ó áp7]c @sÙ dZddlmZddlZddlmZddlmZd„Z d„Z d„Z d „Z d „Z d „Zd „Zd „Zd„Zd„Zd„ZedkrÕ dZerÈdZejdƒZejjddƒZedd…dd…fZejeeƒeedd…dfZdddgZe e eeeƒƒeejeeƒdd…dfeZ"e e"ƒe eeejeeƒdeƒƒe eejeeƒƒnddlm#Z#m$Z$d„Z%d„Z&d„Z'd„Z(d„Z)d„Z*er[e dƒe e$j+e%dd d!d"dd#d=d$dƒƒe d%edddedƒƒe d&edddedƒƒe e$j+e%d'd d(d"dd#d>d$dƒƒe d&ed'dded)ƒƒe e$j+e%d'd d(d"dd#d?d$dƒƒe d&ed'dded)ƒƒe e$j+e%d'd d(d"dd#d@d$dƒƒe d&ed'dded)ƒƒe e$j+e&dd d(d"dd#dAd$dƒƒe e$j+e'dd d(d"dd#dBd$dƒƒdC\ZZ,Z-Z.e d&eee,e-ed)ƒƒe e$j+e&e,d d(d"dd#ee-e.fd$dƒƒe e$j+e'e-d d(d"dd#ee,e.fd$dƒƒe d*ƒe e$j+e(dd d!d"dd#dDd$dƒƒe edddeƒƒdE\ZZ,Z-e d&eee,e-eƒƒe e$j+e)e,d d(d"dd#ee-fd$dƒƒe e$j+e*e-d d(d"dd#ee,fd$dƒƒdF\ZZ,Z-e d&eee,e-eƒƒe e$j+e)e,d d(d"dd#ee-fd$dƒƒe e$j+e*e-d d(d"dd#ee,fd$dƒƒe d+ƒe e dej/ddgƒd,ƒe%dddd,ƒd-ƒe e dej/ddgƒdƒe%ddddƒd-ƒe edej/ddgƒdƒe%ddddƒƒe e dej/ddGd0gƒdƒe%d.dd.dƒƒe edej/ddgƒd,ƒe%dddd,ƒƒe e dej/ddgƒdƒe%ddd.ej0d1d/ƒdƒƒndd2l1m2Z2dHdIdJdKgZej3d5d4d6ƒZ4x½eD]µ\Z,Z-e$j+e)e,d d(d"dd#e4e-fd$dƒZ5e$j+e*e-d d(d"dd#e4e,fd$dƒZ6ee4e,e-eƒZ7e2e5e7dd7d8d9ƒe2e6e7dd7d8d:ƒq™Wx€ddd,gD]lZ.xceD][\Z,Z-e$j+e&e,d d(d"dd#e4e-e.fd$dƒZ5e$j+e'e-d d(d"dd#e4e,e.fd$dƒZ6ee4e,e-ee.ƒZ7e2e5e7dd;d8d9ƒe2e6e7dd;d8d:ƒe2ee4ej/e,e-dgƒe.ƒe%e4e,e-e.ƒd7d8d<ƒe2e e4ej/e,e-dgƒe.ƒe%e4e,e-ej0e.d4e.ƒe.ƒd7d8d<ƒqoWqbWndS(Lsgradient/Jacobian of normal and t loglikelihood use chain rule normal derivative wrt mu, sigma and beta new version: loc-scale distributions, derivative wrt loc, scale also includes "standardized" t distribution (for use in GARCH) TODO: * use sympy for derivative of loglike wrt shape parameters it works for df of t distribution dlog(gamma(a))da = polygamma(0,a) check polygamma is available in scipy.special * get loc-scale example to work with mean = X*b * write some full unit test examples A: josef-pktd iÿÿÿÿ(tprint_functionN(tspecial(tgammalncCsJ|j\}}dtjdtjƒtj|ƒ||d|}|S(srnormal loglikelihood given observations and mean mu and variance sigma2 Parameters ---------- y : array, 1d normally distributed random variable params: array, (nobs, 2) array of mean, variance (mu, sigma2) with observations in rows Returns ------- lls : array contribution to loglikelihood for each observation gà¿i(tTtnptlogtpi(tytparamstmutsigma2tlls((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytnorm_llss7cCsS|j\}}|||}||d|dtj|ƒ}tj||fƒS(s=Jacobian of normal loglikelihood wrt mean mu and variance sigma2 Parameters ---------- y : array, 1d normally distributed random variable params: array, (nobs, 2) array of mean, variance (mu, sigma2) with observations in rows Returns ------- grad : array (nobs, 2) derivative of loglikelihood for each observation wrt mean in first column, and wrt variance in second column Notes ----- this is actually the derivative wrt sigma not sigma**2, but evaluated with parameter sigma2 = sigma**2 ii(RRtsqrtt column_stack(RRR R tdllsdmut dllsdsigma2((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyt norm_lls_grad0s#cCs|S(s-gradient/Jacobian for d (x*beta)/ d beta ((txtbeta((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyt mean_gradLsc Cs»|d }|dtjt|ƒdfƒ}t||ƒ}tj||ƒ}tj||fƒ}t||ƒ}tj|dd…dd…f||dd…dd…ffƒ} | S(s®Jacobian of normal loglikelihood wrt mean mu and variance sigma2 Parameters ---------- y : array, 1d normally distributed random variable with mean x*beta, and variance sigma2 x : array, 2d explanatory variables, observation in rows, variables in columns params: array_like, (nvars + 1) array of coefficients and variance (beta, sigma2) Returns ------- grad : array (nobs, 2) derivative of loglikelihood for each observation wrt mean in first column, and wrt scale (sigma) in second column assume params = (beta, sigma2) Notes ----- TODO: for heteroscedasticity need sigma to be a 1d array iÿÿÿÿiN(RtonestlenRtdotRR( RRRRR tdmudbetaR tparams2tdllsdmstgrad((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytnormgradQs #EcCs£|j\}}|d}t|ddƒt|dƒdtj|dtjƒ}||ddtjd||d|d|ƒdtj|ƒ8}|S(sät loglikelihood given observations and mean mu and variance sigma2 = 1 Parameters ---------- y : array, 1d normally distributed random variable params: array, (nobs, 2) array of mean, variance (mu, sigma2) with observations in rows df : integer degrees of freedom of the t distribution Returns ------- lls : array contribution to loglikelihood for each observation Notes ----- parameterized for garch gð?ig@gà?i(RRRRR(RRtdfR R R ((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyttstd_llsus  >HcCs| S(s?derivative of log pdf of standard normal with respect to y ((R((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyt norm_dlldy”scCs†tj|dƒ}tjtj|ddƒtj|dƒƒtj|dtjƒ}|d|d|d|dd:}|S(sIpdf for standardized (not standard) t distribution, variance is one gð?ig@i(RtarraytexpRRR R(RRtrtPx((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyttstd_pdfšsI&cCs«t|||ƒ|j\}}|d}t|ddƒt|dƒdtj|tjƒ}||ddtjd||d||ƒdtj|ƒ8}|S(st loglikelihood given observations and mean mu and variance sigma2 = 1 Parameters ---------- y : array, 1d normally distributed random variable params: array, (nobs, 2) array of mean, variance (mu, sigma2) with observations in rows df : integer degrees of freedom of the t distribution Returns ------- lls : array contribution to loglikelihood for each observation Notes ----- parameterized for garch normalized/rescaled so that sigma2 is the variance >>> df = 10; sigma = 1. >>> stats.t.stats(df, loc=0., scale=sigma.*np.sqrt((df-2.)/df)) (array(0.0), array(1.0)) >>> sigma = np.sqrt(2.) >>> stats.t.stats(df, loc=0., scale=sigma*np.sqrt((df-2.)/df)) (array(0.0), array(2.0)) gð?ig@gà?i(tprintRRRRR(RRRR R R ((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytts_lls¤s  :DcCs+|d}|d |d|d||S(sderivative of log pdf of standard t with respect to y Parameters ---------- y : array_like data points of random variable at which loglike is evaluated df : array_like degrees of freedom,shape parameters of log-likelihood function of t distribution Returns ------- dlldy : array derivative of loglikelihood wrt random variable y evaluated at the points given in y Notes ----- with mean 0 and scale 1, but variance is df/(df-2) gð?ii((RR((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytts_dlldyËs cCs)|d |dd|d|d|S(sderivative of log pdf of standardized t with respect to y Parameters ---------- y : array_like data points of random variable at which loglike is evaluated df : array_like degrees of freedom,shape parameters of log-likelihood function of t distribution Returns ------- dlldy : array derivative of loglikelihood wrt random variable y evaluated at the points given in y Notes ----- parameterized for garch, standardized to variance=1 ig@i((RR((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyt tstd_dlldyæscGsS|||}|||Œ |}d||||Œ|||d}||fS(sËderivative of log-likelihood with respect to location and scale Parameters ---------- y : array_like data points of random variable at which loglike is evaluated loc : float location parameter of distribution scale : float scale parameter of distribution dlldy : function derivative of loglikelihood fuction wrt. random variable x args : array_like shape parameters of log-likelihood function Returns ------- dlldloc : array derivative of loglikelihood wrt location evaluated at the points given in y dlldscale : array derivative of loglikelihood wrt scale evaluated at the points given in y gð¿i((Rtloctscaletdlldytargstysttdlldloct dlldscale((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyt locscale_grads't__main__igš™™™™™¹?ii ii(tstatstmisccCs(tjtjj||d|d|ƒƒS(NR)R*(RRR2tttpdf(RR)R*R((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytllt3scCs(tjtjj||d|d|ƒƒS(NR)R*(RRR2R4R5(R)RR*R((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytlltloc5scCs(tjtjj||d|d|ƒƒS(NR)R*(RRR2R4R5(R*RR)R((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytlltscale7scCs%tjtjj|d|d|ƒƒS(NR)R*(RRR2tnormR5(RR)R*((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytllnorm:scCs%tjtjj|d|d|ƒƒS(NR)R*(RRR2R9R5(R)RR*((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyt llnormloc<scCs%tjtjj|d|d|ƒƒS(NR)R*(RRR2R9R5(R*RR)((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pyt llnormscale>ss gradient of ttdxgíµ ÷Æ°>tnR,torderst ttsgø?g»½×Ùß|Û=is gradient of norms loglike of tidsdifferently standardizedgð?g$@g @i(tassert_almost_equalgg@gÀi iterr_msgs deriv locs deriv scaleitloglike(iii (iii(iii(iii(gø?ii(gø?ii(gø?iii(ii(gø?ii(gø?iig$@(ii(gð?gð?(gg@(gð?g@(8t__doc__t __future__RtnumpyRtscipyRt scipy.specialRR RRRRRR$R&R'R(R0t__name__tverbosetsigRRtrandomtrandntrvsRRRRR%tNonet dllfdbetaR2R3R6R7R8R:R;R<t derivativeR)R*RR R t numpy.testingRAtlinspacetyttdlldlotdlldsctgr(((sClib/python2.7/site-packages/statsmodels/sandbox/regression/tools.pytsº     $   '    ** %       ......77 .4444 ::7;7K0033%%-