def psi(self, t):
"""DOG wavelet as described in Torrence and Compo (1998)
The derivative of a Gaussian of order n can be determined using
the probabilistic Hermite polynomials. They are explicitly
written as:
Hn(x) = 2 ** (-n / s) * n! * sum ((-1) ** m) * (2 ** 0.5 *
x) ** (n - 2 * m) / (m! * (n - 2*m)!)
or in the recursive form:
Hn(x) = x * Hn(x) - nHn-1(x)
Source: http://www.ask.com/wiki/Hermite_polynomials
"""
p = hermitenorm(self.m)
return (-1) ** (self.m + 1) * polyval(p, t) * np.exp(-t ** 2 / 2) / np.sqrt(gamma(self.m + 0.5))
def psi(self, t):
"""DOG wavelet as described in Torrence and Compo (1998)
The derivative of a Gaussian of order n can be determined using
the probabilistic Hermite polynomials. They are explicitly
written as:
Hn(x) = 2 ** (-n / s) * n! * sum ((-1) ** m) * (2 ** 0.5 *
x) ** (n - 2 * m) / (m! * (n - 2*m)!)
or in the recursive form:
Hn(x) = x * Hn(x) - nHn-1(x)
Source: http://www.ask.com/wiki/Hermite_polynomials
"""
p = hermitenorm(self.m)
return ((-1) ** (self.m + 1) * polyval(p, t) * exp(-t ** 2 / 2) / sqrt(gamma(self.m + 0.5)))
def test_he_roots():
rootf = orth.he_roots
evalf = orth.eval_hermitenorm
weightf = orth.hermitenorm(5).weight_func
verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 5)
verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 25, atol=1e-13)
verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 100, atol=1e-12)
x, w = orth.he_roots(5, False)
y, v, m = orth.he_roots(5, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
muI, muI_err = integrate.quad(weightf, -np.inf, np.inf)
assert_allclose(m, muI, rtol=muI_err)
assert_raises(ValueError, orth.he_roots, 0)
assert_raises(ValueError, orth.he_roots, 3.3)
def test_roots_hermitenorm():
rootf = sc.roots_hermitenorm
evalf = orth.eval_hermitenorm
weightf = orth.hermitenorm(5).weight_func
verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 5)
verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 25, atol=1e-13)
verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 100, atol=1e-12)
x, w = sc.roots_hermitenorm(5, False)
y, v, m = sc.roots_hermitenorm(5, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
muI, muI_err = integrate.quad(weightf, -np.inf, np.inf)
assert_allclose(m, muI, rtol=muI_err)
assert_raises(ValueError, sc.roots_hermitenorm, 0)
assert_raises(ValueError, sc.roots_hermitenorm, 3.3)
def test_hermitenorm(self):
# He_n(x) = 2**(-n/2) H_n(x/sqrt(2))
psub = np.poly1d([1.0 / sqrt(2), 0])
H0 = orth.hermitenorm(0)
H1 = orth.hermitenorm(1)
H2 = orth.hermitenorm(2)
H3 = orth.hermitenorm(3)
H4 = orth.hermitenorm(4)
H5 = orth.hermitenorm(5)
he0 = orth.hermite(0)(psub)
he1 = orth.hermite(1)(psub) / sqrt(2)
he2 = orth.hermite(2)(psub) / 2.0
he3 = orth.hermite(3)(psub) / (2 * sqrt(2))
he4 = orth.hermite(4)(psub) / 4.0
he5 = orth.hermite(5)(psub) / (4.0 * sqrt(2))
assert_array_almost_equal(H0.c, he0.c, 13)
assert_array_almost_equal(H1.c, he1.c, 13)
assert_array_almost_equal(H2.c, he2.c, 13)
assert_array_almost_equal(H3.c, he3.c, 13)
assert_array_almost_equal(H4.c, he4.c, 13)
assert_array_almost_equal(H5.c, he5.c, 13)
def test_hermitenorm(self):
# He_n(x) = 2**(-n/2) H_n(x/sqrt(2))
psub = np.poly1d([1.0/sqrt(2),0])
H0 = orth.hermitenorm(0)
H1 = orth.hermitenorm(1)
H2 = orth.hermitenorm(2)
H3 = orth.hermitenorm(3)
H4 = orth.hermitenorm(4)
H5 = orth.hermitenorm(5)
he0 = orth.hermite(0)(psub)
he1 = orth.hermite(1)(psub) / sqrt(2)
he2 = orth.hermite(2)(psub) / 2.0
he3 = orth.hermite(3)(psub) / (2*sqrt(2))
he4 = orth.hermite(4)(psub) / 4.0
he5 = orth.hermite(5)(psub) / (4.0*sqrt(2))
assert_array_almost_equal(H0.c,he0.c,13)
assert_array_almost_equal(H1.c,he1.c,13)
assert_array_almost_equal(H2.c,he2.c,13)
assert_array_almost_equal(H3.c,he3.c,13)
assert_array_almost_equal(H4.c,he4.c,13)
assert_array_almost_equal(H5.c,he5.c,13)
def wavelet_inverse(wave, scale, dt, dj=0.25, mother="MORLET", param=-1):
"""Inverse continuous wavelet transform
Torrence and Compo (1998), eq. (11)
INPUTS
waves (array like):
WAVE is the WAVELET transform. This is a complex array.
scale (array like):
the vector of scale indices
dt (float) :
amount of time between each original value, i.e. the sampling time.
dj (float, optional) :
the spacing between discrete scales. Default is 0.25.
A smaller # will give better scale resolution, but be slower to plot.
mother (string, optional) :
the mother wavelet function.
The choices are 'MORLET', 'PAUL', or 'DOG'
PARAM = the mother wavelet parameter.
For 'MORLET' this is k0 (wavenumber), default is 6.
For 'PAUL' this is m (order), default is 4.
For 'DOG' this is m (m-th derivative), default is 2.
OUTPUTS
iwave (array like) :
Inverse wavelet transform.
"""
import numpy as np
j1, n = wave.shape
J1 = len(scale)
if not j1 == J1:
print j1, n, J1
raise Exception("Input array are inconsistent")
sj = np.dot(scale.reshape(len(scale), 1), np.ones((1, n)))
#
mother = mother.upper()
# psi0 comes from Table 1,2 Torrence and Compo (1998)
# Cdelta comes from Table 2 Torrence and Compo (1998)
if mother == 'MORLET': #----------------------------------- Morlet
if (param == -1): param = 6.
psi0 = np.pi**(-0.25)
if param == 6.:
Cdelta = 0.776
elif mother == 'PAUL': #-------------------------------- Paul
if (param == -1): param = 4.
m = param
psi0 = np.real(2.**m * 1j**m * np.prod(np.arange(2, m + 1)) /
np.sqrt(np.pi * np.prod(np.arange(2, 2 * m + 1))) *
(1**(-(m + 1))))
if m == 4.:
Cdelta = 1.132
elif mother == 'DOG': #-------------------------------- DOG
if (param == -1): param = 2.
m = param
from scipy.special import gamma
from numpy.lib.polynomial import polyval
from scipy.special.orthogonal import hermitenorm
p = hermitenorm(m)
psi0 = (-1)**(m + 1) / np.sqrt(gamma(m + 0.5)) * polyval(p, 0)
print psi0
if m == 2.:
Cdelta = 3.541
if m == 6.:
Cdelta = 1.966
else:
raise Exception("Mother must be one of MORLET,PAUL,DOG")
#eq. (11) in Torrence and Compo (1998)
iwave = dj * np.sqrt(dt) / Cdelta / psi0 * (np.real(wave) /
sj**0.5).sum(axis=0)
return iwave
def wavelet_inverse(wave, scale, dt, dj=0.25, mother="MORLET",param=-1):
"""Inverse continuous wavelet transform
Torrence and Compo (1998), eq. (11)
INPUTS
waves (array like):
WAVE is the WAVELET transform. This is a complex array.
scale (array like):
the vector of scale indices
dt (float) :
amount of time between each original value, i.e. the sampling time.
dj (float, optional) :
the spacing between discrete scales. Default is 0.25.
A smaller # will give better scale resolution, but be slower to plot.
mother (string, optional) :
the mother wavelet function.
The choices are 'MORLET', 'PAUL', or 'DOG'
PARAM = the mother wavelet parameter.
For 'MORLET' this is k0 (wavenumber), default is 6.
For 'PAUL' this is m (order), default is 4.
For 'DOG' this is m (m-th derivative), default is 2.
OUTPUTS
iwave (array like) :
Inverse wavelet transform.
"""
import numpy as np
j1, n = wave.shape
J1 = len(scale)
if not j1 == J1:
print j1,n,J1
raise Exception("Input array are inconsistent")
sj = np.dot(scale.reshape(len(scale),1),np.ones((1,n)))
#
mother = mother.upper()
# psi0 comes from Table 1,2 Torrence and Compo (1998)
# Cdelta comes from Table 2 Torrence and Compo (1998)
if mother=='MORLET': #----------------------------------- Morlet
if (param == -1): param = 6.
psi0=np.pi**(-0.25)
if param==6.:
Cdelta = 0.776
elif mother=='PAUL': #-------------------------------- Paul
if (param == -1): param = 4.
m = param
psi0=np.real(2.**m*1j**m*np.prod(np.arange(2, m + 1))/np.sqrt(np.pi*np.prod(np.arange(2,2*m+1)))*(1**(-(m+1))))
if m==4.:
Cdelta = 1.132
elif mother=='DOG': #-------------------------------- DOG
if (param == -1): param = 2.
m = param
from scipy.special import gamma
from numpy.lib.polynomial import polyval
from scipy.special.orthogonal import hermitenorm
p = hermitenorm(m)
psi0=(-1)**(m+1)/np.sqrt(gamma(m+0.5))*polyval(p, 0)
print psi0
if m==2.:
Cdelta=3.541
if m==6.:
Cdelta=1.966
else:
raise Exception("Mother must be one of MORLET,PAUL,DOG")
#eq. (11) in Torrence and Compo (1998)
iwave = dj * np.sqrt(dt) / Cdelta /psi0 * (np.real(wave) / sj**0.5).sum(axis=0)
return iwave
