src/wafo/kdetools/kernels.py
'''
Created on 15. des. 2016
@author: pab
'''
from abc import ABCMeta, abstractmethod
import warnings
import numpy as np
from numpy import pi, sqrt, exp, percentile
from numpy.fft import fft, ifft
from scipy import optimize
from scipy.special import gamma
from scipy.linalg import sqrtm
from wafo.misc import tranproc # , trangood
from wafo.kdetools.gridding import gridcount
from wafo.dctpack import dct
from wafo.testing import test_docstrings
__all__ = ['Kernel', 'sphere_volume', 'qlevels', 'iqrange', 'percentile']
def _assert(cond, msg):
if not cond:
raise ValueError(msg)
def _assert_warn(cond, msg):
if not cond:
warnings.warn(msg)
# stats = (mu2, R, Rdd) where
# mu2 : 2'nd order moment, i.e.,int(x^2*kernel(x))
# R : integral of squared kernel, i.e., int(kernel(x)^2)
# Rdd : int( (kernel''(x))^2 ).
_stats_epan = (1. / 5, 3. / 5, np.inf)
_stats_biwe = (1. / 7, 5. / 7, 45. / 2)
_stats_triw = (1. / 9, 350. / 429, np.inf)
_stats_rect = (1. / 3, 1. / 2, np.inf)
_stats_tria = (1. / 6, 2. / 3, np.inf)
_stats_lapl = (2, 1. / 4, np.inf)
_stats_logi = (pi ** 2 / 3, 1. / 6, 1 / 42)
_stats_gaus = (1, 1. / (2 * sqrt(pi)), 3. / (8 * sqrt(pi)))
def qlevels(pdf, p=(10, 30, 50, 70, 90, 95, 99, 99.9), xi=(), indexing='xy'):
"""QLEVELS Calculates quantile levels which encloses P% of pdf.
Parameters
----------
pdf: array-like
joint point density function given as array or vector
p : float in range of [0,100] (or sequence of floats)
Percentage to compute which must be between 0 and 100 inclusive.
xi : tuple
input arguments to the pdf, i.e., (x0, x1,...., xn)
indexing : {'xy', 'ij'}, optional
Cartesian ('xy', default) or matrix ('ij') indexing of pdf.
See numpy.meshgrid for more details.
Returns
------
levels: array-like
discrete levels which encloses P% of pdf
QLEVELS numerically integrates PDF by decreasing height and find the
quantile levels which encloses P% of the distribution.
If Xi is unspecified it is assumed that dX0, dX1,..., and dXn is constant.
NB! QLEVELS normalizes the integral of PDF to n/(n+0.001) before
calculating 'levels' in order to reflect the sampling of PDF is finite.
Examples
--------
>>> import wafo.stats as ws
>>> x = np.linspace(-8,8,2001);
>>> PL = np.r_[10:90:20, 90, 95, 99, 99.9]
>>> qlevels(ws.norm.pdf(x),p=PL, xi=(x,));
array([ 0.39591707, 0.37058719, 0.31830968, 0.23402133, 0.10362052,
0.05862129, 0.01449505, 0.00178806])
# compared with the exact values
>>> ws.norm.pdf(ws.norm.ppf((100-PL)/200))
array([ 0.39580488, 0.370399 , 0.31777657, 0.23315878, 0.10313564,
0.05844507, 0.01445974, 0.00177719])
See also
--------
qlevels2, tranproc
"""
def _dx(x):
dx = np.diff(x.ravel()) * 0.5
return np.r_[0, dx] + np.r_[dx, 0]
def _init(pdf, xi, indexing):
if not xi:
return pdf.ravel()
if not isinstance(xi, tuple):
xi = (xi,)
dx = np.meshgrid(*[_dx(x) for x in xi], sparse=True, indexing=indexing)
dxij = np.ones((1))
for dxi in dx:
dxij = dxij * dxi
_assert(dxij.shape == pdf.shape,
'Shape of pdf does not match the arguments')
return (pdf * dxij).ravel()
def _check_levels(levels, pdf):
_assert_warn(not np.any(levels >= max(pdf.ravel())),
'The lowest percent level is too close to 0%')
_assert_warn(not np.any(levels <= min(pdf.ravel())),
'The given pdf is too sparsely sampled or the highest '
'percent level is too close to 100%')
pdf, p = np.atleast_1d(pdf, p)
_assert(not any(pdf.ravel() < 0),
'This is not a pdf since one or more values of pdf is negative')
_assert(not np.any((p < 0) | (100 < p)), 'PL must satisfy 0 <= PL <= 100')
if min(pdf.shape) == 0:
return []
ind = np.argsort(pdf.ravel()) # sort by height of pdf
ind = ind[::-1]
sorted_pdf = pdf.flat[ind]
pdf_dx = _init(pdf, xi, indexing=indexing)
# integration in the order of decreasing height of pdf
cdf = np.cumsum(pdf_dx[ind])
n = pdf_dx.size
# normalize cdf to make sure int pdf dx1 dx2 approx 1
cdf = cdf / cdf[-1] * n / (n + 1.5e-8)
# make sure cdf is strictly increasing by not considering duplicate values
ind, = np.where(np.diff(np.r_[cdf, 1]) > 0)
# calculating the inverse of cdf to find the levels
levels = tranproc(cdf[ind], sorted_pdf[ind], p / 100.0)
_check_levels(levels, pdf)
levels[levels < 0] = 0.0
return levels
def qlevels2(data, p=(10, 30, 50, 70, 90, 95, 99, 99.9), method=1):
"""QLEVELS2 Calculates quantile levels which encloses P% of data.
CALL: [ql PL] = qlevels2(data,PL,method);
ql = the discrete quantile levels, size D X Np
Parameters
----------
data : data matrix, size D x N (D = # of dimensions)
p : percent level vector, length Np (default [10:20:90 95 99 99.9])
method : integer
1 Interpolation so that F(X_[k]) == k/(n-1). (linear default)
2 Interpolation so that F(X_[k]) == (k+0.5)/n. (midpoint)
3 Interpolation so that F(X_[k]) == (k+1)/n. (lower)
4 Interpolation so that F(X_[k]) == k/n. (higher)
Returns
-------
QLEVELS2 sort the columns of data in ascending order and find the
quantile levels for each column which encloses P% of the data.
Examples : Finding quantile levels enclosing P% of data:
--------
>>> import wafo.stats as ws
>>> PL = np.r_[10:90:20, 90, 95, 99, 99.9]
>>> xs = ws.norm.rvs(size=2500000)
>>> np.allclose(qlevels2(ws.norm.pdf(xs), p=PL),
... [0.3958, 0.3704, 0.3179, 0.2331, 0.1031, 0.05841, 0.01451, 0.001751],
... rtol=1e-1)
True
# compared with the exact values
>>> ws.norm.pdf(ws.norm.ppf((100-PL)/200))
array([ 0.39580488, 0.370399 , 0.31777657, 0.23315878, 0.10313564,
0.05844507, 0.01445974, 0.00177719])
# Finding the median of xs:
>>> '%2.2f' % np.abs(qlevels2(xs,50)[0])
'0.00'
See also
--------
qlevels
"""
_assert(0 < method < 5,
'Method must be between 1 to 4. Got method={}.'.format(method))
interpolation = ['', 'linear', 'midpoint', 'lower', 'higher'][method]
q = 100 - np.atleast_1d(p)
return percentile(data, q, axis=-1, interpolation=interpolation)
def iqrange(data, axis=None):
"""Returns the Inter Quartile Range of data.
Parameters
----------
data : array-like
Input array or object that can be converted to an array.
axis : {None, int}, optional
Axis along which the percentiles are computed. The default (axis=None)
is to compute the median along a flattened version of the array.
Returns
-------
r : array-like
abs(np.percentile(data, 75, axis)-np.percentile(data, 25, axis))
Notes
-----
IQRANGE is a robust measure of spread. The use of interquartile range
guards against outliers if the distribution have heavy tails.
Examples
--------
>>> a = np.arange(101)
>>> iqrange(a)
50.0
See also
--------
np.std
"""
return np.abs(np.percentile(data, 75, axis=axis) -
np.percentile(data, 25, axis=axis))
def sphere_volume(d, r=1.0):
"""
Returns volume of d-dimensional sphere with radius r
Parameters
----------
d : scalar or array_like
dimension of sphere
r : scalar or array_like
radius of sphere (default 1)
Examples
--------
>>> sphere_volume(2., r=2.)
12.566370614359172
>>> sphere_volume(2., r=1.)
3.1415926535897931
References
----------
Wand,M.P. and Jones, M.C. (1995)
'Kernel smoothing'
Chapman and Hall, pp 105
"""
return (r ** d) * 2.0 * pi ** (d / 2.0) / (d * gamma(d / 2.0))
class _Kernel(object):
__metaclass__ = ABCMeta
def __init__(self, r=1.0, stats=None, name=''):
self.r = r # radius of effective support of kernel
self.stats = stats
if not name:
name = self.__class__.__name__.replace('_Kernel', '')
self._name = name
@property
def name(self):
return self._name
def norm_factor(self, d=1, n=None):
_assert(0 < d, "D")
_assert(0 < n, "Number of samples too few (n={})".format(n))
return 1.0
@abstractmethod
def _kernel(self, x):
pass
def norm_kernel(self, x):
X = np.atleast_2d(x)
return self._kernel(X) / self.norm_factor(*X.shape)
def kernel(self, x):
return self._kernel(np.atleast_2d(x))
# def deriv4_6_8_10(self, t, numout=4):
# raise NotImplementedError('Method not implemented for this kernel!')
def get_ste_constant(self, n):
mu2, R = self.stats[:2]
return R / (n * mu2 ** 2)
def get_amise_constant(self, n):
# R= int(mkernel(x)^2), mu2= int(x^2*mkernel(x))
mu2, R = self.stats[:2]
return (8 * sqrt(pi) * R / (3 * mu2 ** 2 * n)) ** (1. / 5)
def effective_support(self):
"""Return the effective support of kernel.
The kernel must be symmetric and compactly supported on [-tau tau]
if the kernel has infinite support then the kernel must have the
effective support in [-tau tau], i.e., be negligible outside the range
"""
return self._effective_support()
def _effective_support(self):
return -self.r, self.r
__call__ = kernel
class _KernelMulti(_Kernel):
"""
p=0; Sphere = rect for 1D
p=1; Multivariate Epanechnikov kernel.
p=2; Multivariate Bi-weight Kernel
p=3; Multivariate Tri-weight Kernel
p=4; Multivariate Four-weight Kernel
"""
def __init__(self, r=1.0, p=1, stats=None, name=''):
self.p = p
super(_KernelMulti, self).__init__(r, stats, name)
def norm_factor(self, d=1, n=None):
r = self.r
p = self.p
c = 2 ** p * np.prod(np.r_[1:p + 1]) * sphere_volume(d, r) / np.prod(
np.r_[(d + 2):(2 * p + d + 1):2]) # normalizing constant
return c
def _kernel(self, x):
r = self.r
p = self.p
x2 = x ** 2
return ((1.0 - x2.sum(axis=0) / r ** 2).clip(min=0.0)) ** p
mkernel_epanechnikov = _KernelMulti(p=1, stats=_stats_epan, name='epanechnikov')
mkernel_biweight = _KernelMulti(p=2, stats=_stats_biwe, name='biweight')
mkernel_triweight = _KernelMulti(p=3, stats=_stats_triw, name='triweight')
class _KernelProduct(_KernelMulti):
"""
p=0; rectangular
p=1; 1D product Epanechnikov kernel.
p=2; 1D product Bi-weight Kernel
p=3; 1D product Tri-weight Kernel
p=4; 1D product Four-weight Kernel
"""
def norm_factor(self, d=1, n=None):
r = self.r
p = self.p
c = (2 ** p * np.prod(np.r_[1:p + 1]) * sphere_volume(1, r) /
np.prod(np.r_[(1 + 2):(2 * p + 2):2]))
return c ** d
def _kernel(self, x):
r = self.r # radius
pdf = (1 - (x / r) ** 2).clip(min=0.0) ** self.p
return pdf.prod(axis=0)
mkernel_p1epanechnikov = _KernelProduct(p=1, stats=_stats_epan, name='p1epanechnikov')
mkernel_p1biweight = _KernelProduct(p=2, stats=_stats_biwe, name='p1biweight')
mkernel_p1triweight = _KernelProduct(p=3, stats=_stats_triw,
name='p1triweight')
class _KernelRectangular(_Kernel):
def _kernel(self, x):
return np.where(np.all(np.abs(x) <= self.r, axis=0), 1, 0.0)
def norm_factor(self, d=1, n=None):
r = self.r
return (2 * r) ** d
mkernel_rectangular = _KernelRectangular(stats=_stats_rect)
class _KernelTriangular(_Kernel):
def _kernel(self, x):
pdf = (1 - np.abs(x)).clip(min=0.0)
return pdf.prod(axis=0)
mkernel_triangular = _KernelTriangular(stats=_stats_tria)
class _KernelGaussian(_Kernel):
def _kernel(self, x):
sigma = self.r / 4.0
x2 = (x / sigma) ** 2
return exp(-0.5 * x2.sum(axis=0))
def norm_factor(self, d=1, n=None):
sigma = self.r / 4.0
return (2 * pi * sigma) ** (d / 2.0)
def deriv4_6_8_10(self, t, numout=4):
"""Returns 4th, 6th, 8th and 10th derivatives of the kernel function.
"""
phi0 = exp(-0.5 * t ** 2) / sqrt(2 * pi)
pn = [1, 0, -6, 0, 3]
out = [np.polyval(pn, t) * phi0]
for _i in range(numout - 1):
pnp1 = np.polyadd(-np.r_[pn, 0], np.polyder(pn))
pnp2 = np.polyadd(-np.r_[pnp1, 0], np.polyder(pnp1))
out.append(np.polyval(pnp2, t) * phi0)
pn = pnp2
return tuple(out)
def psi(self, r, sigma=1):
"""Eq. 3.7 in Wand and Jones (1995)"""
rd2 = r // 2
psi_r = (-1) ** rd2 * np.prod(np.r_[rd2 + 1:r + 1]
) / (sqrt(pi) * (2 * sigma) ** (r + 1))
return psi_r
mkernel_gaussian = _KernelGaussian(r=4.0, stats=_stats_gaus)
_GAUSS_KERNEL = mkernel_gaussian
class _KernelLaplace(_Kernel):
def _kernel(self, x):
absX = np.abs(x)
return exp(-absX.sum(axis=0))
def norm_factor(self, d=1, n=None):
return 2 ** d
mkernel_laplace = _KernelLaplace(r=7.0, stats=_stats_lapl)
class _KernelLogistic(_Kernel):
def _kernel(self, x):
s = exp(x)
return np.prod(s / (s + 1) ** 2, axis=0)
mkernel_logistic = _KernelLogistic(r=7.0, stats=_stats_logi)
_MKERNEL_DICT = dict(
epan=mkernel_epanechnikov,
biwe=mkernel_biweight,
triw=mkernel_triweight,
p1ep=mkernel_p1epanechnikov,
p1bi=mkernel_p1biweight,
p1tr=mkernel_p1triweight,
rect=mkernel_rectangular,
tria=mkernel_triangular,
lapl=mkernel_laplace,
logi=mkernel_logistic,
gaus=mkernel_gaussian)
_KERNEL_EXPONENT_DICT = dict(
re=0, sp=0, ep=1, bi=2, tr=3, fo=4, fi=5, si=6, se=7)
class Kernel(object):
"""Multivariate kernel.
Parameters
----------
name : string
defining the kernel. Valid options are:
'epanechnikov' - Epanechnikov kernel.
'biweight' - Bi-weight kernel.
'triweight' - Tri-weight kernel.
'p1epanechnikov' - product of 1D Epanechnikov kernel.
'p1biweight' - product of 1D Bi-weight kernel.
'p1triweight' - product of 1D Tri-weight kernel.
'triangular' - Triangular kernel.
'gaussian' - Gaussian kernel
'rectangular' - Rectangular kernel.
'laplace' - Laplace kernel.
'logistic' - Logistic kernel.
Note that only the first 4 letters of the kernel name is needed.
fun : string
defining smoothing function/bandwidth.
Examples
--------
N = 20
data = np.random.rayleigh(1, size=(N,))
>>> data = np.array([
... 0.75355792, 0.72779194, 0.94149169, 0.07841119, 2.32291887,
... 1.10419995, 0.77055114, 0.60288273, 1.36883635, 1.74754326,
... 1.09547561, 1.01671133, 0.73211143, 0.61891719, 0.75903487,
... 1.8919469 , 0.72433808, 1.92973094, 0.44749838, 1.36508452])
>>> import wafo.kdetools as wk
>>> gauss = wk.Kernel('gaussian')
>>> gauss.stats()
(1, 0.28209479177387814, 0.21157109383040862)
>>> np.allclose(gauss.hscv(data), 0.21555043)
True
>>> np.allclose(gauss.hstt(data), 0.16341135)
True
>>> np.allclose(gauss.hste(data), 0.1968276)
True
>>> np.allclose(gauss.hldpi(data), 0.22502733)
True
>>> wk.Kernel('laplace').stats()
(2, 0.25, inf)
>>> triweight = wk.Kernel('triweight')
>>> np.allclose(triweight.stats(),
... (0.1111111111111111, 0.81585081585081587, np.inf))
True
>>> np.allclose(triweight(np.linspace(-1,1,11)),
... [ 0., 0.046656, 0.262144, 0.592704, 0.884736, 1.,
... 0.884736, 0.592704, 0.262144, 0.046656, 0.])
True
>>> np.allclose(triweight.hns(data), 0.82, rtol=1e-2)
True
>>> np.allclose(triweight.hos(data), 0.88, rtol=1e-2)
True
>>> np.allclose(triweight.hste(data), 0.588, rtol=1e-2)
True
>>> np.allclose(triweight.hscv(data), 0.648, rtol=1e-2)
True
See also
--------
mkernel
References
----------
B. W. Silverman (1986)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp. 43, 76
Wand, M. P. and Jones, M. C. (1995)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp 31, 103, 175
"""
def __init__(self, name, fun='hste'):
self.kernel = _MKERNEL_DICT[name[:4]]
self.get_smoothing = getattr(self, fun)
@property
def name(self):
return self.kernel.name
def stats(self):
"""Return some 1D statistics of the kernel.
Returns
-------
mu2 : real scalar
2'nd order moment, i.e.,int(x^2*kernel(x))
R : real scalar
integral of squared kernel, i.e., int(kernel(x)^2)
Rdd : real scalar
integral of squared double derivative of kernel,
i.e., int( (kernel''(x))^2 ).
References
----------
Wand,M.P. and Jones, M.C. (1995)
'Kernel smoothing'
Chapman and Hall, pp 176.
"""
return self.kernel.stats
def effective_support(self):
return self.kernel.effective_support()
def hns(self, data):
"""Returns Normal Scale Estimate of Smoothing Parameter.
Parameter
---------
data : 2D array
shape d x n (d = # dimensions )
Returns
-------
h : array-like
one dimensional optimal value for smoothing parameter
given the data and kernel. size D
HNS only gives an optimal value with respect to mean integrated
square error, when the true underlying distribution
is Gaussian. This works reasonably well if the data resembles a
Gaussian distribution. However if the distribution is asymmetric,
multimodal or have long tails then HNS may return a to large
smoothing parameter, i.e., the KDE may be oversmoothed and mask
important features of the data. (=> large bias).
One way to remedy this is to reduce H by multiplying with a constant
factor, e.g., 0.85. Another is to try different values for H and make a
visual check by eye.
Examples
--------
>>> import numpy as np
>>> import wafo.kdetools as wk
>>> import wafo.stats as ws
>>> kernel = wk.Kernel('epan')
>>> data = ws.norm.rvs(0, 1, size=(1,20))
>>> h = kernel.hns(data)
See also:
---------
hste, hbcv, hboot, hos, hldpi, hlscv, hscv, hstt, kde
References
----------
B. W. Silverman (1986)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp 43-48
Wand,M.P. and Jones, M.C. (1995)
'Kernel smoothing'
Chapman and Hall, pp 60--63
"""
a = np.atleast_2d(data)
n = a.shape[1]
amise_constant = self.kernel.get_amise_constant(n)
iqr = iqrange(a, axis=1) # interquartile range
std_a = np.std(a, axis=1, ddof=1)
# use of interquartile range guards against outliers.
# the use of interquartile range is better if
# the distribution is skew or have heavy tails
# This lessen the chance of oversmoothing.
sigma = np.where(iqr > 0, np.minimum(std_a, iqr / 1.349), std_a)
return sigma * amise_constant
def hos(self, data):
"""Returns Oversmoothing Parameter.
Parameter
---------
data = data matrix, size N x D (D = # dimensions )
Returns
-------
h : vector size 1 x D
one dimensional maximum smoothing value for smoothing parameter
given the data and kernel.
The oversmoothing or maximal smoothing principle relies on the fact
that there is a simple upper bound for the AMISE-optimal bandwidth for
estimation of densities with a fixed value of a particular scale
measure. While HOS will give too large bandwidth for optimal estimation
of a general density it provides an excellent starting point for
subjective choice of bandwidth. A sensible strategy is to plot an
estimate with bandwidth HOS and then sucessively look at plots based on
convenient fractions of HOS to see what features are present in the
data for various amount of smoothing. The relation to HNS is given by:
HOS = HNS/0.93
Examples
--------
data = rndnorm(0, 1,20,1)
h = hos(data,'epan');
See also hste, hbcv, hboot, hldpi, hlscv, hscv, hstt, kde, kdefun
References
----------
B. W. Silverman (1986)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp 43-48
Wand,M.P. and Jones, M.C. (1986)
'Kernel smoothing'
Chapman and Hall, pp 60--63
"""
return self.hns(data) / 0.93
def _hmns_scale(self, n, d):
name = self.name
short_name = name[:4].lower()
scale_dict = dict(epan=(8.0 * (d + 4.0) * (2 * sqrt(pi)) ** d /
sphere_volume(d)) ** (1. / (4.0 + d)),
biwe=2.7779, triw=3.12,
gaus=(4.0 / (d + 2.0)) ** (1. / (d + 4.0)))
if short_name not in scale_dict:
raise NotImplementedError('Hmns bandwidth not implemented for '
'kernel {}.'.format(name))
if d > 2 and short_name in ['biwe', 'triw']:
raise NotImplementedError('Not implemented for d>2 and '
'kernel {}'.format(name))
return scale_dict[short_name] * n ** (-1. / (d + 4))
def hmns(self, data):
"""Returns Multivariate Normal Scale Estimate of Smoothing Parameter.
CALL: h = hmns(data,kernel)
h = M dimensional optimal value for smoothing parameter
given the data and kernel. size D x D
data = data matrix, size D x N (D = # dimensions )
kernel = 'epanechnikov' - Epanechnikov kernel.
'biweight' - Bi-weight kernel.
'triweight' - Tri-weight kernel.
'gaussian' - Gaussian kernel
HMNS only gives a optimal value with respect to mean integrated
square error, when the true underlying distribution is Multivariate
Gaussian. This works reasonably well if the data resembles a
Multivariate Gaussian distribution. However if the distribution is
asymmetric, multimodal or have long tails then HNS is maybe more
appropriate.
Examples
--------
data = rndnorm(0, 1,20,2)
h = hmns(data,'epan')
See also
--------
hns, hste, hbcv, hboot, hos, hldpi, hlscv, hscv, hstt
References
----------
B. W. Silverman (1986)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp 43-48, 87
Wand,M.P. and Jones, M.C. (1995)
'Kernel smoothing'
Chapman and Hall, pp 60--63, 86--88
"""
a = np.atleast_2d(data)
d, n = a.shape
if d == 1:
return self.hns(data)
return self._hmns_scale(n, d) * np.real(sqrtm(np.cov(a)))
@staticmethod
def _get_g(k_order_2, mu2, psi_order, n, order):
return (-2. * k_order_2 / (mu2 * psi_order * n)) ** (1. / (order + 1))
def hste(self, data, h0=None, inc=128, maxit=100, releps=0.01, abseps=0.0):
'''HSTE 2-Stage Solve the Equation estimate of smoothing parameter.
CALL: hs = hste(data,kernel,h0)
hs = one dimensional value for smoothing parameter
given the data and kernel. size 1 x D
data = data matrix, size N x D (D = # dimensions )
kernel = 'gaussian' - Gaussian kernel (default)
( currently the only supported kernel)
h0 = initial starting guess for hs (default h0=hns(A,kernel))
Examples
--------
x = rndnorm(0,1,50,1);
hs = hste(x,'gauss');
See also
--------
hbcv, hboot, hos, hldpi, hlscv, hscv, hstt, kde, kdefun
References
----------
B. W. Silverman (1986)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp 57--61
Wand,M.P. and Jones, M.C. (1986)
'Kernel smoothing'
Chapman and Hall, pp 74--75
'''
A = np.atleast_2d(data)
d, n = A.shape
amise_constant = self.kernel.get_amise_constant(n)
ste_constant = self.kernel.get_ste_constant(n)
sigmaA = self.hns(A) / amise_constant
if h0 is None:
h0 = sigmaA * amise_constant
h = np.asarray(h0, dtype=float)
ax1, bx1 = self._get_grid_limits(A)
mu2, R = _GAUSS_KERNEL.stats[:2]
ste_constant2 = _GAUSS_KERNEL.get_ste_constant(n)
for dim in range(d):
s = sigmaA[dim]
ax = ax1[dim]
bx = bx1[dim]
xa = np.linspace(ax, bx, inc)
xn = np.linspace(0, bx - ax, inc)
c = gridcount(A[dim], xa)
psi6NS = _GAUSS_KERNEL.psi(6, s)
psi8NS = _GAUSS_KERNEL.psi(8, s)
k40, k60 = _GAUSS_KERNEL.deriv4_6_8_10(0, numout=2)
g1 = self._get_g(k40, mu2, psi6NS, n, order=6)
g2 = self._get_g(k60, mu2, psi8NS, n, order=8)
psi4 = self._estimate_psi(c, xn, g1, n, order=4)
psi6 = self._estimate_psi(c, xn, g2, n, order=6)
h1 = h[dim]
h_old = 0
count = 0
while ((abs(h_old - h1) > max(releps * h1, abseps)) and
(count < maxit)):
count += 1
h_old = h1
gamma_ = ((2 * k40 * mu2 * psi4 * h1 ** 5) /
(-psi6 * R)) ** (1.0 / 7)
psi4Gamma = self._estimate_psi(c, xn, gamma_, n, order=4)
h1 = (ste_constant2 / psi4Gamma) ** (1.0 / 5)
# Kernel other than Gaussian scale bandwidth
h1 = h1 * (ste_constant / ste_constant2) ** (1.0 / 5)
_assert_warn(count < maxit, 'The obtained value did not converge.')
h[dim] = h1
# end for dim loop
return h
def hisj(self, data, inc=512, L=7):
'''
HISJ Improved Sheather-Jones estimate of smoothing parameter.
Unlike many other implementations, this one is immune to problems
caused by multimodal densities with widely separated modes. The
estimation does not deteriorate for multimodal densities, because
it do not assume a parametric model for the data.
Parameters
----------
data - a vector of data from which the density estimate is constructed
inc - the number of mesh points used in the uniform discretization
Returns
-------
bandwidth - the optimal bandwidth
References
----------
Z. I. Botev, J. F. Grotowski, and D. P. Kroese (2010)
"Kernel density estimation via diffusion"
Annals of Statistics, Volume 38, Number 5, pages 2916-2957.
'''
A = np.atleast_2d(data)
d, n = A.shape
ste_constant = self.kernel.get_ste_constant(n)
ax1, bx1 = self._get_grid_limits(A)
ste_constant2 = _GAUSS_KERNEL.get_ste_constant(n)
def fixed_point(t, N, I, a2):
''' this implements the function t-zeta*gamma^[L](t)'''
prod = np.prod
# L = 7
logI = np.log(I)
def fun(s, time):
return (2 * pi ** (2 * s) *
(a2 * exp(s * logI - I * pi ** 2 * time)).sum())
f = fun(L, t)
for s in range(L - 1, 1, -1):
K0 = prod(np.r_[1:2 * s:2]) / sqrt(2 * pi)
const = (1 + (1. / 2) ** (s + 1. / 2)) / 3
time = (2 * const * K0 / N / f) ** (2. / (3 + 2 * s))
f = fun(s, time)
return t - (2 * N * sqrt(pi) * f) ** (-2. / 5)
h = np.empty(d)
for dim in range(d):
ax, bx = ax1[dim], bx1[dim]
xa = np.linspace(ax, bx, inc)
R = bx - ax
c = gridcount(A[dim], xa)
N = len(set(A[dim]))
a = dct(c / len(A[dim]), norm=None)
# now compute the optimal bandwidth^2 using the referenced method
I = np.asfarray(np.arange(1, inc)) ** 2
a2 = (a[1:] / 2) ** 2
x = np.linspace(0, 0.1, 150)
ai = x[0]
bi = x[1]
f0 = fixed_point(ai, N, I, a2)
for xi in x[1:]:
bi = xi
f1 = fixed_point(bi, N, I, a2)
if f1 * f0 <= 0:
# print('ai = %g, bi = %g' % (ai,bi))
break
else:
ai = xi
# use fzero to solve the equation t=zeta*gamma^[5](t)
try:
t_star = optimize.brentq(lambda t: fixed_point(t, N, I, a2),
a=ai, b=bi)
except Exception as err:
t_star = 0.28 * N ** (-2. / 5)
warnings.warn('Failure in obtaining smoothing parameter'
' ({})'.format(str(err)))
# smooth the discrete cosine transform of initial data using t_star
# a_t = a*exp(-np.arange(inc)**2*pi**2*t_star/2)
# now apply the inverse discrete cosine transform
# density = idct(a_t)/R;
# take the rescaling of the data into account
bandwidth = sqrt(t_star) * R
# Kernel other than Gaussian scale bandwidth
h[dim] = bandwidth * (ste_constant / ste_constant2) ** (1.0 / 5)
# end for dim loop
return h
def hstt(self, data, h0=None, inc=128, maxit=100, releps=0.01, abseps=0.0):
'''HSTT Scott-Tapia-Thompson estimate of smoothing parameter.
CALL: hs = hstt(data,kernel)
hs = one dimensional value for smoothing parameter
given the data and kernel. size 1 x D
data = data matrix, size N x D (D = # dimensions )
kernel = 'epanechnikov' - Epanechnikov kernel. (default)
'biweight' - Bi-weight kernel.
'triweight' - Tri-weight kernel.
'triangular' - Triangular kernel.
'gaussian' - Gaussian kernel
'rectangular' - Rectangular kernel.
'laplace' - Laplace kernel.
'logistic' - Logistic kernel.
HSTT returns Scott-Tapia-Thompson (STT) estimate of smoothing
parameter. This is a Solve-The-Equation rule (STE).
Simulation studies shows that the STT estimate of HS
is a good choice under a variety of models. A comparison with
likelihood cross-validation (LCV) indicates that LCV performs slightly
better for short tailed densities.
However, STT method in contrast to LCV is insensitive to outliers.
Examples
--------
x = rndnorm(0,1,50,1);
hs = hstt(x,'gauss');
See also
--------
hste, hbcv, hboot, hos, hldpi, hlscv, hscv, kde, kdebin
References
----------
B. W. Silverman (1986)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp 57--61
'''
A = np.atleast_2d(data)
d, n = A.shape
amise_constant = self.kernel.get_amise_constant(n)
ste_constant = self.kernel.get_ste_constant(n)
sigmaA = self.hns(A) / amise_constant
if h0 is None:
h0 = sigmaA * amise_constant
h = np.asarray(h0, dtype=float)
ax1, bx1 = self._get_grid_limits(A)
for dim in range(d):
s = sigmaA[dim]
datan = A[dim] / s
ax = ax1[dim] / s
bx = bx1[dim] / s
xa = np.linspace(ax, bx, inc)
xn = np.linspace(0, bx - ax, inc)
c = gridcount(datan, xa)
count = 1
h_old = 0
h1 = h[dim] / s
delta = (bx - ax) / (inc - 1)
while ((abs(h_old - h1) > max(releps * h1, abseps)) and
(count < maxit)):
count += 1
h_old = h1
kw4 = self.kernel(xn / h1) / (n * h1 * self.norm_factor(d=1))
kw = np.r_[kw4, 0, kw4[-1:0:-1]] # Apply 'fftshift' to kw.
f = np.real(ifft(fft(c, 2 * inc) * fft(kw))) # convolution.
# Estimate psi4=R(f'') using simple finite differences and
# quadrature.
ix = np.arange(1, inc - 1)
z = ((f[ix + 1] - 2 * f[ix] + f[ix - 1]) / delta ** 2) ** 2
psi4 = delta * z.sum()
h1 = (ste_constant / psi4) ** (1. / 5)
_assert_warn(count < maxit, 'The obtained value did not converge.')
h[dim] = h1 * s
# end # for dim loop
return h
@staticmethod
def _estimate_psi(c, xn, gi, n, order=4):
# order = numout*2+2
inc = len(xn)
kw0 = _GAUSS_KERNEL.deriv4_6_8_10(xn / gi, numout=(order - 2) // 2)[-1]
kw = np.r_[kw0, 0, kw0[-1:0:-1]] # Apply fftshift to kw.
z = np.real(ifft(fft(c, 2 * inc) * fft(kw))) # convolution.
return np.sum(c * z[:inc]) / (n ** 2 * gi ** (order + 1))
def hscv(self, data, hvec=None, inc=128, maxit=100, fulloutput=False):
'''
HSCV Smoothed cross-validation estimate of smoothing parameter.
Parameters
----------
data = data vector
hvec = vector defining possible values of hs
(default linspace(0.25*h0,h0,100), h0=0.62)
inc = length of estimated kerneldensity estimate
maxit = maximum number of iterations
fulloutput = True if fulloutput is wanted
Returns
-------
hs = smoothing parameter
hvec = vector defining possible values of hs
score = score vector
Examples
--------
>>> import wafo.kdetools as wk
>>> import wafo.stats as ws
>>> data = ws.norm.rvs(0,1, size=(1,20))
>>> kernel = wk.Kernel('epan')
>>> hs0 = kernel.hscv(data, fulloutput=False)
>>> hs, hvec, score = kernel.hscv(data, fulloutput=True)
>>> np.allclose(hs, hs0)
True
import matplotlib.pyplot as plt
plt.plot(hvec, score)
See also:
hste, hbcv, hboot, hos, hldpi, hlscv, hstt, kde, kdefun
References
----------
Wand,M.P. and Jones, M.C. (1986)
'Kernel smoothing'
Chapman and Hall, pp 75--79
'''
A = np.atleast_2d(data)
d, n = A.shape
amise_constant = self.kernel.get_amise_constant(n)
ste_constant = self.kernel.get_ste_constant(n)
sigmaA = self.hns(A) / amise_constant
if hvec is None:
H = amise_constant / 0.93
hvec = np.linspace(0.25 * H, H, maxit)
hvec = np.asarray(hvec, dtype=float)
steps = len(hvec)
score = np.zeros(steps)
ax1, bx1 = self._get_grid_limits(A)
ste_constant2 = _GAUSS_KERNEL.get_ste_constant(n)
h = np.zeros(d)
hvec = hvec * (ste_constant2 / ste_constant) ** (1. / 5.)
k40, k60, k80, k100 = _GAUSS_KERNEL.deriv4_6_8_10(0, numout=4)
mu2 = _GAUSS_KERNEL.stats[0]
# psi8 = _GAUSS_KERNEL.psi(8)
# psi12 = _GAUSS_KERNEL.psi(12)
psi8 = 105 / (32 * sqrt(pi))
psi12 = 3465. / (512 * sqrt(pi))
g1 = self._get_g(k60, mu2, psi8, n, order=8)
g2 = self._get_g(k100, mu2, psi12, n, order=12)
for dim in range(d):
s = sigmaA[dim]
ax = ax1[dim] / s
bx = bx1[dim] / s
datan = A[dim] / s
xa = np.linspace(ax, bx, inc)
xn = np.linspace(0, bx - ax, inc)
c = gridcount(datan, xa)
psi6 = self._estimate_psi(c, xn, g1, n, order=6)
psi10 = self._estimate_psi(c, xn, g2, n, order=10)
g3 = self._get_g(k40, mu2, psi6, n, order=6)
g4 = self._get_g(k80, mu2, psi10, n, order=10)
psi4 = self._estimate_psi(c, xn, g3, n, order=4)
psi8 = self._estimate_psi(c, xn, g4, n, order=8)
const = ((441. / (64 * pi)) ** (1. / 18.) *
(4 * pi) ** (-1. / 5.) *
psi4 ** (-2. / 5.) * psi8 ** (-1. / 9.))
M = np.atleast_2d(datan)
Y = (M - M.T).ravel()
for i in range(steps):
g = const * n ** (-23. / 45) * hvec[i] ** (-2)
sig1 = sqrt(2 * hvec[i] ** 2 + 2 * g ** 2)
sig2 = sqrt(hvec[i] ** 2 + 2 * g ** 2)
sig3 = sqrt(2 * g ** 2)
term2 = np.sum(_GAUSS_KERNEL(Y / sig1) / sig1 -
2 * _GAUSS_KERNEL(Y / sig2) / sig2 +
_GAUSS_KERNEL(Y / sig3) / sig3)
score[i] = 1. / (n * hvec[i] * 2. * sqrt(pi)) + term2 / n ** 2
idx = score.argmin()
# Kernel other than Gaussian scale bandwidth
h[dim] = hvec[idx] * (ste_constant / ste_constant2) ** (1 / 5)
_assert_warn(0 < idx,
"Optimum is probably lower than "
"hs={0:g} for dim={1:d}".format(h[dim] * s, dim))
_assert_warn(idx < maxit - 1,
"Optimum is probably higher than "
"hs={0:g} for dim={1:d}".format(h[dim] * s, dim))
hvec = hvec * (ste_constant / ste_constant2) ** (1 / 5)
if fulloutput:
return h * sigmaA, score, hvec
return h * sigmaA
def _get_grid_limits(self, data):
min_a, max_a = data.min(axis=1), data.max(axis=1)
offset = (max_a - min_a) / 8.0
return min_a - offset, max_a + offset
def hldpi(self, data, L=2, inc=128):
'''HLDPI L-stage Direct Plug-In estimate of smoothing parameter.
CALL: hs = hldpi(data,kernel,L)
hs = one dimensional value for smoothing parameter
given the data and kernel. size 1 x D
data = data matrix, size N x D (D = # dimensions )
kernel = 'epanechnikov' - Epanechnikov kernel.
'biweight' - Bi-weight kernel.
'triweight' - Tri-weight kernel.
'triangluar' - Triangular kernel.
'gaussian' - Gaussian kernel
'rectangular' - Rectanguler kernel.
'laplace' - Laplace kernel.
'logistic' - Logistic kernel.
L = 0,1,2,3,... (default 2)
Note that only the first 4 letters of the kernel name is needed.
Examples
--------
x = rndnorm(0,1,50,1);
hs = hldpi(x,'gauss',1);
See also
--------
hste, hbcv, hboot, hos, hlscv, hscv, hstt, kde, kdefun
References
----------
Wand,M.P. and Jones, M.C. (1995)
'Kernel smoothing'
Chapman and Hall, pp 67--74
'''
A = np.atleast_2d(data)
d, n = A.shape
amise_constant = self.kernel.get_amise_constant(n)
ste_constant = self.kernel.get_ste_constant(n)
sigmaA = self.hns(A) / amise_constant
ax1, bx1 = self._get_grid_limits(A)
mu2 = _GAUSS_KERNEL.stats[0]
h = np.zeros(d)
for dim in range(d):
s = sigmaA[dim]
datan = A[dim] # / s
ax = ax1[dim] # / s
bx = bx1[dim] # / s
xa = np.linspace(ax, bx, inc)
xn = np.linspace(0, bx - ax, inc)
c = gridcount(datan, xa)
psi = _GAUSS_KERNEL.psi(r=2 * L + 4, sigma=s)
if L > 0:
# High order derivatives of the Gaussian kernel
Kd = _GAUSS_KERNEL.deriv4_6_8_10(0, numout=L)
# L-stage iterations to estimate PSI_4
for ix in range(L, 0, -1):
gi = self._get_g(Kd[ix - 1], mu2, psi, n, order=2 * ix + 4)
psi = self._estimate_psi(c, xn, gi, n, order=2 * ix + 2)
h[dim] = (ste_constant / psi) ** (1. / 5)
return h
def norm_factor(self, d=1, n=None):
return self.kernel.norm_factor(d, n)
def eval_points(self, points):
return self.kernel(np.atleast_2d(points))
__call__ = eval_points
def mkernel(X, kernel):
"""MKERNEL Multivariate Kernel Function.
Paramaters
----------
X : array-like
matrix size d x n (d = # dimensions, n = # evaluation points)
kernel : string
defining kernel
'epanechnikov' - Epanechnikov kernel.
'biweight' - Bi-weight kernel.
'triweight' - Tri-weight kernel.
'p1epanechnikov' - product of 1D Epanechnikov kernel.
'p1biweight' - product of 1D Bi-weight kernel.
'p1triweight' - product of 1D Tri-weight kernel.
'triangular' - Triangular kernel.
'gaussian' - Gaussian kernel
'rectangular' - Rectangular kernel.
'laplace' - Laplace kernel.
'logistic' - Logistic kernel.
Note that only the first 4 letters of the kernel name is needed.
Returns
-------
z : ndarray
kernel function values evaluated at X
See also
--------
KDE
References
----------
B. W. Silverman (1986)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp. 43, 76
Wand, M. P. and Jones, M. C. (1995)
'Density estimation for statistics and data analysis'
Chapman and Hall, pp 31, 103, 175
"""
fun = _MKERNEL_DICT[kernel[:4]]
return fun(np.atleast_2d(X))
if __name__ == '__main__':
test_docstrings(__file__)