0.0, 1.0, 0.0, 0.0 def _entropy(self): return 0.5*(np.log(2*np.pi)+1) @replace_notes_in_docstring(rv_continuous, notes="""\ This function uses explicit formulas for the maximum likelihood estimation of the normal distribution parameters, so the `optimizer` argument is ignored.\n\n""") def fit(self, data, **kwds): floc = kwds.pop('floc', None) fscale = kwds.pop('fscale', None) _remove_optimizer_parameters(kwds) if floc is not None and fscale is not None: # This check is for consistency with `rv_continuous.fit`. # Without this check, this function would just return the # parameters that were given. raise ValueError("All parameters fixed. There is nothing to " "optimize.") data = np.asarray(data) if not np.isfinite(data).all(): raise RuntimeError("The data contains non-finite values.") if floc is None: loc = data.mean() else: loc = floc if fscale is None: scale = np.sqrt(((data - loc)**2).mean()) else: scale = fscale return loc, scale def _munp(self, n): """ @returns Moments of standard normal distribution for integer n >= 0 See eq. 16 of https://arxiv.org/abs/1209.4340v2 """ if n % 2 == 0: return sc.factorial2(n - 1) else: return 0. norm = norm_gen(name='norm') class alpha_gen(rv_continuous): r"""An alpha continuous random variable. %(before_notes)s Notes ----- The probability density function for `alpha` ([1]_, [2]_) is: .. math:: f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} * \exp(-\frac{1}{2} (a-1/x)^2) where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`. `alpha` takes ``a`` as a shape parameter. %(after_notes)s References ---------- .. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate Distributions, Volume 1", Second Edition, John Wiley and Sons, p. 173 (1994). .. [2] Anthony A. Salvia, "Reliability applications of the Alpha Distribution", IEEE Transactions on Reliability, Vol. R-34, No. 3, pp. 251-252 (1985). %(example)s """ _support_mask = rv_continuous._open_support_mask def _pdf(self, x, a): # alpha.pdf(x, a) = 1/(x**2*Phi(a)*sqrt(2*pi)) * exp(-1/2 * (a-1/x)**2) return 1.0/(x**2)/_norm_cdf(a)*_norm_pdf(a-1.0/x) def _logpdf(self, x, a): return -2*np.log(x) + _norm_logpdf(a-1.0/x) - np.log(_norm_cdf(a)) def _cdf(self, x, a): return _norm_cdf(a-1.0/x) / _norm_cdf(a) def _ppf(self, q, a): return 1.0/np.asarray(a-sc.ndtri(q*_norm_cdf(a))) def _stats(self, a): return [np.inf]*2 + [np.nan]*2 alpha = alpha_gen(a=0.0, name='alpha') class anglit_gen(rv_continuous): r"""An anglit continuous random variable. %(before_notes)s Notes ----- The probability density function for `anglit` is: .. math:: f(x) = \sin(2x + \pi/2) = \cos(2x) for :math:`-\pi/4 \le x \le \pi/4`. %(after_notes)s %(example)s """ def _pdf(self, x): # anglit.pdf(x) = sin(2*x + \pi/2) = cos(2*x) return np.cos(2*x) def _cdf(self, x): return np.sin(x+np.pi/4)**2.0 def _ppf(self, q): return np.arcsin(np.sqrt(q))-np.pi/4 def _stats(self): return 0.0, np.pi*np.pi/16-0.5, 0.0, -2*(np.pi**4 - 96)/(np.pi*np.pi-8)**2 def _entropy(self): return 1-np.log(2) anglit = anglit_gen(a=-np.pi/4, b=np.pi/4, name='anglit') class arcsine_gen(rv_continuous): r"""An arcsine continuous random variable. %(before_notes)s Notes ----- The probability density function for `arcsine` is: .. math:: f(x) = \frac{1}{\pi \sqrt{x (1-x)}} for :math:`0 < x < 1`. %(after_notes)s %(example)s """ def _pdf(self, x): # arcsine.pdf(x) = 1/(pi*sqrt(x*(1-x))) return 1.0/np.pi/np.sqrt(x*(1-x)) def _cdf(self, x): return 2.0/np.pi*np.arcsin(np.sqrt(x)) def _ppf(self, q): return np.sin(np.pi/2.0*q)**2.0 def _stats(self): mu = 0.5 mu2 = 1.0/8 g1 = 0 g2 = -3.0/2.0 return mu, mu2, g1, g2 def _entropy(self): return -0.24156447527049044468 arcsine = arcsine_gen(a=0.0, b=1.0, name='arcsine') class FitDataError(ValueError): # This exception is raised by, for example, beta_gen.fit when both floc # and fscale are fixed and there are values in the data not in the open # interval (floc, floc+fscale). def __init__(self, distr, lower, upper): self.args = ( "Invalid values in `data`. Maximum likelihood " "estimation with {distr!r} requires that {lower!r} < x " "< {upper!r} for each x in `data`.".format( distr=distr, lower=lower, upper=upper), ) class FitSolverError(RuntimeError): # This exception is raised by, for example, beta_gen.fit when # optimize.fsolve returns with ier != 1. def __init__(self, mesg): emsg = "Solver for the MLE equations failed to converge: " emsg += mesg.replace('\n', '') self.args = (emsg,) def _beta_mle_a(a, b, n, s1): # The zeros of this function give the MLE for `a`, with # `b`, `n` and `s1` given. `s1` is the sum of the logs of # the data. `n` is the number of data points. psiab = sc.psi(a + b) func = s1 - n * (-psiab + sc.psi(a)) return func def _beta_mle_ab(theta, n, s1, s2): # Zeros of this function are critical points of # the maximum likelihood function. Solving this system # for theta (which contains a and b) gives the MLE for a and b # given `n`, `s1` and `s2`. `s1` is the sum of the logs of the data, # and `s2` is the sum of the logs of 1 - data. `n` is the number # of data points. a, b = theta psiab = sc.psi(a + b) func = [s1 - n * (-psiab + sc.psi(a)), s2 - n * (-psiab + sc.psi(b))] return func class beta_gen(rv_continuous): r"""A beta continuous random variable. %(before_notes)s Notes ----- The probability density function for `beta` is: .. math:: f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}} {\Gamma(a) \Gamma(b)} for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `beta` takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s %(example)s """ def _rvs(self, a, b, size=None, random_state=None): return random_state.beta(a, b, size) def _pdf(self, x, a, b): # gamma(a+b) * x**(a-1) * (1-x)**(b-1) # beta.pdf(x, a, b) = ------------------------------------ # gamma(a)*gamma(b) return np.exp(self._logpdf(x, a, b)) def _logpdf(self, x, a, b): lPx = sc.xlog1py(b - 1.0, -x) + sc.xlogy(a - 1.0, x) lPx -= sc.betaln(a, b) return lPx def _cdf(self, x, a, b): return sc.btdtr(a, b, x) def _ppf(self, q, a, b): return sc.btdtri(a, b, q) def _stats(self, a, b): mn = a*1.0 / (a + b) var = (a*b*1.0)/(a+b+1.0)/(a+b)**2.0 g1 = 2.0*(b-a)*np.sqrt((1.0+a+b)/(a*b)) / (2+a+b) g2 = 6.0*(a**3 + a**2*(1-2*b) + b**2*(1+b) - 2*a*b*(2+b)) g2 /= a*b*(a+b+2)*(a+b+3) return mn, var, g1, g2 def _fitstart(self, data): g1 = _skew(data) g2 = _kurtosis(data) def func(x): a, b = x sk = 2*(b-a)*np.sqrt(a + b + 1) / (a + b + 2) / np.sqrt(a*b) ku = a**3 - a**2*(2*b-1) + b**2*(b+1) - 2*a*b*(b+2) ku /= a*b*(a+b+2)*(a+b+3) ku *= 6 return [sk-g1, ku-g2] a, b = optimize.fsolve(func, (1.0, 1.0)) return super(beta_gen, self)._fitstart(data, args=(a, b)) @extend_notes_in_docstring(rv_continuous, notes="""\ In the special case where both `floc` and `fscale` are given, a `ValueError` is raised if any value `x` in `data` does not satisfy `floc < x < floc + fscale`.\n\n""") def fit(self, data, *args, **kwds): # Override rv_continuous.fit, so we can more efficiently handle the # case where floc and fscale are given. floc = kwds.get('floc', None) fscale = kwds.get('fscale', None) if floc is None or fscale is None: # do general fit return super(beta_gen, self).fit(data, *args, **kwds) # We already got these from kwds, so just pop them. kwds.pop('floc', None) kwds.pop('fscale', None) f0 = _get_fixed_fit_value(kwds, ['f0', 'fa', 'fix_a']) f1 = _get_fixed_fit_value(kwds, ['f1', 'fb', 'fix_b']) _remove_optimizer_parameters(kwds) if f0 is not None and f1 is not None: # This check is for consistency with `rv_continuous.fit`. raise ValueError("All parameters fixed. There is nothing to " "optimize.") # Special case: loc and scale are constrained, so we are fitting # just the shape parameters. This can be done much more efficiently # than the method used in `rv_continuous.fit`. (See the subsection # "Two unknown parameters" in the section "Maximum likelihood" of # the Wikipedia article on the Beta distribution for the formulas.) if not np.isfinite(data).all(): raise RuntimeError("The data contains non-finite values.") # Normalize the data to the interval [0, 1]. data = (np.ravel(data) - floc) / fscale if np.any(data <= 0) or np.any(data >= 1): raise FitDataError("beta", lower=floc, upper=floc + fscale) xbar = data.mean() if f0 is not None or f1 is not None: # One of the shape parameters is fixed. if f0 is not None: # The shape parameter a is fixed, so swap the parameters # and flip the data. We always solve for `a`. The result # will be swapped back before returning. b = f0 data = 1 - data xbar = 1 - xbar else: b = f1 # Initial guess for a. Use the formula for the mean of the beta # distribution, E[x] = a / (a + b), to generate a reasonable # starting point based on the mean of the data and the given # value of b. a = b * xbar / (1 - xbar) # Compute the MLE for `a` by solving _beta_mle_a. theta, info, ier, mesg = optimize.fsolve( _beta_mle_a, a, args=(b, len(data), np.log(data).sum()), full_output=True ) if ier != 1: raise FitSolverError(mesg=mesg) a = theta[0] if f0 is not None: # The shape parameter a was fixed, so swap back the # parameters. a, b = b, a else: # Neither of the shape parameters is fixed. # s1 and s2 are used in the extra arguments passed to _beta_mle_ab # by optimize.fsolve. s1 = np.log(data).sum() s2 = sc.log1p(-data).sum() # Use the "method of moments" to estimate the initial # guess for a and b. fac = xbar * (1 - xbar) / data.var(ddof=0) - 1 a = xbar * fac b = (1 - xbar) * fac # Compute the MLE for a and b by solving _beta_mle_ab. theta, info, ier, mesg = optimize.fsolve( _beta_mle_ab, [a, b], args=(len(data), s1, s2), full_output=True ) if ier != 1: raise FitSolverError(mesg=mesg) a, b = theta return a, b, floc, fscale beta = beta_gen(a=0.0, b=1.0, name='beta') class betaprime_gen(rv_continuous): r"""A beta prime continuous random variable. %(before_notes)s Notes ----- The probability density function for `betaprime` is: .. math:: f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)} for :math:`x >= 0`, :math:`a > 0`, :math:`b > 0`, where :math:`\beta(a, b)` is the beta function (see `scipy.special.beta`). `betaprime` takes ``a`` and ``b`` as shape parameters. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _rvs(self, a, b, size=None, random_state=None): u1 = gamma.rvs(a, size=size, random_state=random_state) u2 = gamma.rvs(b, size=size, random_state=random_state) return u1 / u2 def _pdf(self, x, a, b): # betaprime.pdf(x, a, b) = x**(a-1) * (1+x)**(-a-b) / beta(a, b) return np.exp(self._logpdf(x, a, b)) def _logpdf(self, x, a, b): return sc.xlogy(a - 1.0, x) - sc.xlog1py(a + b, x) - sc.betaln(a, b) def _cdf(self, x, a, b): return sc.betainc(a, b, x/(1.+x)) def _munp(self, n, a, b): if n == 1.0: return np.where(b > 1, a/(b-1.0), np.inf) elif n == 2.0: return np.where(b > 2, a*(a+1.0)/((b-2.0)*(b-1.0)), np.inf) elif n == 3.0: return np.where(b > 3, a*(a+1.0)*(a+2.0)/((b-3.0)*(b-2.0)*(b-1.0)), np.inf) elif n == 4.0: return np.where(b > 4, (a*(a + 1.0)*(a + 2.0)*(a + 3.0) / ((b - 4.0)*(b - 3.0)*(b - 2.0)*(b - 1.0))), np.inf) else: raise NotImplementedError betaprime = betaprime_gen(a=0.0, name='betaprime') class bradford_gen(rv_continuous): r"""A Bradford continuous random variable. %(before_notes)s Notes ----- The probability density function for `bradford` is: .. math:: f(x, c) = \frac{c}{\log(1+c) (1+cx)} for :math:`0 <= x <= 1` and :math:`c > 0`. `bradford` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _pdf(self, x, c): # bradford.pdf(x, c) = c / (k * (1+c*x)) return c / (c*x + 1.0) / sc.log1p(c) def _cdf(self, x, c): return sc.log1p(c*x) / sc.log1p(c) def _ppf(self, q, c): return sc.expm1(q * sc.log1p(c)) / c def _stats(self, c, moments='mv'): k = np.log(1.0+c) mu = (c-k)/(c*k) mu2 = ((c+2.0)*k-2.0*c)/(2*c*k*k) g1 = None g2 = None if 's' in moments: g1 = np.sqrt(2)*(12*c*c-9*c*k*(c+2)+2*k*k*(c*(c+3)+3)) g1 /= np.sqrt(c*(c*(k-2)+2*k))*(3*c*(k-2)+6*k) if 'k' in moments: g2 = (c**3*(k-3)*(k*(3*k-16)+24)+12*k*c*c*(k-4)*(k-3) + 6*c*k*k*(3*k-14) + 12*k**3) g2 /= 3*c*(c*(k-2)+2*k)**2 return mu, mu2, g1, g2 def _entropy(self, c): k = np.log(1+c) return k/2.0 - np.log(c/k) bradford = bradford_gen(a=0.0, b=1.0, name='bradford') class burr_gen(rv_continuous): r"""A Burr (Type III) continuous random variable. %(before_notes)s See Also -------- fisk : a special case of either `burr` or `burr12` with ``d=1`` burr12 : Burr Type XII distribution mielke : Mielke Beta-Kappa / Dagum distribution Notes ----- The probability density function for `burr` is: .. math:: f(x, c, d) = c d x^{-c - 1} / (1 + x^{-c})^{d + 1} for :math:`x >= 0` and :math:`c, d > 0`. `burr` takes :math:`c` and :math:`d` as shape parameters. This is the PDF corresponding to the third CDF given in Burr's list; specifically, it is equation (11) in Burr's paper [1]_. The distribution is also commonly referred to as the Dagum distribution [2]_. If the parameter :math:`c < 1` then the mean of the distribution does not exist and if :math:`c < 2` the variance does not exist [2]_. The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 1`. %(after_notes)s References ---------- .. [1] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). .. [2] https://en.wikipedia.org/wiki/Dagum_distribution .. [3] Kleiber, Christian. "A guide to the Dagum distributions." Modeling Income Distributions and Lorenz Curves pp 97-117 (2008). %(example)s """ # Do not set _support_mask to rv_continuous._open_support_mask # Whether the left-hand endpoint is suitable for pdf evaluation is dependent # on the values of c and d: if c*d >= 1, the pdf is finite, otherwise infinite. def _pdf(self, x, c, d): # burr.pdf(x, c, d) = c * d * x**(-c-1) * (1+x**(-c))**(-d-1) output = _lazywhere(x == 0, [x, c, d], lambda x_, c_, d_: c_ * d_ * (x_**(c_*d_-1)) / (1 + x_**c_), f2 = lambda x_, c_, d_: (c_ * d_ * (x_ ** (-c_ - 1.0)) / ((1 + x_ ** (-c_)) ** (d_ + 1.0)))) if output.ndim == 0: return output[()] return output def _logpdf(self, x, c, d): output = _lazywhere( x == 0, [x, c, d], lambda x_, c_, d_: (np.log(c_) + np.log(d_) + sc.xlogy(c_*d_ - 1, x_) - (d_+1) * sc.log1p(x_**(c_))), f2 = lambda x_, c_, d_: (np.log(c_) + np.log(d_) + sc.xlogy(-c_ - 1, x_) - sc.xlog1py(d_+1, x_**(-c_)))) if output.ndim == 0: return output[()] return output def _cdf(self, x, c, d): return (1 + x**(-c))**(-d) def _logcdf(self, x, c, d): return sc.log1p(x**(-c)) * (-d) def _sf(self, x, c, d): return np.exp(self._logsf(x, c, d)) def _logsf(self, x, c, d): return np.log1p(- (1 + x**(-c))**(-d)) def _ppf(self, q, c, d): return (q**(-1.0/d) - 1)**(-1.0/c) def _stats(self, c, d): nc = np.arange(1, 5).reshape(4,1) / c #ek is the kth raw moment, e1 is the mean e2-e1**2 variance etc. e1, e2, e3, e4 = sc.beta(d + nc, 1. - nc) * d mu = np.where(c > 1.0, e1, np.nan) mu2_if_c = e2 - mu**2 mu2 = np.where(c > 2.0, mu2_if_c, np.nan) g1 = _lazywhere( c > 3.0, (c, e1, e2, e3, mu2_if_c), lambda c, e1, e2, e3, mu2_if_c: (e3 - 3*e2*e1 + 2*e1**3) / np.sqrt((mu2_if_c)**3), fillvalue=np.nan) g2 = _lazywhere( c > 4.0, (c, e1, e2, e3, e4, mu2_if_c), lambda c, e1, e2, e3, e4, mu2_if_c: ( ((e4 - 4*e3*e1 + 6*e2*e1**2 - 3*e1**4) / mu2_if_c**2) - 3), fillvalue=np.nan) return mu, mu2, g1, g2 def _munp(self, n, c, d): def __munp(n, c, d): nc = 1. * n / c return d * sc.beta(1.0 - nc, d + nc) n, c, d = np.asarray(n), np.asarray(c), np.asarray(d) return _lazywhere((c > n) & (n == n) & (d == d), (c, d, n), lambda c, d, n: __munp(n, c, d), np.nan) burr = burr_gen(a=0.0, name='burr') class burr12_gen(rv_continuous): r"""A Burr (Type XII) continuous random variable. %(before_notes)s See Also -------- fisk : a special case of either `burr` or `burr12` with ``d=1`` burr : Burr Type III distribution Notes ----- The probability density function for `burr` is: .. math:: f(x, c, d) = c d x^{c-1} / (1 + x^c)^{d + 1} for :math:`x >= 0` and :math:`c, d > 0`. `burr12` takes ``c`` and ``d`` as shape parameters for :math:`c` and :math:`d`. This is the PDF corresponding to the twelfth CDF given in Burr's list; specifically, it is equation (20) in Burr's paper [1]_. %(after_notes)s The Burr type 12 distribution is also sometimes referred to as the Singh-Maddala distribution from NIST [2]_. References ---------- .. [1] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). .. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm .. [3] "Burr distribution", https://en.wikipedia.org/wiki/Burr_distribution %(example)s """ def _pdf(self, x, c, d): # burr12.pdf(x, c, d) = c * d * x**(c-1) * (1+x**(c))**(-d-1) return np.exp(self._logpdf(x, c, d)) def _logpdf(self, x, c, d): return np.log(c) + np.log(d) + sc.xlogy(c - 1, x) + sc.xlog1py(-d-1, x**c) def _cdf(self, x, c, d): return -sc.expm1(self._logsf(x, c, d)) def _logcdf(self, x, c, d): return sc.log1p(-(1 + x**c)**(-d)) def _sf(self, x, c, d): return np.exp(self._logsf(x, c, d)) def _logsf(self, x, c, d): return sc.xlog1py(-d, x**c) def _ppf(self, q, c, d): # The following is an implementation of # ((1 - q)**(-1.0/d) - 1)**(1.0/c) # that does a better job handling small values of q. return sc.expm1(-1/d * sc.log1p(-q))**(1/c) def _munp(self, n, c, d): nc = 1. * n / c return d * sc.beta(1.0 + nc, d - nc) burr12 = burr12_gen(a=0.0, name='burr12') class fisk_gen(burr_gen): r"""A Fisk continuous random variable. The Fisk distribution is also known as the log-logistic distribution. %(before_notes)s Notes ----- The probability density function for `fisk` is: .. math:: f(x, c) = c x^{-c-1} (1 + x^{-c})^{-2} for :math:`x >= 0` and :math:`c > 0`. `fisk` takes ``c`` as a shape parameter for :math:`c`. `fisk` is a special case of `burr` or `burr12` with ``d=1``. %(after_notes)s See Also -------- burr %(example)s """ def _pdf(self, x, c): # fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2) return burr._pdf(x, c, 1.0) def _cdf(self, x, c): return burr._cdf(x, c, 1.0) def _sf(self, x, c): return burr._sf(x, c, 1.0) def _logpdf(self, x, c): # fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2) return burr._logpdf(x, c, 1.0) def _logcdf(self, x, c): return burr._logcdf(x, c, 1.0) def _logsf(self, x, c): return burr._logsf(x, c, 1.0) def _ppf(self, x, c): return burr._ppf(x, c, 1.0) def _munp(self, n, c): return burr._munp(n, c, 1.0) def _stats(self, c): return burr._stats(c, 1.0) def _entropy(self, c): return 2 - np.log(c) fisk = fisk_gen(a=0.0, name='fisk') # median = loc class cauchy_gen(rv_continuous): r"""A Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `cauchy` is .. math:: f(x) = \frac{1}{\pi (1 + x^2)} for a real number :math:`x`. %(after_notes)s %(example)s """ def _pdf(self, x): # cauchy.pdf(x) = 1 / (pi * (1 + x**2)) return 1.0/np.pi/(1.0+x*x) def _cdf(self, x): return 0.5 + 1.0/np.pi*np.arctan(x) def _ppf(self, q): return np.tan(np.pi*q-np.pi/2.0) def _sf(self, x): return 0.5 - 1.0/np.pi*np.arctan(x) def _isf(self, q): return np.tan(np.pi/2.0-np.pi*q) def _stats(self): return np.nan, np.nan, np.nan, np.nan def _entropy(self): return np.log(4*np.pi) def _fitstart(self, data, args=None): # Initialize ML guesses using quartiles instead of moments. p25, p50, p75 = np.percentile(data, [25, 50, 75]) return p50, (p75 - p25)/2 cauchy = cauchy_gen(name='cauchy') class chi_gen(rv_continuous): r"""A chi continuous random variable. %(before_notes)s Notes ----- The probability density function for `chi` is: .. math:: f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)} x^{k-1} \exp \left( -x^2/2 \right) for :math:`x >= 0` and :math:`k > 0` (degrees of freedom, denoted ``df`` in the implementation). :math:`\Gamma` is the gamma function (`scipy.special.gamma`). Special cases of `chi` are: - ``chi(1, loc, scale)`` is equivalent to `halfnorm` - ``chi(2, 0, scale)`` is equivalent to `rayleigh` - ``chi(3, 0, scale)`` is equivalent to `maxwell` `chi` takes ``df`` as a shape parameter. %(after_notes)s %(example)s """ def _rvs(self, df, size=None, random_state=None): return np.sqrt(chi2.rvs(df, size=size, random_state=random_state)) def _pdf(self, x, df): # x**(df-1) * exp(-x**2/2) # chi.pdf(x, df) = ------------------------- # 2**(df/2-1) * gamma(df/2) return np.exp(self._logpdf(x, df)) def _logpdf(self, x, df): l = np.log(2) - .5*np.log(2)*df - sc.gammaln(.5*df) return l + sc.xlogy(df - 1., x) - .5*x**2 def _cdf(self, x, df): return sc.gammainc(.5*df, .5*x**2) def _ppf(self, q, df): return np.sqrt(2*sc.gammaincinv(.5*df, q)) def _stats(self, df): mu = np.sqrt(2)*sc.gamma(df/2.0+0.5)/sc.gamma(df/2.0) mu2 = df - mu*mu g1 = (2*mu**3.0 + mu*(1-2*df))/np.asarray(np.power(mu2, 1.5)) g2 = 2*df*(1.0-df)-6*mu**4 + 4*mu**2 * (2*df-1) g2 /= np.asarray(mu2**2.0) return mu, mu2, g1, g2 chi = chi_gen(a=0.0, name='chi') ## Chi-squared (gamma-distributed with loc=0 and scale=2 and shape=df/2) class chi2_gen(rv_continuous): r"""A chi-squared continuous random variable. %(before_notes)s Notes ----- The probability density function for `chi2` is: .. math:: f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)} x^{k/2-1} \exp \left( -x/2 \right) for :math:`x > 0` and :math:`k > 0` (degrees of freedom, denoted ``df`` in the implementation). `chi2` takes ``df`` as a shape parameter. %(after_notes)s %(example)s """ def _rvs(self, df, size=None, random_state=None): return random_state.chisquare(df, size) def _pdf(self, x, df): # chi2.pdf(x, df) = 1 / (2*gamma(df/2)) * (x/2)**(df/2-1) * exp(-x/2) return np.exp(self._logpdf(x, df)) def _logpdf(self, x, df): return sc.xlogy(df/2.-1, x) - x/2. - sc.gammaln(df/2.) - (np.log(2)*df)/2. def _cdf(self, x, df): return sc.chdtr(df, x) def _sf(self, x, df): return sc.chdtrc(df, x) def _isf(self, p, df): return sc.chdtri(df, p) def _ppf(self, p, df): return 2*sc.gammaincinv(df/2, p) def _stats(self, df): mu = df mu2 = 2*df g1 = 2*np.sqrt(2.0/df) g2 = 12.0/df return mu, mu2, g1, g2 chi2 = chi2_gen(a=0.0, name='chi2') class cosine_gen(rv_continuous): r"""A cosine continuous random variable. %(before_notes)s Notes ----- The cosine distribution is an approximation to the normal distribution. The probability density function for `cosine` is: .. math:: f(x) = \frac{1}{2\pi} (1+\cos(x)) for :math:`-\pi \le x \le \pi`. %(after_notes)s %(example)s """ def _pdf(self, x): # cosine.pdf(x) = 1/(2*pi) * (1+cos(x)) return 1.0/2/np.pi*(1+np.cos(x)) def _cdf(self, x): return 1.0/2/np.pi*(np.pi + x + np.sin(x)) def _stats(self): return 0.0, np.pi*np.pi/3.0-2.0, 0.0, -6.0*(np.pi**4-90)/(5.0*(np.pi*np.pi-6)**2) def _entropy(self): return np.log(4*np.pi)-1.0 cosine = cosine_gen(a=-np.pi, b=np.pi, name='cosine') class dgamma_gen(rv_continuous): r"""A double gamma continuous random variable. %(before_notes)s Notes ----- The probability density function for `dgamma` is: .. math:: f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|) for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `dgamma` takes ``a`` as a shape parameter for :math:`a`. %(after_notes)s %(example)s """ def _rvs(self, a, size=None, random_state=None): u = random_state.uniform(size=size) gm = gamma.rvs(a, size=size, random_state=random_state) return gm * np.where(u >= 0.5, 1, -1) def _pdf(self, x, a): # dgamma.pdf(x, a) = 1 / (2*gamma(a)) * abs(x)**(a-1) * exp(-abs(x)) ax = abs(x) return 1.0/(2*sc.gamma(a))*ax**(a-1.0) * np.exp(-ax) def _logpdf(self, x, a): ax = abs(x) return sc.xlogy(a - 1.0, ax) - ax - np.log(2) - sc.gammaln(a) def _cdf(self, x, a): fac = 0.5*sc.gammainc(a, abs(x)) return np.where(x > 0, 0.5 + fac, 0.5 - fac) def _sf(self, x, a): fac = 0.5*sc.gammainc(a, abs(x)) return np.where(x > 0, 0.5-fac, 0.5+fac) def _ppf(self, q, a): fac = sc.gammainccinv(a, 1-abs(2*q-1)) return np.where(q > 0.5, fac, -fac) def _stats(self, a): mu2 = a*(a+1.0) return 0.0, mu2, 0.0, (a+2.0)*(a+3.0)/mu2-3.0 dgamma = dgamma_gen(name='dgamma') class dweibull_gen(rv_continuous): r"""A double Weibull continuous random variable. %(before_notes)s Notes ----- The probability density function for `dweibull` is given by .. math:: f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c) for a real number :math:`x` and :math:`c > 0`. `dweibull` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _rvs(self, c, size=None, random_state=None): u = random_state.uniform(size=size) w = weibull_min.rvs(c, size=size, random_state=random_state) return w * (np.where(u >= 0.5, 1, -1)) def _pdf(self, x, c): # dweibull.pdf(x, c) = c / 2 * abs(x)**(c-1) * exp(-abs(x)**c) ax = abs(x) Px = c / 2.0 * ax**(c-1.0) * np.exp(-ax**c) return Px def _logpdf(self, x, c): ax = abs(x) return np.log(c) - np.log(2.0) + sc.xlogy(c - 1.0, ax) - ax**c def _cdf(self, x, c): Cx1 = 0.5 * np.exp(-abs(x)**c) return np.where(x > 0, 1 - Cx1, Cx1) def _ppf(self, q, c): fac = 2. * np.where(q <= 0.5, q, 1. - q) fac = np.power(-np.log(fac), 1.0 / c) return np.where(q > 0.5, fac, -fac) def _munp(self, n, c): return (1 - (n % 2)) * sc.gamma(1.0 + 1.0 * n / c) # since we know that all odd moments are zeros, return them at once. # returning Nones from _stats makes the public stats call _munp # so overall we're saving one or two gamma function evaluations here. def _stats(self, c): return 0, None, 0, None dweibull = dweibull_gen(name='dweibull') ## Exponential (gamma distributed with a=1.0, loc=loc and scale=scale) class expon_gen(rv_continuous): r"""An exponential continuous random variable. %(before_notes)s Notes ----- The probability density function for `expon` is: .. math:: f(x) = \exp(-x) for :math:`x \ge 0`. %(after_notes)s A common parameterization for `expon` is in terms of the rate parameter ``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This parameterization corresponds to using ``scale = 1 / lambda``. %(example)s """ def _rvs(self, size=None, random_state=None): return random_state.standard_exponential(size) def _pdf(self, x): # expon.pdf(x) = exp(-x) return np.exp(-x) def _logpdf(self, x): return -x def _cdf(self, x): return -sc.expm1(-x) def _ppf(self, q): return -sc.log1p(-q) def _sf(self, x): return np.exp(-x) def _logsf(self, x): return -x def _isf(self, q): return -np.log(q) def _stats(self): return 1.0, 1.0, 2.0, 6.0 def _entropy(self): return 1.0 @replace_notes_in_docstring(rv_continuous, notes="""\ This function uses explicit formulas for the maximum likelihood estimation of the exponential distribution parameters, so the `optimizer`, `loc` and `scale` keyword arguments are ignored.\n\n""") def fit(self, data, *args, **kwds): if len(args) > 0: raise TypeError("Too many arguments.") floc = kwds.pop('floc', None) fscale = kwds.pop('fscale', None) _remove_optimizer_parameters(kwds) if floc is not None and fscale is not None: # This check is for consistency with `rv_continuous.fit`. raise ValueError("All parameters fixed. There is nothing to " "optimize.") data = np.asarray(data) if not np.isfinite(data).all(): raise RuntimeError("The data contains non-finite values.") data_min = data.min() if floc is None: # ML estimate of the location is the minimum of the data. loc = data_min else: loc = floc if data_min < loc: # There are values that are less than the specified loc. raise FitDataError("expon", lower=floc, upper=np.inf) if fscale is None: # ML estimate of the scale is the shifted mean. scale = data.mean() - loc else: scale = fscale # We expect the return values to be floating point, so ensure it # by explicitly converting to float. return float(loc), float(scale) expon = expon_gen(a=0.0, name='expon') ## Exponentially Modified Normal (exponential distribution ## convolved with a Normal). ## This is called an exponentially modified gaussian on wikipedia class exponnorm_gen(rv_continuous): r"""An exponentially modified Normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `exponnorm` is: .. math:: f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right) \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right) where :math:`x` is a real number and :math:`K > 0`. It can be thought of as the sum of a standard normal random variable and an independent exponentially distributed random variable with rate ``1/K``. %(after_notes)s An alternative parameterization of this distribution (for example, in `Wikipedia `_) involves three parameters, :math:`\mu`, :math:`\lambda` and :math:`\sigma`. In the present parameterization this corresponds to having ``loc`` and ``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and shape parameter :math:`K = 1/(\sigma\lambda)`. .. versionadded:: 0.16.0 %(example)s """ def _rvs(self, K, size=None, random_state=None): expval = random_state.standard_exponential(size) * K gval = random_state.standard_normal(size) return expval + gval def _pdf(self, x, K): # exponnorm.pdf(x, K) = # 1/(2*K) exp(1/(2 * K**2)) exp(-x / K) * erfc-(x - 1/K) / sqrt(2)) invK = 1.0 / K exparg = 0.5 * invK**2 - invK * x # Avoid overflows; setting np.exp(exparg) to the max float works # all right here expval = _lazywhere(exparg < _LOGXMAX, (exparg,), np.exp, _XMAX) return 0.5 * invK * (expval * sc.erfc(-(x - invK) / np.sqrt(2))) def _logpdf(self, x, K): invK = 1.0 / K exparg = 0.5 * invK**2 - invK * x return exparg + np.log(0.5 * invK * sc.erfc(-(x - invK) / np.sqrt(2))) def _cdf(self, x, K): invK = 1.0 / K expval = invK * (0.5 * invK - x) return _norm_cdf(x) - np.exp(expval) * _norm_cdf(x - invK) def _sf(self, x, K): invK = 1.0 / K expval = invK * (0.5 * invK - x) return _norm_cdf(-x) + np.exp(expval) * _norm_cdf(x - invK) def _stats(self, K): K2 = K * K opK2 = 1.0 + K2 skw = 2 * K**3 * opK2**(-1.5) krt = 6.0 * K2 * K2 * opK2**(-2) return K, opK2, skw, krt exponnorm = exponnorm_gen(name='exponnorm') class exponweib_gen(rv_continuous): r"""An exponentiated Weibull continuous random variable. %(before_notes)s See Also -------- weibull_min, numpy.random.RandomState.weibull Notes ----- The probability density function for `exponweib` is: .. math:: f(x, a, c) = a c [1-\exp(-x^c)]^{a-1} \exp(-x^c) x^{c-1} and its cumulative distribution function is: .. math:: F(x, a, c) = [1-\exp(-x^c)]^a for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`. `exponweib` takes :math:`a` and :math:`c` as shape parameters: * :math:`a` is the exponentiation parameter, with the special case :math:`a=1` corresponding to the (non-exponentiated) Weibull distribution `weibull_min`. * :math:`c` is the shape parameter of the non-exponentiated Weibull law. %(after_notes)s References ---------- https://en.wikipedia.org/wiki/Exponentiated_Weibull_distribution %(example)s """ def _pdf(self, x, a, c): # exponweib.pdf(x, a, c) = # a * c * (1-exp(-x**c))**(a-1) * exp(-x**c)*x**(c-1) return np.exp(self._logpdf(x, a, c)) def _logpdf(self, x, a, c): negxc = -x**c exm1c = -sc.expm1(negxc) logp = (np.log(a) + np.log(c) + sc.xlogy(a - 1.0, exm1c) + negxc + sc.xlogy(c - 1.0, x)) return logp def _cdf(self, x, a, c): exm1c = -sc.expm1(-x**c) return exm1c**a def _ppf(self, q, a, c): return (-sc.log1p(-q**(1.0/a)))**np.asarray(1.0/c) exponweib = exponweib_gen(a=0.0, name='exponweib') class exponpow_gen(rv_continuous): r"""An exponential power continuous random variable. %(before_notes)s Notes ----- The probability density function for `exponpow` is: .. math:: f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b)) for :math:`x \ge 0`, :math:`b > 0`. Note that this is a different distribution from the exponential power distribution that is also known under the names "generalized normal" or "generalized Gaussian". `exponpow` takes ``b`` as a shape parameter for :math:`b`. %(after_notes)s References ---------- http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf %(example)s """ def _pdf(self, x, b): # exponpow.pdf(x, b) = b * x**(b-1) * exp(1 + x**b - exp(x**b)) return np.exp(self._logpdf(x, b)) def _logpdf(self, x, b): xb = x**b f = 1 + np.log(b) + sc.xlogy(b - 1.0, x) + xb - np.exp(xb) return f def _cdf(self, x, b): return -sc.expm1(-sc.expm1(x**b)) def _sf(self, x, b): return np.exp(-sc.expm1(x**b)) def _isf(self, x, b): return (sc.log1p(-np.log(x)))**(1./b) def _ppf(self, q, b): return pow(sc.log1p(-sc.log1p(-q)), 1.0/b) exponpow = exponpow_gen(a=0.0, name='exponpow') class fatiguelife_gen(rv_continuous): r"""A fatigue-life (Birnbaum-Saunders) continuous random variable. %(before_notes)s Notes ----- The probability density function for `fatiguelife` is: .. math:: f(x, c) = \frac{x+1}{2c\sqrt{2\pi x^3}} \exp(-\frac{(x-1)^2}{2x c^2}) for :math:`x >= 0` and :math:`c > 0`. `fatiguelife` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- .. [1] "Birnbaum-Saunders distribution", https://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution %(example)s """ _support_mask = rv_continuous._open_support_mask def _rvs(self, c, size=None, random_state=None): z = random_state.standard_normal(size) x = 0.5*c*z x2 = x*x t = 1.0 + 2*x2 + 2*x*np.sqrt(1 + x2) return t def _pdf(self, x, c): # fatiguelife.pdf(x, c) = # (x+1) / (2*c*sqrt(2*pi*x**3)) * exp(-(x-1)**2/(2*x*c**2)) return np.exp(self._logpdf(x, c)) def _logpdf(self, x, c): return (np.log(x+1) - (x-1)**2 / (2.0*x*c**2) - np.log(2*c) - 0.5*(np.log(2*np.pi) + 3*np.log(x))) def _cdf(self, x, c): return _norm_cdf(1.0 / c * (np.sqrt(x) - 1.0/np.sqrt(x))) def _ppf(self, q, c): tmp = c*sc.ndtri(q) return 0.25 * (tmp + np.sqrt(tmp**2 + 4))**2 def _stats(self, c): # NB: the formula for kurtosis in wikipedia seems to have an error: # it's 40, not 41. At least it disagrees with the one from Wolfram # Alpha. And the latter one, below, passes the tests, while the wiki # one doesn't So far I didn't have the guts to actually check the # coefficients from the expressions for the raw moments. c2 = c*c mu = c2 / 2.0 + 1.0 den = 5.0 * c2 + 4.0 mu2 = c2*den / 4.0 g1 = 4 * c * (11*c2 + 6.0) / np.power(den, 1.5) g2 = 6 * c2 * (93*c2 + 40.0) / den**2.0 return mu, mu2, g1, g2 fatiguelife = fatiguelife_gen(a=0.0, name='fatiguelife') class foldcauchy_gen(rv_continuous): r"""A folded Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `foldcauchy` is: .. math:: f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)} for :math:`x \ge 0`. `foldcauchy` takes ``c`` as a shape parameter for :math:`c`. %(example)s """ def _rvs(self, c, size=None, random_state=None): return abs(cauchy.rvs(loc=c, size=size, random_state=random_state)) def _pdf(self, x, c): # foldcauchy.pdf(x, c) = 1/(pi*(1+(x-c)**2)) + 1/(pi*(1+(x+c)**2)) return 1.0/np.pi*(1.0/(1+(x-c)**2) + 1.0/(1+(x+c)**2)) def _cdf(self, x, c): return 1.0/np.pi*(np.arctan(x-c) + np.arctan(x+c)) def _stats(self, c): return np.inf, np.inf, np.nan, np.nan foldcauchy = foldcauchy_gen(a=0.0, name='foldcauchy') class f_gen(rv_continuous): r"""An F continuous random variable. %(before_notes)s Notes ----- The probability density function for `f` is: .. math:: f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}} {(df_2+df_1 x)^{(df_1+df_2)/2} B(df_1/2, df_2/2)} for :math:`x > 0`. `f` takes ``dfn`` and ``dfd`` as shape parameters. %(after_notes)s %(example)s """ def _rvs(self, dfn, dfd, size=None, random_state=None): return random_state.f(dfn, dfd, size) def _pdf(self, x, dfn, dfd): # df2**(df2/2) * df1**(df1/2) * x**(df1/2-1) # F.pdf(x, df1, df2) = -------------------------------------------- # (df2+df1*x)**((df1+df2)/2) * B(df1/2, df2/2) return np.exp(self._logpdf(x, dfn, dfd)) def _logpdf(self, x, dfn, dfd): n = 1.0 * dfn m = 1.0 * dfd lPx = m/2 * np.log(m) + n/2 * np.log(n) + sc.xlogy(n/2 - 1, x) lPx -= ((n+m)/2) * np.log(m + n*x) + sc.betaln(n/2, m/2) return lPx def _cdf(self, x, dfn, dfd): return sc.fdtr(dfn, dfd, x) def _sf(self, x, dfn, dfd): return sc.fdtrc(dfn, dfd, x) def _ppf(self, q, dfn, dfd): return sc.fdtri(dfn, dfd, q) def _stats(self, dfn, dfd): v1, v2 = 1. * dfn, 1. * dfd v2_2, v2_4, v2_6, v2_8 = v2 - 2., v2 - 4., v2 - 6., v2 - 8. mu = _lazywhere( v2 > 2, (v2, v2_2), lambda v2, v2_2: v2 / v2_2, np.inf) mu2 = _lazywhere( v2 > 4, (v1, v2, v2_2, v2_4), lambda v1, v2, v2_2, v2_4: 2 * v2 * v2 * (v1 + v2_2) / (v1 * v2_2**2 * v2_4), np.inf) g1 = _lazywhere( v2 > 6, (v1, v2_2, v2_4, v2_6), lambda v1, v2_2, v2_4, v2_6: (2 * v1 + v2_2) / v2_6 * np.sqrt(v2_4 / (v1 * (v1 + v2_2))), np.nan) g1 *= np.sqrt(8.) g2 = _lazywhere( v2 > 8, (g1, v2_6, v2_8), lambda g1, v2_6, v2_8: (8 + g1 * g1 * v2_6) / v2_8, np.nan) g2 *= 3. / 2. return mu, mu2, g1, g2 f = f_gen(a=0.0, name='f') ## Folded Normal ## abs(Z) where (Z is normal with mu=L and std=S so that c=abs(L)/S) ## ## note: regress docs have scale parameter correct, but first parameter ## he gives is a shape parameter A = c * scale ## Half-normal is folded normal with shape-parameter c=0. class foldnorm_gen(rv_continuous): r"""A folded normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `foldnorm` is: .. math:: f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2}) for :math:`c \ge 0`. `foldnorm` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _argcheck(self, c): return c >= 0 def _rvs(self, c, size=None, random_state=None): return abs(random_state.standard_normal(size) + c) def _pdf(self, x, c): # foldnormal.pdf(x, c) = sqrt(2/pi) * cosh(c*x) * exp(-(x**2+c**2)/2) return _norm_pdf(x + c) + _norm_pdf(x-c) def _cdf(self, x, c): return _norm_cdf(x-c) + _norm_cdf(x+c) - 1.0 def _stats(self, c): # Regina C. Elandt, Technometrics 3, 551 (1961) # https://www.jstor.org/stable/1266561 # c2 = c*c expfac = np.exp(-0.5*c2) / np.sqrt(2.*np.pi) mu = 2.*expfac + c * sc.erf(c/np.sqrt(2)) mu2 = c2 + 1 - mu*mu g1 = 2. * (mu*mu*mu - c2*mu - expfac) g1 /= np.power(mu2, 1.5) g2 = c2 * (c2 + 6.) + 3 + 8.*expfac*mu g2 += (2. * (c2 - 3.) - 3. * mu**2) * mu**2 g2 = g2 / mu2**2.0 - 3. return mu, mu2, g1, g2 foldnorm = foldnorm_gen(a=0.0, name='foldnorm') class weibull_min_gen(rv_continuous): r"""Weibull minimum continuous random variable. The Weibull Minimum Extreme Value distribution, from extreme value theory (Fisher-Gnedenko theorem), is also often simply called the Weibull distribution. It arises as the limiting distribution of the rescaled minimum of iid random variables. %(before_notes)s See Also -------- weibull_max, numpy.random.RandomState.weibull, exponweib Notes ----- The probability density function for `weibull_min` is: .. math:: f(x, c) = c x^{c-1} \exp(-x^c) for :math:`x > 0`, :math:`c > 0`. `weibull_min` takes ``c`` as a shape parameter for :math:`c`. (named :math:`k` in Wikipedia article and :math:`a` in ``numpy.random.weibull``). Special shape values are :math:`c=1` and :math:`c=2` where Weibull distribution reduces to the `expon` and `rayleigh` distributions respectively. %(after_notes)s References ---------- https://en.wikipedia.org/wiki/Weibull_distribution https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem %(example)s """ def _pdf(self, x, c): # frechet_r.pdf(x, c) = c * x**(c-1) * exp(-x**c) return c*pow(x, c-1)*np.exp(-pow(x, c)) def _logpdf(self, x, c): return np.log(c) + sc.xlogy(c - 1, x) - pow(x, c) def _cdf(self, x, c): return -sc.expm1(-pow(x, c)) def _sf(self, x, c): return np.exp(-pow(x, c)) def _logsf(self, x, c): return -pow(x, c) def _ppf(self, q, c): return pow(-sc.log1p(-q), 1.0/c) def _munp(self, n, c): return sc.gamma(1.0+n*1.0/c) def _entropy(self, c): return -_EULER / c - np.log(c) + _EULER + 1 weibull_min = weibull_min_gen(a=0.0, name='weibull_min') class weibull_max_gen(rv_continuous): r"""Weibull maximum continuous random variable. The Weibull Maximum Extreme Value distribution, from extreme value theory (Fisher-Gnedenko theorem), is the limiting distribution of rescaled maximum of iid random variables. This is the distribution of -X if X is from the `weibull_min` function. %(before_notes)s See Also -------- weibull_min Notes ----- The probability density function for `weibull_max` is: .. math:: f(x, c) = c (-x)^{c-1} \exp(-(-x)^c) for :math:`x < 0`, :math:`c > 0`. `weibull_max` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- https://en.wikipedia.org/wiki/Weibull_distribution https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem %(example)s """ def _pdf(self, x, c): # frechet_l.pdf(x, c) = c * (-x)**(c-1) * exp(-(-x)**c) return c*pow(-x, c-1)*np.exp(-pow(-x, c)) def _logpdf(self, x, c): return np.log(c) + sc.xlogy(c-1, -x) - pow(-x, c) def _cdf(self, x, c): return np.exp(-pow(-x, c)) def _logcdf(self, x, c): return -pow(-x, c) def _sf(self, x, c): return -sc.expm1(-pow(-x, c)) def _ppf(self, q, c): return -pow(-np.log(q), 1.0/c) def _munp(self, n, c): val = sc.gamma(1.0+n*1.0/c) if int(n) % 2: sgn = -1 else: sgn = 1 return sgn * val def _entropy(self, c): return -_EULER / c - np.log(c) + _EULER + 1 weibull_max = weibull_max_gen(b=0.0, name='weibull_max') # Public methods to be deprecated in frechet_r and frechet_l: # ['__call__', 'cdf', 'entropy', 'expect', 'fit', 'fit_loc_scale', 'freeze', # 'interval', 'isf', 'logcdf', 'logpdf', 'logsf', 'mean', 'median', 'moment', # 'nnlf', 'pdf', 'ppf', 'rvs', 'sf', 'stats', 'std', 'var'] _frechet_r_deprec_msg = """\ The distribution `frechet_r` is a synonym for `weibull_min`; this historical usage is deprecated because of possible confusion with the (quite different) Frechet distribution. To preserve the existing behavior of the program, use `scipy.stats.weibull_min`. For the Frechet distribution (i.e. the Type II extreme value distribution), use `scipy.stats.invweibull`.""" class frechet_r_gen(weibull_min_gen): """A Frechet right (or Weibull minimum) continuous random variable. %(before_notes)s See Also -------- weibull_min : The same distribution as `frechet_r`. Notes ----- %(after_notes)s %(example)s """ @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def __call__(self, *args, **kwargs): return weibull_min_gen.__call__(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def cdf(self, *args, **kwargs): return weibull_min_gen.cdf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def entropy(self, *args, **kwargs): return weibull_min_gen.entropy(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def expect(self, *args, **kwargs): return weibull_min_gen.expect(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def fit(self, *args, **kwargs): return weibull_min_gen.fit(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def fit_loc_scale(self, *args, **kwargs): return weibull_min_gen.fit_loc_scale(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def freeze(self, *args, **kwargs): return weibull_min_gen.freeze(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def interval(self, *args, **kwargs): return weibull_min_gen.interval(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def isf(self, *args, **kwargs): return weibull_min_gen.isf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def logcdf(self, *args, **kwargs): return weibull_min_gen.logcdf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def logpdf(self, *args, **kwargs): return weibull_min_gen.logpdf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def logsf(self, *args, **kwargs): return weibull_min_gen.logsf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def mean(self, *args, **kwargs): return weibull_min_gen.mean(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def median(self, *args, **kwargs): return weibull_min_gen.median(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def moment(self, *args, **kwargs): return weibull_min_gen.moment(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def nnlf(self, *args, **kwargs): return weibull_min_gen.nnlf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def pdf(self, *args, **kwargs): return weibull_min_gen.pdf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def ppf(self, *args, **kwargs): return weibull_min_gen.ppf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def rvs(self, *args, **kwargs): return weibull_min_gen.rvs(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def sf(self, *args, **kwargs): return weibull_min_gen.sf(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def stats(self, *args, **kwargs): return weibull_min_gen.stats(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def std(self, *args, **kwargs): return weibull_min_gen.std(self, *args, **kwargs) @np.deprecate(old_name='frechet_r', message=_frechet_r_deprec_msg) def var(self, *args, **kwargs): return weibull_min_gen.var(self, *args, **kwargs) frechet_r = frechet_r_gen(a=0.0, name='frechet_r') _frechet_l_deprec_msg = """\ The distribution `frechet_l` is a synonym for `weibull_max`; this historical usage is deprecated because of possible confusion with the (quite different) Frechet distribution. To preserve the existing behavior of the program, use `scipy.stats.weibull_max`. For the Frechet distribution (i.e. the Type II extreme value distribution), use `scipy.stats.invweibull`.""" class frechet_l_gen(weibull_max_gen): """A Frechet left (or Weibull maximum) continuous random variable. %(before_notes)s See Also -------- weibull_max : The same distribution as `frechet_l`. Notes ----- %(after_notes)s %(example)s """ @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def __call__(self, *args, **kwargs): return weibull_max_gen.__call__(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def cdf(self, *args, **kwargs): return weibull_max_gen.cdf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def entropy(self, *args, **kwargs): return weibull_max_gen.entropy(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def expect(self, *args, **kwargs): return weibull_max_gen.expect(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def fit(self, *args, **kwargs): return weibull_max_gen.fit(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def fit_loc_scale(self, *args, **kwargs): return weibull_max_gen.fit_loc_scale(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def freeze(self, *args, **kwargs): return weibull_max_gen.freeze(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def interval(self, *args, **kwargs): return weibull_max_gen.interval(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def isf(self, *args, **kwargs): return weibull_max_gen.isf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def logcdf(self, *args, **kwargs): return weibull_max_gen.logcdf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def logpdf(self, *args, **kwargs): return weibull_max_gen.logpdf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def logsf(self, *args, **kwargs): return weibull_max_gen.logsf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def mean(self, *args, **kwargs): return weibull_max_gen.mean(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def median(self, *args, **kwargs): return weibull_max_gen.median(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def moment(self, *args, **kwargs): return weibull_max_gen.moment(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def nnlf(self, *args, **kwargs): return weibull_max_gen.nnlf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def pdf(self, *args, **kwargs): return weibull_max_gen.pdf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def ppf(self, *args, **kwargs): return weibull_max_gen.ppf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def rvs(self, *args, **kwargs): return weibull_max_gen.rvs(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def sf(self, *args, **kwargs): return weibull_max_gen.sf(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def stats(self, *args, **kwargs): return weibull_max_gen.stats(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def std(self, *args, **kwargs): return weibull_max_gen.std(self, *args, **kwargs) @np.deprecate(old_name='frechet_l', message=_frechet_l_deprec_msg) def var(self, *args, **kwargs): return weibull_max_gen.var(self, *args, **kwargs) frechet_l = frechet_l_gen(b=0.0, name='frechet_l') class genlogistic_gen(rv_continuous): r"""A generalized logistic continuous random variable. %(before_notes)s Notes ----- The probability density function for `genlogistic` is: .. math:: f(x, c) = c \frac{\exp(-x)} {(1 + \exp(-x))^{c+1}} for :math:`x >= 0`, :math:`c > 0`. `genlogistic` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _pdf(self, x, c): # genlogistic.pdf(x, c) = c * exp(-x) / (1 + exp(-x))**(c+1) return np.exp(self._logpdf(x, c)) def _logpdf(self, x, c): return np.log(c) - x - (c+1.0)*sc.log1p(np.exp(-x)) def _cdf(self, x, c): Cx = (1+np.exp(-x))**(-c) return Cx def _ppf(self, q, c): vals = -np.log(pow(q, -1.0/c)-1) return vals def _stats(self, c): mu = _EULER + sc.psi(c) mu2 = np.pi*np.pi/6.0 + sc.zeta(2, c) g1 = -2*sc.zeta(3, c) + 2*_ZETA3 g1 /= np.power(mu2, 1.5) g2 = np.pi**4/15.0 + 6*sc.zeta(4, c) g2 /= mu2**2.0 return mu, mu2, g1, g2 genlogistic = genlogistic_gen(name='genlogistic') class genpareto_gen(rv_continuous): r"""A generalized Pareto continuous random variable. %(before_notes)s Notes ----- The probability density function for `genpareto` is: .. math:: f(x, c) = (1 + c x)^{-1 - 1/c} defined for :math:`x \ge 0` if :math:`c \ge 0`, and for :math:`0 \le x \le -1/c` if :math:`c < 0`. `genpareto` takes ``c`` as a shape parameter for :math:`c`. For :math:`c=0`, `genpareto` reduces to the exponential distribution, `expon`: .. math:: f(x, 0) = \exp(-x) For :math:`c=-1`, `genpareto` is uniform on ``[0, 1]``: .. math:: f(x, -1) = 1 %(after_notes)s %(example)s """ def _argcheck(self, c): return np.isfinite(c) def _get_support(self, c): c = np.asarray(c) b = _lazywhere(c < 0, (c,), lambda c: -1. / c, np.inf) a = np.where(c >= 0, self.a, self.a) return a, b def _pdf(self, x, c): # genpareto.pdf(x, c) = (1 + c * x)**(-1 - 1/c) return np.exp(self._logpdf(x, c)) def _logpdf(self, x, c): return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: -sc.xlog1py(c + 1., c*x) / c, -x) def _cdf(self, x, c): return -sc.inv_boxcox1p(-x, -c) def _sf(self, x, c): return sc.inv_boxcox(-x, -c) def _logsf(self, x, c): return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: -sc.log1p(c*x) / c, -x) def _ppf(self, q, c): return -sc.boxcox1p(-q, -c) def _isf(self, q, c): return -sc.boxcox(q, -c) def _stats(self, c, moments='mv'): if 'm' not in moments: m = None else: m = _lazywhere(c < 1, (c,), lambda xi: 1/(1 - xi), np.inf) if 'v' not in moments: v = None else: v = _lazywhere(c < 1/2, (c,), lambda xi: 1 / (1 - xi)**2 / (1 - 2*xi), np.nan) if 's' not in moments: s = None else: s = _lazywhere(c < 1/3, (c,), lambda xi: 2 * (1 + xi) * np.sqrt(1 - 2*xi) / (1 - 3*xi), np.nan) if 'k' not in moments: k = None else: k = _lazywhere(c < 1/4, (c,), lambda xi: 3 * (1 - 2*xi) * (2*xi**2 + xi + 3) / (1 - 3*xi) / (1 - 4*xi) - 3, np.nan) return m, v, s, k def _munp(self, n, c): def __munp(n, c): val = 0.0 k = np.arange(0, n + 1) for ki, cnk in zip(k, sc.comb(n, k)): val = val + cnk * (-1) ** ki / (1.0 - c * ki) return np.where(c * n < 1, val * (-1.0 / c) ** n, np.inf) return _lazywhere(c != 0, (c,), lambda c: __munp(n, c), sc.gamma(n + 1)) def _entropy(self, c): return 1. + c genpareto = genpareto_gen(a=0.0, name='genpareto') class genexpon_gen(rv_continuous): r"""A generalized exponential continuous random variable. %(before_notes)s Notes ----- The probability density function for `genexpon` is: .. math:: f(x, a, b, c) = (a + b (1 - \exp(-c x))) \exp(-a x - b x + \frac{b}{c} (1-\exp(-c x))) for :math:`x \ge 0`, :math:`a, b, c > 0`. `genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters. %(after_notes)s References ---------- H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential Distribution", Journal of the American Statistical Association, 1993. N. Balakrishnan, "The Exponential Distribution: Theory, Methods and Applications", Asit P. Basu. %(example)s """ def _pdf(self, x, a, b, c): # genexpon.pdf(x, a, b, c) = (a + b * (1 - exp(-c*x))) * \ # exp(-a*x - b*x + b/c * (1-exp(-c*x))) return (a + b*(-sc.expm1(-c*x)))*np.exp((-a-b)*x + b*(-sc.expm1(-c*x))/c) def _cdf(self, x, a, b, c): return -sc.expm1((-a-b)*x + b*(-sc.expm1(-c*x))/c) def _logpdf(self, x, a, b, c): return np.log(a+b*(-sc.expm1(-c*x))) + (-a-b)*x+b*(-sc.expm1(-c*x))/c genexpon = genexpon_gen(a=0.0, name='genexpon') class genextreme_gen(rv_continuous): r"""A generalized extreme value continuous random variable. %(before_notes)s See Also -------- gumbel_r Notes ----- For :math:`c=0`, `genextreme` is equal to `gumbel_r`. The probability density function for `genextreme` is: .. math:: f(x, c) = \begin{cases} \exp(-\exp(-x)) \exp(-x) &\text{for } c = 0\\ \exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1} &\text{for } x \le 1/c, c > 0 \end{cases} Note that several sources and software packages use the opposite convention for the sign of the shape parameter :math:`c`. `genextreme` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _argcheck(self, c): return np.where(abs(c) == np.inf, 0, 1) def _get_support(self, c): _b = np.where(c > 0, 1.0 / np.maximum(c, _XMIN), np.inf) _a = np.where(c < 0, 1.0 / np.minimum(c, -_XMIN), -np.inf) return _a, _b def _loglogcdf(self, x, c): return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: sc.log1p(-c*x)/c, -x) def _pdf(self, x, c): # genextreme.pdf(x, c) = # exp(-exp(-x))*exp(-x), for c==0 # exp(-(1-c*x)**(1/c))*(1-c*x)**(1/c-1), for x \le 1/c, c > 0 return np.exp(self._logpdf(x, c)) def _logpdf(self, x, c): cx = _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: c*x, 0.0) logex2 = sc.log1p(-cx) logpex2 = self._loglogcdf(x, c) pex2 = np.exp(logpex2) # Handle special cases np.putmask(logpex2, (c == 0) & (x == -np.inf), 0.0) logpdf = np.where((cx == 1) | (cx == -np.inf), -np.inf, -pex2+logpex2-logex2) np.putmask(logpdf, (c == 1) & (x == 1), 0.0) return logpdf def _logcdf(self, x, c): return -np.exp(self._loglogcdf(x, c)) def _cdf(self, x, c): return np.exp(self._logcdf(x, c)) def _sf(self, x, c): return -sc.expm1(self._logcdf(x, c)) def _ppf(self, q, c): x = -np.log(-np.log(q)) return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: -sc.expm1(-c * x) / c, x) def _isf(self, q, c): x = -np.log(-sc.log1p(-q)) return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: -sc.expm1(-c * x) / c, x) def _stats(self, c): g = lambda n: sc.gamma(n*c + 1) g1 = g(1) g2 = g(2) g3 = g(3) g4 = g(4) g2mg12 = np.where(abs(c) < 1e-7, (c*np.pi)**2.0/6.0, g2-g1**2.0) gam2k = np.where(abs(c) < 1e-7, np.pi**2.0/6.0, sc.expm1(sc.gammaln(2.0*c+1.0)-2*sc.gammaln(c + 1.0))/c**2.0) eps = 1e-14 gamk = np.where(abs(c) < eps, -_EULER, sc.expm1(sc.gammaln(c + 1))/c) m = np.where(c < -1.0, np.nan, -gamk) v = np.where(c < -0.5, np.nan, g1**2.0*gam2k) # skewness sk1 = _lazywhere(c >= -1./3, (c, g1, g2, g3, g2mg12), lambda c, g1, g2, g3, g2gm12: np.sign(c)*(-g3 + (g2 + 2*g2mg12)*g1)/g2mg12**1.5, fillvalue=np.nan) sk = np.where(abs(c) <= eps**0.29, 12*np.sqrt(6)*_ZETA3/np.pi**3, sk1) # kurtosis ku1 = _lazywhere(c >= -1./4, (g1, g2, g3, g4, g2mg12), lambda g1, g2, g3, g4, g2mg12: (g4 + (-4*g3 + 3*(g2 + g2mg12)*g1)*g1)/g2mg12**2, fillvalue=np.nan) ku = np.where(abs(c) <= (eps)**0.23, 12.0/5.0, ku1-3.0) return m, v, sk, ku def _fitstart(self, data): # This is better than the default shape of (1,). g = _skew(data) if g < 0: a = 0.5 else: a = -0.5 return super(genextreme_gen, self)._fitstart(data, args=(a,)) def _munp(self, n, c): k = np.arange(0, n+1) vals = 1.0/c**n * np.sum( sc.comb(n, k) * (-1)**k * sc.gamma(c*k + 1), axis=0) return np.where(c*n > -1, vals, np.inf) def _entropy(self, c): return _EULER*(1 - c) + 1 genextreme = genextreme_gen(name='genextreme') def _digammainv(y): # Inverse of the digamma function (real positive arguments only). # This function is used in the `fit` method of `gamma_gen`. # The function uses either optimize.fsolve or optimize.newton # to solve `sc.digamma(x) - y = 0`. There is probably room for # improvement, but currently it works over a wide range of y: # >>> y = 64*np.random.randn(1000000) # >>> y.min(), y.max() # (-311.43592651416662, 351.77388222276869) # x = [_digammainv(t) for t in y] # np.abs(sc.digamma(x) - y).max() # 1.1368683772161603e-13 # _em = 0.5772156649015328606065120 func = lambda x: sc.digamma(x) - y if y > -0.125: x0 = np.exp(y) + 0.5 if y < 10: # Some experimentation shows that newton reliably converges # must faster than fsolve in this y range. For larger y, # newton sometimes fails to converge. value = optimize.newton(func, x0, tol=1e-10) return value elif y > -3: x0 = np.exp(y/2.332) + 0.08661 else: x0 = 1.0 / (-y - _em) value, info, ier, mesg = optimize.fsolve(func, x0, xtol=1e-11, full_output=True) if ier != 1: raise RuntimeError("_digammainv: fsolve failed, y = %r" % y) return value[0] ## Gamma (Use MATLAB and MATHEMATICA (b=theta=scale, a=alpha=shape) definition) ## gamma(a, loc, scale) with a an integer is the Erlang distribution ## gamma(1, loc, scale) is the Exponential distribution ## gamma(df/2, 0, 2) is the chi2 distribution with df degrees of freedom. class gamma_gen(rv_continuous): r"""A gamma continuous random variable. %(before_notes)s See Also -------- erlang, expon Notes ----- The probability density function for `gamma` is: .. math:: f(x, a) = \frac{x^{a-1} \exp(-x)}{\Gamma(a)} for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the gamma function. `gamma` takes ``a`` as a shape parameter for :math:`a`. When :math:`a` is an integer, `gamma` reduces to the Erlang distribution, and when :math:`a=1` to the exponential distribution. %(after_notes)s %(example)s """ def _rvs(self, a, size=None, random_state=None): return random_state.standard_gamma(a, size) def _pdf(self, x, a): # gamma.pdf(x, a) = x**(a-1) * exp(-x) / gamma(a) return np.exp(self._logpdf(x, a)) def _logpdf(self, x, a): return sc.xlogy(a-1.0, x) - x - sc.gammaln(a) def _cdf(self, x, a): return sc.gammainc(a, x) def _sf(self, x, a): return sc.gammaincc(a, x) def _ppf(self, q, a): return sc.gammaincinv(a, q) def _stats(self, a): return a, a, 2.0/np.sqrt(a), 6.0/a def _entropy(self, a): return sc.psi(a)*(1-a) + a + sc.gammaln(a) def _fitstart(self, data): # The skewness of the gamma distribution is `4 / np.sqrt(a)`. # We invert that to estimate the shape `a` using the skewness # of the data. The formula is regularized with 1e-8 in the # denominator to allow for degenerate data where the skewness # is close to 0. a = 4 / (1e-8 + _skew(data)**2) return super(gamma_gen, self)._fitstart(data, args=(a,)) @extend_notes_in_docstring(rv_continuous, notes="""\ When the location is fixed by using the argument `floc`, this function uses explicit formulas or solves a simpler numerical problem than the full ML optimization problem. So in that case, the `optimizer`, `loc` and `scale` arguments are ignored.\n\n""") def fit(self, data, *args, **kwds): floc = kwds.get('floc', None) if floc is None: # loc is not fixed. Use the default fit method. return super(gamma_gen, self).fit(data, *args, **kwds) # We already have this value, so just pop it from kwds. kwds.pop('floc', None) f0 = _get_fixed_fit_value(kwds, ['f0', 'fa', 'fix_a']) fscale = kwds.pop('fscale', None) _remove_optimizer_parameters(kwds) # Special case: loc is fixed. if f0 is not None and fscale is not None: # This check is for consistency with `rv_continuous.fit`. # Without this check, this function would just return the # parameters that were given. raise ValueError("All parameters fixed. There is nothing to " "optimize.") # Fixed location is handled by shifting the data. data = np.asarray(data) if not np.isfinite(data).all(): raise RuntimeError("The data contains non-finite values.") if np.any(data <= floc): raise FitDataError("gamma", lower=floc, upper=np.inf) if floc != 0: # Don't do the subtraction in-place, because `data` might be a # view of the input array. data = data - floc xbar = data.mean() # Three cases to handle: # * shape and scale both free # * shape fixed, scale free # * shape free, scale fixed if fscale is None: # scale is free if f0 is not None: # shape is fixed a = f0 else: # shape and scale are both free. # The MLE for the shape parameter `a` is the solution to: # np.log(a) - sc.digamma(a) - np.log(xbar) + # np.log(data).mean() = 0 s = np.log(xbar) - np.log(data).mean() func = lambda a: np.log(a) - sc.digamma(a) - s aest = (3-s + np.sqrt((s-3)**2 + 24*s)) / (12*s) xa = aest*(1-0.4) xb = aest*(1+0.4) a = optimize.brentq(func, xa, xb, disp=0) # The MLE for the scale parameter is just the data mean # divided by the shape parameter. scale = xbar / a else: # scale is fixed, shape is free # The MLE for the shape parameter `a` is the solution to: # sc.digamma(a) - np.log(data).mean() + np.log(fscale) = 0 c = np.log(data).mean() - np.log(fscale) a = _digammainv(c) scale = fscale return a, floc, scale gamma = gamma_gen(a=0.0, name='gamma') class erlang_gen(gamma_gen): """An Erlang continuous random variable. %(before_notes)s See Also -------- gamma Notes ----- The Erlang distribution is a special case of the Gamma distribution, with the shape parameter `a` an integer. Note that this restriction is not enforced by `erlang`. It will, however, generate a warning the first time a non-integer value is used for the shape parameter. Refer to `gamma` for examples. """ def _argcheck(self, a): allint = np.all(np.floor(a) == a) if not allint: # An Erlang distribution shouldn't really have a non-integer # shape parameter, so warn the user. warnings.warn( 'The shape parameter of the erlang distribution ' 'has been given a non-integer value %r.' % (a,), RuntimeWarning) return a > 0 def _fitstart(self, data): # Override gamma_gen_fitstart so that an integer initial value is # used. (Also regularize the division, to avoid issues when # _skew(data) is 0 or close to 0.) a = int(4.0 / (1e-8 + _skew(data)**2)) return super(gamma_gen, self)._fitstart(data, args=(a,)) # Trivial override of the fit method, so we can monkey-patch its # docstring. def fit(self, data, *args, **kwds): return super(erlang_gen, self).fit(data, *args, **kwds) if fit.__doc__: fit.__doc__ = (rv_continuous.fit.__doc__ + """ Notes ----- The Erlang distribution is generally defined to have integer values for the shape parameter. This is not enforced by the `erlang` class. When fitting the distribution, it will generally return a non-integer value for the shape parameter. By using the keyword argument `f0=`, the fit method can be constrained to fit the data to a specific integer shape parameter. """) erlang = erlang_gen(a=0.0, name='erlang') class gengamma_gen(rv_continuous): r"""A generalized gamma continuous random variable. %(before_notes)s Notes ----- The probability density function for `gengamma` is: .. math:: f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)} for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `gengamma` takes :math:`a` and :math:`c` as shape parameters. %(after_notes)s %(example)s """ def _argcheck(self, a, c): return (a > 0) & (c != 0) def _pdf(self, x, a, c): # gengamma.pdf(x, a, c) = abs(c) * x**(c*a-1) * exp(-x**c) / gamma(a) return np.exp(self._logpdf(x, a, c)) def _logpdf(self, x, a, c): return np.log(abs(c)) + sc.xlogy(c*a - 1, x) - x**c - sc.gammaln(a) def _cdf(self, x, a, c): xc = x**c val1 = sc.gammainc(a, xc) val2 = sc.gammaincc(a, xc) return np.where(c > 0, val1, val2) def _sf(self, x, a, c): xc = x**c val1 = sc.gammainc(a, xc) val2 = sc.gammaincc(a, xc) return np.where(c > 0, val2, val1) def _ppf(self, q, a, c): val1 = sc.gammaincinv(a, q) val2 = sc.gammainccinv(a, q) return np.where(c > 0, val1, val2)**(1.0/c) def _isf(self, q, a, c): val1 = sc.gammaincinv(a, q) val2 = sc.gammainccinv(a, q) return np.where(c > 0, val2, val1)**(1.0/c) def _munp(self, n, a, c): # Pochhammer symbol: sc.pocha,n) = gamma(a+n)/gamma(a) return sc.poch(a, n*1.0/c) def _entropy(self, a, c): val = sc.psi(a) return a*(1-val) + 1.0/c*val + sc.gammaln(a) - np.log(abs(c)) gengamma = gengamma_gen(a=0.0, name='gengamma') class genhalflogistic_gen(rv_continuous): r"""A generalized half-logistic continuous random variable. %(before_notes)s Notes ----- The probability density function for `genhalflogistic` is: .. math:: f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2} for :math:`0 \le x \le 1/c`, and :math:`c > 0`. `genhalflogistic` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _argcheck(self, c): return c > 0 def _get_support(self, c): return self.a, 1.0/c def _pdf(self, x, c): # genhalflogistic.pdf(x, c) = # 2 * (1-c*x)**(1/c-1) / (1+(1-c*x)**(1/c))**2 limit = 1.0/c tmp = np.asarray(1-c*x) tmp0 = tmp**(limit-1) tmp2 = tmp0*tmp return 2*tmp0 / (1+tmp2)**2 def _cdf(self, x, c): limit = 1.0/c tmp = np.asarray(1-c*x) tmp2 = tmp**(limit) return (1.0-tmp2) / (1+tmp2) def _ppf(self, q, c): return 1.0/c*(1-((1.0-q)/(1.0+q))**c) def _entropy(self, c): return 2 - (2*c+1)*np.log(2) genhalflogistic = genhalflogistic_gen(a=0.0, name='genhalflogistic') class gompertz_gen(rv_continuous): r"""A Gompertz (or truncated Gumbel) continuous random variable. %(before_notes)s Notes ----- The probability density function for `gompertz` is: .. math:: f(x, c) = c \exp(x) \exp(-c (e^x-1)) for :math:`x \ge 0`, :math:`c > 0`. `gompertz` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _pdf(self, x, c): # gompertz.pdf(x, c) = c * exp(x) * exp(-c*(exp(x)-1)) return np.exp(self._logpdf(x, c)) def _logpdf(self, x, c): return np.log(c) + x - c * sc.expm1(x) def _cdf(self, x, c): return -sc.expm1(-c * sc.expm1(x)) def _ppf(self, q, c): return sc.log1p(-1.0 / c * sc.log1p(-q)) def _entropy(self, c): return 1.0 - np.log(c) - np.exp(c)*sc.expn(1, c) gompertz = gompertz_gen(a=0.0, name='gompertz') class gumbel_r_gen(rv_continuous): r"""A right-skewed Gumbel continuous random variable. %(before_notes)s See Also -------- gumbel_l, gompertz, genextreme Notes ----- The probability density function for `gumbel_r` is: .. math:: f(x) = \exp(-(x + e^{-x})) The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett distribution. It is also related to the extreme value distribution, log-Weibull and Gompertz distributions. %(after_notes)s %(example)s """ def _pdf(self, x): # gumbel_r.pdf(x) = exp(-(x + exp(-x))) return np.exp(self._logpdf(x)) def _logpdf(self, x): return -x - np.exp(-x) def _cdf(self, x): return np.exp(-np.exp(-x)) def _logcdf(self, x): return -np.exp(-x) def _ppf(self, q): return -np.log(-np.log(q)) def _stats(self): return _EULER, np.pi*np.pi/6.0, 12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5 def _entropy(self): # https://en.wikipedia.org/wiki/Gumbel_distribution return _EULER + 1. gumbel_r = gumbel_r_gen(name='gumbel_r') class gumbel_l_gen(rv_continuous): r"""A left-skewed Gumbel continuous random variable. %(before_notes)s See Also -------- gumbel_r, gompertz, genextreme Notes ----- The probability density function for `gumbel_l` is: .. math:: f(x) = \exp(x - e^x) The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett distribution. It is also related to the extreme value distribution, log-Weibull and Gompertz distributions. %(after_notes)s %(example)s """ def _pdf(self, x): # gumbel_l.pdf(x) = exp(x - exp(x)) return np.exp(self._logpdf(x)) def _logpdf(self, x): return x - np.exp(x) def _cdf(self, x): return -sc.expm1(-np.exp(x)) def _ppf(self, q): return np.log(-sc.log1p(-q)) def _logsf(self, x): return -np.exp(x) def _sf(self, x): return np.exp(-np.exp(x)) def _isf(self, x): return np.log(-np.log(x)) def _stats(self): return -_EULER, np.pi*np.pi/6.0, \ -12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5 def _entropy(self): return _EULER + 1. gumbel_l = gumbel_l_gen(name='gumbel_l') class halfcauchy_gen(rv_continuous): r"""A Half-Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `halfcauchy` is: .. math:: f(x) = \frac{2}{\pi (1 + x^2)} for :math:`x \ge 0`. %(after_notes)s %(example)s """ def _pdf(self, x): # halfcauchy.pdf(x) = 2 / (pi * (1 + x**2)) return 2.0/np.pi/(1.0+x*x) def _logpdf(self, x): return np.log(2.0/np.pi) - sc.log1p(x*x) def _cdf(self, x): return 2.0/np.pi*np.arctan(x) def _ppf(self, q): return np.tan(np.pi/2*q) def _stats(self): return np.inf, np.inf, np.nan, np.nan def _entropy(self): return np.log(2*np.pi) halfcauchy = halfcauchy_gen(a=0.0, name='halfcauchy') class halflogistic_gen(rv_continuous): r"""A half-logistic continuous random variable. %(before_notes)s Notes ----- The probability density function for `halflogistic` is: .. math:: f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 } = \frac{1}{2} \text{sech}(x/2)^2 for :math:`x \ge 0`. %(after_notes)s %(example)s """ def _pdf(self, x): # halflogistic.pdf(x) = 2 * exp(-x) / (1+exp(-x))**2 # = 1/2 * sech(x/2)**2 return np.exp(self._logpdf(x)) def _logpdf(self, x): return np.log(2) - x - 2. * sc.log1p(np.exp(-x)) def _cdf(self, x): return np.tanh(x/2.0) def _ppf(self, q): return 2*np.arctanh(q) def _munp(self, n): if n == 1: return 2*np.log(2) if n == 2: return np.pi*np.pi/3.0 if n == 3: return 9*_ZETA3 if n == 4: return 7*np.pi**4 / 15.0 return 2*(1-pow(2.0, 1-n))*sc.gamma(n+1)*sc.zeta(n, 1) def _entropy(self): return 2-np.log(2) halflogistic = halflogistic_gen(a=0.0, name='halflogistic') class halfnorm_gen(rv_continuous): r"""A half-normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `halfnorm` is: .. math:: f(x) = \sqrt{2/\pi} \exp(-x^2 / 2) for :math:`x >= 0`. `halfnorm` is a special case of `chi` with ``df=1``. %(after_notes)s %(example)s """ def _rvs(self, size=None, random_state=None): return abs(random_state.standard_normal(size=size)) def _pdf(self, x): # halfnorm.pdf(x) = sqrt(2/pi) * exp(-x**2/2) return np.sqrt(2.0/np.pi)*np.exp(-x*x/2.0) def _logpdf(self, x): return 0.5 * np.log(2.0/np.pi) - x*x/2.0 def _cdf(self, x): return _norm_cdf(x)*2-1.0 def _ppf(self, q): return sc.ndtri((1+q)/2.0) def _stats(self): return (np.sqrt(2.0/np.pi), 1-2.0/np.pi, np.sqrt(2)*(4-np.pi)/(np.pi-2)**1.5, 8*(np.pi-3)/(np.pi-2)**2) def _entropy(self): return 0.5*np.log(np.pi/2.0)+0.5 halfnorm = halfnorm_gen(a=0.0, name='halfnorm') class hypsecant_gen(rv_continuous): r"""A hyperbolic secant continuous random variable. %(before_notes)s Notes ----- The probability density function for `hypsecant` is: .. math:: f(x) = \frac{1}{\pi} \text{sech}(x) for a real number :math:`x`. %(after_notes)s %(example)s """ def _pdf(self, x): # hypsecant.pdf(x) = 1/pi * sech(x) return 1.0/(np.pi*np.cosh(x)) def _cdf(self, x): return 2.0/np.pi*np.arctan(np.exp(x)) def _ppf(self, q): return np.log(np.tan(np.pi*q/2.0)) def _stats(self): return 0, np.pi*np.pi/4, 0, 2 def _entropy(self): return np.log(2*np.pi) hypsecant = hypsecant_gen(name='hypsecant') class gausshyper_gen(rv_continuous): r"""A Gauss hypergeometric continuous random variable. %(before_notes)s Notes ----- The probability density function for `gausshyper` is: .. math:: f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c} for :math:`0 \le x \le 1`, :math:`a > 0`, :math:`b > 0`, and :math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`. :math:`F[2, 1]` is the Gauss hypergeometric function `scipy.special.hyp2f1`. `gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape parameters. %(after_notes)s %(example)s """ def _argcheck(self, a, b, c, z): return (a > 0) & (b > 0) & (c == c) & (z == z) def _pdf(self, x, a, b, c, z): # gausshyper.pdf(x, a, b, c, z) = # C * x**(a-1) * (1-x)**(b-1) * (1+z*x)**(-c) Cinv = sc.gamma(a)*sc.gamma(b)/sc.gamma(a+b)*sc.hyp2f1(c, a, a+b, -z) return 1.0/Cinv * x**(a-1.0) * (1.0-x)**(b-1.0) / (1.0+z*x)**c def _munp(self, n, a, b, c, z): fac = sc.beta(n+a, b) / sc.beta(a, b) num = sc.hyp2f1(c, a+n, a+b+n, -z) den = sc.hyp2f1(c, a, a+b, -z) return fac*num / den gausshyper = gausshyper_gen(a=0.0, b=1.0, name='gausshyper') class invgamma_gen(rv_continuous): r"""An inverted gamma continuous random variable. %(before_notes)s Notes ----- The probability density function for `invgamma` is: .. math:: f(x, a) = \frac{x^{-a-1}}{\Gamma(a)} \exp(-\frac{1}{x}) for :math:`x >= 0`, :math:`a > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `invgamma` takes ``a`` as a shape parameter for :math:`a`. `invgamma` is a special case of `gengamma` with ``c=-1``. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _pdf(self, x, a): # invgamma.pdf(x, a) = x**(-a-1) / gamma(a) * exp(-1/x) return np.exp(self._logpdf(x, a)) def _logpdf(self, x, a): return -(a+1) * np.log(x) - sc.gammaln(a) - 1.0/x def _cdf(self, x, a): return sc.gammaincc(a, 1.0 / x) def _ppf(self, q, a): return 1.0 / sc.gammainccinv(a, q) def _sf(self, x, a): return sc.gammainc(a, 1.0 / x) def _isf(self, q, a): return 1.0 / sc.gammaincinv(a, q) def _stats(self, a, moments='mvsk'): m1 = _lazywhere(a > 1, (a,), lambda x: 1. / (x - 1.), np.inf) m2 = _lazywhere(a > 2, (a,), lambda x: 1. / (x - 1.)**2 / (x - 2.), np.inf) g1, g2 = None, None if 's' in moments: g1 = _lazywhere( a > 3, (a,), lambda x: 4. * np.sqrt(x - 2.) / (x - 3.), np.nan) if 'k' in moments: g2 = _lazywhere( a > 4, (a,), lambda x: 6. * (5. * x - 11.) / (x - 3.) / (x - 4.), np.nan) return m1, m2, g1, g2 def _entropy(self, a): return a - (a+1.0) * sc.psi(a) + sc.gammaln(a) invgamma = invgamma_gen(a=0.0, name='invgamma') # scale is gamma from DATAPLOT and B from Regress class invgauss_gen(rv_continuous): r"""An inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `invgauss` is: .. math:: f(x, \mu) = \frac{1}{\sqrt{2 \pi x^3}} \exp(-\frac{(x-\mu)^2}{2 x \mu^2}) for :math:`x >= 0` and :math:`\mu > 0`. `invgauss` takes ``mu`` as a shape parameter for :math:`\mu`. %(after_notes)s When :math:`\mu` is too small, evaluating the cumulative distribution function will be inaccurate due to ``cdf(mu -> 0) = inf * 0``. NaNs are returned for :math:`\mu \le 0.0028`. %(example)s """ _support_mask = rv_continuous._open_support_mask def _rvs(self, mu, size=None, random_state=None): return random_state.wald(mu, 1.0, size=size) def _pdf(self, x, mu): # invgauss.pdf(x, mu) = # 1 / sqrt(2*pi*x**3) * exp(-(x-mu)**2/(2*x*mu**2)) return 1.0/np.sqrt(2*np.pi*x**3.0)*np.exp(-1.0/(2*x)*((x-mu)/mu)**2) def _logpdf(self, x, mu): return -0.5*np.log(2*np.pi) - 1.5*np.log(x) - ((x-mu)/mu)**2/(2*x) def _cdf(self, x, mu): fac = np.sqrt(1.0/x) # Numerical accuracy for small `mu` is bad. See #869. C1 = _norm_cdf(fac*(x-mu)/mu) C1 += np.exp(1.0/mu) * _norm_cdf(-fac*(x+mu)/mu) * np.exp(1.0/mu) return C1 def _stats(self, mu): return mu, mu**3.0, 3*np.sqrt(mu), 15*mu invgauss = invgauss_gen(a=0.0, name='invgauss') class geninvgauss_gen(rv_continuous): r"""A Generalized Inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `geninvgauss` is: .. math:: f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b)) where `x > 0`, and the parameters `p, b` satisfy `b > 0` ([1]_). :math:`K_p` is the modified Bessel function of second kind of order `p` (`scipy.special.kv`). %(after_notes)s The inverse Gaussian distribution `stats.invgauss(mu)` is a special case of `geninvgauss` with `p = -1/2`, `b = 1 / mu` and `scale = mu`. Generating random variates is challenging for this distribution. The implementation is based on [2]_. References ---------- .. [1] O. Barndorff-Nielsen, P. Blaesild, C. Halgreen, "First hitting time models for the generalized inverse gaussian distribution", Stochastic Processes and their Applications 7, pp. 49--54, 1978. .. [2] W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian random variates", Statistics and Computing, 24(4), p. 547--557, 2014. %(example)s """ def _argcheck(self, p, b): return (p == p) & (b > 0) def _logpdf(self, x, p, b): # kve instead of kv works better for large values of b # warn if kve produces infinite values and replace by nan # otherwise c = -inf and the results are often incorrect @np.vectorize def logpdf_single(x, p, b): return _stats.geninvgauss_logpdf(x, p, b) z = logpdf_single(x, p, b) if np.isnan(z).any(): msg = ("Infinite values encountered in scipy.special.kve(p, b). " "Values replaced by NaN to avoid incorrect results.") warnings.warn(msg, RuntimeWarning) return z def _pdf(self, x, p, b): # relying on logpdf avoids overflow of x**(p-1) for large x and p return np.exp(self._logpdf(x, p, b)) def _cdf(self, x, *args): _a, _b = self._get_support(*args) @np.vectorize def _cdf_single(x, *args): p, b = args user_data = np.array([p, b], float).ctypes.data_as(ctypes.c_void_p) llc = LowLevelCallable.from_cython(_stats, '_geninvgauss_pdf', user_data) return integrate.quad(llc, _a, x)[0] return _cdf_single(x, *args) def _logquasipdf(self, x, p, b): # log of the quasi-density (w/o normalizing constant) used in _rvs return _lazywhere(x > 0, (x, p, b), lambda x, p, b: (p - 1)*np.log(x) - b*(x + 1/x)/2, -np.inf) def _rvs(self, p, b, size=None, random_state=None): # if p and b are scalar, use _rvs_scalar, otherwise need to create # output by iterating over parameters if np.isscalar(p) and np.isscalar(b): out = self._rvs_scalar(p, b, size, random_state) elif p.size == 1 and b.size == 1: out = self._rvs_scalar(p.item(), b.item(), size, random_state) else: # When this method is called, size will be a (possibly empty) # tuple of integers. It will not be None; if `size=None` is passed # to `rvs()`, size will be the empty tuple (). p, b = np.broadcast_arrays(p, b) # p and b now have the same shape. # `shp` is the shape of the blocks of random variates that are # generated for each combination of parameters associated with # broadcasting p and b. # bc is a tuple the same lenth as size. The values # in bc are bools. If bc[j] is True, it means that # entire axis is filled in for a given combination of the # broadcast arguments. shp, bc = _check_shape(p.shape, size) # `numsamples` is the total number of variates to be generated # for each combination of the input arguments. numsamples = int(np.prod(shp)) # `out` is the array to be returned. It is filled in in the # loop below. out = np.empty(size) it = np.nditer([p, b], flags=['multi_index'], op_flags=[['readonly'], ['readonly']]) while not it.finished: # Convert the iterator's multi_index into an index into the # `out` array where the call to _rvs_scalar() will be stored. # Where bc is True, we use a full slice; otherwise we use the # index value from it.multi_index. len(it.multi_index) might # be less than len(bc), and in that case we want to align these # two sequences to the right, so the loop variable j runs from # -len(size) to 0. This doesn't cause an IndexError, as # bc[j] will be True in those cases where it.multi_index[j] # would cause an IndexError. idx = tuple((it.multi_index[j] if not bc[j] else slice(None)) for j in range(-len(size), 0)) out[idx] = self._rvs_scalar(it[0], it[1], numsamples, random_state).reshape(shp) it.iternext() if size == (): out = out.item() return out def _rvs_scalar(self, p, b, numsamples, random_state): # following [2], the quasi-pdf is used instead of the pdf for the # generation of rvs invert_res = False if not(numsamples): numsamples = 1 if p < 0: # note: if X is geninvgauss(p, b), then 1/X is geninvgauss(-p, b) p = -p invert_res = True m = self._mode(p, b) # determine method to be used following [2] ratio_unif = True if p >= 1 or b > 1: # ratio of uniforms with mode shift below mode_shift = True elif b >= min(0.5, 2 * np.sqrt(1 - p) / 3): # ratio of uniforms without mode shift below mode_shift = False else: # new algorithm in [2] ratio_unif = False # prepare sampling of rvs size1d = tuple(np.atleast_1d(numsamples)) N = np.prod(size1d) # number of rvs needed, reshape upon return x = np.zeros(N) simulated = 0 if ratio_unif: # use ratio of uniforms method if mode_shift: a2 = -2 * (p + 1) / b - m a1 = 2 * m * (p - 1) / b - 1 # find roots of x**3 + a2*x**2 + a1*x + m (Cardano's formula) p1 = a1 - a2**2 / 3 q1 = 2 * a2**3 / 27 - a2 * a1 / 3 + m phi = np.arccos(-q1 * np.sqrt(-27 / p1**3) / 2) s1 = -np.sqrt(-4 * p1 / 3) root1 = s1 * np.cos(phi / 3 + np.pi / 3) - a2 / 3 root2 = -s1 * np.cos(phi / 3) - a2 / 3 # root3 = s1 * np.cos(phi / 3 - np.pi / 3) - a2 / 3 # if g is the quasipdf, rescale: g(x) / g(m) which we can write # as exp(log(g(x)) - log(g(m))). This is important # since for large values of p and b, g cannot be evaluated. # denote the rescaled quasipdf by h lm = self._logquasipdf(m, p, b) d1 = self._logquasipdf(root1, p, b) - lm d2 = self._logquasipdf(root2, p, b) - lm # compute the bounding rectangle w.r.t. h. Note that # np.exp(0.5*d1) = np.sqrt(g(root1)/g(m)) = np.sqrt(h(root1)) vmin = (root1 - m) * np.exp(0.5 * d1) vmax = (root2 - m) * np.exp(0.5 * d2) umax = 1 # umax = sqrt(h(m)) = 1 logqpdf = lambda x: self._logquasipdf(x, p, b) - lm c = m else: # ratio of uniforms without mode shift # compute np.sqrt(quasipdf(m)) umax = np.exp(0.5*self._logquasipdf(m, p, b)) xplus = ((1 + p) + np.sqrt((1 + p)**2 + b**2))/b vmin = 0 # compute xplus * np.sqrt(quasipdf(xplus)) vmax = xplus * np.exp(0.5 * self._logquasipdf(xplus, p, b)) c = 0 logqpdf = lambda x: self._logquasipdf(x, p, b) if vmin >= vmax: raise ValueError("vmin must be smaller than vmax.") if umax <= 0: raise ValueError("umax must be positive.") i = 1 while simulated < N: k = N - simulated # simulate uniform rvs on [0, umax] and [vmin, vmax] u = umax * random_state.uniform(size=k) v = random_state.uniform(size=k) v = vmin + (vmax - vmin) * v rvs = v / u + c # rewrite acceptance condition u**2 <= pdf(rvs) by taking logs accept = (2*np.log(u) <= logqpdf(rvs)) num_accept = np.sum(accept) if num_accept > 0: x[simulated:(simulated + num_accept)] = rvs[accept] simulated += num_accept if (simulated == 0) and (i*N >= 50000): msg = ("Not a single random variate could be generated " "in {} attempts. Sampling does not appear to " "work for the provided parameters.".format(i*N)) raise RuntimeError(msg) i += 1 else: # use new algorithm in [2] x0 = b / (1 - p) xs = np.max((x0, 2 / b)) k1 = np.exp(self._logquasipdf(m, p, b)) A1 = k1 * x0 if x0 < 2 / b: k2 = np.exp(-b) if p > 0: A2 = k2 * ((2 / b)**p - x0**p) / p else: A2 = k2 * np.log(2 / b**2) else: k2, A2 = 0, 0 k3 = xs**(p - 1) A3 = 2 * k3 * np.exp(-xs * b / 2) / b A = A1 + A2 + A3 # [2]: rejection constant is < 2.73; so expected runtime is finite while simulated < N: k = N - simulated h, rvs = np.zeros(k), np.zeros(k) # simulate uniform rvs on [x1, x2] and [0, y2] u = random_state.uniform(size=k) v = A * random_state.uniform(size=k) cond1 = v <= A1 cond2 = np.logical_not(cond1) & (v <= A1 + A2) cond3 = np.logical_not(cond1 | cond2) # subdomain (0, x0) rvs[cond1] = x0 * v[cond1] / A1 h[cond1] = k1 # subdomain (x0, 2 / b) if p > 0: rvs[cond2] = (x0**p + (v[cond2] - A1) * p / k2)**(1 / p) else: rvs[cond2] = b * np.exp((v[cond2] - A1) * np.exp(b)) h[cond2] = k2 * rvs[cond2]**(p - 1) # subdomain (xs, infinity) z = np.exp(-xs * b / 2) - b * (v[cond3] - A1 - A2) / (2 * k3) rvs[cond3] = -2 / b * np.log(z) h[cond3] = k3 * np.exp(-rvs[cond3] * b / 2) # apply rejection method accept = (np.log(u * h) <= self._logquasipdf(rvs, p, b)) num_accept = sum(accept) if num_accept > 0: x[simulated:(simulated + num_accept)] = rvs[accept] simulated += num_accept rvs = np.reshape(x, size1d) if invert_res: rvs = 1 / rvs return rvs def _mode(self, p, b): # distinguish cases to avoid catastrophic cancellation (see [2]) if p < 1: return b / (np.sqrt((p - 1)**2 + b**2) + 1 - p) else: return (np.sqrt((1 - p)**2 + b**2) - (1 - p)) / b def _munp(self, n, p, b): num = sc.kve(p + n, b) denom = sc.kve(p, b) inf_vals = np.isinf(num) | np.isinf(denom) if inf_vals.any(): msg = ("Infinite values encountered in the moment calculation " "involving scipy.special.kve. Values replaced by NaN to " "avoid incorrect results.") warnings.warn(msg, RuntimeWarning) m = np.full_like(num, np.nan, dtype=np.double) m[~inf_vals] = num[~inf_vals] / denom[~inf_vals] else: m = num / denom return m geninvgauss = geninvgauss_gen(a=0.0, name="geninvgauss") class norminvgauss_gen(rv_continuous): r"""A Normal Inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `norminvgauss` is: .. math:: f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \, \exp(\sqrt{a^2 - b^2} + b x) where :math:`x` is a real number, the parameter :math:`a` is the tail heaviness and :math:`b` is the asymmetry parameter satisfying :math:`a > 0` and :math:`|b| <= a`. :math:`K_1` is the modified Bessel function of second kind (`scipy.special.k1`). %(after_notes)s A normal inverse Gaussian random variable `Y` with parameters `a` and `b` can be expressed as a normal mean-variance mixture: `Y = b * V + sqrt(V) * X` where `X` is `norm(0,1)` and `V` is `invgauss(mu=1/sqrt(a**2 - b**2))`. This representation is used to generate random variates. References ---------- O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3), pp. 151-157, 1978. O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling", Scandinavian Journal of Statistics, Vol. 24, pp. 1-13, 1997. %(example)s """ _support_mask = rv_continuous._open_support_mask def _argcheck(self, a, b): return (a > 0) & (np.absolute(b) < a) def _pdf(self, x, a, b): gamma = np.sqrt(a**2 - b**2) fac1 = a / np.pi * np.exp(gamma) sq = np.hypot(1, x) # reduce overflows return fac1 * sc.k1e(a * sq) * np.exp(b*x - a*sq) / sq def _rvs(self, a, b, size=None, random_state=None): # note: X = b * V + sqrt(V) * X is norminvgaus(a,b) if X is standard # normal and V is invgauss(mu=1/sqrt(a**2 - b**2)) gamma = np.sqrt(a**2 - b**2) ig = invgauss.rvs(mu=1/gamma, size=size, random_state=random_state) return b * ig + np.sqrt(ig) * norm.rvs(size=size, random_state=random_state) def _stats(self, a, b): gamma = np.sqrt(a**2 - b**2) mean = b / gamma variance = a**2 / gamma**3 skewness = 3.0 * b / (a * np.sqrt(gamma)) kurtosis = 3.0 * (1 + 4 * b**2 / a**2) / gamma return mean, variance, skewness, kurtosis norminvgauss = norminvgauss_gen(name="norminvgauss") class invweibull_gen(rv_continuous): u"""An inverted Weibull continuous random variable. This distribution is also known as the Fréchet distribution or the type II extreme value distribution. %(before_notes)s Notes ----- The probability density function for `invweibull` is: .. math:: f(x, c) = c x^{-c-1} \\exp(-x^{-c}) for :math:`x > 0`, :math:`c > 0`. `invweibull` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011. %(example)s """ _support_mask = rv_continuous._open_support_mask def _pdf(self, x, c): # invweibull.pdf(x, c) = c * x**(-c-1) * exp(-x**(-c)) xc1 = np.power(x, -c - 1.0) xc2 = np.power(x, -c) xc2 = np.exp(-xc2) return c * xc1 * xc2 def _cdf(self, x, c): xc1 = np.power(x, -c) return np.exp(-xc1) def _ppf(self, q, c): return np.power(-np.log(q), -1.0/c) def _munp(self, n, c): return sc.gamma(1 - n / c) def _entropy(self, c): return 1+_EULER + _EULER / c - np.log(c) invweibull = invweibull_gen(a=0, name='invweibull') class johnsonsb_gen(rv_continuous): r"""A Johnson SB continuous random variable. %(before_notes)s See Also -------- johnsonsu Notes ----- The probability density function for `johnsonsb` is: .. math:: f(x, a, b) = \frac{b}{x(1-x)} \phi(a + b \log \frac{x}{1-x} ) for :math:`0 <= x < =1` and :math:`a, b > 0`, and :math:`\phi` is the normal pdf. `johnsonsb` takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _argcheck(self, a, b): return (b > 0) & (a == a) def _pdf(self, x, a, b): # johnsonsb.pdf(x, a, b) = b / (x*(1-x)) * phi(a + b * log(x/(1-x))) trm = _norm_pdf(a + b*np.log(x/(1.0-x))) return b*1.0/(x*(1-x))*trm def _cdf(self, x, a, b): return _norm_cdf(a + b*np.log(x/(1.0-x))) def _ppf(self, q, a, b): return 1.0 / (1 + np.exp(-1.0 / b * (_norm_ppf(q) - a))) johnsonsb = johnsonsb_gen(a=0.0, b=1.0, name='johnsonsb') class johnsonsu_gen(rv_continuous): r"""A Johnson SU continuous random variable. %(before_notes)s See Also -------- johnsonsb Notes ----- The probability density function for `johnsonsu` is: .. math:: f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}} \phi(a + b \log(x + \sqrt{x^2 + 1})) for all :math:`x, a, b > 0`, and :math:`\phi` is the normal pdf. `johnsonsu` takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s %(example)s """ def _argcheck(self, a, b): return (b > 0) & (a == a) def _pdf(self, x, a, b): # johnsonsu.pdf(x, a, b) = b / sqrt(x**2 + 1) * # phi(a + b * log(x + sqrt(x**2 + 1))) x2 = x*x trm = _norm_pdf(a + b * np.log(x + np.sqrt(x2+1))) return b*1.0/np.sqrt(x2+1.0)*trm def _cdf(self, x, a, b): return _norm_cdf(a + b * np.log(x + np.sqrt(x*x + 1))) def _ppf(self, q, a, b): return np.sinh((_norm_ppf(q) - a) / b) johnsonsu = johnsonsu_gen(name='johnsonsu') class laplace_gen(rv_continuous): r"""A Laplace continuous random variable. %(before_notes)s Notes ----- The probability density function for `laplace` is .. math:: f(x) = \frac{1}{2} \exp(-|x|) for a real number :math:`x`. %(after_notes)s %(example)s """ def _rvs(self, size=None, random_state=None): return random_state.laplace(0, 1, size=size) def _pdf(self, x): # laplace.pdf(x) = 1/2 * exp(-abs(x)) return 0.5*np.exp(-abs(x)) def _cdf(self, x): return np.where(x > 0, 1.0-0.5*np.exp(-x), 0.5*np.exp(x)) def _ppf(self, q): return np.where(q > 0.5, -np.log(2*(1-q)), np.log(2*q)) def _stats(self): return 0, 2, 0, 3 def _entropy(self): return np.log(2)+1 @replace_notes_in_docstring(rv_continuous, notes="""\ This function uses explicit formulas for the maximum likelihood estimation of the Laplace distribution parameters, so the keyword arguments `loc`, `scale`, and `optimizer` are ignored.\n\n""") def fit(self, data, *args, **kwds): floc = kwds.pop('floc', None) fscale = kwds.pop('fscale', None) _check_fit_input_parameters(data, args, kwds, fixed_param=(floc, fscale)) # MLE for the laplace distribution if floc is None: loc = np.median(data) else: loc = floc if fscale is None: scale = (np.sum(np.abs(data - loc))) / len(data) else: scale = fscale # Source: Statistical Distributions, 3rd Edition. Evans, Hastings, # and Peacock (2000), Page 124 return loc, scale laplace = laplace_gen(name='laplace') def _check_fit_input_parameters(data, args, kwds, fixed_param): if len(args) > 0: raise TypeError("Too many arguments.") _remove_optimizer_parameters(kwds) if None not in fixed_param: # This check is for consistency with `rv_continuous.fit`. # Without this check, this function would just return the # parameters that were given. raise RuntimeError("All parameters fixed. There is nothing to " "optimize.") data = np.asarray(data) if not np.isfinite(data).all(): raise RuntimeError("The data contains non-finite values.") class levy_gen(rv_continuous): r"""A Levy continuous random variable. %(before_notes)s See Also -------- levy_stable, levy_l Notes ----- The probability density function for `levy` is: .. math:: f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right) for :math:`x >= 0`. This is the same as the Levy-stable distribution with :math:`a=1/2` and :math:`b=1`. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _pdf(self, x): # levy.pdf(x) = 1 / (x * sqrt(2*pi*x)) * exp(-1/(2*x)) return 1 / np.sqrt(2*np.pi*x) / x * np.exp(-1/(2*x)) def _cdf(self, x): # Equivalent to 2*norm.sf(np.sqrt(1/x)) return sc.erfc(np.sqrt(0.5 / x)) def _ppf(self, q): # Equivalent to 1.0/(norm.isf(q/2)**2) or 0.5/(erfcinv(q)**2) val = -sc.ndtri(q/2) return 1.0 / (val * val) def _stats(self): return np.inf, np.inf, np.nan, np.nan levy = levy_gen(a=0.0, name="levy") class levy_l_gen(rv_continuous): r"""A left-skewed Levy continuous random variable. %(before_notes)s See Also -------- levy, levy_stable Notes ----- The probability density function for `levy_l` is: .. math:: f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp{ \left(-\frac{1}{2|x|} \right)} for :math:`x <= 0`. This is the same as the Levy-stable distribution with :math:`a=1/2` and :math:`b=-1`. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _pdf(self, x): # levy_l.pdf(x) = 1 / (abs(x) * sqrt(2*pi*abs(x))) * exp(-1/(2*abs(x))) ax = abs(x) return 1/np.sqrt(2*np.pi*ax)/ax*np.exp(-1/(2*ax)) def _cdf(self, x): ax = abs(x) return 2 * _norm_cdf(1 / np.sqrt(ax)) - 1 def _ppf(self, q): val = _norm_ppf((q + 1.0) / 2) return -1.0 / (val * val) def _stats(self): return np.inf, np.inf, np.nan, np.nan levy_l = levy_l_gen(b=0.0, name="levy_l") class levy_stable_gen(rv_continuous): r"""A Levy-stable continuous random variable. %(before_notes)s See Also -------- levy, levy_l Notes ----- The distribution for `levy_stable` has characteristic function: .. math:: \varphi(t, \alpha, \beta, c, \mu) = e^{it\mu -|ct|^{\alpha}(1-i\beta \operatorname{sign}(t)\Phi(\alpha, t))} where: .. math:: \Phi = \begin{cases} \tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\ -{\frac {2}{\pi }}\log |t|&\alpha =1 \end{cases} The probability density function for `levy_stable` is: .. math:: f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty \varphi(t)e^{-ixt}\,dt where :math:`-\infty < t < \infty`. This integral does not have a known closed form. For evaluation of pdf we use either Zolotarev :math:`S_0` parameterization with integration, direct integration of standard parameterization of characteristic function or FFT of characteristic function. If set to other than None and if number of points is greater than ``levy_stable.pdf_fft_min_points_threshold`` (defaults to None) we use FFT otherwise we use one of the other methods. The default method is 'best' which uses Zolotarev's method if alpha = 1 and integration of characteristic function otherwise. The default method can be changed by setting ``levy_stable.pdf_default_method`` to either 'zolotarev', 'quadrature' or 'best'. To increase accuracy of FFT calculation one can specify ``levy_stable.pdf_fft_grid_spacing`` (defaults to 0.001) and ``pdf_fft_n_points_two_power`` (defaults to a value that covers the input range * 4). Setting ``pdf_fft_n_points_two_power`` to 16 should be sufficiently accurate in most cases at the expense of CPU time. For evaluation of cdf we use Zolatarev :math:`S_0` parameterization with integration or integral of the pdf FFT interpolated spline. The settings affecting FFT calculation are the same as for pdf calculation. Setting the threshold to ``None`` (default) will disable FFT. For cdf calculations the Zolatarev method is superior in accuracy, so FFT is disabled by default. Fitting estimate uses quantile estimation method in [MC]. MLE estimation of parameters in fit method uses this quantile estimate initially. Note that MLE doesn't always converge if using FFT for pdf calculations; so it's best that ``pdf_fft_min_points_threshold`` is left unset. .. warning:: For pdf calculations implementation of Zolatarev is unstable for values where alpha = 1 and beta != 0. In this case the quadrature method is recommended. FFT calculation is also considered experimental. For cdf calculations FFT calculation is considered experimental. Use Zolatarev's method instead (default). %(after_notes)s References ---------- .. [MC] McCulloch, J., 1986. Simple consistent estimators of stable distribution parameters. Communications in Statistics - Simulation and Computation 15, 11091136. .. [MS] Mittnik, S.T. Rachev, T. Doganoglu, D. Chenyao, 1999. Maximum likelihood estimation of stable Paretian models, Mathematical and Computer Modelling, Volume 29, Issue 10, 1999, Pages 275-293. .. [BS] Borak, S., Hardle, W., Rafal, W. 2005. Stable distributions, Economic Risk. %(example)s """ def _rvs(self, alpha, beta, size=None, random_state=None): def alpha1func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W): return (2/np.pi*(np.pi/2 + bTH)*tanTH - beta*np.log((np.pi/2*W*cosTH)/(np.pi/2 + bTH))) def beta0func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W): return (W/(cosTH/np.tan(aTH) + np.sin(TH)) * ((np.cos(aTH) + np.sin(aTH)*tanTH)/W)**(1.0/alpha)) def otherwise(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W): # alpha is not 1 and beta is not 0 val0 = beta*np.tan(np.pi*alpha/2) th0 = np.arctan(val0)/alpha val3 = W/(cosTH/np.tan(alpha*(th0 + TH)) + np.sin(TH)) res3 = val3*((np.cos(aTH) + np.sin(aTH)*tanTH - val0*(np.sin(aTH) - np.cos(aTH)*tanTH))/W)**(1.0/alpha) return res3 def alphanot1func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W): res = _lazywhere(beta == 0, (alpha, beta, TH, aTH, bTH, cosTH, tanTH, W), beta0func, f2=otherwise) return res alpha = np.broadcast_to(alpha, size) beta = np.broadcast_to(beta, size) TH = uniform.rvs(loc=-np.pi/2.0, scale=np.pi, size=size, random_state=random_state) W = expon.rvs(size=size, random_state=random_state) aTH = alpha*TH bTH = beta*TH cosTH = np.cos(TH) tanTH = np.tan(TH) res = _lazywhere(alpha == 1, (alpha, beta, TH, aTH, bTH, cosTH, tanTH, W), alpha1func, f2=alphanot1func) return res def _argcheck(self, alpha, beta): return (alpha > 0) & (alpha <= 2) & (beta <= 1) & (beta >= -1) @staticmethod def _cf(t, alpha, beta): Phi = lambda alpha, t: np.tan(np.pi*alpha/2) if alpha != 1 else -2.0*np.log(np.abs(t))/np.pi return np.exp(-(np.abs(t)**alpha)*(1-1j*beta*np.sign(t)*Phi(alpha, t))) @staticmethod def _pdf_from_cf_with_fft(cf, h=0.01, q=9): """Calculates pdf from cf using fft. Using region around 0 with N=2**q points separated by distance h. As suggested by [MS]. """ N = 2**q n = np.arange(1,N+1) density = ((-1)**(n-1-N/2))*np.fft.fft(((-1)**(n-1))*cf(2*np.pi*(n-1-N/2)/h/N))/h/N x = (n-1-N/2)*h return (x, density) @staticmethod def _pdf_single_value_best(x, alpha, beta): if alpha != 1. or (alpha == 1. and beta == 0.): return levy_stable_gen._pdf_single_value_zolotarev(x, alpha, beta) else: return levy_stable_gen._pdf_single_value_cf_integrate(x, alpha, beta) @staticmethod def _pdf_single_value_cf_integrate(x, alpha, beta): cf = lambda t: levy_stable_gen._cf(t, alpha, beta) return integrate.quad(lambda t: np.real(np.exp(-1j*t*x)*cf(t)), -np.inf, np.inf, limit=1000)[0]/np.pi/2 @staticmethod def _pdf_single_value_zolotarev(x, alpha, beta): """Calculate pdf using Zolotarev's methods as detailed in [BS]. """ zeta = -beta*np.tan(np.pi*alpha/2.) if alpha != 1: x0 = x + zeta # convert to S_0 parameterization xi = np.arctan(-zeta)/alpha def V(theta): return np.cos(alpha*xi)**(1/(alpha-1)) * \ (np.cos(theta)/np.sin(alpha*(xi+theta)))**(alpha/(alpha-1)) * \ (np.cos(alpha*xi+(alpha-1)*theta)/np.cos(theta)) if x0 > zeta: def g(theta): return (V(theta) * np.real(np.complex128(x0-zeta)**(alpha/(alpha-1)))) def f(theta): return g(theta) * np.exp(-g(theta)) # spare calculating integral on null set # use isclose as macos has fp differences if np.isclose(-xi, np.pi/2, rtol=1e-014, atol=1e-014): return 0. with np.errstate(all="ignore"): intg_max = optimize.minimize_scalar(lambda theta: -f(theta), bounds=[-xi, np.pi/2]) intg_kwargs = {} # windows quadpack less forgiving with points out of bounds if intg_max.success and not np.isnan(intg_max.fun)\ and intg_max.x > -xi and intg_max.x < np.pi/2: intg_kwargs["points"] = [intg_max.x] intg = integrate.quad(f, -xi, np.pi/2, **intg_kwargs)[0] return alpha * intg / np.pi / np.abs(alpha-1) / (x0-zeta) elif x0 == zeta: return sc.gamma(1+1/alpha)*np.cos(xi)/np.pi/((1+zeta**2)**(1/alpha/2)) else: return levy_stable_gen._pdf_single_value_zolotarev(-x, alpha, -beta) else: # since location zero, no need to reposition x for S_0 parameterization xi = np.pi/2 if beta != 0: warnings.warn('Density calculation unstable for alpha=1 and beta!=0.' + ' Use quadrature method instead.', RuntimeWarning) def V(theta): expr_1 = np.pi/2+beta*theta return 2. * expr_1 * np.exp(expr_1*np.tan(theta)/beta) / np.cos(theta) / np.pi def g(theta): return np.exp(-np.pi * x / 2. / beta) * V(theta) def f(theta): return g(theta) * np.exp(-g(theta)) with np.errstate(all="ignore"): intg_max = optimize.minimize_scalar(lambda theta: -f(theta), bounds=[-np.pi/2, np.pi/2]) intg = integrate.fixed_quad(f, -np.pi/2, intg_max.x)[0] + integrate.fixed_quad(f, intg_max.x, np.pi/2)[0] return intg / np.abs(beta) / 2. else: return 1/(1+x**2)/np.pi @staticmethod def _cdf_single_value_zolotarev(x, alpha, beta): """Calculate cdf using Zolotarev's methods as detailed in [BS]. """ zeta = -beta*np.tan(np.pi*alpha/2.) if alpha != 1: x0 = x + zeta # convert to S_0 parameterization xi = np.arctan(-zeta)/alpha def V(theta): return np.cos(alpha*xi)**(1/(alpha-1)) * \ (np.cos(theta)/np.sin(alpha*(xi+theta)))**(alpha/(alpha-1)) * \ (np.cos(alpha*xi+(alpha-1)*theta)/np.cos(theta)) if x0 > zeta: c_1 = 1 if alpha > 1 else .5 - xi/np.pi def f(theta): z = np.complex128(x0 - zeta) return np.exp(-V(theta) * np.real(z**(alpha/(alpha-1)))) with np.errstate(all="ignore"): # spare calculating integral on null set # use isclose as macos has fp differences if np.isclose(-xi, np.pi/2, rtol=1e-014, atol=1e-014): intg = 0 else: intg = integrate.quad(f, -xi, np.pi/2)[0] return c_1 + np.sign(1-alpha) * intg / np.pi elif x0 == zeta: return .5 - xi/np.pi else: return 1 - levy_stable_gen._cdf_single_value_zolotarev(-x, alpha, -beta) else: # since location zero, no need to reposition x for S_0 parameterization xi = np.pi/2 if beta > 0: def V(theta): expr_1 = np.pi/2+beta*theta return 2. * expr_1 * np.exp(expr_1*np.tan(theta)/beta) / np.cos(theta) / np.pi with np.errstate(all="ignore"): expr_1 = np.exp(-np.pi*x/beta/2.) int_1 = integrate.quad(lambda theta: np.exp(-expr_1 * V(theta)), -np.pi/2, np.pi/2)[0] return int_1 / np.pi elif beta == 0: return .5 + np.arctan(x)/np.pi else: return 1 - levy_stable_gen._cdf_single_value_zolotarev(-x, 1, -beta) def _pdf(self, x, alpha, beta): x = np.asarray(x).reshape(1, -1)[0,:] x, alpha, beta = np.broadcast_arrays(x, alpha, beta) data_in = np.dstack((x, alpha, beta))[0] data_out = np.empty(shape=(len(data_in),1)) pdf_default_method_name = getattr(self, 'pdf_default_method', 'best') if pdf_default_method_name == 'best': pdf_single_value_method = levy_stable_gen._pdf_single_value_best elif pdf_default_method_name == 'zolotarev': pdf_single_value_method = levy_stable_gen._pdf_single_value_zolotarev else: pdf_single_value_method = levy_stable_gen._pdf_single_value_cf_integrate fft_min_points_threshold = getattr(self, 'pdf_fft_min_points_threshold', None) fft_grid_spacing = getattr(self, 'pdf_fft_grid_spacing', 0.001) fft_n_points_two_power = getattr(self, 'pdf_fft_n_points_two_power', None) # group data in unique arrays of alpha, beta pairs uniq_param_pairs = np.vstack(list({tuple(row) for row in data_in[:, 1:]})) for pair in uniq_param_pairs: data_mask = np.all(data_in[:,1:] == pair, axis=-1) data_subset = data_in[data_mask] if fft_min_points_threshold is None or len(data_subset) < fft_min_points_threshold: data_out[data_mask] = np.array([pdf_single_value_method(_x, _alpha, _beta) for _x, _alpha, _beta in data_subset]).reshape(len(data_subset), 1) else: warnings.warn('Density calculations experimental for FFT method.' + ' Use combination of zolatarev and quadrature methods instead.', RuntimeWarning) _alpha, _beta = pair _x = data_subset[:,(0,)] # need enough points to "cover" _x for interpolation h = fft_grid_spacing q = np.ceil(np.log(2*np.max(np.abs(_x))/h)/np.log(2)) + 2 if fft_n_points_two_power is None else int(fft_n_points_two_power) density_x, density = levy_stable_gen._pdf_from_cf_with_fft(lambda t: levy_stable_gen._cf(t, _alpha, _beta), h=h, q=q) f = interpolate.interp1d(density_x, np.real(density)) data_out[data_mask] = f(_x) return data_out.T[0] def _cdf(self, x, alpha, beta): x = np.asarray(x).reshape(1, -1)[0,:] x, alpha, beta = np.broadcast_arrays(x, alpha, beta) data_in = np.dstack((x, alpha, beta))[0] data_out = np.empty(shape=(len(data_in),1)) fft_min_points_threshold = getattr(self, 'pdf_fft_min_points_threshold', None) fft_grid_spacing = getattr(self, 'pdf_fft_grid_spacing', 0.001) fft_n_points_two_power = getattr(self, 'pdf_fft_n_points_two_power', None) # group data in unique arrays of alpha, beta pairs uniq_param_pairs = np.vstack( list({tuple(row) for row in data_in[:,1:]})) for pair in uniq_param_pairs: data_mask = np.all(data_in[:,1:] == pair, axis=-1) data_subset = data_in[data_mask] if fft_min_points_threshold is None or len(data_subset) < fft_min_points_threshold: data_out[data_mask] = np.array([levy_stable._cdf_single_value_zolotarev(_x, _alpha, _beta) for _x, _alpha, _beta in data_subset]).reshape(len(data_subset), 1) else: warnings.warn(u'FFT method is considered experimental for ' + u'cumulative distribution function ' + u'evaluations. Use Zolotarev’s method instead).', RuntimeWarning) _alpha, _beta = pair _x = data_subset[:,(0,)] # need enough points to "cover" _x for interpolation h = fft_grid_spacing q = 16 if fft_n_points_two_power is None else int(fft_n_points_two_power) density_x, density = levy_stable_gen._pdf_from_cf_with_fft(lambda t: levy_stable_gen._cf(t, _alpha, _beta), h=h, q=q) f = interpolate.InterpolatedUnivariateSpline(density_x, np.real(density)) data_out[data_mask] = np.array([f.integral(self.a, x_1) for x_1 in _x]).reshape(data_out[data_mask].shape) return data_out.T[0] def _fitstart(self, data): # We follow McCullock 1986 method - Simple Consistent Estimators # of Stable Distribution Parameters # Table III and IV nu_alpha_range = [2.439, 2.5, 2.6, 2.7, 2.8, 3, 3.2, 3.5, 4, 5, 6, 8, 10, 15, 25] nu_beta_range = [0, 0.1, 0.2, 0.3, 0.5, 0.7, 1] # table III - alpha = psi_1(nu_alpha, nu_beta) alpha_table = [ [2.000, 2.000, 2.000, 2.000, 2.000, 2.000, 2.000], [1.916, 1.924, 1.924, 1.924, 1.924, 1.924, 1.924], [1.808, 1.813, 1.829, 1.829, 1.829, 1.829, 1.829], [1.729, 1.730, 1.737, 1.745, 1.745, 1.745, 1.745], [1.664, 1.663, 1.663, 1.668, 1.676, 1.676, 1.676], [1.563, 1.560, 1.553, 1.548, 1.547, 1.547, 1.547], [1.484, 1.480, 1.471, 1.460, 1.448, 1.438, 1.438], [1.391, 1.386, 1.378, 1.364, 1.337, 1.318, 1.318], [1.279, 1.273, 1.266, 1.250, 1.210, 1.184, 1.150], [1.128, 1.121, 1.114, 1.101, 1.067, 1.027, 0.973], [1.029, 1.021, 1.014, 1.004, 0.974, 0.935, 0.874], [0.896, 0.892, 0.884, 0.883, 0.855, 0.823, 0.769], [0.818, 0.812, 0.806, 0.801, 0.780, 0.756, 0.691], [0.698, 0.695, 0.692, 0.689, 0.676, 0.656, 0.597], [0.593, 0.590, 0.588, 0.586, 0.579, 0.563, 0.513]] # table IV - beta = psi_2(nu_alpha, nu_beta) beta_table = [ [0, 2.160, 1.000, 1.000, 1.000, 1.000, 1.000], [0, 1.592, 3.390, 1.000, 1.000, 1.000, 1.000], [0, 0.759, 1.800, 1.000, 1.000, 1.000, 1.000], [0, 0.482, 1.048, 1.694, 1.000, 1.000, 1.000], [0, 0.360, 0.760, 1.232, 2.229, 1.000, 1.000], [0, 0.253, 0.518, 0.823, 1.575, 1.000, 1.000], [0, 0.203, 0.410, 0.632, 1.244, 1.906, 1.000], [0, 0.165, 0.332, 0.499, 0.943, 1.560, 1.000], [0, 0.136, 0.271, 0.404, 0.689, 1.230, 2.195], [0, 0.109, 0.216, 0.323, 0.539, 0.827, 1.917], [0, 0.096, 0.190, 0.284, 0.472, 0.693, 1.759], [0, 0.082, 0.163, 0.243, 0.412, 0.601, 1.596], [0, 0.074, 0.147, 0.220, 0.377, 0.546, 1.482], [0, 0.064, 0.128, 0.191, 0.330, 0.478, 1.362], [0, 0.056, 0.112, 0.167, 0.285, 0.428, 1.274]] # Table V and VII alpha_range = [2, 1.9, 1.8, 1.7, 1.6, 1.5, 1.4, 1.3, 1.2, 1.1, 1, 0.9, 0.8, 0.7, 0.6, 0.5] beta_range = [0, 0.25, 0.5, 0.75, 1] # Table V - nu_c = psi_3(alpha, beta) nu_c_table = [ [1.908, 1.908, 1.908, 1.908, 1.908], [1.914, 1.915, 1.916, 1.918, 1.921], [1.921, 1.922, 1.927, 1.936, 1.947], [1.927, 1.930, 1.943, 1.961, 1.987], [1.933, 1.940, 1.962, 1.997, 2.043], [1.939, 1.952, 1.988, 2.045, 2.116], [1.946, 1.967, 2.022, 2.106, 2.211], [1.955, 1.984, 2.067, 2.188, 2.333], [1.965, 2.007, 2.125, 2.294, 2.491], [1.980, 2.040, 2.205, 2.435, 2.696], [2.000, 2.085, 2.311, 2.624, 2.973], [2.040, 2.149, 2.461, 2.886, 3.356], [2.098, 2.244, 2.676, 3.265, 3.912], [2.189, 2.392, 3.004, 3.844, 4.775], [2.337, 2.634, 3.542, 4.808, 6.247], [2.588, 3.073, 4.534, 6.636, 9.144]] # Table VII - nu_zeta = psi_5(alpha, beta) nu_zeta_table = [ [0, 0.000, 0.000, 0.000, 0.000], [0, -0.017, -0.032, -0.049, -0.064], [0, -0.030, -0.061, -0.092, -0.123], [0, -0.043, -0.088, -0.132, -0.179], [0, -0.056, -0.111, -0.170, -0.232], [0, -0.066, -0.134, -0.206, -0.283], [0, -0.075, -0.154, -0.241, -0.335], [0, -0.084, -0.173, -0.276, -0.390], [0, -0.090, -0.192, -0.310, -0.447], [0, -0.095, -0.208, -0.346, -0.508], [0, -0.098, -0.223, -0.380, -0.576], [0, -0.099, -0.237, -0.424, -0.652], [0, -0.096, -0.250, -0.469, -0.742], [0, -0.089, -0.262, -0.520, -0.853], [0, -0.078, -0.272, -0.581, -0.997], [0, -0.061, -0.279, -0.659, -1.198]] psi_1 = interpolate.interp2d(nu_beta_range, nu_alpha_range, alpha_table, kind='linear') psi_2 = interpolate.interp2d(nu_beta_range, nu_alpha_range, beta_table, kind='linear') psi_2_1 = lambda nu_beta, nu_alpha: psi_2(nu_beta, nu_alpha) if nu_beta > 0 else -psi_2(-nu_beta, nu_alpha) phi_3 = interpolate.interp2d(beta_range, alpha_range, nu_c_table, kind='linear') phi_3_1 = lambda beta, alpha: phi_3(beta, alpha) if beta > 0 else phi_3(-beta, alpha) phi_5 = interpolate.interp2d(beta_range, alpha_range, nu_zeta_table, kind='linear') phi_5_1 = lambda beta, alpha: phi_5(beta, alpha) if beta > 0 else -phi_5(-beta, alpha) # quantiles p05 = np.percentile(data, 5) p50 = np.percentile(data, 50) p95 = np.percentile(data, 95) p25 = np.percentile(data, 25) p75 = np.percentile(data, 75) nu_alpha = (p95 - p05)/(p75 - p25) nu_beta = (p95 + p05 - 2*p50)/(p95 - p05) if nu_alpha >= 2.439: alpha = np.clip(psi_1(nu_beta, nu_alpha)[0], np.finfo(float).eps, 2.) beta = np.clip(psi_2_1(nu_beta, nu_alpha)[0], -1., 1.) else: alpha = 2.0 beta = np.sign(nu_beta) c = (p75 - p25) / phi_3_1(beta, alpha)[0] zeta = p50 + c*phi_5_1(beta, alpha)[0] delta = np.clip(zeta-beta*c*np.tan(np.pi*alpha/2.) if alpha == 1. else zeta, np.finfo(float).eps, np.inf) return (alpha, beta, delta, c) def _stats(self, alpha, beta): mu = 0 if alpha > 1 else np.nan mu2 = 2 if alpha == 2 else np.inf g1 = 0. if alpha == 2. else np.NaN g2 = 0. if alpha == 2. else np.NaN return mu, mu2, g1, g2 levy_stable = levy_stable_gen(name='levy_stable') class logistic_gen(rv_continuous): r"""A logistic (or Sech-squared) continuous random variable. %(before_notes)s Notes ----- The probability density function for `logistic` is: .. math:: f(x) = \frac{\exp(-x)} {(1+\exp(-x))^2} `logistic` is a special case of `genlogistic` with ``c=1``. %(after_notes)s %(example)s """ def _rvs(self, size=None, random_state=None): return random_state.logistic(size=size) def _pdf(self, x): # logistic.pdf(x) = exp(-x) / (1+exp(-x))**2 return np.exp(self._logpdf(x)) def _logpdf(self, x): return -x - 2. * sc.log1p(np.exp(-x)) def _cdf(self, x): return sc.expit(x) def _ppf(self, q): return sc.logit(q) def _sf(self, x): return sc.expit(-x) def _isf(self, q): return -sc.logit(q) def _stats(self): return 0, np.pi*np.pi/3.0, 0, 6.0/5.0 def _entropy(self): # https://en.wikipedia.org/wiki/Logistic_distribution return 2.0 logistic = logistic_gen(name='logistic') class loggamma_gen(rv_continuous): r"""A log gamma continuous random variable. %(before_notes)s Notes ----- The probability density function for `loggamma` is: .. math:: f(x, c) = \frac{\exp(c x - \exp(x))} {\Gamma(c)} for all :math:`x, c > 0`. Here, :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `loggamma` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _rvs(self, c, size=None, random_state=None): return np.log(random_state.gamma(c, size=size)) def _pdf(self, x, c): # loggamma.pdf(x, c) = exp(c*x-exp(x)) / gamma(c) return np.exp(c*x-np.exp(x)-sc.gammaln(c)) def _cdf(self, x, c): return sc.gammainc(c, np.exp(x)) def _ppf(self, q, c): return np.log(sc.gammaincinv(c, q)) def _stats(self, c): # See, for example, "A Statistical Study of Log-Gamma Distribution", by # Ping Shing Chan (thesis, McMaster University, 1993). mean = sc.digamma(c) var = sc.polygamma(1, c) skewness = sc.polygamma(2, c) / np.power(var, 1.5) excess_kurtosis = sc.polygamma(3, c) / (var*var) return mean, var, skewness, excess_kurtosis loggamma = loggamma_gen(name='loggamma') class loglaplace_gen(rv_continuous): r"""A log-Laplace continuous random variable. %(before_notes)s Notes ----- The probability density function for `loglaplace` is: .. math:: f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1} &\text{for } 0 < x < 1\\ \frac{c}{2} x^{-c-1} &\text{for } x \ge 1 \end{cases} for :math:`c > 0`. `loglaplace` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s References ---------- T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model", The Mathematical Scientist, vol. 28, pp. 49-60, 2003. %(example)s """ def _pdf(self, x, c): # loglaplace.pdf(x, c) = c / 2 * x**(c-1), for 0 < x < 1 # = c / 2 * x**(-c-1), for x >= 1 cd2 = c/2.0 c = np.where(x < 1, c, -c) return cd2*x**(c-1) def _cdf(self, x, c): return np.where(x < 1, 0.5*x**c, 1-0.5*x**(-c)) def _ppf(self, q, c): return np.where(q < 0.5, (2.0*q)**(1.0/c), (2*(1.0-q))**(-1.0/c)) def _munp(self, n, c): return c**2 / (c**2 - n**2) def _entropy(self, c): return np.log(2.0/c) + 1.0 loglaplace = loglaplace_gen(a=0.0, name='loglaplace') def _lognorm_logpdf(x, s): return _lazywhere(x != 0, (x, s), lambda x, s: -np.log(x)**2 / (2*s**2) - np.log(s*x*np.sqrt(2*np.pi)), -np.inf) class lognorm_gen(rv_continuous): r"""A lognormal continuous random variable. %(before_notes)s Notes ----- The probability density function for `lognorm` is: .. math:: f(x, s) = \frac{1}{s x \sqrt{2\pi}} \exp\left(-\frac{\log^2(x)}{2s^2}\right) for :math:`x > 0`, :math:`s > 0`. `lognorm` takes ``s`` as a shape parameter for :math:`s`. %(after_notes)s A common parametrization for a lognormal random variable ``Y`` is in terms of the mean, ``mu``, and standard deviation, ``sigma``, of the unique normally distributed random variable ``X`` such that exp(X) = Y. This parametrization corresponds to setting ``s = sigma`` and ``scale = exp(mu)``. %(example)s """ _support_mask = rv_continuous._open_support_mask def _rvs(self, s, size=None, random_state=None): return np.exp(s * random_state.standard_normal(size)) def _pdf(self, x, s): # lognorm.pdf(x, s) = 1 / (s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2) return np.exp(self._logpdf(x, s)) def _logpdf(self, x, s): return _lognorm_logpdf(x, s) def _cdf(self, x, s): return _norm_cdf(np.log(x) / s) def _logcdf(self, x, s): return _norm_logcdf(np.log(x) / s) def _ppf(self, q, s): return np.exp(s * _norm_ppf(q)) def _sf(self, x, s): return _norm_sf(np.log(x) / s) def _logsf(self, x, s): return _norm_logsf(np.log(x) / s) def _stats(self, s): p = np.exp(s*s) mu = np.sqrt(p) mu2 = p*(p-1) g1 = np.sqrt((p-1))*(2+p) g2 = np.polyval([1, 2, 3, 0, -6.0], p) return mu, mu2, g1, g2 def _entropy(self, s): return 0.5 * (1 + np.log(2*np.pi) + 2 * np.log(s)) @extend_notes_in_docstring(rv_continuous, notes="""\ When the location parameter is fixed by using the `floc` argument, this function uses explicit formulas for the maximum likelihood estimation of the log-normal shape and scale parameters, so the `optimizer`, `loc` and `scale` keyword arguments are ignored.\n\n""") def fit(self, data, *args, **kwds): floc = kwds.get('floc', None) if floc is None: # loc is not fixed. Use the default fit method. return super(lognorm_gen, self).fit(data, *args, **kwds) f0 = (kwds.get('f0', None) or kwds.get('fs', None) or kwds.get('fix_s', None)) fscale = kwds.get('fscale', None) if len(args) > 1: raise TypeError("Too many input arguments.") for name in ['f0', 'fs', 'fix_s', 'floc', 'fscale', 'loc', 'scale', 'optimizer']: kwds.pop(name, None) if kwds: raise TypeError("Unknown arguments: %s." % kwds) # Special case: loc is fixed. Use the maximum likelihood formulas # instead of the numerical solver. if f0 is not None and fscale is not None: # This check is for consistency with `rv_continuous.fit`. raise ValueError("All parameters fixed. There is nothing to " "optimize.") data = np.asarray(data) if not np.isfinite(data).all(): raise RuntimeError("The data contains non-finite values.") floc = float(floc) if floc != 0: # Shifting the data by floc. Don't do the subtraction in-place, # because `data` might be a view of the input array. data = data - floc if np.any(data <= 0): raise FitDataError("lognorm", lower=floc, upper=np.inf) lndata = np.log(data) # Three cases to handle: # * shape and scale both free # * shape fixed, scale free # * shape free, scale fixed if fscale is None: # scale is free. scale = np.exp(lndata.mean()) if f0 is None: # shape is free. shape = lndata.std() else: # shape is fixed. shape = float(f0) else: # scale is fixed, shape is free scale = float(fscale) shape = np.sqrt(((lndata - np.log(scale))**2).mean()) return shape, floc, scale lognorm = lognorm_gen(a=0.0, name='lognorm') class gilbrat_gen(rv_continuous): r"""A Gilbrat continuous random variable. %(before_notes)s Notes ----- The probability density function for `gilbrat` is: .. math:: f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2) `gilbrat` is a special case of `lognorm` with ``s=1``. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _rvs(self, size=None, random_state=None): return np.exp(random_state.standard_normal(size)) def _pdf(self, x): # gilbrat.pdf(x) = 1/(x*sqrt(2*pi)) * exp(-1/2*(log(x))**2) return np.exp(self._logpdf(x)) def _logpdf(self, x): return _lognorm_logpdf(x, 1.0) def _cdf(self, x): return _norm_cdf(np.log(x)) def _ppf(self, q): return np.exp(_norm_ppf(q)) def _stats(self): p = np.e mu = np.sqrt(p) mu2 = p * (p - 1) g1 = np.sqrt((p - 1)) * (2 + p) g2 = np.polyval([1, 2, 3, 0, -6.0], p) return mu, mu2, g1, g2 def _entropy(self): return 0.5 * np.log(2 * np.pi) + 0.5 gilbrat = gilbrat_gen(a=0.0, name='gilbrat') class maxwell_gen(rv_continuous): r"""A Maxwell continuous random variable. %(before_notes)s Notes ----- A special case of a `chi` distribution, with ``df=3``, ``loc=0.0``, and given ``scale = a``, where ``a`` is the parameter used in the Mathworld description [1]_. The probability density function for `maxwell` is: .. math:: f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2) for :math:`x >= 0`. %(after_notes)s References ---------- .. [1] http://mathworld.wolfram.com/MaxwellDistribution.html %(example)s """ def _rvs(self, size=None, random_state=None): return chi.rvs(3.0, size=size, random_state=random_state) def _pdf(self, x): # maxwell.pdf(x) = sqrt(2/pi)x**2 * exp(-x**2/2) return _SQRT_2_OVER_PI*x*x*np.exp(-x*x/2.0) def _logpdf(self, x): return _LOG_SQRT_2_OVER_PI + 2*np.log(x) - 0.5*x*x def _cdf(self, x): return sc.gammainc(1.5, x*x/2.0) def _ppf(self, q): return np.sqrt(2*sc.gammaincinv(1.5, q)) def _stats(self): val = 3*np.pi-8 return (2*np.sqrt(2.0/np.pi), 3-8/np.pi, np.sqrt(2)*(32-10*np.pi)/val**1.5, (-12*np.pi*np.pi + 160*np.pi - 384) / val**2.0) def _entropy(self): return _EULER + 0.5*np.log(2*np.pi)-0.5 maxwell = maxwell_gen(a=0.0, name='maxwell') class mielke_gen(rv_continuous): r"""A Mielke Beta-Kappa / Dagum continuous random variable. %(before_notes)s Notes ----- The probability density function for `mielke` is: .. math:: f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}} for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes called Dagum distribution ([2]_). It was already defined in [3]_, called a Burr Type III distribution (`burr` with parameters ``c=s`` and ``d=k/s``). `mielke` takes ``k`` and ``s`` as shape parameters. %(after_notes)s References ---------- .. [1] Mielke, P.W., 1973 "Another Family of Distributions for Describing and Analyzing Precipitation Data." J. Appl. Meteor., 12, 275-280 .. [2] Dagum, C., 1977 "A new model for personal income distribution." Economie Appliquee, 33, 327-367. .. [3] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). %(example)s """ def _argcheck(self, k, s): return (k > 0) & (s > 0) def _pdf(self, x, k, s): return k*x**(k-1.0) / (1.0+x**s)**(1.0+k*1.0/s) def _logpdf(self, x, k, s): return np.log(k) + np.log(x)*(k-1.0) - np.log1p(x**s)*(1.0+k*1.0/s) def _cdf(self, x, k, s): return x**k / (1.0+x**s)**(k*1.0/s) def _ppf(self, q, k, s): qsk = pow(q, s*1.0/k) return pow(qsk/(1.0-qsk), 1.0/s) def _munp(self, n, k, s): def nth_moment(n, k, s): # n-th moment is defined for -k < n < s return sc.gamma((k+n)/s)*sc.gamma(1-n/s)/sc.gamma(k/s) return _lazywhere(n < s, (n, k, s), nth_moment, np.inf) mielke = mielke_gen(a=0.0, name='mielke') class kappa4_gen(rv_continuous): r"""Kappa 4 parameter distribution. %(before_notes)s Notes ----- The probability density function for kappa4 is: .. math:: f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1} if :math:`h` and :math:`k` are not equal to 0. If :math:`h` or :math:`k` are zero then the pdf can be simplified: h = 0 and k != 0:: kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)* exp(-(1.0 - k*x)**(1.0/k)) h != 0 and k = 0:: kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0) h = 0 and k = 0:: kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x)) kappa4 takes :math:`h` and :math:`k` as shape parameters. The kappa4 distribution returns other distributions when certain :math:`h` and :math:`k` values are used. +------+-------------+----------------+------------------+ | h | k=0.0 | k=1.0 | -inf<=k<=inf | +======+=============+================+==================+ | -1.0 | Logistic | | Generalized | | | | | Logistic(1) | | | | | | | | logistic(x) | | | +------+-------------+----------------+------------------+ | 0.0 | Gumbel | Reverse | Generalized | | | | Exponential(2) | Extreme Value | | | | | | | | gumbel_r(x) | | genextreme(x, k) | +------+-------------+----------------+------------------+ | 1.0 | Exponential | Uniform | Generalized | | | | | Pareto | | | | | | | | expon(x) | uniform(x) | genpareto(x, -k) | +------+-------------+----------------+------------------+ (1) There are at least five generalized logistic distributions. Four are described here: https://en.wikipedia.org/wiki/Generalized_logistic_distribution The "fifth" one is the one kappa4 should match which currently isn't implemented in scipy: https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html (2) This distribution is currently not in scipy. References ---------- J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College, (August, 2004), https://digitalcommons.lsu.edu/gradschool_dissertations/3672 J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res. Develop. 38 (3), 25 1-258 (1994). B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao Site in the Chi River Basin, Thailand", Journal of Water Resource and Protection, vol. 4, 866-869, (2012). https://doi.org/10.4236/jwarp.2012.410101 C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March 2000). http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf %(after_notes)s %(example)s """ def _argcheck(self, h, k): return h == h def _get_support(self, h, k): condlist = [np.logical_and(h > 0, k > 0), np.logical_and(h > 0, k == 0), np.logical_and(h > 0, k < 0), np.logical_and(h <= 0, k > 0), np.logical_and(h <= 0, k == 0), np.logical_and(h <= 0, k < 0)] def f0(h, k): return (1.0 - float_power(h, -k))/k def f1(h, k): return np.log(h) def f3(h, k): a = np.empty(np.shape(h)) a[:] = -np.inf return a def f5(h, k): return 1.0/k _a = _lazyselect(condlist, [f0, f1, f0, f3, f3, f5], [h, k], default=np.nan) def f0(h, k): return 1.0/k def f1(h, k): a = np.empty(np.shape(h)) a[:] = np.inf return a _b = _lazyselect(condlist, [f0, f1, f1, f0, f1, f1], [h, k], default=np.nan) return _a, _b def _pdf(self, x, h, k): # kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)* # (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1) return np.exp(self._logpdf(x, h, k)) def _logpdf(self, x, h, k): condlist = [np.logical_and(h != 0, k != 0), np.logical_and(h == 0, k != 0), np.logical_and(h != 0, k == 0), np.logical_and(h == 0, k == 0)] def f0(x, h, k): '''pdf = (1.0 - k*x)**(1.0/k - 1.0)*( 1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0) logpdf = ... ''' return (sc.xlog1py(1.0/k - 1.0, -k*x) + sc.xlog1py(1.0/h - 1.0, -h*(1.0 - k*x)**(1.0/k))) def f1(x, h, k): '''pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-( 1.0 - k*x)**(1.0/k)) logpdf = ... ''' return sc.xlog1py(1.0/k - 1.0, -k*x) - (1.0 - k*x)**(1.0/k) def f2(x, h, k): '''pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0) logpdf = ... ''' return -x + sc.xlog1py(1.0/h - 1.0, -h*np.exp(-x)) def f3(x, h, k): '''pdf = np.exp(-x-np.exp(-x)) logpdf = ... ''' return -x - np.exp(-x) return _lazyselect(condlist, [f0, f1, f2, f3], [x, h, k], default=np.nan) def _cdf(self, x, h, k): return np.exp(self._logcdf(x, h, k)) def _logcdf(self, x, h, k): condlist = [np.logical_and(h != 0, k != 0), np.logical_and(h == 0, k != 0), np.logical_and(h != 0, k == 0), np.logical_and(h == 0, k == 0)] def f0(x, h, k): '''cdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h) logcdf = ... ''' return (1.0/h)*sc.log1p(-h*(1.0 - k*x)**(1.0/k)) def f1(x, h, k): '''cdf = np.exp(-(1.0 - k*x)**(1.0/k)) logcdf = ... ''' return -(1.0 - k*x)**(1.0/k) def f2(x, h, k): '''cdf = (1.0 - h*np.exp(-x))**(1.0/h) logcdf = ... ''' return (1.0/h)*sc.log1p(-h*np.exp(-x)) def f3(x, h, k): '''cdf = np.exp(-np.exp(-x)) logcdf = ... ''' return -np.exp(-x) return _lazyselect(condlist, [f0, f1, f2, f3], [x, h, k], default=np.nan) def _ppf(self, q, h, k): condlist = [np.logical_and(h != 0, k != 0), np.logical_and(h == 0, k != 0), np.logical_and(h != 0, k == 0), np.logical_and(h == 0, k == 0)] def f0(q, h, k): return 1.0/k*(1.0 - ((1.0 - (q**h))/h)**k) def f1(q, h, k): return 1.0/k*(1.0 - (-np.log(q))**k) def f2(q, h, k): '''ppf = -np.log((1.0 - (q**h))/h) ''' return -sc.log1p(-(q**h)) + np.log(h) def f3(q, h, k): return -np.log(-np.log(q)) return _lazyselect(condlist, [f0, f1, f2, f3], [q, h, k], default=np.nan) def _stats(self, h, k): if h >= 0 and k >= 0: maxr = 5 elif h < 0 and k >= 0: maxr = int(-1.0/h*k) elif k < 0: maxr = int(-1.0/k) else: maxr = 5 outputs = [None if r < maxr else np.nan for r in range(1, 5)] return outputs[:] kappa4 = kappa4_gen(name='kappa4') class kappa3_gen(rv_continuous): r"""Kappa 3 parameter distribution. %(before_notes)s Notes ----- The probability density function for `kappa3` is: .. math:: f(x, a) = a (a + x^a)^{-(a + 1)/a} for :math:`x > 0` and :math:`a > 0`. `kappa3` takes ``a`` as a shape parameter for :math:`a`. References ---------- P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum Likelihood and Likelihood Ratio Tests", Methods in Weather Research, 701-707, (September, 1973), https://doi.org/10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2 B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2, 415-419 (2012), https://doi.org/10.4236/ojs.2012.24050 %(after_notes)s %(example)s """ def _argcheck(self, a): return a > 0 def _pdf(self, x, a): # kappa3.pdf(x, a) = a*(a + x**a)**(-(a + 1)/a), for x > 0 return a*(a + x**a)**(-1.0/a-1) def _cdf(self, x, a): return x*(a + x**a)**(-1.0/a) def _ppf(self, q, a): return (a/(q**-a - 1.0))**(1.0/a) def _stats(self, a): outputs = [None if i < a else np.nan for i in range(1, 5)] return outputs[:] kappa3 = kappa3_gen(a=0.0, name='kappa3') class moyal_gen(rv_continuous): r"""A Moyal continuous random variable. %(before_notes)s Notes ----- The probability density function for `moyal` is: .. math:: f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi} for a real number :math:`x`. %(after_notes)s This distribution has utility in high-energy physics and radiation detection. It describes the energy loss of a charged relativistic particle due to ionization of the medium [1]_. It also provides an approximation for the Landau distribution. For an in depth description see [2]_. For additional description, see [3]_. References ---------- .. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol 46, 263-280, (1955). :doi:`10.1080/14786440308521076` (gated) .. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution", International Journal of Research and Reviews in Applied Sciences, vol 10, 171-192, (2012). http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf .. [3] C. Walck, "Handbook on Statistical Distributions for Experimentalists; International Report SUF-PFY/96-01", Chapter 26, University of Stockholm: Stockholm, Sweden, (2007). http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf .. versionadded:: 1.1.0 %(example)s """ def _rvs(self, size=None, random_state=None): u1 = gamma.rvs(a = 0.5, scale = 2, size=size, random_state=random_state) return -np.log(u1) def _pdf(self, x): return np.exp(-0.5 * (x + np.exp(-x))) / np.sqrt(2*np.pi) def _cdf(self, x): return sc.erfc(np.exp(-0.5 * x) / np.sqrt(2)) def _sf(self, x): return sc.erf(np.exp(-0.5 * x) / np.sqrt(2)) def _ppf(self, x): return -np.log(2 * sc.erfcinv(x)**2) def _stats(self): mu = np.log(2) + np.euler_gamma mu2 = np.pi**2 / 2 g1 = 28 * np.sqrt(2) * sc.zeta(3) / np.pi**3 g2 = 4. return mu, mu2, g1, g2 def _munp(self, n): if n == 1.0: return np.log(2) + np.euler_gamma elif n == 2.0: return np.pi**2 / 2 + (np.log(2) + np.euler_gamma)**2 elif n == 3.0: tmp1 = 1.5 * np.pi**2 * (np.log(2)+np.euler_gamma) tmp2 = (np.log(2)+np.euler_gamma)**3 tmp3 = 14 * sc.zeta(3) return tmp1 + tmp2 + tmp3 elif n == 4.0: tmp1 = 4 * 14 * sc.zeta(3) * (np.log(2) + np.euler_gamma) tmp2 = 3 * np.pi**2 * (np.log(2) + np.euler_gamma)**2 tmp3 = (np.log(2) + np.euler_gamma)**4 tmp4 = 7 * np.pi**4 / 4 return tmp1 + tmp2 + tmp3 + tmp4 else: # return generic for higher moments # return rv_continuous._mom1_sc(self, n, b) return self._mom1_sc(n) moyal = moyal_gen(name="moyal") class nakagami_gen(rv_continuous): r"""A Nakagami continuous random variable. %(before_notes)s Notes ----- The probability density function for `nakagami` is: .. math:: f(x, \nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2) for :math:`x >= 0`, :math:`\nu > 0`. `nakagami` takes ``nu`` as a shape parameter for :math:`\nu`. %(after_notes)s %(example)s """ def _pdf(self, x, nu): # nakagami.pdf(x, nu) = 2 * nu**nu / gamma(nu) * # x**(2*nu-1) * exp(-nu*x**2) return 2*nu**nu/sc.gamma(nu)*(x**(2*nu-1.0))*np.exp(-nu*x*x) def _cdf(self, x, nu): return sc.gammainc(nu, nu*x*x) def _ppf(self, q, nu): return np.sqrt(1.0/nu*sc.gammaincinv(nu, q)) def _stats(self, nu): mu = sc.gamma(nu+0.5)/sc.gamma(nu)/np.sqrt(nu) mu2 = 1.0-mu*mu g1 = mu * (1 - 4*nu*mu2) / 2.0 / nu / np.power(mu2, 1.5) g2 = -6*mu**4*nu + (8*nu-2)*mu**2-2*nu + 1 g2 /= nu*mu2**2.0 return mu, mu2, g1, g2 nakagami = nakagami_gen(a=0.0, name="nakagami") class ncx2_gen(rv_continuous): r"""A non-central chi-squared continuous random variable. %(before_notes)s Notes ----- The probability density function for `ncx2` is: .. math:: f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2) (x/\lambda)^{(k-2)/4} I_{(k-2)/2}(\sqrt{\lambda x}) for :math:`x >= 0` and :math:`k, \lambda > 0`. :math:`k` specifies the degrees of freedom (denoted ``df`` in the implementation) and :math:`\lambda` is the non-centrality parameter (denoted ``nc`` in the implementation). :math:`I_\nu` denotes the modified Bessel function of first order of degree :math:`\nu` (`scipy.special.iv`). `ncx2` takes ``df`` and ``nc`` as shape parameters. %(after_notes)s %(example)s """ def _argcheck(self, df, nc): return (df > 0) & (nc >= 0) def _rvs(self, df, nc, size=None, random_state=None): return random_state.noncentral_chisquare(df, nc, size) def _logpdf(self, x, df, nc): cond = np.ones_like(x, dtype=bool) & (nc != 0) return _lazywhere(cond, (x, df, nc), f=_ncx2_log_pdf, f2=chi2.logpdf) def _pdf(self, x, df, nc): # ncx2.pdf(x, df, nc) = exp(-(nc+x)/2) * 1/2 * (x/nc)**((df-2)/4) # * I[(df-2)/2](sqrt(nc*x)) cond = np.ones_like(x, dtype=bool) & (nc != 0) return _lazywhere(cond, (x, df, nc), f=_ncx2_pdf, f2=chi2.pdf) def _cdf(self, x, df, nc): cond = np.ones_like(x, dtype=bool) & (nc != 0) return _lazywhere(cond, (x, df, nc), f=_ncx2_cdf, f2=chi2.cdf) def _ppf(self, q, df, nc): cond = np.ones_like(q, dtype=bool) & (nc != 0) return _lazywhere(cond, (q, df, nc), f=sc.chndtrix, f2=chi2.ppf) def _stats(self, df, nc): val = df + 2.0*nc return (df + nc, 2*val, np.sqrt(8)*(val+nc)/val**1.5, 12.0*(val+2*nc)/val**2.0) ncx2 = ncx2_gen(a=0.0, name='ncx2') class ncf_gen(rv_continuous): r"""A non-central F distribution continuous random variable. %(before_notes)s Notes ----- The probability density function for `ncf` is: .. math:: f(x, n_1, n_2, \lambda) = \exp\left(\frac{\lambda}{2} + \lambda n_1 \frac{x}{2(n_1 x + n_2)} \right) n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\ (n_2 + n_1 x)^{-(n_1 + n_2)/2} \gamma(n_1/2) \gamma(1 + n_2/2) \\ \frac{L^{\frac{n_1}{2}-1}_{n_2/2} \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)} {B(n_1/2, n_2/2) \gamma\left(\frac{n_1 + n_2}{2}\right)} for :math:`n_1, n_2 > 0`, :math:`\lambda\geq 0`. Here :math:`n_1` is the degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in the denominator, :math:`\lambda` the non-centrality parameter, :math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a generalized Laguerre polynomial and :math:`B` is the beta function. `ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters. If ``nc=0``, the distribution becomes equivalent to the Fisher distribution. %(after_notes)s See Also -------- scipy.stats.f : Fisher distribution %(example)s """ def _argcheck(self, df1, df2, nc): return (df1 > 0) & (df2 > 0) & (nc >= 0) def _rvs(self, dfn, dfd, nc, size=None, random_state=None): return random_state.noncentral_f(dfn, dfd, nc, size) def _pdf_skip(self, x, dfn, dfd, nc): # ncf.pdf(x, df1, df2, nc) = exp(nc/2 + nc*df1*x/(2*(df1*x+df2))) * # df1**(df1/2) * df2**(df2/2) * x**(df1/2-1) * # (df2+df1*x)**(-(df1+df2)/2) * # gamma(df1/2)*gamma(1+df2/2) * # L^{v1/2-1}^{v2/2}(-nc*v1*x/(2*(v1*x+v2))) / # (B(v1/2, v2/2) * gamma((v1+v2)/2)) n1, n2 = dfn, dfd term = -nc/2+nc*n1*x/(2*(n2+n1*x)) + sc.gammaln(n1/2.)+sc.gammaln(1+n2/2.) term -= sc.gammaln((n1+n2)/2.0) Px = np.exp(term) Px *= n1**(n1/2) * n2**(n2/2) * x**(n1/2-1) Px *= (n2+n1*x)**(-(n1+n2)/2) Px *= sc.assoc_laguerre(-nc*n1*x/(2.0*(n2+n1*x)), n2/2, n1/2-1) Px /= sc.beta(n1/2, n2/2) # This function does not have a return. Drop it for now, the generic # function seems to work OK. def _cdf(self, x, dfn, dfd, nc): return sc.ncfdtr(dfn, dfd, nc, x) def _ppf(self, q, dfn, dfd, nc): return sc.ncfdtri(dfn, dfd, nc, q) def _munp(self, n, dfn, dfd, nc): val = (dfn * 1.0/dfd)**n term = sc.gammaln(n+0.5*dfn) + sc.gammaln(0.5*dfd-n) - sc.gammaln(dfd*0.5) val *= np.exp(-nc / 2.0+term) val *= sc.hyp1f1(n+0.5*dfn, 0.5*dfn, 0.5*nc) return val def _stats(self, dfn, dfd, nc): # Note: the rv_continuous class ensures that dfn > 0 when this function # is called, so we don't have to check for division by zero with dfn # in the following. mu_num = dfd * (dfn + nc) mu_den = dfn * (dfd - 2) mu = np.full_like(mu_num, dtype=np.float64, fill_value=np.inf) np.true_divide(mu_num, mu_den, where=dfd > 2, out=mu) mu2_num = 2*((dfn + nc)**2 + (dfn + 2*nc)*(dfd - 2))*(dfd/dfn)**2 mu2_den = (dfd - 2)**2 * (dfd - 4) mu2 = np.full_like(mu2_num, dtype=np.float64, fill_value=np.inf) np.true_divide(mu2_num, mu2_den, where=dfd > 4, out=mu2) return mu, mu2, None, None ncf = ncf_gen(a=0.0, name='ncf') class t_gen(rv_continuous): r"""A Student's t continuous random variable. %(before_notes)s Notes ----- The probability density function for `t` is: .. math:: f(x, \nu) = \frac{\Gamma((\nu+1)/2)} {\sqrt{\pi \nu} \Gamma(\nu/2)} (1+x^2/\nu)^{-(\nu+1)/2} where :math:`x` is a real number and the degrees of freedom parameter :math:`\nu` (denoted ``df`` in the implementation) satisfies :math:`\nu > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). %(after_notes)s %(example)s """ def _argcheck(self, df): return df > 0 def _rvs(self, df, size=None, random_state=None): return random_state.standard_t(df, size=size) def _pdf(self, x, df): # gamma((df+1)/2) # t.pdf(x, df) = --------------------------------------------------- # sqrt(pi*df) * gamma(df/2) * (1+x**2/df)**((df+1)/2) r = np.asarray(df*1.0) Px = np.exp(sc.gammaln((r+1)/2)-sc.gammaln(r/2)) Px /= np.sqrt(r*np.pi)*(1+(x**2)/r)**((r+1)/2) return Px def _logpdf(self, x, df): r = df*1.0 lPx = sc.gammaln((r+1)/2)-sc.gammaln(r/2) lPx -= 0.5*np.log(r*np.pi) + (r+1)/2*np.log(1+(x**2)/r) return lPx def _cdf(self, x, df): return sc.stdtr(df, x) def _sf(self, x, df): return sc.stdtr(df, -x) def _ppf(self, q, df): return sc.stdtrit(df, q) def _isf(self, q, df): return -sc.stdtrit(df, q) def _stats(self, df): mu = np.where(df > 1, 0.0, np.inf) mu2 = _lazywhere(df > 2, (df,), lambda df: df / (df-2.0), np.inf) mu2 = np.where(df <= 1, np.nan, mu2) g1 = np.where(df > 3, 0.0, np.nan) g2 = _lazywhere(df > 4, (df,), lambda df: 6.0 / (df-4.0), np.inf) g2 = np.where(df <= 2, np.nan, g2) return mu, mu2, g1, g2 t = t_gen(name='t') class nct_gen(rv_continuous): r"""A non-central Student's t continuous random variable. %(before_notes)s Notes ----- If :math:`Y` is a standard normal random variable and :math:`V` is an independent chi-square random variable (`chi2`) with :math:`k` degrees of freedom, then .. math:: X = \frac{Y + c}{\sqrt{V/k}} has a non-central Student's t distribution on the real line. The degrees of freedom parameter :math:`k` (denoted ``df`` in the implementation) satisfies :math:`k > 0` and the noncentrality parameter :math:`c` (denoted ``nc`` in the implementation) is a real number. %(after_notes)s %(example)s """ def _argcheck(self, df, nc): return (df > 0) & (nc == nc) def _rvs(self, df, nc, size=None, random_state=None): n = norm.rvs(loc=nc, size=size, random_state=random_state) c2 = chi2.rvs(df, size=size, random_state=random_state) return n * np.sqrt(df) / np.sqrt(c2) def _pdf(self, x, df, nc): n = df*1.0 nc = nc*1.0 x2 = x*x ncx2 = nc*nc*x2 fac1 = n + x2 trm1 = n/2.*np.log(n) + sc.gammaln(n+1) trm1 -= n*np.log(2)+nc*nc/2.+(n/2.)*np.log(fac1)+sc.gammaln(n/2.) Px = np.exp(trm1) valF = ncx2 / (2*fac1) trm1 = np.sqrt(2)*nc*x*sc.hyp1f1(n/2+1, 1.5, valF) trm1 /= np.asarray(fac1*sc.gamma((n+1)/2)) trm2 = sc.hyp1f1((n+1)/2, 0.5, valF) trm2 /= np.asarray(np.sqrt(fac1)*sc.gamma(n/2+1)) Px *= trm1+trm2 return Px def _cdf(self, x, df, nc): return sc.nctdtr(df, nc, x) def _ppf(self, q, df, nc): return sc.nctdtrit(df, nc, q) def _stats(self, df, nc, moments='mv'): # # See D. Hogben, R.S. Pinkham, and M.B. Wilk, # 'The moments of the non-central t-distribution' # Biometrika 48, p. 465 (2961). # e.g. https://www.jstor.org/stable/2332772 (gated) # mu, mu2, g1, g2 = None, None, None, None gfac = sc.gamma(df/2.-0.5) / sc.gamma(df/2.) c11 = np.sqrt(df/2.) * gfac c20 = df / (df-2.) c22 = c20 - c11*c11 mu = np.where(df > 1, nc*c11, np.inf) mu2 = np.where(df > 2, c22*nc*nc + c20, np.inf) if 's' in moments: c33t = df * (7.-2.*df) / (df-2.) / (df-3.) + 2.*c11*c11 c31t = 3.*df / (df-2.) / (df-3.) mu3 = (c33t*nc*nc + c31t) * c11*nc g1 = np.where(df > 3, mu3 / np.power(mu2, 1.5), np.nan) # kurtosis if 'k' in moments: c44 = df*df / (df-2.) / (df-4.) c44 -= c11*c11 * 2.*df*(5.-df) / (df-2.) / (df-3.) c44 -= 3.*c11**4 c42 = df / (df-4.) - c11*c11 * (df-1.) / (df-3.) c42 *= 6.*df / (df-2.) c40 = 3.*df*df / (df-2.) / (df-4.) mu4 = c44 * nc**4 + c42*nc**2 + c40 g2 = np.where(df > 4, mu4/mu2**2 - 3., np.nan) return mu, mu2, g1, g2 nct = nct_gen(name="nct") class pareto_gen(rv_continuous): r"""A Pareto continuous random variable. %(before_notes)s Notes ----- The probability density function for `pareto` is: .. math:: f(x, b) = \frac{b}{x^{b+1}} for :math:`x \ge 1`, :math:`b > 0`. `pareto` takes ``b`` as a shape parameter for :math:`b`. %(after_notes)s %(example)s """ def _pdf(self, x, b): # pareto.pdf(x, b) = b / x**(b+1) return b * x**(-b-1) def _cdf(self, x, b): return 1 - x**(-b) def _ppf(self, q, b): return pow(1-q, -1.0/b) def _sf(self, x, b): return x**(-b) def _stats(self, b, moments='mv'): mu, mu2, g1, g2 = None, None, None, None if 'm' in moments: mask = b > 1 bt = np.extract(mask, b) mu = valarray(np.shape(b), value=np.inf) np.place(mu, mask, bt / (bt-1.0)) if 'v' in moments: mask = b > 2 bt = np.extract(mask, b) mu2 = valarray(np.shape(b), value=np.inf) np.place(mu2, mask, bt / (bt-2.0) / (bt-1.0)**2) if 's' in moments: mask = b > 3 bt = np.extract(mask, b) g1 = valarray(np.shape(b), value=np.nan) vals = 2 * (bt + 1.0) * np.sqrt(bt - 2.0) / ((bt - 3.0) * np.sqrt(bt)) np.place(g1, mask, vals) if 'k' in moments: mask = b > 4 bt = np.extract(mask, b) g2 = valarray(np.shape(b), value=np.nan) vals = (6.0*np.polyval([1.0, 1.0, -6, -2], bt) / np.polyval([1.0, -7.0, 12.0, 0.0], bt)) np.place(g2, mask, vals) return mu, mu2, g1, g2 def _entropy(self, c): return 1 + 1.0/c - np.log(c) pareto = pareto_gen(a=1.0, name="pareto") class lomax_gen(rv_continuous): r"""A Lomax (Pareto of the second kind) continuous random variable. %(before_notes)s Notes ----- The probability density function for `lomax` is: .. math:: f(x, c) = \frac{c}{(1+x)^{c+1}} for :math:`x \ge 0`, :math:`c > 0`. `lomax` takes ``c`` as a shape parameter for :math:`c`. `lomax` is a special case of `pareto` with ``loc=-1.0``. %(after_notes)s %(example)s """ def _pdf(self, x, c): # lomax.pdf(x, c) = c / (1+x)**(c+1) return c*1.0/(1.0+x)**(c+1.0) def _logpdf(self, x, c): return np.log(c) - (c+1)*sc.log1p(x) def _cdf(self, x, c): return -sc.expm1(-c*sc.log1p(x)) def _sf(self, x, c): return np.exp(-c*sc.log1p(x)) def _logsf(self, x, c): return -c*sc.log1p(x) def _ppf(self, q, c): return sc.expm1(-sc.log1p(-q)/c) def _stats(self, c): mu, mu2, g1, g2 = pareto.stats(c, loc=-1.0, moments='mvsk') return mu, mu2, g1, g2 def _entropy(self, c): return 1+1.0/c-np.log(c) lomax = lomax_gen(a=0.0, name="lomax") class pearson3_gen(rv_continuous): r"""A pearson type III continuous random variable. %(before_notes)s Notes ----- The probability density function for `pearson3` is: .. math:: f(x, skew) = \frac{|\beta|}{\Gamma(\alpha)} (\beta (x - \zeta))^{\alpha - 1} \exp(-\beta (x - \zeta)) where: .. math:: \beta = \frac{2}{skew stddev} \alpha = (stddev \beta)^2 \zeta = loc - \frac{\alpha}{\beta} :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `pearson3` takes ``skew`` as a shape parameter for :math:`skew`. %(after_notes)s %(example)s References ---------- R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water Resources Research, Vol.27, 3149-3158 (1991). L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist., Vol.1, 191-198 (1930). "Using Modern Computing Tools to Fit the Pearson Type III Distribution to Aviation Loads Data", Office of Aviation Research (2003). """ def _preprocess(self, x, skew): # The real 'loc' and 'scale' are handled in the calling pdf(...). The # local variables 'loc' and 'scale' within pearson3._pdf are set to # the defaults just to keep them as part of the equations for # documentation. loc = 0.0 scale = 1.0 # If skew is small, return _norm_pdf. The divide between pearson3 # and norm was found by brute force and is approximately a skew of # 0.000016. No one, I hope, would actually use a skew value even # close to this small. norm2pearson_transition = 0.000016 ans, x, skew = np.broadcast_arrays([1.0], x, skew) ans = ans.copy() # mask is True where skew is small enough to use the normal approx. mask = np.absolute(skew) < norm2pearson_transition invmask = ~mask beta = 2.0 / (skew[invmask] * scale) alpha = (scale * beta)**2 zeta = loc - alpha / beta transx = beta * (x[invmask] - zeta) return ans, x, transx, mask, invmask, beta, alpha, zeta def _argcheck(self, skew): # The _argcheck function in rv_continuous only allows positive # arguments. The skew argument for pearson3 can be zero (which I want # to handle inside pearson3._pdf) or negative. So just return True # for all skew args. return np.ones(np.shape(skew), dtype=bool) def _stats(self, skew): _, _, _, _, _, beta, alpha, zeta = ( self._preprocess([1], skew)) m = zeta + alpha / beta v = alpha / (beta**2) s = 2.0 / (alpha**0.5) * np.sign(beta) k = 6.0 / alpha return m, v, s, k def _pdf(self, x, skew): # pearson3.pdf(x, skew) = abs(beta) / gamma(alpha) * # (beta * (x - zeta))**(alpha - 1) * exp(-beta*(x - zeta)) # Do the calculation in _logpdf since helps to limit # overflow/underflow problems ans = np.exp(self._logpdf(x, skew)) if ans.ndim == 0: if np.isnan(ans): return 0.0 return ans ans[np.isnan(ans)] = 0.0 return ans def _logpdf(self, x, skew): # PEARSON3 logpdf GAMMA logpdf # np.log(abs(beta)) # + (alpha - 1)*np.log(beta*(x - zeta)) + (a - 1)*np.log(x) # - beta*(x - zeta) - x # - sc.gammalnalpha) - sc.gammalna) ans, x, transx, mask, invmask, beta, alpha, _ = ( self._preprocess(x, skew)) ans[mask] = np.log(_norm_pdf(x[mask])) ans[invmask] = np.log(abs(beta)) + gamma._logpdf(transx, alpha) return ans def _cdf(self, x, skew): ans, x, transx, mask, invmask, _, alpha, _ = ( self._preprocess(x, skew)) ans[mask] = _norm_cdf(x[mask]) ans[invmask] = gamma._cdf(transx, alpha) return ans def _rvs(self, skew, size=None, random_state=None): skew = np.broadcast_to(skew, size) ans, _, _, mask, invmask, beta, alpha, zeta = ( self._preprocess([0], skew)) nsmall = mask.sum() nbig = mask.size - nsmall ans[mask] = random_state.standard_normal(nsmall) ans[invmask] = random_state.standard_gamma(alpha, nbig)/beta + zeta if size == (): ans = ans[0] return ans def _ppf(self, q, skew): ans, q, _, mask, invmask, beta, alpha, zeta = ( self._preprocess(q, skew)) ans[mask] = _norm_ppf(q[mask]) ans[invmask] = sc.gammaincinv(alpha, q[invmask])/beta + zeta return ans pearson3 = pearson3_gen(name="pearson3") class powerlaw_gen(rv_continuous): r"""A power-function continuous random variable. %(before_notes)s Notes ----- The probability density function for `powerlaw` is: .. math:: f(x, a) = a x^{a-1} for :math:`0 \le x \le 1`, :math:`a > 0`. `powerlaw` takes ``a`` as a shape parameter for :math:`a`. %(after_notes)s `powerlaw` is a special case of `beta` with ``b=1``. %(example)s """ def _pdf(self, x, a): # powerlaw.pdf(x, a) = a * x**(a-1) return a*x**(a-1.0) def _logpdf(self, x, a): return np.log(a) + sc.xlogy(a - 1, x) def _cdf(self, x, a): return x**(a*1.0) def _logcdf(self, x, a): return a*np.log(x) def _ppf(self, q, a): return pow(q, 1.0/a) def _stats(self, a): return (a / (a + 1.0), a / (a + 2.0) / (a + 1.0) ** 2, -2.0 * ((a - 1.0) / (a + 3.0)) * np.sqrt((a + 2.0) / a), 6 * np.polyval([1, -1, -6, 2], a) / (a * (a + 3.0) * (a + 4))) def _entropy(self, a): return 1 - 1.0/a - np.log(a) powerlaw = powerlaw_gen(a=0.0, b=1.0, name="powerlaw") class powerlognorm_gen(rv_continuous): r"""A power log-normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `powerlognorm` is: .. math:: f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s) (\Phi(-\log(x)/s))^{c-1} where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf, and :math:`x > 0`, :math:`s, c > 0`. `powerlognorm` takes :math:`c` and :math:`s` as shape parameters. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _pdf(self, x, c, s): # powerlognorm.pdf(x, c, s) = c / (x*s) * phi(log(x)/s) * # (Phi(-log(x)/s))**(c-1), return (c/(x*s) * _norm_pdf(np.log(x)/s) * pow(_norm_cdf(-np.log(x)/s), c*1.0-1.0)) def _cdf(self, x, c, s): return 1.0 - pow(_norm_cdf(-np.log(x)/s), c*1.0) def _ppf(self, q, c, s): return np.exp(-s * _norm_ppf(pow(1.0 - q, 1.0 / c))) powerlognorm = powerlognorm_gen(a=0.0, name="powerlognorm") class powernorm_gen(rv_continuous): r"""A power normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `powernorm` is: .. math:: f(x, c) = c \phi(x) (\Phi(-x))^{c-1} where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf, and :math:`x >= 0`, :math:`c > 0`. `powernorm` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _pdf(self, x, c): # powernorm.pdf(x, c) = c * phi(x) * (Phi(-x))**(c-1) return c*_norm_pdf(x) * (_norm_cdf(-x)**(c-1.0)) def _logpdf(self, x, c): return np.log(c) + _norm_logpdf(x) + (c-1)*_norm_logcdf(-x) def _cdf(self, x, c): return 1.0-_norm_cdf(-x)**(c*1.0) def _ppf(self, q, c): return -_norm_ppf(pow(1.0 - q, 1.0 / c)) powernorm = powernorm_gen(name='powernorm') class rdist_gen(rv_continuous): r"""An R-distributed (symmetric beta) continuous random variable. %(before_notes)s Notes ----- The probability density function for `rdist` is: .. math:: f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)} for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the symmetric beta distribution: if B has a `beta` distribution with parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with parameter c. `rdist` takes ``c`` as a shape parameter for :math:`c`. This distribution includes the following distribution kernels as special cases:: c = 2: uniform c = 3: `semicircular` c = 4: Epanechnikov (parabolic) c = 6: quartic (biweight) c = 8: triweight %(after_notes)s %(example)s """ # use relation to the beta distribution for pdf, cdf, etc def _pdf(self, x, c): return 0.5*beta._pdf((x + 1)/2, c/2, c/2) def _logpdf(self, x, c): return -np.log(2) + beta._logpdf((x + 1)/2, c/2, c/2) def _cdf(self, x, c): return beta._cdf((x + 1)/2, c/2, c/2) def _ppf(self, q, c): return 2*beta._ppf(q, c/2, c/2) - 1 def _rvs(self, c, size=None, random_state=None): return 2 * random_state.beta(c/2, c/2, size) - 1 def _munp(self, n, c): numerator = (1 - (n % 2)) * sc.beta((n + 1.0) / 2, c / 2.0) return numerator / sc.beta(1. / 2, c / 2.) rdist = rdist_gen(a=-1.0, b=1.0, name="rdist") class rayleigh_gen(rv_continuous): r"""A Rayleigh continuous random variable. %(before_notes)s Notes ----- The probability density function for `rayleigh` is: .. math:: f(x) = x \exp(-x^2/2) for :math:`x \ge 0`. `rayleigh` is a special case of `chi` with ``df=2``. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _rvs(self, size=None, random_state=None): return chi.rvs(2, size=size, random_state=random_state) def _pdf(self, r): # rayleigh.pdf(r) = r * exp(-r**2/2) return np.exp(self._logpdf(r)) def _logpdf(self, r): return np.log(r) - 0.5 * r * r def _cdf(self, r): return -sc.expm1(-0.5 * r**2) def _ppf(self, q): return np.sqrt(-2 * sc.log1p(-q)) def _sf(self, r): return np.exp(self._logsf(r)) def _logsf(self, r): return -0.5 * r * r def _isf(self, q): return np.sqrt(-2 * np.log(q)) def _stats(self): val = 4 - np.pi return (np.sqrt(np.pi/2), val/2, 2*(np.pi-3)*np.sqrt(np.pi)/val**1.5, 6*np.pi/val-16/val**2) def _entropy(self): return _EULER/2.0 + 1 - 0.5*np.log(2) rayleigh = rayleigh_gen(a=0.0, name="rayleigh") class reciprocal_gen(rv_continuous): r"""A loguniform or reciprocal continuous random variable. %(before_notes)s Notes ----- The probability density function for this class is: .. math:: f(x, a, b) = \frac{1}{x \log(b/a)} for :math:`a \le x \le b`, :math:`b > a > 0`. This class takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s %(example)s This doesn't show the equal probability of ``0.01``, ``0.1`` and ``1``. This is best when the x-axis is log-scaled: >>> import numpy as np >>> fig, ax = plt.subplots(1, 1) >>> ax.hist(np.log10(r)) >>> ax.set_ylabel("Frequency") >>> ax.set_xlabel("Value of random variable") >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0])) >>> ticks = ["$10^{{ {} }}$".format(i) for i in [-2, -1, 0]] >>> ax.set_xticklabels(ticks) # doctest: +SKIP >>> plt.show() This random variable will be log-uniform regardless of the base chosen for ``a`` and ``b``. Let's specify with base ``2`` instead: >>> rvs = %(name)s(2**-2, 2**0).rvs(size=1000) Values of ``1/4``, ``1/2`` and ``1`` are equally likely with this random variable. Here's the histogram: >>> fig, ax = plt.subplots(1, 1) >>> ax.hist(np.log2(rvs)) >>> ax.set_ylabel("Frequency") >>> ax.set_xlabel("Value of random variable") >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0])) >>> ticks = ["$2^{{ {} }}$".format(i) for i in [-2, -1, 0]] >>> ax.set_xticklabels(ticks) # doctest: +SKIP >>> plt.show() """ def _argcheck(self, a, b): return (a > 0) & (b > a) def _get_support(self, a, b): return a, b def _pdf(self, x, a, b): # reciprocal.pdf(x, a, b) = 1 / (x*log(b/a)) return 1.0 / (x * np.log(b * 1.0 / a)) def _logpdf(self, x, a, b): return -np.log(x) - np.log(np.log(b * 1.0 / a)) def _cdf(self, x, a, b): return (np.log(x)-np.log(a)) / np.log(b * 1.0 / a) def _ppf(self, q, a, b): return a*pow(b*1.0/a, q) def _munp(self, n, a, b): return 1.0/np.log(b*1.0/a) / n * (pow(b*1.0, n) - pow(a*1.0, n)) def _entropy(self, a, b): return 0.5*np.log(a*b)+np.log(np.log(b*1.0/a)) loguniform = reciprocal_gen(name="loguniform") reciprocal = reciprocal_gen(name="reciprocal") class rice_gen(rv_continuous): r"""A Rice continuous random variable. %(before_notes)s Notes ----- The probability density function for `rice` is: .. math:: f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I_0(x b) for :math:`x >= 0`, :math:`b > 0`. :math:`I_0` is the modified Bessel function of order zero (`scipy.special.i0`). `rice` takes ``b`` as a shape parameter for :math:`b`. %(after_notes)s The Rice distribution describes the length, :math:`r`, of a 2-D vector with components :math:`(U+u, V+v)`, where :math:`U, V` are constant, :math:`u, v` are independent Gaussian random variables with standard deviation :math:`s`. Let :math:`R = \sqrt{U^2 + V^2}`. Then the pdf of :math:`r` is ``rice.pdf(x, R/s, scale=s)``. %(example)s """ def _argcheck(self, b): return b >= 0 def _rvs(self, b, size=None, random_state=None): # https://en.wikipedia.org/wiki/Rice_distribution t = b/np.sqrt(2) + random_state.standard_normal(size=(2,) + size) return np.sqrt((t*t).sum(axis=0)) def _cdf(self, x, b): return sc.chndtr(np.square(x), 2, np.square(b)) def _ppf(self, q, b): return np.sqrt(sc.chndtrix(q, 2, np.square(b))) def _pdf(self, x, b): # rice.pdf(x, b) = x * exp(-(x**2+b**2)/2) * I[0](x*b) # # We use (x**2 + b**2)/2 = ((x-b)**2)/2 + xb. # The factor of np.exp(-xb) is then included in the i0e function # in place of the modified Bessel function, i0, improving # numerical stability for large values of xb. return x * np.exp(-(x-b)*(x-b)/2.0) * sc.i0e(x*b) def _munp(self, n, b): nd2 = n/2.0 n1 = 1 + nd2 b2 = b*b/2.0 return (2.0**(nd2) * np.exp(-b2) * sc.gamma(n1) * sc.hyp1f1(n1, 1, b2)) rice = rice_gen(a=0.0, name="rice") # FIXME: PPF does not work. class recipinvgauss_gen(rv_continuous): r"""A reciprocal inverse Gaussian continuous random variable. %(before_notes)s Notes ----- The probability density function for `recipinvgauss` is: .. math:: f(x, \mu) = \frac{1}{\sqrt{2\pi x}} \exp\left(\frac{-(1-\mu x)^2}{2\mu^2x}\right) for :math:`x \ge 0`. `recipinvgauss` takes ``mu`` as a shape parameter for :math:`\mu`. %(after_notes)s %(example)s """ def _pdf(self, x, mu): # recipinvgauss.pdf(x, mu) = # 1/sqrt(2*pi*x) * exp(-(1-mu*x)**2/(2*x*mu**2)) return 1.0/np.sqrt(2*np.pi*x)*np.exp(-(1-mu*x)**2.0 / (2*x*mu**2.0)) def _logpdf(self, x, mu): return -(1-mu*x)**2.0 / (2*x*mu**2.0) - 0.5*np.log(2*np.pi*x) def _cdf(self, x, mu): trm1 = 1.0/mu - x trm2 = 1.0/mu + x isqx = 1.0/np.sqrt(x) return 1.0-_norm_cdf(isqx*trm1)-np.exp(2.0/mu)*_norm_cdf(-isqx*trm2) def _rvs(self, mu, size=None, random_state=None): return 1.0/random_state.wald(mu, 1.0, size=size) recipinvgauss = recipinvgauss_gen(a=0.0, name='recipinvgauss') class semicircular_gen(rv_continuous): r"""A semicircular continuous random variable. %(before_notes)s Notes ----- The probability density function for `semicircular` is: .. math:: f(x) = \frac{2}{\pi} \sqrt{1-x^2} for :math:`-1 \le x \le 1`. The distribution is a special case of `rdist` with `c = 3`. %(after_notes)s See Also -------- rdist References ---------- .. [1] "Wigner semicircle distribution", https://en.wikipedia.org/wiki/Wigner_semicircle_distribution %(example)s """ def _pdf(self, x): return 2.0/np.pi*np.sqrt(1-x*x) def _logpdf(self, x): return np.log(2/np.pi) + 0.5*np.log1p(-x*x) def _cdf(self, x): return 0.5+1.0/np.pi*(x*np.sqrt(1-x*x) + np.arcsin(x)) def _ppf(self, q): return rdist._ppf(q, 3) def _rvs(self, size=None, random_state=None): # generate values uniformly distributed on the area under the pdf # (semi-circle) by randomly generating the radius and angle r = np.sqrt(random_state.uniform(size=size)) a = np.cos(np.pi * random_state.uniform(size=size)) return r * a def _stats(self): return 0, 0.25, 0, -1.0 def _entropy(self): return 0.64472988584940017414 semicircular = semicircular_gen(a=-1.0, b=1.0, name="semicircular") class skew_norm_gen(rv_continuous): r"""A skew-normal random variable. %(before_notes)s Notes ----- The pdf is:: skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x) `skewnorm` takes a real number :math:`a` as a skewness parameter When ``a = 0`` the distribution is identical to a normal distribution (`norm`). `rvs` implements the method of [1]_. %(after_notes)s %(example)s References ---------- .. [1] A. Azzalini and A. Capitanio (1999). Statistical applications of the multivariate skew-normal distribution. J. Roy. Statist. Soc., B 61, 579-602. https://arxiv.org/abs/0911.2093 """ def _argcheck(self, a): return np.isfinite(a) def _pdf(self, x, a): return 2.*_norm_pdf(x)*_norm_cdf(a*x) def _cdf_single(self, x, *args): _a, _b = self._get_support(*args) if x <= 0: cdf = integrate.quad(self._pdf, _a, x, args=args)[0] else: t1 = integrate.quad(self._pdf, _a, 0, args=args)[0] t2 = integrate.quad(self._pdf, 0, x, args=args)[0] cdf = t1 + t2 if cdf > 1: # Presumably numerical noise, e.g. 1.0000000000000002 cdf = 1.0 return cdf def _sf(self, x, a): return self._cdf(-x, -a) def _rvs(self, a, size=None, random_state=None): u0 = random_state.normal(size=size) v = random_state.normal(size=size) d = a/np.sqrt(1 + a**2) u1 = d*u0 + v*np.sqrt(1 - d**2) return np.where(u0 >= 0, u1, -u1) def _stats(self, a, moments='mvsk'): output = [None, None, None, None] const = np.sqrt(2/np.pi) * a/np.sqrt(1 + a**2) if 'm' in moments: output[0] = const if 'v' in moments: output[1] = 1 - const**2 if 's' in moments: output[2] = ((4 - np.pi)/2) * (const/np.sqrt(1 - const**2))**3 if 'k' in moments: output[3] = (2*(np.pi - 3)) * (const**4/(1 - const**2)**2) return output skewnorm = skew_norm_gen(name='skewnorm') class trapz_gen(rv_continuous): r"""A trapezoidal continuous random variable. %(before_notes)s Notes ----- The trapezoidal distribution can be represented with an up-sloping line from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)`` and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``. This defines the trapezoid base from ``loc`` to ``(loc+scale)`` and the flat top from ``c`` to ``d`` proportional to the position along the base with ``0 <= c <= d <= 1``. When ``c=d``, this is equivalent to `triang` with the same values for `loc`, `scale` and `c`. The method of [1]_ is used for computing moments. `trapz` takes :math:`c` and :math:`d` as shape parameters. %(after_notes)s The standard form is in the range [0, 1] with c the mode. The location parameter shifts the start to `loc`. The scale parameter changes the width from 1 to `scale`. %(example)s References ---------- .. [1] Kacker, R.N. and Lawrence, J.F. (2007). Trapezoidal and triangular distributions for Type B evaluation of standard uncertainty. Metrologia 44, 117–127. https://doi.org/10.1088/0026-1394/44/2/003 """ def _argcheck(self, c, d): return (c >= 0) & (c <= 1) & (d >= 0) & (d <= 1) & (d >= c) def _pdf(self, x, c, d): u = 2 / (d-c+1) return _lazyselect([x < c, (c <= x) & (x <= d), x > d], [lambda x, c, d, u: u * x / c, lambda x, c, d, u: u, lambda x, c, d, u: u * (1-x) / (1-d)], (x, c, d, u)) def _cdf(self, x, c, d): return _lazyselect([x < c, (c <= x) & (x <= d), x > d], [lambda x, c, d: x**2 / c / (d-c+1), lambda x, c, d: (c + 2 * (x-c)) / (d-c+1), lambda x, c, d: 1-((1-x) ** 2 / (d-c+1) / (1-d))], (x, c, d)) def _ppf(self, q, c, d): qc, qd = self._cdf(c, c, d), self._cdf(d, c, d) condlist = [q < qc, q <= qd, q > qd] choicelist = [np.sqrt(q * c * (1 + d - c)), 0.5 * q * (1 + d - c) + 0.5 * c, 1 - np.sqrt((1 - q) * (d - c + 1) * (1 - d))] return np.select(condlist, choicelist) def _munp(self, n, c, d): # Using the parameterization from Kacker, 2007, with # a=bottom left, c=top left, d=top right, b=bottom right, then # E[X^n] = h/(n+1)/(n+2) [(b^{n+2}-d^{n+2})/(b-d) # - ((c^{n+2} - a^{n+2})/(c-a)] # with h = 2/((b-a) - (d-c)). The corresponding parameterization # in scipy, has a'=loc, c'=loc+c*scale, d'=loc+d*scale, b'=loc+scale, # which for standard form reduces to a'=0, b'=1, c'=c, d'=d. # Substituting into E[X^n] gives the bd' term as (1 - d^{n+2})/(1 - d) # and the ac' term as c^{n-1} for the standard form. The bd' term has # numerical difficulties near d=1, so replace (1 - d^{n+2})/(1-d) # with expm1((n+2)*log(d))/(d-1). # Testing with n=18 for c=(1e-30,1-eps) shows that this is stable. # We still require an explicit test for d=1 to prevent divide by zero, # and now a test for d=0 to prevent log(0). ab_term = c**(n+1) dc_term = _lazyselect( [d == 0.0, (0.0 < d) & (d < 1.0), d == 1.0], [lambda d: 1.0, lambda d: np.expm1((n+2) * np.log(d)) / (d-1.0), lambda d: n+2], [d]) val = 2.0 / (1.0+d-c) * (dc_term - ab_term) / ((n+1) * (n+2)) return val def _entropy(self, c, d): # Using the parameterization from Wikipedia (van Dorp, 2003) # with a=bottom left, c=top left, d=top right, b=bottom right # gives a'=loc, b'=loc+c*scale, c'=loc+d*scale, d'=loc+scale, # which for loc=0, scale=1 is a'=0, b'=c, c'=d, d'=1. # Substituting into the entropy formula from Wikipedia gives # the following result. return 0.5 * (1.0-d+c) / (1.0+d-c) + np.log(0.5 * (1.0+d-c)) trapz = trapz_gen(a=0.0, b=1.0, name="trapz") class triang_gen(rv_continuous): r"""A triangular continuous random variable. %(before_notes)s Notes ----- The triangular distribution can be represented with an up-sloping line from ``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)`` to ``(loc + scale)``. `triang` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s The standard form is in the range [0, 1] with c the mode. The location parameter shifts the start to `loc`. The scale parameter changes the width from 1 to `scale`. %(example)s """ def _rvs(self, c, size=None, random_state=None): return random_state.triangular(0, c, 1, size) def _argcheck(self, c): return (c >= 0) & (c <= 1) def _pdf(self, x, c): # 0: edge case where c=0 # 1: generalised case for x < c, don't use x <= c, as it doesn't cope # with c = 0. # 2: generalised case for x >= c, but doesn't cope with c = 1 # 3: edge case where c=1 r = _lazyselect([c == 0, x < c, (x >= c) & (c != 1), c == 1], [lambda x, c: 2 - 2 * x, lambda x, c: 2 * x / c, lambda x, c: 2 * (1 - x) / (1 - c), lambda x, c: 2 * x], (x, c)) return r def _cdf(self, x, c): r = _lazyselect([c == 0, x < c, (x >= c) & (c != 1), c == 1], [lambda x, c: 2*x - x*x, lambda x, c: x * x / c, lambda x, c: (x*x - 2*x + c) / (c-1), lambda x, c: x * x], (x, c)) return r def _ppf(self, q, c): return np.where(q < c, np.sqrt(c * q), 1-np.sqrt((1-c) * (1-q))) def _stats(self, c): return ((c+1.0)/3.0, (1.0-c+c*c)/18, np.sqrt(2)*(2*c-1)*(c+1)*(c-2) / (5*np.power((1.0-c+c*c), 1.5)), -3.0/5.0) def _entropy(self, c): return 0.5-np.log(2) triang = triang_gen(a=0.0, b=1.0, name="triang") class truncexpon_gen(rv_continuous): r"""A truncated exponential continuous random variable. %(before_notes)s Notes ----- The probability density function for `truncexpon` is: .. math:: f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)} for :math:`0 <= x <= b`. `truncexpon` takes ``b`` as a shape parameter for :math:`b`. %(after_notes)s %(example)s """ def _argcheck(self, b): return b > 0 def _get_support(self, b): return self.a, b def _pdf(self, x, b): # truncexpon.pdf(x, b) = exp(-x) / (1-exp(-b)) return np.exp(-x)/(-sc.expm1(-b)) def _logpdf(self, x, b): return -x - np.log(-sc.expm1(-b)) def _cdf(self, x, b): return sc.expm1(-x)/sc.expm1(-b) def _ppf(self, q, b): return -sc.log1p(q*sc.expm1(-b)) def _munp(self, n, b): # wrong answer with formula, same as in continuous.pdf # return sc.gamman+1)-sc.gammainc1+n, b) if n == 1: return (1-(b+1)*np.exp(-b))/(-sc.expm1(-b)) elif n == 2: return 2*(1-0.5*(b*b+2*b+2)*np.exp(-b))/(-sc.expm1(-b)) else: # return generic for higher moments # return rv_continuous._mom1_sc(self, n, b) return self._mom1_sc(n, b) def _entropy(self, b): eB = np.exp(b) return np.log(eB-1)+(1+eB*(b-1.0))/(1.0-eB) truncexpon = truncexpon_gen(a=0.0, name='truncexpon') TRUNCNORM_TAIL_X = 30 TRUNCNORM_MAX_BRENT_ITERS = 40 def _truncnorm_get_delta_scalar(a, b): if (a > TRUNCNORM_TAIL_X) or (b < -TRUNCNORM_TAIL_X): return 0 if a > 0: delta = _norm_sf(a) - _norm_sf(b) else: delta = _norm_cdf(b) - _norm_cdf(a) delta = max(delta, 0) return delta def _truncnorm_get_delta(a, b): if np.isscalar(a) and np.isscalar(b): return _truncnorm_get_delta_scalar(a, b) a, b = np.atleast_1d(a), np.atleast_1d(b) if a.size == 1 and b.size == 1: return _truncnorm_get_delta_scalar(a.item(), b.item()) delta = np.zeros(np.shape(a)) condinner = (a <= TRUNCNORM_TAIL_X) & (b >= -TRUNCNORM_TAIL_X) conda = (a > 0) & condinner condb = (a <= 0) & condinner if np.any(conda): np.place(delta, conda, _norm_sf(a[conda]) - _norm_sf(b[conda])) if np.any(condb): np.place(delta, condb, _norm_cdf(b[condb]) - _norm_cdf(a[condb])) delta[delta < 0] = 0 return delta def _truncnorm_get_logdelta_scalar(a, b): if (a <= TRUNCNORM_TAIL_X) and (b >= -TRUNCNORM_TAIL_X): if a > 0: delta = _norm_sf(a) - _norm_sf(b) else: delta = _norm_cdf(b) - _norm_cdf(a) delta = max(delta, 0) if delta > 0: return np.log(delta) if b < 0 or (np.abs(a) >= np.abs(b)): nla, nlb = _norm_logcdf(a), _norm_logcdf(b) logdelta = nlb + np.log1p(-np.exp(nla - nlb)) else: sla, slb = _norm_logsf(a), _norm_logsf(b) logdelta = sla + np.log1p(-np.exp(slb - sla)) return logdelta def _truncnorm_logpdf_scalar(x, a, b): with np.errstate(invalid='ignore'): if np.isscalar(x): if x < a: return -np.inf if x > b: return -np.inf shp = np.shape(x) x = np.atleast_1d(x) out = np.full_like(x, np.nan, dtype=np.double) condlta, condgtb = (x < a), (x > b) if np.any(condlta): np.place(out, condlta, -np.inf) if np.any(condgtb): np.place(out, condgtb, -np.inf) cond_inner = ~condlta & ~condgtb if np.any(cond_inner): _logdelta = _truncnorm_get_logdelta_scalar(a, b) np.place(out, cond_inner, _norm_logpdf(x[cond_inner]) - _logdelta) return (out[0] if (shp == ()) else out) def _truncnorm_pdf_scalar(x, a, b): with np.errstate(invalid='ignore'): if np.isscalar(x): if x < a: return 0.0 if x > b: return 0.0 shp = np.shape(x) x = np.atleast_1d(x) out = np.full_like(x, np.nan, dtype=np.double) condlta, condgtb = (x < a), (x > b) if np.any(condlta): np.place(out, condlta, 0.0) if np.any(condgtb): np.place(out, condgtb, 0.0) cond_inner = ~condlta & ~condgtb if np.any(cond_inner): delta = _truncnorm_get_delta_scalar(a, b) if delta > 0: np.place(out, cond_inner, _norm_pdf(x[cond_inner]) / delta) else: np.place(out, cond_inner, np.exp(_truncnorm_logpdf_scalar(x[cond_inner], a, b))) return (out[0] if (shp == ()) else out) def _truncnorm_logcdf_scalar(x, a, b): with np.errstate(invalid='ignore'): if np.isscalar(x): if x <= a: return -np.inf if x >= b: return 0 shp = np.shape(x) x = np.atleast_1d(x) out = np.full_like(x, np.nan, dtype=np.double) condlea, condgeb = (x <= a), (x >= b) if np.any(condlea): np.place(out, condlea, -np.inf) if np.any(condgeb): np.place(out, condgeb, 0.0) cond_inner = ~condlea & ~condgeb if np.any(cond_inner): delta = _truncnorm_get_delta_scalar(a, b) if delta > 0: np.place(out, cond_inner, np.log((_norm_cdf(x[cond_inner]) - _norm_cdf(a)) / delta)) else: with np.errstate(divide='ignore'): if a < 0: nla, nlb = _norm_logcdf(a), _norm_logcdf(b) tab = np.log1p(-np.exp(nla - nlb)) nlx = _norm_logcdf(x[cond_inner]) tax = np.log1p(-np.exp(nla - nlx)) np.place(out, cond_inner, nlx + tax - (nlb + tab)) else: sla = _norm_logsf(a) slb = _norm_logsf(b) np.place(out, cond_inner, np.log1p(-np.exp(_norm_logsf(x[cond_inner]) - sla)) - np.log1p(-np.exp(slb - sla))) return (out[0] if (shp == ()) else out) def _truncnorm_cdf_scalar(x, a, b): with np.errstate(invalid='ignore'): if np.isscalar(x): if x <= a: return -0 if x >= b: return 1 shp = np.shape(x) x = np.atleast_1d(x) out = np.full_like(x, np.nan, dtype=np.double) condlea, condgeb = (x <= a), (x >= b) if np.any(condlea): np.place(out, condlea, 0) if np.any(condgeb): np.place(out, condgeb, 1.0) cond_inner = ~condlea & ~condgeb if np.any(cond_inner): delta = _truncnorm_get_delta_scalar(a, b) if delta > 0: np.place(out, cond_inner, (_norm_cdf(x[cond_inner]) - _norm_cdf(a)) / delta) else: with np.errstate(divide='ignore'): np.place(out, cond_inner, np.exp(_truncnorm_logcdf_scalar(x[cond_inner], a, b))) return (out[0] if (shp == ()) else out) def _truncnorm_logsf_scalar(x, a, b): with np.errstate(invalid='ignore'): if np.isscalar(x): if x <= a: return 0.0 if x >= b: return -np.inf shp = np.shape(x) x = np.atleast_1d(x) out = np.full_like(x, np.nan, dtype=np.double) condlea, condgeb = (x <= a), (x >= b) if np.any(condlea): np.place(out, condlea, 0) if np.any(condgeb): np.place(out, condgeb, -np.inf) cond_inner = ~condlea & ~condgeb if np.any(cond_inner): delta = _truncnorm_get_delta_scalar(a, b) if delta > 0: np.place(out, cond_inner, np.log((_norm_sf(x[cond_inner]) - _norm_sf(b)) / delta)) else: with np.errstate(divide='ignore'): if b < 0: nla, nlb = _norm_logcdf(a), _norm_logcdf(b) np.place(out, cond_inner, np.log1p(-np.exp(_norm_logcdf(x[cond_inner]) - nlb)) - np.log1p(-np.exp(nla - nlb))) else: sla, slb = _norm_logsf(a), _norm_logsf(b) tab = np.log1p(-np.exp(slb - sla)) slx = _norm_logsf(x[cond_inner]) tax = np.log1p(-np.exp(slb - slx)) np.place(out, cond_inner, slx + tax - (sla + tab)) return (out[0] if (shp == ()) else out) def _truncnorm_sf_scalar(x, a, b): with np.errstate(invalid='ignore'): if np.isscalar(x): if x <= a: return 1.0 if x >= b: return 0.0 shp = np.shape(x) x = np.atleast_1d(x) out = np.full_like(x, np.nan, dtype=np.double) condlea, condgeb = (x <= a), (x >= b) if np.any(condlea): np.place(out, condlea, 1.0) if np.any(condgeb): np.place(out, condgeb, 0.0) cond_inner = ~condlea & ~condgeb if np.any(cond_inner): delta = _truncnorm_get_delta_scalar(a, b) if delta > 0: np.place(out, cond_inner, (_norm_sf(x[cond_inner]) - _norm_sf(b)) / delta) else: np.place(out, cond_inner, np.exp(_truncnorm_logsf_scalar(x[cond_inner], a, b))) return (out[0] if (shp == ()) else out) def _norm_logcdfprime(z): # derivative of special.log_ndtr (See special/cephes/ndtr.c) # Differentiate formula for log Phi(z)_truncnorm_ppf # log Phi(z) = -z^2/2 - log(-z) - log(2pi)/2 + log(1 + sum (-1)^n (2n-1)!! / z^(2n)) # Convergence of series is slow for |z| < 10, but can use d(log Phi(z))/dz = dPhi(z)/dz / Phi(z) # Just take the first 10 terms because that is sufficient for use in _norm_ilogcdf assert np.all(z <= -10) lhs = -z - 1/z denom_cons = 1/z**2 numerator = 1 pwr = 1.0 denom_total, numerator_total = 0, 0 sign = -1 for i in range(1, 11): pwr *= denom_cons numerator *= 2 * i - 1 term = sign * numerator * pwr denom_total += term numerator_total += term * (2 * i) / z sign = -sign return lhs - numerator_total / (1 + denom_total) def _norm_ilogcdf(y): """Inverse function to _norm_logcdf==sc.log_ndtr.""" # Apply approximate Newton-Raphson # Only use for very negative values of y. # At minimum requires y <= -(log(2pi)+2^2)/2 ~= -2.9 # Much better convergence for y <= -10 z = -np.sqrt(-2 * (y + np.log(2*np.pi)/2)) for _ in range(4): z = z - (_norm_logcdf(z) - y) / _norm_logcdfprime(z) return z def _truncnorm_ppf_scalar(q, a, b): shp = np.shape(q) q = np.atleast_1d(q) out = np.zeros(np.shape(q)) condle0, condge1 = (q <= 0), (q >= 1) if np.any(condle0): out[condle0] = a if np.any(condge1): out[condge1] = b delta = _truncnorm_get_delta_scalar(a, b) cond_inner = ~condle0 & ~condge1 if np.any(cond_inner): qinner = q[cond_inner] if delta > 0: if a > 0: sa, sb = _norm_sf(a), _norm_sf(b) np.place(out, cond_inner, _norm_isf(qinner * sb + sa * (1.0 - qinner))) else: na, nb = _norm_cdf(a), _norm_cdf(b) np.place(out, cond_inner, _norm_ppf(qinner * nb + na * (1.0 - qinner))) elif np.isinf(b): np.place(out, cond_inner, -_norm_ilogcdf(np.log1p(-qinner) + _norm_logsf(a))) elif np.isinf(a): np.place(out, cond_inner, _norm_ilogcdf(np.log(q) + _norm_logcdf(b))) else: if b < 0: # Solve norm_logcdf(x) = norm_logcdf(a) + log1p(q * (expm1(norm_logcdf(b) - norm_logcdf(a))) # = nla + log1p(q * expm1(nlb - nla)) # = nlb + log(q) + log1p((1-q) * exp(nla - nlb)/q) def _f_cdf(x, c): return _norm_logcdf(x) - c nla, nlb = _norm_logcdf(a), _norm_logcdf(b) values = nlb + np.log(q[cond_inner]) C = np.exp(nla - nlb) if C: one_minus_q = (1 - q)[cond_inner] values += np.log1p(one_minus_q * C / q[cond_inner]) x = [optimize.zeros.brentq(_f_cdf, a, b, args=(c,), maxiter=TRUNCNORM_MAX_BRENT_ITERS)for c in values] np.place(out, cond_inner, x) else: # Solve norm_logsf(x) = norm_logsf(b) + log1p((1-q) * (expm1(norm_logsf(a) - norm_logsf(b))) # = slb + log1p((1-q)[cond_inner] * expm1(sla - slb)) # = sla + log(1-q) + log1p(q * np.exp(slb - sla)/(1-q)) def _f_sf(x, c): return _norm_logsf(x) - c sla, slb = _norm_logsf(a), _norm_logsf(b) one_minus_q = (1-q)[cond_inner] values = sla + np.log(one_minus_q) C = np.exp(slb - sla) if C: values += np.log1p(q[cond_inner] * C / one_minus_q) x = [optimize.zeros.brentq(_f_sf, a, b, args=(c,), maxiter=TRUNCNORM_MAX_BRENT_ITERS) for c in values] np.place(out, cond_inner, x) out[out < a] = a out[out > b] = b return (out[0] if (shp == ()) else out) class truncnorm_gen(rv_continuous): r"""A truncated normal continuous random variable. %(before_notes)s Notes ----- The standard form of this distribution is a standard normal truncated to the range [a, b] --- notice that a and b are defined over the domain of the standard normal. To convert clip values for a specific mean and standard deviation, use:: a, b = (myclip_a - my_mean) / my_std, (myclip_b - my_mean) / my_std `truncnorm` takes :math:`a` and :math:`b` as shape parameters. %(after_notes)s %(example)s """ def _argcheck(self, a, b): return a < b def _get_support(self, a, b): return a, b def _pdf(self, x, a, b): if np.isscalar(a) and np.isscalar(b): return _truncnorm_pdf_scalar(x, a, b) a, b = np.atleast_1d(a), np.atleast_1d(b) if a.size == 1 and b.size == 1: return _truncnorm_pdf_scalar(x, a.item(), b.item()) it = np.nditer([x, a, b, None], [], [['readonly'], ['readonly'], ['readonly'], ['writeonly','allocate']]) for (_x, _a, _b, _ld) in it: _ld[...] = _truncnorm_pdf_scalar(_x, _a, _b) return it.operands[3] def _logpdf(self, x, a, b): if np.isscalar(a) and np.isscalar(b): return _truncnorm_logpdf_scalar(x, a, b) a, b = np.atleast_1d(a), np.atleast_1d(b) if a.size == 1 and b.size == 1: return _truncnorm_logpdf_scalar(x, a.item(), b.item()) it = np.nditer([x, a, b, None], [], [['readonly'], ['readonly'], ['readonly'], ['writeonly','allocate']]) for (_x, _a, _b, _ld) in it: _ld[...] = _truncnorm_logpdf_scalar(_x, _a, _b) return it.operands[3] def _cdf(self, x, a, b): if np.isscalar(a) and np.isscalar(b): return _truncnorm_cdf_scalar(x, a, b) a, b = np.atleast_1d(a), np.atleast_1d(b) if a.size == 1 and b.size == 1: return _truncnorm_cdf_scalar(x, a.item(), b.item()) out = None it = np.nditer([x, a, b, out], [], [['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']]) for (_x, _a, _b, _p) in it: _p[...] = _truncnorm_cdf_scalar(_x, _a, _b) return it.operands[3] def _logcdf(self, x, a, b): if np.isscalar(a) and np.isscalar(b): return _truncnorm_logcdf_scalar(x, a, b) a, b = np.atleast_1d(a), np.atleast_1d(b) if a.size == 1 and b.size == 1: return _truncnorm_logcdf_scalar(x, a.item(), b.item()) it = np.nditer([x, a, b, None], [], [['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']]) for (_x, _a, _b, _p) in it: _p[...] = _truncnorm_logcdf_scalar(_x, _a, _b) return it.operands[3] def _sf(self, x, a, b): if np.isscalar(a) and np.isscalar(b): return _truncnorm_sf_scalar(x, a, b) a, b = np.atleast_1d(a), np.atleast_1d(b) if a.size == 1 and b.size == 1: return _truncnorm_sf_scalar(x, a.item(), b.item()) out = None it = np.nditer([x, a, b, out], [], [['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']]) for (_x, _a, _b, _p) in it: _p[...] = _truncnorm_sf_scalar(_x, _a, _b) return it.operands[3] def _logsf(self, x, a, b): if np.isscalar(a) and np.isscalar(b): return _truncnorm_logsf_scalar(x, a, b) a, b = np.atleast_1d(a), np.atleast_1d(b) if a.size == 1 and b.size == 1: return _truncnorm_logsf_scalar(x, a.item(), b.item()) out = None it = np.nditer([x, a, b, out], [], [['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']]) for (_x, _a, _b, _p) in it: _p[...] = _truncnorm_logsf_scalar(_x, _a, _b) return it.operands[3] def _ppf(self, q, a, b): if np.isscalar(a) and np.isscalar(b): return _truncnorm_ppf_scalar(q, a, b) a, b = np.atleast_1d(a), np.atleast_1d(b) if a.size == 1 and b.size == 1: return _truncnorm_ppf_scalar(q, a.item(), b.item()) out = None it = np.nditer([q, a, b, out], [], [['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']]) for (_q, _a, _b, _x) in it: _x[...] = _truncnorm_ppf_scalar(_q, _a, _b) return it.operands[3] def _munp(self, n, a, b): def n_th_moment(n, a, b): """ Returns n-th moment. Defined only if n >= 0. Function cannot broadcast due to the loop over n """ pA, pB = self._pdf([a, b], a, b) probs = [pA, -pB] moments = [0, 1] for k in range(1, n+1): # a or b might be infinite, and the corresponding pdf value # is 0 in that case, but nan is returned for the # multiplication. However, as b->infinity, pdf(b)*b**k -> 0. # So it is safe to use _lazywhere to avoid the nan. vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x * y**(k-1), fillvalue=0) mk = np.sum(vals) + (k-1) * moments[-2] moments.append(mk) return moments[-1] return _lazywhere((n >= 0) & (a == a) & (b == b), (n, a, b), np.vectorize(n_th_moment, otypes=[np.float64]), np.nan) def _stats(self, a, b, moments='mv'): pA, pB = self._pdf(np.array([a, b]), a, b) m1 = pA - pB mu = m1 # use _lazywhere to avoid nan (See detailed comment in _munp) probs = [pA, -pB] vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y, fillvalue=0) m2 = 1 + np.sum(vals) vals = _lazywhere(probs, [probs, [a-mu, b-mu]], lambda x, y: x*y, fillvalue=0) # mu2 = m2 - mu**2, but not as numerically stable as: # mu2 = (a-mu)*pA - (b-mu)*pB + 1 mu2 = 1 + np.sum(vals) vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y**2, fillvalue=0) m3 = 2*m1 + np.sum(vals) vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y**3, fillvalue=0) m4 = 3*m2 + np.sum(vals) mu3 = m3 + m1 * (-3*m2 + 2*m1**2) g1 = mu3 / np.power(mu2, 1.5) mu4 = m4 + m1*(-4*m3 + 3*m1*(2*m2 - m1**2)) g2 = mu4 / mu2**2 - 3 return mu, mu2, g1, g2 def _rvs(self, a, b, size=None, random_state=None): # if a and b are scalar, use _rvs_scalar, otherwise need to create # output by iterating over parameters if np.isscalar(a) and np.isscalar(b): out = self._rvs_scalar(a, b, size, random_state=random_state) elif a.size == 1 and b.size == 1: out = self._rvs_scalar(a.item(), b.item(), size, random_state=random_state) else: # When this method is called, size will be a (possibly empty) # tuple of integers. It will not be None; if `size=None` is passed # to `rvs()`, size will be the empty tuple (). a, b = np.broadcast_arrays(a, b) # a and b now have the same shape. # `shp` is the shape of the blocks of random variates that are # generated for each combination of parameters associated with # broadcasting a and b. # bc is a tuple the same length as size. The values # in bc are bools. If bc[j] is True, it means that # entire axis is filled in for a given combination of the # broadcast arguments. shp, bc = _check_shape(a.shape, size) # `numsamples` is the total number of variates to be generated # for each combination of the input arguments. numsamples = int(np.prod(shp)) # `out` is the array to be returned. It is filled in in the # loop below. out = np.empty(size) it = np.nditer([a, b], flags=['multi_index'], op_flags=[['readonly'], ['readonly']]) while not it.finished: # Convert the iterator's multi_index into an index into the # `out` array where the call to _rvs_scalar() will be stored. # Where bc is True, we use a full slice; otherwise we use the # index value from it.multi_index. len(it.multi_index) might # be less than len(bc), and in that case we want to align these # two sequences to the right, so the loop variable j runs from # -len(size) to 0. This doesn't cause an IndexError, as # bc[j] will be True in those cases where it.multi_index[j] # would cause an IndexError. idx = tuple((it.multi_index[j] if not bc[j] else slice(None)) for j in range(-len(size), 0)) out[idx] = self._rvs_scalar(it[0], it[1], numsamples, random_state).reshape(shp) it.iternext() if size == (): out = out.item() return out def _rvs_scalar(self, a, b, numsamples=None, random_state=None): if not numsamples: numsamples = 1 # prepare sampling of rvs size1d = tuple(np.atleast_1d(numsamples)) N = np.prod(size1d) # number of rvs needed, reshape upon return # Calculate some rvs U = random_state.uniform(low=0, high=1, size=N) x = self._ppf(U, a, b) rvs = np.reshape(x, size1d) return rvs truncnorm = truncnorm_gen(name='truncnorm', momtype=1) # FIXME: RVS does not work. class tukeylambda_gen(rv_continuous): r"""A Tukey-Lamdba continuous random variable. %(before_notes)s Notes ----- A flexible distribution, able to represent and interpolate between the following distributions: - Cauchy (:math:`lambda = -1`) - logistic (:math:`lambda = 0`) - approx Normal (:math:`lambda = 0.14`) - uniform from -1 to 1 (:math:`lambda = 1`) `tukeylambda` takes a real number :math:`lambda` (denoted ``lam`` in the implementation) as a shape parameter. %(after_notes)s %(example)s """ def _argcheck(self, lam): return np.ones(np.shape(lam), dtype=bool) def _pdf(self, x, lam): Fx = np.asarray(sc.tklmbda(x, lam)) Px = Fx**(lam-1.0) + (np.asarray(1-Fx))**(lam-1.0) Px = 1.0/np.asarray(Px) return np.where((lam <= 0) | (abs(x) < 1.0/np.asarray(lam)), Px, 0.0) def _cdf(self, x, lam): return sc.tklmbda(x, lam) def _ppf(self, q, lam): return sc.boxcox(q, lam) - sc.boxcox1p(-q, lam) def _stats(self, lam): return 0, _tlvar(lam), 0, _tlkurt(lam) def _entropy(self, lam): def integ(p): return np.log(pow(p, lam-1)+pow(1-p, lam-1)) return integrate.quad(integ, 0, 1)[0] tukeylambda = tukeylambda_gen(name='tukeylambda') class FitUniformFixedScaleDataError(FitDataError): def __init__(self, ptp, fscale): self.args = ( "Invalid values in `data`. Maximum likelihood estimation with " "the uniform distribution and fixed scale requires that " "data.ptp() <= fscale, but data.ptp() = %r and fscale = %r." % (ptp, fscale), ) class uniform_gen(rv_continuous): r"""A uniform continuous random variable. In the standard form, the distribution is uniform on ``[0, 1]``. Using the parameters ``loc`` and ``scale``, one obtains the uniform distribution on ``[loc, loc + scale]``. %(before_notes)s %(example)s """ def _rvs(self, size=None, random_state=None): return random_state.uniform(0.0, 1.0, size) def _pdf(self, x): return 1.0*(x == x) def _cdf(self, x): return x def _ppf(self, q): return q def _stats(self): return 0.5, 1.0/12, 0, -1.2 def _entropy(self): return 0.0 def fit(self, data, *args, **kwds): """ Maximum likelihood estimate for the location and scale parameters. `uniform.fit` uses only the following parameters. Because exact formulas are used, the parameters related to optimization that are available in the `fit` method of other distributions are ignored here. The only positional argument accepted is `data`. Parameters ---------- data : array_like Data to use in calculating the maximum likelihood estimate. floc : float, optional Hold the location parameter fixed to the specified value. fscale : float, optional Hold the scale parameter fixed to the specified value. Returns ------- loc, scale : float Maximum likelihood estimates for the location and scale. Notes ----- An error is raised if `floc` is given and any values in `data` are less than `floc`, or if `fscale` is given and `fscale` is less than ``data.max() - data.min()``. An error is also raised if both `floc` and `fscale` are given. Examples -------- >>> from scipy.stats import uniform We'll fit the uniform distribution to `x`: >>> x = np.array([2, 2.5, 3.1, 9.5, 13.0]) For a uniform distribution MLE, the location is the minimum of the data, and the scale is the maximum minus the minimum. >>> loc, scale = uniform.fit(x) >>> loc 2.0 >>> scale 11.0 If we know the data comes from a uniform distribution where the support starts at 0, we can use `floc=0`: >>> loc, scale = uniform.fit(x, floc=0) >>> loc 0.0 >>> scale 13.0 Alternatively, if we know the length of the support is 12, we can use `fscale=12`: >>> loc, scale = uniform.fit(x, fscale=12) >>> loc 1.5 >>> scale 12.0 In that last example, the support interval is [1.5, 13.5]. This solution is not unique. For example, the distribution with ``loc=2`` and ``scale=12`` has the same likelihood as the one above. When `fscale` is given and it is larger than ``data.max() - data.min()``, the parameters returned by the `fit` method center the support over the interval ``[data.min(), data.max()]``. """ if len(args) > 0: raise TypeError("Too many arguments.") floc = kwds.pop('floc', None) fscale = kwds.pop('fscale', None) _remove_optimizer_parameters(kwds) if floc is not None and fscale is not None: # This check is for consistency with `rv_continuous.fit`. raise ValueError("All parameters fixed. There is nothing to " "optimize.") data = np.asarray(data) if not np.isfinite(data).all(): raise RuntimeError("The data contains non-finite values.") # MLE for the uniform distribution # -------------------------------- # The PDF is # # f(x, loc, scale) = {1/scale for loc <= x <= loc + scale # {0 otherwise} # # The likelihood function is # L(x, loc, scale) = (1/scale)**n # where n is len(x), assuming loc <= x <= loc + scale for all x. # The log-likelihood is # l(x, loc, scale) = -n*log(scale) # The log-likelihood is maximized by making scale as small as possible, # while keeping loc <= x <= loc + scale. So if neither loc nor scale # are fixed, the log-likelihood is maximized by choosing # loc = x.min() # scale = x.ptp() # If loc is fixed, it must be less than or equal to x.min(), and then # the scale is # scale = x.max() - loc # If scale is fixed, it must not be less than x.ptp(). If scale is # greater than x.ptp(), the solution is not unique. Note that the # likelihood does not depend on loc, except for the requirement that # loc <= x <= loc + scale. All choices of loc for which # x.max() - scale <= loc <= x.min() # have the same log-likelihood. In this case, we choose loc such that # the support is centered over the interval [data.min(), data.max()]: # loc = x.min() = 0.5*(scale - x.ptp()) if fscale is None: # scale is not fixed. if floc is None: # loc is not fixed, scale is not fixed. loc = data.min() scale = data.ptp() else: # loc is fixed, scale is not fixed. loc = floc scale = data.max() - loc if data.min() < loc: raise FitDataError("uniform", lower=loc, upper=loc + scale) else: # loc is not fixed, scale is fixed. ptp = data.ptp() if ptp > fscale: raise FitUniformFixedScaleDataError(ptp=ptp, fscale=fscale) # If ptp < fscale, the ML estimate is not unique; see the comments # above. We choose the distribution for which the support is # centered over the interval [data.min(), data.max()]. loc = data.min() - 0.5*(fscale - ptp) scale = fscale # We expect the return values to be floating point, so ensure it # by explicitly converting to float. return float(loc), float(scale) uniform = uniform_gen(a=0.0, b=1.0, name='uniform') class vonmises_gen(rv_continuous): r"""A Von Mises continuous random variable. %(before_notes)s Notes ----- The probability density function for `vonmises` and `vonmises_line` is: .. math:: f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) } for :math:`-\pi \le x \le \pi`, :math:`\kappa > 0`. :math:`I_0` is the modified Bessel function of order zero (`scipy.special.i0`). `vonmises` is a circular distribution which does not restrict the distribution to a fixed interval. Currently, there is no circular distribution framework in scipy. The ``cdf`` is implemented such that ``cdf(x + 2*np.pi) == cdf(x) + 1``. `vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]` on the real line. This is a regular (i.e. non-circular) distribution. `vonmises` and `vonmises_line` take ``kappa`` as a shape parameter. %(after_notes)s %(example)s """ def _rvs(self, kappa, size=None, random_state=None): return random_state.vonmises(0.0, kappa, size=size) def _pdf(self, x, kappa): # vonmises.pdf(x, \kappa) = exp(\kappa * cos(x)) / (2*pi*I[0](\kappa)) return np.exp(kappa * np.cos(x)) / (2*np.pi*sc.i0(kappa)) def _cdf(self, x, kappa): return _stats.von_mises_cdf(kappa, x) def _stats_skip(self, kappa): return 0, None, 0, None def _entropy(self, kappa): return (-kappa * sc.i1(kappa) / sc.i0(kappa) + np.log(2 * np.pi * sc.i0(kappa))) vonmises = vonmises_gen(name='vonmises') vonmises_line = vonmises_gen(a=-np.pi, b=np.pi, name='vonmises_line') class wald_gen(invgauss_gen): r"""A Wald continuous random variable. %(before_notes)s Notes ----- The probability density function for `wald` is: .. math:: f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x }) for :math:`x >= 0`. `wald` is a special case of `invgauss` with ``mu=1``. %(after_notes)s %(example)s """ _support_mask = rv_continuous._open_support_mask def _rvs(self, size=None, random_state=None): return random_state.wald(1.0, 1.0, size=size) def _pdf(self, x): # wald.pdf(x) = 1/sqrt(2*pi*x**3) * exp(-(x-1)**2/(2*x)) return invgauss._pdf(x, 1.0) def _logpdf(self, x): return invgauss._logpdf(x, 1.0) def _cdf(self, x): return invgauss._cdf(x, 1.0) def _stats(self): return 1.0, 1.0, 3.0, 15.0 wald = wald_gen(a=0.0, name="wald") class wrapcauchy_gen(rv_continuous): r"""A wrapped Cauchy continuous random variable. %(before_notes)s Notes ----- The probability density function for `wrapcauchy` is: .. math:: f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))} for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`. `wrapcauchy` takes ``c`` as a shape parameter for :math:`c`. %(after_notes)s %(example)s """ def _argcheck(self, c): return (c > 0) & (c < 1) def _pdf(self, x, c): # wrapcauchy.pdf(x, c) = (1-c**2) / (2*pi*(1+c**2-2*c*cos(x))) return (1.0-c*c)/(2*np.pi*(1+c*c-2*c*np.cos(x))) def _cdf(self, x, c): output = np.zeros(x.shape, dtype=x.dtype) val = (1.0+c)/(1.0-c) c1 = x < np.pi c2 = 1-c1 xp = np.extract(c1, x) xn = np.extract(c2, x) if np.any(xn): valn = np.extract(c2, np.ones_like(x)*val) xn = 2*np.pi - xn yn = np.tan(xn/2.0) on = 1.0-1.0/np.pi*np.arctan(valn*yn) np.place(output, c2, on) if np.any(xp): valp = np.extract(c1, np.ones_like(x)*val) yp = np.tan(xp/2.0) op = 1.0/np.pi*np.arctan(valp*yp) np.place(output, c1, op) return output def _ppf(self, q, c): val = (1.0-c)/(1.0+c) rcq = 2*np.arctan(val*np.tan(np.pi*q)) rcmq = 2*np.pi-2*np.arctan(val*np.tan(np.pi*(1-q))) return np.where(q < 1.0/2, rcq, rcmq) def _entropy(self, c): return np.log(2*np.pi*(1-c*c)) wrapcauchy = wrapcauchy_gen(a=0.0, b=2*np.pi, name='wrapcauchy') class gennorm_gen(rv_continuous): r"""A generalized normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `gennorm` is [1]_: .. math:: f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta) :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `gennorm` takes ``beta`` as a shape parameter for :math:`\beta`. For :math:`\beta = 1`, it is identical to a Laplace distribution. For :math:`\beta = 2`, it is identical to a normal distribution (with ``scale=1/sqrt(2)``). See Also -------- laplace : Laplace distribution norm : normal distribution References ---------- .. [1] "Generalized normal distribution, Version 1", https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1 %(example)s """ def _pdf(self, x, beta): return np.exp(self._logpdf(x, beta)) def _logpdf(self, x, beta): return np.log(0.5*beta) - sc.gammaln(1.0/beta) - abs(x)**beta def _cdf(self, x, beta): c = 0.5 * np.sign(x) # evaluating (.5 + c) first prevents numerical cancellation return (0.5 + c) - c * sc.gammaincc(1.0/beta, abs(x)**beta) def _ppf(self, x, beta): c = np.sign(x - 0.5) # evaluating (1. + c) first prevents numerical cancellation return c * sc.gammainccinv(1.0/beta, (1.0 + c) - 2.0*c*x)**(1.0/beta) def _sf(self, x, beta): return self._cdf(-x, beta) def _isf(self, x, beta): return -self._ppf(x, beta) def _stats(self, beta): c1, c3, c5 = sc.gammaln([1.0/beta, 3.0/beta, 5.0/beta]) return 0., np.exp(c3 - c1), 0., np.exp(c5 + c1 - 2.0*c3) - 3. def _entropy(self, beta): return 1. / beta - np.log(.5 * beta) + sc.gammaln(1. / beta) gennorm = gennorm_gen(name='gennorm') class halfgennorm_gen(rv_continuous): r"""The upper half of a generalized normal continuous random variable. %(before_notes)s Notes ----- The probability density function for `halfgennorm` is: .. math:: f(x, \beta) = \frac{\beta}{\Gamma(1/\beta)} \exp(-|x|^\beta) for :math:`x > 0`. :math:`\Gamma` is the gamma function (`scipy.special.gamma`). `gennorm` takes ``beta`` as a shape parameter for :math:`\beta`. For :math:`\beta = 1`, it is identical to an exponential distribution. For :math:`\beta = 2`, it is identical to a half normal distribution (with ``scale=1/sqrt(2)``). See Also -------- gennorm : generalized normal distribution expon : exponential distribution halfnorm : half normal distribution References ---------- .. [1] "Generalized normal distribution, Version 1", https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1 %(example)s """ def _pdf(self, x, beta): # beta # halfgennorm.pdf(x, beta) = ------------- exp(-|x|**beta) # gamma(1/beta) return np.exp(self._logpdf(x, beta)) def _logpdf(self, x, beta): return np.log(beta) - sc.gammaln(1.0/beta) - x**beta def _cdf(self, x, beta): return sc.gammainc(1.0/beta, x**beta) def _ppf(self, x, beta): return sc.gammaincinv(1.0/beta, x)**(1.0/beta) def _sf(self, x, beta): return sc.gammaincc(1.0/beta, x**beta) def _isf(self, x, beta): return sc.gammainccinv(1.0/beta, x)**(1.0/beta) def _entropy(self, beta): return 1.0/beta - np.log(beta) + sc.gammaln(1.0/beta) halfgennorm = halfgennorm_gen(a=0, name='halfgennorm') class crystalball_gen(rv_continuous): r""" Crystalball distribution %(before_notes)s Notes ----- The probability density function for `crystalball` is: .. math:: 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} where :math:`A = (m / |\beta|)^n \exp(-\beta^2 / 2)`, :math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant. `crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape parameters. :math:`\beta` defines the point where the pdf changes from a power-law to a Gaussian distribution. :math:`m` is the power of the power-law tail. References ---------- .. [1] "Crystal Ball Function", https://en.wikipedia.org/wiki/Crystal_Ball_function %(after_notes)s .. versionadded:: 0.19.0 %(example)s """ def _pdf(self, x, beta, m): """ Return PDF of the crystalball function. -- | exp(-x**2 / 2), for x > -beta crystalball.pdf(x, beta, m) = N * | | A * (B - x)**(-m), for x <= -beta -- """ N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) + _norm_pdf_C * _norm_cdf(beta)) def rhs(x, beta, m): return np.exp(-x**2 / 2) def lhs(x, beta, m): return ((m/beta)**m * np.exp(-beta**2 / 2.0) * (m/beta - beta - x)**(-m)) return N * _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs) def _logpdf(self, x, beta, m): """ Return the log of the PDF of the crystalball function. """ N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) + _norm_pdf_C * _norm_cdf(beta)) def rhs(x, beta, m): return -x**2/2 def lhs(x, beta, m): return m*np.log(m/beta) - beta**2/2 - m*np.log(m/beta - beta - x) return np.log(N) + _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs) def _cdf(self, x, beta, m): """ Return CDF of the crystalball function """ N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) + _norm_pdf_C * _norm_cdf(beta)) def rhs(x, beta, m): return ((m/beta) * np.exp(-beta**2 / 2.0) / (m-1) + _norm_pdf_C * (_norm_cdf(x) - _norm_cdf(-beta))) def lhs(x, beta, m): return ((m/beta)**m * np.exp(-beta**2 / 2.0) * (m/beta - beta - x)**(-m+1) / (m-1)) return N * _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs) def _ppf(self, p, beta, m): N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) + _norm_pdf_C * _norm_cdf(beta)) pbeta = N * (m/beta) * np.exp(-beta**2/2) / (m - 1) def ppf_less(p, beta, m): eb2 = np.exp(-beta**2/2) C = (m/beta) * eb2 / (m-1) N = 1/(C + _norm_pdf_C * _norm_cdf(beta)) return (m/beta - beta - ((m - 1)*(m/beta)**(-m)/eb2*p/N)**(1/(1-m))) def ppf_greater(p, beta, m): eb2 = np.exp(-beta**2/2) C = (m/beta) * eb2 / (m-1) N = 1/(C + _norm_pdf_C * _norm_cdf(beta)) return _norm_ppf(_norm_cdf(-beta) + (1/_norm_pdf_C)*(p/N - C)) return _lazywhere(p < pbeta, (p, beta, m), f=ppf_less, f2=ppf_greater) def _munp(self, n, beta, m): """ Returns the n-th non-central moment of the crystalball function. """ N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) + _norm_pdf_C * _norm_cdf(beta)) def n_th_moment(n, beta, m): """ Returns n-th moment. Defined only if n+1 < m Function cannot broadcast due to the loop over n """ A = (m/beta)**m * np.exp(-beta**2 / 2.0) B = m/beta - beta rhs = (2**((n-1)/2.0) * sc.gamma((n+1)/2) * (1.0 + (-1)**n * sc.gammainc((n+1)/2, beta**2 / 2))) lhs = np.zeros(rhs.shape) for k in range(n + 1): lhs += (sc.binom(n, k) * B**(n-k) * (-1)**k / (m - k - 1) * (m/beta)**(-m + k + 1)) return A * lhs + rhs return N * _lazywhere(n + 1 < m, (n, beta, m), np.vectorize(n_th_moment, otypes=[np.float64]), np.inf) def _argcheck(self, beta, m): """ Shape parameter bounds are m > 1 and beta > 0. """ return (m > 1) & (beta > 0) crystalball = crystalball_gen(name='crystalball', longname="A Crystalball Function") def _argus_phi(chi): """ Utility function for the argus distribution used in the CDF and norm of the Argus Funktion """ return _norm_cdf(chi) - chi * _norm_pdf(chi) - 0.5 class argus_gen(rv_continuous): r""" Argus distribution %(before_notes)s Notes ----- The probability density function for `argus` is: .. math:: f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2} \exp(-\chi^2 (1 - x^2)/2) for :math:`0 < x < 1` and :math:`\chi > 0`, where .. math:: \Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2 with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard normal distribution, respectively. `argus` takes :math:`\chi` as shape a parameter. References ---------- .. [1] "ARGUS distribution", https://en.wikipedia.org/wiki/ARGUS_distribution %(after_notes)s .. versionadded:: 0.19.0 %(example)s """ def _pdf(self, x, chi): y = 1.0 - x**2 A = chi**3 / (_norm_pdf_C * _argus_phi(chi)) return A * x * np.sqrt(y) * np.exp(-chi**2 * y / 2) def _cdf(self, x, chi): return 1.0 - self._sf(x, chi) def _sf(self, x, chi): return _argus_phi(chi * np.sqrt(1 - x**2)) / _argus_phi(chi) def _rvs(self, chi, size=None, random_state=None): chi = np.asarray(chi) if chi.size == 1: out = self._rvs_scalar(chi, numsamples=size, random_state=random_state) else: shp, bc = _check_shape(chi.shape, size) numsamples = int(np.prod(shp)) out = np.empty(size) it = np.nditer([chi], flags=['multi_index'], op_flags=[['readonly']]) while not it.finished: idx = tuple((it.multi_index[j] if not bc[j] else slice(None)) for j in range(-len(size), 0)) r = self._rvs_scalar(it[0], numsamples=numsamples, random_state=random_state) out[idx] = r.reshape(shp) it.iternext() if size == (): out = out[()] return out def _rvs_scalar(self, chi, numsamples=None, random_state=None): # if chi <= 2.611: # use rejection method, see Devroye: # Non-Uniform Random Variate Generation, 1986, section II.3.2. # write: self.pdf = c * g(x) * h(x), where # h is [0,1]-valued and g is a density # g(x) = d1 * chi**2 * x * exp(-chi**2 * (1 - x**2) / 2), 0 <= x <= 1 # h(x) = sqrt(1 - x**2), 0 <= x <= 1 # Integrating g, we get: # G(x) = d1 * exp(-chi**2 * (1 - x**2) / 2) - d2 # d1 and d2 are determined by G(0) = 0 and G(1) = 1 # d1 = 1 / (1 - exp(-0.5 * chi**2)) # d2 = 1 / (exp(0.5 * chi**2) - 1) # => G(x) = (exp(chi**2 * x**2 /2) - 1) / (exp(chi**2 / 2) - 1) # expected number of iterations is c with # c = -np.expm1(-0.5 * chi**2) * chi / (_norm_pdf_C * _argus_phi(chi)) # note that G can be inverted easily, so we can sample # rvs from this distribution # G_inv(y) = sqrt(2 * log(1 + (exp(chi**2 / 2) - 1) * y) / chi**2) # to avoid an overflow of exp(chi**2 / 2), it is convenient to write # G_inv(y) = sqrt(1 + 2 * log(exp(-chi**2 / 2) * (1-y) + y) / chi**2) # # if chi > 2.611: # use ratio of uniforms method applied to a transformed variable of X # (X is ARGUS with parameter chi): # Y = chi * sqrt(1 - X**2) has density proportional to # u**2 * exp(-u**2 / 2) on [0, chi] (Maxwell distribution conditioned # on [0, chi]). Apply ratio of uniforms to this density to generate # samples of Y and convert back to X # # The expected number of iterations using the rejection method # increases with increasing chi, whereas the expected number of # iterations using the ratio of uniforms method decreases with # increasing chi. The crossover occurs where # chi*(1 - exp(-0.5*chi**2)) = 8*sqrt(2)*exp(-1.5) => chi ~ 2.611 # Switching algorithms at chi=2.611 means that the expected number of # iterations is always below 2.2. if chi <= 2.611: # use rejection method size1d = tuple(np.atleast_1d(numsamples)) N = int(np.prod(size1d)) x = np.zeros(N) echi = np.exp(-chi**2 / 2) simulated = 0 while simulated < N: k = N - simulated u = random_state.uniform(size=k) v = random_state.uniform(size=k) # acceptance condition: u <= h(G_inv(v)). This simplifies to z = 2 * np.log(echi * (1 - v) + v) / chi**2 accept = (u**2 + z <= 0) num_accept = np.sum(accept) if num_accept > 0: # rvs follow a distribution with density g: rvs = G_inv(v) rvs = np.sqrt(1 + z[accept]) x[simulated:(simulated + num_accept)] = rvs simulated += num_accept return np.reshape(x, size1d) else: # use ratio of uniforms method def f(x): return np.where((x >= 0) & (x <= chi), np.exp(2*np.log(x) - x**2/2), 0) umax = np.sqrt(2) / np.exp(0.5) vmax = 4 / np.exp(1) z = rvs_ratio_uniforms(f, umax, 0, vmax, size=numsamples, random_state=random_state) return np.sqrt(1 - z*z / chi**2) def _stats(self, chi): chi2 = chi**2 phi = _argus_phi(chi) m = np.sqrt(np.pi/8) * chi * sc.ive(1, chi2/4) / phi v = (1 - 3 / chi2 + chi * _norm_pdf(chi) / phi) - m**2 return m, v, None, None argus = argus_gen(name='argus', longname="An Argus Function", a=0.0, b=1.0) class rv_histogram(rv_continuous): """ Generates a distribution given by a histogram. This is useful to generate a template distribution from a binned datasample. As a subclass of the `rv_continuous` class, `rv_histogram` inherits from it a collection of generic methods (see `rv_continuous` for the full list), and implements them based on the properties of the provided binned datasample. Parameters ---------- histogram : tuple of array_like Tuple containing two array_like objects The first containing the content of n bins The second containing the (n+1) bin boundaries In particular the return value np.histogram is accepted Notes ----- There are no additional shape parameters except for the loc and scale. The pdf is defined as a stepwise function from the provided histogram The cdf is a linear interpolation of the pdf. .. versionadded:: 0.19.0 Examples -------- Create a scipy.stats distribution from a numpy histogram >>> import scipy.stats >>> import numpy as np >>> data = scipy.stats.norm.rvs(size=100000, loc=0, scale=1.5, random_state=123) >>> hist = np.histogram(data, bins=100) >>> hist_dist = scipy.stats.rv_histogram(hist) Behaves like an ordinary scipy rv_continuous distribution >>> hist_dist.pdf(1.0) 0.20538577847618705 >>> hist_dist.cdf(2.0) 0.90818568543056499 PDF is zero above (below) the highest (lowest) bin of the histogram, defined by the max (min) of the original dataset >>> hist_dist.pdf(np.max(data)) 0.0 >>> hist_dist.cdf(np.max(data)) 1.0 >>> hist_dist.pdf(np.min(data)) 7.7591907244498314e-05 >>> hist_dist.cdf(np.min(data)) 0.0 PDF and CDF follow the histogram >>> import matplotlib.pyplot as plt >>> X = np.linspace(-5.0, 5.0, 100) >>> plt.title("PDF from Template") >>> plt.hist(data, density=True, bins=100) >>> plt.plot(X, hist_dist.pdf(X), label='PDF') >>> plt.plot(X, hist_dist.cdf(X), label='CDF') >>> plt.show() """ _support_mask = rv_continuous._support_mask def __init__(self, histogram, *args, **kwargs): """ Create a new distribution using the given histogram Parameters ---------- histogram : tuple of array_like Tuple containing two array_like objects The first containing the content of n bins The second containing the (n+1) bin boundaries In particular the return value np.histogram is accepted """ self._histogram = histogram if len(histogram) != 2: raise ValueError("Expected length 2 for parameter histogram") self._hpdf = np.asarray(histogram[0]) self._hbins = np.asarray(histogram[1]) if len(self._hpdf) + 1 != len(self._hbins): raise ValueError("Number of elements in histogram content " "and histogram boundaries do not match, " "expected n and n+1.") self._hbin_widths = self._hbins[1:] - self._hbins[:-1] self._hpdf = self._hpdf / float(np.sum(self._hpdf * self._hbin_widths)) self._hcdf = np.cumsum(self._hpdf * self._hbin_widths) self._hpdf = np.hstack([0.0, self._hpdf, 0.0]) self._hcdf = np.hstack([0.0, self._hcdf]) # Set support kwargs['a'] = self.a = self._hbins[0] kwargs['b'] = self.b = self._hbins[-1] super(rv_histogram, self).__init__(*args, **kwargs) def _pdf(self, x): """ PDF of the histogram """ return self._hpdf[np.searchsorted(self._hbins, x, side='right')] def _cdf(self, x): """ CDF calculated from the histogram """ return np.interp(x, self._hbins, self._hcdf) def _ppf(self, x): """ Percentile function calculated from the histogram """ return np.interp(x, self._hcdf, self._hbins) def _munp(self, n): """Compute the n-th non-central moment.""" integrals = (self._hbins[1:]**(n+1) - self._hbins[:-1]**(n+1)) / (n+1) return np.sum(self._hpdf[1:-1] * integrals) def _entropy(self): """Compute entropy of distribution""" res = _lazywhere(self._hpdf[1:-1] > 0.0, (self._hpdf[1:-1],), np.log, 0.0) return -np.sum(self._hpdf[1:-1] * res * self._hbin_widths) def _updated_ctor_param(self): """ Set the histogram as additional constructor argument """ dct = super(rv_histogram, self)._updated_ctor_param() dct['histogram'] = self._histogram return dct # Collect names of classes and objects in this module. pairs = list(globals().copy().items()) _distn_names, _distn_gen_names = get_distribution_names(pairs, rv_continuous) __all__ = _distn_names + _distn_gen_names + ['rv_histogram']