o 0Gf@sddlZddlZddlmZddlmZddlmZddlm Z dggdgdggdgddgd ggd gddgd gd dgd ggd Z ddZ ddZ e dejZddZddZddZGdddeZdS)N)HermiteE) factorial) rv_continuous)r)r)rr)r)rr)r)rr)rr)rrrrcCs6|dkr td|zt|WStytdw)a< 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 : list 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) rz+Expected a positive integer; got %s insteadz'Higher order terms not yet implemented.) ValueError_faa_di_bruno_cacheKeyErrorNotImplementedError)nrClib/python3.10/site-packages/statsmodels/distributions/edgeworth.py_faa_di_bruno_partitionss   rcCs|dkr td|t||krtd||t|fd}t|D]6}tdd|D}d|dt|d}|D]\}}|t||dt||t|9}q:||7}q!|t|9}|S)auCompute 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 : int which cumulant to calculate (must be >1) Returns ------- kappa : float n-th cumulant. rz,Expected a positive integer. Got %s instead.z0%s-th cumulant requires %s moments, only got %s.cs|]\}}|VqdSNr.0mkrrr Pz(cumulant_from_moments..)r lenrsumrnppower)Zmomtr Zkappaprtermrrrrrcumulant_from_moments8s     *  r"rcCst|d dtS)Nrg@)rZexp _norm_pdf_Cxrrr _norm_pdf[sr&cCs t|SrspecialZndtrr$rrr _norm_cdf^s r)cCs t| Srr'r$rrr_norm_sfas r*csBeZdZdZd fdd ZddZddZd d Zd d ZZ S)ExpandedNormalavConstruct 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.special 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. .. [*] https://en.wikipedia.org/wiki/Edgeworth_series .. [*] S. Blinnikov and R. Moessner, Expansions for nearly Gaussian distributions, Astron. Astrophys. Suppl. Ser. 130, 193 (1998) Edgeworth expanded normalc st|dkr td||\|_|_|_t|j|_|jjdkr-t|jdd |_ ndd|_ t |j }||j|j}|t |dkt |dk@r^d|}t|t||dd tjd i|dS) Nrz"At least two cumulants are needed.rcSsdS)Nrrr$rrrsz)ExpandedNormal.__init__..rrzPDF has zeros at %s )nameZmomtyper)rr _compute_coefs_pdfZ_coef_mu_sigmar _herm_pdfsize _herm_cdfrZ real_if_closerootsimagabsanywarningswarnRuntimeWarningupdatesuper__init__)selfcumr.kwdsr Zmesg __class__rrr>s    $ zExpandedNormal.__init__cCs(||j|j}||t||jSr)r0r1r2r&r?r%yrrr_pdfszExpandedNormal._pdfcCs*||j|j}t|||t|Sr)r0r1r)r4r&rDrrr_cdfzExpandedNormal._cdfcCs*||j|j}t|||t|Sr)r0r1r*r4r&rDrrr_sfrHzExpandedNormal._sfc Cs |dt|d}}t|}t|D]\}}|||d|<qt|jdd}d|d<t|jdD]E}t|dD]<} ||d} | D]\} } | t|| dt | d| t | 9} qMt dd| D} ||dd| | 7<qCq;|||fS) Nrrrg?rcsrrrrrrrrrz4ExpandedNormal._compute_coefs_pdf..) rsqrtZasarray enumerateZzerosr3rangerrrr)r?r@ZmuZsigmaZlamjlZcoefsrr!rrr rrrr/s   . z!ExpandedNormal._compute_coefs_pdf)r,) __name__ __module__ __qualname____doc__r>rFrGrIr/ __classcell__rrrBrr+es2r+)r9ZnumpyrZnumpy.polynomial.hermite_erZ scipy.specialrZ scipy.statsrr(r rr"rKZpir#r&r)r*r+rrrrs$     ""