from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
from numpy.polynomial.hermite_e import HermiteE
from scipy.misc import factorial
from scipy.stats import rv_continuous
import scipy.special as special
# TODO:
# * actually solve (31) of Blinnikov & Moessner
# * numerical stability: multiply factorials in logspace?
# * ppf & friends: Cornish & Fisher series, or tabulate/solve
_faa_di_bruno_cache = {
1: [[(1, 1)]],
2: [[(1, 2)], [(2, 1)]],
3: [[(1, 3)], [(2, 1), (1, 1)], [(3, 1)]],
4: [[(1, 4)], [(1, 2), (2, 1)], [(2, 2)], [(3, 1), (1, 1)], [(4, 1)]]}
def _faa_di_bruno_partitions(n):
""" Return all non-negative integer solutions of the diophantine equation::
n*k_n + ... + 2*k_2 + 1*k_1 = n (1)
Parameters
----------
n: int
the r.h.s. of Eq. (1)
Returns
-------
partitions: a list of solutions of (1). Each solution is itself
a list of the form `[(m, k_m), ...]` for non-zero `k_m`.
Notice that the index `m` is 1-based.
Examples:
---------
>>> _faa_di_bruno_partitions(2)
[[(1, 2)], [(2, 1)]]
>>> for p in faa_di_bruno_partitions(4):
... assert 4 == sum(m * k for (m, k) in p)
"""
if n < 1:
raise ValueError("Expected a positive integer; got %s instead" % n)
try:
return _faa_di_bruno_cache[n]
except KeyError:
# TODO: higher order terms
# solve Eq. (31) from Blinninkov & Moessner here
raise NotImplementedError('Higher order terms not yet implemented.')
[docs]def cumulant_from_moments(momt, n):
"""Compute n-th cumulant given moments.
Parameters
----------
momt: array_like
`momt[j]` contains `(j+1)`-th moment.
These can be raw moments around zero, or central moments
(in which case, `momt[0]` == 0).
n: integer
which cumulant to calculate (must be >1)
Returns
-------
kappa: float
n-th cumulant.
"""
if n < 1:
raise ValueError("Expected a positive integer. Got %s instead." % n)
if len(momt) < n:
raise ValueError("%s-th cumulant requires %s moments, "
"only got %s." % (n, n, len(momt)))
kappa = 0.
for p in _faa_di_bruno_partitions(n):
r = sum(k for (m, k) in p)
term = (-1)**(r - 1) * factorial(r - 1)
for (m, k) in p:
term *= np.power(momt[m - 1] / factorial(m), k) / factorial(k)
kappa += term
kappa *= factorial(n)
return kappa
## copied from scipy.stats.distributions to avoid the overhead of
## the public methods
_norm_pdf_C = np.sqrt(2*np.pi)
def _norm_pdf(x):
return np.exp(-x**2/2.0) / _norm_pdf_C
def _norm_cdf(x):
return special.ndtr(x)
def _norm_sf(x):
return special.ndtr(-x)
[docs]class ExpandedNormal(rv_continuous):
"""Construct the Edgeworth expansion pdf given cumulants.
Parameters
----------
cum: array_like
`cum[j]` contains `(j+1)`-th cumulant: cum[0] is the mean,
cum[1] is the variance and so on.
Notes
-----
This is actually an asymptotic rather than convergent series, hence
higher orders of the expansion may or may not improve the result.
In a strongly non-Gaussian case, it is possible that the density
becomes negative, especially far out in the tails.
Examples
--------
Construct the 4th order expansion for the chi-square distribution using
the known values of the cumulants:
>>> from scipy.misc import factorial
>>> df = 12
>>> chi2_c = [2**(j-1) * factorial(j-1) * df for j in range(1, 5)]
>>> edgw_chi2 = ExpandedNormal(chi2_c, name='edgw_chi2', momtype=0)
Calculate several moments:
>>> m, v = edgw_chi2.stats(moments='mv')
>>> np.allclose([m, v], [df, 2 * df])
True
Plot the density function:
>>> mu, sigma = df, np.sqrt(2*df)
>>> x = np.linspace(mu - 3*sigma, mu + 3*sigma)
>>> plt.plot(x, stats.chi2.pdf(x, df=df), 'g-', lw=4, alpha=0.5)
>>> plt.plot(x, stats.norm.pdf(x, mu, sigma), 'b--', lw=4, alpha=0.5)
>>> plt.plot(x, edgw_chi2.pdf(x), 'r-', lw=2)
>>> plt.show()
References
----------
.. [1] E.A. Cornish and R.A. Fisher, Moments and cumulants in the
specification of distributions, Revue de l'Institut Internat.
de Statistique. 5: 307 (1938), reprinted in
R.A. Fisher, Contributions to Mathematical Statistics. Wiley, 1950.
.. [2] http://en.wikipedia.org/wiki/Edgeworth_series
.. [3] S. Blinnikov and R. Moessner, Expansions for nearly Gaussian
distributions, Astron. Astrophys. Suppl. Ser. 130, 193 (1998)
"""
[docs] def __init__(self, cum, name='Edgeworth expanded normal', **kwds):
if len(cum) < 2:
raise ValueError("At least two cumulants are needed.")
self._coef, self._mu, self._sigma = self._compute_coefs_pdf(cum)
self._herm_pdf = HermiteE(self._coef)
if self._coef.size > 2:
self._herm_cdf = HermiteE(-self._coef[1:])
else:
self._herm_cdf = lambda x: 0.
# warn if pdf(x) < 0 for some values of x within 4 sigma
r = np.real_if_close(self._herm_pdf.roots())
r = (r - self._mu) / self._sigma
if r[(np.imag(r) == 0) & (np.abs(r) < 4)].any():
mesg = 'PDF has zeros at %s ' % r
warnings.warn(mesg, UserWarning)
kwds.update({'name': name,
'momtype': 0}) # use pdf, not ppf in self.moment()
super(ExpandedNormal, self).__init__(**kwds)
def _pdf(self, x):
y = (x - self._mu) / self._sigma
return self._herm_pdf(y) * _norm_pdf(y) / self._sigma
def _cdf(self, x):
y = (x - self._mu) / self._sigma
return (_norm_cdf(y) +
self._herm_cdf(y) * _norm_pdf(y))
def _sf(self, x):
y = (x - self._mu) / self._sigma
return (_norm_sf(y) -
self._herm_cdf(y) * _norm_pdf(y))
def _compute_coefs_pdf(self, cum):
# scale cumulants by \sigma
mu, sigma = cum[0], np.sqrt(cum[1])
lam = np.asarray(cum)
for j, l in enumerate(lam):
lam[j] /= cum[1]**j
coef = np.zeros(lam.size * 3 - 5)
coef[0] = 1.
for s in range(lam.size - 2):
for p in _faa_di_bruno_partitions(s+1):
term = sigma**(s+1)
for (m, k) in p:
term *= np.power(lam[m+1] / factorial(m+2), k) / factorial(k)
r = sum(k for (m, k) in p)
coef[s + 1 + 2*r] += term
return coef, mu, sigma
if __name__ == "__main__":
cum =[1, 1, 1, 1]
en = ExpandedNormal(cum)