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Simplified Kernel

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master
Per A Brodtkorb 8 years ago
parent b0ad9c7438
commit 526d0e896f
1 changed files with 50 additions and 103 deletions
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153
wafo/kdetools/kernels.py
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@ -8,6 +8,7 @@ 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, linalg
from scipy.special import gamma
from wafo.misc import tranproc # , trangood
@ -564,6 +565,12 @@ class Kernel(object):
def effective_support(self):
return self.kernel.effective_support()
def get_amise_constant(self, n):
# R= int(mkernel(x)^2), mu2= int(x^2*mkernel(x))
mu2, R = self.stats()[:2]
amise_constant = (8 * sqrt(pi) * R / (3 * mu2 ** 2 * n)) ** (1. / 5)
return amise_constant
def hns(self, data):
"""Returns Normal Scale Estimate of Smoothing Parameter.
@ -615,9 +622,7 @@ class Kernel(object):
a = np.atleast_2d(data)
n = a.shape[1]
# R= int(mkernel(x)^2), mu2= int(x^2*mkernel(x))
mu2, R, _Rdd = self.stats()
amise_constant = (8 * sqrt(pi) * R / (3 * mu2 ** 2 * n)) ** (1. / 5)
amise_constant = self.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.
@ -737,6 +742,10 @@ class Kernel(object):
cov_a = np.cov(a)
return scale * linalg.sqrtm(cov_a).real * n ** (-1. / (d + 4))
def get_ste_constant(self, n):
mu2, R = self.stats()[:2]
return R / (mu2 ** (2) * n)
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.
@ -765,17 +774,12 @@ class Kernel(object):
'Kernel smoothing'
Chapman and Hall, pp 74--75
'''
# TODO: NB: this routine can be made faster:
# TODO: replace the iteration in the end with a Newton Raphson scheme
A = np.atleast_2d(data)
d, n = A.shape
# R = int(mkernel(x)^2), mu2 = int(x^2*mkernel(x))
mu2, R, _Rdd = self.stats()
amise_constant = (8 * sqrt(pi) * R / (3 * mu2 ** 2 * n)) ** (1. / 5)
ste_constant = R / (mu2 ** (2) * n)
amise_constant = self.get_amise_constant(n)
ste_constant = self.get_ste_constant(n)
sigmaA = self.hns(A) / amise_constant
if h0 is None:
@ -784,21 +788,12 @@ class Kernel(object):
h = np.asarray(h0, dtype=float)
nfft = inc * 2
amin = A.min(axis=1) # Find the minimum value of A.
amax = A.max(axis=1) # Find the maximum value of A.
arange = amax - amin # Find the range of A.
# xa holds the x 'axis' vector, defining a grid of x values where
# the k.d. function will be evaluated.
ax1 = amin - arange / 8.0
bx1 = amax + arange / 8.0
ax1, bx1 = self._get_grid_limits(A)
kernel2 = Kernel('gauss')
mu2, R, _Rdd = kernel2.stats()
ste_constant2 = R / (mu2 ** (2) * n)
fft = np.fft.fft
ifft = np.fft.ifft
ste_constant2 = kernel2.get_ste_constant(n)
for dim in range(d):
s = sigmaA[dim]
@ -822,8 +817,7 @@ class Kernel(object):
# Estimate psi6 given g2.
# kernel weights.
kw4, kw6 = kernel2.deriv4_6_8_10(xn / g2, numout=2)
# Apply fftshift to kw.
kw = np.r_[kw6, 0, kw6[-1:0:-1]]
kw = np.r_[kw6, 0, kw6[-1:0:-1]] # Apply fftshift to kw.
z = np.real(ifft(fft(c, nfft) * fft(kw))) # convolution.
psi6 = np.sum(c * z[:inc]) / (n * (n - 1) * g2 ** 7)
@ -892,24 +886,11 @@ class Kernel(object):
'''
A = np.atleast_2d(data)
d, n = A.shape
# R = int(mkernel(x)^2), mu2 = int(x^2*mkernel(x))
mu2, R, _Rdd = self.stats()
ste_constant = R / (n * mu2 ** 2)
amin = A.min(axis=1) # Find the minimum value of A.
amax = A.max(axis=1) # Find the maximum value of A.
arange = amax - amin # Find the range of A.
# xa holds the x 'axis' vector, defining a grid of x values where
# the k.d. function will be evaluated.
ax1 = amin - arange / 8.0
bx1 = amax + arange / 8.0
ste_constant = self.get_ste_constant(n)
ax1, bx1 = self._get_grid_limits(A)
kernel2 = Kernel('gauss')
mu2, R, _Rdd = kernel2.stats()
ste_constant2 = R / (mu2 ** (2) * n)
ste_constant2 = kernel2.get_ste_constant(n)
def fixed_point(t, N, I, a2):
''' this implements the function t-zeta*gamma^[L](t)'''
@ -929,8 +910,7 @@ class Kernel(object):
h = np.empty(d)
for dim in range(d):
ax = ax1[dim]
bx = bx1[dim]
ax, bx = ax1[dim], bx1[dim]
xa = np.linspace(ax, bx, inc)
R = bx - ax
@ -942,13 +922,11 @@ class Kernel(object):
I = np.asfarray(np.arange(1, inc)) ** 2
a2 = (a[1:] / 2) ** 2
def fun(t):
return fixed_point(t, N, I, a2)
x = np.linspace(0, 0.1, 150)
ai = x[0]
f0 = fun(ai)
f0 = fixed_point(ai, N, I, a2)
for bi in x[1:]:
f1 = fun(bi)
f1 = fixed_point(bi, N, I, a2)
if f1 * f0 <= 0:
# print('ai = %g, bi = %g' % (ai,bi))
break
@ -961,10 +939,12 @@ class Kernel(object):
# use fzero to solve the equation t=zeta*gamma^[5](t)
try:
t_star = optimize.brentq(fun, a=ai, b=bi)
except:
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')
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)
@ -1022,11 +1002,8 @@ class Kernel(object):
A = np.atleast_2d(data)
d, n = A.shape
# R= int(mkernel(x)^2), mu2= int(x^2*mkernel(x))
mu2, R, _Rdd = self.stats()
amise_constant = (8 * sqrt(pi) * R / (3 * mu2 ** 2 * n)) ** (1. / 5)
ste_constant = R / (mu2 ** (2) * n)
amise_constant = self.get_amise_constant(n)
ste_constant = self.get_ste_constant(n)
sigmaA = self.hns(A) / amise_constant
if h0 is None:
@ -1035,18 +1012,9 @@ class Kernel(object):
h = np.asarray(h0, dtype=float)
nfft = inc * 2
amin = A.min(axis=1) # Find the minimum value of A.
amax = A.max(axis=1) # Find the maximum value of A.
arange = amax - amin # Find the range of A.
# xa holds the x 'axis' vector, defining a grid of x values where
# the k.d. function will be evaluated.
ax1 = amin - arange / 8.0
bx1 = amax + arange / 8.0
ax1, bx1 = self._get_grid_limits(A)
fft = np.fft.fft
ifft = np.fft.ifft
for dim in range(d):
s = sigmaA[dim]
datan = A[dim] / s
@ -1078,8 +1046,7 @@ class Kernel(object):
psi4 = delta * z.sum()
h1 = (ste_constant / psi4) ** (1. / 5)
if count >= maxit:
warnings.warn('The obtained value did not converge.')
_assert_warn(count < maxit, 'The obtained value did not converge.')
h[dim] = h1 * s
# end # for dim loop
@ -1132,11 +1099,8 @@ class Kernel(object):
A = np.atleast_2d(data)
d, n = A.shape
# R= int(mkernel(x)^2), mu2= int(x^2*mkernel(x))
mu2, R, _Rdd = self.stats()
amise_constant = (8 * sqrt(pi) * R / (3 * mu2 ** 2 * n)) ** (1. / 5)
ste_constant = R / (mu2 ** (2) * n)
amise_constant = self.get_amise_constant(n)
ste_constant = self.get_ste_constant(n)
sigmaA = self.hns(A) / amise_constant
if hvec is None:
@ -1148,21 +1112,12 @@ class Kernel(object):
score = np.zeros(steps)
nfft = inc * 2
amin = A.min(axis=1) # Find the minimum value of A.
amax = A.max(axis=1) # Find the maximum value of A.
arange = amax - amin # Find the range of A.
# xa holds the x 'axis' vector, defining a grid of x values where
# the k.d. function will be evaluated.
ax1 = amin - arange / 8.0
bx1 = amax + arange / 8.0
ax1, bx1 = self._get_grid_limits(A)
kernel2 = Kernel('gauss')
mu2, R, _Rdd = kernel2.stats()
ste_constant2 = R / (mu2 ** (2) * n)
fft = np.fft.fft
ifft = np.fft.ifft
mu2 = kernel2.stats()[0]
ste_constant2 = kernel2.get_ste_constant(n)
h = np.zeros(d)
hvec = hvec * (ste_constant2 / ste_constant) ** (1. / 5.)
@ -1220,8 +1175,9 @@ class Kernel(object):
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(kernel2(Y / sig1) / sig1 - 2 * kernel2(
Y / sig2) / sig2 + kernel2(Y / sig3) / sig3)
term2 = np.sum(kernel2(Y / sig1) / sig1 -
2 * kernel2(Y / sig2) / sig2 +
kernel2(Y / sig3) / sig3)
score[i] = 1. / (n * hvec[i] * 2. * sqrt(pi)) + term2 / n ** 2
@ -1240,6 +1196,11 @@ class Kernel(object):
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.
@ -1273,31 +1234,17 @@ class Kernel(object):
A = np.atleast_2d(data)
d, n = A.shape
# R= int(mkernel(x)^2), mu2= int(x^2*mkernel(x))
mu2, R, _Rdd = self.stats()
amise_constant = (8 * sqrt(pi) * R / (3 * n * mu2 ** 2)) ** (1. / 5)
ste_constant = R / (n * mu2 ** 2)
amise_constant = self.get_amise_constant(n)
ste_constant = self.get_ste_constant(n)
sigmaA = self.hns(A) / amise_constant
nfft = inc * 2
amin = A.min(axis=1) # Find the minimum value of A.
amax = A.max(axis=1) # Find the maximum value of A.
arange = amax - amin # Find the range of A.
# xa holds the x 'axis' vector, defining a grid of x values where
# the k.d. function will be evaluated.
ax1 = amin - arange / 8.0
bx1 = amax + arange / 8.0
ax1, bx1 = self._get_grid_limits(A)
kernel2 = Kernel('gauss')
mu2, _R, _Rdd = kernel2.stats()
fft = np.fft.fft
ifft = np.fft.ifft
h = np.zeros(d)
for dim in range(d):
s = sigmaA[dim]
@ -1330,8 +1277,8 @@ class Kernel(object):
kw0 = kernel2.deriv4_6_8_10(xn / gi, numout=ix)
if ix > 1:
kw0 = kw0[-1]
# Apply 'fftshift' to kw.
kw = np.r_[kw0, 0, kw0[inc - 1:0:-1]]
kw = np.r_[kw0, 0, kw0[inc - 1:0:-1]] # Apply 'fftshift'
# Perform the convolution.
z = np.real(ifft(fft(c, nfft) * fft(kw)))

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