ó áp7]c@`s=ddlmZmZmZddlZddlZddlmZddl m Z ddl m Z ddl jZidggd6dgdggd6dgddgdggd6dgddgdgddgdggd 6Zd „Zd „ZejdejƒZd „Zd „Zd„Zde fd„ƒYZdS(i(tdivisiontprint_functiontabsolute_importN(tHermiteE(t factorial(t rv_continuousiiiicC`sO|dkrtd|ƒ‚ny t|SWntk rJtdƒ‚nXdS(sN 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) is+Expected a positive integer; got %s insteads'Higher order terms not yet implemented.N(t ValueErrort_faa_di_bruno_cachetKeyErrortNotImplementedError(tn((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyt_faa_di_bruno_partitionss    cC`s |dkrtd|ƒ‚nt|ƒ|krStd||t|ƒfƒ‚nd}x™t|ƒD]‹}td„|Dƒƒ}d|dt|dƒ}xF|D]>\}}|tj||dt|ƒ|ƒt|ƒ9}q¥W||7}qfW|t|ƒ9}|S(swCompute 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. is,Expected a positive integer. Got %s instead.s0%s-th cumulant requires %s moments, only got %s.gcs`s|]\}}|VqdS(N((t.0tmtk((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pys Ssiÿÿÿÿ(RtlenR tsumRtnptpower(tmomtR tkappatptrttermR R((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pytcumulant_from_moments:s 6cC`stj|d dƒtS(Nig@(Rtexpt _norm_pdf_C(tx((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyt _norm_pdf^scC`s tj|ƒS(N(tspecialtndtr(R((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyt _norm_cdfascC`stj| ƒS(N(RR(R((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyt_norm_sfdstExpandedNormalcB`s>eZdZdd„Zd„Zd„Zd„Zd„ZRS(srConstruct 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) sEdgeworth expanded normalcK`s<t|ƒdkr!tdƒ‚n|j|ƒ\|_|_|_t|jƒ|_|jjdkr€t|jd ƒ|_ n d„|_ t j |jj ƒƒ}||j|j}|t j |ƒdkt j|ƒdk@jƒrd|}tj|tƒn|ji|d6dd 6ƒtt|ƒj|dS( Nis"At least two cumulants are needed.icS`sdS(Ng((R((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyt¤tiisPDF has zeros at %s tnametmomtype(RRt_compute_coefs_pdft_coeft_mut_sigmaRt _herm_pdftsizet _herm_cdfRt real_if_closetrootstimagtabstanytwarningstwarntRuntimeWarningtupdatetsuperR!t__init__(tselftcumR$tkwdsRtmesg((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyR7œs! 2  cC`s2||j|j}|j|ƒt|ƒ|jS(N(R(R)R*R(R8Rty((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyt_pdf±scC`s5||j|j}t|ƒ|j|ƒt|ƒS(N(R(R)RR,R(R8RR<((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyt_cdfµs cC`s5||j|j}t|ƒ|j|ƒt|ƒS(N(R(R)R R,R(R8RR<((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyt_sfºs c C`sY|dtj|dƒ}}tj|ƒ}x2t|ƒD]$\}}||c|d|Ís( Rtsqrttasarrayt enumeratetzerosR+trangeR RRR(R8R9tmutsigmatlamtjtltcoeftsRRR RR((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyR&¿s :$(t__name__t __module__t__doc__R7R=R>R?R&(((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyR!hs 3    (ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(t __future__RRRR2tnumpyRtnumpy.polynomial.hermite_eRtstatsmodels.compat.scipyRt scipy.statsRt scipy.specialRRR RR@tpiRRRR R!(((sBlib/python2.7/site-packages/statsmodels/distributions/edgeworth.pyts$   . 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