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"""A collection of bandwidth estimators for Kernel Density Estimation"""
from __future__ import division
from __future__ import print_function
__author__ = "Felix Simkovic"
__date__ = "02 Aug 2017"
__version__ = "1.0"
import abc
import numpy as np
ABC = abc.ABCMeta('ABC', (object,), {})
[docs]class BandwidthBase(ABC):
"""Abstract class for bandwidth calculations"""
@abc.abstractproperty
def bandwidth(self):
return 0.0
# REM: abstractproperty requires us to re-declare it in every child class
@property
def bw(self):
return self.bandwidth
[docs]class AmiseBW(BandwidthBase):
"""Asymptotic Mean Integrated Squared Error (AMISE)
This particular choice of bandwidth recovers all the important features whilst maintaining smoothness.
It is a direct implementation of the method used by [#]_.
.. [#] Sadowski, M.I. (2013). Prediction of protein domain boundaries from inverse covariances.
"""
def __init__(self, data, niterations=25, eps=1e-3):
"""Instantiate a new bandwith calculator"""
self._data = np.asarray(data)
self._niterations = niterations
self._eps = eps
@property
def bandwidth(self):
data = self._data
x0 = BowmanBW(data).bandwidth
y0 = AmiseBW.optimal_bandwidth_equation(data, x0)
x = 0.8 * x0
y = AmiseBW.optimal_bandwidth_equation(data, x)
for _ in np.arange(self._niterations):
x -= y * (x0 - x) / (y0 - y)
y = AmiseBW.optimal_bandwidth_equation(data, x)
if abs(y) < (self._eps * y0):
break
return x
[docs] @staticmethod
def curvature(p, x, w):
z = (x - p) / w
y = (1 * (z ** 2 - 1.0) * np.exp(-0.5 * z * z) / (w * np.sqrt(2. * np.pi)) / w ** 2).sum()
return y / p.shape[0]
[docs] @staticmethod
def extended_range(mn, mx, bw, ext=3):
return mn - ext * bw, mx + ext * bw
[docs] @staticmethod
def optimal_bandwidth_equation(p, default_bw):
alpha = 1. / (2. * np.sqrt(np.pi))
sigma = 1.0
n = p.shape[0]
q = AmiseBW.stiffness_integral(p, default_bw)
return default_bw - ((n * q * sigma ** 4) / alpha) ** (-1.0 / (p.shape[1] + 4))
[docs] @staticmethod
def stiffness_integral(p, default_bw, eps=1e-4):
mn, mx = AmiseBW.extended_range(p.min(), p.max(), default_bw, ext=3)
n = 1
dx = (mx - mn) / n
yy = 0.5 * dx * (AmiseBW.curvature(p, mn, default_bw) ** 2 +
AmiseBW.curvature(p, mx, default_bw) ** 2)
# The trapezoidal rule guarantees a relative error of better than eps
# for some number of steps less than maxn.
maxn = (mx - mn) / np.sqrt(eps)
# Cap the total computation spent
maxn = 2048 if maxn > 2048 else maxn
n = 2
while n <= maxn:
dx /= 2.
y = 0
for i in np.arange(1, n, 2):
y += AmiseBW.curvature(p, mn + i * dx, default_bw) ** 2
yy = 0.5 * yy + y * dx
if n > 8 and abs(y * dx - 0.5 * yy) < eps * yy:
break
n *= 2
return yy
[docs]class BowmanBW(BandwidthBase):
"""Bowman & Azzalini [#]_ bandwidth calculation
.. math::
\\sqrt{\\frac{\\sum{X}^2}{n}-(\\frac{\\sum{X}}{n})^2}*(\\frac{(d+2)*n}{4})^\\frac{-1}{d+4}
.. [#] Bowman, A.W. & Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis.
"""
def __init__(self, data):
"""Instantiate a new bandwith calculator"""
self._data = np.asarray(data)
@property
def bandwidth(self):
data = self._data
sigma = np.sqrt((data ** 2).sum() / data.shape[0] - (data.sum() / data.shape[0]) ** 2)
return sigma * ((((data.shape[1] + 2) * data.shape[0]) / 4.) ** (-1. / (data.shape[1] + 4)))
[docs]class LinearBW(BandwidthBase):
"""Linear [#]_ implementation
.. math::
\\frac{N_{max}}{t}
.. [#] Sadowski, M.I. (2013). Prediction of protein domain boundaries from inverse covariances.
"""
def __init__(self, data, threshold=15):
self._data = np.asarray(data)
self._threshold = threshold
@property
def bandwidth(self):
return float(self._data.max() / self._threshold)
[docs]class ScottBW(BandwidthBase):
"""Scott's [#]_ implementation
.. math::
1.059*\\sigma*n^\\frac{-1}{d+4}
.. [#] Scott, D.W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization.
"""
def __init__(self, data):
"""Instantiate a new bandwith calculator"""
self._data = np.asarray(data)
@property
def bandwidth(self):
data = self._data
sigma = np.minimum(np.std(data, axis=0, ddof=1), (np.percentile(data, 75) - np.percentile(data, 25)) / 1.349)[0]
return 1.059 * sigma * data.shape[0] ** (-1. / (data.shape[1] + 4))
[docs]class SilvermanBW(BandwidthBase):
"""Silverman's [#]_ implementation
.. math::
0.9*\\sigma*(n*\\frac{d+2}{4})^\\frac{-1}{d+4}
.. [#] Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis.
"""
def __init__(self, data):
"""Instantiate a new bandwith calculator"""
self._data = np.asarray(data)
@property
def bandwidth(self):
data = self._data
sigma = np.minimum(np.std(data, axis=0, ddof=1), (np.percentile(data, 75) - np.percentile(data, 25)) / 1.349)[0]
return 0.9 * sigma * (data.shape[0] * (data.shape[1] + 2) / 4.) ** (-1. / (data.shape[1] + 4))
[docs]def bandwidth_factory(method):
"""Obtain the bandwidth as defined by user method"""
if method == "amise":
return AmiseBW
elif method == "bowman":
return BowmanBW
elif method == "linear":
return LinearBW
elif method == "scott":
return ScottBW
elif method == "silverman":
return SilvermanBW
else:
msg = "Undefined bandwidth method: {0}".format(method)
raise ValueError(msg)
