U Ãmœdã@sÌddlZddlZddlmZddlmZddlmZddlm Z dggdgdggdgddgd ggd gddgd gd dgd ggd œZ dd„Z dd„Z e  dej¡Zdd„Zdd„Zdd„ZGdd„deƒZdS)éN)ÚHermiteE)Ú factorial)Ú rv_continuous)ér)ré)rr)ré)rr)ré)rr)rr)rrrrcCs@|dkrtd|ƒ‚z t|WStk r:tdƒ‚YnXdS)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.N)Ú ValueErrorÚ_faa_di_bruno_cacheÚKeyErrorÚNotImplementedError)Ún©rú\/home/sam/Atlas/atlas_env/lib/python3.8/site-packages/statsmodels/distributions/edgeworth.pyÚ_faa_di_bruno_partitionss   rcCsÀ|dkrtd|ƒ‚t|ƒ|kr6td||t|ƒfƒ‚d}t|ƒD]l}tdd„|Dƒƒ}d|dt|dƒ}|D]0\}}|t ||dt|ƒ|¡t|ƒ9}qt||7}qB|t|ƒ9}|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.çcss|]\}}|VqdS©Nr©Ú.0ÚmÚkrrrÚ Qsz(cumulant_from_moments..éÿÿÿÿ)r ÚlenrÚsumrÚnpÚpower)Zmomtr ÚkappaÚpÚrÚtermrrrrrÚcumulant_from_moments9s   ÿ  *  r!rcCst |d d¡tS)Nrg@)rÚexpÚ _norm_pdf_C©ÚxrrrÚ _norm_pdf\sr&cCs t |¡Sr©ÚspecialZndtrr$rrrÚ _norm_cdf_sr)cCs t | ¡Srr'r$rrrÚ_norm_sfbsr*csBeZdZdZd ‡fdd„ Zdd„Zdd„Zd d „Zd d „Z‡Z S)ÚExpandedNormalavConstruct 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) úEdgeworth expanded normalc sât|ƒdkrtdƒ‚| |¡\|_|_|_t|jƒ|_|jjdkrZt|jdd… ƒ|_ n dd„|_ t   |j  ¡¡}||j|j}|t   |¡dkt  |¡dk@ ¡r¼d|}t |t¡| |dd œ¡tt|ƒjf|ŽdS) Nrz"At least two cumulants are needed.rcSsdS)Nrrr$rrrÚ¡óz)ExpandedNormal.__init__..rrzPDF has zeros at %s )ÚnameZmomtype)rr Ú_compute_coefs_pdfZ_coefÚ_muÚ_sigmarÚ _herm_pdfÚsizeÚ _herm_cdfrZ real_if_closeÚrootsÚimagÚabsÚanyÚwarningsÚwarnÚRuntimeWarningÚupdateÚsuperr+Ú__init__)ÚselfÚcumr/ÚkwdsrZmesg©Ú __class__rrr?™s     $ ÿzExpandedNormal.__init__cCs(||j|j}| |¡t|ƒ|jSr)r1r2r3r&©r@r%ÚyrrrÚ_pdf®szExpandedNormal._pdfcCs*||j|j}t|ƒ| |¡t|ƒSr)r1r2r)r5r&rErrrÚ_cdf²sÿzExpandedNormal._cdfcCs*||j|j}t|ƒ| |¡t|ƒSr)r1r2r*r5r&rErrrÚ_sf·sÿzExpandedNormal._sfc Cs |dt |d¡}}t |¡}t|ƒD] \}}|||d|<q*t |jdd¡}d|d<t|jdƒD]Š}t|dƒD]x} ||d} | D]4\} } | t || dt | dƒ| ¡t | ƒ9} qšt dd„| Dƒƒ} ||dd| | 7<q†qv|||fS) Nrrrégð?rcss|]\}}|VqdSrrrrrrrÊsz4ExpandedNormal._compute_coefs_pdf..) rÚsqrtZasarrayÚ enumerateZzerosr4Úrangerrrr)r@rAÚmuÚsigmaZlamÚjÚlZcoefÚsrr rrrrrrr0¼s   . z!ExpandedNormal._compute_coefs_pdf)r,) Ú__name__Ú __module__Ú __qualname__Ú__doc__r?rGrHrIr0Ú __classcell__rrrCrr+fs 2r+)r:ÚnumpyrZnumpy.polynomial.hermite_erZ scipy.specialrZ scipy.statsrr(r rr!rKÚpir#r&r)r*r+rrrrÚs"     ü""