# Source code for pyqt_fit.kde_methods

r"""
:Author: Pierre Barbier de Reuille <pierre.barbierdereuille@gmail.com>

This module contains a set of methods to compute univariate KDEs. See the
objects in the :py:mod:pyqt_fit.kde module for more details on these methods.

These methods provide various variations on :math:\hat{K}(x;X,h,L,U), the
modified kernel evaluated on the point :math:x based on the estimation points
:math:X, a bandwidth :math:h and on the domain :math:[L,U].

The definitions of the methods rely on the following definitions:

.. math::

\begin{array}{rcl}
a_0(l,u) &=& \int_l^u K(z) dz\\
a_1(l,u) &=& \int_l^u zK(z) dz\\
a_2(l,u) &=& \int_l^u z^2K(z) dz
\end{array}

These definitions correspond to:

- :math:a_0(l,u) -- The partial cumulative distribution function
- :math:a_1(l,u) -- The partial first moment of the distribution. In
particular, :math:a_1(-\infty, \infty) is the mean of the kernel (i.e. and
should be 0).
- :math:a_2(l,u) -- The partial second moment of the distribution. In
particular, :math:a_2(-\infty, \infty) is the variance of the kernel (i.e.
which should be close to 1, unless using higher order kernel).

References:

.. [1] Jones, M. C. 1993. Simple boundary correction for kernel density
estimation. Statistics and Computing 3: 135--146.
"""

from __future__ import division, absolute_import, print_function
import numpy as np
from scipy import fftpack, integrate, optimize
from .compat import irange
from .utils import make_ufunc, namedtuple, numpy_trans_idx
from .binning import fast_bin

[docs]def generate_grid(kde, N=None, cut=None):
r"""
Helper method returning a regular grid on the domain of the KDE.

:param KDE1D kde: Object describing the KDE computation. The object must
have been fitted!
:param int N: Number of points in the grid
:param float cut: for unbounded domains, how far past the maximum should
the grid extend to, in term of KDE bandwidth

:return: A vector of N regularly spaced points
"""
if N is None:
N = 2 ** 10
if cut is None:
cut = kde.kernel.cut
if kde.lower == -np.inf:
lower = np.min(kde.xdata) - cut * kde.bandwidth
else:
lower = kde.lower
if kde.upper == np.inf:
upper = np.max(kde.xdata) + cut * kde.bandwidth
else:
upper = kde.upper
return np.linspace(lower, upper, N)

[docs]def compute_bandwidth(kde):
"""
Compute the bandwidth and covariance for the model, based of its xdata attribute
"""
if kde.bandwidth_function:
bw = float(kde.bandwidth_function(kde.xdata, model=kde))
cov = bw * bw
elif kde.covariance_function:
cov = float(kde.covariance_function(kde.xdata, model=kde))
bw = np.sqrt(cov)
else:
return kde.bandwidth, kde.covariance
return bw, cov

[docs]class KDE1DMethod(object):
"""
Base class providing a default grid method and a default method for
unbounded evaluation of the PDF and CDF. It also provides default methods
for the other metrics, based on PDF and CDF calculations.

:Note:
- It is expected that all grid methods will return the same grid if
used with the same arguments.
- It is fair to assume all array-like arguments will be at least 1D
arrays.

"""

name = 'unbounded'

[docs]    def pdf(self, kde, points, out):
"""
Compute the PDF of the estimated distribution.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param ndarray points: Points to evaluate the distribution on
:param ndarray out: Result object. If must have the same shapes as points
:rtype: ndarray
:return: Returns the out variable, updated with the PDF.
:Default: Direct implementation of the formula for unbounded pdf
computation.
"""
xdata = kde.xdata
points = points[..., np.newaxis]

bw = kde.bandwidth * kde.lambdas

z = (points - xdata) / bw

kernel = kde.kernel

terms = kernel(z)

terms *= kde.weights / bw

terms.sum(axis=-1, out=out)
out /= kde.total_weights

return out

[docs]    def fit(self, kde):
"""
Method called by the KDE1D object right after fitting to allow for
one-time calculation.

:param pyqt_fit.kde.KDE1D kde: KDE object being fitted
:Default: Compute the bandwidth and covariance if specified as functions
"""
kde.compute_bandwidth()

[docs]    def __call__(self, kde, points, out):
"""
Call the :py:meth:pdf method.
"""
return self.pdf(kde, points, out)

[docs]    def cdf(self, kde, points, out):
r"""
Compute the CDF of the estimated distribution, defined as:

.. math::

cdf(x) = P(X \leq x) = \int_l^x p(t) dt

where :math:l is the lower bound of the distribution domain and
:math:p the density of probability

:param pyqt_fit.kde.KDE1D kde: KDE object
:param ndarray points: Points to evaluate the CDF on
:param ndarray out: Result object. If must have the same shapes as points
:rtype: ndarray
:return: Returns the out variable, updated with the CDF.
:Default: Direct implementation of the formula for unbounded CDF
computation.
"""
xdata = kde.xdata
points = points[..., np.newaxis]
bw = kde.bandwidth * kde.lambdas

z = (points - xdata) / bw

kernel = kde.kernel

terms = kernel.cdf(z)
terms *= kde.weights

terms.sum(axis=-1, out=out)
out /= kde.total_weights

return out

[docs]    def icdf(self, kde, points, out):
r"""
Compute the inverse cumulative distribution (quantile) function,
defined as:

.. math::

icdf(p) = \inf\left\{x\in\mathbb{R} : cdf(x) \geq p\right\}

:param pyqt_fit.kde.KDE1D kde: KDE object
:param ndarray points: Points to evaluate the iCDF on
:param ndarray out: Result object. If must have the same shapes as points
:rtype: ndarray
:return: Returns the out variable, updated with the iCDF.
:Default: First approximate the result using linear interpolation on
the CDF and refine the result numerically using the Newton method.
"""
xs, ys = self.cdf_grid(kde)
coarse_result = np.interp(points, ys, xs, kde.lower, kde.upper)
lower = kde.lower
upper = kde.upper
cdf = self.cdf
pdf_out = np.empty(1, dtype=float)
def pdf(x):
if x <= lower:
return 0
if x >= upper:
return 0
return self.pdf(kde, np.atleast_1d(x), pdf_out)

@make_ufunc()
def find_inverse(p, approx):
if p > 1-1e-10:
return upper
if p < 1e-10:
return lower
if approx >= xs[-1] or approx <= xs[0]:
return approx
cdf_out = np.empty(1, dtype=float)
def f(x):
if x <= lower:
return -p
elif x >= upper:
return 1-p
return cdf(kde, np.atleast_1d(x), cdf_out) - p
return optimize.newton(f, approx, fprime=pdf, tol=1e-6)

return find_inverse(points, coarse_result, out=out)

[docs]    def sf(self, kde, points, out):
r"""
Compute the survival function, defined as:

.. math::

sf(x) = P(X \geq x) = \int_x^u p(t) dt = 1 - cdf(x)

:param pyqt_fit.kde.KDE1D kde: KDE object
:param ndarray points: Points to evaluate the survival function on
:param ndarray out: Result object. If must have the same shapes as points
:rtype: ndarray
:return: Returns the out variable, updated with the survival function.
:Default: Compute explicitly :math:1 - cdf(x)
"""
self.cdf(kde, points, out)
out -= 1
out *= -1
return out

[docs]    def isf(self, kde, points, out):
r"""
Compute the inverse survival function, defined as:

.. math::

isf(p) = \sup\left\{x\in\mathbb{R} : sf(x) \leq p\right\}

:param pyqt_fit.kde.KDE1D kde: KDE object
:param ndarray points: Points to evaluate the iSF on
:param ndarray out: Result object. If must have the same shapes as points
:rtype: ndarray
:return: Returns the out variable, updated with the inverse survival function.
:Default: Compute :math:icdf(1-p)
"""
return self.icdf(kde, 1-points, out)

[docs]    def hazard(self, kde, points, out):
r"""
Compute the hazard function evaluated on the points.

The hazard function is defined as:

.. math::

h(x) = \frac{p(x)}{sf(x)}

where :math:p(x) is the probability density function and
:math:sf(x) is the survival function.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param ndarray points: Points to evaluate the hazard function on
:param ndarray out: Result object. If must have the same shapes as points
:rtype: ndarray
:return: Returns the out variable, updated with the hazard function
:Default: Compute explicitly :math:pdf(x) / sf(x)
"""
self.pdf(kde, points, out=out)
sf = np.empty(out.shape, dtype=out.dtype)
self.sf(kde, points, sf)
sf[sf < 0] = 0 # Some methods can produce negative sf
out /= sf
return out

[docs]    def cumhazard(self, kde, points, out):
r"""
Compute the cumulative hazard function evaluated on the points.

The hazard function is defined as:

.. math::

ch(x) = \int_l^x h(t) dt = -\ln sf(x)

where :math:l is the lower bound of the domain, :math:h the hazard
function and :math:sf the survival function.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param ndarray points: Points to evaluate the cumulative hazard function on
:param ndarray out: Result object. If must have the same shapes as points
:rtype: ndarray
:return: Returns the out variable, updated with the cumulative hazard function
:Default: Compute explicitly :math:-\ln sf(x)
"""
self.sf(kde, points, out)
out[out < 0] = 0 # Some methods can produce negative sf
np.log(out, out=out)
out *= -1
return out

[docs]    def grid(self, kde, N=None, cut=None):
"""
Evaluate the PDF of the distribution on a regular grid with at least
N elements.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param int N: minimum number of element in the returned grid. Most
methods will want to round it to the next power of 2.
:param float cut: for unbounded domains, how far from the last data
point should the grid go, as a fraction of the bandwidth.
:rtype: (ndarray, ndarray)
:returns: The array of positions the PDF has been estimated on, and the
estimations.
:Default: Evaluate :math:pdf(x) on a grid generated using
:py:func:generate_grid
"""
N = self.grid_size(N)
g = generate_grid(kde, N, cut)
out = np.empty(g.shape, dtype=float)
return g, self.pdf(kde, g, out)

[docs]    def cdf_grid(self, kde, N=None, cut=None):
"""
Evaluate the CDF of the distribution on a regular grid with at least
N elements.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param int N: minimum number of element in the returned grid. Most
methods will want to round it to the next power of 2.
:param float cut: for unbounded domains, how far from the last data
point should the grid go, as a fraction of the bandwidth.
:rtype: (ndarray, ndarray)
:returns: The array of positions the CDF has been estimated on, and the
estimations.
:Default: Evaluate :math:cdf(x) on a grid generated using
:py:func:generate_grid
"""
N = self.grid_size(N)
g = generate_grid(kde, N, cut)
out = np.empty(g.shape, dtype=float)
return g, self.cdf(kde, g, out)

[docs]    def icdf_grid(self, kde, N=None, cut=None):
"""
Compute the inverse cumulative distribution (quantile) function on
a grid.

:Note: The default implementation is not as good an approximation as
the plain icdf default method.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param int N: minimum number of element in the returned grid. Most
methods will want to round it to the next power of 2.
:param float cut: for unbounded domains, how far from the last data
point should the grid go, as a fraction of the bandwidth.
:rtype: (ndarray, ndarray)
:returns: The array of positions the CDF has been estimated on, and the
estimations.
:Default: Linear interpolation of the inverse CDF on a grid
"""
xs, ys = self.cdf_grid(kde, N, cut)
N = len(xs)
points = np.linspace(0, 1, N)
icdf = np.interp(points, ys, xs, kde.lower, kde.upper)
return points, icdf

[docs]    def sf_grid(self, kde, N=None, cut=None):
r"""
Compute the survival function on a grid.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param int N: minimum number of element in the returned grid. Most
methods will want to round it to the next power of 2.
:param float cut: for unbounded domains, how far from the last data
point should the grid go, as a fraction of the bandwidth.
:rtype: (ndarray, ndarray)
:returns: The array of positions the survival function has been
estimated on, and the estimations.
:Default: Compute explicitly :math:1 - cdf(x)
"""
points, out = self.cdf_grid(kde, N, cut)
out -= 1
out *= -1
return points, out

[docs]    def isf_grid(self, kde, N=None, cut=None):
"""
Compute the inverse survival function on a grid.

:Note: The default implementation is not as good an approximation as
the plain isf default method.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param int N: minimum number of element in the returned grid. Most
methods will want to round it to the next power of 2.
:param float cut: for unbounded domains, how far from the last data
point should the grid go, as a fraction of the bandwidth.
:rtype: (ndarray, ndarray)
:returns: The array of positions the CDF has been estimated on, and the
estimations.
:Default: Linear interpolation of the inverse survival function on a grid
"""
xs, ys = self.sf_grid(kde, N, cut)
N = len(xs)
points = np.linspace(0, 1, N)
isf = np.interp(points, ys[::-1], xs[::-1], kde.upper, kde.lower)
return points, isf

[docs]    def hazard_grid(self, kde, N=None, cut=None):
r"""
Compute the hazard function on a grid.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param int N: minimum number of element in the returned grid. Most
methods will want to round it to the next power of 2.
:param float cut: for unbounded domains, how far from the last data
point should the grid go, as a fraction of the bandwidth.
:rtype: (ndarray, ndarray)
:returns: The array of positions the hazard function has been
estimated on, and the estimations.
:Default: Compute explicitly :math:pdf(x) / sf(x)
"""
points, out = self.grid(kde, N, cut)
_, sf = self.sf_grid(kde, N, cut)
sf[sf < 0] = 0 # Some methods can produce negative sf
out /= sf
return points, out

[docs]    def cumhazard_grid(self, kde, N=None, cut=None):
r"""
Compute the hazard function on a grid.

:param pyqt_fit.kde.KDE1D kde: KDE object
:param int N: minimum number of element in the returned grid. Most
methods will want to round it to the next power of 2.
:param float cut: for unbounded domains, how far from the last data
point should the grid go, as a fraction of the bandwidth.
:rtype: (ndarray, ndarray)
:returns: The array of positions the hazard function has been
estimated on, and the estimations.
:Default: Compute explicitly :math:-\ln sf(x)
"""
points, out = self.sf_grid(kde, N, cut)
out[out < 0] = 0 # Some methods can produce negative sf
np.log(out, out=out)
out *= -1
return points, out

def __str__(self):
"""
Return the name of the method
"""
return self.name

[docs]    def numeric_cdf(self, kde, points, out):
"""
Provide a numeric approximation of the CDF based on integrating the pdf
using :py:func:scipy.integrate.quad.
"""
pts = points.ravel()

pts[pts < kde.lower] = kde.lower
pts[pts > kde.upper] = kde.upper

ix = pts.argsort()

sp = pts[ix]

pdf_out = np.empty((1,), dtype=float)
def pdf(x):
return self.pdf(kde, np.array([x]), pdf_out)

@make_ufunc()
def comp_cdf(i):
low = kde.lower if i == 0 else sp[i-1]

parts = np.empty(sp.shape, dtype=float)
comp_cdf(np.arange(len(sp)), out=parts)

ints = parts.cumsum()

out.put(ix, ints)
return out

[docs]    def numeric_cdf_grid(self, kde, N=None, cut=None):
"""
Compute the CDF on a grid using a trivial, but fast, numeric
integration of the pdf.
"""
pts, pdf = self.grid(kde, N, cut)
cdf = integrate.cumtrapz(pdf, pts, initial=0)
return pts, cdf

def grid_size(self, N=None):
if N is None:
return 2**10
return N

unbounded = KDE1DMethod()

[docs]class RenormalizationMethod(KDE1DMethod):
r"""
This method consists in using the normal kernel method, but renormalize
to only take into account the part of the kernel within the domain of the
density [1]_.

The kernel is then replaced with:

.. math::

\hat{K}(x;X,h,L,U) \triangleq \frac{1}{a_0(u,l)} K(z)

where:

.. math::

"""

name = 'renormalization'

def pdf(self, kde, points, out):
if not kde.bounded:
return KDE1DMethod.pdf(self, kde, points, out)

xdata = kde.xdata
points = points[..., np.newaxis]

bw = kde.bandwidth * kde.lambdas

l = (points - kde.lower) / bw
u = (points - kde.upper) / bw
z = (points - xdata) / bw

kernel = kde.kernel

a1 = (kernel.cdf(l) - kernel.cdf(u))

terms = kernel(z) * ((kde.weights / bw) / a1)

terms.sum(axis=-1, out=out)
out /= kde.total_weights

return out

def cdf(self, kde, points, out):
if not kde.bounded:
return KDE1DMethod.cdf(self, kde, points, out)
return self.numeric_cdf(kde, points, out)

def cdf_grid(self, kde, N=None, cut=None):
if N is None:
N = 2**10
if not kde.bounded or N <= 2**11:
return KDE1DMethod.cdf_grid(self, kde, N, cut)
return self.numeric_cdf_grid(kde, N, cut)

renormalization = RenormalizationMethod()

[docs]class ReflectionMethod(KDE1DMethod):
r"""
This method consist in simulating the reflection of the data left and
right of the boundaries. If one of the boundary is infinite, then the
data is not reflected in that direction. To this purpose, the kernel is
replaced with:

.. math::

\hat{K}(x; X, h, L, U) \triangleq K(z)
+ K\left(\frac{x+X-2L}{h}\right)
+ K\left(\frac{x+X-2U}{h}\right)

where:

.. math::

z = \frac{x-X}{h}

See the :py:mod:pyqt_fit.kde_methods for a description of the various
symbols.

When computing grids, if the bandwidth is constant, the result is computing
using CDT.
"""

name = 'reflection'

def pdf(self, kde, points, out):
if not kde.bounded:
return KDE1DMethod.pdf(self, kde, points, out)

xdata = kde.xdata
points = points[..., np.newaxis]

# Make sure points are between the bounds, with reflection if needed
if any(points < kde.lower) or any(points > kde.upper):
span = kde.upper - kde.lower
points = points - (kde.lower + span)
points %= 2*span
points -= kde.lower + span
points = np.abs(points)

bw = kde.bandwidth * kde.lambdas

z = (points - xdata) / bw
z1 = (points + xdata) / bw
L = kde.lower
U = kde.upper

kernel = kde.kernel

terms = kernel(z)

if L > -np.inf:
terms += kernel(z1 - (2 * L / bw))

if U < np.inf:
terms += kernel(z1 - (2 * U / bw))

terms *= kde.weights / bw
terms.sum(axis=-1, out=out)
out /= kde.total_weights

return out

def cdf(self, kde, points, out):
if not kde.bounded:
return KDE1DMethod.cdf(self, kde, points, out)

xdata = kde.xdata
points = points[..., np.newaxis]

# Make sure points are between the bounds, with reflection if needed
if any(points < kde.lower) or any(points > kde.upper):
span = kde.upper - kde.lower
points = points - (kde.lower + span)
points %= 2*span
points -= kde.lower + span
points = np.abs(points)

bw = kde.bandwidth * kde.lambdas

z = (points - xdata) / bw
z1 = (points + xdata) / bw
L = kde.lower
U = kde.upper

kernel = kde.kernel

terms = kernel.cdf(z)

if L > -np.inf:
terms -= kernel.cdf((L - xdata) / bw) # Remove the truncated part on the left
terms += kernel.cdf(z1 - (2 * L / bw)) # Add the reflected part
terms -= kernel.cdf((xdata - L) / bw) # Remove the truncated part from the reflection

if U < np.inf:
terms += kernel.cdf(z1 - (2 * U / bw)) # Add the reflected part

terms *= kde.weights
terms.sum(axis=-1, out=out)
out /= kde.total_weights

return out

def grid(self, kde, N=None, cut=None):
"""
DCT-based estimation of KDE estimation, i.e. with reflection boundary
conditions. This works only for fixed bandwidth (i.e. lambdas = 1) and
gaussian kernel.

For open domains, the grid is taken with 3 times the bandwidth as extra
space to remove the boundary problems.
"""
if kde.lambdas.shape:
return KDE1DMethod.grid(self, kde, N, cut)

bw = kde.bandwidth * kde.lambdas
data = kde.xdata
N = self.grid_size(N)

if cut is None:
cut = kde.kernel.cut

if kde.lower == -np.inf:
lower = np.min(data) - cut * kde.bandwidth
else:
lower = kde.lower
if kde.upper == np.inf:
upper = np.max(data) + cut * kde.bandwidth
else:
upper = kde.upper

R = upper - lower

# Histogram the data to get a crude first approximation of the density
weights = kde.weights
if not weights.shape:
weights = None

DataHist, mesh = fast_bin(data, lower, upper, N, weights=weights, cyclic=False)

DataHist = DataHist / kde.total_weights
DCTData = fftpack.dct(DataHist, norm=None)

t_star = bw / R
gp = np.arange(N) * np.pi * t_star
smth = kde.kernel.dct(gp)

# Smooth the DCTransformed data using t_star
SmDCTData = DCTData * smth
# Inverse DCT to get density
density = fftpack.idct(SmDCTData, norm=None) / (2 * R)

return mesh, density

def grid_size(self, N=None):
if N is None:
return 2**14
return 2 ** int(np.ceil(np.log2(N)))

reflection = ReflectionMethod()

[docs]class LinearCombinationMethod(KDE1DMethod):
r"""
This method uses the linear combination correction published in [1]_.

The estimation is done with a modified kernel given by:

.. math::

\hat{K}(x;X,h,L,U) \triangleq \frac{a_2(l,u) - a_1(-u, -l) z}{a_2(l,u)a_0(l,u)
- a_1(-u,-l)^2} K(z)

where:

.. math::

"""

name = 'linear combination'

def pdf(self, kde, points, out):
if not kde.bounded:
return KDE1DMethod.pdf(self, kde, points, out)

xdata = kde.xdata
points = np.atleast_1d(points)[..., np.newaxis]

bw = kde.bandwidth * kde.lambdas

l = (kde.lower - points) / bw
u = (kde.upper - points) / bw
z = (points - xdata) / bw

kernel = kde.kernel

a0 = kernel.cdf(u) - kernel.cdf(l)
a1 = kernel.pm1(-l) - kernel.pm1(-u)
a2 = kernel.pm2(u) - kernel.pm2(l)

denom = a2 * a0 - a1 * a1
upper = a2 - a1 * z

upper /= denom
upper *= (kde.weights / bw) * kernel(z)

upper.sum(axis=-1, out=out)
out /= kde.total_weights

return out

def cdf(self, kde, points, out):
if not kde.bounded:
return KDE1DMethod.cdf(self, kde, points, out)
return self.numeric_cdf(kde, points, out)

def cdf_grid(self, kde, N=None, cut=None):
if N is None:
N = 2**10
if not kde.bounded or N <= 2**11:
return KDE1DMethod.cdf_grid(self, kde, N, cut)
return self.numeric_cdf_grid(kde, N, cut)

linear_combination = LinearCombinationMethod()

[docs]class CyclicMethod(KDE1DMethod):
r"""
This method assumes cyclic boundary conditions and works only for closed
boundaries.

The estimation is done with a modified kernel given by:

.. math::

\hat{K}(x; X, h, L, U) \triangleq K(z)
+ K\left(z - \frac{U-L}{h}\right)
+ K\left(z + \frac{U-L}{h}\right)

where:

.. math::

z = \frac{x-X}{h}

When computing grids, if the bandwidth is constant, the result is computing
using FFT.
"""

name = 'cyclic'

def pdf(self, kde, points, out):
if not kde.bounded:
return KDE1DMethod.pdf(self, kde, points, out)
if not kde.closed:
raise ValueError("Cyclic boundary conditions can only be used with "
"closed or un-bounded domains.")

xdata = kde.xdata
points = np.atleast_1d(points)[..., np.newaxis]

# Make sure points are between the bounds
if any(points < kde.lower) or any(points > kde.upper):
points = points - kde.lower
points %= kde.upper - kde.lower
points += kde.lower

bw = kde.bandwidth * kde.lambdas

z = (points - xdata) / bw
L = kde.lower
U = kde.upper

span = (U - L) / bw

kernel = kde.kernel

terms = kernel(z)
terms += kernel(z + span) # Add points to the left
terms += kernel(z - span) # Add points to the right

terms *= kde.weights / bw
terms.sum(axis=-1, out=out)
out /= kde.total_weights

return out

def cdf(self, kde, points, out):
if not kde.bounded:
return KDE1DMethod.cdf(self, kde, points, out)
if not kde.closed:
raise ValueError("Cyclic boundary conditions can only be used with "
"closed or unbounded domains.")

xdata = kde.xdata
points = np.atleast_1d(points)[..., np.newaxis]

# Make sure points are between the bounds
if any(points < kde.lower) or any(points > kde.upper):
points = points - kde.lower
points %= kde.upper - kde.lower
points += kde.lower

bw = kde.bandwidth * kde.lambdas

z = (points - xdata) / bw
L = kde.lower
U = kde.upper

span = (U - L) / bw

kernel = kde.kernel

terms = kernel.cdf(z)
terms -= kernel.cdf((L - xdata) / bw) # Remove the parts left of the lower bound

terms += kernel.cdf(z + span) # Repeat on the left
terms -= kernel.cdf((L - xdata) / bw + span) # Remove parts left of lower bounds

terms += kernel.cdf(z - span) # Repeat on the right

terms *= kde.weights
terms.sum(axis=-1, out=out)
out /= kde.total_weights

return out

def grid(self, kde, N=None, cut=None):
"""
FFT-based estimation of KDE estimation, i.e. with cyclic boundary
conditions. This works only for closed domains, fixed bandwidth
(i.e. lambdas = 1) and gaussian kernel.
"""
if kde.lambdas.shape:
return KDE1DMethod.grid(self, kde, N, cut)
if kde.bounded and not kde.closed:
raise ValueError("Error, cyclic boundary conditions require "
"a closed or un-bounded domain.")
bw = kde.bandwidth * kde.lambdas
data = kde.xdata
N = self.grid_size(N)

lower = kde.lower
upper = kde.upper

if upper == np.inf:
lower = np.min(data) - cut * kde.bandwidth
upper = np.max(data) + cut * kde.bandwidth

R = upper - lower
weights = kde.weights
if not weights.shape:
weights = None

DataHist, mesh = fast_bin(data, lower, upper, N, weights=weights, cyclic=True)
DataHist = DataHist / kde.total_weights
FFTData = np.fft.rfft(DataHist)

t_star = (2 * bw / R)
gp = np.arange(len(FFTData)) * np.pi * t_star
smth = kde.kernel.fft(gp)

SmoothFFTData = FFTData * smth
density = np.fft.irfft(SmoothFFTData, len(DataHist)) / (mesh[1] - mesh[0])
return mesh, density

def cdf_grid(self, kde, N=None, cut=None):
if kde.lambdas.shape:
return KDE1DMethod.cdf_grid(self, kde, N, cut)
if not kde.closed:
raise ValueError("Error, cyclic boundary conditions require "
"a closed domain.")
N = self.grid_size(N)

if N <= 2**12:
return KDE1DMethod.cdf_grid(self, kde, N, cut)
return self.numeric_cdf_grid(kde, N, cut)

def grid_size(self, N=None):
if N is None:
return 2**14
return 2 ** int(np.ceil(np.log2(N)))

cyclic = CyclicMethod()

Transform = namedtuple('Tranform', ['__call__', 'inv', 'Dinv'])

def _inverse(x, out=None):
return np.divide(1, x, out)

LogTransform = Transform(np.log, np.exp, np.exp)
ExpTransform = Transform(np.exp, np.log, _inverse)

def transform_distribution(xs, ys, Dinv, out):
r"""
Transform a distribution into another one by a change a variable.

:param ndarray xs: Evaluation points of the distribution
:param ndarray ys: Distribution value on the points xs
:param func Dinv: Function evaluating the derivative of the inverse transformation  function
:param ndarray out: Array in which to store the result
:rtype: ndarray
:returns: The variable out, updated wih the transformed distribution

Given a random variable :math:X of distribution :math:f_X, the random
variable :math:Y = g(X) has a distribution :math:f_Y given by:

.. math::

f_Y(y) = \left| \frac{1}{g'(g^{-1}(y))} \right| \cdot f_X(g^{-1}(y))

"""
Dinv(xs, out=out)
np.abs(out, out=out)
_inverse(out, out=out)
np.multiply(out, ys, out=out)
return out

[docs]def create_transform(obj, inv=None, Dinv=None):
"""
Create a transform object.

:param fun obj: This can be either simple a function, or a function-object with an 'inv' and/or 'Dinv' attributes
containing the inverse function and its derivative (respectively)
:param fun inv: If provided, inverse of the main function
:param fun Dinv: If provided, derivative of the inverse function
:rtype: Transform
:returns: A transform object with function, inverse and derivative of the inverse

The inverse function must be provided, either as argument or as attribute to the object. The derivative of the
inverse will be estimated numerically if not provided.

:Note: All the functions should accept an out argument to store the result.
"""
if isinstance(obj, Transform):
return obj
fct = obj.__call__
if inv is None:
if not hasattr(obj, 'inv'):
raise AttributeError("Error, transform object must have a 'inv' "
"attribute or you must specify the 'inv' argument")
inv = obj.inv
if Dinv is None:
if hasattr(obj, 'Dinv'):
Dinv = obj.Dinv
else:
@numpy_trans_idx
def Dinv(x):
dx = x * 1e-9
dx[x == 0] = np.min(dx[x != 0])
return (inv(x + dx) - inv(x - dx)) / (2 * dx)
return Transform(fct, inv, Dinv)

class _fakeKDE(object):
def __init__(self, method):
self.method = method.method

[docs]class TransformKDE1DMethod(KDE1DMethod):
r"""
Compute the Kernel Density Estimate of a dataset, transforming it first to
a domain where distances are "more meaningful".

Often, KDE is best estimated in a different domain. This object takes
a KDE1D object (or one compatible), and a transformation function.

Given a random variable :math:X of distribution :math:f_X, the random
variable :math:Y = g(X) has a distribution :math:f_Y given by:

.. math::

f_Y(y) = \left| \frac{1}{g'(g^{-1}(y))} \right| \cdot f_X(g^{-1}(y))

In our term, :math:Y is the random variable the user is interested in,
and :math:X the random variable we can estimate using the KDE. In this
case, :math:g is the transform from :math:Y to :math:X.

So to estimate the distribution on a set of points given in :math:x, we
need a total of three functions:

- Direct function: transform from the original space to the one in
which the KDE will be perform (i.e. :math:g^{-1}: y \mapsto x)
- Invert function: transform from the KDE space to the original one
(i.e. :math:g: x \mapsto y)
- Derivative of the invert function

If the derivative is not provided, it will be estimated numerically.

:param trans: Either a simple function, or a function object with
attributes inv and Dinv to use in case they are not provided as
arguments. The helper :py:func:create_transform will provide numeric
approximation of the derivative if required.
:param method: instance of KDE1DMethod used in the transformed domain.
Default is :py:class:pyqt_fit.kde_methods.KDE1DMethod
:param inv: Invert of the function. If not provided, trans must have
it as attribute.
:param Dinv: Derivative of the invert function.

:Note: all given functions should accept an optional out argument to
get a pre-allocated array to store its result. Also the out
parameter may be one of the input argument.
"""
def __init__(self, trans, method=None, inv=None, Dinv=None):
self.trans = create_transform(trans, inv, Dinv)
if method is None:
method = KDE1DMethod()
self.method = method

def fit(self, kde):
"""
Method called by the KDE1D object right after fitting to allow for
one-time calculation.

This method copy, and transform, the various attributes of the KDE.

:param pyqt_fit.kde.KDE1D kde: KDE object being fitted
"""
fake_kde = _fakeKDE(self)
fake_kde.lower = self.trans(kde.lower)
fake_kde.upper = self.trans(kde.upper)
fake_kde.xdata = self.trans(kde.xdata)

copy_attrs = [ 'weights', 'lambdas', 'kernel'
, 'bandwidth', 'covariance'
, 'bandwidth_function'
, 'covariance_function'
, 'total_weights' ]

for attr in copy_attrs:
setattr(fake_kde, attr, getattr(kde, attr))

# Compute the bandwidth for the fake KDE and update the KDE itself
bw, cov = compute_bandwidth(fake_kde)
fake_kde.bandwidth = kde.bandwidth = bw
fake_kde.covariance = kde.covariance = cov

self.fake_kde = fake_kde

def pdf(self, kde, points, out):
trans = self.trans
pts = trans(points)
self.method(self.fake_kde, pts, out)
return transform_distribution(pts, out, trans.Dinv, out=out)

def grid(self, kde, N=None, cut=None):
xs, ys = self.method.grid(self.fake_kde, N, cut)
trans = self.trans
transform_distribution(xs, ys, trans.Dinv, out=ys)
trans.inv(xs, out=xs)
return xs, ys

def cdf(self, kde, points, out):
return self.method.cdf(self.fake_kde, self.trans(points), out)

def cdf_grid(self, kde, N=None, cut=None):
xs, ys = self.method.cdf_grid(self.fake_kde, N, cut)
self.trans.inv(xs, out=xs)
return xs, ys

def sf(self, kde, points, out):
return self.method.sf(self.fake_kde, self.trans(points), out)

def sf_grid(self, kde, N=None, cut=None):
xs, ys = self.method.sf_grid(self.fake_kde, N, cut)
return self.trans.inv(xs), ys

def icdf(self, kde, points, out):
self.method.icdf(self.fake_kde, points, out)
self.trans.inv(out, out=out)
return out

def icdf_grid(self, kde, N=None, cut=None):
xs, ys = self.method.icdf_grid(self.fake_kde, N, cut)
self.trans.inv(ys, out=ys)
return xs, ys

def isf(self, kde, points, out):
self.method.isf(self.fake_kde, points, out)
self.trans.inv(out, out=out)
return out

def isf_grid(self, kde, N=None, cut=None):
xs, ys = self.method.isf_grid(self.fake_kde, N, cut)
self.trans.inv(ys, out=ys)
return xs, ys

[docs]def transformKDE1D(trans, method=None, inv=None, Dinv=None):
"""
Creates an instance of :py:class:TransformKDE1DMethod
"""
return TransformKDE1DMethod(trans, method, inv, Dinv)

default_method = reflection
`