# Source code for statsmodels.robust.norms

import numpy as np

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

def _cabs(x):
"""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.
"""
sign = (x.real >= 0) * 2 - 1
return sign * x

[docs]class RobustNorm:
"""
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.

--------
statsmodels.rlm

Notes
-----
Currently only M-estimators are available.

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.
"""

[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):
"""
Derivative of psi.  Used to obtain robust covariance matrix.

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.

--------
statsmodels.robust.norms.RobustNorm
"""

[docs]    def rho(self, z):
"""
The least squares estimator rho function

Parameters
----------
z : ndarray
1d array

Returns
-------
rho : ndarray
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 : ndarray
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 : ndarray
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 : ndarray
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.

--------
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.abs(z), self.t)

[docs]    def rho(self, z):
r"""
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
"""
z = np.asarray(z)
test = self._subset(z)
return (test * 0.5 * z**2 +
(1 - test) * (np.abs(z) * self.t - 0.5 * self.t**2))

[docs]    def psi(self, z):
r"""
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
"""
z = np.asarray(z)
test = self._subset(z)
return test * z + (1 - test) * self.t * np.sign(z)

[docs]    def weights(self, z):
r"""
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
"""
z_isscalar = np.isscalar(z)
z = np.atleast_1d(z)

test = self._subset(z)
absz = np.abs(z)
absz[test] = 1.0
v = test + (1 - test) * self.t / absz

if z_isscalar:
v = v[0]
return v

[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.abs(z), self.t).astype(float)

# 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.

--------
statsmodels.robust.norms.RobustNorm
"""

def __init__(self, a=.3):
self.a = a

[docs]    def rho(self, z):
r"""
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\|))
"""
z = np.asarray(z)
return (1 - np.exp(-self.a * np.abs(z)) *
(1 + self.a * np.abs(z))) / self.a**2

[docs]    def psi(self, z):
r"""
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\|)
"""
z = np.asarray(z)
return z * np.exp(-self.a * np.abs(z))

[docs]    def weights(self, z):
r"""
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\|)
"""

z = np.asarray(z)
return np.exp(-self.a * np.abs(z))

[docs]    def psi_deriv(self, z):
"""
The derivative of Ramsay's Ea psi function.

Notes
-----
Used to estimate the robust covariance matrix.
"""
a = self.a
x = np.exp(-a * np.abs(z))
dx = -a * x * np.sign(z)
y = z
dy = 1
return x * dy + y * dx

[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.

--------
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.abs(z), self.a * np.pi)

[docs]    def rho(self, z):
r"""
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
"""

a = self.a
z = np.asarray(z)
test = self._subset(z)
return (test * a**2 * (1 - np.cos(z / a)) +
(1 - test) * a**2 * 2)

[docs]    def psi(self, z):
r"""
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
"""

a = self.a
z = np.asarray(z)
test = self._subset(z)
return test * a * np.sin(z / a)

[docs]    def weights(self, z):
r"""
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
"""
a = self.a
z = np.asarray(z)
test = self._subset(z)
ratio = z / a
small = np.abs(ratio) < np.finfo(np.double).eps
if np.any(small):
weights = np.ones_like(ratio)
large = ~small
ratio = ratio[large]
weights[large] = test[large] * np.sin(ratio) / ratio
else:
weights = test * np.sin(ratio) / ratio
return weights

[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)

# 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.

--------
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.abs(z), self.c)

[docs]    def rho(self, z):
r"""
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
"""

z = np.asarray(z)
test = self._subset(z)
return test * z**2 * 0.5 + (1 - test) * self.c**2 * 0.5

[docs]    def psi(self, z):
r"""
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
"""
z = np.asarray(z)
test = self._subset(z)
return test * z

[docs]    def weights(self, z):
r"""
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
"""
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.

--------
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.abs(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):
r"""
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
"""
a, b, c = self.a, self.b, self.c

z_isscalar = np.isscalar(z)
z = np.atleast_1d(z)

t1, t2, t3 = self._subset(z)
t34 = ~(t1 | t2)
dt = np.promote_types(z.dtype, "float")
v = np.zeros(z.shape, dtype=dt)
z = np.abs(z)
v[t1] = z[t1]**2 * 0.5
# v[t2] = (a * (z[t2] - a) + a**2 * 0.5)
v[t2] = (a * z[t2] - a**2 * 0.5)
v[t3] = a * (c - z[t3])**2 / (c - b) * (-0.5)
v[t34] += a * (b + c - a) * 0.5

if z_isscalar:
v = v[0]

return v

[docs]    def psi(self, z):
r"""
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
"""
a, b, c = self.a, self.b, self.c

z_isscalar = np.isscalar(z)
z = np.atleast_1d(z)

t1, t2, t3 = self._subset(z)
dt = np.promote_types(z.dtype, "float")
v = np.zeros(z.shape, dtype=dt)
s = np.sign(z)
za = np.abs(z)

v[t1] = z[t1]
v[t2] = a * s[t2]
v[t3] = a * s[t3] * (c - za[t3]) / (c - b)

if z_isscalar:
v = v[0]
return v

[docs]    def weights(self, z):
r"""
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
"""
a, b, c = self.a, self.b, self.c

z_isscalar = np.isscalar(z)
z = np.atleast_1d(z)

t1, t2, t3 = self._subset(z)

dt = np.promote_types(z.dtype, "float")
v = np.zeros(z.shape, dtype=dt)
v[t1] = 1.0
abs_z = np.abs(z)
v[t2] = a / abs_z[t2]
abs_zt3 = abs_z[t3]
v[t3] = a * (c - abs_zt3) / (abs_zt3 * (c - b))

if z_isscalar:
v = v[0]
return v

[docs]    def psi_deriv(self, z):
"""Derivative of psi function, second derivative of rho function.
"""
a, b, c = self.a, self.b, self.c

z_isscalar = np.isscalar(z)
z = np.atleast_1d(z)

t1, _, t3 = self._subset(z)

dt = np.promote_types(z.dtype, "float")
d = np.zeros(z.shape, dtype=dt)
d[t1] = 1.0
zt3 = z[t3]
d[t3] = -(a * np.sign(zt3) * zt3) / (np.abs(zt3) * (c - b))

if z_isscalar:
d = d[0]
return d

[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.abs(np.asarray(z))
return np.less_equal(z, self.c)

[docs]    def rho(self, z):
r"""
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
"""
subset = self._subset(z)
factor = self.c**2 / 6.
return -(1 - (z / self.c)**2)**3 * subset * factor + factor

[docs]    def psi(self, z):
r"""
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
"""

z = np.asarray(z)
subset = self._subset(z)
return z * (1 - (z / self.c)**2)**2 * subset

[docs]    def weights(self, z):
r"""
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
"""

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]class MQuantileNorm(RobustNorm):
"""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.
"""

def __init__(self, q, base_norm):
self.q = q
self.base_norm = base_norm

def _get_q(self, z):

nobs = len(z)
mask_neg = (z < 0)  # if self.q < 0.5 else (z <= 0)  # maybe symmetric
qq = np.empty(nobs)
return qq

[docs]    def rho(self, z):
"""
The robust criterion function for MQuantileNorm.

Parameters
----------
z : array_like
1d array

Returns
-------
rho : ndarray
"""
qq = self._get_q(z)
return qq * self.base_norm.rho(z)

[docs]    def psi(self, z):
"""
The psi function for MQuantileNorm estimator.

The analytic derivative of rho

Parameters
----------
z : array_like
1d array

Returns
-------
psi : ndarray
"""
qq = self._get_q(z)
return qq * self.base_norm.psi(z)

[docs]    def weights(self, z):
"""
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
"""
qq = self._get_q(z)
return qq * self.base_norm.weights(z)

[docs]    def psi_deriv(self, z):
'''
The derivative of MQuantileNorm function

Parameters
----------
z : array_like
1d array

Returns
-------
psi_deriv : ndarray

Notes
-----
Used to estimate the robust covariance matrix.
'''
qq = self._get_q(z)
return qq * self.base_norm.psi_deriv(z)

def __call__(self, z):
"""
Returns the value of estimator rho applied to an input
"""
return self.rho(z)

[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 : 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
"""
if norm is None:
norm = HuberT()

if initial is None:
mu = np.median(a, axis)
else:
mu = initial

for _ in range(maxiter):
W = norm.weights((a-mu)/scale)
nmu = np.sum(W*a, axis) / np.sum(W, axis)
if np.all(np.less(np.abs(mu - nmu), scale * tol)):
return nmu
else:
mu = nmu
raise ValueError("location estimator failed to converge in %d iterations"
% maxiter)


Last update: May 05, 2023