;ς e‘Fc@sqdZdklZdklZdkZdkZdeefd„ƒYZ[[[[defd„ƒYZdS(sGamma distributions. Any time your data are of some kind guaranteed positive (e.g. how tall human beings are), a suitable gamma distribution is more apt than a gaussian - though a lognormal distribution may be more apt (see gauss.py). This module provides implementations of suitable interfaces for the gamma distribution. Gamma distributions have the general form p = (: exp(-beta*x)*(beta*x)**(alpha-1) ←x :{positives})*beta/gamma(alpha) for some positive beta and alpha. The normalisation factor gamma(alpha) generalises factorial; gamma(1+n) = n! for natural n. It is defined by gamma(alpha) = integral(: exp(-t) * t**(alpha-1) ←t :{positives}) for which one can show that gamma(1+a) = a*gamma(a), corresponding to (1+n)! = gamma(2+n) = (1+n).gamma(1+n) = (1+n).n! Various tweaks (see stirling.lngamma) are available to enable accurate calculation of gamma(). One might plausibly hope to be able to do a bit of trickery with the integrals to determine gamma's derivative; but all this actually ends up yielding is what could be inferred from: d((x+1)!)/dx = d(x!.(x+1))/dx = x! +(x+1).d(x!)/dx (If I could integrate log(t), log(t)*t**n or exp(-t)*log(t), I could do the partial differentiation the other way and maybe something interesting would drop out ...) For alpha < 1, the distribution has a pole at 0; for sufficiently small beta*X, integral(: p(x) ←x; 0 0, is just (beta*X)**alpha / (beta * alpha), making our integral of p up to X pretty close to (beta*X)**alpha / alpha / gamma(alpha) $Id: gamma.py,v 1.4 2007/03/24 15:14:20 eddy Exp $ (s Integrator(sLazyNsGammacBs‘tZeiZd„ZdkZeid„Z[ed„Z[e i e i e i d„Ze i d„Ze id„ZeiZd„Zd„Zd „ZRS( NcCsΥy |dWn"tj ot|dƒ‚nX||f\|_|_|djo d|_n;|djo d|_n!|djot|dƒ‚n|i|id|ƒ|i||ƒd||_ dS(Nis<Gamma distribution's order parameter should be dimensionlessis2Gamma function cannot be normalised for this alpha( salphas TypeErrorsbetasselfs_Gamma__atzeros ValueErrors_Gamma__upinits _Gamma__ps _Gamma__setups _Gamma__zero(sselfsalphasbeta((s'/home/eddy/.sys/py/study/maths/gamma.pys__init__5s    cCs||ddƒSdS(Ni(sgsa(sasg((s'/home/eddy/.sys/py/study/maths/gamma.pys__genHscCs||iƒ|iSdS(N(sgsselfsalphasbeta(sselfsg((s'/home/eddy/.sys/py/study/maths/gamma.pyssampleKsc Csχd}y|i|ƒ}Wntj o||ƒ}nXt}y/|i|i ft |iƒ o t‚nWn,tj o ||d||d„}n!Xt}||d||d„}|o|||d„|_n|||d„|_dS(sΜFiddly bits of setup dealing with whether alpha, beta are Quantity()s. Overall goal is to set self.__ep to the function (: (beta * x)**(alpha-1) * exp(-beta * x) ← x :) * beta / gamma(alpha) i.e. the density function for our Gamma variate. However, assorted complications raise their heads: we only have gamma as log&on;gamma, evaluation of which, along with that of log and gamma, should be done via .evaluate() if pertinent values are Quantity()s (in which case we need a scalar to apply .evaluate() to). Note that self.__ep() should not be used directly, only via self.__p(), which performs necessary checks against invalid input. icCs$||}|||ƒ||SdS(N(sbsxsbxsaslnsn(sxsbsaslnsnsbx((s'/home/eddy/.sys/py/study/maths/gamma.pyslogpjs cCs'||}||i|ƒ||SdS(N(sbsxsbxsasevaluateslnsn(sxsbsaslnsnsbx((s'/home/eddy/.sys/py/study/maths/gamma.pyslogpos cCs|||ƒƒ|S(N(seslpsxsb(sxseslpsb((s'/home/eddy/.sys/py/study/maths/gamma.pyssscCs||ƒi|ƒ|S(N(slpsxsevaluatesesb(sxseslpsb((s'/home/eddy/.sys/py/study/maths/gamma.pystsN(sscalarsalphasevaluateslngamslognormsAttributeErrorsNonesbetaswidthsbestscallableslogslogpsexpsselfs _Gamma__ep( sselfsalphasbetaslogsexpslngamslognormslogpsscalar((s'/home/eddy/.sys/py/study/maths/gamma.pys__setupQs$ cCsr||ijo.y |iSWq>tj otd‚q>Xn||ijot|dƒ‚n|i|ƒSdS(sThe probability distributions*Gamma, with alpha < 1, is infinite at zeros-Gamma distribution only defined on positives.N(sxsselfs _Gamma__zeros_Gamma__atzerosAttributeErrors OverflowErrors ValueErrors _Gamma__ep(sselfsxsexp((s'/home/eddy/.sys/py/study/maths/gamma.pys__pvs cCs‚||idjod|i}n|d}y |i}Wntj onX|i||i|i||iƒ|fSdS(sApproximates integration from zero to some small value < end. This involves chosing an X < end for which exp(beta*X) differs negligibly from 1 = exp(0), then using eps = (beta*X)**alpha / alpha / gamma(alpha) as estimate of the integral. Returns a twople, (eps, X). if1.0f9.9999999999999995e-07N(sendsselfsbetasbestsAttributeErrorsalphasgam(sselfsendsgam((s'/home/eddy/.sys/py/study/maths/gamma.pys__slivers  cOsO|| jod|}n|| jod|}n||jod|iSn||jo"t|i||f||ƒ Sn|idjo |djoŽ|djpt ‚|i |ƒ\}}y|d}Wntj o||d 0 :) / gamma(alpha) = integrate(: u**alpha * exp(-u) ← u; u > 0 :) / beta / gamma(alpha) whose integrand is d(-exp(-u) * u**alpha) +alpha * u**(alpha-1) * exp(-u) the differential term in which is zero at both 0 and infinity, for alpha > 0, so we can integrate by parts to obtain the mean as: integrate(: u**(alpha-1) * exp(-u) ← u; u > 0 :) * alpha / beta / gamma(alpha) which is just alpha / beta, since the ingegral is gamma(alpha). N(sselfsalphasbeta(sselfsignored((s'/home/eddy/.sys/py/study/maths/gamma.pys_lazy_get_mean_¦s cCs|i|idSdS(s’Variance of the gamma distribution. This is the expected value of the variate's square, minus the square of the mean. The first term is: integrate(: (beta * x)**(alpha+1) * exp(-beta * x) ← x; x > 0 :) / beta / gamma(alpha) = integrate(: u**(alpha+1) * exp(-u) ← u; u > 0 :) / beta**2 / gamma(alpha) whose integrand is d(-exp(-u) * u**(alpha+1)) + (alpha+1) * u**alpha * exp(-u) = d(-exp(-u) * (u**(alpha+1) + (alpha+1) * u**alpha)) + (alpha+1) * alpha * u**(alpha-1) * exp(-u) again with its differential terms zero at 0 and infinity, for alpha > 0, enabling us to integrate by parts, as for the mean, to get the expected square as alpha * (alpha+1) / beta**2 from which we subtract (alpha/beta)**2 to get alpha / beta**2. iN(sselfsalphasbeta(sselfsignored((s'/home/eddy/.sys/py/study/maths/gamma.pys_lazy_get_variance_Άs(s__name__s __module__s Integrators__init__s_Gamma__upinitsrandoms gammavariates _Gamma__genssamplesmathslogsexpsstirlingslngammas _Gamma__setups _Gamma__psgammas_Gamma__sliversbetweens_Gamma__betweens_lazy_get_mean_s_lazy_get_variance_(((s'/home/eddy/.sys/py/study/maths/gamma.pysGamma3s    %    sGammaByMeanVarcBstZeiZd„ZRS(NcCsŠy|d|Wn=tj ot||dƒ‚nttfj onX||f\|_|_||}|i|||ƒdS(sΒInitialises a Gamma using mean and variance as inputs. This simply infers alpha and beta from the formulae used, above, to compute mean and variance lazily from alpha and beta. is?Incompatible quantities for mean and variance of a distributionN(smeansvariances TypeErrors OverflowErrorsOverflowWarningsselfsbetas_GammaByMeanVar__upinit(sselfsmeansvariancesbeta((s'/home/eddy/.sys/py/study/maths/gamma.pys__init__Οs   (s__name__s __module__sGammas__init__s_GammaByMeanVar__upinit(((s'/home/eddy/.sys/py/study/maths/gamma.pysGammaByMeanVarΝs ( s__doc__s integrates Integratorsstudy.value.lazysLazysmathsstirlingsGammasGammaByMeanVar(sGammaByMeanVarsLazys IntegratorsstirlingsmathsGamma((s'/home/eddy/.sys/py/study/maths/gamma.pys?-s   ˜