ó áp7]c@sèddlmZddlZdefd„ƒYZdefd„ƒYZdefd„ƒYZd efd „ƒYZd efd „ƒYZ d efd„ƒYZ defd„ƒYZ defd„ƒYZ dddddd„ZdS(iÿÿÿÿ(trangeNt RobustNormcBs;eZdZd„Zd„Zd„Zd„Zd„ZRS(s¼ The parent class for the norms used for robust regression. Lays out the methods expected of the robust norms to be used by statsmodels.RLM. Parameters ---------- None : Some subclasses have optional tuning constants. References ---------- PJ Huber. 'Robust Statistics' John Wiley and Sons, Inc., New York, 1981. DC Montgomery, EA Peck. 'Introduction to Linear Regression Analysis', John Wiley and Sons, Inc., New York, 2001. R Venables, B Ripley. 'Modern Applied Statistics in S' Springer, New York, 2002. See Also -------- statsmodels.rlm Notes ----- Currently only M-estimators are available. cCs t‚dS(s| The robust criterion estimator function. Abstract method: -2 loglike used in M-estimator N(tNotImplementedError(tselftz((s7lib/python2.7/site-packages/statsmodels/robust/norms.pytrho&scCs t‚dS(s„ Derivative of rho. Sometimes referred to as the influence function. Abstract method: psi = rho' N(R(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pytpsi0scCs t‚dS(s_ Returns the value of psi(z) / z Abstract method: psi(z) / z N(R(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pytweights:scCs t‚dS(sµ Deriative of psi. Used to obtain robust covariance matrix. See statsmodels.rlm for more information. Abstract method: psi_derive = psi' N(R(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyt psi_derivDs cCs |j|ƒS(sH Returns the value of estimator rho applied to an input (R(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyt__call__Ps(t__name__t __module__t__doc__RRRRR (((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRs  t LeastSquarescBs2eZdZd„Zd„Zd„Zd„ZRS(sŠ Least squares rho for M-estimation and its derived functions. See Also -------- statsmodels.robust.norms.RobustNorm cCs |ddS(sÞ The least squares estimator rho function Parameters ---------- z : array 1d array Returns ------- rho : array rho(z) = (1/2.)*z**2 igà?((RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRascCs tj|ƒS(s  The psi function for the least squares estimator The analytic derivative of rho Parameters ---------- z : array-like 1d array Returns ------- psi : array psi(z) = z (tnptasarray(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRrscCs%tj|ƒ}tj|jtjƒS(s> The least squares estimator weighting function for the IRLS algorithm. The psi function scaled by the input z Parameters ---------- z : array-like 1d array Returns ------- weights : array weights(z) = np.ones(z.shape) (RRtonestshapetfloat64(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR…scCstj|jtjƒS(sì The derivative of the least squares psi function. Returns ------- psi_deriv : array ones(z.shape) Notes ----- Used to estimate the robust covariance matrix. (RRRR(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR™s (R R R RRRR(((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR Ws    tHuberTcBsGeZdZdd„Zd„Zd„Zd„Zd„Zd„ZRS(s÷ Huber's T for M estimation. Parameters ---------- t : float, optional The tuning constant for Huber's t function. The default value is 1.345. See Also -------- statsmodels.robust.norms.RobustNorm g…ëQ¸…õ?cCs ||_dS(N(tt(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyt__init__¸scCs+tj|ƒ}tjtj|ƒ|jƒS(sE Huber's T is defined piecewise over the range for z (RRt less_equaltfabsR(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyt_subset»scCsYtj|ƒ}|j|ƒ}|d|dd|tj|ƒ|jd|jdS(s6 The robust criterion function for Huber's t. Parameters ---------- z : array-like 1d array Returns ------- rho : array rho(z) = .5*z**2 for \|z\| <= t rho(z) = \|z\|*t - .5*t**2 for \|z\| > t gà?ii(RRRRR(RRttest((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRÂscCsBtj|ƒ}|j|ƒ}||d||jtj|ƒS(sC The psi function for Huber's t estimator The analytic derivative of rho Parameters ---------- z : array-like 1d array Returns ------- psi : array psi(z) = z for \|z\| <= t psi(z) = sign(z)*t for \|z\| > t i(RRRRtsign(RRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR×scCsNtj|ƒ}|j|ƒ}tj|ƒ}d||<|d||j|S(s_ Huber's t weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array-like 1d array Returns ------- weights : array weights(z) = 1 for \|z\| <= t weights(z) = t/\|z\| for \|z\| > t gð?i(RRRRR(RRRtabsz((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRís  cCstjtj|ƒ|jƒS(sŽ The derivative of Huber's t psi function Notes ----- Used to estimate the robust covariance matrix. (RRRR(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRs( R R R RRRRRR(((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR©s      tRamsayEcBs>eZdZdd„Zd„Zd„Zd„Zd„ZRS(sú Ramsay's Ea for M estimation. Parameters ---------- a : float, optional The tuning constant for Ramsay's Ea function. The default value is 0.3. See Also -------- statsmodels.robust.norms.RobustNorm g333333Ó?cCs ||_dS(N(ta(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR scCsTtj|ƒ}dtj|j tj|ƒƒd|jtj|ƒ|jdS(s The robust criterion function for Ramsay's Ea. Parameters ---------- z : array-like 1d array Returns ------- rho : array rho(z) = a**-2 * (1 - exp(-a*\|z\|)*(1 + a*\|z\|)) ii(RRtexpRR(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR#s cCs1tj|ƒ}|tj|j tj|ƒƒS(s The psi function for Ramsay's Ea estimator The analytic derivative of rho Parameters ---------- z : array-like 1d array Returns ------- psi : array psi(z) = z*exp(-a*\|z\|) (RRRRR(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR5scCs-tj|ƒ}tj|j tj|ƒƒS(s  Ramsay's Ea weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array-like 1d array Returns ------- weights : array weights(z) = exp(-a*\|z\|) (RRRRR(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRHscCsYtj|j tj|ƒƒ|dtj|j tj|ƒƒ|j tj|ƒS(s‘ The derivative of Ramsay's Ea psi function. Notes ----- Used to estimate the robust covariance matrix. i(RRRR(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR\s (R R R RRRRR(((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRs      t AndrewWavecBsGeZdZdd„Zd„Zd„Zd„Zd„Zd„ZRS(s Andrew's wave for M estimation. Parameters ---------- a : float, optional The tuning constant for Andrew's Wave function. The default value is 1.339. See Also -------- statsmodels.robust.norms.RobustNorm g•C‹lõ?cCs ||_dS(N(R(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRxscCs2tj|ƒ}tjtj|ƒ|jtjƒS(sI Andrew's wave is defined piecewise over the range of z. (RRRRRtpi(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR{scCsT|j}tj|ƒ}|j|ƒ}||dtj||ƒd|d|S(s> The robust criterion function for Andrew's wave. Parameters ---------- z : array-like 1d array Returns ------- rho : array rho(z) = a*(1-cos(z/a)) for \|z\| <= a*pi rho(z) = 2*a for \|z\| > a*pi ii(RRRRtcos(RRRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR‚s  cCs<|j}tj|ƒ}|j|ƒ}|tj||ƒS(sP The psi function for Andrew's wave The analytic derivative of rho Parameters ---------- z : array-like 1d array Returns ------- psi : array psi(z) = sin(z/a) for \|z\| <= a*pi psi(z) = 0 for \|z\| > a*pi (RRRRtsin(RRRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR™s cCsD|j}tj|ƒ}|j|ƒ}|tj||ƒ||S(sw Andrew's wave weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array-like 1d array Returns ------- weights : array weights(z) = sin(z/a)/(z/a) for \|z\| <= a*pi weights(z) = 0 for \|z\| > a*pi (RRRRR"(RRRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR±s cCs.|j|ƒ}|tj||jƒ|jS(s’ The derivative of Andrew's wave psi function Notes ----- Used to estimate the robust covariance matrix. (RRR!R(RRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRÈs ( R R R RRRRRR(((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRis     t TrimmedMeancBsGeZdZdd„Zd„Zd„Zd„Zd„Zd„ZRS(s Trimmed mean function for M-estimation. Parameters ---------- c : float, optional The tuning constant for Ramsay's Ea function. The default value is 2.0. See Also -------- statsmodels.robust.norms.RobustNorm g@cCs ||_dS(N(tc(RR$((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRåscCs+tj|ƒ}tjtj|ƒ|jƒS(sN Least trimmed mean is defined piecewise over the range of z. (RRRRR$(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRèscCs.tj|ƒ}|j|ƒ}||ddS(s5 The robust criterion function for least trimmed mean. Parameters ---------- z : array-like 1d array Returns ------- rho : array rho(z) = (1/2.)*z**2 for \|z\| <= c rho(z) = 0 for \|z\| > c igà?(RRR(RRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRðscCs&tj|ƒ}|j|ƒ}||S(sP The psi function for least trimmed mean The analytic derivative of rho Parameters ---------- z : array-like 1d array Returns ------- psi : array psi(z) = z for \|z\| <= c psi(z) = 0 for \|z\| > c (RRR(RRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRscCs"tj|ƒ}|j|ƒ}|S(sm Least trimmed mean weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array-like 1d array Returns ------- weights : array weights(z) = 1 for \|z\| <= c weights(z) = 0 for \|z\| > c (RRR(RRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRscCs|j|ƒ}|S(s— The derivative of least trimmed mean psi function Notes ----- Used to estimate the robust covariance matrix. (R(RRR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR3s( R R R RRRRRR(((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR#Ös      tHampelcBsMeZdZdddd„Zd„Zd„Zd„Zd„Zd „ZRS( s; Hampel function for M-estimation. Parameters ---------- a : float, optional b : float, optional c : float, optional The tuning constants for Hampel's function. The default values are a,b,c = 2, 4, 8. See Also -------- statsmodels.robust.norms.RobustNorm g@g@g @cCs||_||_||_dS(N(RtbR$(RRR&R$((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRQs  cCsŠtjtj|ƒƒ}tj||jƒ}tj||jƒtj||jƒ}tj||jƒtj||jƒ}|||fS(sL Hampel's function is defined piecewise over the range of z (RRRRRR&tgreaterR$(RRtt1tt2tt3((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRVs ((c CsÄtj|ƒ}|j}|j}|j}|j|ƒ\}}}||dd||||dd|||||dd||d|ddd|||||||}|S(sè The robust criterion function for Hampel's estimator Parameters ---------- z : array-like 1d array Returns ------- rho : array rho(z) = (1/2.)*z**2 for \|z\| <= a rho(z) = a*\|z\| - 1/2.*a**2 for a < \|z\| <= b rho(z) = a*(c*\|z\|-(1/2.)*z**2)/(c-b) for b < \|z\| <= c rho(z) = a*(b + c - a) for \|z\| > c igà?ig@i(RRRR&R$R( RRRR&R$R(R)R*tv((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR`s   [#c Csštj|ƒ}|j}|j}|j}|j|ƒ\}}}tj|ƒ}tj|ƒ}|||||||||||||} | S(sù The psi function for Hampel's estimator The analytic derivative of rho Parameters ---------- z : array-like 1d array Returns ------- psi : array psi(z) = z for \|z\| <= a psi(z) = a*sign(z) for a < \|z\| <= b psi(z) = a*sign(z)*(c - \|z\|)/(c-b) for b < \|z\| <= c psi(z) = 0 for \|z\| > c (RRRR&R$RRR( RRRR&R$R(R)R*tsR+((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR€s    c Cs«tj|ƒ}|j}|j}|j}|j|ƒ\}}}|||tj|ƒ|||tj|ƒtj|ƒ||}d|tjtj|ƒƒ<|S(s Hampel weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array-like 1d array Returns ------- weights : array weights(z) = 1 for \|z\| <= a weights(z) = a/\|z\| for a < \|z\| <= b weights(z) = a*(c - \|z\|)/(\|z\|*(c-b)) for b < \|z\| <= c weights(z) = 0 for \|z\| > c gð?( RRRR&R$RRtwheretisnan( RRRR&R$R(R)R*R+((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR¢s   1cCsS|j|ƒ\}}}|||jtj|ƒ|tj|ƒ|j|jS(N(RRRRRR$R&(RRR(R)R*((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRÄs( R R R RRRRRR(((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR%?s " "t TukeyBiweightcBsGeZdZdd„Zd„Zd„Zd„Zd„Zd„ZRS(s Tukey's biweight function for M-estimation. Parameters ---------- c : float, optional The tuning constant for Tukey's Biweight. The default value is c = 4.685. Notes ----- Tukey's biweight is sometime's called bisquare. g= ×£p½@cCs ||_dS(N(R$(RR$((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRÙscCs+tjtj|ƒƒ}tj||jƒS(sK Tukey's biweight is defined piecewise over the range of z (RRRRR$(RR((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRÜscCs:|j|ƒ}d||jdd ||jddS(s\ The robust criterion function for Tukey's biweight estimator Parameters ---------- z : array-like 1d array Returns ------- rho : array rho(z) = -(1 - (z/c)**2)**3 * c**2/6. for \|z\| <= R rho(z) = 0 for \|z\| > R iiig@(RR$(RRtsubset((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRãscCs=tj|ƒ}|j|ƒ}|d||jdd|S(sp The psi function for Tukey's biweight estimator The analytic derivative of rho Parameters ---------- z : array-like 1d array Returns ------- psi : array psi(z) = z*(1 - (z/c)**2)**2 for \|z\| <= R psi(z) = 0 for \|z\| > R ii(RRRR$(RRR0((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyRöscCs*|j|ƒ}d||jdd|S(s| Tukey's biweight weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array-like 1d array Returns ------- weights : array psi(z) = (1 - (z/c)**2)**2 for \|z\| <= R psi(z) = 0 for \|z\| > R ii(RR$(RRR0((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR scCsT|j|ƒ}|d||jddd|d|jdd||jdS(s• The derivative of Tukey's biweight psi function Notes ----- Used to estimate the robust covariance matrix. iii(RR$(RRR0((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR#s/( R R R RRRRRR(((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyR/És     iigíµ ÷Æ°>c CsÝ|dkrtƒ}n|dkr9tj||ƒ}n|}x‡t|ƒD]y}|j|||ƒ} tj| ||ƒtj| |ƒ} tjtjtj || ƒ||ƒƒr¿| S| }qLWt d|ƒ‚dS(s¤ M-estimator of location using self.norm and a current estimator of scale. This iteratively finds a solution to norm.psi((a-mu)/scale).sum() == 0 Parameters ---------- a : array Array over which the location parameter is to be estimated scale : array Scale parameter to be used in M-estimator norm : RobustNorm, optional Robust norm used in the M-estimator. The default is HuberT(). axis : int, optional Axis along which to estimate the location parameter. The default is 0. initial : array, optional Initial condition for the location parameter. Default is None, which uses the median of a. niter : int, optional Maximum number of iterations. The default is 30. tol : float, optional Toleration for convergence. The default is 1e-06. Returns ------- mu : array Estimate of location s6location estimator failed to converge in %d iterationsN( tNoneRRtmedianRRtsumtalltruetlessRt ValueError( RtscaletnormtaxistinitialtmaxiterttoltmutitertWtnmu((s7lib/python2.7/site-packages/statsmodels/robust/norms.pytestimate_location/s!   &, (tstatsmodels.compat.pythonRtnumpyRtobjectRR RRRR#R%R/R1RA(((s7lib/python2.7/site-packages/statsmodels/robust/norms.pyts PRhXmiŠf