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pywafo/wafo/transform/models.py

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'''
Transform Gaussian models
-------------------------
TrHermite
TrOchi
TrLinear
'''
# !/usr/bin/env python
from __future__ import division, absolute_import
from scipy.optimize import brentq # @UnresolvedImport
from numpy import (sqrt, atleast_1d, abs, imag, sign, where, cos, arccos, ceil,
expm1, log1p, pi)
import numpy as np
import warnings
from .core import TrCommon, TrData
__all__ = ['TrHermite', 'TrLinear', 'TrOchi']
_EPS = np.finfo(float).eps
_example = '''
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> import wafo.transform.models as tm
>>> std = 7./4
>>> g = tm.<generic>(sigma=std, ysigma=std)
Simulate a Transformed Gaussian process:
>>> Sj = sm.Jonswap(Hm0=4*std, Tp=11)
>>> w = np.linspace(0,4,256)
>>> S = Sj.tospecdata(w) # Make spectrum object from numerical values
>>> ys = S.sim(ns=15000) # Simulated in the Gaussian world
>>> me, va, sk, ku = S.stats_nl(moments='mvsk')
>>> g2 = tm.<generic>(mean=me, var=va, skew=sk, kurt=ku, ysigma=std)
>>> xs = g2.gauss2dat(ys[:,1:]) # Transformed to the real world
'''
def _assert(cond, msg):
if not cond:
raise ValueError(msg)
def _assert_warn(cond, msg):
if not cond:
warnings.warn(msg)
class TrCommon2(TrCommon):
__doc__ = TrCommon.__doc__ # @ReservedAssignment
def trdata(self, x=None, xnmin=-5, xnmax=5, n=513):
"""
Return a discretized transformation model.
Parameters
----------
x : vector (default sigma*linspace(xnmin,xnmax,n)+mean)
xnmin : real, scalar
minimum on normalized scale
xnmax : real, scalar
maximum on normalized scale
n : integer, scalar
number of evaluation points
Returns
-------
t0 : real, scalar
a measure of departure from the Gaussian model calculated as
trapz((xn-g(x))**2., xn) where int. limits is given by X.
"""
if x is None:
xn = np.linspace(xnmin, xnmax, n)
x = self.sigma * xn + self.mean
else:
xn = (x - self.mean) / self.sigma
yn = (self._dat2gauss(x) - self.ymean) / self.ysigma
return TrData(yn, x, mean=self.mean, sigma=self.sigma)
class TrHermite(TrCommon2):
__doc__ = TrCommon2.__doc__.replace('<generic>', 'Hermite'
) + """
pardef : scalar, integer
1 Winterstein et. al. (1994) parametrization [1]_ (default)
2 Winterstein (1988) parametrization [2]_
Description
-----------
The hermite transformation model is monotonic cubic polynomial, calibrated
such that the first 4 moments of the transformed model G(y)=g^-1(y) match
the moments of the true process. The model is given as:
g(x) = xn - c3(xn**2-1) - c4*(xn**3-3*xn)
for kurt<3 (hardening model) where
xn = (x-mean)/sigma
c3 = skew/6
c4 = (kurt-3)/24.
or
G(y) = mean + K*sigma*[ y + c3(y**2-1) + c4*(y**3-3*y) ]
for kurt>=3 (softening model) where
y = g(x) = G**-1(x)
K = 1/sqrt(1+2*c3^2+6*c4^2)
If pardef = 1 :
c3 = skew/6*(1-0.015*abs(skew)+0.3*skew^2)/(1+0.2*(kurt-3))
c4 = 0.1*((1+1.25*(kurt-3))^(1/3)-1)*c41
c41 = (1-1.43*skew^2/(kurt-3))^(1-0.1*(kurt)^0.8)
If pardef = 2 :
c3 = skew/(6*(1+6*c4))
c4 = [sqrt(1+1.5*(kurt-3))-1]/18
Example:
--------
""" + _example.replace('<generic>', 'TrHermite') + """
>>> np.allclose(g.dist2gauss(), 0.88230868748851499)
True
>>> np.allclose(g2.dist2gauss(), 1.1411663205144991)
True
See also
--------
SpecData1d.stats_nl
wafo.transform.TrOchi
wafo.objects.LevelCrossings.trdata
wafo.objects.TimeSeries.trdata
References
----------
.. [1] Winterstein, S.R, Ude, T.C. and Kleiven, G. (1994)
"Springing and slow drift responses:
predicted extremes and fatigue vs. simulation"
In Proc. 7th International behaviour of Offshore structures, (BOSS)
Vol. 3, pp.1-15
.. [2] Winterstein, S.R. (1988)
'Nonlinear vibration models for extremes and fatigue.'
J. Engng. Mech., ASCE, Vol 114, No 10, pp 1772-1790
"""
def __init__(self, *args, **kwds):
super(TrHermite, self).__init__(*args, **kwds)
self._c3 = None
self._c4 = None
self._forward = None
self._backward = None
self._x_limit = None
self.pardef = kwds.get('pardef', 1)
self.set_poly()
@property
def pardef(self):
return self._pardef
@pardef.setter
def pardef(self, pardef):
self._pardef = pardef
if pardef == 2:
self._softening_parameters = self._winterstein1988
else:
self._softening_parameters = self._winterstein1994
def _check_c3_c4(self, c3, c4):
_assert(np.isfinite(c3) and np.isfinite(c4),
'Unable to calculate the polynomial')
if abs(c4) < sqrt(_EPS):
c4 = 0.0
return c4
def _winterstein1988(self, skew, excess_kurtosis):
"""Winterstein 1988 parametrization"""
_assert_warn(skew ** 2 <= 8 * (excess_kurtosis + 3.) / 9,
'Kurtosis too low compared to the skewness')
c4 = (sqrt(1. + 1.5 * excess_kurtosis) - 1.) / 18.
c3 = skew / (6. * (1 + 6. * c4))
c4 = self._check_c3_c4(c3, c4)
return c3, c4
def _winterstein1994(self, skew, excess_kurtosis):
"""Winterstein et. al. 1994 parametrization
intended to apply for the range:
0 <= excess_kurtosis < 12 and 0<= skew^2 < 2*excess_kurtosis/3
"""
_assert_warn(skew ** 2 <= 2 * (excess_kurtosis) / 3,
'Kurtosis too low compared to the skewness')
_assert_warn(0 <= excess_kurtosis < 12,
'Kurtosis must be between 0 and 12')
c3 = (skew / 6 * (1 - 0.015 * abs(skew) + 0.3 * skew ** 2) /
(1 + 0.2 * excess_kurtosis))
if excess_kurtosis == 0.:
c4 = 0.0
else:
expon = 1. - 0.1 * (excess_kurtosis + 3.) ** 0.8
c41 = (1. - 1.43 * skew ** 2. / excess_kurtosis) ** (expon)
c4 = 0.1 * ((1. + 1.25 * excess_kurtosis) ** (1. / 3.) - 1.) * c41
c4 = self._check_c3_c4(c3, c4)
return c3, c4
def _hardening_parameters(self, skew, excess_kurtosis):
c4 = excess_kurtosis / 24.
c3 = skew / 6.
c4 = self._check_c3_c4(c3, c4)
return c3, c4
def _set_x_limit(self, root, polynom):
"""Compute where it is possible to invert the polynomial"""
if self.kurt <= 3.:
self._x_limit = root
else:
self._x_limit = self.sigma * polynom(root) + self.mean
txt1 = '''
The polynomial is not a strictly increasing function.
The derivative of g(x) is infinite at x = %g''' % self._x_limit
warnings.warn(txt1)
def _check_monotonicity(self, p):
dp = p.deriv(m=1) # derivative
roots = dp.r # roots of the derivative
roots = roots[where(abs(imag(roots)) < _EPS)] # Keep only real roots
if roots.size > 0:
self._set_x_limit(roots, p)
def _set_hardening_model(self):
skew, excess_kurtosis = self.skew, self.kurt - 3.0
c3, c4 = self._hardening_parameters(skew, excess_kurtosis)
p = np.poly1d([-c4, -c3, 1. + 3. * c4, c3])
self._forward = p
self._backward = lambda yn: self._poly_inv(self._forward, yn)
# Check if it is a strictly increasing function.
self._check_monotonicity(p)
def _set_softening_model(self):
skew, excess_kurtosis = self.skew, self.kurt - 3.0
c3, c4 = self._softening_parameters(skew, excess_kurtosis)
Km1 = np.sqrt(1. + 2. * c3 ** 2 + 6 * c4 ** 2)
# backward G
p = np.poly1d(np.r_[c4, c3, 1. - 3. * c4, -c3] / Km1)
self._backward = p
self._forward = lambda yn: self._poly_inv(self._backward, yn)
# Check if it is a strictly increasing function.
self._check_monotonicity(p)
def set_poly(self):
'''
Set poly function from stats (i.e., mean, sigma, skew and kurt)
'''
if self.kurt <= 3.0:
self._set_hardening_model()
else:
self._set_softening_model()
def check_forward(self, x):
if self._x_limit is not None:
x00 = self._x_limit
txt2 = 'for the given interval x = [%g, %g]' % (x[0], x[-1])
if any(np.logical_and(x[0] <= x00, x00 <= x[-1])):
cdef = 1
else:
cdef = sum(np.logical_xor(x00 <= x[0], x00 <= x[-1]))
if np.mod(cdef, 2):
errtxt = 'Unable to invert the polynomial \n %s' % txt2
raise ValueError(errtxt)
np.disp(
'However, successfully inverted the polynomial\n %s' % txt2)
def _dat2gauss(self, x, *xi):
if len(xi) > 0:
raise ValueError('Transforming derivatives is not implemented!')
xn = atleast_1d(x)
self.check_forward(xn)
xn = (xn - self.mean) / self.sigma
if self._forward is None:
# Inverting the polynomial
yn = self._poly_inv(self._backward, xn)
else:
yn = self._forward(xn)
return yn * self.ysigma + self.ymean
def _gauss2dat(self, y, *yi):
if len(yi) > 0:
raise ValueError('Transforming derivatives is not implemented!')
yn = (atleast_1d(y) - self.ymean) / self.ysigma
# self.check_forward(y)
if self._backward is None:
# Inverting the polynomial
xn = self._poly_inv(self._forward, yn)
else:
xn = self._backward(yn)
return self.sigma * xn + self.mean
def _solve_quadratic(self, p, xn):
# Quadratic: Solve a*u**2+b*u+c = xn
coefs = p.coeffs
a = coefs[0]
b = coefs[1]
c = coefs[2] - xn
t = 0.5 * (b + sign(b) * sqrt(b ** 2 - 4 * a * c))
# so1 = t/a # largest solution
so2 = -c / t # smallest solution
return so2
def _poly_inv(self, p, xn):
'''
Invert polynomial
'''
if p.order < 2:
return xn
elif p.order == 2:
return self._solve_quadratic(p, xn)
elif p.order == 3:
return self._solve_third_order(p, xn)
def _solve_third_order(self, p, xn):
# Solve
# K*(c4*u^3+c3*u^2+(1-3*c4)*u-c3) = xn = (x-ma)/sa
# -c4*xn^3-c3*xn^2+(1+3*c4)*xn+c3 = u
coefs = p.coeffs[1::] / p.coeffs[0]
a = coefs[0]
b = coefs[1]
c = coefs[2] - xn / p.coeffs[0]
x0 = a / 3.
# substitue xn = z-x0 and divide by c4 => z^3 + 3*p1*z+2*q0 = 0
p1 = b / 3 - x0 ** 2
# p1 = (b-a**2/3)/3
# q0 = (c + x0*(2.*x0/3.-b))/2.
# q0 = x0**3 -a*b/6 +c/2
q0 = x0 * (x0 ** 2 - b / 2) + c / 2
# z^3+3*p1*z+2*q0=0
# c3 = self._c3
# c4 = self._c4
# b1 = 1./(3.*c4)
# x0 = c3*b1
# % substitue u = z-x0 and divide by c4 => z^3 + 3*c*z+2*q0 = 0
# p1 = b1-1.-x0**2.
# Km1 = np.sqrt(1.+2.*c3**2+6*c4**2)
# q0 = x0**3-1.5*b1*(x0+xn*Km1)
# q0 = x0**3-1.5*b1*(x0+xn)
if self._x_limit is not None: # % Three real roots
d = sqrt(-p1)
theta1 = arccos(-q0 / d ** 3) / 3
th2 = np.r_[0, -2 * pi / 3, 2 * pi / 3]
x1 = abs(2 * d * cos(theta1[ceil(len(xn) / 2)] + th2) - x0)
ix = x1.argmin() # choose the smallest solution
return 2. * d * cos(theta1 + th2[ix]) - x0
else: # Only one real root exist
q1 = sqrt((q0) ** 2 + p1 ** 3)
# Find the real root of the monic polynomial
A0 = (q1 - q0) ** (1. / 3.)
B0 = -(q1 + q0) ** (1. / 3.)
return A0 + B0 - x0 # real root
# The other complex roots are given by
# x= -(A0+B0)/2+(A0-B0)*sqrt(3)/2-x0
# x=-(A0+B0)/2+(A0-B0)*sqrt(-3)/2-x0
class TrLinear(TrCommon2):
__doc__ = TrCommon2.__doc__.replace('<generic>', 'Linear'
) + """
Description
-----------
The linear transformation model is monotonic linear polynomial, calibrated
such that the first 2 moments of the transformed model G(y)=g^-1(y) match
the moments of the true process.
Example:
--------
""" + _example.replace('<generic>', 'TrLinear') + """
>>> np.allclose(g.dist2gauss(), 0)
True
>>> np.allclose(g2.dist2gauss(), 0)
True
See also
--------
TrOchi
TrHermite
SpecData1D.stats_nl
LevelCrossings.trdata
TimeSeries.trdata
spec2skew, ochitr, lc2tr, dat2tr
"""
def _dat2gauss(self, x, *xi):
sratio = atleast_1d(self.ysigma / self.sigma)
y = (atleast_1d(x) - self.mean) * sratio + self.ymean
if len(xi) > 0:
y = [y, ] + [ix * sratio for ix in xi]
return y
def _gauss2dat(self, y, *yi):
sratio = atleast_1d(self.sigma / self.ysigma)
x = (atleast_1d(y) - self.ymean) * sratio + self.mean
if len(yi) > 0:
x = [x, ] + [iy * sratio for iy in yi]
return x
class TrOchi(TrCommon2):
__doc__ = TrCommon2.__doc__.replace('<generic>', 'Ochi'
) + """
Description
-----------
The Ochi transformation model is a monotonic exponential function,
calibrated such that the first 3 moments of the transformed model
G(y)=g^-1(y) match the moments of the true process. However, the
skewness is limited by ABS(SKEW)<2.82. According to Ochi it is
appropriate for a process with very strong non-linear characteristics.
The model is given as:
g(x) = ((1-exp(-gamma*(x-mean)/sigma))/gamma-mean2)/sigma2
where
gamma = 1.28*a for x>=mean
3*a otherwise
mean,
sigma = standard deviation and mean, respectively, of the process.
mean2,
sigma2 = normalizing parameters in the transformed world, i.e., to
make the gaussian process in the transformed world is N(0,1).
The unknown parameters a, mean2 and sigma2 are found by solving the
following non-linear equations:
a*(sigma2^2+mean2^2)+mean2 = 0
sigma2^2-2*a^2*sigma2^4 = 1
2*a*sigma2^4*(3-8*a^2*sigma2^2) = skew
Note
----
Transformation, g, does not have continous derivatives of 2'nd order or
higher.
Example
-------
""" + _example.replace('<generic>', 'TrOchi') + """
>>> np.allclose(g.dist2gauss(), 1.410698801056657)
True
>>> np.allclose(g2.dist2gauss(), 1.988807188766706)
True
See also
--------
spec2skew, hermitetr, lc2tr, dat2tr
References
----------
Ochi, M.K. and Ahn, K. (1994)
'Non-Gaussian probability distribution of coastal waves.'
In Proc. 24th Conf. Coastal Engng, Vol. 1, pp 482-496
Michel K. Ochi (1998),
"OCEAN WAVES, The stochastic approach",
OCEAN TECHNOLOGY series 6, Cambridge, pp 255-275.
"""
def __init__(self, *args, **kwds):
super(TrOchi, self).__init__(*args, **kwds)
self.kurt = None
self._phat = None
self._par_from_stats()
def _par_from_stats(self):
skew = self.skew
if abs(skew) > 2.82842712474619:
raise ValueError('Skewness must be less than 2.82842')
mean1 = self.mean
sigma1 = self.sigma
if skew == 0:
self._phat = [sigma1, mean1, 0, 0, 1, 0]
return
# Solve the equations to obtain the gamma parameters:
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# a*(sig2^2+ma2^2)+ma2 = 0
# sig2^2-2*a^2*sig2^4 = E(y^2) % =1
# 2*a*sig2^4*(3-8*a^2*sig2^2) = E(y^3) % = skew
# Let x = [a sig2^2 ]
# Set up the 2D non-linear equations for a and sig2^2:
# g1='[x(2)-2.*x(1).^2.*x(2).^2-P1,
# 2.*x(1).*x(2).^2.*(3-8.*x(1).^2.*x(2))-P2 ]'
# Or solve the following 1D non-linear equation for sig2^2:
def g2(x):
return (-sqrt(abs(x - 1) * 2) * (3. * x - 4 * abs(x - 1)) +
abs(skew))
a1 = 1. # Start interval where sig2^2 is located.
a2 = 2.
sig22 = brentq(g2, a1, a2) # % smallest solution for sig22
a = sign(skew) * sqrt(abs(sig22 - 1) / 2) / sig22
gam_a = 1.28 * a
gam_b = 3 * a
sigma2 = sqrt(sig22)
# Solve the following 2nd order equation to obtain ma2
# a*(sig2^2+ma2^2)+ma2 = 0
my2 = (-1. - sqrt(1. - 4. * a ** 2 * sig22)) / a # % Largest mean
mean2 = a * sig22 / my2 # % choose the smallest mean
self._phat = [sigma1, mean1, gam_a, gam_b, sigma2, mean2]
return
def _get_par(self):
'''
Returns ga, gb, sigma2, mean2
'''
if (self._phat is None or self.sigma != self._phat[0] or
self.mean != self._phat[1]):
self._par_from_stats()
# sigma1 = self._phat[0]
# mean1 = self._phat[1]
ga = self._phat[2]
gb = self._phat[3]
sigma2 = self._phat[4]
mean2 = self._phat[5]
return ga, gb, sigma2, mean2
def _dat2gauss(self, x, *xi):
if len(xi) > 0:
raise ValueError('Transforming derivatives is not implemented!')
ga, gb, sigma2, mean2 = self._get_par()
mean = self.mean
sigma = self.sigma
xn = atleast_1d(x)
shape0 = xn.shape
xn = (xn.ravel() - mean) / sigma
igp, = where(0 <= xn)
igm, = where(xn < 0)
g = xn.copy()
if ga != 0:
np.put(g, igp, (-expm1(-ga * xn[igp])) / ga)
if gb != 0:
np.put(g, igm, (-expm1(-gb * xn[igm])) / gb)
g.shape = shape0
return (g - mean2) * self.ysigma / sigma2 + self.ymean
def _gauss2dat(self, y, *yi):
if len(yi) > 0:
raise ValueError('Transforming derivatives is not implemented!')
ga, gb, sigma2, mean2 = self._get_par()
mean = self.mean
sigma = self.sigma
yn = (atleast_1d(y) - self.ymean) / self.ysigma
xn = sigma2 * yn.ravel() + mean2
igp, = where(0 <= xn)
igm, = where(xn < 0)
if ga != 0:
np.put(xn, igp, -log1p(-ga * xn[igp]) / ga)
if gb != 0:
np.put(xn, igm, -log1p(-gb * xn[igm]) / gb)
xn.shape = yn.shape
return sigma * xn + mean
def main():
import pylab
g = TrHermite(skew=0.1, kurt=3.01)
g.dist2gauss()
# g = TrOchi(skew=0.56)
x = np.linspace(-5, 5)
y = g(x)
pylab.plot(np.abs(x - g.gauss2dat(y)))
# pylab.plot(x,y,x,x,':',g.gauss2dat(y),y,'r')
pylab.show()
np.disp('finito')
if __name__ == '__main__':
if True: # False: #
import doctest
doctest.testmod()
else:
main()
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