# -*- coding: utf-8 -*-
"""
Definition of a hierarchy of classes for kernel functions to be used
in convolution, e.g., for data smoothing (low pass filtering) or
firing rate estimation.
Base kernel classes
~~~~~~~~~~~~~~~~~~~
.. autosummary::
:toctree: kernels/
Kernel
SymmetricKernel
Symmetric kernels
~~~~~~~~~~~~~~~~~
.. autosummary::
:toctree: kernels/
RectangularKernel
TriangularKernel
EpanechnikovLikeKernel
GaussianKernel
LaplacianKernel
Asymmetric kernels
~~~~~~~~~~~~~~~~~~
.. autosummary::
:toctree: kernels/
ExponentialKernel
AlphaKernel
Examples
--------
>>> import quantities as pq
>>> kernel1 = GaussianKernel(sigma=100*pq.ms)
>>> kernel2 = ExponentialKernel(sigma=8*pq.mm, invert=True)
:copyright: Copyright 2016 by the Elephant team, see `doc/authors.rst`.
:license: Modified BSD, see LICENSE.txt for details.
"""
from __future__ import division, print_function, unicode_literals
import numpy as np
import quantities as pq
import scipy.special
[docs]class Kernel(object):
r"""
This is the base class for commonly used kernels.
**General definition of kernel:**
A function :math:`K(x, y)` is called a kernel function if
:math:`\int{K(x, y) g(x) g(y) \textrm{d}x \textrm{d}y} \ \geq 0 \quad
\forall g \in L_2`
**Currently implemented kernels are:**
* rectangular
* triangular
* epanechnikovlike
* gaussian
* laplacian
* exponential (asymmetric)
* alpha function (asymmetric)
In neuroscience, a popular application of kernels is in performing
smoothing operations via convolution. In this case, the kernel has the
properties of a probability density, i.e., it is positive and normalized
to one. Popular choices are the rectangular or Gaussian kernels.
Exponential and alpha kernels may also be used to represent the
postynaptic current/potentials in a linear (current-based) model.
Parameters
----------
sigma : pq.Quantity
Standard deviation of the kernel.
invert: bool, optional
If True, asymmetric kernels (e.g., exponential or alpha kernels) are
inverted along the time axis.
Default: False.
Raises
------
TypeError
If `sigma` is not `pq.Quantity`.
If `sigma` is negative.
If `invert` is not `bool`.
"""
def __init__(self, sigma, invert=False):
if not (isinstance(sigma, pq.Quantity)):
raise TypeError("sigma must be a quantity!")
if sigma.magnitude < 0:
raise ValueError("sigma cannot be negative!")
if not isinstance(invert, bool):
raise ValueError("invert must be bool!")
self.sigma = sigma
self.invert = invert
[docs] def __call__(self, t):
"""
Evaluates the kernel at all points in the array `t`.
Parameters
----------
t : pq.Quantity
Vector with the interval on which the kernel is evaluated,
not necessarily a time interval.
Returns
-------
pq.Quantity
Vector with the result of the kernel evaluations.
Raises
------
TypeError
If `t` is not `pq.Quantity`.
If the dimensionality of `t` and :attr:`sigma` are different.
"""
if not (isinstance(t, pq.Quantity)):
raise TypeError("The argument of the kernel callable must be "
"of type quantity!")
if t.dimensionality.simplified != self.sigma.dimensionality.simplified:
raise TypeError("The dimensionality of sigma and the input array "
"to the callable kernel object must be the same. "
"Otherwise a normalization to 1 of the kernel "
"cannot be performed.")
self._sigma_scaled = self.sigma.rescale(t.units)
# A hidden variable _sigma_scaled is introduced here in order to avoid
# accumulation of floating point errors of sigma upon multiple
# usages of the __call__ - function for the same Kernel instance.
return self._evaluate(t)
def _evaluate(self, t):
"""
Evaluates the kernel.
Parameters
----------
t : pq.Quantity
Vector with the interval on which the kernel is evaluated, not
necessarily a time interval.
Returns
-------
pq.Quantity
Vector with the result of the kernel evaluation.
"""
raise NotImplementedError(
"The Kernel class should not be used directly, "
"instead the subclasses for the single kernels.")
[docs] def boundary_enclosing_area_fraction(self, fraction):
"""
Calculates the boundary :math:`b` so that the integral from
:math:`-b` to :math:`b` encloses a certain fraction of the
integral over the complete kernel.
By definition the returned value is hence non-negative, even if the
whole probability mass of the kernel is concentrated over negative
support for inverted kernels.
Parameter
---------
fraction : float
Fraction of the whole area which has to be enclosed.
Returns
-------
pq.Quantity
Boundary of the kernel containing area `fraction` under the
kernel density.
Raises
------
ValueError
If `fraction` was chosen too close to one, such that in
combination with integral approximation errors the calculation of
a boundary was not possible.
"""
self._check_fraction(fraction)
sigma_division = 500 # arbitrary choice
interval = self.sigma / sigma_division
self._sigma_scaled = self.sigma
area = 0
counter = 0
while area < fraction:
area += (self._evaluate((counter + 1) * interval) +
self._evaluate(counter * interval)) * interval / 2
area += (self._evaluate(-1 * (counter + 1) * interval) +
self._evaluate(-1 * counter * interval)) * interval / 2
counter += 1
if(counter > 250000):
raise ValueError("fraction was chosen too close to one such "
"that in combination with integral "
"approximation errors the calculation of a "
"boundary was not possible.")
return counter * interval
def _check_fraction(self, fraction):
"""
Checks the input variable of the method
:attr:`boundary_enclosing_area_fraction` for validity of type and
value.
Parameters
----------
fraction : float or int
Fraction of the area under the kernel function.
Raises
------
TypeError
If `fraction` is neither a float nor an int.
If `fraction` is not in the interval [0, 1).
"""
if not isinstance(fraction, (float, int)):
raise TypeError("`fraction` must be float or integer!")
if not 0 <= fraction < 1:
raise ValueError("`fraction` must be in the interval [0, 1)!")
def median_index(self, t):
"""
Estimates the index of the Median of the kernel.
This parameter is not mandatory for symmetrical kernels but it is
required when asymmetrical kernels have to be aligned at their median.
Parameters
----------
t : pq.Quantity
Vector with the interval on which the kernel is evaluated.
Returns
-------
int
Index of the estimated value of the kernel median.
Notes
-----
The formula in this method using retrieval of the sampling interval
from `t` only works for `t` with equidistant intervals!
The formula calculates the Median slightly wrong by the potentially
ignored probability in the distribution corresponding to lower values
than the minimum in the array `t`.
"""
return np.nonzero(self(t).cumsum() *
(t[len(t) - 1] - t[0]) / (len(t) - 1) >= 0.5)[0].min()
[docs] def is_symmetric(self):
"""
In the case of symmetric kernels, this method is overwritten in the
class `SymmetricKernel`, where it returns True, hence leaving the
here returned value False for the asymmetric kernels.
Returns
-------
bool
True in classes `SymmetricKernel`, `RectangularKernel`,
`TriangularKernel`, `EpanechnikovLikeKernel`, `GaussianKernel`,
and `LaplacianKernel`.
False in classes `Kernel`, `ExponentialKernel`, and `AlphaKernel`.
"""
return False
@property
def min_cutoff(self):
"""
Half width of the kernel.
Returns
-------
float
The returned value varies according to the kernel type.
"""
raise NotImplementedError
[docs]class SymmetricKernel(Kernel):
"""
Base class for symmetric kernels.
"""
[docs] def is_symmetric(self):
return True
[docs]class RectangularKernel(SymmetricKernel):
r"""
Class for rectangular kernels.
.. math::
K(t) = \left\{\begin{array}{ll} \frac{1}{2 \tau}, & |t| < \tau \\\
0, & |t| \geq \tau \end{array} \right.
with :math:`\tau = \sqrt{3} \sigma` corresponding to the half width
of the kernel.
Besides the standard deviation `sigma`, for consistency of interfaces the
parameter `invert` needed for asymmetric kernels also exists without
having any effect in the case of symmetric kernels.
Attributes
----------
min_cutoff : float
"""
@property
def min_cutoff(self):
min_cutoff = np.sqrt(3.0)
return min_cutoff
def _evaluate(self, t):
return (0.5 / (np.sqrt(3.0) * self._sigma_scaled)) * \
(np.absolute(t) < np.sqrt(3.0) * self._sigma_scaled)
[docs] def boundary_enclosing_area_fraction(self, fraction):
self._check_fraction(fraction)
return np.sqrt(3.0) * self.sigma * fraction
[docs]class TriangularKernel(SymmetricKernel):
"""
Class for triangular kernels.
.. math::
K(t) = \\left\\{ \\begin{array}{ll} \\frac{1}{\\tau} (1
- \\frac{|t|}{\\tau}), & |t| < \\tau \\\\
0, & |t| \\geq \\tau \\end{array} \\right.
with :math:`\\tau = \\sqrt{6} \\sigma` corresponding to the half width of
the kernel.
Besides the standard deviation `sigma`, for consistency of interfaces the
parameter `invert` needed for asymmetric kernels also exists without
having any effect in the case of symmetric kernels.
Attributes
----------
min_cutoff : float
"""
@property
def min_cutoff(self):
min_cutoff = np.sqrt(6.0)
return min_cutoff
def _evaluate(self, t):
return (1.0 / (np.sqrt(6.0) * self._sigma_scaled)) * np.maximum(
0.0,
(1.0 - (np.absolute(t) /
(np.sqrt(6.0) * self._sigma_scaled)).magnitude))
[docs] def boundary_enclosing_area_fraction(self, fraction):
self._check_fraction(fraction)
return np.sqrt(6.0) * self.sigma * (1 - np.sqrt(1 - fraction))
[docs]class EpanechnikovLikeKernel(SymmetricKernel):
"""
Class for epanechnikov-like kernels.
.. math::
K(t) = \\left\\{\\begin{array}{ll} (3 /(4 d)) (1 - (t / d)^2),
& |t| < d \\\\
0, & |t| \\geq d \\end{array} \\right.
with :math:`d = \\sqrt{5} \\sigma` being the half width of the kernel.
The Epanechnikov kernel under full consideration of its axioms has a half
width of :math:`\\sqrt{5}`. Ignoring one axiom also the respective kernel
with half width = 1 can be called Epanechnikov kernel [1]_.
However, arbitrary width of this type of kernel is here preferred to be
called 'Epanechnikov-like' kernel.
Besides the standard deviation `sigma`, for consistency of interfaces the
parameter `invert` needed for asymmetric kernels also exists without
having any effect in the case of symmetric kernels.
Attributes
----------
min_cutoff : float
References
----------
.. [1] https://de.wikipedia.org/wiki/Epanechnikov-Kern
"""
@property
def min_cutoff(self):
min_cutoff = np.sqrt(5.0)
return min_cutoff
def _evaluate(self, t):
return (3.0 / (4.0 * np.sqrt(5.0) * self._sigma_scaled)) * np.maximum(
0.0,
1 - (t / (np.sqrt(5.0) * self._sigma_scaled)).magnitude ** 2)
[docs] def boundary_enclosing_area_fraction(self, fraction):
"""
Refer to :func:`Kernel.boundary_enclosing_area_fraction` for the
documentation.
Notes
-----
For Epanechnikov-like kernels, integration of its density within
the boundaries 0 and :math:`b`, and then solving for :math:`b` leads
to the problem of finding the roots of a polynomial of third order.
The implemented formulas are based on the solution of this problem
given in [1]_, where the following 3 solutions are given:
* :math:`u_1 = 1`, solution on negative side;
* :math:`u_2 = \\frac{-1 + i\\sqrt{3}}{2}`, solution for larger
values than zero crossing of the density;
* :math:`u_3 = \\frac{-1 - i\\sqrt{3}}{2}`, solution for smaller
values than zero crossing of the density.
The solution :math:`u_3` is the relevant one for the problem at hand,
since it involves only positive area contributions.
References
----------
.. [1] https://en.wikipedia.org/wiki/Cubic_function
"""
self._check_fraction(fraction)
# Python's complex-operator cannot handle quantities, hence the
# following construction on quantities is necessary:
Delta_0 = complex(1.0 / (5.0 * self.sigma.magnitude**2), 0) / \
self.sigma.units**2
Delta_1 = complex(2.0 * np.sqrt(5.0) * fraction /
(25.0 * self.sigma.magnitude**3), 0) / \
self.sigma.units**3
C = ((Delta_1 + (Delta_1**2.0 - 4.0 * Delta_0**3.0)**(1.0 / 2.0)) /
2.0)**(1.0 / 3.0)
u_3 = complex(-1.0 / 2.0, -np.sqrt(3.0) / 2.0)
b = -5.0 * self.sigma**2 * (u_3 * C + Delta_0 / (u_3 * C))
return b.real
[docs]class GaussianKernel(SymmetricKernel):
"""
Class for gaussian kernels.
.. math::
K(t) = (\\frac{1}{\\sigma \\sqrt{2 \\pi}})
\\exp(-\\frac{t^2}{2 \\sigma^2})
with :math:`\\sigma` being the standard deviation.
Besides the standard deviation `sigma`, for consistency of interfaces the
parameter `invert` needed for asymmetric kernels also exists without
having any effect in the case of symmetric kernels.
Attributes
----------
min_cutoff : float
"""
@property
def min_cutoff(self):
min_cutoff = 3.0
return min_cutoff
def _evaluate(self, t):
return (1.0 / (np.sqrt(2.0 * np.pi) * self._sigma_scaled)) * np.exp(
-0.5 * (t / self._sigma_scaled).magnitude ** 2)
[docs] def boundary_enclosing_area_fraction(self, fraction):
self._check_fraction(fraction)
return self.sigma * np.sqrt(2.0) * scipy.special.erfinv(fraction)
[docs]class LaplacianKernel(SymmetricKernel):
"""
Class for laplacian kernels.
.. math::
K(t) = \\frac{1}{2 \\tau} \\exp(-|\\frac{t}{\\tau}|)
with :math:`\\tau = \\sigma / \\sqrt{2}`.
Besides the standard deviation `sigma`, for consistency of interfaces the
parameter `invert` needed for asymmetric kernels also exists without
having any effect in the case of symmetric kernels.
Attributes
----------
min_cutoff : float
"""
@property
def min_cutoff(self):
min_cutoff = 3.0
return min_cutoff
def _evaluate(self, t):
return (1 / (np.sqrt(2.0) * self._sigma_scaled)) * np.exp(
-(np.absolute(t) * np.sqrt(2.0) / self._sigma_scaled).magnitude)
[docs] def boundary_enclosing_area_fraction(self, fraction):
self._check_fraction(fraction)
return -self.sigma * np.log(1.0 - fraction) / np.sqrt(2.0)
# Potential further symmetric kernels from Wiki Kernels (statistics):
# Quartic (biweight), Triweight, Tricube, Cosine, Logistics, Silverman
[docs]class ExponentialKernel(Kernel):
"""
Class for exponential kernels.
.. math::
K(t) = \\left\\{\\begin{array}{ll} (1 / \\tau) \\exp{(-t / \\tau)},
& t > 0 \\\\
0, & t \\leq 0 \\end{array} \\right.
with :math:`\\tau = \\sigma`.
Attributes
----------
min_cutoff : float
"""
@property
def min_cutoff(self):
min_cutoff = 3.0
return min_cutoff
def _evaluate(self, t):
if not self.invert:
kernel = (t >= 0) * (1. / self._sigma_scaled.magnitude) *\
np.exp((-t / self._sigma_scaled).magnitude) / t.units
elif self.invert:
kernel = (t <= 0) * (1. / self._sigma_scaled.magnitude) *\
np.exp((t / self._sigma_scaled).magnitude) / t.units
return kernel
[docs] def boundary_enclosing_area_fraction(self, fraction):
self._check_fraction(fraction)
return -self.sigma * np.log(1.0 - fraction)
[docs]class AlphaKernel(Kernel):
"""
Class for alpha kernels.
.. math::
K(t) = \\left\\{\\begin{array}{ll} (1 / \\tau^2)
\\ t\\ \\exp{(-t / \\tau)}, & t > 0 \\\\
0, & t \\leq 0 \\end{array} \\right.
with :math:`\\tau = \\sigma / \\sqrt{2}`.
For the alpha kernel an analytical expression for the boundary of the
integral as a function of the area under the alpha kernel function
cannot be given. Hence in this case the value of the boundary is
determined by kernel-approximating numerical integration, inherited
from the `Kernel` class.
Attributes
----------
min_cutoff : float
"""
@property
def min_cutoff(self):
min_cutoff = 3.0
return min_cutoff
def _evaluate(self, t):
if not self.invert:
kernel = (t >= 0) * 2. * (t / self._sigma_scaled**2).magnitude *\
np.exp((
-t * np.sqrt(2.) / self._sigma_scaled).magnitude) / t.units
elif self.invert:
kernel = (t <= 0) * -2. * (t / self._sigma_scaled**2).magnitude *\
np.exp((
t * np.sqrt(2.) / self._sigma_scaled).magnitude) / t.units
return kernel
