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# # Author: Travis Oliphant 2002-2011 with contributions from # SciPy Developers 2004-2011 #
replace_notes_in_docstring)
tukeylambda_kurtosis as _tlkurt) _lazyselect, _lazywhere, _ncx2_cdf, _ncx2_log_pdf, _ncx2_pdf, rv_continuous, _skew, valarray)
# In numpy 1.12 and above, np.power refuses to raise integers to negative # powers, and `np.float_power` is a new replacement. except AttributeError: float_power = np.power
## Kolmogorov-Smirnov one-sided and two-sided test statistics """General Kolmogorov-Smirnov one-sided test.
%(default)s
""" return 1.0 - sc.smirnov(n, x)
return sc.smirnovi(n, 1.0 - q)
"""Kolmogorov-Smirnov two-sided test for large N.
%(default)s
""" return 1.0 - sc.kolmogorov(x)
return sc.kolmogorov(x)
return sc.kolmogi(1.0 - q)
## Normal distribution
# loc = mu, scale = std # Keep these implementations out of the class definition so they can be reused # by other distributions.
return np.exp(-x**2/2.0) / _norm_pdf_C
return -x**2 / 2.0 - _norm_pdf_logC
return sc.ndtr(x)
return sc.log_ndtr(x)
return sc.ndtri(q)
return _norm_cdf(-x)
return _norm_logcdf(-x)
return -_norm_ppf(q)
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}}
The survival function, ``norm.sf``, is also referred to as the Q-function in some contexts (see, e.g., `Wikipedia's <https://en.wikipedia.org/wiki/Q-function>`_ definition).
%(after_notes)s
%(example)s
""" return self._random_state.standard_normal(self._size)
# norm.pdf(x) = exp(-x**2/2)/sqrt(2*pi) return _norm_pdf(x)
return _norm_logpdf(x)
return _norm_cdf(x)
return _norm_logcdf(x)
return _norm_sf(x)
return _norm_logsf(x)
return _norm_ppf(q)
return _norm_isf(q)
return 0.0, 1.0, 0.0, 0.0
return 0.5*(np.log(2*np.pi)+1)
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.get('floc', None) fscale = kwds.get('fscale', None)
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 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
r"""An alpha continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `alpha` is:
.. math::
f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} * \exp(-\frac{1}{2} (a-1/x)^2)
where ``Phi(alpha)`` is the normal CDF, ``x > 0``, and ``a > 0``.
`alpha` takes ``a`` as a shape parameter.
%(after_notes)s
%(example)s
"""
# 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)
return -2*np.log(x) + _norm_logpdf(a-1.0/x) - np.log(_norm_cdf(a))
return _norm_cdf(a-1.0/x) / _norm_cdf(a)
return 1.0/np.asarray(a-sc.ndtri(q*_norm_cdf(a)))
return [np.inf]*2 + [np.nan]*2
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
""" # anglit.pdf(x) = sin(2*x + \pi/2) = cos(2*x) return np.cos(2*x)
return np.sin(x+np.pi/4)**2.0
return np.arcsin(np.sqrt(q))-np.pi/4
return 0.0, np.pi*np.pi/16-0.5, 0.0, -2*(np.pi**4 - 96)/(np.pi*np.pi-8)**2
return 1-np.log(2)
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 \le x \le 1`.
%(after_notes)s
%(example)s
""" # arcsine.pdf(x) = 1/(pi*sqrt(x*(1-x))) return 1.0/np.pi/np.sqrt(x*(1-x))
return 2.0/np.pi*np.arcsin(np.sqrt(x))
return np.sin(np.pi/2.0*q)**2.0
mu = 0.5 mu2 = 1.0/8 g1 = 0 g2 = -3.0/2.0 return mu, mu2, g1, g2
return -0.24156447527049044468
# 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). 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), )
# This exception is raised by, for example, beta_gen.fit when # optimize.fsolve returns with ier != 1. emsg = "Solver for the MLE equations failed to converge: " emsg += mesg.replace('\n', '') self.args = (emsg,)
# 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
# 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
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(z)` is the gamma function (`scipy.special.gamma`).
`beta` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
""" return self._random_state.beta(a, b, self._size)
# 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))
lPx = sc.xlog1py(b - 1.0, -x) + sc.xlogy(a - 1.0, x) lPx -= sc.betaln(a, b) return lPx
return sc.btdtr(a, b, x)
return sc.btdtri(a, b, q)
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
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.
f0 = (kwds.get('f0', None) or kwds.get('fa', None) or kwds.get('fix_a', None)) f1 = (kwds.get('f1', None) or kwds.get('fb', None) or kwds.get('fix_b', None)) 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)
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.)
# 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
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 ``x > 0``, ``a > 0``, ``b > 0``, where ``beta(a, b)`` is the beta function (see `scipy.special.beta`).
`betaprime` takes ``a`` and ``b`` as shape parameters.
%(after_notes)s
%(example)s
"""
sz, rndm = self._size, self._random_state u1 = gamma.rvs(a, size=sz, random_state=rndm) u2 = gamma.rvs(b, size=sz, random_state=rndm) return u1 / u2
# betaprime.pdf(x, a, b) = x**(a-1) * (1+x)**(-a-b) / beta(a, b) return np.exp(self._logpdf(x, a, b))
return sc.xlogy(a - 1.0, x) - sc.xlog1py(a + b, x) - sc.betaln(a, b)
return sc.betainc(a, b, x/(1.+x))
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
r"""A Bradford continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `bradford` is:
.. math::
f(x, c) = \frac{c}{k (1+cx)}
for :math:`0 < x < 1`, :math:`c > 0` and :math:`k = \log(1+c)`.
`bradford` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" # bradford.pdf(x, c) = c / (k * (1+c*x)) return c / (c*x + 1.0) / sc.log1p(c)
return sc.log1p(c*x) / sc.log1p(c)
return sc.expm1(q * sc.log1p(c)) / c
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
k = np.log(1+c) return k/2.0 - np.log(c/k)
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
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`.
`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]_.
%(after_notes)s
References ---------- .. [1] Burr, I. W. "Cumulative frequency functions", Annals of Mathematical Statistics, 13(2), pp 215-232 (1942).
%(example)s
"""
# burr.pdf(x, c, d) = c * d * x**(-c-1) * (1+x**(-c))**(-d-1) return c * d * (x**(-c - 1.0)) * ((1 + x**(-c))**(-d - 1.0))
return (1 + x**(-c))**(-d)
return (q**(-1.0/d) - 1)**(-1.0/c)
nc = 1. * n / c return d * sc.beta(1.0 - nc, d + nc)
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`.
`burr12` takes :math:`c` and :math:`d` as shape parameters.
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] http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm
%(example)s
"""
# burr12.pdf(x, c, d) = c * d * x**(c-1) * (1+x**(c))**(-d-1) return np.exp(self._logpdf(x, c, d))
return np.log(c) + np.log(d) + sc.xlogy(c - 1, x) + sc.xlog1py(-d-1, x**c)
return -sc.expm1(self._logsf(x, c, d))
return sc.log1p(-(1 + x**c)**(-d))
return np.exp(self._logsf(x, c, d))
return sc.xlog1py(-d, x**c)
# 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)
nc = 1. * n / c return d * sc.beta(1.0 + nc, d - nc)
r"""A Fisk continuous random variable.
The Fisk distribution is also known as the log-logistic distribution, and equals the Burr distribution with ``d == 1``.
`fisk` takes :math:`c` as a shape parameter.
%(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`.
`fisk` takes :math:`c` as a shape parameters.
%(after_notes)s
See Also -------- burr
%(example)s
""" # fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2) return burr_gen._pdf(self, x, c, 1.0)
return burr_gen._cdf(self, x, c, 1.0)
return burr_gen._ppf(self, x, c, 1.0)
return burr_gen._munp(self, n, c, 1.0)
return 2 - np.log(c)
# median = loc 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)}
%(after_notes)s
%(example)s
""" # cauchy.pdf(x) = 1 / (pi * (1 + x**2)) return 1.0/np.pi/(1.0+x*x)
return 0.5 + 1.0/np.pi*np.arctan(x)
return np.tan(np.pi*q-np.pi/2.0)
return 0.5 - 1.0/np.pi*np.arctan(x)
return np.tan(np.pi/2.0-np.pi*q)
return np.nan, np.nan, np.nan, np.nan
return np.log(4*np.pi)
# Initialize ML guesses using quartiles instead of moments. p25, p50, p75 = np.percentile(data, [25, 50, 75]) return p50, (p75 - p25)/2
r"""A chi continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `chi` is:
.. math::
f(x, df) = \frac{x^{df-1} \exp(-x^2/2)}{2^{df/2-1} \gamma(df/2)}
for :math:`x > 0`.
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
"""
sz, rndm = self._size, self._random_state return np.sqrt(chi2.rvs(df, size=sz, random_state=rndm))
# 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))
l = np.log(2) - .5*np.log(2)*df - sc.gammaln(.5*df) return l + sc.xlogy(df - 1., x) - .5*x**2
return sc.gammainc(.5*df, .5*x**2)
return np.sqrt(2*sc.gammaincinv(.5*df, q))
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-squared (gamma-distributed with loc=0 and scale=2 and shape=df/2) r"""A chi-squared continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `chi2` is:
.. math::
f(x, df) = \frac{1}{(2 \gamma(df/2)} (x/2)^{df/2-1} \exp(-x/2)
`chi2` takes ``df`` as a shape parameter.
%(after_notes)s
%(example)s
""" return self._random_state.chisquare(df, self._size)
# chi2.pdf(x, df) = 1 / (2*gamma(df/2)) * (x/2)**(df/2-1) * exp(-x/2) return np.exp(self._logpdf(x, df))
return sc.xlogy(df/2.-1, x) - x/2. - sc.gammaln(df/2.) - (np.log(2)*df)/2.
return sc.chdtr(df, x)
return sc.chdtrc(df, x)
return sc.chdtri(df, p)
return self._isf(1.0-p, df)
mu = df mu2 = 2*df g1 = 2*np.sqrt(2.0/df) g2 = 12.0/df return mu, mu2, g1, g2
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
""" # cosine.pdf(x) = 1/(2*pi) * (1+cos(x)) return 1.0/2/np.pi*(1+np.cos(x))
return 1.0/2/np.pi*(np.pi + x + np.sin(x))
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)
return np.log(4*np.pi)-1.0
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 :math:`a > 0`.
`dgamma` takes :math:`a` as a shape parameter.
%(after_notes)s
%(example)s
""" sz, rndm = self._size, self._random_state u = rndm.random_sample(size=sz) gm = gamma.rvs(a, size=sz, random_state=rndm) return gm * np.where(u >= 0.5, 1, -1)
# 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)
ax = abs(x) return sc.xlogy(a - 1.0, ax) - ax - np.log(2) - sc.gammaln(a)
fac = 0.5*sc.gammainc(a, abs(x)) return np.where(x > 0, 0.5 + fac, 0.5 - fac)
fac = 0.5*sc.gammainc(a, abs(x)) return np.where(x > 0, 0.5-fac, 0.5+fac)
fac = sc.gammainccinv(a, 1-abs(2*q-1)) return np.where(q > 0.5, fac, -fac)
mu2 = a*(a+1.0) return 0.0, mu2, 0.0, (a+2.0)*(a+3.0)/mu2-3.0
r"""A double Weibull continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `dweibull` is:
.. math::
f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c)
`dweibull` takes :math:`d` as a shape parameter.
%(after_notes)s
%(example)s
""" sz, rndm = self._size, self._random_state u = rndm.random_sample(size=sz) w = weibull_min.rvs(c, size=sz, random_state=rndm) return w * (np.where(u >= 0.5, 1, -1))
# 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
ax = abs(x) return np.log(c) - np.log(2.0) + sc.xlogy(c - 1.0, ax) - ax**c
Cx1 = 0.5 * np.exp(-abs(x)**c) return np.where(x > 0, 1 - Cx1, Cx1)
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)
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. return 0, None, 0, None
## Exponential (gamma distributed with a=1.0, loc=loc and scale=scale) 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
""" return self._random_state.standard_exponential(self._size)
# expon.pdf(x) = exp(-x) return np.exp(-x)
return -x
return -sc.expm1(-x)
return -sc.log1p(-q)
return np.exp(-x)
return -x
return -np.log(q)
return 1.0, 1.0, 2.0, 6.0
return 1.0
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)
# Ignore the optimizer-related keyword arguments, if given. kwds.pop('loc', None) kwds.pop('scale', None) kwds.pop('optimizer', None) if kwds: raise TypeError("Unknown arguments: %s." % 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) 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)
## Exponentially Modified Normal (exponential distribution ## convolved with a Normal). ## This is called an exponentially modified gaussian on wikipedia 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}\right) \exp(-x / K) \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)
where the shape parameter :math:`K > 0`.
It can be thought of as the sum of a normally distributed random value with mean ``loc`` and sigma ``scale`` and an exponentially distributed random number with a pdf proportional to ``exp(-lambda * x)`` where ``lambda = (K * scale)**(-1)``.
%(after_notes)s
An alternative parameterization of this distribution (for example, in `Wikipedia <http://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution>`_) 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
""" expval = self._random_state.standard_exponential(self._size) * K gval = self._random_state.standard_normal(self._size) return expval + gval
# 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))
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)))
invK = 1.0 / K expval = invK * (0.5 * invK - x) return _norm_cdf(x) - np.exp(expval) * _norm_cdf(x - invK)
invK = 1.0 / K expval = invK * (0.5 * invK - x) return _norm_cdf(-x) + np.exp(expval) * _norm_cdf(x - invK)
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
r"""An exponentiated Weibull continuous random variable.
%(before_notes)s
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}
for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`.
`exponweib` takes :math:`a` and :math:`c` as shape parameters.
%(after_notes)s
%(example)s
""" # 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))
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
exm1c = -sc.expm1(-x**c) return exm1c**a
return (-sc.log1p(-q**(1.0/a)))**np.asarray(1.0/c)
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 :math:`b` as a shape parameter.
%(after_notes)s
References ---------- http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf
%(example)s
""" # exponpow.pdf(x, b) = b * x**(b-1) * exp(1 + x**b - exp(x**b)) return np.exp(self._logpdf(x, b))
xb = x**b f = 1 + np.log(b) + sc.xlogy(b - 1.0, x) + xb - np.exp(xb) return f
return -sc.expm1(-sc.expm1(x**b))
return np.exp(-sc.expm1(x**b))
return (sc.log1p(-np.log(x)))**(1./b)
return pow(sc.log1p(-sc.log1p(-q)), 1.0/b)
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`.
`fatiguelife` takes :math:`c` as a shape parameter.
%(after_notes)s
References ---------- .. [1] "Birnbaum-Saunders distribution", http://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution
%(example)s
"""
z = self._random_state.standard_normal(self._size) x = 0.5*c*z x2 = x*x t = 1.0 + 2*x2 + 2*x*np.sqrt(1 + x2) return t
# 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))
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)))
return _norm_cdf(1.0 / c * (np.sqrt(x) - 1.0/np.sqrt(x)))
tmp = c*sc.ndtri(q) return 0.25 * (tmp + np.sqrt(tmp**2 + 4))**2
# 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
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 :math:`c` as a shape parameter.
%(example)s
""" return abs(cauchy.rvs(loc=c, size=self._size, random_state=self._random_state))
# 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))
return 1.0/np.pi*(np.arctan(x-c) + np.arctan(x+c))
return np.inf, np.inf, np.nan, np.nan
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
""" return self._random_state.f(dfn, dfd, self._size)
# 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))
n = 1.0 * dfn m = 1.0 * dfd lPx = m/2 * np.log(m) + n/2 * np.log(n) + (n/2 - 1) * np.log(x) lPx -= ((n+m)/2) * np.log(m + n*x) + sc.betaln(n/2, m/2) return lPx
return sc.fdtr(dfn, dfd, x)
return sc.fdtrc(dfn, dfd, x)
return sc.fdtri(dfn, dfd, q)
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
## 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.
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 :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" return c >= 0
return abs(self._random_state.standard_normal(self._size) + 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)
return _norm_cdf(x-c) + _norm_cdf(x+c) - 1.0
# Regina C. Elandt, Technometrics 3, 551 (1961) # http://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
r"""Weibull minimum continuous random variable.
%(before_notes)s
See Also -------- weibull_max
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.
%(after_notes)s
%(example)s
"""
# frechet_r.pdf(x, c) = c * x**(c-1) * exp(-x**c) return c*pow(x, c-1)*np.exp(-pow(x, c))
return np.log(c) + sc.xlogy(c - 1, x) - pow(x, c)
return -sc.expm1(-pow(x, c))
return np.exp(-pow(x, c))
return -pow(x, c)
return pow(-sc.log1p(-q), 1.0/c)
return sc.gamma(1.0+n*1.0/c)
return -_EULER / c - np.log(c) + _EULER + 1
r"""Weibull maximum continuous random variable.
%(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.
%(after_notes)s
%(example)s
""" # frechet_l.pdf(x, c) = c * (-x)**(c-1) * exp(-(-x)**c) return c*pow(-x, c-1)*np.exp(-pow(-x, c))
return np.log(c) + sc.xlogy(c-1, -x) - pow(-x, c)
return np.exp(-pow(-x, c))
return -pow(-x, c)
return -sc.expm1(-pow(-x, c))
return -pow(-np.log(q), 1.0/c)
val = sc.gamma(1.0+n*1.0/c) if int(n) % 2: sgn = -1 else: sgn = 1 return sgn * val
return -_EULER / c - np.log(c) + _EULER + 1
# 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']
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`."""
def __call__(self, *args, **kwargs): return weibull_min_gen.__call__(self, *args, **kwargs)
def cdf(self, *args, **kwargs): return weibull_min_gen.cdf(self, *args, **kwargs)
def entropy(self, *args, **kwargs): return weibull_min_gen.entropy(self, *args, **kwargs)
def expect(self, *args, **kwargs): return weibull_min_gen.expect(self, *args, **kwargs)
def fit(self, *args, **kwargs): return weibull_min_gen.fit(self, *args, **kwargs)
def fit_loc_scale(self, *args, **kwargs): return weibull_min_gen.fit_loc_scale(self, *args, **kwargs)
def freeze(self, *args, **kwargs): return weibull_min_gen.freeze(self, *args, **kwargs)
def interval(self, *args, **kwargs): return weibull_min_gen.interval(self, *args, **kwargs)
def isf(self, *args, **kwargs): return weibull_min_gen.isf(self, *args, **kwargs)
def logcdf(self, *args, **kwargs): return weibull_min_gen.logcdf(self, *args, **kwargs)
def logpdf(self, *args, **kwargs): return weibull_min_gen.logpdf(self, *args, **kwargs)
def logsf(self, *args, **kwargs): return weibull_min_gen.logsf(self, *args, **kwargs)
def mean(self, *args, **kwargs): return weibull_min_gen.mean(self, *args, **kwargs)
def median(self, *args, **kwargs): return weibull_min_gen.median(self, *args, **kwargs)
def moment(self, *args, **kwargs): return weibull_min_gen.moment(self, *args, **kwargs)
def nnlf(self, *args, **kwargs): return weibull_min_gen.nnlf(self, *args, **kwargs)
def pdf(self, *args, **kwargs): return weibull_min_gen.pdf(self, *args, **kwargs)
def ppf(self, *args, **kwargs): return weibull_min_gen.ppf(self, *args, **kwargs)
def rvs(self, *args, **kwargs): return weibull_min_gen.rvs(self, *args, **kwargs)
def sf(self, *args, **kwargs): return weibull_min_gen.sf(self, *args, **kwargs)
def stats(self, *args, **kwargs): return weibull_min_gen.stats(self, *args, **kwargs)
def std(self, *args, **kwargs): return weibull_min_gen.std(self, *args, **kwargs)
def var(self, *args, **kwargs): return weibull_min_gen.var(self, *args, **kwargs)
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`."""
def __call__(self, *args, **kwargs): return weibull_max_gen.__call__(self, *args, **kwargs)
def cdf(self, *args, **kwargs): return weibull_max_gen.cdf(self, *args, **kwargs)
def entropy(self, *args, **kwargs): return weibull_max_gen.entropy(self, *args, **kwargs)
def expect(self, *args, **kwargs): return weibull_max_gen.expect(self, *args, **kwargs)
def fit(self, *args, **kwargs): return weibull_max_gen.fit(self, *args, **kwargs)
def fit_loc_scale(self, *args, **kwargs): return weibull_max_gen.fit_loc_scale(self, *args, **kwargs)
def freeze(self, *args, **kwargs): return weibull_max_gen.freeze(self, *args, **kwargs)
def interval(self, *args, **kwargs): return weibull_max_gen.interval(self, *args, **kwargs)
def isf(self, *args, **kwargs): return weibull_max_gen.isf(self, *args, **kwargs)
def logcdf(self, *args, **kwargs): return weibull_max_gen.logcdf(self, *args, **kwargs)
def logpdf(self, *args, **kwargs): return weibull_max_gen.logpdf(self, *args, **kwargs)
def logsf(self, *args, **kwargs): return weibull_max_gen.logsf(self, *args, **kwargs)
def mean(self, *args, **kwargs): return weibull_max_gen.mean(self, *args, **kwargs)
def median(self, *args, **kwargs): return weibull_max_gen.median(self, *args, **kwargs)
def moment(self, *args, **kwargs): return weibull_max_gen.moment(self, *args, **kwargs)
def nnlf(self, *args, **kwargs): return weibull_max_gen.nnlf(self, *args, **kwargs)
def pdf(self, *args, **kwargs): return weibull_max_gen.pdf(self, *args, **kwargs)
def ppf(self, *args, **kwargs): return weibull_max_gen.ppf(self, *args, **kwargs)
def rvs(self, *args, **kwargs): return weibull_max_gen.rvs(self, *args, **kwargs)
def sf(self, *args, **kwargs): return weibull_max_gen.sf(self, *args, **kwargs)
def stats(self, *args, **kwargs): return weibull_max_gen.stats(self, *args, **kwargs)
def std(self, *args, **kwargs): return weibull_max_gen.std(self, *args, **kwargs)
def var(self, *args, **kwargs): return weibull_max_gen.var(self, *args, **kwargs)
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 :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" # genlogistic.pdf(x, c) = c * exp(-x) / (1 + exp(-x))**(c+1) return np.exp(self._logpdf(x, c))
return np.log(c) - x - (c+1.0)*sc.log1p(np.exp(-x))
Cx = (1+np.exp(-x))**(-c) return Cx
vals = -np.log(pow(q, -1.0/c)-1) return vals
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
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 :math:`c` as a shape parameter.
For ``c == 0``, `genpareto` reduces to the exponential distribution, `expon`:
.. math::
f(x, c=0) = \exp(-x)
For ``c == -1``, `genpareto` is uniform on ``[0, 1]``:
.. math::
f(x, c=-1) = x
%(after_notes)s
%(example)s
""" c = np.asarray(c) self.b = _lazywhere(c < 0, (c,), lambda c: -1. / c, np.inf) return True
# genpareto.pdf(x, c) = (1 + c * x)**(-1 - 1/c) return np.exp(self._logpdf(x, c))
return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: -sc.xlog1py(c + 1., c*x) / c, -x)
return -sc.inv_boxcox1p(-x, -c)
return sc.inv_boxcox(-x, -c)
return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: -sc.log1p(c*x) / c, -x)
return -sc.boxcox1p(-q, -c)
return -sc.boxcox(q, -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))
return 1. + c
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
""" # 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)
return -sc.expm1((-a-b)*x + b*(-sc.expm1(-c*x))/c)
return np.log(a+b*(-sc.expm1(-c*x))) + (-a-b)*x+b*(-sc.expm1(-c*x))/c
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 :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" self.b = np.where(c > 0, 1.0 / np.maximum(c, _XMIN), np.inf) self.a = np.where(c < 0, 1.0 / np.minimum(c, -_XMIN), -np.inf) return np.where(abs(c) == np.inf, 0, 1)
return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: sc.log1p(-c*x)/c, -x)
# 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))
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
return -np.exp(self._loglogcdf(x, c))
return np.exp(self._logcdf(x, c))
return -sc.expm1(self._logcdf(x, c))
x = -np.log(-np.log(q)) return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: -sc.expm1(-c * x) / c, x)
x = -np.log(-sc.log1p(-q)) return _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: -sc.expm1(-c * x) / c, x)
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
# 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,))
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)
return _EULER*(1 - c) + 1
# 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.
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` has a shape parameter `a` which needs to be set explicitly.
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
""" return self._random_state.standard_gamma(a, self._size)
# gamma.pdf(x, a) = x**(a-1) * exp(-x) / gamma(a) return np.exp(self._logpdf(x, a))
return sc.xlogy(a-1.0, x) - x - sc.gammaln(a)
return sc.gammainc(a, x)
return sc.gammaincc(a, x)
return sc.gammaincinv(a, q)
return a, a, 2.0/np.sqrt(a), 6.0/a
return sc.psi(a)*(1-a) + a + sc.gammaln(a)
# 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,))
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): f0 = (kwds.get('f0', None) or kwds.get('fa', None) or kwds.get('fix_a', None)) floc = kwds.get('floc', None) fscale = kwds.get('fscale', None)
if floc is None: # loc is not fixed. Use the default fit method. return super(gamma_gen, self).fit(data, *args, **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 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
"""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.
"""
allint = np.all(np.floor(a) == a) allpos = np.all(a > 0) 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 allpos
# 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. return super(erlang_gen, self).fit(data, *args, **kwds)
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=<integer>`, the fit method can be constrained to fit the data to a specific integer shape parameter. """)
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`.
`gengamma` takes :math:`a` and :math:`c` as shape parameters.
%(after_notes)s
%(example)s
""" return (a > 0) & (c != 0)
# gengamma.pdf(x, a, c) = abs(c) * x**(c*a-1) * exp(-x**c) / gamma(a) return np.exp(self._logpdf(x, a, c))
return np.log(abs(c)) + sc.xlogy(c*a - 1, x) - x**c - sc.gammaln(a)
xc = x**c val1 = sc.gammainc(a, xc) val2 = sc.gammaincc(a, xc) return np.where(c > 0, val1, val2)
xc = x**c val1 = sc.gammainc(a, xc) val2 = sc.gammaincc(a, xc) return np.where(c > 0, val2, val1)
val1 = sc.gammaincinv(a, q) val2 = sc.gammainccinv(a, q) return np.where(c > 0, val1, val2)**(1.0/c)
val1 = sc.gammaincinv(a, q) val2 = sc.gammainccinv(a, q) return np.where(c > 0, val2, val1)**(1.0/c)
# Pochhammer symbol: sc.pocha,n) = gamma(a+n)/gamma(a) return sc.poch(a, n*1.0/c)
val = sc.psi(a) return a*(1-val) + 1.0/c*val + sc.gammaln(a) - np.log(abs(c))
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 :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" self.b = 1.0 / c return c > 0
# 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
limit = 1.0/c tmp = np.asarray(1-c*x) tmp2 = tmp**(limit) return (1.0-tmp2) / (1+tmp2)
return 1.0/c*(1-((1.0-q)/(1.0+q))**c)
return 2 - (2*c+1)*np.log(2)
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 :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" # gompertz.pdf(x, c) = c * exp(x) * exp(-c*(exp(x)-1)) return np.exp(self._logpdf(x, c))
return np.log(c) + x - c * sc.expm1(x)
return -sc.expm1(-c * sc.expm1(x))
return sc.log1p(-1.0 / c * sc.log1p(-q))
return 1.0 - np.log(c) - np.exp(c)*sc.expn(1, c)
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
""" # gumbel_r.pdf(x) = exp(-(x + exp(-x))) return np.exp(self._logpdf(x))
return -x - np.exp(-x)
return np.exp(-np.exp(-x))
return -np.exp(-x)
return -np.log(-np.log(q))
return _EULER, np.pi*np.pi/6.0, 12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5
# http://en.wikipedia.org/wiki/Gumbel_distribution return _EULER + 1.
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
""" # gumbel_l.pdf(x) = exp(x - exp(x)) return np.exp(self._logpdf(x))
return x - np.exp(x)
return -sc.expm1(-np.exp(x))
return np.log(-sc.log1p(-q))
return -np.exp(x)
return np.exp(-np.exp(x))
return np.log(-np.log(x))
return -_EULER, np.pi*np.pi/6.0, \ -12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5
return _EULER + 1.
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
""" # halfcauchy.pdf(x) = 2 / (pi * (1 + x**2)) return 2.0/np.pi/(1.0+x*x)
return np.log(2.0/np.pi) - sc.log1p(x*x)
return 2.0/np.pi*np.arctan(x)
return np.tan(np.pi/2*q)
return np.inf, np.inf, np.nan, np.nan
return np.log(2*np.pi)
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} sech(x/2)^2
for :math:`x \ge 0`.
%(after_notes)s
%(example)s
""" # halflogistic.pdf(x) = 2 * exp(-x) / (1+exp(-x))**2 # = 1/2 * sech(x/2)**2 return np.exp(self._logpdf(x))
return np.log(2) - x - 2. * sc.log1p(np.exp(-x))
return np.tanh(x/2.0)
return 2*np.arctanh(q)
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)
return 2-np.log(2)
r"""A half-normal continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `halfnorm` is:
.. math::
f(x) = \sqrt{2/\pi} e^{-\frac{x^2}{2}}
for :math:`x > 0`.
`halfnorm` is a special case of :math`\chi` with ``df == 1``.
%(after_notes)s
%(example)s
""" return abs(self._random_state.standard_normal(size=self._size))
# halfnorm.pdf(x) = sqrt(2/pi) * exp(-x**2/2) return np.sqrt(2.0/np.pi)*np.exp(-x*x/2.0)
return 0.5 * np.log(2.0/np.pi) - x*x/2.0
return _norm_cdf(x)*2-1.0
return sc.ndtri((1+q)/2.0)
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)
return 0.5*np.log(np.pi/2.0)+0.5
r"""A hyperbolic secant continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `hypsecant` is:
.. math::
f(x) = \frac{1}{\pi} sech(x)
%(after_notes)s
%(example)s
""" # hypsecant.pdf(x) = 1/pi * sech(x) return 1.0/(np.pi*np.cosh(x))
return 2.0/np.pi*np.arctan(np.exp(x))
return np.log(np.tan(np.pi*q/2.0))
return 0, np.pi*np.pi/4, 0, 2
return np.log(2*np.pi)
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)}`
`gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape parameters.
%(after_notes)s
%(example)s
""" return (a > 0) & (b > 0) & (c == c) & (z == 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
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
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`.
`invgamma` takes :math:`a` as a shape parameter.
`invgamma` is a special case of `gengamma` with ``c == -1``.
%(after_notes)s
%(example)s
"""
# invgamma.pdf(x, a) = x**(-a-1) / gamma(a) * exp(-1/x) return np.exp(self._logpdf(x, a))
return -(a+1) * np.log(x) - sc.gammaln(a) - 1.0/x
return sc.gammaincc(a, 1.0 / x)
return 1.0 / sc.gammainccinv(a, q)
return sc.gammainc(a, 1.0 / x)
return 1.0 / sc.gammaincinv(a, q)
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
return a - (a+1.0) * sc.psi(a) + sc.gammaln(a)
# scale is gamma from DATAPLOT and B from Regress 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`.
`invgauss` takes :math:`\mu` as a shape parameter.
%(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
"""
return self._random_state.wald(mu, 1.0, size=self._size)
# 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)
return -0.5*np.log(2*np.pi) - 1.5*np.log(x) - ((x-mu)/mu)**2/(2*x)
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
return mu, mu**3.0, 3*np.sqrt(mu), 15*mu
r"""A Normal Inverse Gaussian continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `norminvgauss` is:
.. math::
f(x; a, b) = (a \exp(\sqrt{a^2 - b^2} + b x)) / (\pi \sqrt{1 + x^2} \, K_1(a * \sqrt{1 + x^2}))
where `x` is a real number, the parameter `a` is the tail heaviness and `b` is the asymmetry parameter satisfying `a > 0` and `abs(b) <= a`. `K_1` is the modified Bessel function of second kind (`scipy.special.k1`).
%(after_notes)s
A normal inverse Gaussian random variable with parameters `a` and `b` can be expressed as `X = 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
"""
return (a > 0) & (np.absolute(b) < a)
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
# 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) sz, rndm = self._size, self._random_state ig = invgauss.rvs(mu=1/gamma, size=sz, random_state=rndm) return b * ig + np.sqrt(ig) * norm.rvs(size=sz, random_state=rndm)
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
r"""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 :math:`c`` as a shape parameter.
%(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
"""
# 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
xc1 = np.power(x, -c) return np.exp(-xc1)
return np.power(-np.log(q), -1.0/c)
return sc.gamma(1 - n / c)
return 1+_EULER + _EULER / c - np.log(c)
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
"""
return (b > 0) & (a == a)
# 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
return _norm_cdf(a + b*np.log(x/(1.0-x)))
return 1.0 / (1 + np.exp(-1.0 / b * (_norm_ppf(q) - a)))
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
""" return (b > 0) & (a == a)
# 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
return _norm_cdf(a + b * np.log(x + np.sqrt(x*x + 1)))
return np.sinh((_norm_ppf(q) - a) / b)
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|)
%(after_notes)s
%(example)s
""" return self._random_state.laplace(0, 1, size=self._size)
# laplace.pdf(x) = 1/2 * exp(-abs(x)) return 0.5*np.exp(-abs(x))
return np.where(x > 0, 1.0-0.5*np.exp(-x), 0.5*np.exp(x))
return np.where(q > 0.5, -np.log(2*(1-q)), np.log(2*q))
return 0, 2, 0, 3
return np.log(2)+1
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}{x \sqrt{2\pi x}) \exp(-\frac{1}{2x}}
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
"""
# 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))
# Equivalent to 2*norm.sf(np.sqrt(1/x)) return sc.erfc(np.sqrt(0.5 / x))
# 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)
return np.inf, np.inf, np.nan, np.nan
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(-\frac{1}{2 |x|})
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
"""
# 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))
ax = abs(x) return 2 * _norm_cdf(1 / np.sqrt(ax)) - 1
val = _norm_ppf((q + 1.0) / 2) return -1.0 / (val * val)
return np.inf, np.inf, np.nan, np.nan
r"""A Levy-stable continuous random variable.
%(before_notes)s
See Also -------- levy, levy_l
Notes ----- Levy-stable distribution (only random variates available -- ignore other docs)
"""
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
sz = self._size alpha = broadcast_to(alpha, sz) beta = broadcast_to(beta, sz) TH = uniform.rvs(loc=-np.pi/2.0, scale=np.pi, size=sz, random_state=self._random_state) W = expon.rvs(size=sz, random_state=self._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
return (alpha > 0) & (alpha <= 2) & (beta <= 1) & (beta >= -1)
raise NotImplementedError
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
""" return self._random_state.logistic(size=self._size)
# logistic.pdf(x) = exp(-x) / (1+exp(-x))**2 return np.exp(self._logpdf(x))
return -x - 2. * sc.log1p(np.exp(-x))
return sc.expit(x)
return sc.logit(q)
return sc.expit(-x)
return -sc.logit(q)
return 0, np.pi*np.pi/3.0, 0, 6.0/5.0
# http://en.wikipedia.org/wiki/Logistic_distribution return 2.0
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`.
`loggamma` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" return np.log(self._random_state.gamma(c, size=self._size))
# loggamma.pdf(x, c) = exp(c*x-exp(x)) / gamma(c) return np.exp(c*x-np.exp(x)-sc.gammaln(c))
return sc.gammainc(c, np.exp(x))
return np.log(sc.gammaincinv(c, q))
# 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
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 ``c > 0``.
`loglaplace` takes ``c`` as a shape parameter.
%(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
""" # 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)
return np.where(x < 1, 0.5*x**c, 1-0.5*x**(-c))
return np.where(q < 0.5, (2.0*q)**(1.0/c), (2*(1.0-q))**(-1.0/c))
return c**2 / (c**2 - n**2)
return np.log(2.0/c) + 1.0
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)
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(-\frac{1}{2} (\frac{\log(x)}{s})^2)
for ``x > 0``, ``s > 0``.
`lognorm` takes ``s`` as a shape parameter.
%(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
"""
return np.exp(s * self._random_state.standard_normal(self._size))
# lognorm.pdf(x, s) = 1 / (s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2) return np.exp(self._logpdf(x, s))
return _lognorm_logpdf(x, s)
return _norm_cdf(np.log(x) / s)
return _norm_logcdf(np.log(x) / s)
return np.exp(s * _norm_ppf(q))
return _norm_sf(np.log(x) / s)
return _norm_logsf(np.log(x) / 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
return 0.5 * (1 + np.log(2*np.pi) + 2 * np.log(s))
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) 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
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
"""
return np.exp(self._random_state.standard_normal(self._size))
# gilbrat.pdf(x) = 1/(x*sqrt(2*pi)) * exp(-1/2*(log(x))**2) return np.exp(self._logpdf(x))
return _lognorm_logpdf(x, 1.0)
return _norm_cdf(np.log(x))
return np.exp(_norm_ppf(q))
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
return 0.5 * np.log(2 * np.pi) + 0.5
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 ``x > 0``.
%(after_notes)s
References ---------- .. [1] http://mathworld.wolfram.com/MaxwellDistribution.html
%(example)s """ return chi.rvs(3.0, size=self._size, random_state=self._random_state)
# maxwell.pdf(x) = sqrt(2/pi)x**2 * exp(-x**2/2) return np.sqrt(2.0/np.pi)*x*x*np.exp(-x*x/2.0)
return sc.gammainc(1.5, x*x/2.0)
return np.sqrt(2*sc.gammaincinv(1.5, q))
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)
return _EULER + 0.5*np.log(2*np.pi)-0.5
r"""A Mielke's Beta-Kappa 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 ``x > 0``.
`mielke` takes ``k`` and ``s`` as shape parameters.
%(after_notes)s
%(example)s
""" # mielke.pdf(x, k, s) = k * x**(k-1) / (1+x**s)**(1+k/s) return k*x**(k-1.0) / (1.0+x**s)**(1.0+k*1.0/s)
return x**k / (1.0+x**s)**(k*1.0/s)
qsk = pow(q, s*1.0/k) return pow(qsk/(1.0-qsk), 1.0/s)
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 http://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), http://digitalcommons.lsu.edu/cgi/viewcontent.cgi?article=4671&context=gradschool_dissertations
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). http://file.scirp.org/pdf/JWARP20121000009_14676002.pdf
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
""" 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
self.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
self.b = _lazyselect(condlist, [f0, f1, f1, f0, f1, f1], [h, k], default=np.nan) return h == h
# 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))
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)
return np.exp(self._logcdf(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)
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)
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[:]
r"""Kappa 3 parameter distribution.
%(before_notes)s
Notes ----- The probability density function for `kappa` is:
.. math::
f(x, a) = \begin{cases} a [a + x^a]^{-(a + 1)/a}, &\text{for } x > 0\\ 0.0, &\text{for } x \le 0 \end{cases}
`kappa3` takes :math:`a` as a shape parameter and :math:`a > 0`.
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), http://docs.lib.noaa.gov/rescue/mwr/101/mwr-101-09-0701.pdf
B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2, 415-419 (2012) http://file.scirp.org/pdf/OJS20120400011_95789012.pdf
%(after_notes)s
%(example)s
""" return a > 0
# kappa3.pdf(x, a) = # a*[a + x**a]**(-(a + 1)/a), for x > 0 # 0.0, for x <= 0 return a*(a + x**a)**(-1.0/a-1)
return x*(a + x**a)**(-1.0/a)
return (a/(q**-a - 1.0))**(1.0/a)
outputs = [None if i < a else np.nan for i in range(1, 5)] return outputs[:]
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}
%(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). https://doi.org/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). www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf
.. versionadded:: 1.1.0
%(example)s
""" sz, rndm = self._size, self._random_state u1 = gamma.rvs(a = 0.5, scale = 2, size=sz, random_state=rndm) return -np.log(u1)
return np.exp(-0.5 * (x + np.exp(-x))) / np.sqrt(2*np.pi)
return sc.erfc(np.exp(-0.5 * x) / np.sqrt(2))
return sc.erf(np.exp(-0.5 * x) / np.sqrt(2))
return -np.log(2 * sc.erfcinv(x)**2)
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
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)
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 ``x > 0``, ``nu > 0``.
`nakagami` takes ``nu`` as a shape parameter.
%(after_notes)s
%(example)s
""" # 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)
return sc.gammainc(nu, nu*x*x)
return np.sqrt(1.0/nu*sc.gammaincinv(nu, q))
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
r"""A non-central chi-squared continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `ncx2` is:
.. math::
f(x, df, nc) = \exp(-\frac{nc+x}{2}) \frac{1}{2} (x/nc)^{(df-2)/4} I[(df-2)/2](\sqrt{nc x})
for :math:`x > 0`.
`ncx2` takes ``df`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
""" return self._random_state.noncentral_chisquare(df, nc, self._size)
return _ncx2_log_pdf(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)) return _ncx2_pdf(x, df, nc)
return _ncx2_cdf(x, df, nc)
return sc.chndtrix(q, 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)
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(\frac{\lambda}{2} + \lambda n_1 \frac{x}{2(n_1 x+n_2)}) 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{v_1}{2}-1}_{v_2/2} (-\lambda v_1 \frac{x}{2(v_1 x+v_2)})} {(B(v_1/2, v_2/2) \gamma(\frac{v_1+v_2}{2})}
for :math:`n_1 > 1`, :math:`n_2, \lambda > 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.
%(after_notes)s
%(example)s
""" return self._random_state.noncentral_f(dfn, dfd, nc, self._size)
# 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.
return sc.ncfdtr(dfn, dfd, nc, x)
return sc.ncfdtri(dfn, dfd, nc, q)
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
mu = np.where(dfd <= 2, np.inf, dfd / (dfd-2.0)*(1+nc*1.0/dfn)) mu2 = np.where(dfd <= 4, np.inf, 2*(dfd*1.0/dfn)**2.0 * ((dfn+nc/2.0)**2.0 + (dfn+nc)*(dfd-2.0)) / ((dfd-2.0)**2.0 * (dfd-4.0))) return mu, mu2, None, None
r"""A Student's T continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `t` is:
.. math::
f(x, df) = \frac{\gamma((df+1)/2)} {\sqrt{\pi*df} \gamma(df/2) (1+x^2/df)^{(df+1)/2}}
for ``df > 0``.
`t` takes ``df`` as a shape parameter.
%(after_notes)s
%(example)s
""" return self._random_state.standard_t(df, size=self._size)
# 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
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
return sc.stdtr(df, x)
return sc.stdtrit(df, q)
return -sc.stdtrit(df, q)
mu2 = _lazywhere(df > 2, (df,), lambda df: df / (df-2.0), np.inf) g1 = np.where(df > 3, 0.0, np.nan) g2 = _lazywhere(df > 4, (df,), lambda df: 6.0 / (df-4.0), np.nan) return 0, mu2, g1, g2
r"""A non-central Student's T continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `nct` is:
.. math::
f(x, df, nc) = \frac{df^{df/2} \gamma(df+1)}{2^{df} \exp(nc^2 / 2) (df+x^2)^{df/2} \gamma(df/2)}
for ``df > 0``.
`nct` takes ``df`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
""" return (df > 0) & (nc == nc)
sz, rndm = self._size, self._random_state n = norm.rvs(loc=nc, size=sz, random_state=rndm) c2 = chi2.rvs(df, size=sz, random_state=rndm) return n * np.sqrt(df) / np.sqrt(c2)
# nct.pdf(x, df, nc) = # df**(df/2) * gamma(df+1) # ---------------------------------------------------- # 2**df*exp(nc**2/2) * (df+x**2)**(df/2) * gamma(df/2) 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
return sc.nctdtr(df, nc, x)
return sc.nctdtrit(df, nc, q)
# # 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. http://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
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 :math:`b` as a shape parameter.
%(after_notes)s
%(example)s
""" # pareto.pdf(x, b) = b / x**(b+1) return b * x**(-b-1)
return 1 - x**(-b)
return pow(1-q, -1.0/b)
return x**(-b)
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
return 1 + 1.0/c - np.log(c)
r"""A Lomax (Pareto of the second kind) continuous random variable.
%(before_notes)s
Notes ----- The Lomax distribution is a special case of the Pareto distribution, with (loc=-1.0).
The probability density function for `lomax` is:
.. math::
f(x, c) = \frac{c}{(1+x)^{c+1}}
for :math:`x \ge 0`, ``c > 0``.
`lomax` takes :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" # lomax.pdf(x, c) = c / (1+x)**(c+1) return c*1.0/(1.0+x)**(c+1.0)
return np.log(c) - (c+1)*sc.log1p(x)
return -sc.expm1(-c*sc.log1p(x))
return np.exp(-c*sc.log1p(x))
return -c*sc.log1p(x)
return sc.expm1(-sc.log1p(-q)/c)
mu, mu2, g1, g2 = pareto.stats(c, loc=-1.0, moments='mvsk') return mu, mu2, g1, g2
return 1+1.0/c-np.log(c)
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}
`pearson3` takes ``skew`` as a shape parameter.
%(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).
""" # 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
# 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)
_, _, _, _, _, 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
# 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
# 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
ans, x, transx, mask, invmask, _, alpha, _ = ( self._preprocess(x, skew))
ans[mask] = _norm_cdf(x[mask]) ans[invmask] = gamma._cdf(transx, alpha) return ans
skew = broadcast_to(skew, self._size) ans, _, _, mask, invmask, beta, alpha, zeta = ( self._preprocess([0], skew))
nsmall = mask.sum() nbig = mask.size - nsmall ans[mask] = self._random_state.standard_normal(nsmall) ans[invmask] = (self._random_state.standard_gamma(alpha, nbig)/beta + zeta)
if self._size == (): ans = ans[0] return ans
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
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 :math:`a` as a shape parameter.
%(after_notes)s
`powerlaw` is a special case of `beta` with ``b == 1``.
%(example)s
""" # powerlaw.pdf(x, a) = a * x**(a-1) return a*x**(a-1.0)
return np.log(a) + sc.xlogy(a - 1, x)
return x**(a*1.0)
return a*np.log(x)
return pow(q, 1.0/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)))
return 1 - 1.0/a - np.log(a)
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
"""
# 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))
return 1.0 - pow(_norm_cdf(-np.log(x)/s), c*1.0)
return np.exp(-s * _norm_ppf(pow(1.0 - q, 1.0 / c)))
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 :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" # powernorm.pdf(x, c) = c * phi(x) * (Phi(-x))**(c-1) return c*_norm_pdf(x) * (_norm_cdf(-x)**(c-1.0))
return np.log(c) + _norm_logpdf(x) + (c-1)*_norm_logcdf(-x)
return 1.0-_norm_cdf(-x)**(c*1.0)
return -_norm_ppf(pow(1.0 - q, 1.0 / c))
r"""An R-distributed 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` takes :math:`c` as a shape parameter.
This distribution includes the following distribution kernels as special cases::
c = 2: uniform c = 4: Epanechnikov (parabolic) c = 6: quartic (biweight) c = 8: triweight
%(after_notes)s
%(example)s
""" # rdist.pdf(x, c) = (1-x**2)**(c/2-1) / B(1/2, c/2) return np.power((1.0 - x**2), c / 2.0 - 1) / sc.beta(0.5, c / 2.0)
term1 = x / sc.beta(0.5, c / 2.0) res = 0.5 + term1 * sc.hyp2f1(0.5, 1 - c / 2.0, 1.5, x**2) # There's an issue with hyp2f1, it returns nans near x = +-1, c > 100. # Use the generic implementation in that case. See gh-1285 for # background. if np.any(np.isnan(res)): return rv_continuous._cdf(self, x, c) return res
numerator = (1 - (n % 2)) * sc.beta((n + 1.0) / 2, c / 2.0) return numerator / sc.beta(1. / 2, c / 2.)
r"""A Rayleigh continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `rayleigh` is:
.. math::
f(r) = r \exp(-r^2/2)
for :math:`x \ge 0`.
`rayleigh` is a special case of `chi` with ``df == 2``.
%(after_notes)s
%(example)s
"""
return chi.rvs(2, size=self._size, random_state=self._random_state)
# rayleigh.pdf(r) = r * exp(-r**2/2) return np.exp(self._logpdf(r))
return np.log(r) - 0.5 * r * r
return -sc.expm1(-0.5 * r**2)
return np.sqrt(-2 * sc.log1p(-q))
return np.exp(self._logsf(r))
return -0.5 * r * r
return np.sqrt(-2 * np.log(q))
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)
return _EULER/2.0 + 1 - 0.5*np.log(2)
r"""A reciprocal continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `reciprocal` is:
.. math::
f(x, a, b) = \frac{1}{x \log(b/a)}
for :math:`a \le x \le b`, :math:`a, b > 0`.
`reciprocal` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
""" self.a = a self.b = b self.d = np.log(b*1.0 / a) return (a > 0) & (b > 0) & (b > a)
# reciprocal.pdf(x, a, b) = 1 / (x*log(b/a)) return 1.0 / (x * self.d)
return -np.log(x) - np.log(self.d)
return (np.log(x)-np.log(a)) / self.d
return a*pow(b*1.0/a, q)
return 1.0/self.d / n * (pow(b*1.0, n) - pow(a*1.0, n))
return 0.5*np.log(a*b)+np.log(np.log(b/a))
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`.
`rice` takes :math:`b` as a shape parameter.
%(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
""" return b >= 0
# http://en.wikipedia.org/wiki/Rice_distribution t = b/np.sqrt(2) + self._random_state.standard_normal(size=(2,) + self._size) return np.sqrt((t*t).sum(axis=0))
return sc.chndtr(np.square(x), 2, np.square(b))
return np.sqrt(sc.chndtrix(q, 2, np.square(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)
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))
# FIXME: PPF does not work. 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}} \frac{\exp(-(1-\mu x)^2}{2x\mu^2)}
for :math:`x \ge 0`.
`recipinvgauss` takes :math:`\mu` as a shape parameter.
%(after_notes)s
%(example)s
"""
# 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))
return -(1-mu*x)**2.0 / (2*x*mu**2.0) - 0.5*np.log(2*np.pi*x)
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)
return 1.0/self._random_state.wald(mu, 1.0, size=self._size)
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`.
%(after_notes)s
%(example)s
""" # semicircular.pdf(x) = 2/pi * sqrt(1-x**2) return 2.0/np.pi*np.sqrt(1-x*x)
return 0.5+1.0/np.pi*(x*np.sqrt(1-x*x) + np.arcsin(x))
return 0, 0.25, 0, -1.0
return 0.64472988584940017414
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 :math:`a` as a skewness parameter When ``a = 0`` the distribution is identical to a normal distribution. 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. http://azzalini.stat.unipd.it/SN/faq-r.html
""" return np.isfinite(a)
return 2.*_norm_pdf(x)*_norm_cdf(a*x)
if x <= 0: cdf = integrate.quad(self._pdf, self.a, x, args=args)[0] else: t1 = integrate.quad(self._pdf, self.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
return self._cdf(-x, -a)
u0 = self._random_state.normal(size=self._size) v = self._random_state.normal(size=self._size) d = a/np.sqrt(1 + a**2) u1 = d*u0 + v*np.sqrt(1 - d**2) return np.where(u0 >= 0, u1, -u1)
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
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)``.
`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
""" return (c >= 0) & (c <= 1) & (d >= 0) & (d <= 1) & (d >= c)
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))
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))
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)
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 :math:`c` as a shape parameter.
%(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
""" return self._random_state.triangular(0, c, 1, self._size)
return (c >= 0) & (c <= 1)
# 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
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
return np.where(q < c, np.sqrt(c * q), 1-np.sqrt((1-c) * (1-q)))
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)
return 0.5-np.log(2)
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 :math:`b` as a shape parameter.
%(after_notes)s
%(example)s
""" self.b = b return b > 0
# truncexpon.pdf(x, b) = exp(-x) / (1-exp(-b)) return np.exp(-x)/(-sc.expm1(-b))
return -x - np.log(-sc.expm1(-b))
return sc.expm1(-x)/sc.expm1(-b)
return -sc.log1p(q*sc.expm1(-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)
eB = np.exp(b) return np.log(eB-1)+(1+eB*(b-1.0))/(1.0-eB)
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
""" self.a = a self.b = b self._nb = _norm_cdf(b) self._na = _norm_cdf(a) self._sb = _norm_sf(b) self._sa = _norm_sf(a) self._delta = np.where(self.a > 0, -(self._sb - self._sa), self._nb - self._na) self._logdelta = np.log(self._delta) return a != b
return _norm_pdf(x) / self._delta
return _norm_logpdf(x) - self._logdelta
return (_norm_cdf(x) - self._na) / self._delta
# XXX Use _lazywhere... ppf = np.where(self.a > 0, _norm_isf(q*self._sb + self._sa*(1.0-q)), _norm_ppf(q*self._nb + self._na*(1.0-q))) return ppf
nA, nB = self._na, self._nb d = nB - nA pA, pB = _norm_pdf(a), _norm_pdf(b) mu = (pA - pB) / d # correction sign mu2 = 1 + (a*pA - b*pB) / d - mu*mu return mu, mu2, None, None
# FIXME: RVS does not work. r"""A Tukey-Lamdba continuous random variable.
%(before_notes)s
Notes ----- A flexible distribution, able to represent and interpolate between the following distributions:
- Cauchy (lam=-1) - logistic (lam=0.0) - approx Normal (lam=0.14) - u-shape (lam = 0.5) - uniform from -1 to 1 (lam = 1)
`tukeylambda` takes ``lam`` as a shape parameter.
%(after_notes)s
%(example)s
""" return np.ones(np.shape(lam), dtype=bool)
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)
return sc.tklmbda(x, lam)
return sc.boxcox(q, lam) - sc.boxcox1p(-q, lam)
return 0, _tlvar(lam), 0, _tlkurt(lam)
def integ(p): return np.log(pow(p, lam-1)+pow(1-p, lam-1)) return integrate.quad(integ, 0, 1)[0]
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), )
r"""A uniform continuous random variable.
This distribution is constant between `loc` and ``loc + scale``.
%(before_notes)s
%(example)s
""" return self._random_state.uniform(0.0, 1.0, self._size)
return 1.0*(x == x)
return x
return q
return 0.5, 1.0/12, 0, -1.2
return 0.0
""" 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)
# Ignore the optimizer-related keyword arguments, if given. kwds.pop('loc', None) kwds.pop('scale', None) kwds.pop('optimizer', None) if kwds: raise TypeError("Unknown arguments: %s." % 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)
# 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)
r"""A Von Mises continuous random variable.
%(before_notes)s
Notes ----- If `x` is not in range or `loc` is not in range it assumes they are angles and converts them to [-\pi, \pi] equivalents.
The probability density function for `vonmises` 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`.
`vonmises` takes :math:`\kappa` as a shape parameter.
%(after_notes)s
See Also -------- vonmises_line : The same distribution, defined on a [-\pi, \pi] segment of the real line.
%(example)s
""" return self._random_state.vonmises(0.0, kappa, size=self._size)
# 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))
return _stats.von_mises_cdf(kappa, x)
return 0, None, 0, None
return (-kappa * sc.i1(kappa) / sc.i0(kappa) + np.log(2 * np.pi * sc.i0(kappa)))
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 """
return self._random_state.wald(1.0, 1.0, size=self._size)
# wald.pdf(x) = 1/sqrt(2*pi*x**3) * exp(-(x-1)**2/(2*x)) return invgauss._pdf(x, 1.0)
return invgauss._logpdf(x, 1.0)
return invgauss._cdf(x, 1.0)
return 1.0, 1.0, 3.0, 15.0
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 :math:`c` as a shape parameter.
%(after_notes)s
%(example)s
""" return (c > 0) & (c < 1)
# 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)))
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
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)
return np.log(2*np.pi*(1-c*c))
r"""A generalized normal continuous random variable.
%(before_notes)s
Notes ----- The probability density function for `gennorm` is [1]_::
beta gennorm.pdf(x, beta) = --------------- exp(-|x|**beta) 2 gamma(1/beta)
`gennorm` takes :math:`\beta` as a shape parameter. For :math:`\beta = 1`, it is identical to a Laplace distribution. For ``\beta = 2``, it is identical to a normal distribution (with :math:`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
"""
return np.exp(self._logpdf(x, beta))
return np.log(0.5*beta) - sc.gammaln(1.0/beta) - abs(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)
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)
return self._cdf(-x, beta)
return -self._ppf(x, 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.
return 1. / beta - np.log(.5 * beta) + sc.gammaln(1. / beta)
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)
`gennorm` takes :math:`\beta` as a shape parameter. For :math:`\beta = 1`, it is identical to an exponential distribution. For :math:`\beta = 2`, it is identical to a half normal distribution (with :math:`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
"""
# beta # halfgennorm.pdf(x, beta) = ------------- exp(-|x|**beta) # gamma(1/beta) return np.exp(self._logpdf(x, beta))
return np.log(beta) - sc.gammaln(1.0/beta) - x**beta
return sc.gammainc(1.0/beta, x**beta)
return sc.gammaincinv(1.0/beta, x)**(1.0/beta)
return sc.gammaincc(1.0/beta, x**beta)
return sc.gammainccinv(1.0/beta, x)**(1.0/beta)
return 1.0/beta - np.log(beta) + sc.gammaln(1.0/beta)
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` and :math:`m` as shape parameters. :math:`\beta` defines the point where the pdf changes from a power-law to a gaussian distribution :math:`m` is 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 """ """ 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)) rhs = lambda x, beta, m: np.exp(-x**2 / 2) lhs = lambda x, beta, m: (m/beta)**m * np.exp(-beta**2 / 2.0) * (m/beta - beta - x)**(-m) return N * _lazywhere(np.atleast_1d(x > -beta), (x, beta, m), f=rhs, f2=lhs)
""" 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)) rhs = lambda x, beta, m: (m/beta) * np.exp(-beta**2 / 2.0) / (m-1) + _norm_pdf_C * (_norm_cdf(x) - _norm_cdf(-beta)) lhs = lambda x, beta, m: (m/beta)**m * np.exp(-beta**2 / 2.0) * (m/beta - beta - x)**(-m+1) / (m-1) return N * _lazywhere(np.atleast_1d(x > -beta), (x, beta, m), f=rhs, f2=lhs)
""" 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(np.atleast_1d(n + 1 < m), (n, beta, m), np.vectorize(n_th_moment, otypes=[np.float]), np.inf)
""" In HEP crystal-ball is also defined for m = 1 (see plot on wikipedia) But the function doesn't have a finite integral in this corner case, and isn't a PDF anymore (but can still be used on a finite range). Here we restrict the function to m > 1. In addition we restrict beta to be positive """ return (m > 1) & (beta > 0)
""" 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
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(- 0.5 \chi^2 (1 - x^2))
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 """ """ Return PDF of the argus function
argus.pdf(x, chi) = chi**3 / (sqrt(2*pi) * Psi(chi)) * x * sqrt(1-x**2) * exp(- 0.5 * chi**2 * (1 - x**2)) """ y = 1.0 - x**2 return chi**3 / (_norm_pdf_C * _argus_phi(chi)) * x * np.sqrt(y) * np.exp(-chi**2 * y / 2)
""" Return CDF of the argus function """ return 1.0 - self._sf(x, chi)
""" Return survival function of the argus function """ return _argus_phi(chi * np.sqrt(1 - x**2)) / _argus_phi(chi)
""" 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()
"""
""" 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._hbins[0] kwargs['b'] = self._hbins[-1] super(rv_histogram, self).__init__(*args, **kwargs)
""" PDF of the histogram """ return self._hpdf[np.searchsorted(self._hbins, x, side='right')]
""" CDF calculated from the histogram """ return np.interp(x, self._hbins, self._hcdf)
""" Percentile function calculated from the histogram """ return np.interp(x, self._hcdf, self._hbins)
"""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)
"""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)
""" 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.
|