U Kv f’aã@s¬ddlZdd„ZGdd„dƒZGdd„deƒZGdd „d eƒZGd d „d eƒZGd d „d eƒZGdd„deƒZGdd„deƒZ Gdd„deƒZ Gdd„deƒZ ddd„Z dS)éNcCs|jdkdd}||S)z®absolute value function that changes complex sign based on real sign This could be useful for complex step derivatives of functions that need abs. Not yet used. réé)Úreal)ÚxÚsign©rúszRobustNorm.weightscCst‚dS)z¶ Derivative of psi. Used to obtain robust covariance matrix. See statsmodels.rlm for more information. Abstract method: psi_derive = psi' Nr r rrrÚ psi_derivHs zRobustNorm.psi_derivcCs | |¡S©zH Returns the value of estimator rho applied to an input ©rr rrrÚ__call__TszRobustNorm.__call__N) Ú__name__Ú __module__Ú __qualname__Ú__doc__rrrrrrrrrr s     r c@s0eZdZdZdd„Zdd„Zdd„Zdd „Zd S) Ú LeastSquareszŠ Least squares rho for M-estimation and its derived functions. See Also -------- statsmodels.robust.norms.RobustNorm cCs |ddS)zâ The least squares estimator rho function Parameters ---------- z : ndarray 1d array Returns ------- rho : ndarray rho(z) = (1/2.)*z**2 rçà?rr rrrrdszLeastSquares.rhocCs t |¡S)a  The psi function for the least squares estimator The analytic derivative of rho Parameters ---------- z : array_like 1d array Returns ------- psi : ndarray psi(z) = z )ÚnpÚasarrayr rrrruszLeastSquares.psicCst |¡}t |jtj¡S)a@ 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 : ndarray weights(z) = np.ones(z.shape) )rrÚonesÚshapeÚfloat64r rrrrˆs zLeastSquares.weightscCst |jtj¡S)zî The derivative of the least squares psi function. Returns ------- psi_deriv : ndarray ones(z.shape) Notes ----- Used to estimate the robust covariance matrix. )rrr r!r rrrrœs zLeastSquares.psi_derivN)rrrrrrrrrrrrr rc@sBeZdZdZddd„Zdd„Zdd„Zd d „Zd d „Zd d„Z dS)ÚHuberTz÷ 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 ç…ëQ¸…õ?cCs ||_dS©N)Út)rr%rrrÚ__init__»szHuberT.__init__cCst |¡}t t |¡|j¡S)zE Huber's T is defined piecewise over the range for z )rrÚ less_equalÚabsr%r rrrÚ_subset¾s zHuberT._subsetcCsJt |¡}| |¡}|d|dd|t |¡|jd|jdS)a8 The robust criterion function for Huber's t. Parameters ---------- z : array_like 1d array Returns ------- rho : ndarray rho(z) = .5*z**2 for \|z\| <= t rho(z) = \|z\|*t - .5*t**2 for \|z\| > t rrr)rrr)r(r%©rrÚtestrrrrÅs   $ÿz HuberT.rhocCs4t |¡}| |¡}||d||jt |¡S)aE The psi function for Huber's t estimator The analytic derivative of rho Parameters ---------- z : array_like 1d array Returns ------- psi : ndarray psi(z) = z for \|z\| <= t psi(z) = sign(z)*t for \|z\| > t r)rrr)r%rr*rrrrÚs  z HuberT.psicCsVt |¡}t |¡}| |¡}t |¡}d||<|d||j|}|rR|d}|S)aa Huber's t weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array_like 1d array Returns ------- weights : ndarray weights(z) = 1 for \|z\| <= t weights(z) = t/\|z\| for \|z\| > t çð?rr)rÚisscalarÚ atleast_1dr)r(r%)rrÚ z_isscalarr+ZabszÚvrrrrðs    zHuberT.weightscCst t |¡|j¡ t¡S)zŽ The derivative of Huber's t psi function Notes ----- Used to estimate the robust covariance matrix. )rr'r(r%ZastypeÚfloatr rrrrszHuberT.psi_derivN)r#© rrrrr&r)rrrrrrrrr"¬s r"c@s:eZdZdZddd„Zdd„Zdd„Zd d „Zd d „Zd S)ÚRamsayEzú Ramsay's Ea for M estimation. Parameters ---------- a : float, optional The tuning constant for Ramsay's Ea function. The default value is 0.3. See Also -------- statsmodels.robust.norms.RobustNorm ç333333Ó?cCs ||_dSr$©Úa©rr6rrrr&)szRamsayE.__init__cCsDt |¡}dt |j t |¡¡d|jt |¡|jdS)a  The robust criterion function for Ramsay's Ea. Parameters ---------- z : array_like 1d array Returns ------- rho : ndarray rho(z) = a**-2 * (1 - exp(-a*\|z\|)*(1 + a*\|z\|)) rr©rrÚexpr6r(r rrrr,s  ÿÿz RamsayE.rhocCs&t |¡}|t |j t |¡¡S)a The psi function for Ramsay's Ea estimator The analytic derivative of rho Parameters ---------- z : array_like 1d array Returns ------- psi : ndarray psi(z) = z*exp(-a*\|z\|) r8r rrrr>s z RamsayE.psicCs"t |¡}t |j t |¡¡S)a" Ramsay's Ea weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array_like 1d array Returns ------- weights : ndarray weights(z) = exp(-a*\|z\|) r8r rrrrQs zRamsayE.weightscCsH|j}t | t |¡¡}| |t |¡}|}d}||||S)z‘ The derivative of Ramsay's Ea psi function. Notes ----- Used to estimate the robust covariance matrix. r)r6rr9r(r)rrr6rZdxÚyZdyrrrres zRamsayE.psi_derivN)r4) rrrrr&rrrrrrrrr3s  r3c@sBeZdZdZddd„Zdd„Zdd„Zd d „Zd d „Zd d„Z dS)Ú AndrewWavea 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 ç•C‹lõ?cCs ||_dSr$r5r7rrrr&ƒszAndrewWave.__init__cCs$t |¡}t t |¡|jtj¡S)zI Andrew's wave is defined piecewise over the range of z. )rrr'r(r6Úpir rrrr)†s zAndrewWave._subsetcCsL|j}t |¡}| |¡}||ddt ||¡d||ddS)a{ The robust criterion function for Andrew's wave. Parameters ---------- z : array_like 1d array Returns ------- rho : ndarray The elements of rho are defined as: .. math:: rho(z) & = a^2 *(1-cos(z/a)), |z| \leq a\pi \\ rho(z) & = 2a, |z|>q\pi rr)r6rrr)Úcos©rrr6r+rrrrs   ÿzAndrewWave.rhocCs0|j}t |¡}| |¡}||t ||¡S)aR The psi function for Andrew's wave The analytic derivative of rho Parameters ---------- z : array_like 1d array Returns ------- psi : ndarray psi(z) = a * sin(z/a) for \|z\| <= a*pi psi(z) = 0 for \|z\| > a*pi )r6rrr)Úsinr?rrrr§s  zAndrewWave.psicCsŽ|j}t |¡}| |¡}||}t |¡t tj¡jk}t |¡rxt  |¡}|}||}||t  |¡|||<n|t  |¡|}|S)a} Andrew's wave weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array_like 1d array Returns ------- weights : ndarray weights(z) = sin(z/a) / (z/a) for \|z\| <= a*pi weights(z) = 0 for \|z\| > a*pi ) r6rrr)r(ZfinfoÚdoubleZepsÚanyZ ones_liker@)rrr6r+ZratioÚsmallrZlargerrrr¿s    zAndrewWave.weightscCs| |¡}|t ||j¡S)z’ The derivative of Andrew's wave psi function Notes ----- Used to estimate the robust covariance matrix. )r)rr>r6r*rrrrßs zAndrewWave.psi_derivN)r<r2rrrrr;us  r;c@sBeZdZdZddd„Zdd„Zdd„Zd d „Zd d „Zd d„Z dS)Ú TrimmedMeana 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 ç@cCs ||_dSr$©Úc©rrGrrrr&üszTrimmedMean.__init__cCst |¡}t t |¡|j¡S)zN Least trimmed mean is defined piecewise over the range of z. )rrr'r(rGr rrrr)ÿs zTrimmedMean._subsetcCs:t |¡}| |¡}||ddd||jddS)aA The robust criterion function for least trimmed mean. Parameters ---------- z : array_like 1d array Returns ------- rho : ndarray rho(z) = (1/2.)*z**2 for \|z\| <= c rho(z) = (1/2.)*c**2 for \|z\| > c rrr©rrr)rGr*rrrrs  zTrimmedMean.rhocCst |¡}| |¡}||S)aQ The psi function for least trimmed mean The analytic derivative of rho Parameters ---------- z : array_like 1d array Returns ------- psi : ndarray psi(z) = z for \|z\| <= c psi(z) = 0 for \|z\| > c ©rrr)r*rrrrs  zTrimmedMean.psicCst |¡}| |¡}|S)an Least trimmed mean weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array_like 1d array Returns ------- weights : ndarray weights(z) = 1 for \|z\| <= c weights(z) = 0 for \|z\| > c rJr*rrrr2s  zTrimmedMean.weightscCs| |¡}|S)z— The derivative of least trimmed mean psi function Notes ----- Used to estimate the robust covariance matrix. )r)r*rrrrHs zTrimmedMean.psi_derivN)rEr2rrrrrDís rDc@sBeZdZdZddd„Zdd„Zd d „Zd d „Zd d„Zdd„Z dS)ÚHampela; Hampel function for M-estimation. Parameters ---------- a : float, optional b : float, optional c : float, optional The tuning constants for Hampel's function. The default values are a,b,c = 2, 4, 8. See Also -------- statsmodels.robust.norms.RobustNorm rEç@ç @cCs||_||_||_dSr$)r6ÚbrG)rr6rNrGrrrr&fszHampel.__init__cCs`t t |¡¡}t ||j¡}t ||j¡t ||j¡}t ||j¡t ||j¡}|||fS)zL Hampel's function is defined piecewise over the range of z )rr(rr'r6rNZgreaterrG)rrÚt1Út2Út3rrrr)ks zHampel._subsetc Csð|j|j|j}}}t |¡}t |¡}| |¡\}}}||B} t |jd¡} tj |j | d} t  |¡}||dd| |<||||dd| |<||||d||d| |<| | ||||d7<|rì| d} | S)aí The robust criterion function for Hampel's estimator Parameters ---------- z : array_like 1d array Returns ------- rho : ndarray rho(z) = z**2 / 2 for \|z\| <= a rho(z) = a*\|z\| - 1/2.*a**2 for a < \|z\| <= b rho(z) = a*(c - \|z\|)**2 / (c - b) / 2 for b < \|z\| <= c rho(z) = a*(b + c - a) / 2 for \|z\| > c r1©Údtyperrgà¿r© r6rNrGrr-r.r)Ú promote_typesrSÚzerosr r() rrr6rNrGr/rOrPrQZt34Údtr0rrrrus    $ z Hampel.rhoc Cs¼|j|j|j}}}t |¡}t |¡}| |¡\}}}t |jd¡} tj |j | d} t  |¡} t  |¡} ||| |<|| || |<|| ||| |||| |<|r¸| d} | S)aû The psi function for Hampel's estimator The analytic derivative of rho Parameters ---------- z : array_like 1d array Returns ------- psi : ndarray 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 r1rRr© r6rNrGrr-r.r)rUrSrVr rr() rrr6rNrGr/rOrPrQrWr0ÚsZzarrrržs     $z Hampel.psic Cs®|j|j|j}}}t |¡}t |¡}| |¡\}}}t |jd¡} tj |j | d} d| |<t  |¡} || || |<| |} ||| | ||| |<|rª| d} | S)a$ Hampel weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array_like 1d array Returns ------- weights : ndarray 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 r1rRr,rrT) rrr6rNrGr/rOrPrQrWr0Zabs_zZabs_zt3rrrrÇs   zHampel.weightsc Cs¢|j|j|j}}}t |¡}t |¡}| |¡\}}}t |jd¡} tj |j | d} d| |<||} |t  | ¡|  t  | ¡||| |<|rž| d} | S)zGDerivative of psi function, second derivative of rho function. r1rRr,rrX) rrr6rNrGr/rOÚ_rQrWÚdZzt3rrrrðs  *zHampel.psi_derivN)rErLrMr2rrrrrKTs  )))rKc@sBeZdZdZddd„Zdd„Zdd„Zd d „Zd d „Zd d„Z dS)Ú TukeyBiweighta 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. ç= ×£p½@cCs ||_dSr$rFrHrrrr&szTukeyBiweight.__init__cCst t |¡¡}t ||j¡S)zK Tukey's biweight is defined piecewise over the range of z )rr(rr'rGr rrrr)szTukeyBiweight._subsetcCs<| |¡}|jdd}d||jdd |||S)a^ The robust criterion function for Tukey's biweight estimator Parameters ---------- z : array_like 1d array Returns ------- rho : ndarray rho(z) = -(1 - (z/c)**2)**3 * c**2/6. for \|z\| <= R rho(z) = 0 for \|z\| > R rg@ré©r)rG)rrÚsubsetZfactorrrrrs zTukeyBiweight.rhocCs2t |¡}| |¡}|d||jdd|S)ar The psi function for Tukey's biweight estimator The analytic derivative of rho Parameters ---------- z : array_like 1d array Returns ------- psi : ndarray psi(z) = z*(1 - (z/c)**2)**2 for \|z\| <= R psi(z) = 0 for \|z\| > R rrrI©rrr`rrrr3s  zTukeyBiweight.psicCs$| |¡}d||jdd|S)a~ Tukey's biweight weighting function for the IRLS algorithm The psi function scaled by z Parameters ---------- z : array_like 1d array Returns ------- weights : ndarray psi(z) = (1 - (z/c)**2)**2 for \|z\| <= R psi(z) = 0 for \|z\| > R rrr_rarrrrJs zTukeyBiweight.weightscCsL| |¡}|d||jddd|d|jdd||jdS)z• The derivative of Tukey's biweight psi function Notes ----- Used to estimate the robust covariance matrix. rrér_rarrrr`s &ÿzTukeyBiweight.psi_derivN)r]r2rrrrr\s r\c@sHeZdZdZdd„Zdd„Zdd„Zdd „Zd d „Zd d „Z dd„Z dS)Ú MQuantileNormu•M-quantiles objective function based on a base norm This norm has the same asymmetric structure as the objective function in QuantileRegression but replaces the L1 absolute value by a chosen base norm. rho_q(u) = abs(q - I(q < 0)) * rho_base(u) or, equivalently, rho_q(u) = q * rho_base(u) if u >= 0 rho_q(u) = (1 - q) * rho_base(u) if u < 0 Parameters ---------- q : float M-quantile, must be between 0 and 1 base_norm : RobustNorm instance basic norm that is transformed into an asymmetric M-quantile norm Notes ----- This is mainly for base norms that are not redescending, like HuberT or LeastSquares. (See Jones for the relationship of M-quantiles to quantiles in the case of non-redescending Norms.) Expectiles are M-quantiles with the LeastSquares as base norm. References ---------- .. [*] Bianchi, Annamaria, and Nicola Salvati. 2015. “Asymptotic Properties and Variance Estimators of the M-Quantile Regression Coefficients Estimators.†Communications in Statistics - Theory and Methods 44 (11): 2416–29. doi:10.1080/03610926.2013.791375. .. [*] Breckling, Jens, and Ray Chambers. 1988. “M-Quantiles.†Biometrika 75 (4): 761–71. doi:10.2307/2336317. .. [*] Jones, M. C. 1994. “Expectiles and M-Quantiles Are Quantiles.†Statistics & Probability Letters 20 (2): 149–53. doi:10.1016/0167-7152(94)90031-0. .. [*] Newey, Whitney K., and James L. Powell. 1987. “Asymmetric Least Squares Estimation and Testing.†Econometrica 55 (4): 819–47. doi:10.2307/1911031. cCs||_||_dSr$)ÚqÚ base_norm)rrdrerrrr&ŸszMQuantileNorm.__init__cCs8t|ƒ}|dk}t |¡}d|j||<|j||<|S)Nrr)ÚlenrÚemptyrd)rrZnobsZmask_negÚqqrrrÚ_get_q£s   zMQuantileNorm._get_qcCs| |¡}||j |¡S)zÌ The robust criterion function for MQuantileNorm. Parameters ---------- z : array_like 1d array Returns ------- rho : ndarray )rirer©rrrhrrrr¬s zMQuantileNorm.rhocCs| |¡}||j |¡S)zñ The psi function for MQuantileNorm estimator. The analytic derivative of rho Parameters ---------- z : array_like 1d array Returns ------- psi : ndarray )rirerrjrrrr¼s zMQuantileNorm.psicCs| |¡}||j |¡S)a  MQuantileNorm weighting function for the IRLS algorithm The psi function scaled by z, psi(z) / z Parameters ---------- z : array_like 1d array Returns ------- weights : ndarray )rirerrjrrrrÎs zMQuantileNorm.weightscCs| |¡}||j |¡S)a The derivative of MQuantileNorm function Parameters ---------- z : array_like 1d array Returns ------- psi_deriv : ndarray Notes ----- Used to estimate the robust covariance matrix. )rirerrjrrrràs zMQuantileNorm.psi_derivcCs | |¡Srrr rrrrôszMQuantileNorm.__call__N) rrrrr&rirrrrrrrrrrcms1 rcéçíµ ÷Æ°>c Cs |dkrtƒ}|dkr$t ||¡}n|}t|ƒD]^}| |||¡} t | ||¡t | |¡} t t t || ¡||¡¡rŠ| S| }q0t d|ƒ‚dS)a¬ 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 : ndarray Array over which the location parameter is to be estimated scale : ndarray 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 : ndarray, 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 : ndarray Estimate of location Nz6location estimator failed to converge in %d iterations) r"rZmedianÚrangerÚsumÚallZlessr(Ú ValueError) r6ZscaleZnormZaxisÚinitialÚmaxiterZtolÚmurZÚWZnmurrrÚestimate_locationûs!  ÿru)NrNrkrl) Znumpyrr r rr"r3r;rDrKr\rcrurrrrÚs  KQn[xg2hÿ