Source code for statsmodels.robust.norms

from statsmodels.compat.python import range
import numpy as np

#TODO: add plots to weighting functions for online docs.

[docs]class RobustNorm(object): """ 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 for more information on how the estimators are used and the inputs for the methods of RobustNorm and subclasses. Notes ----- Currently only M-estimators are available. """
[docs] def rho(self, z): """ The robust criterion estimator function. Abstract method: -2 loglike used in M-estimator """ raise NotImplementedError
[docs] def psi(self, z): """ Derivative of rho. Sometimes referred to as the influence function. Abstract method: psi = rho' """ raise NotImplementedError
[docs] def weights(self, z): """ Returns the value of psi(z) / z Abstract method: psi(z) / z """ raise NotImplementedError
[docs] def psi_deriv(self, z): ''' Deriative of psi. Used to obtain robust covariance matrix. See statsmodels.rlm for more information. Abstract method: psi_derive = psi' ''' raise NotImplementedError
def __call__(self, z): """ Returns the value of estimator rho applied to an input """ return self.rho(z)
[docs]class LeastSquares(RobustNorm): """ Least squares rho for M-estimation and its derived functions. See also -------- statsmodels.robust.norms.RobustNorm for the methods. """
[docs] def rho(self, z): """ The least squares estimator rho function Parameters ----------- z : array 1d array Returns ------- rho : array rho(z) = (1/2.)*z**2 """ return z**2 * 0.5
[docs] def psi(self, z): """ 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 """ return np.asarray(z)
[docs] def weights(self, z): """ 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) """ z = np.asarray(z) return np.ones(z.shape, np.float64)
[docs] def psi_deriv(self, z): """ The derivative of the least squares psi function. Returns ------- psi_deriv : array ones(z.shape) Notes ----- Used to estimate the robust covariance matrix. """ return np.ones(z.shape, np.float64)
[docs]class HuberT(RobustNorm): """ 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 """ def __init__(self, t=1.345): self.t = t def _subset(self, z): """ Huber's T is defined piecewise over the range for z """ z = np.asarray(z) return np.less_equal(np.fabs(z), self.t)
[docs] def rho(self, z): """ 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 """ z = np.asarray(z) test = self._subset(z) return (test * 0.5 * z**2 + (1 - test) * (np.fabs(z) * self.t - 0.5 * self.t**2))
[docs] def psi(self, z): """ 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 """ z = np.asarray(z) test = self._subset(z) return test * z + (1 - test) * self.t * np.sign(z)
[docs] def weights(self, z): """ 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 """ z = np.asarray(z) test = self._subset(z) absz = np.fabs(z) absz[test] = 1.0 return test + (1 - test) * self.t / absz
[docs] def psi_deriv(self, z): """ The derivative of Huber's t psi function Notes ----- Used to estimate the robust covariance matrix. """ return np.less_equal(np.fabs(z), self.t) #TODO: untested, but looks right. RamsayE not available in R or SAS?
[docs]class RamsayE(RobustNorm): """ 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 """ def __init__(self, a = .3): self.a = a
[docs] def rho(self, z): """ 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\|)) """ z = np.asarray(z) return (1 - np.exp(-self.a * np.fabs(z)) * (1 + self.a * np.fabs(z))) / self.a**2
[docs] def psi(self, z): """ 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\|) """ z = np.asarray(z) return z * np.exp(-self.a * np.fabs(z))
[docs] def weights(self, z): """ 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\|) """ z = np.asarray(z) return np.exp(-self.a * np.fabs(z))
[docs] def psi_deriv(self, z): """ The derivative of Ramsay's Ea psi function. Notes ----- Used to estimate the robust covariance matrix. """ return np.exp(-self.a * np.fabs(z)) + z**2*\ np.exp(-self.a*np.fabs(z))*-self.a/np.fabs(z)
[docs]class AndrewWave(RobustNorm): """ 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 """ def __init__(self, a = 1.339): self.a = a def _subset(self, z): """ Andrew's wave is defined piecewise over the range of z. """ z = np.asarray(z) return np.less_equal(np.fabs(z), self.a * np.pi)
[docs] def rho(self, z): """ 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 """ a = self.a z = np.asarray(z) test = self._subset(z) return (test * a * (1 - np.cos(z / a)) + (1 - test) * 2 * a)
[docs] def psi(self, z): """ 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 """ a = self.a z = np.asarray(z) test = self._subset(z) return test * np.sin(z / a)
[docs] def weights(self, z): """ 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 """ a = self.a z = np.asarray(z) test = self._subset(z) return test * np.sin(z / a) / (z / a)
[docs] def psi_deriv(self, z): """ The derivative of Andrew's wave psi function Notes ----- Used to estimate the robust covariance matrix. """ test = self._subset(z) return test*np.cos(z / self.a)/self.a #TODO: this is untested
[docs]class TrimmedMean(RobustNorm): """ 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 """ def __init__(self, c=2.): self.c = c def _subset(self, z): """ Least trimmed mean is defined piecewise over the range of z. """ z = np.asarray(z) return np.less_equal(np.fabs(z), self.c)
[docs] def rho(self, z): """ 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 """ z = np.asarray(z) test = self._subset(z) return test * z**2 * 0.5
[docs] def psi(self, z): """ 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 """ z = np.asarray(z) test = self._subset(z) return test * z
[docs] def weights(self, z): """ 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 """ z = np.asarray(z) test = self._subset(z) return test
[docs] def psi_deriv(self, z): """ The derivative of least trimmed mean psi function Notes ----- Used to estimate the robust covariance matrix. """ test = self._subset(z) return test
[docs]class Hampel(RobustNorm): """ 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 """ def __init__(self, a = 2., b = 4., c = 8.): self.a = a self.b = b self.c = c def _subset(self, z): """ Hampel's function is defined piecewise over the range of z """ z = np.fabs(np.asarray(z)) t1 = np.less_equal(z, self.a) t2 = np.less_equal(z, self.b) * np.greater(z, self.a) t3 = np.less_equal(z, self.c) * np.greater(z, self.b) return t1, t2, t3
[docs] def rho(self, z): """ 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 """ z = np.fabs(z) a = self.a; b = self.b; c = self.c t1, t2, t3 = self._subset(z) v = (t1 * z**2 * 0.5 + t2 * (a * z - a**2 * 0.5) + t3 * (a * (c * z - z**2 * 0.5) / (c - b) - 7 * a**2 / 6.) + (1 - t1 + t2 + t3) * a * (b + c - a)) return v
[docs] def psi(self, z): """ 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 """ z = np.asarray(z) a = self.a; b = self.b; c = self.c t1, t2, t3 = self._subset(z) s = np.sign(z) z = np.fabs(z) v = s * (t1 * z + t2 * a*s + t3 * a*s * (c - z) / (c - b)) return v
[docs] def weights(self, z): """ 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 """ z = np.asarray(z) a = self.a; b = self.b; c = self.c t1, t2, t3 = self._subset(z) v = (t1 + t2 * a/np.fabs(z) + t3 * a*(c-np.fabs(z))/(np.fabs(z)*(c-b))) v[np.where(np.isnan(v))]=1. # for some reason 0 returns a nan? return v
[docs] def psi_deriv(self, z): t1, t2, t3 = self._subset(z) return t1 + t3 * (self.a*np.sign(z)*z)/(np.fabs(z)*(self.c-self.b))
[docs]class TukeyBiweight(RobustNorm): """ 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. """ def __init__(self, c = 4.685): self.c = c def _subset(self, z): """ Tukey's biweight is defined piecewise over the range of z """ z = np.fabs(np.asarray(z)) return np.less_equal(z, self.c)
[docs] def rho(self, z): """ 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 """ subset = self._subset(z) return -(1 - (z / self.c)**2)**3 * subset * self.c**2 / 6.
[docs] def psi(self, z): """ 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 """ z = np.asarray(z) subset = self._subset(z) return z * (1 - (z / self.c)**2)**2 * subset
[docs] def weights(self, z): """ 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 """ subset = self._subset(z) return (1 - (z / self.c)**2)**2 * subset
[docs] def psi_deriv(self, z): """ The derivative of Tukey's biweight psi function Notes ----- Used to estimate the robust covariance matrix. """ subset = self._subset(z) return subset*((1 - (z/self.c)**2)**2 - (4*z**2/self.c**2) *\ (1-(z/self.c)**2))
[docs]def estimate_location(a, scale, norm=None, axis=0, initial=None, maxiter=30, tol=1.0e-06): """ 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 """ if norm is None: norm = HuberT() if initial is None: mu = np.median(a, axis) else: mu = initial for iter in range(maxiter): W = norm.weights((a-mu)/scale) nmu = np.sum(W*a, axis) / np.sum(W, axis) if np.alltrue(np.less(np.fabs(mu - nmu), scale * tol)): return nmu else: mu = nmu raise ValueError("location estimator failed to converge in %d iterations"\ % maxiter)