"""
Support and standalone functions for Robust Linear Models
References
----------
PJ Huber. 'Robust Statistics' John Wiley and Sons, Inc., New York, 1981.
R Venables, B Ripley. 'Modern Applied Statistics in S'
Springer, New York, 2002.
C Croux, PJ Rousseeuw, 'Time-efficient algorithms for two highly robust
estimators of scale' Computational statistics. Physica, Heidelberg, 1992.
"""
import numpy as np
from scipy.stats import norm as Gaussian
from statsmodels.tools import tools
from statsmodels.tools.validation import array_like, float_like
from . import norms
from ._qn import _qn
[docs]def mad(a, c=Gaussian.ppf(3 / 4.0), axis=0, center=np.median):
"""
The Median Absolute Deviation along given axis of an array
Parameters
----------
a : array_like
Input array.
c : float, optional
The normalization constant. Defined as scipy.stats.norm.ppf(3/4.),
which is approximately 0.6745.
axis : int, optional
The default is 0. Can also be None.
center : callable or float
If a callable is provided, such as the default `np.median` then it
is expected to be called center(a). The axis argument will be applied
via np.apply_over_axes. Otherwise, provide a float.
Returns
-------
mad : float
`mad` = median(abs(`a` - center))/`c`
"""
a = array_like(a, "a", ndim=None)
c = float_like(c, "c")
if not a.size:
center_val = 0.0
elif callable(center):
if axis is not None:
center_val = np.apply_over_axes(center, a, axis)
else:
center_val = center(a.ravel())
else:
center_val = float_like(center, "center")
err = (np.abs(a - center_val)) / c
if not err.size:
if axis is None or err.ndim == 1:
return np.nan
else:
shape = list(err.shape)
shape.pop(axis)
return np.empty(shape)
return np.median(err, axis=axis)
[docs]def iqr(a, c=Gaussian.ppf(3 / 4) - Gaussian.ppf(1 / 4), axis=0):
"""
The normalized interquartile range along given axis of an array
Parameters
----------
a : array_like
Input array.
c : float, optional
The normalization constant, used to get consistent estimates of the
standard deviation at the normal distribution. Defined as
scipy.stats.norm.ppf(3/4.) - scipy.stats.norm.ppf(1/4.), which is
approximately 1.349.
axis : int, optional
The default is 0. Can also be None.
Returns
-------
The normalized interquartile range
"""
a = array_like(a, "a", ndim=None)
c = float_like(c, "c")
if a.ndim == 0:
raise ValueError("a should have at least one dimension")
elif a.size == 0:
return np.nan
else:
quantiles = np.quantile(a, [0.25, 0.75], axis=axis)
return np.squeeze(np.diff(quantiles, axis=0) / c)
[docs]def qn_scale(a, c=1 / (np.sqrt(2) * Gaussian.ppf(5 / 8)), axis=0):
"""
Computes the Qn robust estimator of scale
The Qn scale estimator is a more efficient alternative to the MAD.
The Qn scale estimator of an array a of length n is defined as
c * {abs(a[i] - a[j]): i<j}_(k), for k equal to [n/2] + 1 choose 2. Thus,
the Qn estimator is the k-th order statistic of the absolute differences
of the array. The optional constant is used to normalize the estimate
as explained below. The implementation follows the algorithm described
in Croux and Rousseeuw (1992).
Parameters
----------
a : array_like
Input array.
c : float, optional
The normalization constant. The default value is used to get consistent
estimates of the standard deviation at the normal distribution.
axis : int, optional
The default is 0.
Returns
-------
{float, ndarray}
The Qn robust estimator of scale
"""
a = array_like(
a, "a", ndim=None, dtype=np.float64, contiguous=True, order="C"
)
c = float_like(c, "c")
if a.ndim == 0:
raise ValueError("a should have at least one dimension")
elif a.size == 0:
return np.nan
else:
out = np.apply_along_axis(_qn, axis=axis, arr=a, c=c)
if out.ndim == 0:
return float(out)
return out
def _qn_naive(a, c=1 / (np.sqrt(2) * Gaussian.ppf(5 / 8))):
"""
A naive implementation of the Qn robust estimator of scale, used solely
to test the faster, more involved one
Parameters
----------
a : array_like
Input array.
c : float, optional
The normalization constant, used to get consistent estimates of the
standard deviation at the normal distribution. Defined as
1/(np.sqrt(2) * scipy.stats.norm.ppf(5/8)), which is 2.219144.
Returns
-------
The Qn robust estimator of scale
"""
a = np.squeeze(a)
n = a.shape[0]
if a.size == 0:
return np.nan
else:
h = int(n // 2 + 1)
k = int(h * (h - 1) / 2)
idx = np.triu_indices(n, k=1)
diffs = np.abs(a[idx[0]] - a[idx[1]])
output = np.partition(diffs, kth=k - 1)[k - 1]
output = c * output
return output
[docs]class Huber(object):
"""
Huber's proposal 2 for estimating location and scale jointly.
Parameters
----------
c : float, optional
Threshold used in threshold for chi=psi**2. Default value is 1.5.
tol : float, optional
Tolerance for convergence. Default value is 1e-08.
maxiter : int, optional0
Maximum number of iterations. Default value is 30.
norm : statsmodels.robust.norms.RobustNorm, optional
A robust norm used in M estimator of location. If None,
the location estimator defaults to a one-step
fixed point version of the M-estimator using Huber's T.
call
Return joint estimates of Huber's scale and location.
Examples
--------
>>> import numpy as np
>>> import statsmodels.api as sm
>>> chem_data = np.array([2.20, 2.20, 2.4, 2.4, 2.5, 2.7, 2.8, 2.9, 3.03,
... 3.03, 3.10, 3.37, 3.4, 3.4, 3.4, 3.5, 3.6, 3.7, 3.7, 3.7, 3.7,
... 3.77, 5.28, 28.95])
>>> sm.robust.scale.huber(chem_data)
(array(3.2054980819923693), array(0.67365260010478967))
"""
def __init__(self, c=1.5, tol=1.0e-08, maxiter=30, norm=None):
self.c = c
self.maxiter = maxiter
self.tol = tol
self.norm = norm
tmp = 2 * Gaussian.cdf(c) - 1
self.gamma = tmp + c ** 2 * (1 - tmp) - 2 * c * Gaussian.pdf(c)
def __call__(self, a, mu=None, initscale=None, axis=0):
"""
Compute Huber's proposal 2 estimate of scale, using an optional
initial value of scale and an optional estimate of mu. If mu
is supplied, it is not reestimated.
Parameters
----------
a : ndarray
1d array
mu : float or None, optional
If the location mu is supplied then it is not reestimated.
Default is None, which means that it is estimated.
initscale : float or None, optional
A first guess on scale. If initscale is None then the standardized
median absolute deviation of a is used.
Notes
-----
`Huber` minimizes the function
sum(psi((a[i]-mu)/scale)**2)
as a function of (mu, scale), where
psi(x) = np.clip(x, -self.c, self.c)
"""
a = np.asarray(a)
if mu is None:
n = a.shape[0] - 1
mu = np.median(a, axis=axis)
est_mu = True
else:
n = a.shape[0]
mu = mu
est_mu = False
if initscale is None:
scale = mad(a, axis=axis)
else:
scale = initscale
scale = tools.unsqueeze(scale, axis, a.shape)
mu = tools.unsqueeze(mu, axis, a.shape)
return self._estimate_both(a, scale, mu, axis, est_mu, n)
def _estimate_both(self, a, scale, mu, axis, est_mu, n):
"""
Estimate scale and location simultaneously with the following
pseudo_loop:
while not_converged:
mu, scale = estimate_location(a, scale, mu), estimate_scale(a, scale, mu)
where estimate_location is an M-estimator and estimate_scale implements
the check used in Section 5.5 of Venables & Ripley
""" # noqa:E501
for _ in range(self.maxiter):
# Estimate the mean along a given axis
if est_mu:
if self.norm is None:
# This is a one-step fixed-point estimator
# if self.norm == norms.HuberT
# It should be faster than using norms.HuberT
nmu = (
np.clip(
a, mu - self.c * scale, mu + self.c * scale
).sum(axis)
/ a.shape[axis]
)
else:
nmu = norms.estimate_location(
a, scale, self.norm, axis, mu, self.maxiter, self.tol
)
else:
# Effectively, do nothing
nmu = mu.squeeze()
nmu = tools.unsqueeze(nmu, axis, a.shape)
subset = np.less_equal(np.abs((a - mu) / scale), self.c)
card = subset.sum(axis)
scale_num = np.sum(subset * (a - nmu) ** 2, axis)
scale_denom = n * self.gamma - (a.shape[axis] - card) * self.c ** 2
nscale = np.sqrt(scale_num / scale_denom)
nscale = tools.unsqueeze(nscale, axis, a.shape)
test1 = np.alltrue(
np.less_equal(np.abs(scale - nscale), nscale * self.tol)
)
test2 = np.alltrue(
np.less_equal(np.abs(mu - nmu), nscale * self.tol)
)
if not (test1 and test2):
mu = nmu
scale = nscale
else:
return nmu.squeeze(), nscale.squeeze()
raise ValueError(
"joint estimation of location and scale failed "
"to converge in %d iterations" % self.maxiter
)
huber = Huber()
[docs]class HuberScale:
r"""
Huber's scaling for fitting robust linear models.
Huber's scale is intended to be used as the scale estimate in the
IRLS algorithm and is slightly different than the `Huber` class.
Parameters
----------
d : float, optional
d is the tuning constant for Huber's scale. Default is 2.5
tol : float, optional
The convergence tolerance
maxiter : int, optiona
The maximum number of iterations. The default is 30.
Methods
-------
call
Return's Huber's scale computed as below
Notes
-----
Huber's scale is the iterative solution to
scale_(i+1)**2 = 1/(n*h)*sum(chi(r/sigma_i)*sigma_i**2
where the Huber function is
chi(x) = (x**2)/2 for \|x\| < d
chi(x) = (d**2)/2 for \|x\| >= d
and the Huber constant h = (n-p)/n*(d**2 + (1-d**2)*
scipy.stats.norm.cdf(d) - .5 - d*sqrt(2*pi)*exp(-0.5*d**2)
"""
def __init__(self, d=2.5, tol=1e-08, maxiter=30):
self.d = d
self.tol = tol
self.maxiter = maxiter
def __call__(self, df_resid, nobs, resid):
h = (
df_resid
/ nobs
* (
self.d ** 2
+ (1 - self.d ** 2) * Gaussian.cdf(self.d)
- 0.5
- self.d / (np.sqrt(2 * np.pi)) * np.exp(-0.5 * self.d ** 2)
)
)
s = mad(resid)
def subset(x):
return np.less(np.abs(resid / x), self.d)
def chi(s):
return subset(s) * (resid / s) ** 2 / 2 + (1 - subset(s)) * (
self.d ** 2 / 2
)
scalehist = [np.inf, s]
niter = 1
while (
np.abs(scalehist[niter - 1] - scalehist[niter]) > self.tol
and niter < self.maxiter
):
nscale = np.sqrt(
1
/ (nobs * h)
* np.sum(chi(scalehist[-1]))
* scalehist[-1] ** 2
)
scalehist.append(nscale)
niter += 1
# TODO: raise on convergence failure?
return scalehist[-1]
hubers_scale = HuberScale()