B ¤ŠãZ‹ã@süddlmZmZmZddlZddlZddlmZddl m Z ddl m Z ddl mZdggdgdggd gddgd ggd gddgd gd dgd ggdœZdd„Zdd„Ze dej¡Zdd„Zdd„Zdd„ZGdd„de ƒZedkrøddddgZeeƒZdS)é)ÚdivisionÚprint_functionÚabsolute_importN)ÚHermiteE)Ú factorial)Ú rv_continuous)ér)ré)r r)ré)r r)ré)r r )r r)rr r r cCs>|dkrtd|ƒ‚yt|Stk r8tdƒ‚YnXdS)aN 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) rz+Expected a positive integer; got %s insteadz'Higher order terms not yet implemented.N)Ú ValueErrorÚ_faa_di_bruno_cacheÚKeyErrorÚNotImplementedError)Ún©rúBlib/python3.7/site-packages/statsmodels/distributions/edgeworth.pyÚ_faa_di_bruno_partitionss  rcCsÈ|dkrtd|ƒ‚t|ƒ|kr6td||t|ƒfƒ‚d}x|t|ƒD]p}tdd„|Dƒƒ}d|dt|dƒ}x8|D]0\}}|t ||dt|ƒ|¡t|ƒ9}qxW||7}qDW|t|ƒ9}|S)awCompute 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. rz,Expected a positive integer. Got %s instead.z0%s-th cumulant requires %s moments, only got %s.gcss|]\}}|VqdS)Nr)Ú.0ÚmÚkrrrú Ssz(cumulant_from_moments..éÿÿÿÿ)r ÚlenrÚsumrÚnpÚpower)ZmomtrZkappaÚpÚrÚtermrrrrrÚcumulant_from_moments:s  ,  r r cCst |d d¡tS)Nr g@)rZexpÚ _norm_pdf_C)ÚxrrrÚ _norm_pdf^sr#cCs t |¡S)N)ÚspecialÚndtr)r"rrrÚ _norm_cdfasr&cCs t | ¡S)N)r$r%)r"rrrÚ_norm_sfdsr'csBeZdZdZd ‡fdd„ Zdd„Zdd„Zd d „Zd d „Z‡Z S)ÚExpandedNormalarConstruct 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) 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) Nr z"At least two cumulants are needed.rcSsdS)Ngr)r"rrrÚ¤sz)ExpandedNormal.__init__..rr zPDF 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;œs    $  zExpandedNormal.__init__cCs(||j|j}| |¡t|ƒ|jS)N)r-r.r/r#)r<r"ÚyrrrÚ_pdf±szExpandedNormal._pdfcCs*||j|j}t|ƒ| |¡t|ƒS)N)r-r.r&r1r#)r<r"r@rrrÚ_cdfµszExpandedNormal._cdfcCs*||j|j}t|ƒ| |¡t|ƒS)N)r-r.r'r1r#)r<r"r@rrrÚ_sfºszExpandedNormal._sfc Cs|dt |d¡}}t |¡}x,t|ƒD] \}}|||d|<q,Wt |jdd¡}d|d<x¤t|jdƒD]’}xŒt|dƒD]|} ||d} x<| D]4\} } | t || dt | dƒ| ¡t | ƒ9} q¤Wt dd„| Dƒƒ} ||dd| | 7<qŽWq|W|||fS) Nrrr égð?r css|]\}}|VqdS)Nr)rrrrrrrÍsz4ExpandedNormal._compute_coefs_pdf..) rÚsqrtZasarrayÚ enumerateZzerosr0Úrangerrrr)r<r=ZmuZsigmaZlamÚjÚlZcoefÚsrrrrrrrrr,¿s  0$z!ExpandedNormal._compute_coefs_pdf)r)) Ú__name__Ú __module__Ú __qualname__Ú__doc__r;rArBrCr,Ú __classcell__rr)r?rr(hs 3r(Ú__main__r)Z __future__rrrr6ZnumpyrZnumpy.polynomial.hermite_erZ scipy.miscrZ scipy.statsrZ scipy.specialr$r rr rEZpir!r#r&r'r(rKr=ZenrrrrÚs(      "#j