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fixed a bug in Kernel.name and simplified glevels and made it general.

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master
Per A Brodtkorb 8 years ago
parent 40bfa22d46
commit 669fadc9a5
3 changed files with 174 additions and 129 deletions
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264
wafo/kdetools/kernels.py
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@ -22,6 +22,12 @@ 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)
@ -36,31 +42,39 @@ _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), x1=None, x2=None):
"""QLEVELS Calculates quantile levels which encloses P% of PDF.
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.
CALL: [ql PL] = qlevels(pdf,PL,x1,x2);
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.
ql = the discrete quantile levels.
pdf = joint point density function matrix or vector
PL = percent level (default [10:20:90 95 99 99.9])
x1,x2 = vectors of the spacing of the variables
(Default unit spacing)
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 X1 and
(or) X2 is unspecified it is assumed that dX1 and dX2 is constant.
NB! QLEVELS normalizes the integral of PDF to N/(N+0.001) before
calculating QL in order to reflect the sampling of PDF is finite.
Currently only able to handle 1D and 2D PDF's if dXi is not constant
(i=1,2).
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.
Example
-------
>>> 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, x1=x);
>>> 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])
@ -74,73 +88,58 @@ def qlevels(pdf, p=(10, 30, 50, 70, 90, 95, 99, 99.9), x1=None, x2=None):
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')
norm = 1 # normalize cdf to unity
pdf = np.atleast_1d(pdf)
_assert(not any(pdf.ravel() < 0), 'This is not a pdf since one or more '
'values of pdf is negative')
fsiz = pdf.shape
fsizmin = min(fsiz)
if fsizmin == 0:
if min(pdf.shape) == 0:
return []
N = np.prod(fsiz)
d = len(fsiz)
if x1 is None or ((x2 is None) and d > 2):
fdfi = pdf.ravel()
else:
if d == 1: # pdf in one dimension
dx22 = np.ones(1)
else: # % pdf in two dimensions
dx2 = np.diff(x2.ravel()) * 0.5
dx22 = np.r_[0, dx2] + np.r_[dx2, 0]
dx1 = np.diff(x1.ravel()) * 0.5
dx11 = np.r_[0, dx1] + np.r_[dx1, 0]
dx1x2 = dx22[:, None] * dx11
fdfi = (pdf * dx1x2).ravel()
p = np.atleast_1d(p)
_assert(not np.any((p < 0) | (100 < p)), 'PL must satisfy 0 <= PL <= 100')
p2 = p / 100.0
ind = np.argsort(pdf.ravel()) # sort by height of pdf
ind = ind[::-1]
fi = pdf.flat[ind]
sorted_pdf = pdf.flat[ind]
pdf_dx = _init(pdf, xi, indexing=indexing)
# integration in the order of decreasing height of pdf
Fi = np.cumsum(fdfi[ind])
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)
if norm: # normalize Fi to make sure int pdf dx1 dx2 approx 1
Fi = Fi / Fi[-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)
maxFi = np.max(Fi)
if maxFi > 1:
warnings.warn('this is not a pdf since cdf>1! normalizing')
# calculating the inverse of cdf to find the levels
levels = tranproc(cdf[ind], sorted_pdf[ind], p / 100.0)
Fi = Fi / Fi[-1] * N / (N + 1.5e-8)
elif maxFi < .95:
msg = '''The given pdf is too sparsely sampled since cdf<.95.
Thus QL is questionable'''
warnings.warn(msg)
# make sure Fi is strictly increasing by not considering duplicate values
ind, = np.where(np.diff(np.r_[Fi, 1]) > 0)
# calculating the inverse of Fi to find the index
ui = tranproc(Fi[ind], fi[ind], p2)
if np.any(ui >= max(pdf.ravel())):
warnings.warn('The lowest percent level is too close to 0%')
if np.any(ui <= min(pdf.ravel())):
msg = '''The given pdf is too sparsely sampled or
the highest percent level is too close to 100%'''
warnings.warn(msg)
ui[ui < 0] = 0.0
return ui
_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):
@ -262,9 +261,16 @@ def sphere_volume(d, r=1.0):
class _Kernel(object):
__metaclass__ = ABCMeta
def __init__(self, r=1.0, stats=None):
self.r = r # radius of kernel
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")
@ -305,12 +311,12 @@ class _KernelMulti(_Kernel):
p=0; Sphere = rect for 1D
p=1; Multivariate Epanechnikov kernel.
p=2; Multivariate Bi-weight Kernel
p=3; Multi variate Tri-weight Kernel
p=4; Multi variate Four-weight Kernel
p=3; Multivariate Tri-weight Kernel
p=4; Multivariate Four-weight Kernel
"""
def __init__(self, r=1.0, p=1, stats=None):
def __init__(self, r=1.0, p=1, stats=None, name=''):
self.p = p
super(_KernelMulti, self).__init__(r, stats)
super(_KernelMulti, self).__init__(r, stats, name)
def norm_factor(self, d=1, n=None):
r = self.r
@ -325,9 +331,10 @@ class _KernelMulti(_Kernel):
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)
mkernel_biweight = _KernelMulti(p=2, stats=_stats_biwe)
mkernel_triweight = _KernelMulti(p=3, stats=_stats_triw)
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):
@ -350,9 +357,11 @@ class _KernelProduct(_KernelMulti):
pdf = (1 - (x / r) ** 2).clip(min=0.0) ** self.p
return pdf.prod(axis=0)
mkernel_p1epanechnikov = _KernelProduct(p=1, stats=_stats_epan)
mkernel_p1biweight = _KernelProduct(p=2, stats=_stats_biwe)
mkernel_p1triweight = _KernelProduct(p=3, stats=_stats_triw)
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):
@ -404,11 +413,6 @@ class _KernelGaussian(_Kernel):
mkernel_gaussian = _KernelGaussian(r=4.0, stats=_stats_gaus)
# def mkernel_gaussian(X):
# x2 = X ** 2
# d = X.shape[0]
# return (2 * pi) ** (-d / 2) * exp(-0.5 * x2.sum(axis=0))
class _KernelLaplace(_Kernel):
@ -439,8 +443,8 @@ _MKERNEL_DICT = dict(
tria=mkernel_triangular,
lapl=mkernel_laplace,
logi=mkernel_logistic,
gaus=mkernel_gaussian
)
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)
@ -530,7 +534,7 @@ class Kernel(object):
@property
def name(self):
return self.kernel.__class__.__name__.replace('_Kernel', '').title()
return self.kernel.name
def stats(self):
"""Return some 1D statistics of the kernel.
@ -586,8 +590,13 @@ class Kernel(object):
visual check by eye.
Example:
data = rndnorm(0, 1,20,1)
h = hns(data,'epan')
-------
>>> 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:
---------
@ -601,7 +610,6 @@ class Kernel(object):
Wand,M.P. and Jones, M.C. (1995)
'Kernel smoothing'
Chapman and Hall, pp 60--63
"""
a = np.atleast_2d(data)
@ -611,13 +619,13 @@ class Kernel(object):
mu2, R, _Rdd = self.stats()
amise_constant = (8 * sqrt(pi) * R / (3 * mu2 ** 2 * n)) ** (1. / 5)
iqr = iqrange(a, axis=1) # interquartile range
stdA = np.std(a, axis=1, ddof=1)
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.
return np.where(iqr > 0,
np.minimum(stdA, iqr / 1.349), stdA) * amise_constant
np.minimum(std_a, iqr / 1.349), std_a) * amise_constant
def hos(self, data):
"""Returns Oversmoothing Parameter.
@ -680,7 +688,8 @@ class Kernel(object):
elif name == 'gaus': # Gaussian kernel
a = (4.0 / (d + 2.0)) ** (1. / (d + 4.0))
else:
raise ValueError('Unknown kernel.')
raise NotImplementedError('Hmns bandwidth not implemented for '
'kernel {}.'.format(name))
return a
def hmns(self, data):
@ -696,7 +705,6 @@ class Kernel(object):
'triweight' - Tri-weight kernel.
'gaussian' - Gaussian kernel
Note that only the first 4 letters of the kernel name is needed.
HMNS only gives a optimal value with respect to mean integrated
square error, when the true underlying distribution is Multivariate
@ -725,7 +733,6 @@ class Kernel(object):
Chapman and Hall, pp 60--63, 86--88
"""
# TODO: implement more kernels
a = np.atleast_2d(data)
d, n = a.shape
@ -1081,35 +1088,53 @@ class Kernel(object):
warnings.warn('The obtained value did not converge.')
h[dim] = h1 * s
# end % for dim loop
# end # for dim loop
return h
def hscv(self, data, hvec=None, inc=128, maxit=100, fulloutput=False):
'''
HSCV Smoothed cross-validation estimate of smoothing parameter.
CALL: [hs,hvec,score] = hscv(data,kernel,hvec)
hs = 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
data = data vector
kernel = 'gaussian' - Gaussian kernel the only supported
Note that only the first 4 letters of the kernel name is needed.
Example
------
>>> 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
Example:
data = rndnorm(0,1,20,1)
[hs hvec score] = hscv(data,'epan');
plot(hvec,score)
See also hste, hbcv, hboot, hos, hldpi, hlscv, hstt, kde, kdefun
import matplotlib.pyplot as plt
plt.plot(hvec,score)
See also:
hste, hbcv, hboot, hos, hldpi, hlscv, hstt, kde, kdefun
Reference
---------
Wand,M.P. and Jones, M.C. (1986)
'Kernel smoothing'
Chapman and Hall, pp 75--79
'''
# TODO: Add support for other kernels than Gaussian
A = np.atleast_2d(data)
d, n = A.shape
@ -1209,17 +1234,16 @@ class Kernel(object):
idx = score.argmin()
# Kernel other than Gaussian scale bandwidth
h[dim] = hvec[idx] * (ste_constant / ste_constant2) ** (1 / 5)
if idx == 0:
warnings.warn("Optimum is probably lower than "
_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))
elif idx == maxit - 1:
msg = "Optimum is probably higher than hs={0:g] for dim={1:d}"
warnings.warn(msg.format(h[dim] * s, dim))
hvec = hvec * (ste_constant / ste_constant2) ** (1 / 5)
if fulloutput:
return h * sigmaA, score, hvec, sigmaA
else:
return h * sigmaA, score, hvec
return h * sigmaA
def hldpi(self, data, L=2, inc=128):
@ -1361,7 +1385,7 @@ def mkernel(X, kernel):
See also
--------
kde, kdefun, kdebin
KDE
References
----------

2
wafo/spectrum/core.py
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@ -1131,7 +1131,7 @@ class SpecData1D(PlotData):
title='Joint density of maximum and minimum')
try:
pl = [10, 30, 50, 70, 90, 95, 99, 99.9]
mmpdf.cl = qlevels(uvdens, pl, h, h)
mmpdf.cl = qlevels(uvdens, pl, xi=(h, h))
mmpdf.pl = pl
except:
pass

21
wafo/tests/test_kdetools.py
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@ -449,6 +449,27 @@ class TestSmoothing(unittest.TestCase):
assert_allclose(hs, [[3.25196193e-01, -2.68892467e-02, 3.18932448e-04],
[-2.68892467e-02, 3.91283306e-01, 2.38654678e-02],
[3.18932448e-04, 2.38654678e-02, 4.05123874e-01]])
hs = self.gauss.hmns(self.data[0])
assert_allclose(hs, self.gauss.hns(self.data[0]))
hs = wk.Kernel('epan').hmns(self.data)
assert_allclose(hs,
[[8.363847e-01, -6.915749e-02, 8.202747e-04],
[-6.915749e-02, 1.006357e+00, 6.138052e-02],
[8.202747e-04, 6.138052e-02, 1.041954e+00]],
rtol=1e-5)
hs = wk.Kernel('biwe').hmns(self.data[:2])
assert_allclose(hs, [[0.868428, -0.071705],
[-0.071705, 1.04685]], rtol=1e-5)
hs = wk.Kernel('triwe').hmns(self.data[:2])
assert_allclose(hs, [[0.975375, -0.080535],
[-0.080535, 1.17577]], rtol=1e-5)
self.assertRaises(NotImplementedError,
wk.Kernel('biwe').hmns, self.data)
self.assertRaises(NotImplementedError,
wk.Kernel('triwe').hmns, self.data)
self.assertRaises(NotImplementedError,
wk.Kernel('triangular').hmns, self.data)
def test_hscv(self):
hs = self.gauss.hscv(self.data)

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