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, RuntimeWarning)
kwds.update({'name': name,
'momtype': 0}) # use pdf, not ppf in self.moment()
super(ExpandedNormal, self).__init__(**kwds)
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)
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:
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> 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)
>>> fig1 = plt.plot(x, stats.chi2.pdf(x, df=df), 'g-', lw=4, alpha=0.5)
>>> fig2 = plt.plot(x, stats.norm.pdf(x, mu, sigma), 'b--', lw=4, alpha=0.5)
>>> fig3 = plt.plot(x, edgw_chi2.pdf(x), 'r-', lw=2)
>>> plt.show()
References
----------
.. [*] 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.
.. [*] http://en.wikipedia.org/wiki/Edgeworth_series
.. [*] S. Blinnikov and R. Moessner, Expansions for nearly Gaussian
distributions, Astron. Astrophys. Suppl. Ser. 130, 193 (1998)
"""
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, RuntimeWarning)
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
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)
"""
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