ó áp7]c@sªdZddlZddlZejeƒjZde fd„ƒYZ de fd„ƒYZ de fd„ƒYZ d e fd „ƒYZ d e fd „ƒYZd e fd„ƒYZde fd„ƒYZde fd„ƒYZde fd„ƒYZdefd„ƒYZde fd„ƒYZdefd„ƒYZdefd„ƒYZde fd„ƒYZdefd „ƒYZd!e fd"„ƒYZd#efd$„ƒYZdS(%sB Defines the link functions to be used with GLM and GEE families. iÿÿÿÿNtLinkcBsDeZdZd„Zd„Zd„Zd„Zd„Zd„ZRS(s” A generic link function for one-parameter exponential family. `Link` does nothing, but lays out the methods expected of any subclass. cCstS(s Return the value of the link function. This is just a placeholder. Parameters ---------- p : array-like Probabilities Returns ------- g(p) : array-like The value of the link function g(p) = z (tNotImplementedError(tselftp((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyt__call__scCstS(s~ Inverse of the link function. Just a placeholder. Parameters ---------- z : array-like `z` is usually the linear predictor of the transformed variable in the IRLS algorithm for GLM. Returns ------- g^(-1)(z) : array The value of the inverse of the link function g^(-1)(z) = p (R(Rtz((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pytinverse!scCstS(s Derivative of the link function g'(p). Just a placeholder. Parameters ---------- p : array-like Returns ------- g'(p) : array The value of the derivative of the link function g'(p) (R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pytderiv4s cCs)ddlm}tj|||jƒƒS(smSecond derivative of the link function g''(p) implemented through numerical differentiation iÿÿÿÿ(tapprox_fprime_cs(tstatsmodels.tools.numdiffRtnptdiagR(RRR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pytderiv2CscCsd|j|j|ƒƒS(sð Derivative of the inverse link function g^(-1)(z). Parameters ---------- z : array-like `z` is usually the linear predictor for a GLM or GEE model. Returns ------- g'^(-1)(z) : array The value of the derivative of the inverse of the link function Notes ----- This reference implementation gives the correct result but is inefficient, so it can be overriden in subclasses. i(RR(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyt inverse_derivLscCs.|j|ƒ}|j|ƒ |j|ƒdS(s  Second derivative of the inverse link function g^(-1)(z). Parameters ---------- z : array-like `z` is usually the linear predictor for a GLM or GEE model. Returns ------- g'^(-1)(z) : array The value of the second derivative of the inverse of the link function Notes ----- This reference implementation gives the correct result but is inefficient, so it can be overriden in subclasses. i(RR R(RRtiz((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pytinverse_deriv2as( t__name__t __module__t__doc__RRRR R R(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR s    tLogitcBsDeZdZd„Zd„Zd„Zd„Zd„Zd„ZRS(sË The logit transform Notes ----- call and derivative use a private method _clean to make trim p by machine epsilon so that p is in (0,1) Alias of Logit: logit = Logit() cCstj|tdtƒS(sí Clip logistic values to range (eps, 1-eps) Parameters ---------- p : array-like Probabilities Returns ------- pclip : array Clipped probabilities gð?(R tclipt FLOAT_EPS(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyt_clean†scCs$|j|ƒ}tj|d|ƒS(s The logit transform Parameters ---------- p : array-like Probabilities Returns ------- z : array Logit transform of `p` Notes ----- g(p) = log(p / (1 - p)) gð?(RR tlog(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR–scCs+tj|ƒ}tj| ƒ}dd|S(s2 Inverse of the logit transform Parameters ---------- z : array-like The value of the logit transform at `p` Returns ------- p : array Probabilities Notes ----- g^(-1)(z) = exp(z)/(1+exp(z)) gð?(R tasarraytexp(RRtt((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR«scCs|j|ƒ}d|d|S(sr Derivative of the logit transform Parameters ---------- p: array-like Probabilities Returns ------- g'(p) : array Value of the derivative of logit transform at `p` Notes ----- g'(p) = 1 / (p * (1 - p)) Alias for `Logit`: logit = Logit() gð?i(R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRÁscCstj|ƒ}|d|dS(sR Derivative of the inverse of the logit transform Parameters ---------- z : array-like `z` is usually the linear predictor for a GLM or GEE model. Returns ------- g'^(-1)(z) : array The value of the derivative of the inverse of the logit function ii(R R(RRR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR ÚscCs"|d|}d|d|dS(s Second derivative of the logit function. Parameters ---------- p : array-like probabilities Returns ------- g''(z) : array The value of the second derivative of the logit function ii((RRtv((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR ìs( RRRRRRRR R (((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRys      tlogitcBseZRS((RR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRþstPowercBsPeZdZdd„Zd„Zd„Zd„Zd„Zd„Zd„Z RS( s$ The power transform Parameters ---------- power : float The exponent of the power transform Notes ----- Aliases of Power: inverse = Power(power=-1) sqrt = Power(power=.5) inverse_squared = Power(power=-2.) identity = Power(power=1.) gð?cCs ||_dS(N(tpower(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyt__init__scCs*|jdkr|Stj||jƒSdS(s Power transform link function Parameters ---------- p : array-like Mean parameters Returns ------- z : array-like Power transform of x Notes ----- g(p) = x**self.power iN(RR (RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRscCs.|jdkr|Stj|d|jƒSdS(sN Inverse of the power transform link function Parameters ---------- `z` : array-like Value of the transformed mean parameters at `p` Returns ------- `p` : array Mean parameters Notes ----- g^(-1)(z`) = `z`**(1/`power`) igð?N(RR (RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR.scCs>|jdkrtj|ƒS|jtj||jdƒSdS(sA Derivative of the power transform Parameters ---------- p : array-like Mean parameters Returns ------- g'(p) : array Derivative of power transform of `p` Notes ----- g'(`p`) = `power` * `p`**(`power` - 1) iN(RR t ones_like(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyREs cCsI|jdkrtj|ƒS|j|jdtj||jdƒSdS(se Second derivative of the power transform Parameters ---------- p : array-like Mean parameters Returns ------- g''(p) : array Second derivative of the power transform of `p` Notes ----- g''(`p`) = `power` * (`power` - 1) * `p`**(`power` - 2) iiN(RR t zeros_like(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR \s cCsE|jdkrtj|ƒStj|d|j|jƒ|jSdS(sc Derivative of the inverse of the power transform Parameters ---------- z : array-like `z` is usually the linear predictor for a GLM or GEE model. Returns ------- g^(-1)'(z) : array The value of the derivative of the inverse of the power transform function iN(RR R (RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR ss cCsX|jdkrtj|ƒSd|jtj|dd|j|jƒ|jdSdS(sj Second derivative of the inverse of the power transform Parameters ---------- z : array-like `z` is usually the linear predictor for a GLM or GEE model. Returns ------- g^(-1)'(z) : array The value of the derivative of the inverse of the power transform function iiN(RR R!(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR‡s ( RRRRRRRR R R(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRs      t inverse_powercBseZdZd„ZRS(s{ The inverse transform Notes ----- g(p) = 1/p Alias of statsmodels.family.links.Power(power=-1.) cCstt|ƒjddƒdS(NRgð¿(tsuperR"R(R((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR§s(RRRR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR"s tsqrtcBseZdZd„ZRS(s† The square-root transform Notes ----- g(`p`) = sqrt(`p`) Alias of statsmodels.family.links.Power(power=.5) cCstt|ƒjddƒdS(NRgà?(R#R$R(R((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRµs(RRRR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR$«s tinverse_squaredcBseZdZd„ZRS(s The inverse squared transform Notes ----- g(`p`) = 1/(`p`\*\*2) Alias of statsmodels.family.links.Power(power=2.) cCstt|ƒjddƒdS(NRgÀ(R#R%R(R((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRÃs(RRRR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR%¹s tidentitycBseZdZd„ZRS(s} The identity transform Notes ----- g(`p`) = `p` Alias of statsmodels.family.links.Power(power=1.) cCstt|ƒjddƒdS(NRgð?(R#R&R(R((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRÑs(RRRR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR&Çs tLogcBsDeZdZd„Zd„Zd„Zd„Zd„Zd„ZRS(s¼ The log transform Notes ----- call and derivative call a private method _clean to trim the data by machine epsilon so that p is in (0,1). log is an alias of Log. cCstj|ttjƒS(N(R RRtinf(Rtx((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRßscKs|j|ƒ}tj|ƒS(s Log transform link function Parameters ---------- x : array-like Mean parameters Returns ------- z : array log(x) Notes ----- g(p) = log(p) (RR R(RRtextraR)((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRâscCs tj|ƒS(sV Inverse of log transform link function Parameters ---------- z : array The inverse of the link function at `p` Returns ------- p : array The mean probabilities given the value of the inverse `z` Notes ----- g^{-1}(z) = exp(z) (R R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR÷scCs|j|ƒ}d|S(s* Derivative of log transform link function Parameters ---------- p : array-like Mean parameters Returns ------- g'(p) : array derivative of log transform of x Notes ----- g'(x) = 1/x gð?(R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR scCs|j|ƒ}d|dS(sA Second derivative of the log transform link function Parameters ---------- p : array-like Mean parameters Returns ------- g''(p) : array Second derivative of log transform of x Notes ----- g''(x) = -1/x^2 gð¿i(R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR scCs tj|ƒS(sh Derivative of the inverse of the log transform link function Parameters ---------- z : array The inverse of the link function at `p` Returns ------- g^(-1)'(z) : array The value of the derivative of the inverse of the log function, the exponential function (R R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR 5s( RRRRRRRR R (((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR'Õs     RcBseZdZRS(sN The log transform Notes ----- log is a an alias of Log. (RRR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRGstCDFLinkcBsMeZdZejjd„Zd„Zd„Zd„Z d„Z d„Z RS(s? The use the CDF of a scipy.stats distribution CDFLink is a subclass of logit in order to use its _clean method for the link and its derivative. Parameters ---------- dbn : scipy.stats distribution Default is dbn=scipy.stats.norm Notes ----- The CDF link is untested. cCs ||_dS(N(tdbn(RR,((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRdscCs|j|ƒ}|jj|ƒS(s CDF link function Parameters ---------- p : array-like Mean parameters Returns ------- z : array (ppf) inverse of CDF transform of p Notes ----- g(`p`) = `dbn`.ppf(`p`) (RR,tppf(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRgscCs|jj|ƒS(sn The inverse of the CDF link Parameters ---------- z : array-like The value of the inverse of the link function at `p` Returns ------- p : array Mean probabilities. The value of the inverse of CDF link of `z` Notes ----- g^(-1)(`z`) = `dbn`.cdf(`z`) (R,tcdf(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR|scCs/|j|ƒ}d|jj|jj|ƒƒS(s9 Derivative of CDF link Parameters ---------- p : array-like mean parameters Returns ------- g'(p) : array The derivative of CDF transform at `p` Notes ----- g'(`p`) = 1./ `dbn`.pdf(`dbn`.ppf(`p`)) gð?(RR,tpdfR-(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRscCs>ddlm}tj|ƒ}tj|||jdtƒƒS(sv Second derivative of the link function g''(p) implemented through numerical differentiation iÿÿÿÿ(t approx_fprimetcentered(R R0R t atleast_1dR RtTrue(RRR0((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR ¥scCsd|j|j|ƒƒS(sI Derivative of the inverse of the CDF transformation link function Parameters ---------- z : array The inverse of the link function at `p` Returns ------- g^(-1)'(z) : array The value of the derivative of the inverse of the logit function i(RR(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR °s( RRRtscipytstatstnormRRRRR R (((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR+Ss    tprobitcBseZdZRS(s The probit (standard normal CDF) transform Notes ----- g(p) = scipy.stats.norm.ppf(p) probit is an alias of CDFLink. (RRR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR7Ás tcauchycBs eZdZd„Zd„ZRS(s­ The Cauchy (standard Cauchy CDF) transform Notes ----- g(p) = scipy.stats.cauchy.ppf(p) cauchy is an alias of CDFLink with dbn=scipy.stats.cauchy cCs#tt|ƒjdtjjƒdS(NR,(R#R8RR4R5(R((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRÙscCsDtj|d}dtjdtj|ƒtj|ƒd}|S(s Second derivative of the Cauchy link function. Parameters ---------- p: array-like Probabilities Returns ------- g''(p) : array Value of the second derivative of Cauchy link function at `p` gà?ii(R tpitsintcos(RRtatd2((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR Üs/(RRRRR (((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR8Îs  tCLogLogcBs;eZdZd„Zd„Zd„Zd„Zd„ZRS(sÎ The complementary log-log transform CLogLog inherits from Logit in order to have access to its _clean method for the link and its derivative. Notes ----- CLogLog is untested. cCs*|j|ƒ}tjtjd|ƒ ƒS(s C-Log-Log transform link function Parameters ---------- p : array Mean parameters Returns ------- z : array The CLogLog transform of `p` Notes ----- g(p) = log(-log(1-p)) i(RR R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRúscCsdtjtj|ƒ ƒS(sY Inverse of C-Log-Log transform link function Parameters ---------- z : array-like The value of the inverse of the CLogLog link function at `p` Returns ------- p : array Mean parameters Notes ----- g^(-1)(`z`) = 1-exp(-exp(`z`)) i(R R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRscCs,|j|ƒ}d|dtjd|ƒS(sX Derivative of C-Log-Log transform link function Parameters ---------- p : array-like Mean parameters Returns ------- g'(p) : array The derivative of the CLogLog transform link function Notes ----- g'(p) = - 1 / ((p-1)*log(1-p)) gð?i(RR R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR$scCsN|j|ƒ}tjd|ƒ}dd|d|}|dd|9}|S(s Second derivative of the C-Log-Log ink function Parameters ---------- p : array-like Mean parameters Returns ------- g''(p) : array The second derivative of the CLogLog link function iiÿÿÿÿi(RR R(RRtflR=((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR 9s cCstj|tj|ƒƒS(s^ Derivative of the inverse of the C-Log-Log transform link function Parameters ---------- z : array-like The value of the inverse of the CLogLog link function at `p` Returns ------- g^(-1)'(z) : array The derivative of the inverse of the CLogLog link function (R R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR Ms(RRRRRRR R (((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR>ïs      tcloglogcBseZdZRS(sž The CLogLog transform link function. Notes ----- g(`p`) = log(-log(1-`p`)) cloglog is an alias for CLogLog cloglog = CLogLog() (RRR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR@^s tNegativeBinomialcBsPeZdZdd„Zd„Zd„Zd„Zd„Zd„Zd„Z RS( s? The negative binomial link function Parameters ---------- alpha : float, optional Alpha is the ancillary parameter of the Negative Binomial link function. It is assumed to be nonstochastic. The default value is 1. Permissible values are usually assumed to be in (.01, 2). gð?cCs ||_dS(N(talpha(RRB((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRxscCstj|ttjƒS(N(R RRR((RR)((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR{scCs+|j|ƒ}tj||d|jƒS(s< Negative Binomial transform link function Parameters ---------- p : array-like Mean parameters Returns ------- z : array The negative binomial transform of `p` Notes ----- g(p) = log(p/(p + 1/alpha)) i(RR RRB(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR~scCsd|jdtj| ƒS(s_ Inverse of the negative binomial transform Parameters ---------- z : array-like The value of the inverse of the negative binomial link at `p`. Returns ------- p : array Mean parameters Notes ----- g^(-1)(z) = exp(z)/(alpha*(1-exp(z))) iÿÿÿÿi(RBR R(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR“scCsd||j|dS(sY Derivative of the negative binomial transform Parameters ---------- p : array-like Mean parameters Returns ------- g'(p) : array The derivative of the negative binomial transform link function Notes ----- g'(x) = 1/(x+alpha*x^2) ii(RB(RR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR§scCs7dd|j| }||j|dd}||S(s‰ Second derivative of the negative binomial link function. Parameters ---------- p : array-like Mean parameters Returns ------- g''(p) : array The second derivative of the negative binomial transform link function Notes ----- g''(x) = -(1+2*alpha*x)/(x+alpha*x^2)^2 ii(RB(RRtnumertdenom((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR »scCs&tj|ƒ}||jd|dS(si Derivative of the inverse of the negative binomial transform Parameters ---------- z : array-like Usually the linear predictor for a GLM or GEE model Returns ------- g^(-1)'(z) : array The value of the derivative of the inverse of the negative binomial link ii(R RRB(RRR((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyR Òs( RRRRRRRRR R (((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyRAls       tnbinomcBseZdZRS(s¸ The negative binomial link function. Notes ----- g(p) = log(p/(p + 1/alpha)) nbinom is an alias of NegativeBinomial. nbinom = NegativeBinomial(alpha=1.) (RRR(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyREås (RtnumpyR t scipy.statsR4tfinfotfloattepsRtobjectRRRRR"R$R%R&R'RR+R7R8R>R@RARE(((s@lib/python2.7/site-packages/statsmodels/genmod/families/links.pyts(  o…›r n !oy