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590 lines
19 KiB
Python
590 lines
19 KiB
Python
'''
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Transform Gaussian models
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-------------------------
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TrHermite
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TrOchi
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TrLinear
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'''
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# !/usr/bin/env python
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from __future__ import division, absolute_import
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from scipy.optimize import brentq # @UnresolvedImport
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from numpy import (sqrt, atleast_1d, abs, imag, sign, where, cos, arccos, ceil,
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expm1, log1p, pi)
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import numpy as np
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import warnings
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from wafo.transform.core import TrCommon, TrData
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__all__ = ['TrHermite', 'TrLinear', 'TrOchi']
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_EPS = np.finfo(float).eps
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_example = '''
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>>> import numpy as np
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>>> import wafo.spectrum.models as sm
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>>> import wafo.transform.models as tm
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>>> std = 7./4
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>>> g = tm.<generic>(sigma=std, ysigma=std)
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Simulate a Transformed Gaussian process:
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>>> Sj = sm.Jonswap(Hm0=4*std, Tp=11)
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>>> w = np.linspace(0,4,256)
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>>> S = Sj.tospecdata(w) # Make spectrum object from numerical values
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>>> ys = S.sim(ns=15000) # Simulated in the Gaussian world
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>>> me, va, sk, ku = S.stats_nl(moments='mvsk')
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>>> g2 = tm.<generic>(mean=me, var=va, skew=sk, kurt=ku, ysigma=std)
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>>> xs = g2.gauss2dat(ys[:,1:]) # Transformed to the real world
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'''
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def _assert(cond, msg):
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if not cond:
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raise ValueError(msg)
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def _assert_warn(cond, msg):
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if not cond:
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warnings.warn(msg)
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class TrCommon2(TrCommon):
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__doc__ = TrCommon.__doc__ # @ReservedAssignment
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def trdata(self, x=None, xnmin=-5, xnmax=5, n=513):
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"""
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Return a discretized transformation model.
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Parameters
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----------
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x : vector (default sigma*linspace(xnmin,xnmax,n)+mean)
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xnmin : real, scalar
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minimum on normalized scale
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xnmax : real, scalar
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maximum on normalized scale
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n : integer, scalar
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number of evaluation points
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Returns
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-------
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t0 : real, scalar
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a measure of departure from the Gaussian model calculated as
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trapz((xn-g(x))**2., xn) where int. limits is given by X.
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"""
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if x is None:
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xn = np.linspace(xnmin, xnmax, n)
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x = self.sigma * xn + self.mean
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else:
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xn = (x - self.mean) / self.sigma
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yn = (self._dat2gauss(x) - self.ymean) / self.ysigma
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return TrData(yn, x, mean=self.mean, sigma=self.sigma)
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class TrHermite(TrCommon2):
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__doc__ = TrCommon2.__doc__.replace('<generic>', 'Hermite'
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) + """
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pardef : scalar, integer
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1 Winterstein et. al. (1994) parametrization [1]_ (default)
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2 Winterstein (1988) parametrization [2]_
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Description
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-----------
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The hermite transformation model is monotonic cubic polynomial, calibrated
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such that the first 4 moments of the transformed model G(y)=g^-1(y) match
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the moments of the true process. The model is given as:
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g(x) = xn - c3(xn**2-1) - c4*(xn**3-3*xn)
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for kurt<3 (hardening model) where
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xn = (x-mean)/sigma
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c3 = skew/6
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c4 = (kurt-3)/24.
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or
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G(y) = mean + K*sigma*[ y + c3(y**2-1) + c4*(y**3-3*y) ]
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for kurt>=3 (softening model) where
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y = g(x) = G**-1(x)
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K = 1/sqrt(1+2*c3^2+6*c4^2)
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If pardef = 1 :
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c3 = skew/6*(1-0.015*abs(skew)+0.3*skew^2)/(1+0.2*(kurt-3))
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c4 = 0.1*((1+1.25*(kurt-3))^(1/3)-1)*c41
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c41 = (1-1.43*skew^2/(kurt-3))^(1-0.1*(kurt)^0.8)
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If pardef = 2 :
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c3 = skew/(6*(1+6*c4))
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c4 = [sqrt(1+1.5*(kurt-3))-1]/18
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Example:
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--------
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""" + _example.replace('<generic>', 'TrHermite') + """
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>>> np.allclose(g.dist2gauss(), 0.88230868748851499)
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True
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>>> np.allclose(g2.dist2gauss(), 1.1411663205144991)
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True
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See also
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--------
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SpecData1d.stats_nl
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wafo.transform.TrOchi
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wafo.objects.LevelCrossings.trdata
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wafo.objects.TimeSeries.trdata
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References
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----------
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.. [1] Winterstein, S.R, Ude, T.C. and Kleiven, G. (1994)
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"Springing and slow drift responses:
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predicted extremes and fatigue vs. simulation"
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In Proc. 7th International behaviour of Offshore structures, (BOSS)
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Vol. 3, pp.1-15
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.. [2] Winterstein, S.R. (1988)
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'Nonlinear vibration models for extremes and fatigue.'
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J. Engng. Mech., ASCE, Vol 114, No 10, pp 1772-1790
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"""
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def __init__(self, *args, **kwds):
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super(TrHermite, self).__init__(*args, **kwds)
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self._c3 = None
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self._c4 = None
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self._forward = None
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self._backward = None
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self._x_limit = None
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self.pardef = kwds.get('pardef', 1)
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self.set_poly()
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@property
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def pardef(self):
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return self._pardef
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@pardef.setter
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def pardef(self, pardef):
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self._pardef = pardef
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if pardef == 2:
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self._softening_parameters = self._winterstein1988
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else:
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self._softening_parameters = self._winterstein1994
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def _check_c3_c4(self, c3, c4):
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_assert(np.isfinite(c3) and np.isfinite(c4),
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'Unable to calculate the polynomial')
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if abs(c4) < sqrt(_EPS):
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c4 = 0.0
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return c4
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def _winterstein1988(self, skew, excess_kurtosis):
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"""Winterstein 1988 parametrization"""
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_assert_warn(skew ** 2 <= 8 * (excess_kurtosis + 3.) / 9,
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'Kurtosis too low compared to the skewness')
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c4 = (sqrt(1. + 1.5 * excess_kurtosis) - 1.) / 18.
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c3 = skew / (6. * (1 + 6. * c4))
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c4 = self._check_c3_c4(c3, c4)
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return c3, c4
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def _winterstein1994(self, skew, excess_kurtosis):
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"""Winterstein et. al. 1994 parametrization
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intended to apply for the range:
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0 <= excess_kurtosis < 12 and 0<= skew^2 < 2*excess_kurtosis/3
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"""
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_assert_warn(skew ** 2 <= 2 * (excess_kurtosis) / 3,
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'Kurtosis too low compared to the skewness')
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_assert_warn(0 <= excess_kurtosis < 12,
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'Kurtosis must be between 0 and 12')
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c3 = (skew / 6 * (1 - 0.015 * abs(skew) + 0.3 * skew ** 2) /
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(1 + 0.2 * excess_kurtosis))
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if excess_kurtosis == 0.:
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c4 = 0.0
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else:
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expon = 1. - 0.1 * (excess_kurtosis + 3.) ** 0.8
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c41 = (1. - 1.43 * skew ** 2. / excess_kurtosis) ** (expon)
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c4 = 0.1 * ((1. + 1.25 * excess_kurtosis) ** (1. / 3.) - 1.) * c41
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c4 = self._check_c3_c4(c3, c4)
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return c3, c4
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def _hardening_parameters(self, skew, excess_kurtosis):
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c4 = excess_kurtosis / 24.
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c3 = skew / 6.
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c4 = self._check_c3_c4(c3, c4)
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return c3, c4
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def _set_x_limit(self, root, polynom):
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"""Compute where it is possible to invert the polynomial"""
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if self.kurt <= 3.:
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self._x_limit = root
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else:
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self._x_limit = self.sigma * polynom(root) + self.mean
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txt1 = '''
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The polynomial is not a strictly increasing function.
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The derivative of g(x) is infinite at x = %g''' % self._x_limit
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warnings.warn(txt1)
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def _check_monotonicity(self, p):
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dp = p.deriv(m=1) # derivative
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roots = dp.r # roots of the derivative
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roots = roots[where(abs(imag(roots)) < _EPS)] # Keep only real roots
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if roots.size > 0:
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self._set_x_limit(roots, p)
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def _set_hardening_model(self):
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skew, excess_kurtosis = self.skew, self.kurt - 3.0
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c3, c4 = self._hardening_parameters(skew, excess_kurtosis)
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p = np.poly1d([-c4, -c3, 1. + 3. * c4, c3])
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self._forward = p
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self._backward = lambda yn: self._poly_inv(self._forward, yn)
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# Check if it is a strictly increasing function.
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self._check_monotonicity(p)
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def _set_softening_model(self):
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skew, excess_kurtosis = self.skew, self.kurt - 3.0
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c3, c4 = self._softening_parameters(skew, excess_kurtosis)
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Km1 = np.sqrt(1. + 2. * c3 ** 2 + 6 * c4 ** 2)
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# backward G
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p = np.poly1d(np.r_[c4, c3, 1. - 3. * c4, -c3] / Km1)
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self._backward = p
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self._forward = lambda yn: self._poly_inv(self._backward, yn)
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# Check if it is a strictly increasing function.
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self._check_monotonicity(p)
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def set_poly(self):
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'''
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Set poly function from stats (i.e., mean, sigma, skew and kurt)
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'''
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if self.kurt <= 3.0:
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self._set_hardening_model()
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else:
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self._set_softening_model()
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def check_forward(self, x):
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if self._x_limit is not None:
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x00 = self._x_limit
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txt2 = 'for the given interval x = [%g, %g]' % (x[0], x[-1])
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if any(np.logical_and(x[0] <= x00, x00 <= x[-1])):
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cdef = 1
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else:
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cdef = sum(np.logical_xor(x00 <= x[0], x00 <= x[-1]))
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if np.mod(cdef, 2):
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errtxt = 'Unable to invert the polynomial \n %s' % txt2
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raise ValueError(errtxt)
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np.disp(
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'However, successfully inverted the polynomial\n %s' % txt2)
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def _dat2gauss(self, x, *xi):
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if len(xi) > 0:
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raise ValueError('Transforming derivatives is not implemented!')
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xn = atleast_1d(x)
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self.check_forward(xn)
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xn = (xn - self.mean) / self.sigma
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yn = self._forward(xn)
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return yn * self.ysigma + self.ymean
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def _gauss2dat(self, y, *yi):
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if len(yi) > 0:
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raise ValueError('Transforming derivatives is not implemented!')
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yn = (atleast_1d(y) - self.ymean) / self.ysigma
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# self.check_forward(y)
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xn = self._backward(yn)
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return self.sigma * xn + self.mean
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def _solve_quadratic(self, p, xn):
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# Quadratic: Solve a*u**2+b*u+c = xn
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coefs = p.coeffs
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a = coefs[0]
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b = coefs[1]
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c = coefs[2] - xn
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t = 0.5 * (b + sign(b) * sqrt(b ** 2 - 4 * a * c))
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# so1 = t/a # largest solution
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so2 = -c / t # smallest solution
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return so2
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def _poly_inv(self, p, xn):
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'''
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Invert polynomial
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'''
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if p.order < 2:
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return xn
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elif p.order == 2:
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return self._solve_quadratic(p, xn)
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elif p.order == 3:
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return self._solve_third_order(p, xn)
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def _solve_third_order(self, p, xn):
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# Solve
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# K*(c4*u^3+c3*u^2+(1-3*c4)*u-c3) = xn = (x-ma)/sa
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# -c4*xn^3-c3*xn^2+(1+3*c4)*xn+c3 = u
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coefs = p.coeffs[1::] / p.coeffs[0]
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a = coefs[0]
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b = coefs[1]
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c = coefs[2] - xn / p.coeffs[0]
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x0 = a / 3.
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# substitue xn = z-x0 and divide by c4 => z^3 + 3*p1*z+2*q0 = 0
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p1 = b / 3 - x0 ** 2
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# p1 = (b-a**2/3)/3
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# q0 = (c + x0*(2.*x0/3.-b))/2.
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# q0 = x0**3 -a*b/6 +c/2
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q0 = x0 * (x0 ** 2 - b / 2) + c / 2
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# z^3+3*p1*z+2*q0=0
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# c3 = self._c3
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# c4 = self._c4
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# b1 = 1./(3.*c4)
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# x0 = c3*b1
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# % substitue u = z-x0 and divide by c4 => z^3 + 3*c*z+2*q0 = 0
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# p1 = b1-1.-x0**2.
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# Km1 = np.sqrt(1.+2.*c3**2+6*c4**2)
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# q0 = x0**3-1.5*b1*(x0+xn*Km1)
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# q0 = x0**3-1.5*b1*(x0+xn)
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if self._x_limit is not None: # % Three real roots
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d = sqrt(-p1)
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theta1 = arccos(-q0 / d ** 3) / 3
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th2 = np.r_[0, -2 * pi / 3, 2 * pi / 3]
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x1 = abs(2 * d * cos(theta1[ceil(len(xn) / 2)] + th2) - x0)
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ix = x1.argmin() # choose the smallest solution
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return 2. * d * cos(theta1 + th2[ix]) - x0
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else: # Only one real root exist
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q1 = sqrt((q0) ** 2 + p1 ** 3)
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# Find the real root of the monic polynomial
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A0 = (q1 - q0) ** (1. / 3.)
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B0 = -(q1 + q0) ** (1. / 3.)
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return A0 + B0 - x0 # real root
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# The other complex roots are given by
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# x= -(A0+B0)/2+(A0-B0)*sqrt(3)/2-x0
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# x=-(A0+B0)/2+(A0-B0)*sqrt(-3)/2-x0
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class TrLinear(TrCommon2):
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__doc__ = TrCommon2.__doc__.replace('<generic>', 'Linear'
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) + """
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Description
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-----------
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The linear transformation model is monotonic linear polynomial, calibrated
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such that the first 2 moments of the transformed model G(y)=g^-1(y) match
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the moments of the true process.
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Example:
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--------
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""" + _example.replace('<generic>', 'TrLinear') + """
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>>> np.allclose(g.dist2gauss(), 0)
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True
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>>> np.allclose(g2.dist2gauss(), 0)
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True
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See also
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--------
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TrOchi
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TrHermite
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SpecData1D.stats_nl
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LevelCrossings.trdata
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TimeSeries.trdata
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spec2skew, ochitr, lc2tr, dat2tr
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"""
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def _transform(self, ymean, ysigma, xmean, xsigma, x, xi):
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sratio = atleast_1d(ysigma / xsigma)
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y = (atleast_1d(x) - xmean) * sratio + ymean
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if len(xi) > 0:
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y = [y] + [ix * sratio for ix in xi]
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return y
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def _dat2gauss(self, x, *xi):
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return self._transform(self.ymean, self.ysigma,
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self.mean, self.sigma, x, xi)
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def _gauss2dat(self, y, *yi):
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return self._transform(self.mean, self.sigma,
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self.ymean, self.ysigma, y, yi)
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class TrOchi(TrCommon2):
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__doc__ = TrCommon2.__doc__.replace('<generic>', 'Ochi'
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) + """
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Description
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-----------
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The Ochi transformation model is a monotonic exponential function,
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calibrated such that the first 3 moments of the transformed model
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G(y)=g^-1(y) match the moments of the true process. However, the
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skewness is limited by ABS(SKEW)<2.82. According to Ochi it is
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appropriate for a process with very strong non-linear characteristics.
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The model is given as:
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g(x) = ((1-exp(-gamma*(x-mean)/sigma))/gamma-mean2)/sigma2
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where
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gamma = 1.28*a for x>=mean
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3*a otherwise
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mean,
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sigma = standard deviation and mean, respectively, of the process.
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mean2,
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sigma2 = normalizing parameters in the transformed world, i.e., to
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make the gaussian process in the transformed world is N(0,1).
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The unknown parameters a, mean2 and sigma2 are found by solving the
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following non-linear equations:
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a*(sigma2^2+mean2^2)+mean2 = 0
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sigma2^2-2*a^2*sigma2^4 = 1
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2*a*sigma2^4*(3-8*a^2*sigma2^2) = skew
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Note
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----
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Transformation, g, does not have continous derivatives of 2'nd order or
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higher.
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Example
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-------
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""" + _example.replace('<generic>', 'TrOchi') + """
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>>> np.allclose(g.dist2gauss(), 1.410698801056657)
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True
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>>> np.allclose(g2.dist2gauss(), 1.988807188766706)
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True
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See also
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--------
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spec2skew, hermitetr, lc2tr, dat2tr
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References
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----------
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Ochi, M.K. and Ahn, K. (1994)
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'Non-Gaussian probability distribution of coastal waves.'
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In Proc. 24th Conf. Coastal Engng, Vol. 1, pp 482-496
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Michel K. Ochi (1998),
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"OCEAN WAVES, The stochastic approach",
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OCEAN TECHNOLOGY series 6, Cambridge, pp 255-275.
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"""
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def __init__(self, *args, **kwds):
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super(TrOchi, self).__init__(*args, **kwds)
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self.kurt = None
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self._phat = None
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self._par_from_stats()
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def _par_from_stats(self):
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skew = self.skew
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if abs(skew) > 2.82842712474619:
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raise ValueError('Skewness must be less than 2.82842')
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mean1 = self.mean
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sigma1 = self.sigma
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if skew == 0:
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self._phat = [sigma1, mean1, 0, 0, 1, 0]
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return
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# Solve the equations to obtain the gamma parameters:
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# a*(sig2^2+ma2^2)+ma2 = 0
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# sig2^2-2*a^2*sig2^4 = E(y^2) % =1
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# 2*a*sig2^4*(3-8*a^2*sig2^2) = E(y^3) % = skew
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# Let x = [a sig2^2 ]
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# Set up the 2D non-linear equations for a and sig2^2:
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# g1='[x(2)-2.*x(1).^2.*x(2).^2-P1,
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# 2.*x(1).*x(2).^2.*(3-8.*x(1).^2.*x(2))-P2 ]'
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# Or solve the following 1D non-linear equation for sig2^2:
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def g2(x):
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return (-sqrt(abs(x - 1) * 2) * (3. * x - 4 * abs(x - 1)) +
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abs(skew))
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a1 = 1. # Start interval where sig2^2 is located.
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a2 = 2.
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sig22 = brentq(g2, a1, a2) # % smallest solution for sig22
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a = sign(skew) * sqrt(abs(sig22 - 1) / 2) / sig22
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gam_a = 1.28 * a
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gam_b = 3 * a
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sigma2 = sqrt(sig22)
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# Solve the following 2nd order equation to obtain ma2
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# a*(sig2^2+ma2^2)+ma2 = 0
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my2 = (-1. - sqrt(1. - 4. * a ** 2 * sig22)) / a # % Largest mean
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mean2 = a * sig22 / my2 # % choose the smallest mean
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self._phat = [sigma1, mean1, gam_a, gam_b, sigma2, mean2]
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def _get_par(self):
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'''
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Returns ga, gb, sigma2, mean2
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'''
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if (self._phat is None or self.sigma != self._phat[0] or
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self.mean != self._phat[1]):
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self._par_from_stats()
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# sigma1 = self._phat[0]
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# mean1 = self._phat[1]
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ga = self._phat[2]
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gb = self._phat[3]
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sigma2 = self._phat[4]
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mean2 = self._phat[5]
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return ga, gb, sigma2, mean2
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def _transform(self, fun, x1, x2, gab, idx):
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if gab != 0:
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np.put(x1, idx, -fun(-gab * x2[idx]) / gab)
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return x1
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def _backward(self, yn):
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ga, gb, sigma2, mean2 = self._get_par()
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y2 = sigma2 * yn.ravel() + mean2
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igp, = where(0 <= y2)
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igm, = where(y2 < 0)
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xn = y2.copy()
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xn = self._transform(log1p, xn, y2, ga, igp)
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xn = self._transform(log1p, xn, y2, gb, igm)
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return xn
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def _forward(self, xn):
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ga, gb, sigma2, mean2 = self._get_par()
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y2 = xn.copy()
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igp, = where(0 <= xn)
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igm, = where(xn < 0)
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y2 = self._transform(expm1, y2, xn, ga, igp)
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y2 = self._transform(expm1, y2, xn, gb, igm)
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return (y2 - mean2) / sigma2
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def _dat2gauss(self, x, *xi):
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return self._transformgauss(self._forward,
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self.ymean, self.ysigma,
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self.mean, self.sigma, x, *xi)
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def _gauss2dat(self, y, *yi):
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return self._transformgauss(self._backward,
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self.mean, self.sigma,
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self.ymean, self.ysigma, y, *yi)
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@staticmethod
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def _transformgauss(trfun, mean, sigma, ymean, ysigma, y, *yi):
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if len(yi) > 0:
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raise ValueError('Transforming derivatives is not implemented!')
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yn = (atleast_1d(y) - ymean) / ysigma
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shape0 = yn.shape
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xn = np.reshape(trfun(yn.ravel()), shape0)
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return xn * sigma + mean
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def main():
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import pylab
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g = TrHermite(skew=0.1, kurt=3.01)
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g.dist2gauss()
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# g = TrOchi(skew=0.56)
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x = np.linspace(-5, 5)
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y = g(x)
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pylab.plot(np.abs(x - g.gauss2dat(y)))
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# pylab.plot(x,y,x,x,':',g.gauss2dat(y),y,'r')
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pylab.show()
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np.disp('finito')
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if __name__ == '__main__':
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if True: # False: #
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from wafo.testing import test_docstrings
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test_docstrings(__file__)
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else:
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main()
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