def __init__(self, cum, **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({'momtype': 0}) # use pdf, not ppf in self.moment() super(ExpandedNormal, self).__init__(**kwds)
class ExpandedNormal(distributions.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 a $\chi^2$ 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, **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({'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) * distributions._norm_pdf(y) / self._sigma def _cdf(self, x): y = (x - self._mu) / self._sigma return (distributions._norm_cdf(y) + self._herm_cdf(y) * distributions._norm_pdf(y)) def _sf(self, x): y = (x - self._mu) / self._sigma return (distributions._norm_sf(y) - self._herm_cdf(y) * distributions._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