def cdf(self, q, lower_tail=True, log_p=False):
"""
The cdf of the distribution.
Parameters
----------
q : array_like
Variates at which to calculate the cdf. If any of `shape`, `a`, `b` or `xmin`
are arrays, `q` must have the same length.
lower_tail : logical, optional
If `True` (default), probabilities are P[X <= q], otherwise, P[X > q].
log_p : logical, optional
If `True`, probabilities *p* are interpreted as log(*p*).
Returns
-------
p : array_like
The integrated probability of a variate being smaller than *q*.
"""
qt = self._xt(q)
p = sp.gammainc(self._z, qt**self.b)/sp.gammainc(self._z, self._xmintb)
return self._cdf_convert(p, lower_tail, log_p)
def raw_moments(self, n):
"""
Calculate the nth raw moment, E[X^n].
Parameters
----------
n : array_like
The order(s) of the moment desired.
Returns
-------
mu_n : array_like
The raw moment(s) corresponding to the order(s) `n`.
Notes
-----
The 1st raw moment is equivalent to the mean.
Examples
--------
>>> from mrpy.stats import TGGD
>>> t = TGGD()
>>> mean = t.raw_moments(1)
>>> ten_moments = t.raw_moments(np.arange(10)) #doctest: +SKIP
"""
zn = (self.a + 1 + n)/self.b
z0 = (self.a + 1)/self.b
x = (self.xmin/self.scale)**self.b
return self.scale**n*sp.gammainc(zn, x)/sp.gammainc(z0, x)
def rho_gtm(m, logHs, alpha, beta, mmin=None, mmax=np.inf, norm="pdf", log=False, **Arhoc_kw):
"""
The mass-weighted integral of the MRP, in reverse (ie. from high to low mass)
%s
"""
t, A = _head(m, logHs, alpha, beta, mmin, norm, log, **Arhoc_kw)
shape = 10**(2*logHs)*sp.gammainc((alpha + 2)/beta, (m/10**logHs)**beta)
if log:
shape = np.log(shape)
return _tail(shape, A, log)
def central_moments(self, n):
"""
Calculate the nth central moment, E[(X-mu)^n].
Parameters
----------
n : integer
The order of the moment desired.
Returns
-------
mu_n : float
The nth central moment.
Notes
-----
The 2nd central moment is equivalent to the variance.
Examples
--------
>>> from mrpy.stats import TGGD
>>> t = TGGD()
>>> variance = t.central_moments(2)
>>> t.raw_moments(10)
199064.8037313875
"""
k = np.arange(n + 1)
sign = (-1)**(n - k)
zk = (self.a + 1 + k)/self.b
z0 = (self.a + 1)/self.b
z1 = (self.a + 2)/self.b
x = (self.xmin/self.scale)**self.b
coeffs = _comb(n, k)
return self.scale**n*np.sum(coeffs*sp.gammainc(z1, x)**(n - k)*
sp.gammainc(zk, x)*sign/sp.gammainc(z0, x)**(n - k + 1))
def _pdf_norm(self, log=False):
a = sp.gammainc(self._z, self._xmintb)
if not log:
return a
else:
return np.log(a)
def _gammainc_zx_(self):
"""
The incomplete gamma function, Gamma(z,x), where z,x are as specified in
this class.
"""
return sp.gammainc(self._z, self._x_)
def gammainc_z1xd(self):
return sp.gammainc(self._zd + self.beta / self.betad, self._xd)
def gammainc_z1x(self):
return sp.gammainc(self._z + 1, self._x)
def gammainc_zxd(self):
return sp.gammainc(self._zd, self._xd)
def _qd_nos(self):
"""
q for the data parameters, without scaling.
"""
return sp.gammainc((self.alphad+1)/self.betad, self._xd)*self.Hsd
def _qd(self):
"""
The normalisation of the MRP (ie. integral of g) (truncation masses)
"""
return sp.gammainc(self._zd, self._xd)*self.Hsd
