r"""
:Author: Pierre Barbier de Reuille <pierre.barbierdereuille@gmail.com>
Module providing a set of kernels for use with either the :py:mod:`pyqt_fit.kde` or the :py:mod:`kernel_smoothing`
module.
Kernels should be created following this template:
"""
from __future__ import division, absolute_import, print_function
import numpy as np
from scipy.special import erf
from scipy import fftpack, integrate
from .utils import make_ufunc
from . import _kernels_py
from .cyth import HAS_CYTHON
kernels_imp = None
def usePython():
"""
Force the use of the Python implementation of the kernels
"""
global kernels_imp
from .import _kernels_py
kernels_imp = _kernels_py
def useCython():
"""
Force the use of the Cython implementation of the kernels, if available
"""
global kernels_imp
if HAS_CYTHON:
from . import _kernels
kernels_imp = _kernels
if HAS_CYTHON:
useCython()
else:
usePython()
import sys
print("Warning, cannot import Cython kernel functions, "
"pure python functions will be used instead", file=sys.stderr)
S2PI = np.sqrt(2 * np.pi)
S2 = np.sqrt(2)
[docs]class Kernel1D(object):
r"""
A 1D kernel :math:`K(z)` is a function with the following properties:
.. math::
\begin{array}{rcl}
\int_\mathbb{R} K(z) &=& 1 \\
\int_\mathbb{R} zK(z)dz &=& 0 \\
\int_\mathbb{R} z^2K(z) dz &<& \infty \quad (\approx 1)
\end{array}
Which translates into the function should have:
- a sum of 1 (i.e. a valid density of probability);
- an average of 0 (i.e. centered);
- a finite variance. It is even recommanded that the variance is close to 1 to give a uniform meaning to the
bandwidth.
.. py:attribute:: cut
:type: float
Cutting point after which there is a negligeable part of the probability. More formally, if :math:`c` is the
cutting point:
.. math::
\int_{-c}^c p(x) dx \approx 1
.. py:attribute:: lower
:type: float
Lower bound of the support of the PDF. Formally, if :math:`l` is the lower bound:
.. math::
\int_{-\infty}^l p(x)dx = 0
.. py:attribute:: upper
:type: float
Upper bound of the support of the PDF. Formally, if :math:`u` is the upper bound:
.. math::
\int_u^\infty p(x)dx = 0
"""
cut = 3.
lower = -np.inf
upper = np.inf
[docs] def pdf(self, z, out=None):
r"""
Returns the density of the kernel on the points `z`. This is the funtion :math:`K(z)` itself.
:param ndarray z: Array of points to evaluate the function on. The method should accept any shape of array.
:param ndarray out: If provided, it will be of the same shape as `z` and the result should be stored in it.
Ideally, it should be used for as many intermediate computation as possible.
"""
raise NotImplementedError()
def __call__(self, z, out=None):
"""
Alias for :py:meth:`Kernel1D.pdf`
"""
return self.pdf(z, out=out)
[docs] def cdf(self, z, out=None):
r"""
Returns the cumulative density function on the points `z`, i.e.:
.. math::
K_0(z) = \int_{-\infty}^z K(t) dt
"""
z = np.asfarray(z)
try:
comp_pdf = self.__comp_pdf
except AttributeError:
def pdf(x):
return self.pdf(np.atleast_1d(x))
lower = self.lower
upper = self.upper
@make_ufunc()
def comp_pdf(x):
if x < lower:
return 0
if x > upper:
x = upper
return integrate.quad(pdf, lower, x)[0]
self.__comp_cdf = comp_pdf
if out is None:
out = np.empty(z.shape, dtype=float)
return comp_pdf(z, out=out)
[docs] def pm1(self, z, out=None):
r"""
Returns the first moment of the density function, i.e.:
.. math::
K_1(z) = \int_{-\infty}^z z K(t) dt
"""
z = np.asfarray(z)
try:
comp_pm1 = self.__comp_pm1
except AttributeError:
lower = self.lower
upper = self.upper
def pm1(x):
return x * self.pdf(np.atleast_1d(x))
@make_ufunc()
def comp_pm1(x):
if x <= lower:
return 0
if x > upper:
x = upper
return integrate.quad(pm1, lower, x)[0]
self.__comp_pm1 = comp_pm1
if out is None:
out = np.empty(z.shape, dtype=float)
return comp_pm1(z, out=out)
[docs] def pm2(self, z, out=None):
r"""
Returns the second moment of the density function, i.e.:
.. math::
K_2(z) = \int_{-\infty}^z z^2 K(t) dt
"""
z = np.asfarray(z)
try:
comp_pm2 = self.__comp_pm2
except AttributeError:
lower = self.lower
upper = self.upper
def pm2(x):
return x * x * self.pdf(np.atleast_1d(x))
@make_ufunc()
def comp_pm2(x):
if x <= lower:
return 0
if x > upper:
x = upper
return integrate.quad(pm2, lower, x)[0]
self.__comp_pm2 = comp_pm2
if out is None:
out = np.empty(z.shape, dtype=float)
return comp_pm2(z, out=out)
[docs] def fft(self, z, out=None):
"""
FFT of the kernel on the points of ``z``. The points will always be provided as a grid with :math:`2^n` points,
representing the whole frequency range to be explored. For convenience, the second half of the points will be
provided as negative values.
"""
z = np.asfarray(z)
t_star = 2*np.pi/(z[1]-z[0])**2 / len(z)
dz = t_star * (z[1] - z[0])
return fftpack.fft(self(z * t_star) * dz).real
[docs] def dct(self, z, out=None):
r"""
DCT of the kernel on the points of ``z``. The points will always be provided as a grid with :math:`2^n` points,
representing the whole frequency range to be explored.
"""
z = np.asfarray(z)
a1 = z[1] - z[0]
gp = (z / a1 + 0.5) * np.pi / (len(z) * a1)
return fftpack.dct(self(gp) * (gp[1] - gp[0])).real
[docs]class normal_kernel1d(Kernel1D):
"""
1D normal density kernel with extra integrals for 1D bounded kernel estimation.
"""
[docs] def pdf(self, z, out=None):
r"""
Return the probability density of the function. The formula used is:
.. math::
\phi(z) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}
:param ndarray xs: Array of any shape
:returns: an array of shape identical to ``xs``
"""
return kernels_imp.norm1d_pdf(z, out)
def _pdf(self, z, out=None):
"""
Full-python implementation of :py:func:`normal_kernel1d.pdf`
"""
z = np.asarray(z)
if out is None:
out = np.empty(z.shape, dtype=z.dtype)
np.multiply(z, z, out)
out *= -0.5
np.exp(out, out)
out /= S2PI
return out
__call__ = pdf
[docs] def fft(self, z, out=None):
"""
Returns the FFT of the normal distribution
"""
z = np.asfarray(z)
out = np.multiply(z, z, out)
out *= -0.5
np.exp(out, out)
return out
[docs] def dct(self, z, out=None):
"""
Returns the DCT of the normal distribution
"""
z = np.asfarray(z)
out = np.multiply(z, z, out)
out *= -0.5
np.exp(out, out)
return out
[docs] def cdf(self, z, out=None):
r"""
Cumulative density of probability. The formula used is:
.. math::
\text{cdf}(z) \triangleq \int_{-\infty}^z \phi(z)
dz = \frac{1}{2}\text{erf}\left(\frac{z}{\sqrt{2}}\right) + \frac{1}{2}
"""
return kernels_imp.norm1d_cdf(z, out)
def _cdf(self, z, out=None):
"""
Full-python implementation of :py:func:`normal_kernel1d.cdf`
"""
z = np.asarray(z)
if out is None:
out = np.empty(z.shape, dtype=z.dtype)
np.divide(z, S2, out)
erf(out, out)
out *= 0.5
out += 0.5
return out
[docs] def pm1(self, z, out=None):
r"""
Partial moment of order 1:
.. math::
\text{pm1}(z) \triangleq \int_{-\infty}^z z\phi(z) dz
= -\frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}
"""
return kernels_imp.norm1d_pm1(z, out)
def _pm1(self, z, out=None):
"""
Full-python implementation of :py:func:`normal_kernel1d.pm1`
"""
z = np.asarray(z)
if out is None:
out = np.empty(z.shape, dtype=z.dtype)
np.multiply(z, z, out)
out *= -0.5
np.exp(out, out)
out /= -S2PI
return out
[docs] def pm2(self, z, out=None):
r"""
Partial moment of order 2:
.. math::
\text{pm2}(z) \triangleq \int_{-\infty}^z z^2\phi(z) dz
= \frac{1}{2}\text{erf}\left(\frac{z}{2}\right) - \frac{z}{\sqrt{2\pi}}
e^{-\frac{z^2}{2}} + \frac{1}{2}
"""
return kernels_imp.norm1d_pm2(z, out)
def _pm2(self, z, out=None):
"""
Full-python implementation of :py:func:`normal_kernel1d.pm2`
"""
z = np.asarray(z, dtype=float)
if out is None:
out = np.empty(z.shape)
np.divide(z, S2, out)
erf(out, out)
out /= 2
if z.shape:
zz = np.isfinite(z)
sz = z[zz]
out[zz] -= sz * np.exp(-0.5 * sz * sz) / S2PI
elif np.isfinite(z):
out -= z * np.exp(-0.5 * z * z) / S2PI
out += 0.5
return out
[docs]class normal_kernel(object):
"""
Returns a function-object for the PDF of a Normal kernel of variance
identity and average 0 in dimension ``dim``.
"""
def __new__(klass, dim):
"""
The __new__ method will automatically select :py:class:`normal_kernel1d` if dim is 1.
"""
if dim == 1:
return normal_kernel1d()
return object.__new__(klass, dim)
def __init__(self, dim):
self.factor = 1 / np.sqrt(2 * np.pi) ** dim
[docs] def pdf(self, xs):
"""
Return the probability density of the function.
:param ndarray xs: Array of shape (D,N) where D is the dimension of the kernel
and N the number of points.
:returns: an array of shape (N,) with the density on each point of ``xs``
"""
xs = np.atleast_2d(xs)
return self.factor * np.exp(-0.5 * np.sum(xs * xs, axis=0))
__call__ = pdf
[docs]class tricube(Kernel1D):
r"""
Return the kernel corresponding to a tri-cube distribution, whose expression is.
The tri-cube function is given by:
.. math::
f_r(x) = \left\{\begin{array}{ll}
\left(1-|x|^3\right)^3 & \text{, if } x \in [-1;1]\\
0 & \text{, otherwise}
\end{array}\right.
As :math:`f_r` is not a probability and is not of variance 1, we use a normalized function:
.. math::
f(x) = a b f_r(ax)
a = \sqrt{\frac{35}{243}}
b = \frac{70}{81}
"""
def pdf(self, z, out=None):
return kernels_imp.tricube_pdf(z, out)
__call__ = pdf
upper = 1. / _kernels_py.tricube_width
lower = -upper
cut = upper
[docs] def cdf(self, z, out=None):
r"""
CDF of the distribution:
.. math::
\text{cdf}(x) = \left\{\begin{array}{ll}
\frac{1}{162} {\left(60 (ax)^{7} - 7 {\left(2 (ax)^{10} + 15 (ax)^{4}\right)}
\mathrm{sgn}\left(ax\right) + 140 ax + 81\right)} & \text{, if}x\in[-1/a;1/a]\\
0 & \text{, if} x < -1/a \\
1 & \text{, if} x > 1/a
\end{array}\right.
"""
return kernels_imp.tricube_cdf(z, out)
[docs] def pm1(self, z, out=None):
r"""
Partial moment of order 1:
.. math::
\text{pm1}(x) = \left\{\begin{array}{ll}
\frac{7}{3564a} {\left(165 (ax)^{8} - 8 {\left(5 (ax)^{11} + 33 (ax)^{5}\right)}
\mathrm{sgn}\left(ax\right) + 220 (ax)^{2} - 81\right)}
& \text{, if} x\in [-1/a;1/a]\\
0 & \text{, otherwise}
\end{array}\right.
"""
return kernels_imp.tricube_pm1(z, out)
[docs] def pm2(self, z, out=None):
r"""
Partial moment of order 2:
.. math::
\text{pm2}(x) = \left\{\begin{array}{ll}
\frac{35}{486a^2} {\left(4 (ax)^{9} + 4 (ax)^{3} - {\left((ax)^{12} + 6 (ax)^{6}\right)}
\mathrm{sgn}\left(ax\right) + 1\right)} & \text{, if} x\in[-1/a;1/a] \\
0 & \text{, if } x < -1/a \\
1 & \text{, if } x > 1/a
\end{array}\right.
"""
return kernels_imp.tricube_pm2(z, out)
[docs]class Epanechnikov(Kernel1D):
r"""
1D Epanechnikov density kernel with extra integrals for 1D bounded kernel estimation.
"""
[docs] def pdf(self, xs, out=None):
r"""
The PDF of the kernel is usually given by:
.. math::
f_r(x) = \left\{\begin{array}{ll}
\frac{3}{4} \left(1-x^2\right) & \text{, if} x \in [-1:1]\\
0 & \text{, otherwise}
\end{array}\right.
As :math:`f_r` is not of variance 1 (and therefore would need adjustments for
the bandwidth selection), we use a normalized function:
.. math::
f(x) = \frac{1}{\sqrt{5}}f\left(\frac{x}{\sqrt{5}}\right)
"""
return kernels_imp.epanechnikov_pdf(xs, out)
__call__ = pdf
upper = 1./_kernels_py.epanechnikov_width
lower = -upper
cut = upper
[docs] def cdf(self, xs, out=None):
r"""
CDF of the distribution. The CDF is defined on the interval :math:`[-\sqrt{5}:\sqrt{5}]` as:
.. math::
\text{cdf}(x) = \left\{\begin{array}{ll}
\frac{1}{2} + \frac{3}{4\sqrt{5}} x - \frac{3}{20\sqrt{5}}x^3
& \text{, if } x\in[-\sqrt{5}:\sqrt{5}] \\
0 & \text{, if } x < -\sqrt{5} \\
1 & \text{, if } x > \sqrt{5}
\end{array}\right.
"""
return kernels_imp.epanechnikov_cdf(xs, out)
[docs] def pm1(self, xs, out=None):
r"""
First partial moment of the distribution:
.. math::
\text{pm1}(x) = \left\{\begin{array}{ll}
-\frac{3\sqrt{5}}{16}\left(1-\frac{2}{5}x^2+\frac{1}{25}x^4\right)
& \text{, if } x\in[-\sqrt{5}:\sqrt{5}] \\
0 & \text{, otherwise}
\end{array}\right.
"""
return kernels_imp.epanechnikov_pm1(xs, out)
[docs] def pm2(self, xs, out=None):
r"""
Second partial moment of the distribution:
.. math::
\text{pm2}(x) = \left\{\begin{array}{ll}
\frac{5}{20}\left(2 + \frac{1}{\sqrt{5}}x^3 - \frac{3}{5^{5/2}}x^5 \right)
& \text{, if } x\in[-\sqrt{5}:\sqrt{5}] \\
0 & \text{, if } x < -\sqrt{5} \\
1 & \text{, if } x > \sqrt{5}
\end{array}\right.
"""
return kernels_imp.epanechnikov_pm2(xs, out)
[docs]class Epanechnikov_order4(Kernel1D):
r"""
Order 4 Epanechnikov kernel. That is:
.. math::
K_{[4]}(x) = \frac{3}{2} K(x) + \frac{1}{2} x K'(x) = -\frac{15}{8}x^2+\frac{9}{8}
where :math:`K` is the non-normalized Epanechnikov kernel.
"""
upper = 1
lower = -upper
cut = upper
def pdf(self, xs, out=None):
return kernels_imp.epanechnikov_o4_pdf(xs, out)
__call__ = pdf
def cdf(self, xs, out=None):
return kernels_imp.epanechnikov_o4_cdf(xs, out)
def pm1(self, xs, out=None):
return kernels_imp.epanechnikov_o4_pm1(xs, out)
def pm2(self, xs, out=None):
return kernels_imp.epanechnikov_o4_pm2(xs, out)
[docs]class normal_order4(Kernel1D):
r"""
Order 4 Normal kernel. That is:
.. math::
\phi_{[4]}(x) = \frac{3}{2} \phi(x) + \frac{1}{2} x \phi'(x) = \frac{1}{2}(3-x^2)\phi(x)
where :math:`\phi` is the normal kernel.
"""
lower = -np.inf
upper = np.inf
cut = 3.
def pdf(self, xs, out=None):
return kernels_imp.normal_o4_pdf(xs, out)
__call__ = pdf
def cdf(self, xs, out=None):
return kernels_imp.normal_o4_cdf(xs, out)
def pm1(self, xs, out=None):
return kernels_imp.normal_o4_pm1(xs, out)
def pm2(self, xs, out=None):
return kernels_imp.normal_o4_pm2(xs, out)
kernels1D = [normal_kernel1d, tricube, Epanechnikov, Epanechnikov_order4, normal_order4]
kernelsnD = [normal_kernel]