src/wafo/spectrum/core.py
import warnings
import os
import numpy as np
from numpy import (pi, inf, zeros, ones, where, nonzero,
flatnonzero, ceil, sqrt, exp, log, arctan2,
tanh, cosh, sinh, random, atleast_1d,
minimum, diff, isnan, r_, conj, mod,
hstack, vstack, interp, ravel, finfo, linspace,
arange, array, nan, newaxis, sign)
from numpy.fft import fft
from scipy.integrate import simps, trapz
from scipy.special import erf
from scipy.linalg import toeplitz
import scipy.interpolate as interpolate
from scipy.interpolate import interp1d, interp2d
from wafo.objects import TimeSeries, mat2timeseries
from wafo.interpolate import stineman_interp
from wafo.wave_theory.dispersion_relation import w2k # , k2w
from wafo.containers import PlotData, now
from wafo.misc import (sub_dict_select, nextpow2, discretize, JITImport,
meshgrid, cart2polar, polar2cart, gravity as _gravity)
from wafo.markov import mctp2rfc, mctp2tc
from wafo.kdetools import qlevels
# from wafo.transform import TrData
from wafo.transform.models import TrLinear
from wafo.plotbackend import plotbackend
try:
from wafo.gaussian import Rind
except ImportError:
Rind = None
try:
from wafo import c_library
except ImportError:
warnings.warn('Compile the c_library.pyd again!')
c_library = None
try:
from wafo import cov2mod
except ImportError:
warnings.warn('Compile the cov2mod.pyd again!')
cov2mod = None
# Trick to avoid error due to circular import
_WAFOCOV = JITImport('wafo.covariance')
__all__ = ['SpecData1D', 'SpecData2D', 'plotspec']
_EPS = np.finfo(float).eps
_TINY = np.finfo(float).tiny
def _set_seed(iseed):
'''Set seed of random generator'''
if iseed is not None:
try:
random.set_state(iseed)
except (KeyError, TypeError):
random.seed(iseed)
def qtf(w, h=inf, g=9.81, method='winterstein', rtol=1e-7, atol=0):
"""
Return Quadratic Transfer Function
Parameters
------------
w : array-like
angular frequencies
h : scalar
water depth
g : scalar
acceleration of gravity
Returns
-------
h_s = sum frequency effects
h_d = difference frequency effects
h_dii = diagonal of h_d
Examples
--------
>>> w = np.r_[0.1, 1./3, 2./3, 1]
>>> hs, hd, hdi = qtf(w, h=np.inf, g=9.81)
>>> np.allclose(hs, [[ 0.00050968, 0.00308642, 0.01158115, 0.02573904],
... [ 0.00308642, 0.00566316, 0.01415789, 0.02831578],
... [ 0.01158115, 0.01415789, 0.02265262, 0.03681051],
... [ 0.02573904, 0.02831578, 0.03681051, 0.0509684 ]])
True
>>> np.allclose(hd, [[-0. , -0.00257674, -0.01107147, -0.02522936],
... [-0.00257674, -0. , -0.00849473, -0.02265262],
... [-0.01107147, -0.00849473, -0. , -0.01415789],
... [-0.02522936, -0.02265262, -0.01415789, -0. ]])
True
>>> hs2, hd2, hdi2 = qtf(w, h=1e+6, g=9.81, method='winterstein')
>>> np.allclose(hs2, [[0.00050968, 0.00308642, 0.01158115, 0.02573904],
... [0.00308642, 0.00566316, 0.01415789, 0.02831578],
... [0.01158115, 0.01415789, 0.02265262, 0.03681051],
... [0.02573904, 0.02831578, 0.03681051, 0.0509684 ]])
True
>>> np.allclose(hd2, [[-2.50061328e-07, 1.38729557e-03, 8.18314621e-03, 2.06421189e-02],
... [1.38729557e-03, -2.50005518e-07, 2.83135545e-03, 1.13261230e-02],
... [8.18314621e-03, 2.83135545e-03, -2.50001380e-07, 2.83133750e-03],
... [2.06421189e-02, 1.13261230e-02, 2.83133750e-03, -2.50000613e-07]])
True
>>> w = np.r_[0, 1e-10, 1e-5, 1e-1]
>>> hs, hd, hdi = qtf(w, h=np.inf, g=9.81)
>>> np.allclose(hs, [[0.00000000e+00, 2.54841998e-22, 2.54841998e-12, 2.54841998e-04],
... [2.54841998e-22, 5.09683996e-22, 2.54841998e-12, 2.54841998e-04],
... [2.54841998e-12, 2.54841998e-12, 5.09683996e-12, 2.54842001e-04],
... [2.54841998e-04, 2.54841998e-04, 2.54842001e-04, 5.09683996e-04]])
True
>>> np.allclose(hd, [[-0.00000000e+00, -2.54841998e-22, -2.54841998e-12, -2.54841998e-04],
... [-2.54841998e-22, -0.00000000e+00, -2.54841998e-12, -2.54841998e-04],
... [-2.54841998e-12, -2.54841998e-12, -0.00000000e+00, -2.54841995e-04],
... [-2.54841998e-04, -2.54841998e-04, -2.54841995e-04, -0.00000000e+00]])
True
>>> hs2, hd2, hdi2 = qtf(w, h=1e+100, g=9.81, method='winterstein')
>>> np.allclose(hs2, [[1.50234572e-63, 2.54841998e-22, 2.54841998e-12, 2.54841998e-04],
... [2.54841998e-22, 5.09683996e-22, 2.54841998e-12, 2.54841998e-04],
... [2.54841998e-12, 2.54841998e-12, 5.09683996e-12, 2.54842001e-04],
... [2.54841998e-04, 2.54841998e-04, 2.54842001e-04, 5.09683996e-04]],
... atol=0)
True
>>> np.allclose(hd2, [[-2.50000000e-101, 2.54841998e-022, 2.54841998e-012, 2.54841998e-004],
... [2.54841998e-022, -2.50000000e-101, 2.54836901e-012, 2.54841997e-004],
... [2.54841998e-012, 2.54836901e-012, -2.50000000e-101, 2.54791032e-004],
... [2.54841998e-004, 2.54841997e-004, 2.54791032e-004, -2.500000e-101]],
... atol=0)
True
References
----------
Langley, RS (1987)
'A statistical analysis of nonlinear random waves'
Ocean Engineering, Vol 14, No 5, pp 389-407
Marthinsen, T. and Winterstein, S.R (1992)
'On the skewness of random surface waves'
In proceedings of the 2nd ISOPE Conference, San Francisco, 14-19 june.
"""
# >>> hs3, hd3, hdi3 = qtf(w, h=200, g=9.81, method='winterstein')
# >>> hs3
#
# >>> hd3
#
# >>> np.allclose(hs3, [[ 0. , 0.00283158, 0.01132631, 0.0254842 ],
# ... [ 0.00283158, 0.00566316, 0.01415789, 0.02831578],
# ... [ 0.01132631, 0.01415789, 0.02265262, 0.03681051],
# ... [ 0.0254842 , 0.02831578, 0.03681051, 0.0509684 ]])
#
# >>> np.allclose(hd3, [[-0. , -0.00283158, -0.01132631, -0.0254842 ],
# ... [-0.00283158, -0. , -0.00849473, -0.02265262],
# ... [-0.01132631, -0.00849473, -0. , -0.01415789],
# ... [-0.0254842 , -0.02265262, -0.01415789, -0. ]])
w = atleast_1d(w)
num_w = w.size
if h == inf: # go here for faster calculations
k_w = w2k(w, theta=0, h=h, g=g, rtol=rtol, atol=atol)[0]
k_1, k_2 = meshgrid(k_w, k_w, sparse=True)
h_s = 0.25 * (abs(k_1) + abs(k_2))
h_d = -0.25 * abs(abs(k_1) - abs(k_2))
h_dii = zeros(num_w)
return h_s, h_d, h_dii
w1 = w + _TINY ** (1. / 10) * (np.sign(w) * np.int_(np.abs(w) < _EPS) + np.int_(w == 0))
w = w1
# k_w += _TINY ** (1./3) * (np.sign(k_w) * np.int_(np.abs(k_w) < _EPS) + np.int_(k_w==0))
k_w = w2k(w, theta=0, h=h, g=g, rtol=rtol, atol=atol)[0]
k_1, k_2 = meshgrid(k_w, k_w, sparse=True)
w_1, w_2 = meshgrid(w, w, sparse=True)
w12 = w_1 * w_2
w1p2 = w_1 + w_2
w1m2 = w_1 - w_2
k12 = k_1 * k_2
k1p2 = k_1 + k_2
k1m2 = abs(k_1 - k_2)
if method.startswith('langley'):
p_1 = (-2 * w1p2 * (k12 * g ** 2. - w12 ** 2.) +
w_1 * (w_2 ** 4. - g ** 2 * k_2 ** 2) +
w_2 * (w_1 ** 4 - g * 2. * k_1 ** 2)) / (4. * w12 + _TINY)
p_2 = w1p2 ** 2. * cosh((k1p2) * h) - g * (k1p2) * sinh((k1p2) * h)
p_2 += _TINY * np.int_(p_2 == 0)
h_s = (-p_1 / p_2 * w1p2 * cosh((k1p2) * h) / g -
(k12 * g ** 2 - w12 ** 2.) / (4 * g * w12 + _TINY) +
(w_1 ** 2 + w_2 ** 2) / (4 * g))
p_3 = (-2 * w1m2 * (k12 * g ** 2 + w12 ** 2) -
w_1 * (w_2 ** 4 - g ** 2 * k_2 ** 2) +
w_2 * (w_1 ** 4 - g ** 2 * k_1 ** 2)) / (4. * w12 + _TINY)
p_4 = w1m2 ** 2. * cosh(k1m2 * h) - g * (k1m2) * sinh((k1m2) * h)
p_4 += _TINY * np.int_(p_4 == 0)
h_d = (-p_3 / p_4 * (w1m2) * cosh((k1m2) * h) / g -
(k12 * g ** 2 + w12 ** 2) / (4 * g * w12 + _TINY) +
(w_1 ** 2. + w_2 ** 2.) / (4. * g))
else: # Marthinsen & Winterstein
tmp1 = 2.0 * g * k12 / (w12 + 0)
tmp2 = (w_1 ** 2. + w_2 ** 2. + w12) / g
h_s = 0.25 * ((tmp1 - tmp2
+ g * (w_1 * k_2 ** 2. + w_2 * k_1 ** 2) / (w12 * w1p2 + 0))
/ (1. - g * h * k1p2 / (w1p2 ** 2. + 0) * tanh(k1p2))
+ tmp2 - 0.5 * tmp1) # OK
tiny_diag = _TINY * np.diag(np.ones(num_w)) # Avoid division by zero on diagonal
tmp3 = (w_1 ** 2 + w_2 ** 2 - w12) / g # OK
numerator = (tmp1 - tmp3 - g * (w_1 * k_2 ** 2
- w_2 * k_1 ** 2) / (w12 * w1m2 + tiny_diag))
h_d = 0.25 * (numerator / (1. - g * h * k1m2 / (w1m2 ** 2. + tiny_diag) * tanh(k1m2))
+ tmp3 - 0.5 * tmp1) # OK
# h_d = 0.25 * ((tmp1 - tmp3
# - g * (w_1 * k_2 ** 2 - w_2 * k_1 ** 2) / (w12 * w1m2 + tiny_diag))
# / (1. - g * h * k1m2 / (w1m2 ** 2. + tiny_diag) * tanh(k1m2))
# + tmp3 - 0.5 * tmp1) # OK
# tmp1 = 2 * g * (k_w./w)^2
# tmp2 = w.^2/g
# Wave group velocity
k_h = k_w * h
c_g = 0.5 * g * (tanh(k_h) + k_h / np.cosh(k_h) ** 2) / w
numerator2 = (g * (k_w / w) ** 2. - w ** 2 / g + 2 * g * k_w / (w * c_g + 0))
h_dii = 0.25 * (numerator2 / (1. - g * h / (c_g ** 2. + 0))
- 2 * k_w / sinh(2 * k_h)) # OK
# c_g = 0.5 * g * (tanh(k_w * h) + k_w * h * (1.0 - tanh(k_w * h) ** 2)) / w
# h_dii = (0.5 * (0.5 * g * (k_w / w) ** 2. - 0.5 * w ** 2 / g + g * k_w / (w * c_g + 0))
# / (1. - g * h / (c_g ** 2. + 0))
# - 0.5 * k_w / sinh(2 * k_w * h)) # OK
h_d.flat[0::num_w + 1] = h_dii
# infinite water
# >>> np.allclose(hs, [[ 0. , 0.00283158, 0.01132631, 0.0254842 ],
# ... [ 0.00283158, 0.00566316, 0.01415789, 0.02831578],
# ... [ 0.01132631, 0.01415789, 0.02265262, 0.03681051],
# ... [ 0.0254842 , 0.02831578, 0.03681051, 0.0509684 ]])
# True
#
# >>> np.allclose(hd, [[-0. , -0.00283158, -0.01132631, -0.0254842 ],
# ... [-0.00283158, -0. , -0.00849473, -0.02265262],
# ... [-0.01132631, -0.00849473, -0. , -0.01415789],
# ... [-0.0254842 , -0.02265262, -0.01415789, -0. ]])
# True
# winterstein
# h_s =
# [[ 0.00000000e+00 -1.64775418e+00 -6.95612056e-01 -4.18817231e-01 -3.15690232e-01]
# [ -1.64775418e+00 -7.98574421e-01 -2.21051428e-01 -6.73808482e-02 -1.69373060e-02]
# [ -6.95612056e-01 -2.21051428e-01 -1.29139936e-01 -4.11797418e-02 1.12063541e-03]
# [ -4.18817231e-01 -6.73808482e-02 -4.11797418e-02 -3.51718594e-03 2.44489725e-02]
# h_d =
# [[ 0. -1.64775418 -0.69561206 -0.41881723 -0.31569023]
# [-1.6103978 0.0130494 -0.12128861 -0.05785645 -0.02806048]
# [-0.65467824 -0.12128861 0.01494093 -0.04402996 -0.02595442]
# [-0.35876732 -0.05785645 -0.04402996 0.01905565 -0.02218373]
# [ inf -0.02806048 -0.02595442 -0.02218373 0.02705736]]
# langley
# h_d = [[ 0.00000000e+00 -8.87390092e+14 -3.87924869e+14 -1.66844106e+15 inf]
# [ -8.87390092e+14 -1.14566397e-02 -1.50113192e-01 -1.11791139e-01 -1.13090565e-01]
# [ -3.87924869e+14 -1.50113192e-01 -8.56987798e-03 -5.10233013e-02 -4.93936523e-02]
# [ -1.66844106e+15 -1.11791139e-01 -5.10233013e-02 -4.72078473e-03 -2.74040590e-02]
# [ inf -1.13090565e-01 -4.93936523e-02 -2.74040590e-02 -1.57316125e-03]]
# h_s =
# [[ 0.00000000e+00 -8.62422934e+14 -3.76136070e+14 -1.61053099e+15 inf]
# [ -8.87390092e+14 2.59936788e-01 1.22409408e-01 7.97392657e-02 6.16999831e-02]
# [ -3.87924869e+14 1.46564082e-01 7.02793126e-02 4.62059958e-02 3.58607610e-02]
# [ -1.66844106e+15 1.18356989e-01 5.82970744e-02 3.92688958e-02 3.13685586e-02]
# [ inf 1.25606419e-01 6.35218804e-02 4.41902963e-02 3.69195895e-02]]
# k = find(w_1==w_2)
# h_d(k) = h_dii
# The NaN's occur due to division by zero. => Set the isnans to zero
h_dii = where(isnan(h_dii), 0, h_dii)
h_d = where(isnan(h_d), 0, h_d)
h_s = where(isnan(h_s), 0, h_s)
return h_s, h_d, h_dii
def plotspec(specdata, linetype='b-', flag=1):
'''
PLOTSPEC Plot a spectral density
Parameters
----------
S : SpecData1D or SpecData2D object
defining spectral density.
linetype : string
defining color and linetype, see plot for possibilities
flag : scalar integer
defining the type of plot
1D:
1 plots the density, S, (default)
2 plot 10log10(S)
3 plots both the above plots
2D:
Directional spectra: S(w,theta), S(f,theta)
1 polar plot S (default)
2 plots spectral density and the directional
spreading, int S(w,theta) dw or int S(f,theta) df
3 plots spectral density and the directional
spreading, int S(w,theta)/S(w) dw or int S(f,theta)/S(f) df
4 mesh of S
5 mesh of S in polar coordinates
6 contour plot of S
7 filled contour plot of S
Wavenumber spectra: S(k1,k2)
1 contour plot of S (default)
2 filled contour plot of S
Examples
--------
>>> import numpy as np
>>> import wafo.spectrum as ws
>>> Sj = ws.models.Jonswap(Hm0=3, Tp=7)
>>> S = Sj.tospecdata()
>>> ws.plotspec(S,flag=1)
S = demospec('dir'); S2 = mkdspec(jonswap,spreading);
plotspec(S,2), hold on
# Same as previous fig. due to frequency independent spreading
plotspec(S,3,'g')
# Not the same as previous figs. due to frequency dependent spreading
plotspec(S2,2,'r')
plotspec(S2,3,'m')
# transform from angular frequency and radians to frequency and degrees
Sf = ttspec(S,'f','d'); clf
plotspec(Sf,2),
See also
dat2spec, createspec, simpson
'''
pass
# # label the contour levels
# txtFlag = 0
# LegendOn = 1
#
# ftype = specdata.freqtype # options are 'f' and 'w' and 'k'
# data = specdata.data
# if data.ndim == 2:
# freq = specdata.args[1]
# theta = specdata.args[0]
# else:
# freq = specdata.args
# if isinstance(specdata.args, (list, tuple)):
#
# if ftype == 'w':
# xlbl_txt = 'Frequency [rad/s]'
# ylbl1_txt = 'S(w) [m^2 s / rad]'
# ylbl3_txt = 'Directional Spectrum'
# zlbl_txt = 'S(w,\theta) [m^2 s / rad^2]'
# funit = ' [rad/s]'
# Sunit = ' [m^2 s / rad]'
# elif ftype == 'f':
# xlbl_txt = 'Frequency [Hz]'
# ylbl1_txt = 'S(f) [m^2 s]'
# ylbl3_txt = 'Directional Spectrum'
# zlbl_txt = 'S(f,\theta) [m^2 s / rad]'
# funit = ' [Hz]'
# Sunit = ' [m^2 s ]'
# elif ftype == 'k':
# xlbl_txt = 'Wave number [rad/m]'
# ylbl1_txt = 'S(k) [m^3/ rad]'
# funit = ' [rad/m]'
# Sunit = ' [m^3 / rad]'
# ylbl4_txt = 'Wave Number Spectrum'
#
# else:
# raise ValueError('Frequency type unknown')
#
#
# if hasattr(specdata, 'norm') and specdata.norm :
# Sunit=[]
# funit = []
# ylbl1_txt = 'Normalized Spectral density'
# ylbl3_txt = 'Normalized Directional Spectrum'
# ylbl4_txt = 'Normalized Wave Number Spectrum'
# if ftype == 'k':
# xlbl_txt = 'Normalized Wave number'
# else:
# xlbl_txt = 'Normalized Frequency'
#
# ylbl2_txt = 'Power spectrum (dB)'
#
# phi = specdata.phi
#
# spectype = specdata.type.lower()
# stype = spectype[-3::]
# if stype in ('enc', 'req', 'k1d') : #1D plot
# Fn = freq[-1] # Nyquist frequency
# indm = findpeaks(data, n=4)
# maxS = data.max()
# if isfield(S,'CI') && ~isempty(S.CI):
# maxS = maxS*S.CI(2)
# txtCI = [num2str(100*S.p), '% CI']
# #end
#
# Fp = freq[indm]# %peak frequency/wave number
#
# if len(indm) == 1:
# txt = [('fp = %0.2g' % Fp) + funit]
# else:
# txt = []
# for i, fp in enumerate(Fp.tolist()):
# txt.append(('fp%d = %0.2g' % (i, fp)) + funit)
#
# txt = ''.join(txt)
# if (flag == 3):
# plotbackend.subplot(2, 1, 1)
# if (flag == 1) or (flag == 3):# Plot in normal scale
# plotbackend.plot(np.vstack([Fp, Fp]),
# np.vstack([zeros(len(indm)), data.take(indm)]),
# ':', label=txt)
# plotbackend.plot(freq, data, linetype)
# specdata.labels.labelfig()
# if isfield(S,'CI'):
# plot(freq,S.S*S.CI(1), 'r:' )
# plot(freq,S.S*S.CI(2), 'r:' )
#
# a = plotbackend.axis()
#
# a1 = Fn
# if (Fp > 0):
# a1 = max(min(Fn, 10 * max(Fp)), a[1])
#
# plotbackend.axis([0, a1 , 0, max(1.01 * maxS, a[3])])
# plotbackend.title('Spectral density')
# plotbackend.xlabel(xlbl_txt)
# plotbackend.ylabel(ylbl1_txt)
#
#
# if (flag == 3):
# plotbackend.subplot(2, 1, 2)
#
# if (flag == 2) or (flag == 3) : # Plot in logaritmic scale
# ind = np.flatnonzero(data > 0)
#
# plotbackend.plot(np.vstack([Fp, Fp]),
# np.vstack((min(10 * log10(data.take(ind) /
# maxS)).repeat(len(Fp)),
# 10 * log10(data.take(indm) / maxS))), ':',label=txt)
# hold on
# if isfield(S,'CI'):
# plot(freq(ind),10*log10(S.S(ind)*S.CI(1)/maxS), 'r:' )
# plot(freq(ind),10*log10(S.S(ind)*S.CI(2)/maxS), 'r:' )
#
# plotbackend.plot(freq[ind], 10 * log10(data[ind] / maxS), linetype)
#
# a = plotbackend.axis()
#
# a1 = Fn
# if (Fp > 0):
# a1 = max(min(Fn, 10 * max(Fp)), a[1])
#
# plotbackend.axis([0, a1 , -20, max(1.01 * 10 * log10(1), a[3])])
#
# specdata.labels.labelfig()
# plotbackend.title('Spectral density')
# plotbackend.xlabel(xlbl_txt)
# plotbackend.ylabel(ylbl2_txt)
#
# if LegendOn:
# plotbackend.legend()
# if isfield(S,'CI'),
# legend(txt{:},txtCI,1)
# else
# legend(txt{:},1)
# end
# end
# case {'k2d'}
# if plotflag==1,
# [c, h] = contour(freq,S.k2,S.S,'b')
# z_level = clevels(c)
#
#
# if txtFlag==1
# textstart_x=0.05; textstart_y=0.94
# cltext1(z_level,textstart_x,textstart_y)
# else
# cltext(z_level,0)
# end
# else
# [c,h] = contourf(freq,S.k2,S.S)
# %clabel(c,h), colorbar(c,h)
# fcolorbar(c) % alternative
# end
# rotate(h,[0 0 1],-phi*180/pi)
#
#
#
# xlabel(xlbl_txt)
# ylabel(xlbl_txt)
# title(ylbl4_txt)
# # return
# km=max([-freq(1) freq(end) S.k2(1) -S.k2(end)])
# axis([-km km -km km])
# hold on
# plot([0 0],[ -km km],':')
# plot([-km km],[0 0],':')
# axis('square')
#
#
# # cltext(z_level)
# # axis('square')
# if ~ih, hold off,end
# case {'dir'}
# thmin = S.theta(1)-phi;thmax=S.theta(end)-phi
# if plotflag==1 % polar plot
# if 0, % alternative but then z_level must be chosen beforehand
# h = polar([0 2*pi],[0 freq(end)])
# delete(h);hold on
# [X,Y]=meshgrid(S.theta,freq)
# [X,Y]=polar2cart(X,Y)
# contour(X,Y,S.S',lintype)
# else
# if (abs(thmax-thmin)<3*pi), % angle given in radians
# theta = S.theta
# else
# theta = S.theta*pi/180 % convert to radians
# phi = phi*pi/180
# end
# c = contours(theta,freq,S.S')%,Nlevel) % calculate levels
# if isempty(c)
# c = contours(theta,freq,S.S)%,Nlevel); % calculate levels
# end
# [z_level c] = clevels(c); % find contour levels
# h = polar(c(1,:),c(2,:),lintype);
# rotate(h,[0 0 1],-phi*180/pi)
# end
# title(ylbl3_txt)
# % label the contour levels
#
# if txtFlag==1
# textstart_x = -0.1; textstart_y=1.00;
# cltext1(z_level,textstart_x,textstart_y);
# else
# cltext(z_level,0)
# end
#
# elseif (plotflag==2) || (plotflag==3),
# %ih = ishold;
#
# subplot(211)
#
# if ih, hold on, end
#
# Sf = spec2spec(S,'freq'); % frequency spectrum
# plotspec(Sf,1,lintype)
#
# subplot(212)
#
# Dtf = S.S;
# [Nt,Nf] = size(S.S);
# Sf = Sf.S(:).';
# ind = find(Sf);
#
# if plotflag==3, %Directional distribution D(theta,freq))
# Dtf(:,ind) = Dtf(:,ind)./Sf(ones(Nt,1),ind);
# end
# Dtheta = simpson(freq,Dtf,2); %Directional spreading, D(theta)
# Dtheta = Dtheta/simpson(S.theta,Dtheta); # int D(theta)dtheta = 1
# [y,ind] = max(Dtheta);
# Wdir = S.theta(ind)-phi; % main wave direction
# txtwdir = ['\theta_p=' num2pistr(Wdir,3)]; % convert to text string
#
# plot([1 1]*S.theta(ind)-phi,[0 Dtheta(ind)],':'), hold on
# if LegendOn
# lh=legend(txtwdir,0);
# end
# plot(S.theta-phi,Dtheta,lintype)
#
# fixthetalabels(thmin,thmax,'x',2)
# ylabel('D(\theta)')
# title('Spreading function')
# if ~ih, hold off, end
# %legend(lh) % refresh current legend
# elseif plotflag==4 % mesh
# mesh(freq,S.theta-phi,S.S)
# xlabel(xlbl_txt);
# fixthetalabels(thmin,thmax,'y',3)
# zlabel(zlbl_txt)
# title(ylbl3_txt)
# elseif plotflag==5 % mesh
# %h=polar([0 2*pi],[0 freq(end)]);
# %delete(h);hold on
# [X,Y]=meshgrid(S.theta-phi,freq);
# [X,Y]=polar2cart(X,Y);
# mesh(X,Y,S.S')
# % display the unit circle beneath the surface
# hold on, mesh(X,Y,zeros(size(S.S'))),hold off
# zlabel(zlbl_txt)
# title(ylbl3_txt)
# set(gca,'xticklabel','','yticklabel','')
# lighting phong
# %lighting gouraud
# %light
# elseif (plotflag==6) || (plotflag==7),
# theta = S.theta-phi;
# [c, h] = contour(freq,theta,S.S); %,Nlevel); % calculate levels
# fixthetalabels(thmin,thmax,'y',2)
# if plotflag==7,
# hold on
# [c,h] = contourf(freq,theta,S.S); %,Nlevel);
# %hold on
# end
#
# title(ylbl3_txt)
# xlabel(xlbl_txt);
# if 0,
# [z_level] = clevels(c); % find contour levels
# % label the contour levels
# if txtFlag==1
# textstart_x = 0.06; textstart_y=0.94;
# cltext1(z_level,textstart_x,textstart_y) #local: cltext
# else
# cltext(z_level)
# end
# else
# colormap('jet')
#
# if plotflag==7,
# fcolorbar(c)
# else
# %clabel(c,h),
# hcb = colorbar;
# end
# grid on
# end
# else
# error('Unknown plot option')
# end
# otherwise, error('unknown spectral type')
# end
#
# if ~ih, hold off, end
#
# # The following two commands install point-and-click editing of
# # all the text objects (title, xlabel, ylabel) of the current figure:
#
# #set(findall(gcf,'type','text'),'buttondownfcn','edtext')
# #set(gcf,'windowbuttondownfcn','edtext(''hide'')')
#
# return
class SpecData1D(PlotData):
"""
Container class for 1D spectrum data objects in WAFO
Member variables
----------------
data : array-like
One sided Spectrum values, size nf
args : array-like
freguency/wave-number-lag values of freqtype, size nf
type : String
spectrum type, one of 'freq', 'k1d', 'enc' (default 'freq')
freqtype : letter
frequency type, one of: 'f', 'w' or 'k' (default 'w')
tr : Transformation function (default (none)).
h : real scalar
Water depth (default inf).
v : real scalar
Ship speed, if type = 'enc'.
norm : bool
Normalization flag, True if S is normalized, False if not
date : string
Date and time of creation or change.
Examples
--------
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=3)
>>> w = np.linspace(0,4,256)
>>> S1 = Sj.tospecdata(w) #Make spectrum object from numerical values
>>> S = sm.SpecData1D(Sj(w),w) # Alternatively do it manually
See also
--------
PlotData
CovData
"""
def __init__(self, *args, **kwds):
super(SpecData1D, self).__init__(*args, **kwds)
self.name_ = kwds.pop('name', 'WAFO Spectrum Object')
self.type = kwds.pop('type', 'freq')
self._freqtype = kwds.pop('freqtype', 'w')
self.angletype = ''
self.h = kwds.pop('h', inf)
self.tr = kwds.pop('tr', None) # TrLinear()
self.phi = kwds.pop('phi', 0.0)
self.v = kwds.pop('v', 0.0)
self.norm = kwds.pop('norm', False)
self.setlabels()
@property
def freqtype(self):
return self._freqtype
@freqtype.setter
def freqtype(self, freqtype):
if self._freqtype == freqtype:
return # do nothind
if freqtype == 'w' and self._freqtype == 'f':
self.args *= 2 * np.pi
self.data /= 2 * np.pi
self._freqtype = 'w'
self.setlabels()
elif freqtype == 'f' and self._freqtype == 'w':
self.args /= 2 * np.pi
self.data *= 2 * np.pi
self._freqtype = 'f'
self.setlabels()
def _get_default_dt_and_rate(self, dt):
dt_old = self.sampling_period()
if dt is None:
return dt_old, 1
rate = max(round(dt_old * 1. / dt), 1.)
return dt, int(rate)
def _check_dt(self, dt):
freq = self.args
checkdt = 1.2 * min(diff(freq)) / 2. / pi
if self.freqtype in 'f':
checkdt *= 2 * pi
if (checkdt < 2. ** -16 / dt):
print('Step dt = %g in computation of the density is ' +
'too small.' % dt)
print('The computed covariance (by FFT(2^K)) may differ from the')
print('theoretical. Solution:')
raise ValueError('use larger dt or sparser grid for spectrum.')
@staticmethod
def _check_cov_matrix(acfmat, nt, dt):
eps0 = 0.0001
if nt + 1 >= 5:
cc2 = acfmat[0, 0] - acfmat[4, 0] * (acfmat[4, 0] / acfmat[0, 0])
if (cc2 < eps0):
warnings.warn('Step dt = %g in computation of the density ' +
'is too small.' % dt)
cc1 = acfmat[0, 0] - acfmat[1, 0] * (acfmat[1, 0] / acfmat[0, 0])
if (cc1 < eps0):
warnings.warn('Step dt = %g is small, and may cause numerical ' +
'inaccuracies.' % dt)
@property
def lagtype(self):
if self.freqtype in 'k': # options are 'f' and 'w' and 'k'
return 'x'
return 't'
def tocov_matrix(self, nr=0, nt=None, dt=None):
'''
Computes covariance function and its derivatives, alternative version
Parameters
----------
nr : scalar integer
number of derivatives in output, nr<=4 (default 0)
nt : scalar integer
number in time grid, i.e., number of time-lags.
(default rate*(n_f-1)) where rate = round(1/(2*f(end)*dt)) or
rate = round(pi/(w(n_f)*dt)) depending on S.
dt : real scalar
time spacing for acfmat
Returns
-------
acfmat : [R0, R1,...Rnr], shape Nt+1 x Nr+1
matrix with autocovariance and its derivatives, i.e., Ri (i=1:nr)
are column vectors with the 1'st to nr'th derivatives of R0.
NB! This routine requires that the spectrum grid is equidistant
starting from zero frequency.
Examples
--------
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap()
>>> S = Sj.tospecdata()
>>> acfmat = S.tocov_matrix(nr=3, nt=256, dt=0.1)
>>> np.round(acfmat[:2,:],3)
array([[ 3.061, 0. , -1.677, 0. ],
[ 3.052, -0.167, -1.668, 0.187]])
See also
--------
cov,
resample,
objects
'''
dt, rate = self._get_default_dt_and_rate(dt)
self._check_dt(dt)
freq = self.args
n_f = len(freq)
if nt is None:
nt = rate * (n_f - 1)
else: # check if Nt is ok
nt = minimum(nt, rate * (n_f - 1))
spec = self.copy()
spec.resample(dt)
acf = spec.tocovdata(nr, nt, rate=1)
acfmat = zeros((nt + 1, nr + 1), dtype=float)
acfmat[:, 0] = acf.data[0:nt + 1]
fieldname = 'R' + self.lagtype * nr
for i in range(1, nr + 1):
fname = fieldname[:i + 1]
r_i = getattr(acf, fname)
acfmat[:, i] = r_i[0:nt + 1]
self._check_cov_matrix(acfmat, nt, dt)
return acfmat
def tocovdata(self, nr=0, nt=None, rate=None):
'''
Computes covariance function and its derivatives
Parameters
----------
nr : number of derivatives in output, nr<=4 (default = 0).
nt : number in time grid, i.e., number of time-lags
(default rate*(length(S.data)-1)).
rate = 1,2,4,8...2**r, interpolation rate for R
(default = 1, no interpolation)
Returns
-------
R : CovData1D
auto covariance function
The input 'rate' with the spectrum gives the time-grid-spacing:
dt=pi/(S.w[-1]*rate),
S.w[-1] is the Nyquist freq.
This results in the time-grid: 0:dt:Nt*dt.
What output is achieved with different S and choices of Nt, Nx and Ny:
1) S.type='freq' or 'dir', Nt set, Nx,Ny not set => R(time) (one-dim)
2) S.type='k1d' or 'k2d', Nt set, Nx,Ny not set: => R(x) (one-dim)
3) Any type, Nt and Nx set => R(x,time); Nt and Ny set => R(y,time)
4) Any type, Nt, Nx and Ny set => R(x,y,time)
5) Any type, Nt not set, Nx and/or Ny set
=> Nt set to default, goto 3) or 4)
NB! This routine requires that the spectrum grid is equidistant
starting from zero frequency.
NB! If you are using a model spectrum, spec, with sharp edges
to calculate covariances then you should probably round off the sharp
edges like this:
Examples
--------
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap()
>>> S = Sj.tospecdata()
>>> S.data[0:40] = 0.0
>>> S.data[100:-1] = 0.0
>>> Nt = len(S.data)-1
>>> acf = S.tocovdata(nr=0, nt=Nt)
>>> S1 = acf.tospecdata()
h = S.plot('r')
h1 = S1.plot('b:')
R = spec2cov(spec,0,Nt)
win = parzen(2*Nt+1)
R.data = R.data.*win(Nt+1:end)
S1 = cov2spec(acf)
R2 = spec2cov(S1)
figure(1)
plotspec(S),hold on, plotspec(S1,'r')
figure(2)
covplot(R), hold on, covplot(R2,[],[],'r')
figure(3)
semilogy(abs(R2.data-R.data)), hold on,
semilogy(abs(S1.data-S.data)+1e-7,'r')
See also
--------
cov2spec
'''
freq = self.args
n_f = len(freq)
if freq[0] > 0:
txt = '''Spectrum does not start at zero frequency/wave number.
Correct it with resample, for example.'''
raise ValueError(txt)
d_w = abs(diff(freq, n=2, axis=0))
if np.any(d_w > 1.0e-8):
txt = '''Not equidistant frequencies/wave numbers in spectrum.
Correct it with resample, for example.'''
raise ValueError(txt)
if rate is None:
rate = 1 # interpolation rate
elif rate > 16:
rate = 16
else: # make sure rate is a power of 2
rate = 2 ** nextpow2(rate)
if nt is None:
nt = int(rate * (n_f - 1))
else: # check if Nt is ok
nt = int(minimum(nt, rate * (n_f - 1)))
spec = self.copy()
lagtype = self.lagtype
d_t = spec.sampling_period()
# normalize spec so that sum(specn)/(n_f-1)=acf(0)=var(X)
specn = spec.data * freq[-1]
if spec.freqtype in 'f':
w = freq * 2 * pi
else:
w = freq
nfft = rate * 2 ** nextpow2(2 * n_f - 2)
# periodogram
rper = r_[
specn, zeros(nfft - (2 * n_f) + 2), conj(specn[n_f - 2:0:-1])]
time = r_[0:nt + 1] * d_t * (2 * n_f - 2) / nfft
r = fft(rper, nfft).real / (2 * n_f - 2)
acf = _WAFOCOV.CovData1D(r[0:nt + 1], time, lagtype=lagtype)
acf.tr = spec.tr
acf.h = spec.h
acf.norm = spec.norm
if nr > 0:
w = r_[w, zeros(nfft - 2 * n_f + 2), -w[n_f - 2:0:-1]]
fieldname = 'R' + lagtype[0] * nr
for i in range(1, nr + 1):
rper = -1j * w * rper
d_acf = fft(rper, nfft).real / (2 * n_f - 2)
setattr(acf, fieldname[0:i + 1], d_acf[0:nt + 1])
return acf
def to_linspec(self, ns=None, dt=None, cases=20, iseed=None,
fn_limit=sqrt(2), gravity=9.81):
'''
Split the linear and non-linear component from the Spectrum
according to 2nd order wave theory
Returns
-------
SL, SN : SpecData1D objects
with linear and non-linear components only, respectively.
Parameters
----------
ns : scalar integer
giving ns load points. (default length(S)-1=n-1).
If np>n-1 it is assummed that S(k)=0 for all k>n-1
cases : scalar integer
number of cases (default=20)
dt : real scalar
step in grid (default dt is defined by the Nyquist freq)
iseed : scalar integer
starting seed number for the random number generator
(default none is set)
fn_limit : real scalar
normalized upper frequency limit of spectrum for 2'nd order
components. The frequency is normalized with
sqrt(gravity*tanh(kbar*water_depth)/a_max)/(2*pi)
(default sqrt(2), i.e., Convergence criterion).
Generally this should be the same as used in the final
non-linear simulation (see example below).
SPEC2LINSPEC separates the linear and non-linear component of the
spectrum according to 2nd order wave theory. This is useful when
simulating non-linear waves because:
If the spectrum does not decay rapidly enough towards zero, the
contribution from the 2nd order wave components at the upper tail can
be very large and unphysical. Another option to ensure convergence of
the perturbation series in the simulation, is to truncate the upper
tail of the spectrum at FNLIMIT in the calculation of the 2nd order
wave components, i.e., in the calculation of sum and difference
frequency effects.
Examples
--------
np = 10000
iseed = 1
pflag = 2
S = jonswap(10)
fnLimit = inf
[SL,SN] = spec2linspec(S,np,[],[],fnLimit)
x0 = spec2nlsdat(SL,8*np,[],iseed,[],fnLimit)
x1 = spec2nlsdat(S,8*np,[],iseed,[],fnLimit)
x2 = spec2nlsdat(S,8*np,[],iseed,[],sqrt(2))
Se0 = dat2spec(x0)
Se1 = dat2spec(x1)
Se2 = dat2spec(x2)
clf
plotspec(SL,'r',pflag), % Linear components
hold on
plotspec(S,'b',pflag) % target spectrum for simulated data
plotspec(Se0,'m',pflag), % approx. same as S
plotspec(Se1,'g',pflag) % unphysical spectrum
plotspec(Se2,'k',pflag) % approx. same as S
axis([0 10 -80 0])
hold off
See also
--------
spec2nlsdat
References
----------
P. A. Brodtkorb (2004),
The probability of Occurrence of dangerous Wave Situations at Sea.
Dr.Ing thesis, Norwegian University of Science and Technolgy, NTNU,
Trondheim, Norway.
Nestegaard, A and Stokka T (1995)
A Third Order Random Wave model.
In proc.ISOPE conf., Vol III, pp 136-142.
R. S Langley (1987)
A statistical analysis of non-linear random waves.
Ocean Engng, Vol 14, pp 389-407
Marthinsen, T. and Winterstein, S.R (1992)
'On the skewness of random surface waves'
In proc. ISOPE Conf., San Francisco, 14-19 june.
'''
# by pab 13.08.2002
# TODO % Can be improved further.
method = 'apstochastic'
trace = 1 # % trace the convergence
max_sim = 30
tolerance = 5e-4
max_lag = 200 # maximum lag size of the window function used in estimate
# ftype = self.freqtype #options are 'f' and 'w' and 'k'
# switch ftype
# case 'f',
# ftype = 'w'
# S = ttspec(S,ftype)
# end
Hm0 = self.characteristic('Hm0')[0]
Tm02 = self.characteristic('Tm02')[0]
if iseed is not None:
_set_seed(iseed) # set the the seed
n = len(self.data)
if ns is None:
ns = max(n - 1, 5000)
if dt is None:
S = self.interp(dt) # interpolate spectrum
else:
S = self.copy()
ns = ns + mod(ns, 2) # make sure np is even
water_depth = abs(self.h)
kbar = w2k(2 * pi / Tm02, 0, water_depth)[0]
# Expected maximum amplitude for 10000 waves seastate
num_waves = 10000 # Typical number of waves in 30 hour seastate
a_max = sqrt(2 * log(num_waves)) * Hm0 / 4
f_limit_lo = sqrt(gravity * tanh(kbar * water_depth) * a_max / water_depth ** 3)
freq = S.args
eps = finfo(float).eps
freq[-1] = freq[-1] - sqrt(eps)
Hw2 = 0
SL = S
ind_zero = nonzero(freq < f_limit_lo)[0]
if len(ind_zero):
SL.data[ind_zero] = 0
max_spec = max(S.data)
# Fs = 2*freq(end)+eps # sampling frequency
for ix in range(max_sim):
x2, x1 = self.sim_nl(ns=ns, cases=cases, dt=None, iseed=iseed,
method=method, fnlimit=fn_limit,
output='timeseries')
x2.data -= x1.data # x2(:,2:end) = x2(:,2:end) -x1(:,2:end)
S2 = x2.tospecdata(max_lag)
S1 = x1.tospecdata(max_lag)
# TODO: Finish spec.to_linspec
# S2 = dat2spec(x2, max_lag)
# S1 = dat2spec(x1, max_lag)
# %[tf21,fi] = tfe(x2(:,2),x1(:,2),1024,Fs,[],512)
# %Hw11 = interp1q(fi,tf21.*conj(tf21),freq)
if True:
Hw1 = exp(interp1d(log(abs(S1.data / (S2.data + 1e-5))), S2.args,
fill_value=0, bounds_error=False)(freq))
else:
# Geometric mean
fun = interp1d(log(abs(S1.data / (S2.data + 1e-5))), S2.args,
fill_value=0, bounds_error=False)
Hw1 = exp((fun(freq) + log(Hw2)) / 2)
# end
# Hw1 = (interp1q( S2.w,abs(S1.S./S2.S),freq)+Hw2)/2
# plot(freq, abs(Hw11-Hw1),'g')
# title('diff')
# pause
# clf
# d1 = interp1q( S2.w,S2.S,freq)
SL.data = (Hw1 * S.data)
if len(ind_zero):
SL.data[ind_zero] = 0
# end
k = nonzero(SL.data < 0)[0]
if len(k): # Make sure that the current guess is larger than zero
# k
# Hw1(k)
Hw1[k] = min(S1.data[k] * 0.9, S.data[k])
SL.data[k] = max(Hw1[k] * S.data[k], eps)
# end
Hw12 = Hw1 - Hw2
maxHw12 = max(abs(Hw12))
if trace == 1:
plotbackend.figure(1)
plotbackend.semilogy(freq, Hw1, 'r')
plotbackend.title('Hw')
plotbackend.figure(2)
plotbackend.semilogy(freq, abs(Hw12), 'r')
plotbackend.title('Hw-HwOld')
# pause(3)
plotbackend.figure(1)
plotbackend.semilogy(freq, Hw1, 'b')
plotbackend.title('Hw')
plotbackend.figure(2)
plotbackend.semilogy(freq, abs(Hw12), 'b')
plotbackend.title('Hw-HwOld')
plotbackend.show('hold')
# figtile
# end
print('Iteration : %d, Hw12 : %g Hw12/max_spec : %g' %
(ix, maxHw12, (maxHw12 / max_spec)))
if (maxHw12 < max_spec * tolerance) and (Hw1[-1] < Hw2[-1]):
break
# end
Hw2 = Hw1
# end
# Hw1(end)
# max_spec*1e-3
# if Hw1[-1]*S.data>max_spec*1e-3,
# warning('The Nyquist frequency of the spectrum may be too low')
# end
SL.date = now() # datestr(now)
# if nargout>1
SN = SL.copy()
SN.data = S.data - SL.data
SN.note = SN.note + ' non-linear component (spec2linspec)'
# end
SL.note = SL.note + ' linear component (spec2linspec)'
return SL, SN
def to_mm_pdf(self, paramt=None, paramu=None, utc=None, nit=2, EPS=5e-5,
EPSS=1e-6, C=4.5, EPS0=1e-5, IAC=1, ISQ=0, verbose=False):
'''
nit = order of numerical integration: 0,1,2,3,4,5.
paramu = parameter vector defining discretization of min/max values.
t = grid of time points between maximum and minimum (to
integrate out). interval between maximum and the following
minimum,
The variable ISQ marks which type of conditioning will be used ISQ=0
means random time where the probability is minimum, ISQ=1 is the time
where the variance of the residual process is minimal(ISQ=1 is faster).
NIT, IAC are described in CROSSPACK paper, EPS0 is the accuracy
constant used in choosing the number of nodes in numerical integrations
(XX1, H1 vectors). The nodes and weights and other parameters are
read in the subroutine INITINTEG from files Z.DAT, H.DAT and ACCUR.DAT.
NIT=0, IAC=1 then one uses RIND0 - subroutine, all other cases
goes through RIND1, ...,RIND5. NIT=0, here means explicite formula
approximation for XIND=E[Y^+1{ HH<BU(I)<0 for all I, I=1,...,N}], where
BU(I) is deterministic function.
NIT=1, leads tp call RIND1, IAC=0 is also explicit form approximation,
while IAC=1 leads to maximum one dimensional integral.
.......
NIT=5, leads tp call RIND5, IAC is maximally 4-dimensional integral,
while IAC=1 leads to maximum 5 dimensional integral.
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=3)
>>> w = np.linspace(0,4,256)
>>> S1 = Sj.tospecdata(w) #Make spectrum object from numerical values
>>> S = sm.SpecData1D(Sj(w),w) # Alternatively do it manually
mm = S.to_mm_pdf()
mm.plot()
mm.plot(plotflag=1)
'''
S = self.copy()
S.normalize()
m = self.moment(nr=4, even=True)[0]
A = sqrt(m[0] / m[1])
if paramt is None:
# (2.5 * mean distance between extremes)
distance_between_extremes = 5 * pi * sqrt(m[1] / m[2])
paramt = [0, distance_between_extremes, 43]
if paramu is None:
paramu = [-5 * sqrt(m[0]), 5 * sqrt(m[0]), 41]
if self.tr is None:
g = TrLinear(var=m[0])
else:
g = self.tr
if utc is None:
utc = g.gauss2dat(0) # most frequent crossed level
# transform reference level into Gaussian level
u = g.dat2gauss(utc)
if verbose:
print('The level u for Gaussian process = %g' % u)
tn, Nt = paramt[1:]
t = linspace(0, tn / A, Nt) # normalized times
# Transform amplitudes to Gaussian levels:
h = linspace(*paramu)
dt = t[1] - t[0]
nr = 4
R = S.tocov_matrix(nr, Nt - 1, dt)
# ulev = linspace(*paramu)
# vlev = linspace(*paramu)
trdata = g.trdata()
Tg = trdata.args
Xg = trdata.data
cov2mod.initinteg(EPS, EPSS, EPS0, C, IAC, ISQ)
uvdens = cov2mod.cov2mmpdfreg(t, R, h, h, Tg, Xg, nit)
uvdens = np.rot90(uvdens, -2)
dh = h[1] - h[0]
uvdens *= dh * dh
mmpdf = PlotData(uvdens, args=(h, h), xlab='max [m]', ylab='min [m]',
title='Joint density of maximum and minimum')
try:
pl = [10, 30, 50, 70, 90, 95, 99, 99.9]
mmpdf.cl = qlevels(uvdens, pl, xi=(h, h))
mmpdf.pl = pl
except Exception:
pass
return mmpdf
def to_t_pdf(self, u=None, kind='Tc', paramt=None, **options):
'''
Density of crest/trough- period or length, version 2.
Parameters
----------
u : real scalar
reference level (default the most frequently crossed level).
kind : string, 'Tc', Tt', 'Lc' or 'Lt'
'Tc', gives half wave period, Tc (default).
'Tt', gives half wave period, Tt
'Lc' and 'Lt' ditto for wave length.
paramt : [t0, tn, nt]
where t0, tn and nt is the first value, last value and the number
of points, respectively, for which the density will be computed.
paramt= [5, 5, 51] implies that the density is computed only for
T=5 and using 51 equidistant points in the interval [0,5].
options : optional parameters
controlling the performance of the integration.
See Rind for details.
Notes
-----
SPEC2TPDF2 calculates pdf of halfperiods Tc, Tt, Lc or Lt
in a stationary Gaussian transform process X(t),
where Y(t) = g(X(t)) (Y zero-mean Gaussian with spectrum given in S).
The transformation, g, can be estimated using LC2TR,
DAT2TR, HERMITETR or OCHITR.
Examples
--------
# The density of Tc is computed by:
>>> import pylab as plb
>>> from wafo.spectrum import models as sm
>>> w = np.linspace(0,3,100)
>>> Sj = sm.Jonswap()
>>> S = Sj.tospecdata()
>>> f = S.to_t_pdf(pdef='Tc', paramt=(0, 10, 51), speed=7)
h = f.plot()
# estimated error bounds
h2 = plb.plot(f.args, f.data+f.err, 'r', f.args, f.data-f.err, 'r')
plb.close('all')
See also
--------
Rind, spec2cov2, specnorm, dat2tr, dat2gaus,
definitions.wave_periods,
definitions.waves
'''
opts = dict(speed=9)
opts.update(options)
if kind[0] in ('l', 'L'):
if self.type != 'k1d':
raise ValueError('Must be spectrum of type: k1d')
elif kind[0] in ('t', 'T'):
if self.type != 'freq':
raise ValueError('Must be spectrum of type: freq')
else:
raise ValueError('pdef must be Tc,Tt or Lc, Lt')
# if strncmpi('l',kind,1)
# spec=spec2spec(spec,'k1d')
# elseif strncmpi('t',kind,1)
# spec=spec2spec(spec,'freq')
# else
# error('Unknown kind')
# end
kind2defnr = dict(tc=1, lc=1, tt=-1, lt=-1)
defnr = kind2defnr[kind.lower()]
S = self.copy()
S.normalize()
m = self.moment(nr=2, even=True)[0]
A = sqrt(m[0] / m[1])
if self.tr is None:
g = TrLinear(var=m[0])
else:
g = self.tr
if u is None:
u = g.gauss2dat(0) # % most frequently crossed level
# transform reference level into Gaussian level
un = g.dat2gauss(u)
# disp(['The level u for Gaussian process = ', num2str(u)])
if paramt is None:
# z2 = u^2/2
z = -sign(defnr) * un / sqrt(2)
expectedMaxPeriod = 2 * \
ceil(2 * pi * A * exp(z) * (0.5 + erf(z) / 2))
paramt = [0, expectedMaxPeriod, 51]
t0 = paramt[0]
tn = paramt[1]
Ntime = paramt[2]
t = linspace(0, tn / A, Ntime) # normalized times
# index to starting point to evaluate
Nstart = max(round(t0 / tn * (Ntime - 1)), 1)
dt = t[1] - t[0]
nr = 2
R = S.tocov_matrix(nr, Ntime - 1, dt)
# R = spec2cov2(S,nr,Ntime-1,dt)
xc = vstack((un, un))
indI = -ones(4, dtype=int)
Nd = 2
Nc = 2
XdInf = 100.e0 * sqrt(-R[0, 2])
XtInf = 100.e0 * sqrt(R[0, 0])
B_up = hstack([un + XtInf, XdInf, 0])
B_lo = hstack([un, 0, -XdInf])
# INFIN = [1 1 0]
# BIG = zeros((Ntime+2,Ntime+2))
ex = zeros(Ntime + 2, dtype=float)
# CC = 2*pi*sqrt(-R(1,1)/R(1,3))*exp(un^2/(2*R(1,1)))
# XcScale = log(CC)
opts['xcscale'] = log(
2 * pi * sqrt(-R[0, 0] / R[0, 2])) + (un ** 2 / (2 * R[0, 0]))
f = zeros(Ntime, dtype=float)
err = zeros(Ntime, dtype=float)
rind = Rind(**opts)
# h11 = fwaitbar(0,[],sprintf('Please wait ...(start at: %s)',
# datestr(now)))
for pt in range(Nstart, Ntime):
Nt = pt - Nd + 1
Ntd = Nt + Nd
Ntdc = Ntd + Nc
indI[1] = Nt - 1
indI[2] = Nt
indI[3] = Ntd - 1
# positive wave period
BIG = self._covinput_t_pdf(pt, R)
tmp = rind(BIG, ex[:Ntdc], B_lo, B_up, indI, xc, Nt)
f[pt], err[pt] = tmp[:2]
# fwaitbar(pt/Ntime,h11,sprintf('%s Ready: %d of %d',
# datestr(now),pt,Ntime))
# end
# close(h11)
titledict = dict(
tc='Density of Tc', tt='Density of Tt', lc='Density of Lc',
lt='Density of Lt')
Htxt = titledict.get(kind.lower())
if kind[0].lower() == 'l':
xtxt = 'wave length [m]'
else:
xtxt = 'period [s]'
Htxt = '%s_{v =%2.5g}' % (Htxt, u)
pdf = PlotData(f / A, t * A, title=Htxt, xlab=xtxt)
pdf.err = err / A
pdf.u = u
pdf.options = opts
return pdf
@staticmethod
def _covinput_t_pdf(pt, R):
"""
Return covariance matrix for Tc or Tt period problems
Parameters
----------
pt : scalar integer
time
R : array-like, shape Ntime x 3
[R0,R1,R2] column vectors with autocovariance and its derivatives,
i.e., R1 and R2 are vectors with the 1'st and 2'nd derivatives of
R0, respectively.
The order of the variables in the covariance matrix are organized as
follows:
For pt>1:
||X(t2)..X(ts),..X(tn-1)|| X'(t1) X'(tn)|| X(t1) X(tn) ||
= [Xt Xd Xc]
where
Xt = time points in the indicator function
Xd = derivatives
Xc=variables to condition on
Computations of all covariances follows simple rules:
Cov(X(t),X(s))=r(t,s),
then Cov(X'(t),X(s))=dr(t,s)/dt. Now for stationary X(t) we have
a function r(tau) such that Cov(X(t),X(s))=r(s-t) (or r(t-s) will give
the same result).
Consequently
Cov(X'(t),X(s)) = -r'(s-t) = -sign(s-t)*r'(|s-t|)
Cov(X'(t),X'(s)) = -r''(s-t) = -r''(|s-t|)
Cov(X''(t),X'(s)) = r'''(s-t) = sign(s-t)*r'''(|s-t|)
Cov(X''(t),X(s)) = r''(s-t) = r''(|s-t|)
Cov(X''(t),X''(s)) = r''''(s-t) = r''''(|s-t|)
"""
# cov(Xd)
Sdd = -toeplitz(R[[0, pt], 2])
# cov(Xc)
Scc = toeplitz(R[[0, pt], 0])
# cov(Xc,Xd)
Scd = array([[0, R[pt, 1]], [-R[pt, 1], 0]])
if pt > 1:
# cov(Xt)
# Cov(X(tn),X(ts)) = r(ts-tn) = r(|ts-tn|)
Stt = toeplitz(R[:pt - 1, 0])
# cov(Xc,Xt)
# Cov(X(tn),X(ts)) = r(ts-tn) = r(|ts-tn|)
Sct = R[1:pt, 0]
Sct = vstack((Sct, Sct[::-1]))
# Cov(Xd,Xt)
# Cov(X'(t1),X(ts)) = -r'(ts-t1) = r(|s-t|)
Sdt = -R[1:pt, 1]
Sdt = vstack((Sdt, -Sdt[::-1]))
# N = pt + 3
big = vstack((hstack((Stt, Sdt.T, Sct.T)),
hstack((Sdt, Sdd, Scd.T)),
hstack((Sct, Scd, Scc))))
else:
# N = 4
big = vstack((hstack((Sdd, Scd.T)),
hstack((Scd, Scc))))
return big
def to_mmt_pdf(self, paramt=None, paramu=None, utc=None, kind='mm',
verbose=False, **options):
''' Returns joint density of Maximum, minimum and period.
Parameters
----------
u = reference level (default the most frequently crossed level).
kind : string
defining density returned
'Mm' : maximum and the following minimum. (M,m) (default)
'rfc' : maximum and the rainflow minimum height.
'AcAt' : (crest,trough) heights.
'vMm' : level v separated Maximum and minimum (M,m)_v
'MmTMm' : maximum, minimum and period between (M,m,TMm)
'vMmTMm': level v separated Maximum, minimum and period
between (M,m,TMm)_v
'MmTMd' : level v separated Maximum, minimum and the period
from Max to level v-down-crossing (M,m,TMd)_v.
'MmTdm' : level v separated Maximum, minimum and the period from
level v-down-crossing to min. (M,m,Tdm)_v
NB! All 'T' above can be replaced by 'L' to get wave length
instead.
paramt : [0 tn Nt]
defines discretization of half period: tn is the longest period
considered while Nt is the number of points, i.e. (Nt-1)/tn is the
sampling frequnecy. paramt= [0 10 51] implies that the halfperiods
are considered at 51 linearly spaced points in the interval [0,10],
i.e. sampling frequency is 5 Hz.
paramu : [u, v, N]
defines discretization of maxima and minima ranges: u is the
lowest minimum considered, v the highest maximum and N is the
number of levels (u,v) included.
options :
rind-options structure containing optional parameters controlling
the performance of the integration. See rindoptset for details.
[] = default values are used.
Returns
-------
f = pdf (density structure) of crests (trough) heights
TO_MMT_PDF calculates densities of wave characteristics in a
stationary Gaussian transform process X(t) where
Y(t) = g(X(t)) (Y zero-mean Gaussian with spectrum given in input spec)
The tr.g can be estimated using lc2tr, dat2tr, hermitetr or ochitr.
Examples
--------
The joint density of zero separated Max2min cycles in time (a);
in space (b); AcAt in time for nonlinear sea model (c):
>>> from wafo.spectrum import models as sm
>>> w = np.linspace(0,3,100)
>>> Sj = sm.Jonswap()
>>> S = Sj.tospecdata()
>>> f = S.to_t_pdf(pdef='Tc', paramt=(0, 10, 51), speed=7)
>>> S = sm.Jonswap(wnc=2, Hm0=7, Tp=11)
Sk = spec2spec(S,'k1d')
L0 = spec2mom(S,1)
paramu = [sqrt(L0)*[-4 4] 41]
ft = spec2mmtpdf(S,0,'vmm',[],paramu); pdfplot(ft) % a)
fs = spec2mmtpdf(Sk,0,'vmm'); figure, pdfplot(fs) % b)
[sk, ku, me]=spec2skew(S)
g = hermitetr([],[sqrt(L0) sk ku me])
Snorm=S; Snorm.S=S.S/L0; Snorm.tr=g
ftg=spec2mmtpdf(Snorm,0,'AcAt',[],paramu); pdfplot(ftg) % c)
See also
--------
rindoptset, dat2tr, datastructures, wavedef, perioddef
References
----------
Podgorski et al. (2000)
"Exact distributions for apparent waves in irregular seas"
Ocean Engineering, Vol 27, no 1, pp979-1016.
P. A. Brodtkorb (2004),
Numerical evaluation of multinormal expectations
In Lund university report series
and in the Dr.Ing thesis:
The probability of Occurrence of dangerous Wave Situations at Sea.
Dr.Ing thesis, Norwegian University of Science and Technolgy, NTNU,
Trondheim, Norway.
P. A. Brodtkorb (2006)
"Evaluating Nearly Singular Multinormal Expectations with Application
to Wave Distributions",
Methodology And Computing In Applied Probability, Volume 8, Number 1,
pp. 65-91(27)
'''
opts = dict(speed=4, nit=2, method=0)
opts.update(**options)
ftype = self.freqtype
kind2defnr = dict(ac=-2, at=-2,
rfc=-1,
mm=0,
mmtmm=1, mmlmm=1,
vmm=2,
vmmtmm=3, vmmlmm=3,
mmtmd=4, vmmtmd=4, mmlmd=4, vmmlmd=4,
mmtdm=5, vmmtdm=5, mmldm=5, vmmldm=5)
defnr = kind2defnr.get(kind, 0)
in_space = (ftype == 'k') # distribution in space or time
if defnr >= 3 or defnr == 1:
in_space = (kind[-2].upper() == 'L')
if in_space:
# spec = spec2spec(spec,'k1d')
ptxt = 'space'
else:
# spec = spec2spec(spec,'freq')
ptxt = 'time'
S = self.copy()
S.normalize()
m, unused_mtxt = self.moment(nr=4, even=True)
L0, L2, L4 = m
A = sqrt(m[0] / m[1])
if paramt is None:
# (2.5 * mean distance between extremes)
distanceBetweenExtremes = 5 * pi * sqrt(m[1] / m[2])
paramt = [0, distanceBetweenExtremes, 43]
if paramu is None:
paramu = [-5 * sqrt(m[0]), 5 * sqrt(m[0]), 41]
if self.tr is None:
g = TrLinear(var=m[0])
else:
g = self.tr
if utc is None:
utc = g.gauss2dat(0) # most frequent crossed level
# transform reference level into Gaussian level
u = g.dat2gauss(utc)
if verbose:
print('The level u for Gaussian process = %g' % u)
t0, tn, Nt = paramt
t = np.linspace(0, tn / A, Nt) # normalized times
# the starting point to evaluate
Nstart = 1 + round(t0 / tn * (Nt - 1))
Nx = paramu[2]
if (defnr > 1):
paramu[0] = max(0, paramu[0])
if (paramu[1] < 0):
raise ValueError(
'Discretization levels must be larger than zero')
# Transform amplitudes to Gaussian levels:
h = linspace(*paramu)
if defnr > 1: # level v separated Max2min densities
hg = np.hstack((utc + h, utc - h))
hg, der = g.dat2gauss(utc + h, ones(Nx))
hg1, der1 = g.dat2gauss(utc - h, ones(Nx))
der, der1 = np.abs(der), np.abs(der1)
hg = np.hstack((hg, hg1))
else: # Max2min densities
hg, der = np.abs(g.dat2gauss(h, ones(Nx)))
der = der1 = np.abs(der)
dt = t[1] - t[0]
nr = 4
R = S.tocov_matrix(nr, Nt - 1, dt)
# NB!!! the spec2XXpdf.exe programmes are very sensitive to how you
# interpolate the covariances, especially where the process is very
# dependent and the covariance matrix is nearly singular. (i.e. for
# small t and high levels of u if Tc and low levels of u if Tt)
# The best is to interpolate the spectrum linearly so that S.S>=0
# This makes sure that the covariance matrix is positive
# semi-definitt, since the circulant spectrum are the eigenvalues of
# the circulant covariance matrix.
# callFortran = 0
# %options.method<0
# if callFortran, % call fortran
# ftmp = cov2mmtpdfexe(R,dt,u,defnr,Nstart,hg,options)
# err = repmat(nan,size(ftmp))
# else
ftmp, err, _terr, options = self._cov2mmtpdf(R, dt, u, defnr, Nstart,
hg, options)
# end
note = ''
if hasattr(self, 'note'):
note = note + self.note
tmp = 'L' if in_space else 'T'
title = ''
labx = ''
laby = ''
args = None
if Nx > 2:
titledict = {
'-2': 'Joint density of (Ac,At) in %s' % ptxt,
'-1': 'Joint density of (M,m_{rfc}) in %s' % ptxt,
'0': 'Joint density of (M,m) in %s' % ptxt,
'1': 'Joint density of (M,m,%sMm) in %s' % (tmp, ptxt),
'2': 'Joint density of (M,m)_{v=%2.5g} in %s' % (utc, ptxt),
'3': 'Joint density of (M,m,%sMm)_{v=%2.5g} in %s' %
(tmp, utc, ptxt),
'4': 'Joint density of (M,m,%sMd)_{v=%2.5g} in %s' %
(tmp, utc, ptxt),
'5': 'Joint density of (M,m,%sdm)_{v=%2.5g} in %s' %
(tmp, utc, ptxt)}
title = titledict[defnr]
labx = 'Max [m]'
laby = 'min [m]'
args = (h, h)
else:
note = note + 'Density is not scaled to unity'
if defnr in (-2, -1, 0, 1):
title_txt = 'Density of (%sMm, M = %2.5g, m = %2.5g)'
title = title_txt % (tmp, h[1], h[0])
elif defnr in (2, 3):
title_txt = 'Density of (%sMm, M = %2.5g, m = %2.5g)_{v=%2.5g}'
title = title_txt % (tmp, h[1], -h[1], utc)
elif defnr == 4:
txt = 'Density of (%sMd, %sMm, M = %2.5g, m = %2.5g)_{v=%2.5g}'
title = txt % (tmp, tmp, h[1], -h[1], utc)
elif defnr == 5:
txt = 'Density of (%sdm, %sMm, M = %2.5g, m = %2.5g)_{v=%2.5g}'
title = txt % (tmp, tmp, h[1], -h[1], utc)
f = PlotData(args=args, title=title, labx=labx, laby=laby)
f.options = options
if defnr > 1 or defnr == -2:
f.u = utc # save level u
if Nx > 2: # amplitude distributions wanted
# f.x{2} = h
# f.labx{2} = 'min [m]'
if defnr > 2 or defnr == 1:
der0 = der1[:, None] * der[None, :]
shape = (Nx, Nx, Nt)
ftmp = np.reshape(ftmp, shape) * der0[:, :, None] / A
err = np.reshape(err, shape) * der0[:, :, None] / A
f.args[2] = t[:] * A
_labz = 'wave length [m]' if in_space else 'period [sec]'
else:
der0 = der[:, None] * der[None, :]
ftmp = np.reshape(ftmp, [Nx, Nx]) * der0
err = np.reshape(err, [Nx, Nx]) * der0
if (defnr == -1):
ftmp0 = np.fliplr(mctp2rfc(np.fliplr(ftmp)))
err = np.abs(ftmp0 -
np.fliplr(mctp2rfc(np.fliplr(ftmp + err))))
ftmp = ftmp0
elif (defnr == -2):
ftmp0 = np.fliplr(mctp2tc(np.fliplr(ftmp), utc,
paramu)) * sqrt(L4 * L0) / L2
err = np.abs(ftmp0 -
np.fliplr(mctp2tc(np.fliplr(ftmp + err),
utc, paramu)) *
sqrt(L4 * L0) / L2)
index1 = np.flatnonzero(f.args[0] > 0)
index2 = np.flatnonzero(f.args[1] < 0)
ftmp = np.flipud(ftmp0[index2, index1])
err = np.flipud(err[index2, index1])
f.args[0] = f.args[0][index1]
f.args[1] = np.abs(np.flipud(f.args[1][index2]))
# end
# end
f.data = ftmp
f.err = err
else: # Only time or wave length distributions wanted
f.data = ftmp / A
f.err = err / A
f.args[0] = A * t
# if def_[0] == 't':
# f.labx{1} = 'period [sec]'
# else:
# f.labx{1} = 'wave length [m]'
# end
if defnr > 3:
f.data = np.reshape(f.data, [Nt, Nt])
f.err = np.reshape(f.err, [Nt, Nt])
f.args[1] = A * t
# if def_[0] == 't':
# f.labx{2} = 'period [sec]'
# else:
# f.labx{2} = 'wave length [m]'
# end
# end
# end
try:
f.cl, f.pl = qlevels(f.data, [10, 30, 50, 70, 90, 95, 99, 99.9],
f.args[0], f.args[1])
except Exception:
warnings.warn('Singularity likely in pdf')
# Test of spec2mmtpdf
# cd f:\matlab\matlab\wafo\source\sp2thpdfalan
# addpath f:\matlab\matlab\wafo ,initwafo,
# addpath f:\matlab\matlab\graphutil
# Hm0=7;Tp=11; S = jonswap(4*pi/Tp,[Hm0 Tp])
# ft = spec2mmtpdf(S,0,'vMmTMm',[0.3,.4,11],[0 .00005 2])
return f
def _cov2mmtpdf(self, R, dt, u, def_nr, Nstart, hg, options):
'''
COV2MMTPDF Joint density of Maximum, minimum and period.
CALL [pdf, err, options] = cov2mmtpdf(R,dt,u,def,Nstart,hg,options)
pdf = calculated pdf size Nx x Ntime
err = error estimate
terr = truncation error
options = requested and actual rindoptions used in integration.
R = [R0,R1,R2,R3,R4] column vectors with autocovariance and its
derivatives, i.e., Ri (i=1:4) are vectors with the 1'st to
4'th derivatives of R0. size Ntime x Nr+1
dt = time spacing between covariance samples, i.e.,
between R0(1),R0(2).
u = crossing level
def = integer defining pdf calculated:
0 : maximum and the following minimum. (M,m) (default)
1 : level v separated Maximum and minimum (M,m)_v
2 : maximum, minimum and period between (M,m,TMm)
3 : level v separated Maximum, minimum and period
between (M,m,TMm)_v
4 : level v separated Maximum, minimum and the period
from Max to level v-down-crossing (M,m,TMd)_v.
5 : level v separated Maximum, minimum and the period from
level v-down-crossing to min. (M,m,Tdm)_v
Nstart = index to where to start calculation, i.e., t0 = t(Nstart)
hg = vector of amplitudes length Nx or 0
options = rind options structure defining the integration parameters
COV2MMTPDF computes joint density of the maximum and the following
minimum or level u separated maxima and minima + period/wavelength
For DEF = 0,1 : (Maxima, Minima and period/wavelength)
= 2,3 : (Level v separated Maxima and Minima and
period/wavelength between them)
If Nx==1 then the conditional density for period/wavelength between
Maxima and Minima given the Max and Min is returned
Y =
X'(t2)..X'(ts)..X'(tn-1)|| X''(t1) X''(tn)|| X'(t1) X'(tn) X(t1) X(tn)
= [ Xt Xd Xc ]
Nt = tn-2, Nd = 2, Nc = 4
Xt = contains Nt time points in the indicator function
Xd = " Nd derivatives in Jacobian
Xc = " Nc variables to condition on
There are 3 (NI=4) regions with constant barriers:
(indI[0]=0); for i in (indI[0],indI[1]] Y[i]<0.
(indI[1]=Nt); for i in (indI[1]+1,indI[2]], Y[i]<0 (deriv. X''(t1))
(indI[2]=Nt+1); for i in (indI[2]+1,indI[3]], Y[i]>0 (deriv. X''(tn))
For DEF = 4,5 (Level v separated Maxima and Minima and
period/wavelength from Max to crossing)
If Nx==1 then the conditional joint density for period/wavelength
between Maxima, Minima and Max to level v crossing given the Max and
the min is returned
Y = [Xt, Xd, Xc]
where
Xt = X'(t2)..X'(ts)..X'(tn-1)
Xd = ||X''(t1) X''(tn) X'(ts)||
Xc = X'(t1) X'(tn) X(t1) X(tn) X(ts)
Nt = tn-2, Nd = 3, Nc = 5
Xt = contains Nt time points in the indicator function
Xd = " Nd derivatives
Xc = " Nc variables to condition on
There are 4 (NI=5) regions with constant barriers:
(indI(1)=0); for i in (indI(1),indI(2)] Y(i)<0.
(indI(2)=Nt) ; for i in (indI(2)+1,indI(3)], Y(i)<0 (deriv. X''(t1))
(indI(3)=Nt+1); for i in (indI(3)+1,indI(4)], Y(i)>0 (deriv. X''(tn))
(indI(4)=Nt+2); for i in (indI(4)+1,indI(5)], Y(i)<0 (deriv. X'(ts))
'''
R0, R2, R4 = R[:, :5:2].T
covinput = self._covinput_mmt_pdf
Ntime = len(R0)
Nx0 = max(1, len(hg))
Nx1 = Nx0
# Nx0 = Nx1 #just plain Mm
if def_nr > 1:
Nx1 = Nx0 // 2
# Nx0 = 2*Nx1 # level v separated max2min densities wanted
# print('def = %d' % def_nr))
# The bound 'infinity' is set to 100*sigma
XdInf = 100.0 * sqrt(R4[0])
XtInf = 100.0 * sqrt(-R2[0])
Nc = 4
NI = 4
Nd = 2
# Mb = 1
# Nj = 0
Nstart = max(2, Nstart)
symmetry = 0
is_odd = np.mod(Nx1, 2)
if def_nr <= 1: # just plain Mm
Nx = Nx1 * (Nx1 - 1) / 2
IJ = (Nx1 + is_odd) / 2
if (hg[0] + hg[Nx1 - 1] == 0 and (hg[IJ - 1] == 0 or
hg[IJ - 1] + hg[IJ] == 0)):
symmetry = 0
print(' Integration region symmetric')
# May save Nx1-is_odd integrations in each time step
# This is not implemented yet.
# Nx = Nx1*(Nx1-1)/2-Nx1+is_odd
# normalizing constant:
# CC = 1/ expected number of zero-up-crossings of X'
# CC = 2*pi*sqrt(-R2[0]/R4[0])
# XcScale = log(CC)
XcScale = log(2 * pi * sqrt(-R2[0] / R4[0]))
else:
# level u separated Mm
Nx = (Nx1 - 1) * (Nx1 - 1)
if (abs(u) <= _EPS and (hg[0] + hg[Nx1] == 0) and
(hg[Nx1 - 1] + hg[2 * Nx1 - 1] == 0)):
symmetry = 0
print(' Integration region symmetric')
# Not implemented for DEF <= 3
# IF (DEF.LE.3) Nx = (Nx1-1)*(Nx1-2)/2
if def_nr > 3:
Nstart = max(Nstart, 3)
Nc = 5
NI = 5
Nd = 3
# CC = 1/ expected number of u-up-crossings of X
# CC = 2*pi*sqrt(-R0(1)/R2(1))*exp(0.5D0*u*u/R0(1))
XcScale = log(2 * pi * sqrt(-R0[0] / R2[0])) + 0.5 * u * u / R0[0]
options['xcscale'] = XcScale
# opt0 = [options[n] for n in ('SCIS', 'XcScale', 'ABSEPS', 'RELEPS',
# 'COVEPS', 'MAXPTS', 'MINPTS', 'seed',
# 'NIT1')]
dt2 = dt ** 2
rind = Rind(**options)
if (Nx > 1):
# (M,m) or (M,m)v distribution wanted
if def_nr in [0, 2]:
asize = [Nx1, Nx1]
else:
# (M,m,TMm), (M,m,TMm)v (M,m,TMd)v or (M,M,Tdm)v
# distributions wanted
asize = [Nx1, Nx1, Ntime]
elif (def_nr > 3):
# Conditional distribution for (TMd,TMm)v or (Tdm,TMm)v given (M,m)
# wanted
asize = [1, Ntime, Ntime]
else:
# Conditional distribution for (TMm) or (TMm)v given (M,m) wanted
asize = [1, 1, Ntime]
# Initialization
pdf = zeros(asize)
err = zeros(asize)
terr = zeros(asize)
BIG = zeros(Ntime + Nc + 1, Ntime + Nc + 1)
ex = zeros(1, Ntime + Nc + 1)
# fxind = zeros(Nx,1)
xc = zeros(Nc, Nx)
indI = zeros(1, NI)
a_up = zeros(1, NI - 1)
a_lo = zeros(1, NI - 1)
# INFIN = INTEGER, array of integration limits flags: size 1 x Nb (in)
# if INFIN(I) < 0, Ith limits are (-infinity, infinity)
# if INFIN(I) = 0, Ith limits are (-infinity, Hup(I)]
# if INFIN(I) = 1, Ith limits are [Hlo(I), infinity)
# if INFIN(I) = 2, Ith limits are [Hlo(I), Hup(I)].
# INFIN = repmat(0,1,NI-1)
# INFIN(3) = 1
a_up[0, 2] = +XdInf
a_lo[0, :2] = [-XtInf, -XdInf]
if (def_nr > 3):
a_lo[0, 3] = -XtInf
IJ = 0
if (def_nr <= 1): # Max2min and period/wavelength
for I in range(1, Nx1):
J = IJ + I
xc[2, IJ:J] = hg[I]
xc[3, IJ:J] = hg[:I].T
IJ = J
else:
# Level u separated Max2min
xc[Nc, :] = u
# Hg(1) = Hg(Nx1+1)= u => start do loop at I=2 since by definition
# we must have: minimum<u-level<Maximum
for i in range(1, Nx1):
J = IJ + Nx1
xc[2, IJ:J] = hg[i] # Max > u
xc[3, IJ:J] = hg[Nx1 + 2: 2 * Nx1].T # Min < u
IJ = J
if (def_nr <= 3):
# h11 = fwaitbar(0,[],sprintf('Please wait ...(start at:
# %s)',datestr(now)))
for Ntd in range(Nstart, Ntime):
# Ntd=tn
Ntdc = Ntd + Nc
Nt = Ntd - Nd
indI[1] = Nt
indI[2] = Nt + 1
indI[3] = Ntd
# positive wave period
# self._covinput_mmt_pdf(BIG, R, tn, ts, tnold)
BIG[:Ntdc, :Ntdc] = covinput(BIG[:Ntdc, :Ntdc], R, Ntd, 0)
fxind, err0, terr0 = rind(BIG[:Ntdc, :Ntdc], ex[:Ntdc],
a_lo, a_up, indI, xc, Nt)
err0 = err0 ** 2
# fxind = CC*rind(BIG(1:Ntdc,1:Ntdc),ex(1:Ntdc),xc,Nt,NIT1,
# speed1,indI,a_lo,a_up)
if (Nx < 2):
# Density of TMm given the Max and the Min. Note that the
# density is not scaled to unity
pdf[0, 0, Ntd] = fxind[0]
err[0, 0, Ntd] = err0[0]
terr[0, 0, Ntd] = terr0[0]
# GOTO 100
else:
IJ = 0
# joint density of (Ac,At),(M,m_rfc) or (M,m).
if def_nr in [-2, -1, 0]:
for i in range(1, Nx1):
J = IJ + i
pdf[:i, i, 0] += fxind[IJ:J].T * dt # *CC
err[:i, i, 0] += err0[IJ + 1:J].T * dt2
terr[:i, i, 0] += terr0[IJ:J].T * dt
IJ = J
elif def_nr == 1: # joint density of (M,m,TMm)
for i in range(1, Nx1):
J = IJ + i
pdf[:i, i, Ntd] = fxind[IJ:J].T # *CC
err[:i, i, Ntd] = err0[IJ:J].T # *CC
terr[:i, i, Ntd] = terr0[IJ:J].T # *CC
IJ = J
# end do
# joint density of level v separated (M,m)v
elif def_nr == 2:
for i in range(1, Nx1):
J = IJ + Nx1
pdf[1:Nx1, i, 0] += fxind[IJ:J].T * dt # *CC
err[1:Nx1, i, 0] += err0[IJ:J].T * dt2
terr[1:Nx1, i, 0] += terr0[IJ:J].T * dt
IJ = J
# end %do
elif def_nr == 3:
# joint density of level v separated (M,m,TMm)v
for i in range(1, Nx1):
J = IJ + Nx1
pdf[1:Nx1, i, Ntd] += fxind[IJ:J].T # %*CC
err[1:Nx1, i, Ntd] += err0[IJ:J].T
terr[1:Nx1, i, Ntd] += terr0[IJ:J].T
IJ = J
# end do
# end SELECT
# end ENDIF
# waitTxt = '%s Ready: %d of %d' % (datestr(now),Ntd,Ntime)
# fwaitbar(Ntd/Ntime,h11,waitTxt)
# end %do
# close(h11)
err = sqrt(err)
# goto 800
else: # def_nr>3
# 200 continue
# waitTxt = sprintf('Please wait ...(start at: %s)',datestr(now))
# h11 = fwaitbar(0,[],waitTxt)
tnold = -1
for tn in range(Nstart, Ntime):
Ntd = tn + 1
Ntdc = Ntd + Nc
Nt = Ntd - Nd
indI[1] = Nt
indI[2] = Nt + 1
indI[3] = Nt + 2
indI[4] = Ntd
if not symmetry: # IF (SYMMETRY) GOTO 300
for ts in range(1, tn - 1): # = 2:tn-1:
# positive wave period
BIG[:Ntdc, :Ntdc] = covinput(BIG[:Ntdc, :Ntdc],
R, tn, ts, tnold)
fxind, err0, terr0 = rind(BIG[:Ntdc, :Ntdc], ex[:Ntdc],
a_lo, a_up, indI, xc, Nt)
err0 = err0 ** 2
# tnold = tn
tns = tn - ts
if def_nr in [3, 4]:
if (Nx == 1):
# Joint density (TMd,TMm) given the Max and min
# Note the density is not scaled to unity
pdf[0, ts, tn] = fxind[0] # *CC
err[0, ts, tn] = err0[0] # *CC
terr[0, ts, tn] = terr0[0] # *CC
else:
# level u separated Max2min and wave period
# from Max to the crossing of level u
# (M,m,TMd).
IJ = 0
for i in range(1, Nx1):
J = IJ + Nx1
pdf[1:Nx1, i, ts] += fxind[IJ:J].T * dt
err[1:Nx1, i, ts] += err0[IJ:J].T * dt2
terr[1:Nx1, i, ts] += terr0[IJ:J].T * dt
IJ = J
# end %do
# end
elif def_nr == 5:
if (Nx == 1):
# Joint density (Tdm,TMm) given the Max and min
# Note the density is not scaled to unity
pdf[0, tns, tn] = fxind[0] # *CC
err[0, tns, tn] = err0[0]
terr[0, tns, tn] = terr0[0]
else:
# level u separated Max2min and wave period
# from the crossing of level u to the
# min (M,m,Tdm).
IJ = 0
for i in range(1, Nx1): # = 2:Nx1
J = IJ + Nx1
# *CC
pdf[1:Nx1, i, tns] += fxind[IJ:J].T * dt
err[1:Nx1, i, tns] += err0[IJ:J].T * dt2
terr[1:Nx1, i, tns] += terr0[IJ:J].T * dt
IJ = J
# end %do
# end
# end % SELECT
# end% enddo
else: # % exploit symmetry
# 300 Symmetry
for ts in range(1, Ntd // 2): # = 2:floor(Ntd//2)
# Using the symmetry since U = 0 and the
# transformation is linear.
# positive wave period
BIG[:Ntdc, :Ntdc] = covinput(BIG[:Ntdc, :Ntdc],
R, tn, ts, tnold)
fxind, err0, terr0 = rind(BIG[:Ntdc, :Ntdc], ex[:Ntdc],
a_lo, a_up, indI, xc, Nt)
# [fxind,err0] = rind(BIG(1:Ntdc,1:Ntdc),ex,a_lo,a_up,
# indI, xc,Nt,opt0{:})
# tnold = tn
tns = tn - ts
if (Nx == 1): # % THEN
# Joint density of (TMd,TMm),(Tdm,TMm) given
# the max and the min.
# Note that the density is not scaled to unity
pdf[0, ts, tn] = fxind[0] # %*CC
err[0, ts, tn] = err0[0]
err[0, ts, tn] = terr0[0]
if (ts < tns): # %THEN
pdf[0, tns, tn] = fxind[0] # *CC
err[0, tns, tn] = err0[0] ** 2
terr[0, tns, tn] = terr0[0]
# end
# GOTO 350
else:
IJ = 0
if def_nr == 4:
# level u separated Max2min and wave period
# from Max to the crossing of level u (M,m,TMd)
for i in range(1, Nx1):
J = IJ + Nx1
# *CC
pdf[1:Nx1, i, ts] += fxind[IJ:J] * dt
err[1:Nx1, i, ts] += err0[IJ:J] * dt2
terr[1:Nx1, i, ts] += terr0[IJ:J] * dt
if (ts < tns):
# exploiting the symmetry
# *CC
pdf[i, 1:Nx1, tns] += fxind[IJ:J] * dt
err[i, 1:Nx1, tns] += err0[IJ:J] * dt2
terr[i, 1:Nx1, tns] += terr0[IJ:J] * dt
# end
IJ = J
# end do
elif def_nr == 5:
# level u separated Max2min and wave period
# from the crossing of level u to min (M,m,Tdm)
for i in range(1, Nx1): # = 2:Nx1,
J = IJ + Nx1
pdf[1:Nx1, i, tns] += fxind[IJ:J] * dt
err[1:Nx1, i, tns] += err0[IJ:J] * dt2
terr[1:Nx1, i, tns] += terr0[IJ:J] * dt
if (ts < tns + 1):
# exploiting the symmetry
pdf[i, 1:Nx1, ts] += fxind[IJ:J] * dt
err[i, 1:Nx1, ts] += err0[IJ:J] * dt2
terr[i, 1:Nx1, ts] += terr0[IJ:J] * dt
# end %ENDIF
IJ = J
# end do
# end %END SELECT
# end
# 350
# end %do
# end
# waitTxt = sprintf('%s Ready: %d of %d',datestr(now),tn,Ntime)
# fwaitbar(tn/Ntime,h11,waitTxt)
# 400 print *,'Ready: ',tn,' of ',Ntime
# end %do
# close(h11)
err = sqrt(err)
# end % if
# Nx1,size(pdf) def Ntime
if (Nx > 1): # % THEN
IJ = 1
if (def_nr > 2 or def_nr == 1):
IJ = Ntime
# end
pdf = pdf[:Nx1, :Nx1, :IJ]
err = err[:Nx1, :Nx1, :IJ]
terr = terr[:Nx1, :Nx1, :IJ]
else:
IJ = 1
if (def_nr > 3):
IJ = Ntime
# end
pdf = np.squeeze(pdf[0, :IJ, :Ntime])
err = np.squeeze(err[0, :IJ, :Ntime])
terr = np.squeeze(terr[0, :IJ, :Ntime])
# end
return pdf, err, terr, options
@staticmethod
def _covinput_mmt_pdf(BIG, R, tn, ts, tnold=-1):
"""
COVINPUT Sets up the covariance matrix
CALL BIG = covinput(BIG, R0,R1,R2,R3,R4,tn,ts)
BIG = covariance matrix for X = [Xt,Xd,Xc] in spec2mmtpdf problems.
The order of the variables in the covariance matrix are organized as
follows:
for ts <= 1:
Xt = X'(t2)..X'(ts),...,X'(tn-1)
Xd = X''(t1), X''(tn), X'(t1), X'(tn)
Xc = X(t1),X(tn)
for ts > =2:
Xt = X'(t2)..X'(ts),...,X'(tn-1)
Xd = X''(t1), X''(tn), X'(ts), X'(t1), X'(tn),
Xc = X(t1),X(tn) X(ts)
where
Xt = time points in the indicator function
Xd = derivatives
Xc = variables to condition on
Computations of all covariances follows simple rules:
Cov(X(t),X(s)) = r(t,s),
then Cov(X'(t),X(s))=dr(t,s)/dt. Now for stationary X(t) we have
a function r(tau) such that Cov(X(t),X(s))=r(s-t) (or r(t-s) will give
the same result).
Consequently Cov(X'(t),X(s)) = -r'(s-t) = -sign(s-t)*r'(|s-t|)
Cov(X'(t),X'(s)) = -r''(s-t) = -r''(|s-t|)
Cov(X''(t),X'(s)) = r'''(s-t) = sign(s-t)*r'''(|s-t|)
Cov(X''(t),X(s)) = r''(s-t) = r''(|s-t|)
Cov(X''(t),X''(s)) = r''''(s-t) = r''''(|s-t|)
"""
R0, R1, R2, R3, R4 = R[:, :5].T
if (ts > 1):
shft = 1
N = tn + 5 + shft
# Cov(Xt,Xc)
# for
i = np.arange(tn - 2) # 1:tn-2
# j = abs(i+1-ts)
# BIG(i,N) = -sign(R1(j+1),R1(j+1)*dble(ts-i-1))
j = i + 1 - ts
tau = abs(j)
# BIG(i,N) = abs(R1(tau)).*sign(R1(tau).*j.')
BIG[i, N] = R1[tau] * sign(j) # cov(X'(ti+1),X(ts))
# end do
# Cov(Xc)
BIG[N, N] = R0[0] # cov(X(ts),X(ts))
BIG[tn + shft + 1, N] = -R1[ts] # cov(X'(t1),X(ts))
BIG[tn + shft + 2, N] = R1[tn - ts] # cov(X'(tn),X(ts))
BIG[tn + shft + 3, N] = R0[ts] # cov(X(t1),X(ts))
BIG[tn + shft + 4, N] = R0[tn - ts] # cov(X(tn),X(ts))
# Cov(Xd,Xc)
BIG[tn - 1, N] = R2[ts] # cov(X''(t1),X(ts))
BIG[tn, N] = R2[tn - ts] # cov(X''(tn),X(ts))
# ADD a level u crossing at ts
# Cov(Xt,Xd)
# for
i = np.arange(tn - 2) # 1:tn-2
j = abs(i + 1 - ts)
BIG[i, tn + shft] = -R2[j] # cov(X'(ti+1),X'(ts))
# end do
# Cov(Xd)
BIG[tn + shft, tn + shft] = -R2[0] # cov(X'(ts),X'(ts))
BIG[tn - 1, tn + shft] = R3[ts] # cov(X''(t1),X'(ts))
BIG[tn, tn + shft] = -R3[tn - ts] # cov(X''(tn),X'(ts))
# Cov(Xd,Xc)
BIG[tn + shft, N] = 0.0 # %cov(X'(ts),X(ts))
# % cov(X'(ts),X'(t1))
BIG[tn + shft, tn + shft + 1] = -R2[ts]
# % cov(X'(ts),X'(tn))
BIG[tn + shft, tn + shft + 2] = -R2[tn - ts]
BIG[tn + shft, tn + shft + 3] = R1[ts] # % cov(X'(ts),X(t1))
# % cov(X'(ts),X(tn))
BIG[tn + shft, tn + shft + 4] = -R1[tn - ts]
if (tnold == tn):
# A previous call to covinput with tn==tnold has been made
# need only to update row and column N and tn+1 of big:
return BIG
# % make lower triangular part equal to upper and then return
# for j=1:tn+shft
# BIG(N,j) = BIG(j,N)
# BIG(tn+shft,j) = BIG(j,tn+shft)
# end
# for j=tn+shft+1:N-1
# BIG(N,j) = BIG(j,N)
# BIG(j,tn+shft) = BIG(tn+shft,j)
# end
# return
# end %if
# %tnold = tn
else:
# N = tn+4
shft = 0
# end %if
if (tn > 2):
# for i=1:tn-2
# cov(Xt)
# for j=i:tn-2
# BIG(i,j) = -R2(j-i+1) % cov(X'(ti+1),X'(tj+1))
# end %do
# % cov(Xt) = % cov(X'(ti+1),X'(tj+1))
BIG[:tn - 2, :tn - 2] = toeplitz(-R2[:tn - 2])
# cov(Xt,Xc)
BIG[:tn - 2, tn + shft] = -R2[1:tn - 1] # cov(X'(ti+1),X'(t1))
# cov(X'(ti+1),X'(tn))
BIG[:tn - 2, tn + shft + 1] = -R2[tn - 2:0:-1]
BIG[:tn - 2, tn + shft + 2] = R1[1:tn - 1] # cov(X'(ti+1),X(t1))
# cov(X'(ti+1),X(tn))
BIG[:tn - 2, tn + shft + 3] = -R1[tn - 2:0:-1]
# Cov(Xt,Xd)
BIG[:tn - 2, tn - 2] = R3[1:tn - 1] # cov(X'(ti+1),X''(t1))
BIG[:tn - 2, tn - 1] = -R3[tn - 2:0:-1] # cov(X'(ti+1),X''(tn))
# end %do
# end
# cov(Xd)
BIG[tn - 2, tn - 2] = R4[0]
BIG[tn - 2, tn - 1] = R4[tn - 1] # cov(X''(t1),X''(tn))
BIG[tn - 1, tn - 1] = R4[0]
# cov(Xc)
BIG[tn + shft + 2, tn + shft + 2] = R0[0] # cov(X(t1),X(t1))
# cov(X(t1),X(tn))
BIG[tn + shft + 2, tn + shft + 3] = R0[tn - 1]
BIG[tn + shft + 1, tn + shft + 2] = 0.0 # cov(X(t1),X'(t1))
# cov(X(t1),X'(tn))
BIG[tn + shft + 1, tn + shft + 2] = R1[tn - 1]
BIG[tn + shft + 3, tn + shft + 3] = R0[0] # cov(X(tn),X(tn))
BIG[tn + shft, tn + shft + 3] = -R1[tn - 1] # cov(X(tn),X'(t1))
BIG[tn + shft + 1, tn + shft + 3] = 0.0 # cov(X(tn),X'(tn))
BIG[tn + shft, tn + shft] = -R2[0] # cov(X'(t1),X'(t1))
BIG[tn + shft, tn + shft + 1] = -R2[tn - 1] # cov(X'(t1),X'(tn))
BIG[tn + shft + 1, tn + shft + 1] = -R2[0] # cov(X'(tn),X'(tn))
# Xc=X(t1),X(tn),X'(t1),X'(tn)
# Xd=X''(t1),X''(tn)
# cov(Xd,Xc)
BIG[tn - 2, tn + shft + 2] = R2[0] # cov(X''(t1),X(t1))
BIG[tn - 2, tn + shft + 3] = R2[tn - 1] # cov(X''(t1),X(tn))
BIG[tn - 2, tn + shft] = 0.0 # cov(X''(t1),X'(t1))
BIG[tn - 2, tn + shft + 1] = R3[tn - 1] # cov(X''(t1),X'(tn))
BIG[tn - 1, tn + shft + 2] = R2[tn - 1] # cov(X''(tn),X(t1))
BIG[tn - 1, tn + shft + 3] = R2[0] # cov(X''(tn),X(tn))
BIG[tn - 1, tn + shft] = -R3[tn - 1] # cov(X''(tn),X'(t1))
BIG[tn - 1, tn + shft + 1] = 0.0 # cov(X''(tn),X'(tn))
# make lower triangular part equal to upper
# for j=1:N-1
# for i=j+1:N
# BIG(i,j) = BIG(j,i)
# end #do
# end #do
# indices to lower triangular part:
lp = np.flatnonzero(np.tril(ones(BIG.shape)))
BIGT = BIG.T
BIG[lp] = BIGT[lp]
return BIG
# END SUBROUTINE COV_INPUT
def _cov2mmtpdfexe(self, R, dt, u, defnr, Nstart, hg, options):
# Write parameters to file
Nx = max(1, len(hg))
if defnr > 1:
Nx = Nx // 2 # level v separated max2min densities wanted
Ntime = R.shape[0]
filenames = ['h.in', 'reflev.in']
self._cleanup(*filenames)
with open('h.in', 'wt') as f:
f.write('%12.10f\n', hg)
# XSPLT = options.xsplit
nit = options.nit
speed = options.speed
seed = options.seed
SCIS = abs(options.method) # method<=0
with open('reflev.in', 'wt') as fid:
fid.write('%2.0f \n', Ntime)
fid.write('%2.0f \n', Nstart)
fid.write('%2.0f \n', nit)
fid.write('%2.0f \n', speed)
fid.write('%2.0f \n', SCIS)
fid.write('%2.0f \n', seed)
fid.write('%2.0f \n', Nx)
fid.write('%12.10E \n', dt)
fid.write('%12.10E \n', u)
fid.write('%2.0f \n', defnr)
filenames2 = self._writecov(R)
print(' Starting Fortran executable.')
# compiled cov2mmtpdf.f with rind70.f
# dos([ wafoexepath 'cov2mmtpdf.exe'])
dens = 1 # load('dens.out')
self._cleanup(*filenames)
self._cleanup(*filenames2)
return dens
@staticmethod
def _cleanup(*files):
'''Removes files from harddisk if they exist'''
for f in files:
if os.path.exists(f):
os.remove(f)
def to_specnorm(self):
S = self.copy()
S.normalize()
return S
def sim(self, ns=None, cases=1, dt=None, iseed=None, method='random',
derivative=False):
''' Simulates a Gaussian process and its derivative from spectrum
Parameters
----------
ns : scalar
number of simulated points. (default length(spec)-1=n-1).
If ns>n-1 it is assummed that acf(k)=0 for all k>n-1
cases : scalar
number of replicates (default=1)
dt : scalar
step in grid (default dt is defined by the Nyquist freq)
iseed : int or state
starting state/seed number for the random number generator
(default none is set)
method : string
if 'exact' : simulation using cov2sdat
if 'random' : random phase and amplitude simulation (default)
derivative : bool
if true : return derivative of simulated signal as well
otherwise
Returns
-------
xs = a cases+1 column matrix ( t,X1(t) X2(t) ...).
xsder = a cases+1 column matrix ( t,X1'(t) X2'(t) ...).
Details
-------
Performs a fast and exact simulation of stationary zero mean
Gaussian process through circulant embedding of the covariance matrix
or by summation of sinus functions with random amplitudes and random
phase angle.
If the spectrum has a non-empty field .tr, then the transformation is
applied to the simulated data, the result is a simulation of a
transformed Gaussian process.
Note: The method 'exact' simulation may give high frequency ripple when
used with a small dt. In this case the method 'random' works better.
Examples
--------
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap();S = Sj.tospecdata()
>>> ns =100; dt = .2
>>> x1 = S.sim(ns,dt=dt)
>>> import numpy as np
>>> import scipy.stats as st
>>> x2 = S.sim(20000,20)
>>> truth1 = [0,np.sqrt(S.moment(1)[0]),0., 0.]
>>> funs = [np.mean,np.std,st.skew,st.kurtosis]
>>> for fun,trueval in zip(funs,truth1):
... res = fun(x2[:,1::],axis=0)
... m = res.mean()
... sa = res.std()
... #trueval, m, sa
... np.abs(m-trueval)<sa
True
array([ True], dtype=bool)
True
True
waveplot(x1,'r',x2,'g',1,1)
See also
--------
cov2sdat, gaus2dat
Reference
-----------
C.S Dietrich and G. N. Newsam (1997)
"Fast and exact simulation of stationary
Gaussian process through circulant embedding
of the Covariance matrix"
SIAM J. SCI. COMPT. Vol 18, No 4, pp. 1088-1107
Hudspeth, S.T. and Borgman, L.E. (1979)
"Efficient FFT simulation of Digital Time sequences"
Journal of the Engineering Mechanics Division, ASCE, Vol. 105, No. EM2,
'''
spec = self.copy()
if dt is not None:
spec.resample(dt)
ftype = spec.freqtype
freq = spec.args
d_t = spec.sampling_period()
Nt = freq.size
if ns is None:
ns = Nt - 1
if method in 'exact':
# nr=0,Nt=None,dt=None
acf = spec.tocovdata(nr=0)
T = Nt * d_t
i = flatnonzero(acf.args > T)
# Trick to avoid adding high frequency noise to the spectrum
if i.size > 0:
acf.data[i[0]::] = 0.0
return acf.sim(ns=ns, cases=cases, iseed=iseed,
derivative=derivative)
_set_seed(iseed)
ns = ns + mod(ns, 2) # make sure it is even
f_i = freq[1:-1]
s_i = spec.data[1:-1]
if ftype in ('w', 'k'):
fact = 2. * pi
s_i = s_i * fact
f_i = f_i / fact
x = zeros((ns, cases + 1))
d_f = 1 / (ns * d_t)
# interpolate for freq. [1:(N/2)-1]*d_f and create 2-sided, uncentered
# spectra
ns2 = ns // 2
f = arange(1, ns2) * d_f
f_u = hstack((0., f_i, d_f * ns2))
s_u = hstack((0., abs(s_i) / 2, 0.))
s_i = interp(f, f_u, s_u)
s_u = hstack((0., s_i, 0, s_i[ns2 - 2::-1]))
del(s_i, f_u)
# Generate standard normal random numbers for the simulations
randn = random.randn
z_r = randn(ns2 + 1, cases)
z_i = vstack(
(zeros((1, cases)), randn(ns2 - 1, cases), zeros((1, cases))))
amp = zeros((ns, cases), dtype=complex)
amp[0:ns2 + 1, :] = z_r - 1j * z_i
del(z_r, z_i)
amp[ns2 + 1:ns, :] = amp[ns2 - 1:0:-1, :].conj()
amp[0, :] = amp[0, :] * sqrt(2.)
amp[ns2, :] = amp[ns2, :] * sqrt(2.)
# Make simulated time series
T = (ns - 1) * d_t
Ssqr = sqrt(s_u * d_f / 2.)
# stochastic amplitude
amp = amp * Ssqr[:, newaxis]
# Deterministic amplitude
# amp =
# sqrt[1]*Ssqr(:,ones(1,cases)) * \
# exp(sqrt(-1)*atan2(imag(amp),real(amp)))
del(s_u, Ssqr)
x[:, 1::] = fft(amp, axis=0).real
x[:, 0] = linspace(0, T, ns) # ' %(0:d_t:(np-1)*d_t).'
if derivative:
xder = zeros(ns, cases + 1)
w = 2. * pi * hstack((0, f, 0., -f[-1::-1]))
amp = -1j * amp * w[:, newaxis]
xder[:, 1:(cases + 1)] = fft(amp, axis=0).real
xder[:, 0] = x[:, 0]
if spec.tr is not None:
# print(' Transforming data.')
g = spec.tr
if derivative:
for i in range(cases):
x[:, i + 1], xder[:, i + 1] = g.gauss2dat(x[:, i + 1],
xder[:, i + 1])
else:
for i in range(cases):
x[:, i + 1] = g.gauss2dat(x[:, i + 1])
if derivative:
return x, xder
else:
return x
# function [x2,x,svec,dvec,amp]=spec2nlsdat(spec,np,dt,iseed,method,
# truncationLimit)
def sim_nl(self, ns=None, cases=1, dt=None, iseed=None, method='random',
fnlimit=1.4142, reltol=1e-3, g=9.81, verbose=False,
output='timeseries'):
"""
Simulates a Randomized 2nd order non-linear wave X(t)
Parameters
----------
ns : scalar
number of simulated points. (default length(spec)-1=n-1).
If ns>n-1 it is assummed that R(k)=0 for all k>n-1
cases : scalar
number of replicates (default=1)
dt : scalar
step in grid (default dt is defined by the Nyquist freq)
iseed : int or state
starting state/seed number for the random number generator
(default none is set)
method : string
'apStochastic' : Random amplitude and phase (default)
'aDeterministic' : Deterministic amplitude and random phase
'apDeterministic' : Deterministic amplitude and phase
fnlimit : scalar
normalized upper frequency limit of spectrum for 2'nd order
components. The frequency is normalized with
sqrt(gravity*tanh(kbar*water_depth)/amp_max)/(2*pi)
(default sqrt(2), i.e., Convergence criterion [1]_).
Other possible values are:
sqrt(1/2) : No bump in trough criterion
sqrt(pi/7) : Wave steepness criterion
reltol : scalar
relative tolerance defining where to truncate spectrum for the
sum and difference frequency effects
Returns
-------
xs2 = a cases+1 column matrix ( t,X1(t) X2(t) ...).
xs1 = a cases+1 column matrix ( t,X1'(t) X2'(t) ...).
Details
-------
Performs a Fast simulation of Randomized 2nd order non-linear
waves by summation of sinus functions with random amplitudes and
phase angles. The extent to which the simulated result are applicable
to real seastates are dependent on the validity of the assumptions:
1. Seastate is unidirectional
2. Surface elevation is adequately represented by 2nd order random
wave theory
3. The first order component of the surface elevation is a Gaussian
random process.
If the spectrum does not decay rapidly enough towards zero, the
contribution from the 2nd order wave components at the upper tail can
be very large and unphysical. To ensure convergence of the perturbation
series, the upper tail of the spectrum is truncated at FNLIMIT in the
calculation of the 2nd order wave components, i.e., in the calculation
of sum and difference frequency effects. This may also be combined with
the elimination of second order effects from the spectrum, i.e.,
extract the linear components from the spectrum. One way to do this is
to use SPEC2LINSPEC.
Examples
--------
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap();S = Sj.tospecdata()
>>> ns =100; dt = .2
>>> x1 = S.sim_nl(ns,dt=dt)
>>> import numpy as np
>>> import scipy.stats as st
>>> x2, x1 = S.sim_nl(ns=20000,cases=20, output='data')
>>> truth1 = [0,np.sqrt(S.moment(1)[0][0])] + S.stats_nl(moments='sk')
>>> truth1[-1] = truth1[-1]-3
>>> np.round(truth1, 3)
array([ 0. , 1.75 , 0.187, 0.062])
>>> funs = [np.mean,np.std,st.skew,st.kurtosis]
>>> for fun,trueval in zip(funs,truth1):
... res = fun(x2[:,1::], axis=0)
... m = res.mean()
... sa = res.std()
... # trueval, m, sa
... np.abs(m-trueval) < 2*sa
True
True
True
True
>>> x = []
>>> for i in range(20):
... x2, x1 = S.sim_nl(ns=20000,cases=1, output='data')
... x.append(x2[:,1::])
>>> x2 = np.hstack(x)
>>> truth1 = [0,np.sqrt(S.moment(1)[0][0])] + S.stats_nl(moments='sk')
>>> truth1[-1] = truth1[-1]-3
>>> np.round(truth1,3)
array([ 0. , 1.75 , 0.187, 0.062])
>>> funs = [np.mean,np.std,st.skew,st.kurtosis]
>>> for fun,trueval in zip(funs,truth1):
... res = fun(x2, axis=0)
... m = res.mean()
... sa = res.std()
... # trueval, m, sa
... np.abs(m-trueval)<sa
True
True
True
True
assert(np.abs(m-trueval)<sa, fun.__name__)
np =100; dt = .2
[x1, x2] = spec2nlsdat(jonswap,np,dt)
waveplot(x1,'r',x2,'g',1,1)
See also
--------
spec2linspec, spec2sdat, cov2sdat
References
----------
.. [1] Nestegaard, amp and Stokka T (1995)
amp Third Order Random Wave model.
In proc.ISOPE conf., Vol III, pp 136-142.
.. [2] R. spec Langley (1987)
amp statistical analysis of non-linear random waves.
Ocean Engng, Vol 14, pp 389-407
.. [3] Marthinsen, T. and Winterstein, spec.R (1992)
'On the skewness of random surface waves'
In proc. ISOPE Conf., San Francisco, 14-19 june.
"""
# TODO % Check the methods: 'apdeterministic' and 'adeterministic'
Hm0, Tm02 = self.characteristic(['Hm0', 'Tm02'])[0].tolist()
_set_seed(iseed)
spec = self.copy()
if dt is not None:
spec.resample(dt)
ftype = spec.freqtype
freq = spec.args
d_t = spec.sampling_period()
Nt = freq.size
if ns is None:
ns = Nt - 1
ns = ns + mod(ns, 2) # make sure it is even
f_i = freq[1:-1]
s_i = spec.data[1:-1]
if ftype in ('w', 'k'):
fact = 2. * pi
s_i = s_i * fact
f_i = f_i / fact
s_max = max(s_i)
water_depth = min(abs(spec.h), 10. ** 30)
x = zeros((ns, cases + 1))
df = 1 / (ns * d_t)
# interpolate for freq. [1:(N/2)-1]*df and create 2-sided, uncentered
# spectra
ns2 = ns // 2
f = arange(1, ns2) * df
f_u = hstack((0., f_i, df * ns2))
w = 2. * pi * hstack((0., f, df * ns2))
kw = w2k(w, 0., water_depth, g)[0]
s_u = hstack((0., abs(s_i) / 2., 0.))
s_i = interp(f, f_u, s_u)
nmin = (s_i > s_max * reltol).argmax()
nmax = flatnonzero(s_i > 0).max()
s_u = hstack((0., s_i, 0, s_i[ns2 - 2::-1]))
del(s_i, f_u)
# Generate standard normal random numbers for the simulations
randn = random.randn
z_r = randn(ns2 + 1, cases)
z_i = vstack((zeros((1, cases)),
randn(ns2 - 1, cases),
zeros((1, cases))))
amp = zeros((ns, cases), dtype=complex)
amp[0:(ns2 + 1), :] = z_r - 1j * z_i
del(z_r, z_i)
amp[(ns2 + 1):ns, :] = amp[ns2 - 1:0:-1, :].conj()
amp[0, :] = amp[0, :] * sqrt(2.)
amp[(ns2), :] = amp[(ns2), :] * sqrt(2.)
# Make simulated time series
T = (ns - 1) * d_t
Ssqr = sqrt(s_u * df / 2.)
if method.startswith('apd'): # apdeterministic
# Deterministic amplitude and phase
amp[1:(ns2), :] = amp[1, 0]
amp[(ns2 + 1):ns, :] = amp[1, 0].conj()
amp = sqrt(2) * Ssqr[:, newaxis] * \
exp(1J * arctan2(amp.imag, amp.real))
elif method.startswith('ade'): # adeterministic
# Deterministic amplitude and random phase
amp = sqrt(2) * Ssqr[:, newaxis] * \
exp(1J * arctan2(amp.imag, amp.real))
else:
# stochastic amplitude
amp = amp * Ssqr[:, newaxis]
# Deterministic amplitude
# amp =
# sqrt(2)*Ssqr(:,ones(1,cases))* \
# exp(sqrt(-1)*atan2(imag(amp),real(amp)))
del(s_u, Ssqr)
x[:, 1::] = fft(amp, axis=0).real
x[:, 0] = linspace(0, T, ns) # ' %(0:d_t:(np-1)*d_t).'
x2 = x.copy()
# If the spectrum does not decay rapidly enough towards zero, the
# contribution from the wave components at the upper tail can be very
# large and unphysical.
# To ensure convergence of the perturbation series, the upper tail of
# the spectrum is truncated in the calculation of sum and difference
# frequency effects.
# Find the critical wave frequency to ensure convergence.
num_waves = 1000. # Typical number of waves in 3 hour seastate
kbar = w2k(2. * pi / Tm02, 0., water_depth)[0]
# Expected maximum amplitude for 1000 waves seastate
amp_max = sqrt(2 * log(num_waves)) * Hm0 / 4
f_limit_up = fnlimit * \
sqrt(g * tanh(kbar * water_depth) / amp_max) / (2 * pi)
f_limit_lo = sqrt(g * tanh(kbar * water_depth) *
amp_max / water_depth) / (2 * pi * water_depth)
nmax = min(flatnonzero(f <= f_limit_up).max(), nmax) + 1
nmin = max(flatnonzero(f_limit_lo <= f).min(), nmin) + 1
# if isempty(nmax),nmax = np/2end
# if isempty(nmin),nmin = 2end % Must always be greater than 1
f_limit_up = df * nmax
f_limit_lo = df * nmin
if verbose:
print('2nd order frequency Limits = %g,%g' %
(f_limit_lo, f_limit_up))
# if nargout>3,
# #compute the sum and frequency effects separately
# [svec, dvec] = disufq((amp.'),w,kw,min(h,10^30),g,nmin,nmax)
# svec = svec.'
# dvec = dvec.'
##
# x2s = fft(svec) % 2'nd order sum frequency component
# x2d = fft(dvec) % 2'nd order difference frequency component
##
# # 1'st order + 2'nd order component.
# x2(:,2:end) =x(:,2:end)+ real(x2s(1:np,:))+real(x2d(1:np,:))
# else
if False:
# TODO: disufq does not work for cases>1
amp = np.array(amp.T).ravel()
rvec, ivec = c_library.disufq(amp.real, amp.imag, w, kw,
water_depth,
g, nmin, nmax, cases, ns)
svec = rvec + 1J * ivec
else:
amp = amp.T
svec = []
for i in range(cases):
rvec, ivec = c_library.disufq(amp[i].real, amp[i].imag, w, kw,
water_depth,
g, nmin, nmax, 1, ns)
svec.append(rvec + 1J * ivec)
svec = np.hstack(svec)
svec.shape = (cases, ns)
x2o = fft(svec, axis=1).T # 2'nd order component
# 1'st order + 2'nd order component.
x2[:, 1::] = x[:, 1::] + x2o[0:ns, :].real
if output == 'timeseries':
xx2 = mat2timeseries(x2)
xx = mat2timeseries(x)
return xx2, xx
return x2, x
def stats_nl(self, h=None, moments='sk', method='approximate', g=9.81):
"""
Statistics of 2'nd order waves to the leading order.
Parameters
----------
h : scalar
water depth (default self.h)
moments : string (default='sk')
composed of letters ['mvsk'] specifying which moments to compute:
'm' = mean,
'v' = variance,
's' = skewness,
'k' = (Pearson's) kurtosis.
method : string
'approximate' method due to Marthinsen & Winterstein (default)
'eigenvalue' method due to Kac and Siegert
Skewness = kurtosis-3 = 0 for a Gaussian process.
The mean, sigma, skewness and kurtosis are determined as follows:
method == 'approximate': due to Marthinsen and Winterstein
mean = 2 * int Hd(w1,w1)*S(w1) dw1
sigma = sqrt(int S(w1) dw1)
skew = 6 * int int [Hs(w1,w2)+Hd(w1,w2)]*S(w1)*S(w2) dw1*dw2/m0^(3/2)
kurt = (4*skew/3)^2
where Hs = sum frequency effects and Hd = difference frequency effects
method == 'eigenvalue'
mean = sum(E)
sigma = sqrt(sum(C^2)+2*sum(E^2))
skew = sum((6*C^2+8*E^2).*E)/sigma^3
kurt = 3+48*sum((C^2+E^2).*E^2)/sigma^4
where
h1 = sqrt(S*dw/2)
C = (ctranspose(V)*[h1;h1])
and E and V is the eigenvalues and eigenvectors, respectively, of the
2'order transfer matrix.
S is the spectrum and dw is the frequency spacing of S.
Examples
--------
# Simulate a Transformed Gaussian process:
>>> import wafo.spectrum.models as sm
>>> import wafo.transform.models as wtm
>>> Hs = 7.
>>> Sj = sm.Jonswap(Hm0=Hs, Tp=11)
>>> S = Sj.tospecdata()
>>> me, va, sk, ku = S.stats_nl(moments='mvsk')
>>> g = wtm.TrHermite(mean=me, sigma=Hs/4, skew=sk, kurt=ku,
... ysigma=Hs/4)
>>> ys = S.sim(15000) # Simulated in the Gaussian world
>>> xs = g.gauss2dat(ys[:,1]) # Transformed to the real world
See also
---------
transform.TrHermite
transform.TrOchi
objects.LevelCrossings.trdata
objects.TimeSeries.trdata
References:
-----------
Langley, RS (1987)
'A statistical analysis of nonlinear random waves'
Ocean Engineering, Vol 14, No 5, pp 389-407
Marthinsen, T. and Winterstein, S.R (1992)
'On the skewness of random surface waves'
In proceedings of the 2nd ISOPE Conference, San Francisco, 14-19 june.
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
"""
# default options
if h is None:
h = self.h
# S = ttspec(S,'w')
w = ravel(self.args)
S = ravel(self.data)
if self.freqtype in ['f', 'w']:
# vari = 't'
if self.freqtype == 'f':
w = 2. * pi * w
S = S / (2. * pi)
# m0 = self.moment(nr=0)
m0 = simps(S, w)
sa = sqrt(m0)
# Nw = w.size
Hs, Hd, Hdii = qtf(w, h, g)
# return
# skew=6/sqrt(m0)^3*simpson(S.w,
# simpson(S.w,(Hs+Hd).*S1(:,ones(1,Nw))).*S1.')
Hspd = trapz(trapz((Hs + Hd) * S[newaxis, :], w) * S, w)
output = []
# %approx : Marthinsen, T. and Winterstein, S.R (1992) method
if method[0] == 'a':
if 'm' in moments:
output.append(2. * trapz(Hdii * S, w))
if 'v' in moments:
output.append(m0)
skew = 6. / sa ** 3 * Hspd
if 's' in moments:
output.append(skew)
if 'k' in moments:
output.append((4. * skew / 3.) ** 2. + 3.)
else:
raise ValueError('Unknown option!')
# elif method[0]== 'q': #, # quasi method
# Fn = self.nyquist_freq()
# dw = Fn/Nw
# tmp1 =sqrt(S[:,newaxis]*S[newaxis,:])*dw
# Hd = Hd*tmp1
# Hs = Hs*tmp1
# k = 6
# stop = 0
# while !stop:
# E = eigs([Hd,Hs;Hs,Hd],[],k)
# %stop = (length(find(abs(E)<1e-4))>0 | k>1200)
# %stop = (any(abs(E(:))<1e-4) | k>1200)
# stop = (any(abs(E(:))<1e-4) | k>=min(2*Nw,1200))
# k = min(2*k,2*Nw)
# end
##
##
# m02=2*sum(E.^2) % variance of 2'nd order contribution
##
# %Hstd = 16*trapz(S.w,(Hdii.*S1).^2)
# %Hstd = trapz(S.w,trapz(S.w,((Hs+Hd)+ 2*Hs.*Hd).*S1(:,ones(1,Nw))).*S1.')
# ma = 2*trapz(S.w,Hdii.*S1)
# %m02 = Hstd-ma^2% variance of second order part
# sa = sqrt(m0+m02)
# skew = 6/sa^3*Hspd
# kurt = (4*skew/3).^2+3
# elif method[0]== 'e': #, % Kac and Siegert eigenvalue analysis
# Fn = self.nyquist_freq()
# dw = Fn/Nw
# tmp1 =sqrt(S[:,newaxis]*S[newaxis,:])*dw
# Hd = Hd*tmp1
# Hs = Hs*tmp1
# k = 6
# stop = 0
##
##
# while (not stop):
# [V,D] = eigs([Hd,HsHs,Hd],[],k)
# E = diag(D)
# %stop = (length(find(abs(E)<1e-4))>0 | k>=min(2*Nw,1200))
# stop = (any(abs(E(:))<1e-4) | k>=min(2*Nw,1200))
# k = min(2*k,2*Nw)
# end
##
##
# h1 = sqrt(S*dw/2)
# C = (ctranspose(V)*[h1;h1])
##
# E2 = E.^2
# C2 = C.^2
##
# ma = sum(E) % mean
# sa = sqrt(sum(C2)+2*sum(E2)) % standard deviation
# skew = sum((6*C2+8*E2).*E)/sa^3 % skewness
# kurt = 3+48*sum((C2+E2).*E2)/sa^4 % kurtosis
return output
def testgaussian(self, ns, test0=None, cases=100, method='nonlinear',
verbose=False, **opt):
'''
TESTGAUSSIAN Test if a stochastic process is Gaussian.
CALL: test1 = testgaussian(S,[ns,Ns],test0,def,options)
Returns
-------
test1 : array,
simulated values of e(g)=int (g(u)-u)^2 du, where int limits is
given by OPTIONS.PARAM.
Parameters
----------
ns : int
# of points simulated
test0 : real scalar
observed value of e(g)=int (g(u)-u)^2 du,
cases : int
# of independent simulations (default 100)
method : string
defines method of estimation of the transform
nonlinear': from smoothed crossing intensity (default)
'mnonlinear': from smoothed marginal distribution
options = options structure defining how the estimation of the
transformation is done. (default troptset('dat2tr'))
TESTGAUSSIAN simulates e(g(u)-u) = int (g(u)-u)^2 du for Gaussian
processes given the spectral density, S. The result is plotted if
test0 is given. This is useful for testing if the process X(t) is
Gaussian. If 95% of TEST1 is less than TEST0 then X(t) is not Gaussian
at a 5% level.
Examples
--------
>>> import wafo.spectrum.models as sm
>>> import wafo.transform.models as wtm
>>> import wafo.objects as wo
>>> Hs = 7
>>> Sj = sm.Jonswap(Hm0=Hs)
>>> S0 = Sj.tospecdata()
>>> ns =100; dt = .2
>>> x1 = S0.sim(ns, dt=dt)
>>> S = S0.copy()
>>> me, va, sk, ku = S.stats_nl(moments='mvsk')
>>> S.tr = wtm.TrHermite(mean=me, sigma=Hs/4, skew=sk, kurt=ku, ysigma=Hs/4)
>>> ys = wo.mat2timeseries(S.sim(ns=2**13))
>>> g0, gemp = ys.trdata()
>>> t0 = g0.dist2gauss()
>>> t1 = S0.testgaussian(ns=2**13, cases=50)
>>> sum(t1 > t0) < 5
True
See also
--------
cov2sdat, dat2tr, troptset
'''
maxsize = 200000 # must divide the computations due to limited memory
# if nargin<5||isempty(opt):
# opt = troptset('dat2tr')
# opt = troptset(opt,'multip',1)
plotflag = False if test0 is None else True
if cases > 50:
print(' ... be patient this may take a while')
rep = int(ns * cases / maxsize) + 1
Nstep = int(cases / rep)
acf = self.tocovdata()
test1 = []
for ix in range(rep):
xs = acf.sim(ns=ns, cases=Nstep)
for iy in range(1, xs.shape[-1]):
ts = TimeSeries(xs[:, iy], xs[:, 0].ravel())
g = ts.trdata(method, **opt)[0]
test1.append(g.dist2gauss())
if verbose:
print('finished %d of %d ' % (ix + 1, rep))
if rep > 1:
xs = acf.sim(ns=ns, cases=np.remainder(cases, rep))
for iy in range(1, xs.shape[-1]):
ts = TimeSeries(xs[:, iy], xs[:, 0].ravel())
g, _tmp = ts.trdata(method, **opt)
test1.append(g.dist2gauss())
if plotflag:
plotbackend.plot(test1, 'o')
plotbackend.plot([1, cases], [test0, test0], '--')
plotbackend.ylabel('e(g(u)-u)')
plotbackend.xlabel('Simulation number')
return test1
def moment(self, nr=2, even=True, j=0):
''' Calculates spectral moments from spectrum
Parameters
----------
nr: int
order of moments (recomended maximum 4)
even : bool
False for all moments,
True for only even orders
j: int
0 or 1
Returns
-------
m : list of moments
mtext : list of strings describing the elements of m, see below
Details
-------
Calculates spectral moments of up to order NR by use of
Simpson-integration.
/ /
mj_t^i = | w^i S(w)^(j+1) dw, or mj_x^i = | k^i S(k)^(j+1) dk
/ /
where k=w^2/gravity, i=0,1,...,NR
The strings in output mtext have the same position in the list
as the corresponding numerical value has in output m
Notation in mtext: 'm0' is the variance,
'm0x' is the first-order moment in x,
'm0xx' is the second-order moment in x,
'm0t' is the first-order moment in t,
etc.
For the calculation of moments see Baxevani et al.
Examples
--------
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=3, Tp=7)
>>> w = np.linspace(0,4,256)
>>> S = SpecData1D(Sj(w),w) #Make spectrum object from numerical values
>>> mom, mom_txt = S.moment()
>>> np.allclose(mom, [0.5616342024616453, 0.7309966918203602])
True
>>> mom_txt == ['m0', 'm0tt']
True
References
----------
Baxevani A. et al. (2001)
Velocities for Random Surfaces
'''
one_dim_spectra = ['freq', 'enc', 'k1d']
if self.type not in one_dim_spectra:
raise ValueError('Unknown spectrum type!')
f = ravel(self.args)
S = ravel(self.data)
if self.freqtype in ['f', 'w']:
vari = 't'
if self.freqtype == 'f':
f = 2. * pi * f
S = S / (2. * pi)
else:
vari = 'x'
S1 = abs(S) ** (j + 1.)
m = [simps(S1, x=f)]
mtxt = 'm%d' % j
mtext = [mtxt]
step = mod(even, 2) + 1
df = f ** step
for i in range(step, nr + 1, step):
S1 = S1 * df
m.append(simps(S1, x=f))
mtext.append(mtxt + vari * i)
return m, mtext
def nyquist_freq(self):
"""
Return Nyquist frequency
Examples
--------
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=5)
>>> S = Sj.tospecdata() #Make spectrum ob
>>> S.nyquist_freq()
3.0
"""
return self.args[-1]
def sampling_period(self):
''' Returns sampling interval from Nyquist frequency of spectrum
Returns
---------
dT : scalar
sampling interval, unit:
[m] if wave number spectrum,
[s] otherwise
Let wm be maximum frequency/wave number in spectrum, then
dT=pi/wm
if angular frequency,
dT=1/(2*wm)
if natural frequency (Hz)
Examples
--------
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=5)
>>> S = Sj.tospecdata() #Make spectrum ob
>>> S.sampling_period()
1.0471975511965976
See also
'''
if self.freqtype == 'f':
wmdt = 0.5 # Nyquist to sampling interval factor
else: # ftype == w og ftype == k
wmdt = pi
wm = self.args[-1] # Nyquist frequency
dt = wmdt / wm # sampling interval = 1/Fs
return dt
def resample(self, dt=None, Nmin=0, Nmax=2 ** 13 + 1, method='stineman'):
'''
Interpolate and zero-padd spectrum to change Nyquist freq.
Parameters
----------
dt : real scalar
wanted sampling interval (default as given by S, see spec2dt)
unit: [s] if frequency-spectrum, [m] if wave number spectrum
Nmin, Nmax : scalar integers
minimum and maximum number of frequencies, respectively.
method : string
interpolation method (options are 'linear', 'cubic' or 'stineman')
To be used before simulation (e.g. spec2sdat) or evaluation of
covariance function (spec2cov) to get the wanted sampling interval.
The input spectrum is interpolated and padded with zeros to reach
the right max-frequency, w[-1]=pi/dt, f(end)=1/(2*dt), or k[-1]=pi/dt.
The objective is that output frequency grid should be at least as dense
as the input grid, have equidistant spacing and length equal to
2^k+1 (>=Nmin). If the max frequency is changed, the number of points
in the spectrum is maximized to 2^13+1.
Note: Also zero-padding down to zero freq, if S does not start there.
If empty input dt, this is the only effect.
See also
--------
spec2cov, spec2sdat, covinterp, spec2dt
'''
ftype = self.freqtype
w = self.args.ravel()
n = w.size
# doInterpolate = 0
# Nyquist to sampling interval factor
Cnf2dt = 0.5 if ftype == 'f' else pi # % ftype == w og ftype == k
wnOld = w[-1] # Old Nyquist frequency
dTold = Cnf2dt / wnOld # sampling interval=1/Fs
# dTold = self.sampling_period()
if dt is None:
dt = dTold
# Find how many points that is needed
nfft = 2 ** nextpow2(max(n - 1, Nmin - 1))
dttest = dTold * (n - 1) / nfft
while (dttest > dt) and (nfft < Nmax - 1):
nfft = nfft * 2
dttest = dTold * (n - 1) / nfft
nfft = nfft + 1
wnNew = Cnf2dt / dt # % New Nyquist frequency
dWn = wnNew - wnOld
doInterpolate = dWn > 0 or w[1] > 0 or (
nfft != n) or dt != dTold or np.any(abs(diff(w, axis=0)) > 1.0e-8)
if doInterpolate > 0:
S1 = self.data
dw = min(diff(w))
if dWn > 0:
# add a zero just above old max-freq, and a zero at new
# max-freq to get correct interpolation there
Nz = 1 + (dWn > dw) # % Number of zeros to add
if Nz == 2:
w = hstack((w, wnOld + dw, wnNew))
else:
w = hstack((w, wnNew))
S1 = hstack((S1, zeros(Nz)))
if w[0] > 0:
# add a zero at freq 0, and, if there is space, a zero just
# below min-freq
Nz = 1 + (w[0] > dw) # % Number of zeros to add
if Nz == 2:
w = hstack((0, w[0] - dw, w))
else:
w = hstack((0, w))
S1 = hstack((zeros(Nz), S1))
# Do a final check on spacing in order to check that the gridding
# is sufficiently dense:
# np1 = S1.size
dwMin = finfo(float).max
# wnc = min(wnNew,wnOld-1e-5)
wnc = wnNew
# specfun = lambda xi : stineman_interp(xi, w, S1)
specfun = interpolate.interp1d(w, S1, kind='cubic')
x = discretize(specfun, 0, wnc)[0]
dwMin = minimum(min(diff(x)), dwMin)
newNfft = 2 ** nextpow2(ceil(wnNew / dwMin)) + 1
if newNfft > nfft:
# if (nfft <= 2 ** 15 + 1) and (newNfft > 2 ** 15 + 1):
# warnings.warn('Spectrum matrix is very large (>33k). ' +
# 'Memory problems may occur.')
nfft = newNfft
self.args = linspace(0, wnNew, nfft)
if method == 'stineman':
self.data = stineman_interp(self.args, w, S1)
else:
intfun = interpolate.interp1d(w, S1, kind=method)
self.data = intfun(self.args)
self.data = self.data.clip(0) # clip negative values to 0
def interp(self, dt):
S = self.copy()
S.resample(dt)
return S
def normalize(self, gravity=9.81):
'''
Normalize a spectral density such that m0=m2=1
Paramter
--------
gravity=9.81
Notes
-----
Normalization performed such that
INT S(freq) dfreq = 1 INT freq^2 S(freq) dfreq = 1
where integration limits are given by freq and S(freq) is the
spectral density; freq can be frequency or wave number.
The normalization is defined by
A=sqrt(m0/m2); B=1/A/m0; freq'=freq*A; S(freq')=S(freq)*B
If S is a directional spectrum then a normalized gravity (.g) is added
to Sn, such that mxx normalizes to 1, as well as m0 and mtt.
(See spec2mom for notation of moments)
If S is complex-valued cross spectral density which has to be
normalized, then m0, m2 (suitable spectral moments) should be given.
Examples
--------
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=5)
>>> S = Sj.tospecdata() #Make spectrum ob
>>> np.allclose(S.moment(2)[0],
... [1.5614600345079888, 0.95567089481941048])
True
>>> Sn = S.copy(); Sn.normalize()
Now the moments should be one
>>> np.allclose(Sn.moment(2)[0], [1.0, 1.0])
True
'''
mom = self.moment(nr=4, even=True)[0]
m0 = mom[0]
m2 = mom[1]
m4 = mom[2]
SM0 = sqrt(m0)
SM2 = sqrt(m2)
A = SM0 / SM2
B = SM2 / (SM0 * m0)
if self.freqtype == 'f':
self.args = self.args * A / 2 / pi
self.data = self.data * B * 2 * pi
elif self.freqtype == 'w':
self.args = self.args * A
self.data = self.data * B
m02 = m4 / gravity ** 2
m20 = m02
self.g = gravity * sqrt(m0 * m20) / m2
self.A = A
self.norm = True
self.date = now()
def bandwidth(self, factors=0):
'''
Return some spectral bandwidth and irregularity factors
Parameters
-----------
factors : array-like
Input vector 'factors' correspondence:
0 alpha=m2/sqrt(m0*m4) (irregularity factor)
1 eps2 = sqrt(m0*m2/m1^2-1) (narrowness factor)
2 eps4 = sqrt(1-m2^2/(m0*m4))=sqrt(1-alpha^2) (broadness factor)
3 Qp=(2/m0^2)int_0^inf f*S(f)^2 df (peakedness factor)
Returns
--------
bw : arraylike
vector of bandwidth factors
Order of output is the same as order in 'factors'
Examples
--------
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=3, Tp=7)
>>> w = np.linspace(0,4,256)
>>> S = SpecData1D(Sj(w),w) #Make spectrum object from numerical values
>>> S.bandwidth([0,'eps2',2,3])
array([ 0.73062845, 0.34476034, 0.68277527, 2.90817052])
'''
m = self.moment(nr=4, even=False)[0]
if isinstance(factors, str):
factors = [factors]
fact_dict = dict(alpha=0, eps2=1, eps4=3, qp=3, Qp=3)
fact = array([fact_dict.get(idx, idx)
for idx in list(factors)], dtype=int)
# fact = atleast_1d(fact)
alpha = m[2] / sqrt(m[0] * m[4])
eps2 = sqrt(m[0] * m[2] / m[1] ** 2. - 1.)
eps4 = sqrt(1. - m[2] ** 2. / m[0] / m[4])
f = self.args
S = self.data
Qp = 2 / m[0] ** 2. * simps(f * S ** 2, x=f)
bw = array([alpha, eps2, eps4, Qp])
return bw[fact]
def characteristic(self, fact='Hm0', T=1200, g=9.81):
"""
Returns spectral characteristics and their covariance
Parameters
----------
fact : vector with factor integers or a string or a list of strings
defining spectral characteristic, see description below.
T : scalar
recording time (sec) (default 1200 sec = 20 min)
g : scalar
acceleration of gravity [m/s^2]
Returns
-------
ch : vector
of spectral characteristics
R : matrix
of the corresponding covariances given T
chtext : a list of strings
describing the elements of ch, see example.
Description
------------
If input spectrum is of wave number type, output are factors for
corresponding 'k1D', else output are factors for 'freq'.
Input vector 'factors' correspondence:
1 Hm0 = 4*sqrt(m0) Significant wave height
2 Tm01 = 2*pi*m0/m1 Mean wave period
3 Tm02 = 2*pi*sqrt(m0/m2) Mean zero-crossing period
4 Tm24 = 2*pi*sqrt(m2/m4) Mean period between maxima
5 Tm_10 = 2*pi*m_1/m0 Energy period
6 Tp = 2*pi/{w | max(S(w))} Peak period
7 Ss = 2*pi*Hm0/(g*Tm02^2) Significant wave steepness
8 Sp = 2*pi*Hm0/(g*Tp^2) Average wave steepness
9 Ka = abs(int S(w)*exp(i*w*Tm02) dw ) /m0 Groupiness parameter
10 Rs = (S(0.092)+S(0.12)+S(0.15)/(3*max(S(w)))
Quality control parameter
11 Tp1 = 2*pi*int S(w)^4 dw Peak Period
------------------ (robust estimate for Tp)
int w*S(w)^4 dw
12 alpha = m2/sqrt(m0*m4) Irregularity factor
13 eps2 = sqrt(m0*m2/m1^2-1) Narrowness factor
14 eps4 = sqrt(1-m2^2/(m0*m4))=sqrt(1-alpha^2) Broadness factor
15 Qp = (2/m0^2)int_0^inf w*S(w)^2 dw Peakedness factor
Order of output is same as order in 'factors'
The covariances are computed with a Taylor expansion technique
and is currently only available for factors 1, 2, and 3. Variances
are also available for factors 4,5,7,12,13,14 and 15
Quality control:
----------------
Critical value for quality control parameter Rs is Rscrit = 0.02
for surface displacement records and Rscrit=0.0001 for records of
surface acceleration or slope. If Rs > Rscrit then probably there
are something wrong with the lower frequency part of S.
Ss may be used as an indicator of major malfunction, by checking that
it is in the range of 1/20 to 1/16 which is the usual range for
locally generated wind seas.
Examples:
---------
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=5)
>>> S = Sj.tospecdata() #Make spectrum ob
>>> S.characteristic(1)
(array([ 8.59007646]), array([[ 0.03040216]]), ['Tm01'])
>>> [ch, R, txt] = S.characteristic([1,2,3]) # fact vector of integers
>>> S.characteristic('Ss') # fact a string
(array([ 0.04963112]), array([[ 2.63624782e-06]]), ['Ss'])
>>> S.characteristic(['Hm0','Tm02']) # fact a list of strings
(array([ 4.99833578, 8.03139757]), array([[ 0.05292989, 0.02511371],
[ 0.02511371, 0.0274645 ]]), ['Hm0', 'Tm02'])
See also
---------
bandwidth,
moment
References
----------
Krogstad, H.E., Wolf, J., Thompson, S.P., and Wyatt, L.R. (1999)
'Methods for intercomparison of wave measurements'
Coastal Enginering, Vol. 37, pp. 235--257
Krogstad, H.E. (1982)
'On the covariance of the periodogram'
Journal of time series analysis, Vol. 3, No. 3, pp. 195--207
Tucker, M.J. (1993)
'Recommended standard for wave data sampling and near-real-time
processing'
Ocean Engineering, Vol.20, No.5, pp. 459--474
Young, I.R. (1999)
"Wind generated ocean waves"
Elsevier Ocean Engineering Book Series, Vol. 2, pp 239
"""
# TODO: Need more checking on computing the variances for Tm24,alpha,
# eps2 and eps4
# TODO: Covariances between Tm24,alpha, eps2 and eps4 variables are
# also needed
tfact = dict(Hm0=0, Tm01=1, Tm02=2, Tm24=3, Tm_10=4, Tp=5, Ss=6, Sp=7,
Ka=8, Rs=9, Tp1=10, Alpha=11, Eps2=12, Eps4=13, Qp=14)
tfact1 = ('Hm0', 'Tm01', 'Tm02', 'Tm24', 'Tm_10', 'Tp', 'Ss', 'Sp',
'Ka', 'Rs', 'Tp1', 'Alpha', 'Eps2', 'Eps4', 'Qp')
if isinstance(fact, str):
fact = list((fact,))
if isinstance(fact, (list, tuple)):
nfact = []
for k in fact:
if isinstance(k, str):
nfact.append(tfact.get(k.capitalize(), 15))
else:
nfact.append(k)
else:
nfact = fact
nfact = atleast_1d(nfact)
if np.any((nfact > 14) | (nfact < 0)):
raise ValueError('Factor outside range (0,...,14)')
# vari = self.freqtype
f = self.args.ravel()
S1 = self.data.ravel()
m, unused_mtxt = self.moment(nr=4, even=False)
# moments corresponding to freq in Hz
for k in range(1, 5):
m[k] = m[k] / (2 * pi) ** k
# pi = np.pi
ind = flatnonzero(f > 0)
m.append(simps(S1[ind] / f[ind], f[ind]) * 2. * pi) # = m_1
m_10 = simps(S1[ind] ** 2 / f[ind], f[ind]) * \
(2 * pi) ** 2 / T # = COV(m_1,m0|T=t0)
m_11 = simps(S1[ind] ** 2. / f[ind] ** 2, f[ind]) * \
(2 * pi) ** 3 / T # = COV(m_1,m_1|T=t0)
# sqrt = np.sqrt
# Hm0 Tm01 Tm02 Tm24 Tm_10
Hm0 = 4. * sqrt(m[0])
Tm01 = m[0] / m[1]
Tm02 = sqrt(m[0] / m[2])
Tm24 = sqrt(m[2] / m[4])
Tm_10 = m[5] / m[0]
Tm12 = m[1] / m[2]
ind = S1.argmax()
maxS = S1[ind]
# [maxS ind] = max(S1)
Tp = 2. * pi / f[ind] # peak period /length
Ss = 2. * pi * Hm0 / g / Tm02 ** 2 # Significant wave steepness
Sp = 2. * pi * Hm0 / g / Tp ** 2 # Average wave steepness
# groupiness factor
Ka = abs(simps(S1 * exp(1J * f * Tm02), f)) / m[0]
# Quality control parameter
# critical value is approximately 0.02 for surface displacement records
# If Rs>0.02 then there are something wrong with the lower frequency
# part of S.
Rs = np.sum(
interp(r_[0.0146, 0.0195, 0.0244] * 2 * pi, f, S1)) / 3. / maxS
Tp2 = 2 * pi * simps(S1 ** 4, f) / simps(f * S1 ** 4, f)
alpha1 = Tm24 / Tm02 # m(3)/sqrt(m(1)*m(5))
eps2 = sqrt(Tm01 / Tm12 - 1.) # sqrt(m(1)*m(3)/m(2)^2-1)
eps4 = sqrt(1. - alpha1 ** 2) # sqrt(1-m(3)^2/m(1)/m(5))
Qp = 2. / m[0] ** 2 * simps(f * S1 ** 2, f)
ch = r_[Hm0, Tm01, Tm02, Tm24, Tm_10, Tp, Ss,
Sp, Ka, Rs, Tp2, alpha1, eps2, eps4, Qp]
# Select the appropriate values
ch = ch[nfact]
chtxt = [tfact1[i] for i in nfact]
# if nargout>1,
# covariance between the moments:
# COV(mi,mj |T=t0) = int f^(i+j)*S(f)^2 df/T
mij = self.moment(nr=8, even=False, j=1)[0]
for ix, tmp in enumerate(mij):
mij[ix] = tmp / T / ((2. * pi) ** (ix - 1.0))
# and the corresponding variances for
# {'hm0', 'tm01', 'tm02', 'tm24', 'tm_10','tp','ss', 'sp', 'ka', 'rs',
# 'tp1','alpha','eps2','eps4','qp'}
R = r_[4 * mij[0] / m[0],
mij[0] / m[1] ** 2. - 2. * m[0] * mij[1] /
m[1] ** 3. + m[0] ** 2. * mij[2] / m[1] ** 4.,
0.25 * (mij[0] / (m[0] * m[2]) - 2. * mij[2] / m[2] ** 2 +
m[0] * mij[4] / m[2] ** 3),
0.25 * (mij[4] / (m[2] * m[4]) - 2 * mij[6] / m[4] ** 2 +
m[2] * mij[8] / m[4] ** 3),
m_11 / m[0] ** 2 + (m[5] / m[0] ** 2) ** 2 *
mij[0] - 2 * m[5] / m[0] ** 3 * m_10,
nan, (8 * pi / g) ** 2 *
(m[2] ** 2 / (4 * m[0] ** 3) *
mij[0] + mij[4] / m[0] - m[2] / m[0] ** 2 * mij[2]),
nan * ones(4),
m[2] ** 2 * mij[0] / (4 * m[0] ** 3 * m[4]) + mij[4] /
(m[0] * m[4]) + mij[8] * m[2] ** 2 / (4 * m[0] * m[4] ** 3) -
m[2] * mij[2] / (m[0] ** 2 * m[4]) + m[2] ** 2 * mij[4] /
(2 * m[0] ** 2 * m[4] ** 2) - m[2] * mij[6] / m[0] / m[4] ** 2,
(m[2] ** 2 * mij[0] / 4 + (m[0] * m[2] / m[1]) ** 2 * mij[2] +
m[0] ** 2 * mij[4] / 4 - m[2] ** 2 * m[0] * mij[1] / m[1] +
m[0] * m[2] * mij[2] / 2 - m[0] ** 2 * m[2] / m[1] * mij[3]) /
eps2 ** 2 / m[1] ** 4,
(m[2] ** 2 * mij[0] / (4 * m[0] ** 2) + mij[4] + m[2] ** 2 *
mij[8] / (4 * m[4] ** 2) - m[2] * mij[2] / m[0] + m[2] ** 2 *
mij[4] / (2 * m[0] * m[4]) - m[2] * mij[6] / m[4]) *
m[2] ** 2 / (m[0] * m[4] * eps4) ** 2,
nan]
# and covariances by a taylor expansion technique:
# Cov(Hm0,Tm01) Cov(Hm0,Tm02) Cov(Tm01,Tm02)
S0 = r_[2. / (sqrt(m[0]) * m[1]) * (mij[0] - m[0] * mij[1] / m[1]),
1. / sqrt(m[2]) * (mij[0] / m[0] - mij[2] / m[2]),
1. / (2 * m[1]) * sqrt(m[0] / m[2])
* (mij[0] / m[0] - mij[2] / m[2] - mij[1] / m[1] + m[0] * mij[3] / (m[1] * m[2]))]
R1 = ones((15, 15))
R1[:, :] = nan
for ix, Ri in enumerate(R):
R1[ix, ix] = Ri
R1[0, 2:4] = S0[:2]
R1[1, 2] = S0[2]
# make lower triangular equal to upper triangular part
for ix in [0, 1]:
R1[ix + 1:, ix] = R1[ix, ix + 1:]
R = R[nfact]
R1 = R1[nfact, :][:, nfact]
# Needs further checking:
# Var(Tm24)= 0.25*(mij[4]/(m[2]*m[4])-
# 2*mij[6]/m[4]**2+m[2]*mij[8]/m[4]**3)
return ch, R1, chtxt
def setlabels(self):
''' Set automatic title, x-,y- and z- labels on SPECDATA object
based on type, angletype, freqtype
'''
N = len(self.type)
if N == 0:
raise ValueError(
'Object does not appear to be initialized, it is empty!')
labels = ['', '', '']
if self.type.endswith('dir'):
title = 'Directional Spectrum'
if self.freqtype.startswith('w'):
labels[0] = 'Frequency [rad/s]'
labels[2] = r'S($\omega$,$\theta$) $[m^2 s / rad^2]$'
else:
labels[0] = 'Frequency [Hz]'
labels[2] = r'S(f,$\theta$) $[m^2 s / rad]$'
if self.angletype.startswith('r'):
labels[1] = 'Wave directions [rad]'
elif self.angletype.startswith('d'):
labels[1] = 'Wave directions [deg]'
elif self.type.endswith('freq'):
title = 'Spectral density'
if self.freqtype.startswith('w'):
labels[0] = 'Frequency [rad/s]'
labels[1] = r'S($\omega$) $[m^2 s/ rad]$'
else:
labels[0] = 'Frequency [Hz]'
labels[1] = r'S(f) $[m^2 s]$'
else:
title = 'Wave Number Spectrum'
labels[0] = 'Wave number [rad/m]'
if self.type.endswith('k1d'):
labels[1] = r'S(k) $[m^3/ rad]$'
elif self.type.endswith('k2d'):
labels[1] = labels[0]
labels[2] = r'S(k1,k2) $[m^4/ rad^2]$'
else:
raise ValueError(
'Object does not appear to be initialized, it is empty!')
if self.norm != 0:
title = 'Normalized ' + title
labels[0] = 'Normalized ' + labels[0].split('[')[0]
if not self.type.endswith('dir'):
labels[1] = labels[1].split('[')[0]
labels[2] = labels[2].split('[')[0]
self.labels.title = title
self.labels.xlab = labels[0]
self.labels.ylab = labels[1]
self.labels.zlab = labels[2]
class SpecData2D(PlotData):
""" Container class for 2D spectrum data objects in WAFO
Member variables
----------------
data : array_like
args : vector for 1D, list of vectors for 2D, 3D, ...
type : string
spectrum type (default 'freq')
freqtype : letter
frequency type (default 'w')
angletype : string
angle type of directional spectrum (default 'radians')
Examples
--------
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=3, Tp=7)
>>> w = np.linspace(0,4,256)
>>> S = SpecData1D(Sj(w),w) #Make spectrum object from numerical values
See also
--------
PlotData
CovData
"""
def __init__(self, *args, **kwds):
super(SpecData2D, self).__init__(*args, **kwds)
self.name = 'WAFO Spectrum Object'
self.type = 'dir'
self.freqtype = 'w'
self.angletype = ''
self.h = inf
self.tr = None
self.phi = 0.
self.v = 0.
self.norm = 0
somekeys = ['angletype', 'phi', 'name', 'h',
'tr', 'freqtype', 'v', 'type', 'norm']
self.__dict__.update(sub_dict_select(kwds, somekeys))
if self.type.endswith('dir') and self.angletype == '':
self.angletype = 'radians'
self.setlabels()
def toacf(self):
pass
def tospecdata(self, type=None): # @ReservedAssignment
pass
def sim(self):
pass
def sim_nl(self):
pass
def rotate(self, phi=0, rotateGrid=False, method='linear'):
'''
Rotate spectrum clockwise around the origin.
Parameters
----------
phi : real scalar
rotation angle (default 0)
rotateGrid : bool
True if rotate grid of Snew physically (thus Snew.phi=0).
False if rotate so that only Snew.phi is changed
(the grid is not physically rotated) (default)
method : string
interpolation method to use when ROTATEGRID==1, (default 'linear')
Rotates the spectrum clockwise around the origin.
This equals a anti-clockwise rotation of the cordinate system (x,y).
The spectrum can be of any of the two-dimensional types.
For spectrum in polar representation:
newtheta = theta-phi, but circulant such that -pi<newtheta<pi
For spectrum in Cartesian representation:
If the grid is rotated physically, the size of it is preserved
(maybe it must be increased such that no nonzero points are
affected, but this is not implemented yet: i.e. corners are cut off)
The spectrum is assumed to be zero outside original grid.
NB! The routine does not change the type of spectrum, use spec2spec
for this.
Examples
--------
S=demospec('dir');
plotspec(S), hold on
plotspec(rotspec(S,pi/2),'r'), hold off
See also
--------
spec2spec
'''
# TODO: Make physical grid rotation of cartesian coordinates more
# robust.
# Snew=S;
self.phi = mod(self.phi + phi + pi, 2 * pi) - pi
stype = self.type.lower()[-3::]
if stype == 'dir':
# any of the directinal types
# Make sure theta is from -pi to pi
theta = self.args[0]
phi0 = theta[0] + pi
self.args[0] = theta - phi0
# make sure -pi<phi<pi
self.phi = mod(self.phi + phi0 + pi, 2 * pi) - pi
if (rotateGrid and (self.phi != 0)):
# Do a physical rotation of spectrum
# theta = Snew.args[0]
ntOld = len(theta)
if (mod(theta[0] - theta[-1], 2 * pi) == 0):
nt = ntOld - 1
else:
nt = ntOld
theta[0:nt] = mod(theta[0:nt] - self.phi + pi, 2 * pi) - pi
self.phi = 0
ind = theta.argsort()
self.data = self.data[ind, :]
self.args[0] = theta[ind]
if (nt < ntOld):
if (self.args[0][0] == -pi):
self.data[ntOld, :] = self.data[0, :]
else:
# ftype = self.freqtype
freq = self.args[1]
theta = linspace(-pi, pi, ntOld)
# [F, T] = meshgrid(freq, theta)
dtheta = self.theta[1] - self.theta[0]
self.theta[nt] = self.theta[nt - 1] + dtheta
self.data[nt, :] = self.data[0, :]
self.data = interp2d(freq,
np.vstack([self.theta[0] - dtheta,
self.theta]),
np.vstack([self.data[nt, :],
self.data]),
kind=method)(freq, theta)
self.args[0] = theta
elif stype == 'k2d':
# any of the 2D wave number types
# Snew.phi = mod(Snew.phi+phi+pi,2*pi)-pi
if (rotateGrid and (self.phi != 0)):
# Do a physical rotation of spectrum
[k, k2] = meshgrid(*self.args)
[th, r] = cart2polar(k, k2)
[k, k2] = polar2cart(th + self.phi, r)
ki1, ki2 = self.args
Sn = interp2d(ki1, ki2, self.data, kind=method)(k, k2)
self.data = np.where(np.isnan(Sn), 0, Sn)
self.phi = 0
else:
raise ValueError('Can only rotate two dimensional spectra')
return
def moment(self, nr=2, vari='xt'):
'''
Calculates spectral moments from spectrum
Parameters
----------
nr : int
order of moments (maximum 4)
vari : string
variables in model, optional when two-dim.spectrum,
string with 'x' and/or 'y' and/or 't'
Returns
-------
m : list of moments
mtext : list of strings describing the elements of m, see below
Details
-------
Calculates spectral moments of up to order four by use of
Simpson-integration.
//
m_jkl=|| k1^j*k2^k*w^l S(w,th) dw dth
//
where k1=w^2/gravity*cos(th-phi), k2=w^2/gravity*sin(th-phi)
and phi is the angle of the rotation in S.phi. If the spectrum
has field .g, gravity is replaced by S.g.
The strings in output mtext have the same position in the cell array
as the corresponding numerical value has in output m
Notation in mtext: 'm0' is the variance,
'mx' is the first-order moment in x,
'mxx' is the second-order moment in x,
'mxt' is the second-order cross moment between x and t,
'myyyy' is the fourth-order moment in y
etc.
For the calculation of moments see Baxevani et al.
Examples
--------
>>> import wafo.spectrum.models as sm
>>> D = sm.Spreading()
>>> SD = D.tospecdata2d(sm.Jonswap().tospecdata(),nt=101)
>>> m,mtext = SD.moment(nr=2,vari='xyt')
>>> np.allclose(np.round(m,3),
... [ 3.061, 0.132, -0. , 2.13 , 0.011, 0.008, 1.677, -0.,
... 0.109, 0.109])
True
>>> mtext == ['m0', 'mx', 'my', 'mt', 'mxx', 'myy', 'mtt', 'mxy',
... 'mxt', 'myt']
True
References
----------
Baxevani A. et al. (2001)
Velocities for Random Surfaces
'''
two_dim_spectra = ['dir', 'encdir', 'k2d']
if self.type not in two_dim_spectra:
raise ValueError('Unknown 2D spectrum type!')
if vari is None and nr <= 1:
vari = 'x'
elif vari is None:
vari = 'xt'
else: # secure the mutual order ('xyt')
vari = ''.join(sorted(vari.lower()))
Nv = len(vari)
if vari[0] == 't' and Nv > 1:
vari = vari[1::] + vari[0]
Nv = len(vari)
if not self.type.endswith('dir'):
S1 = self.tospecdata(self.type[:-2] + 'dir')
else:
S1 = self
w = ravel(S1.args[0])
theta = S1.args[1] - S1.phi
S = S1.data
Sw = simps(S, x=theta, axis=0)
m = [simps(Sw, x=w)]
mtext = ['m0']
if nr > 0:
vec = []
g = np.atleast_1d(S1.__dict__.get('g', _gravity()))
# maybe different normalization in x and y => diff. g
kx = w ** 2 / g[0]
ky = w ** 2 / g[-1]
# nw = w.size
if 'x' in vari:
ct = np.cos(theta[:, None])
Sc = simps(S * ct, x=theta, axis=0)
vec.append(kx * Sc)
mtext.append('mx')
if 'y' in vari:
st = np.sin(theta[:, None])
Ss = simps(S * st, x=theta, axis=0)
vec.append(ky * Ss)
mtext.append('my')
if 't' in vari:
vec.append(w * Sw)
mtext.append('mt')
if nr > 1:
if 'x' in vari:
Sc2 = simps(S * ct ** 2, x=theta, axis=0)
vec.append(kx ** 2 * Sc2)
mtext.append('mxx')
if 'y' in vari:
Ss2 = simps(S * st ** 2, x=theta, axis=0)
vec.append(ky ** 2 * Ss2)
mtext.append('myy')
if 't' in vari:
vec.append(w ** 2 * Sw)
mtext.append('mtt')
if 'x' in vari and 'y' in vari:
Scs = simps(S * ct * st, x=theta, axis=0)
vec.append(kx * ky * Scs)
mtext.append('mxy')
if 'x' in vari and 't' in vari:
vec.append(kx * w * Sc)
mtext.append('mxt')
if 'y' in vari and 't' in vari:
vec.append(ky * w * Sc)
mtext.append('myt')
if nr > 3:
if 'x' in vari:
Sc3 = simps(S * ct ** 3, x=theta, axis=0)
Sc4 = simps(S * ct ** 4, x=theta, axis=0)
vec.append(kx ** 4 * Sc4)
mtext.append('mxxxx')
if 'y' in vari:
Ss3 = simps(S * st ** 3, x=theta, axis=0)
Ss4 = simps(S * st ** 4, x=theta, axis=0)
vec.append(ky ** 4 * Ss4)
mtext.append('myyyy')
if 't' in vari:
vec.append(w ** 4 * Sw)
mtext.append('mtttt')
if 'x' in vari and 'y' in vari:
Sc2s = simps(S * ct ** 2 * st, x=theta, axis=0)
Sc3s = simps(S * ct ** 3 * st, x=theta, axis=0)
Scs2 = simps(S * ct * st ** 2, x=theta, axis=0)
Scs3 = simps(S * ct * st ** 3, x=theta, axis=0)
Sc2s2 = simps(S * ct ** 2 * st ** 2, x=theta, axis=0)
vec.extend((kx ** 3 * ky * Sc3s,
kx ** 2 * ky ** 2 * Sc2s2,
kx * ky ** 3 * Scs3))
mtext.extend(('mxxxy', 'mxxyy', 'mxyyy'))
if 'x' in vari and 't' in vari:
vec.extend((kx ** 3 * w * Sc3,
kx ** 2 * w ** 2 * Sc2, kx * w ** 3 * Sc))
mtext.extend(('mxxxt', 'mxxtt', 'mxttt'))
if 'y' in vari and 't' in vari:
vec.extend((ky ** 3 * w * Ss3, ky ** 2 * w ** 2 * Ss2,
ky * w ** 3 * Ss))
mtext.extend(('myyyt', 'myytt', 'myttt'))
if 'x' in vari and 'y' in vari and 't' in vari:
vec.extend((kx ** 2 * ky * w * Sc2s,
kx * ky ** 2 * w * Scs2,
kx * ky * w ** 2 * Scs))
mtext.extend(('mxxyt', 'mxyyt', 'mxytt'))
# end % if nr>1
m.extend([simps(vals, x=w) for vals in vec])
return np.asarray(m), mtext
def interp(self):
pass
def normalize(self):
pass
def bandwidth(self):
pass
def setlabels(self):
''' Set automatic title, x-,y- and z- labels on SPECDATA object
based on type, angletype, freqtype
'''
N = len(self.type)
if N == 0:
raise ValueError(
'Object does not appear to be initialized, it is empty!')
labels = ['', '', '']
if self.type.endswith('dir'):
title = 'Directional Spectrum'
if self.freqtype.startswith('w'):
labels[0] = 'Frequency [rad/s]'
labels[2] = r'$S(w,\theta) [m**2 s / rad**2]$'
else:
labels[0] = 'Frequency [Hz]'
labels[2] = r'$S(f,\theta) [m**2 s / rad]$'
if self.angletype.startswith('r'):
labels[1] = 'Wave directions [rad]'
elif self.angletype.startswith('d'):
labels[1] = 'Wave directions [deg]'
elif self.type.endswith('freq'):
title = 'Spectral density'
if self.freqtype.startswith('w'):
labels[0] = 'Frequency [rad/s]'
labels[1] = 'S(w) [m**2 s/ rad]'
else:
labels[0] = 'Frequency [Hz]'
labels[1] = 'S(f) [m**2 s]'
else:
title = 'Wave Number Spectrum'
labels[0] = 'Wave number [rad/m]'
if self.type.endswith('k1d'):
labels[1] = 'S(k) [m**3/ rad]'
elif self.type.endswith('k2d'):
labels[1] = labels[0]
labels[2] = 'S(k1,k2) [m**4/ rad**2]'
else:
raise ValueError(
'Object does not appear to be initialized, it is empty!')
if self.norm != 0:
title = 'Normalized ' + title
labels[0] = 'Normalized ' + labels[0].split('[')[0]
if not self.type.endswith('dir'):
labels[1] = labels[1].split('[')[0]
labels[2] = labels[2].split('[')[0]
self.labels.title = title
self.labels.xlab = labels[0]
self.labels.ylab = labels[1]
self.labels.zlab = labels[2]
def main():
import matplotlib
matplotlib.interactive(True)
from wafo.spectrum import models as sm
Sj = sm.Jonswap()
S = Sj.tospecdata()
R = S.tocovdata(nr=1)
Si = R.tospecdata()
ns = 5000
dt = .2
x1 = S.sim_nl(ns=ns, dt=dt)
x2 = TimeSeries(x1[:, 1], x1[:, 0])
R = x2.tocovdata(lag=100)
R.plot()
S.plot('ro')
t = S.moment()
t1 = S.bandwidth([0, 1, 2, 3])
S1 = S.copy()
S1.resample(dt=0.3, method='cubic')
S1.plot('k+')
x = S1.sim(ns=100)
import pylab
pylab.clf()
pylab.plot(x[:, 0], x[:, 1])
pylab.show()
pylab.close('all')
print('done')
def test_mm_pdf():
import wafo.spectrum.models as sm
Sj = sm.Jonswap(Hm0=7, Tp=11)
w = np.linspace(0, 4, 256)
S1 = Sj.tospecdata(w) # Make spectrum object from numerical values
S = sm.SpecData1D(Sj(w), w) # Alternatively do it manually
S0 = S.to_linspec()
mm = S.to_mm_pdf()
mm.plot()
plotbackend.show()
def test_docstrings():
import doctest
doctest.testmod()
if __name__ == '__main__':
test_docstrings()
# test_mm_pdf()
# main()