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pywafo/wafo/data_structures.py

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"""
Class objects for data, spectrum, covariance function and time series in WAFO
To represent data, spectra, covariance functions and time series
in WAFO, the classes WafoData, SpecData1D, CovData1D and TimeSeries is used.
Here follows a list of their attributes:
WafoData
----------
data : array_like
args : vector for 1D, list of vectors for 2D, 3D, ...
date : Date and time of creation or change.
labels : AxisLabels
children : list of WafoData objects
SpecData1D
------------
data : One sided Spectrum values
args : freguency values of freqtype
type :String: 'freq', 'dir', 'k2d', k1d', 'encdir' or 'enc'.
freqtype :'w' OR 'f' OR 'k' Frequency/wave number lag, length nf.
tr : Transformation function (default [] (none)).
h : Water depth (default inf).
norm : Normalization flag, Logical 1 if S is normalized, 0 if not
date :Date and time of creation or change.
v : Ship speed, if .type = 'enc' or 'encdir'.
phi angle of rotation of the coordinate system
(counter-clocwise) e.g. azymuth of a ship.
CovData1D
----------
data : Covariance function values. Size [ny nx nt], all singleton dim. removed.
args : Lag of first space dimension, length nx.
.h Water depth.
.tr Transformation function.
.type 'enc', 'rot' or 'none'.
.v Ship speed, if .type='enc'
.phi Rotation of coordinate system, e.g. direction of ship
.norm Normalization flag, Logical 1 if autocorrelation, 0 if covariance.
.Rx ... .Rtttt Obvious derivatives of .R.
.note Memorandum string.
.date Date and time of creation or change.
"""
#-------------------------------------------------------------------------------
# Name: module1
# Purpose:
#
# Author: pab
#
# Created: 16.09.2008
# Copyright: (c) pab 2008
# Licence: <your licence>
#-------------------------------------------------------------------------------
#!/usr/bin/env python
from __future__ import division
import warnings
import numpy
import numpy as np
from time import gmtime, strftime
from wafo.misc import nextpow2, discretize, findextrema, findrfc, sub_dict_select
#import wafo.spectrum.dispersion_relation as disp_rel
#from scipy import fft
from scipy.integrate import simps, trapz
from pylab import stineman_interp
import scipy.interpolate as interpolate
import warnings
try:
import diffsumfunq
except:
pass
from plotbackend import plotbackend
__all__ = ['SpecData1D','SpecData2D','WafoData', 'AxisLabels','CovData1D',
'TimeSeries','sensortypeid','sensortype']
def empty_copy(obj):
class Empty(obj.__class__):
def __init__(self):
pass
newcopy = Empty()
newcopy.__class__ = obj.__class__
return newcopy
def _set_seed(iseed):
if iseed != None:
try:
np.random.set_state(iseed)
except:
np.random.seed(iseed)
class WafoData(object):
'''Container class for data objects in WAFO
Member variables
----------------
data : array_like
args : vector for 1D, list of vectors for 2D, 3D, ...
labels : AxisLabels
children : list of WafoData objects
Member methods
--------------
plot :
copy :
Example
-------
>>> import numpy as np
>>> x = np.arange(-2,2,0.2)
# Plot 2 objects in one call
>>> d2 = WafoData(np.sin(x),x,xlab='x',ylab='sin',title='sinus')
>>> d2.plot()
% Plot with confidence interval
d3 = wdata(sin(x),x);
d3 = set(d3,'dataCI',[sin(x(:))*0.9 sin(x(:))*1.2]);
plot(d3)
See also
--------
wdata/plot,
specdata,
covdata
'''
def __init__(self,data=None,args=None,**kwds):
self.data = data
self.args = args
self.date = strftime("%a, %d %b %Y %H:%M:%S", gmtime())
self.plotter = None
self.children = None
self.labels = AxisLabels(**kwds)
self.setplotter()
def plot(self,*args,**kwds):
if self.children!=None:
plotbackend.hold('on')
for child in self.children:
tmp = child.plot(*args,**kwds)
tmp2 = self.plotter.plot(self,*args,**kwds)
return tmp2
def copy(self):
newcopy = empty_copy(self)
newcopy.__dict__.update(self.__dict__)
return newcopy
def setplotter(self):
'''
Set plotter based on the data type data_1d, data_2d, data_3d or data_nd
'''
if isinstance(self.args,(list,tuple)): # Multidimensional data
ndim = len(self.args)
if ndim<2:
warnings.warn('Unable to determine plotter-type, because len(self.args)<2.')
print('If the data is 1D, then self.args should be a vector!')
print('If the data is 2D, then length(self.args) should be 2.')
print('If the data is 3D, then length(self.args) should be 3.')
print('Unless you fix this, the plot methods will not work!')
elif ndim==2:
self.plotter = Plotter_2d()
else:
warnings.warn('Plotter method not implemented for ndim>2')
else: #One dimensional data
self.plotter = Plotter_1d()
class AxisLabels:
def __init__(self,title='',xlab='',ylab='',zlab='',**kwds):
self.title = title
self.xlab = xlab
self.ylab = ylab
self.zlab = zlab
def copy(self):
lbkwds = self.labels.__dict__.copy()
labels = AxisLabels(**lbkwds)
return labels
def labelfig(self):
try:
plotbackend.title(self.title)
plotbackend.xlabel(self.xlab)
plotbackend.ylabel(self.ylab)
plotbackend.zlabel(self.zlab)
except:
pass
class Plotter_1d(object):
"""
class comment
bar
barh
loglog
semilogx
semilogy
plot
stem
scatter
"""
def __init__(self,plotmethod='plot'):
self.plotfun = None
self.plotbackend = plotbackend
try:
self.plotfun = plotbackend.__dict__[plotmethod]
except:
pass
def show(self):
plotbackend.show()
def plot(self,wdata,*args,**kwds):
if isinstance(wdata.args,(list,tuple)):
args1 = tuple((wdata.args))+(wdata.data,)+args
else:
args1 = tuple((wdata.args,))+(wdata.data,)+args
self.plotfun(*args1,**kwds)
wdata.labels.labelfig()
class Plotter_2d(Plotter_1d):
"""
class comment
contour
mesh
surf
"""
def __init__(self,plotmethod='contour'):
super(Plotter_2d,self).__init__(plotmethod)
#self.plotfun = plotbackend.__dict__[plotmethod]
class TrGauss(WafoData):
def __init__(self,*args,**kwds):
super(TrGauss, self).__init__(*args,**kwds)
class CycleCount(object):
def __init__(self,M,m):
self.M = M
self.m = m
def amplitudes(self):
return (self.M-self.m)/2.
class SpecData1D(WafoData):
""" Container class for 1D spectrum data objects in WAFO
Member variables
----------------
data : array_like
args : vector for 1D, list of vectors for 2D, 3D, ...
type : string
spectrum type, one of 'freq', 'k1d', 'enc' (default 'freq')
freqtype : letter
frequency type, one of: 'f', 'w' or 'k' (default 'w')
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 = SpecData1D(Sj(w),w) # Alternatively do it manually
See also
--------
WafoData
CovData
"""
def __init__(self,*args,**kwds):
super(SpecData1D, self).__init__(*args,**kwds)
self.name='WAFO Spectrum Object'
self.type='freq'
self.freqtype='w'
self.angletype=''
self.h=np.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))
self.setlabels()
## def copy(self):
## newcopy = empty_copy(self)
## newcopy.update(self.__dict__)
## #kwds = self.__dict__.copy()
## #wdata = SpecData1D(**kwds)
## #wdata.labels = sel.labels.copy()
## return newcopy
def toacf_matrix(self,nr=0,Nt=None,dt=None):
''' Computes covariance function and its derivatives, alternative version
Parameters
----------
Nr = number of derivatives in output, Nr<=4 (default 0)
Nt = number in time grid, i.e., number of time-lags.
(default rate*(n-1)) where rate = round(1/(2*f(end)*dt)) or
rate = round(pi/(w(n)*dt)) depending on S.
dt = time spacing for R
Returns
-------
R = [R0, R1,...Rnr] 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. size Nt+1 x Nr+1
NB! This routine requires that the spectrum grid is equidistant
starting from zero frequency.
Example:
--------
S = jonswap;
dt = 0.1;
R = spec2cov2(S,3,256,dt);
See also
--------
spec2cov, specinterp, datastructures
'''
ftype = self.freqtype #; %options are 'f' and 'w' and 'k'
freq = self.args
n = length(freq)
dTold = self.sampling_period()
if dt is None:
dt = dTold
rate = 1
else:
rate = np.maximum(np.round(dTold*1./dt),1.)
if Nt is None:
Nt = rate*(n-1)
else: #%check if Nt is ok
Nt = np.minimum(Nt,rate*(n-1));
checkdt = 1.2*min(np.diff(freq))/2./np.pi;
if ftype in 'k':
lagtype = 'x'
else:
lagtype = 't'
if ftype in 'f':
checkdt = checkdt*2*np.pi
msg1 = 'The step dt = %g in computation of the density is too small.' % dt
msg2 = 'The step dt = %g step is small, may cause numerical inaccuracies.' % dt
if (checkdt < 2.**-16/dt):
np.disp(msg1)
np.disp('The computed covariance (by FFT(2^K)) may differ from the theoretical.')
np.disp('Solution:')
raise ValueError('use larger dt or sparser grid for spectrum.')
#% Calculating covariances
#%~~~~~~~~~~~~~~~~~~~~~~~~
S2 = self.copy()
S2.resample(dt)
R2 = S2.toacf(nr,Nt,rate=1)
R = np.zeros((Nt+1,nr+1))
R[:,0] = R2.data[0:Nt+1]
fieldname = 'R' + lagtype*nr
for ix in range(1,nr+1):
fn = fieldname[1:ix];
R[:,0:Nt+1] = gettattr(R2,fn)[0:Nt+1];
EPS0 = 0.0001
cc = R[0,0]-R[1,0]*(R[1,0]/R[0,0]);
if Nt+1>=5:
#%cc1=R(1,1)-R(3,1)*(R(3,1)/R(1,1))+R(3,2)*(R(3,2)/R(1,3));
#%cc3=R(1,1)-R(5,1)*(R(5,1)/R(1,1))+R(5,2)*(R(5,2)/R(1,3));
cc2 = R(0,0)-R(4,0)*(R(4,0)/R(0,0));
if (cc2<EPS0):
warnings.warn(msg1)
if (cc<EPS0):
np.disp(msg2)
return R
def toacf(self,nr=0,Nt=None,rate=None):
'''Computes covariance function and its derivatives
Parameters
---------
S = a spectral density structure (See datastructures)
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.S)-1)).
rate = 1,2,4,8...2^r, interpolation rate for R
(default = 1, no interpolation)
Nx,Ny = number in space grid (default = )
dx,dy = space grid step (default: depending on S)
Returns
--------
R = a covariance structure (See datastructures)
The input 'rate' gives together with the spectrum
the t-grid-spacing: dt=pi/(S.w(end)*rate), S.w(end) is the Nyquist freq.
This results in the t-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: then result R(t) (one-dim)
2) S.type='k1d' or 'k2d', Nt set, Nx,Ny not set: then result R(x) (one-dim)
3) Any type, Nt and Nx set =>R(x,t); Nt and Ny set =>R(y,t)
4) Any type, Nt, Nx and Ny set => R(x,y,t)
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, S, with sharp edges
to calculate covariances then you should probably round off the sharp
edges like this:
Example:
>>> 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;
>>> R = S.toacf(nr=0,Nt=Nt)
R = spec2cov(S,0,Nt);
win = parzen(2*Nt+1);
R.R = R.R.*win(Nt+1:end);
S1 = cov2spec(R);
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.R-R.R)), hold on,
semilogy(abs(S1.S-S.S)+1e-7,'r')
See also cov2spec, datastructures
'''
freq = self.args
n = freq.size
if freq[0]>0:
raise ValueError('Spectrum does not start at zero frequency/wave number.\n Correct it with resample, for example.')
dw = np.abs(np.diff(freq,n=2,axis=0))
if np.any(dw>1.0e-8):
raise ValueError('Not equidistant frequencies/wave numbers in spectrum.\n Correct it with resample, for example.')
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 = rate*(n-1)
else: #check if Nt is ok
Nt = np.minimum(Nt,rate*(n-1))
S = self.copy()
if self.freqtype in 'k':
lagtype = 'x'
else:
lagtype = 't'
dT = S.sampling_period()
#normalize spec so that sum(specn)/(n-1)=R(0)=var(X)
specn = S.data*freq[-1]
if S.freqtype in 'f':
w = freq*2*np.pi
else:
w = freq
nfft = rate*2**nextpow2(2*n-2);
Rper = np.r_[specn, np.zeros(nfft-(2*n)+2) , np.conj(specn[n-1:0:-1])] #; % periodogram
t = np.r_[0:Nt+1]*dT*(2*n-2)/nfft
fft = np.fft.fft
r = fft(Rper,nfft).real/(2*n-2)
R = CovData1D(r[0:Nt+1],t,lagtype=lagtype)
R.tr = S.tr;
R.h = S.h;
R.norm = S.norm;
if nr>0:
w = np.r_[w , np.zeros(nfft-2*n+2) ,-w[n-1:0:-1] ]
fieldname = 'R' + lagtype*nr ;
for ix in range(1,nr+1):
Rper = (-1j*w*Rper)
r = fft(Rper,nfft).real/(2*n-2)
setattr(R,fieldname[0:ix+1],r[0:Nt+1])
return R
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(S)-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
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.
Example:
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap();S = Sj.toSpecData();
>>> np =100; dt = .2;
>>> x1 = S.sim(np,dt=dt);
waveplot(x1,'r',x2,'g',1,1)
See also
--------
cov2sdat, gaus2dat
Reference
-----------
C.R 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, R.T. and Borgman, L.E. (1979)
"Efficient FFT simulation of Digital Time sequences"
Journal of the Engineering Mechanics Division, ASCE, Vol. 105, No. EM2,
'''
fft = np.fft.fft
S = self.copy()
if dt is not None:
S.resample(dt)
ftype = S.freqtype
freq = S.args
dT = S.sampling_period()
Nt = freq.size
if ns is None:
ns=Nt-1
if method in 'exact':
#nr=0,Nt=None,dt=None
R = S.toacf(nr=0);
T = Nt*dT
ix = np.flatnonzero(R.args>T);
# Trick to avoid adding high frequency noise to the spectrum
if ix.size>0:
R.data[ix[0]::]=0.0
return R.sim(ns=ns,cases=cases,iseed=iseed,derivative=derivative)
_set_seed(iseed)
ns = ns+np.mod(ns,2) # make sure it is even
fi = freq[1:-1]
Si = S.data[1:-1]
if ftype in ('w','k'):
fact = 2.*np.pi
Si = Si*fact;
fi = fi/fact;
zeros = np.zeros
x = zeros((ns,cases+1));
df = 1/(ns*dT)
# interpolate for freq. [1:(N/2)-1]*df and create 2-sided, uncentered spectra
# ----------------------------------------------------------------------------
f = np.arange(1,ns/2.)*df
Fs = np.hstack((0., fi, df*ns/2.))
Su = np.hstack((0., np.abs(Si)/2., 0.));
Si = np.interp(f,Fs,Su)
Su=np.hstack((0., Si,0, Si[(ns/2)-2::-1]))
del(Si, Fs)
# Generate standard normal random numbers for the simulations
# -----------------------------------------------------------
randn = np.random.randn
Zr = randn((ns/2)+1,cases)
Zi = np.vstack((zeros((1,cases)), randn((ns/2)-1,cases), zeros((1,cases))));
A = zeros((ns,cases),dtype=complex)
A[0:(ns/2+1),:] = Zr - 1j*Zi;
del(Zr, Zi)
A[(ns/2+1):ns,:] = A[ns/2-1:0:-1,:].conj();
A[0,:] = A[0,:]*np.sqrt(2.);
A[(ns/2),:] = A[(ns/2),:]*np.sqrt(2.);
# Make simulated time series
# --------------------------
T = (ns-1)*dT
Ssqr = np.sqrt(Su*df/2.)
# stochastic amplitude
A = A*Ssqr[:,np.newaxis];
# Deterministic amplitude
#A = sqrt[1]*Ssqr(:,ones(1,cases)).*exp(sqrt(-1)*atan2(imag(A),real(A)));
del( Su, Ssqr)
x[:,1::] = fft(A,axis=0).real
x[:,0] = np.linspace(0,T,ns) #'; %(0:dT:(np-1)*dT).';
if derivative:
xder=np.zeros(ns,cases+1)
w = 2.*np.pi*np.hstack((0, f, 0.,-f[-1::-1]))
A = -1j*A*w[:,newaxis]
xder[:,1:(cases+1)] = fft(A,axis=0).real;
xder[:,0] = x[:,0]
if S.tr is not None:
np.disp(' Transforming data.')
g=S.tr;
G=np.fliplr(g); #% the invers of g
if derivative:
for ix in range(cases):
tmp=tranproc(np.hstack((x[:,ix+1], xder[:,ix+1])),G);
x[:,ix+1]=tmp[:,0];
xder[:,ix+1]=tmp[:,1];
else:
for ix in range(cases):
x[:,ix+1]=tranproc(x[:,ix+1],G);
if derivative:
return x,xder
else:
return x
# function [x2,x,svec,dvec,A]=spec2nlsdat(S,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):
""" Simulates a Randomized 2nd order non-linear wave X(t)
Parameters
----------
ns : scalar
number of simulated points. (default length(S)-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*waterDepth)/Amax)/(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
for 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.
Example
--------
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, A and Stokka T (1995)
A Third Order Random Wave model.
In proc.ISOPE conf., Vol III, pp 136-142.
.. [2] R. S Langley (1987)
A statistical analysis of non-linear random waves.
Ocean Engng, Vol 14, pp 389-407
.. [3] Marthinsen, T. and Winterstein, S.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'
from wafo.spectrum import dispersion_relation as sm
Hm0 = self.characteristic('Hm0')[0]
Tm02 = self.characteristic('Tm02')[0]
#Hm0 = spec2char(S,'Hm0');
#Tm02 = spec2char(S,'Tm02');
_set_seed(iseed)
fft = np.fft.fft
S = self.copy()
if dt is not None:
S.resample(dt)
ftype = S.freqtype
freq = S.args
dT = S.sampling_period()
Nt = freq.size
if ns is None:
ns = Nt-1
ns = ns+np.mod(ns,2) # make sure it is even
fi = freq[1:-1]
Si = S.data[1:-1]
if ftype in ('w','k'):
fact = 2.*np.pi
Si = Si*fact;
fi = fi/fact;
Smax = max(Si)
waterDepth = min(abs(S.h),10.**30);
zeros = np.zeros
x = zeros((ns,cases+1));
df = 1/(ns*dT)
# interpolate for freq. [1:(N/2)-1]*df and create 2-sided, uncentered spectra
# ----------------------------------------------------------------------------
f = np.arange(1,ns/2.)*df
Fs = np.hstack((0., fi, df*ns/2.))
w = 2.*np.pi*Fs
kw = sm.w2k(w ,0.,waterDepth,g)[0]
Su = np.hstack((0., np.abs(Si)/2., 0.));
Si = np.interp(f,Fs,Su)
nmin = (Si>Smax*reltol).argmax()
nmax = np.flatnonzero(Si>0).max()
Su = np.hstack((0., Si,0, Si[(ns/2)-2::-1]))
del(Si, Fs)
# Generate standard normal random numbers for the simulations
# -----------------------------------------------------------
randn = np.random.randn
Zr = randn((ns/2)+1,cases)
Zi = np.vstack((zeros((1,cases)), randn((ns/2)-1,cases), zeros((1,cases))));
A = zeros((ns,cases),dtype=complex)
A[0:(ns/2+1),:] = Zr - 1j*Zi;
del(Zr, Zi)
A[(ns/2+1):ns,:] = A[ns/2-1:0:-1,:].conj();
A[0,:] = A[0,:]*np.sqrt(2.);
A[(ns/2),:] = A[(ns/2),:]*np.sqrt(2.);
# Make simulated time series
# --------------------------
T = (ns-1)*dT
Ssqr = np.sqrt(Su*df/2.)
if method.startswith('apd') : # apdeterministic
# Deterministic amplitude and phase
A[1:(ns/2),:] = A[1,0];
A[(ns/2+1):ns,:] = A[1,0].conj();
A = sqrt(2)*Ssqr[:,np.newaxis]*exp(1J*atan2(A.imag,A.real))
elif method.startswith('ade'): # adeterministic
# Deterministic amplitude and random phase
A = sqrt(2)*Ssqr[:,np.newaxis]*exp(1J*atan2(A.imag,A.real));
else:
# stochastic amplitude
A = A*Ssqr[:,np.newaxis];
# Deterministic amplitude
#A = sqrt(2)*Ssqr(:,ones(1,cases)).*exp(sqrt(-1)*atan2(imag(A),real(A)));
del( Su, Ssqr)
x[:,1::] = fft(A,axis=0).real
x[:,0] = np.linspace(0,T,ns) #'; %(0:dT:(np-1)*dT).';
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.
sqrt = np.sqrt
log = np.log
pi = np.pi
tanh = np.tanh
numWaves = 1000. # Typical number of waves in 3 hour seastate
kbar = sm.w2k(2.*np.pi/Tm02,0.,waterDepth)[0];
Amax = sqrt(2*log(numWaves))*Hm0/4; #% Expected maximum amplitude for 1000 waves seastate
fLimitUp = fnlimit*sqrt(g*tanh(kbar*waterDepth)/Amax)/(2*pi);
fLimitLo = sqrt(g*tanh(kbar*waterDepth)*Amax/waterDepth)/(2*pi*waterDepth);
nmax = min(np.flatnonzero(f<=fLimitUp).max(),nmax)+1;
nmin = max(np.flatnonzero(fLimitLo<=f).min(),nmin)+1;
#if isempty(nmax),nmax = np/2;end
#if isempty(nmin),nmin = 2;end % Must always be greater than 1
fLimitUp = df*nmax;
fLimitLo = df*nmin;
print('2nd order frequency Limits = %g,%g'% (fLimitLo, fLimitUp))
## if nargout>3,
## %compute the sum and frequency effects separately
## [svec, dvec] = disufq((A.'),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
rvec,ivec = diffsumfunq.disufq(A.real,A.imag,w,kw,waterDepth,g,nmin,nmax)
svec = rvec + 1J*ivec
x2o = fft(svec) # 2'nd order component
# 1'st order + 2'nd order component.
x2[:,1::] = x[:,1::]+ x2o[0:ns,:].real();
return x2, x
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.
Example:
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=3)
>>> w = np.linspace(0,4,256)
>>> S = SpecData1D(Sj(w),w) #Make spectrum object from numerical values
>>> S.moment()
([0.56220770033914191, 0.35433180985851975], ['m0', 'm0tt'])
References
----------
Baxevani A. et al. (2001)
Velocities for Random Surfaces
'''
pi= np.pi
one_dim_spectra = ['freq','enc','k1d']
if self.type not in one_dim_spectra:
raise ValueError('Unknown spectrum type!')
f = np.ravel(self.args)
S = np.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=np.abs(S)**(j+1.)
m = [simps(S1,x=f)]
mtxt = 'm%d' % j
mtext = [mtxt]
step = np.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 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)
Example
-------
S = jonswap;
dt = spec2dt(S)
See also
'''
if self.freqtype in 'f':
wmdt = 0.5 # Nyquist to sampling interval factor
else: # ftype == w og ftype == k
wmdt = np.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 : scalar
wanted sampling interval (default as given by S, see spec2dt)
unit: [s] if frequency-spectrum, [m] if wave number spectrum
Nmin : scalar
minimum number of frequencies.
Nmax : scalar
minimum number of frequencies
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 directly get wanted sampling interval.
The input spectrum is interpolated and padded with zeros to reach
the right max-frequency, w(end)=pi/dt, f(end)=1/(2*dt), or k(end)=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;
if ftype=='f':
Cnf2dt = 0.5; # Nyquist to sampling interval factor
else: #% ftype == w og ftype == k
Cnf2dt = np.pi
wnOld = w[-1] # Old Nyquist frequency
dTold = Cnf2dt/wnOld; # sampling interval=1/Fs
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 any(abs(np.diff(w,axis=0))>1.0e-8);
if doInterpolate>0:
S1 = self.data
dw = min(np.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 = np.hstack((w, wnOld+dw, wnNew))
else:
w = np.hstack((w, wnNew))
S1 = np.hstack((S1, np.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=np.hstack((0, w[0]-dw, w))
else:
w=np.hstack((0, w))
S1 = np.hstack((zeros(Nz), S1))
#% Do a final check on spacing in order to check that the gridding is
#% sufficiently dense:
#np1 = S1.size
dwMin = np.finfo(float).max
#%wnc = min(wnNew,wnOld-1e-5);
wnc = wnNew;
specfun = lambda xi : stineman_interp(xi,w,S1)
x,y = discretize(specfun,0,wnc)
dwMin = np.minimum(min(np.diff(x)),dwMin)
newNfft = 2**nextpow2(np.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 = np.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 normalize(self):
pass
def bandwidth(self,factors=0):
''' Return some spectral bandwidth and irregularity factors
Returns
--------
bw : arraylike
vector of bandwidth 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)
Order of output is the same as order in 'factors'
Example:
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=3)
>>> w = np.linspace(0,4,256)
>>> S = SpecData1D(Sj(w),w) #Make spectrum object from numerical values
>>> S.bandwidth([0,1,2,3])
array([ 0.65354446, 0.3975428 , 0.75688813, 2.00207912])
'''
if self.freqtype in 'k':
vari = 'k';
else:
vari = 'w';
m,mtxt = self.moment(nr=4,even=False)
sqrt = np.sqrt
fact = np.atleast_1d(factors)
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 = np.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 cellarray of strings, see below.(default [1])
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 numpy as np
>>> 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 a 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
NaN = np.nan
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 = np.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,mtxt = self.moment(nr=4,even=False)
#% moments corresponding to freq in Hz
for k in range(1,5):
m[k] = m[k]/(2*np.pi)**k
pi = np.pi
ind = np.flatnonzero(f>0)
m.append(simps(S1[ind]/f[ind],f[ind])*2.*np.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
Ka = abs(simps(S1*np.exp(1J*f*Tm02),f))/m[0]; #% groupiness factor
#% 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(np.interp(np.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 = np.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,mijtxt = self.moment(nr=8,even=False,j=1)
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 = np.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*np.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 = np.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 = np.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];
for ix in [0,1]: #%make lower triangular equal to upper triangular part
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] = 'S(w,\theta) [m^2 s / rad^2]'
else:
labels[0] = 'Frequency [Hz]'
labels[2] = '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]
class SpecData2D(WafoData):
""" 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)
>>> w = np.linspace(0,4,256)
>>> S = SpecData1D(Sj(w),w) #Make spectrum object from numerical values
See also
--------
WafoData
CovData
"""
def __init__(self,*args,**kwds):
super(SpecData2D, self).__init__(*args,**kwds)
self.name='WAFO Spectrum Object'
self.type='freq'
self.freqtype='w'
self.angletype=''
self.h=np.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 sim(self):
pass
def sim_nl(self):
pass
def rotate(self):
pass
def moment(self,nr=2,vari='xt',even=True):
''' 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'
even : bool
False for all moments,
True for only even orders
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.
Example:
S=demospec('dir')
[m,mtext]=spec2mom(S,2,'xyt')
References
----------
Baxevani A. et al. (2001)
Velocities for Random Surfaces
'''
##% Tested on: Matlab 6.0
##% Tested on: Matlab 5.3
##% History:
##% Revised by I.R. 04.04.2001: Introducing the rotation angle phi.
##% Revised by A.B. 23.05.2001: Correcting 'mxxyy' and introducing
##% 'mxxyt','mxyyt' and 'mxytt'.
##% Revised by A.B. 21.10.2001: Correcting 'mxxyt'.
##% Revised by A.B. 21.10.2001: Adding odd-order moments.
##% By es 27.08.1999
pi= np.pi
two_dim_spectra = ['dir','encdir','k2d']
if self.type not in two_dim_spectra:
raise ValueError('Unknown 2D spectrum type!')
## if (vari==None and nr<=1:
## vari='x'
## elif vari==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.tospec(self.type[:-2]+'dir')
## else:
## S1 = self;
## w = np.ravel(S1.args[0])
## theta = S1.args[1]-S1.phi
## S = S1.data
## Sw = simps(S,x=theta)
## m = [simps(Sw,x=w)]
## mtext=['m0']
##
## if nr>0:
##
## nw=w.size
## if strcmpi(vari(1),'x')
## Sc=simpson(th,S1.S.*(cos(th)*ones(1,nw))).';
## % integral S*cos(th) dth
## end
## if strcmpi(vari(1),'y')
## Ss=simpson(th,S1.S.*(sin(th)*ones(1,nw))).';
## % integral S*sin(th) dth
## if strcmpi(vari(1),'x')
## Sc=simpson(th,S1.S.*(cos(th)*ones(1,nw))).';
## end
## end
## if ~isfield(S1,'g')
## S1.g=gravity;
## end
## kx=w.^2/S1.g(1); % maybe different normalization in x and y => diff. g
## ky=w.^2/S1.g(end);
##
## if Nv>=1
## switch vari
## case 'x'
## vec = kx.*Sc;
## mtext(end+1)={'mx'};
## case 'y'
## vec = ky.*Ss;
## mtext(end+1)={'my'};
## case 't'
## vec = w.*Sw;
## mtext(end+1)={'mt'};
## end
## else
## vec = [kx.*Sc ky.*Ss w*Sw];
## mtext(end+(1:3))={'mx', 'my', 'mt'};
## end
## if nr>1
## if strcmpi(vari(1),'x')
## Sc=simpson(th,S1.S.*(cos(th)*ones(1,nw))).';
## % integral S*cos(th) dth
## Sc2=simpson(th,S1.S.*(cos(th).^2*ones(1,nw))).';
## % integral S*cos(th)^2 dth
## end
## if strcmpi(vari(1),'y')||strcmpi(vari(2),'y')
## Ss=simpson(th,S1.S.*(sin(th)*ones(1,nw))).';
## % integral S*sin(th) dth
## Ss2=simpson(th,S1.S.*(sin(th).^2*ones(1,nw))).';
## % integral S*sin(th)^2 dth
## if strcmpi(vari(1),'x')
## Scs=simpson(th,S1.S.*((cos(th).*sin(th))*ones(1,nw))).';
## % integral S*cos(th)*sin(th) dth
## end
## end
## if ~isfield(S1,'g')
## S1.g=gravity;
## end
##
## if Nv==2
## switch vari
## case 'xy'
## vec=[kx.*Sc ky.*Ss kx.^2.*Sc2 ky.^2.*Ss2 kx.*ky.*Scs];
## mtext(end+(1:5))={'mx','my','mxx', 'myy', 'mxy'};
## case 'xt'
## vec=[kx.*Sc w.*Sw kx.^2.*Sc2 w.^2.*Sw kx.*w.*Sc];
## mtext(end+(1:5))={'mx','mt','mxx', 'mtt', 'mxt'};
## case 'yt'
## vec=[ky.*Ss w.*Sw ky.^2.*Ss2 w.^2.*Sw ky.*w.*Ss];
## mtext(end+(1:5))={'my','mt','myy', 'mtt', 'myt'};
## end
## else
## vec=[kx.*Sc ky.*Ss w.*Sw kx.^2.*Sc2 ky.^2.*Ss2 w.^2.*Sw kx.*ky.*Scs kx.*w.*Sc ky.*w.*Ss];
## mtext(end+(1:9))={'mx','my','mt','mxx', 'myy', 'mtt', 'mxy', 'mxt', 'myt'};
## end
## if nr>3
## if strcmpi(vari(1),'x')
## Sc3=simpson(th,S1.S.*(cos(th).^3*ones(1,nw))).';
## % integral S*cos(th)^3 dth
## Sc4=simpson(th,S1.S.*(cos(th).^4*ones(1,nw))).';
## % integral S*cos(th)^4 dth
## end
## if strcmpi(vari(1),'y')||strcmpi(vari(2),'y')
## Ss3=simpson(th,S1.S.*(sin(th).^3*ones(1,nw))).';
## % integral S*sin(th)^3 dth
## Ss4=simpson(th,S1.S.*(sin(th).^4*ones(1,nw))).';
## % integral S*sin(th)^4 dth
## if strcmpi(vari(1),'x') %both x and y
## Sc2s=simpson(th,S1.S.*((cos(th).^2.*sin(th))*ones(1,nw))).';
## % integral S*cos(th)^2*sin(th) dth
## Sc3s=simpson(th,S1.S.*((cos(th).^3.*sin(th))*ones(1,nw))).';
## % integral S*cos(th)^3*sin(th) dth
## Scs2=simpson(th,S1.S.*((cos(th).*sin(th).^2)*ones(1,nw))).';
## % integral S*cos(th)*sin(th)^2 dth
## Scs3=simpson(th,S1.S.*((cos(th).*sin(th).^3)*ones(1,nw))).';
## % integral S*cos(th)*sin(th)^3 dth
## Sc2s2=simpson(th,S1.S.*((cos(th).^2.*sin(th).^2)*ones(1,nw))).';
## % integral S*cos(th)^2*sin(th)^2 dth
## end
## end
## if Nv==2
## switch vari
## case 'xy'
## vec=[vec kx.^4.*Sc4 ky.^4.*Ss4 kx.^3.*ky.*Sc3s ...
## kx.^2.*ky.^2.*Sc2s2 kx.*ky.^3.*Scs3];
## mtext(end+(1:5))={'mxxxx','myyyy','mxxxy','mxxyy','mxyyy'};
## case 'xt'
## vec=[vec kx.^4.*Sc4 w.^4.*Sw kx.^3.*w.*Sc3 ...
## kx.^2.*w.^2.*Sc2 kx.*w.^3.*Sc];
## mtext(end+(1:5))={'mxxxx','mtttt','mxxxt','mxxtt','mxttt'};
## case 'yt'
## vec=[vec ky.^4.*Ss4 w.^4.*Sw ky.^3.*w.*Ss3 ...
## ky.^2.*w.^2.*Ss2 ky.*w.^3.*Ss];
## mtext(end+(1:5))={'myyyy','mtttt','myyyt','myytt','myttt'};
## end
## else
## vec=[vec kx.^4.*Sc4 ky.^4.*Ss4 w.^4.*Sw kx.^3.*ky.*Sc3s ...
## kx.^2.*ky.^2.*Sc2s2 kx.*ky.^3.*Scs3 kx.^3.*w.*Sc3 ...
## kx.^2.*w.^2.*Sc2 kx.*w.^3.*Sc ky.^3.*w.*Ss3 ...
## ky.^2.*w.^2.*Ss2 ky.*w.^3.*Ss kx.^2.*ky.*w.*Sc2s ...
## kx.*ky.^2.*w.*Scs2 kx.*ky.*w.^2.*Scs];
## mtext(end+(1:15))={'mxxxx','myyyy','mtttt','mxxxy','mxxyy',...
## 'mxyyy','mxxxt','mxxtt','mxttt','myyyt','myytt','myttt','mxxyt','mxyyt','mxytt'};
##
## end % if Nv==2 ... else ...
## end % if nr>3
## end % if nr>1
## m=[m simpson(w,vec)];
## end % if nr>0
## % end; %%if Nv==1... else... to be removed
## end % ... else two-dim spectrum
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] = 'S(w,\theta) [m**2 s / rad**2]'
else:
labels[0] = 'Frequency [Hz]'
labels[2] = '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]
class CovData1D(WafoData):
""" Container class for 1D covariance data objects in WAFO
Member variables
----------------
data : array_like
args : vector for 1D, list of vectors for 2D, 3D, ...
type : string
spectrum type, one of 'freq', 'k1d', 'enc' (default 'freq')
lagtype : letter
lag type, one of: 'x', 'y' or 't' (default 't')
Examples
--------
>>> import numpy as np
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap(Hm0=3)
>>> w = np.linspace(0,4,256)
>>> S = SpecData1D(Sj(w),w) #Make spectrum object from numerical values
See also
--------
WafoData
CovData
"""
def __init__(self,*args,**kwds):
super(CovData1D, self).__init__(*args,**kwds)
self.name='WAFO Covariance Object'
self.type='time'
self.lagtype='t'
self.h=np.inf
self.tr=None
self.phi=0.
self.v=0.
self.norm=0
somekeys = ['phi', 'name', 'h', 'tr', 'lagtype', 'v', 'type', 'norm']
self.__dict__.update(sub_dict_select(kwds,somekeys))
#self.setlabels()
def copy(self):
kwds = self.__dict__.copy()
wdata = CovData1D(**kwds)
return wdata
def tospec(self,rate=None,method='linear',nugget=0.0,trunc=1e-5,fast=True):
'''Computes spectral density from the auto covariance function
Parameters
----------
rate = scalar, int
1,2,4,8...2^r, interpolation rate for f (default 1)
method: string
interpolation method 'stineman', 'linear', 'cubic'
nugget = scalar, real
nugget effect to ensure that round off errors do not result in
negative spectral estimates. Good choice might be 10^-12.
trunc : scalar, real
truncates all spectral values where S/max(S) < trunc
0 <= trunc <1 This is to ensure that high frequency
noise is not added to the spectrum. (default 1e-5)
fast : bool
if True : zero-pad to obtain power of 2 length ACF (default)
otherwise no zero-padding of ACF, slower but more accurate.
Returns
--------
S = SpecData1D object
spectral density
NB! This routine requires that the covariance is evenly spaced
starting from zero lag. Currently only capable of 1D matrices.
Example:
>>> import wafo.spectrum.models as sm
>>> import numpy as np
>>> import scipy.signal.signaltools as st
>>> L = 129
>>> t = np.linspace(0,75,L)
>>> R = np.zeros(L)
>>> win = st.parzen(41)
>>> R[0:20] = win[20:41]
>>> R0 = CovData1D(R,t)
>>> S0 = R0.tospec()
>>> Sj = sm.Jonswap()
>>> S = Sj.toSpecData()
>>> R2 = S.toacf()
>>> S1 = R2.tospec()
>>> assert(all(abs(S1.data-S.data)<1e-4) ,'COV2SPEC')
See also
--------
spec2cov
datastructures
'''
dT = self.sampling_period()
# dT = time-step between data points.(default = T(2)-T1)).
ACF, ti = np.atleast_1d(self.data,self.args)
if self.lagtype in 't':
spectype = 'freq'
ftype = 'w'
else:
spectype = 'k1d'
ftype = 'k'
if rate is None:
rate = 1 #;%interpolation rate
else:
rate = 2**nextpow2(rate) #;%make sure rate is a power of 2
#% add a nugget effect to ensure that round off errors
#% do not result in negative spectral estimates
ACF[0] = ACF[0] +nugget
n = ACF.size
# embedding a circulant vector and Fourier transform
if fast:
nfft = 2**nextpow2(2*n-2)
else:
nfft = 2*n-2
nf = nfft/2 #;% number of frequencies
fft = np.fft.fft
ACF = np.r_[ACF,np.zeros(nfft-2*n+2),ACF[n-1:0:-1]]
Rper = (fft(ACF,nfft).real).clip(0) #% periodogram
RperMax = Rper.max()
Rper = np.where(Rper<trunc*RperMax,0,Rper)
pi = np.pi
S = np.abs(Rper[0:(nf+1)])*dT/pi
w = np.linspace(0,pi/dT,nf+1)
So = SpecData1D(S,w,type=spectype,freqtype=ftype)
So.tr = self.tr
So.h = self.h;
So.norm = self.norm;
if rate>1:
So.args = np.linspace(0,pi/dT,nf*rate);
if method=='stineman':
So.data = stineman_interp(So.args,w,S)
else:
intfun = interpolate.interp1d(w,S,kind=method)
So.data = intfun(So.args)
So.data = So.data.clip(0) # clip negative values to 0
return So
def sampling_period(self):
''' Returns sampling interval
Returns
---------
dT : scalar
sampling interval, unit:
[s] if lagtype=='t'
[m] otherwise
See also
'''
dt1 = self.args[1]-self.args[0]
N = np.size(self.args)-1
T = self.args[-1]-self.args[0]
dt = T/N
return dt
def sim(self,ns=None,cases=1,dt=None,iseed=None,derivative=False):
''' Simulates a Gaussian process and its derivative from ACF
Parameters
----------
ns : scalar
number of simulated points. (default length(S)-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)
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.
If the ACF 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 simulation may give high frequency ripple when used with a
small dt.
Example:
>>> import wafo.spectrum.models as sm
>>> Sj = sm.Jonswap()
>>> S = Sj.toSpecData() #Make spec
>>> R = S.toacf()
>>> x = R.sim(ns=1000,dt=0.2)
See also
--------
spec2sdat, gaus2dat
Reference
-----------
C.R 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
'''
# TODO fix it, it does not work
#% add a nugget effect to ensure that round off errors
#% do not result in negative spectral estimates
nugget=0 #;%10^-12;
_set_seed(iseed)
ACF = self.data.ravel()
n = ACF.size
I = ACF.argmax()
if I!=0:
raise ValueError('ACF does not have a maximum at zero lag')
ACF.shape = (n,1)
dT = self.sampling_period()
fft = np.fft.fft
x=np.zeros((ns,cases+1))
if derivative:
xder=x.copy()
#% add a nugget effect to ensure that round off errors
#% do not result in negative spectral estimates
ACF[0]=ACF[0]+nugget;
#% Fast and exact simulation of simulation of stationary
#% Gaussian process throug circulant embedding of the
#% Covariance matrix
floatinfo = numpy.finfo(float)
if (abs(ACF[-1])>floatinfo.eps): #% assuming ACF(n+1)==0
m2=2*n-1;
nfft=2**nextpow2(max(m2,2*ns));
ACF=np.r_[ACF,np.zeros((nfft-m2,1)),ACF[-1:0:-1,:]]
#disp('Warning: I am now assuming that ACF(k)=0 ')
#disp('for k>MAXLAG.')
else: # % ACF(n)==0
m2=2*n-2;
nfft=2**nextpow2(max(m2,2*ns))
ACF=np.r_[ACF,np.zeros((nfft-m2,1)),ACF[n-1:1:-1,:]]
#%m2=2*n-2;
S=fft(ACF,nfft,axis=0).real #;% periodogram
I=S.argmax()
k=np.flatnonzero(S<0);
if k.size>0:
#disp('Warning: Not able to construct a nonnegative circulant ')
#disp('vector from the ACF. Apply the parzen windowfunction ')
#disp('to the ACF in order to avoid this.')
#disp('The returned result is now only an approximation.')
# truncating negative values to zero to ensure that
# that this noise is not added to the simulated timeseries
S[k] = 0.
ix = np.flatnonzero(k>2*I)
if ix.size>0:
## % truncating all oscillating values above 2 times the peak
## % frequency to zero to ensure that
## % that high frequency noise is not added to
## % the simulated timeseries.
ix0 = k[ix[0]]
S[ix0:-ix0] =0.0
sqrt = np.sqrt
trunc = 1e-5;
maxS = S[I]
k=np.flatnonzero(S[I:-I]<maxS*trunc)
if k.size>0:
S[k+I]=0.
#% truncating small values to zero to ensure that
#% that high frequency noise is not added to
#% the simulated timeseries
cases1 = np.floor(cases/2);
cases2 = np.ceil(cases/2);
#% Generate standard normal random numbers for the simulations
#% -----------------------------------------------------------
randn = np.random.randn
epsi = randn(nfft,cases2)+1j*randn(nfft,cases2);
Ssqr=sqrt(S/(nfft)) #; %sqrt(S(wn)*dw )
ephat=epsi*Ssqr[:,np.newaxis]
y=fft(ephat,nfft);
x[:,1:cases+1]=np.hstack((y[2:ns+2,0:cases2].real, y[2:ns+2,0:cases1].imag))
x[:,0]=np.linspace(0,(ns-1)*dT,ns) #%(0:dT:(dT*(np-1)))';
if derivative:
Ssqr = Ssqr*np.r_[0:(nfft/2+1),-(nfft/2-1):0]*2*np.pi/nfft/dT
ephat = epsi*Ssqr[:,np.newaxis]
y = fft(ephat,nfft);
xder[:,1:(cases+1)]= np.hstack((y[2:ns+2,0:cases2].imag -y[2:ns+2,0:cases1].real))
xder[:,0]=x[:,0]
if self.tr is not None:
np.disp(' Transforming data.')
g=self.tr;
G=np.fliplr(g); #% the invers of g
if derivative:
for ix in range(cases):
tmp=tranproc(np.hstack((x[:,ix+1], xder[:,ix+1])),G);
x[:,ix+1]=tmp[:,0];
xder[:,ix+1]=tmp[:,1];
else:
for ix in range(cases):
x[:,ix+1]=tranproc(x[:,ix+1],G);
if derivative:
return x,xder
else:
return x
class TimeSeries(WafoData):
''' Container class for 1D TimeSeries data objects in WAFO
Member variables
----------------
data : array_like
args : vector for 1D, list of vectors for 2D, 3D, ...
type : integer or string
sensor type for time series (default 'n' : Surface elevation)
see sensortype for more options
position : vector of size 3
instrument position relative to the coordinate system
'''
def __init__(self,*args,**kwds):
super(TimeSeries, self).__init__(*args,**kwds)
self.name='WAFO TimeSeries Object'
self.type='n'
self.position = np.zeros(3)
somekeys = ['type', 'position']
self.__dict__.update(sub_dict_select(kwds,somekeys))
#self.setlabels()
if not any(self.args):
n = len(self.data)
self.args = xrange(0,n)
def sampling_period(self):
''' Returns sampling interval
Returns
---------
dT : scalar
sampling interval, unit:
[s] if lagtype=='t'
[m] otherwise
See also
'''
dt1 = self.args[1]-self.args[0]
N = np.size(self.args)-1
T = self.args[-1]-self.args[0]
dt = T/N
return dt
def acf(self,lag=None,flag='biased',norm=False,dt = None):
''' Return auto covariance function from data.
R = ACF vector length L+1
t = time lags length L+1
stdev = estimated large lag standard deviation of the estimate
assuming x is a Gaussian process:
if R(k)=0 for all lags k>q then an approximation
of the variance for large samples due to Bartlett
var(R(k))=1/N*(R(0)^2+2*R(1)^2+2*R(2)^2+ ..+2*R(q)^2)
for k>q and where N=length(x). Special case is
white noise where it equals R(0)^2/N for k>0
norm = 0 indicating that R is not normalized
x = a column data vector or two column data matrix with sampled times and values.
L = the maximum time-lag for which the ACF is estimated.
(Default L=n-1)
plotflag = 1 then the ACF is plotted vs lag
2 then the ACF is plotted vs lag in seconds
3 then the ACF is plotted vs lag and vs lag (sec
dT = time-step between data points (default xn(2,1)-xn(1,1) or 1 Hz).
flag = 'biased' : scales the raw cross-correlation by 1/n. (default)
'unbiased': scales the raw correlation by 1/(n-abs(k)),
where k is the index into the result.
- x may contain NaN's (i.e. missing values).
Example:
x = load('sea.dat');
rf = dat2cov(x,150,2)
'''
n = len(self.data)
if not lag:
lag = n-1
x = self.data.copy()
indnan = numpy.isnan(x)
if any(indnan):
x = x - numpy.mean(x[1-indnan]) # remove the mean pab 09.10.2000
#indnan = find(indnan);
Ncens = n - sum(indnan)
x[indnan] = 0. # pab 09.10.2000 much faster for censored samples
else:
indnan = None
Ncens = n
x = x - numpy.mean(x)
fft = numpy.fft.fft
nfft = 2**nextpow2(n)
Rper = abs(fft(x,nfft))**2/Ncens # Raw periodogram
R = numpy.real(fft(Rper))/nfft # %ifft=fft/nfft since Rper is real!
lags = np.arange(0,lag+1)
if flag.startswith('unbiased'):
# unbiased result, i.e. divide by n-abs(lag)
R=R[lags]*Ncens/ np.arange(Ncens,Ncens-lag,-1)
#else % biased result, i.e. divide by n
# r=r(1:L+1)*Ncens/Ncens;
#c0 = R[0]
dT = self.sampling_period()
t = numpy.linspace(0,lag*dT,lag+1)
cumsum = numpy.cumsum
acf = CovData1D(R[lags],t)
acf.stdev=numpy.sqrt(numpy.r_[ 0, 1 ,1+2*cumsum(R[1:]**2)]/Ncens)
acf.children = [WafoData(-2.*acf.stdev[lags],t),WafoData(2.*acf.stdev[lags],t)]
return acf
def spec(self):
pass
def turning_points(self,h=0,wavetype=None,output='tp'):
''' Return turning points (tp) from data, optionally rainflowfiltered.
Parameters
----------
h : scalar
a threshold;
if h<0, then tp=x;
if h=0, then tp is a sequence of turning points (default);
if h>0, then all rainflow cycles with height smaller than
h are removed.
wavetype : string
defines the type of wave. Possible options are
'mw' 'Mw' or 'none'.
If None all rainflow filtered min and max
will be returned, otherwise only the rainflow filtered
min and max, which define a wave according to the
wave definition, will be returned.
output : string
'tp' if only returning tp
'all' if tp and ind
Returns
-------
tp : TimeSeries object
with times and turning points.
ind : arraylike
indices to the turning points in the original sequence.
Example:
x = load('sea.dat'); x1 = x(1:200,:);
tp = dat2tp(x1,0,'Mw'); tph = dat2tp(x1,0.3,'Mw');
plot(x1(:,1),x1(:,2),tp(:,1),tp(:,2),'ro',tph(:,1),tph(:,2),'k*')
See also
---------
findcross, findrfc
'''
if h<0:
tp = self.copy()
ind = np.arange(tp.args.size)
if output.startswith('tp'):
return tp
else:
return tp, ind
x2 = self.data
n = len(x2)
ind = findextrema(x2)
if ind.size<2:
tp = None
ind = None
return tp
#% In order to get the exact up-crossing intensity from rfc by
#% mm2lc(tp2mm(rfc)) we have to add the indices
#% to the last value (and also the first if the
#% sequence of turning points does not start with a minimum).
if x2[ind[0]]>x2[ind[1]]:
#% adds indices to first and last value
ind = np.r_[0, ind ,n]
else: # adds index to the last value
ind = np.r_[ind, n]
if h>0:
ind1 = findrfc(x2[ind],h)
ind = ind[ind1]
Nm =ind.size #% number of min and Max
if wavetype in ('mw','Mw'):
xor = lambda a,b : a^b
#% make sure that the first is a Max if wdef == 'Mw'
#% or make sure that the first is a min if wdef == 'mw'
removeFirst = xor((x2[ind[0]]>x2[ind[1]]),wavetype.startswith('Mw'))
if removeFirst:
ind = ind[1::]
Nm = Nm-1;
#% make sure the number of minima and Maxima are according to the wavedef.
#% i.e., make sure Nm=length(ind) is odd
if (mod(Nm,2))!=1:
ind = ind[:-1]
Nm = Nm-1
try:
t = self.args
except:
t = ind
tp = TimeSeries(self.data[ind],t)
return tp
def lc_spectrum(self):
pass
def cycles(self):
pass
def trough_crest(self):
pass
def wave_period(self):
pass
def reconstruct(self):
pass
def plot_wave_idx(self):
''' spwaveplot
'''
pass
def findoutliers(self):
pass
def sensortypeid(*sensortypes):
''' Return ID for sensortype name
Parameter
---------
sensortypes : list of strings defining the sensortype
Returns
-------
sensorids : list of integers defining the sensortype
Valid senor-ids and -types for time series are as follows:
0, 'n' : Surface elevation (n=Eta)
1, 'n_t' : Vertical surface velocity
2, 'n_tt' : Vertical surface acceleration
3, 'n_x' : Surface slope in x-direction
4, 'n_y' : Surface slope in y-direction
5, 'n_xx' : Surface curvature in x-direction
6, 'n_yy' : Surface curvature in y-direction
7, 'n_xy' : Surface curvature in xy-direction
8, 'P' : Pressure fluctuation about static MWL pressure
9, 'U' : Water particle velocity in x-direction
10, 'V' : Water particle velocity in y-direction
11, 'W' : Water particle velocity in z-direction
12, 'U_t' : Water particle acceleration in x-direction
13, 'V_t' : Water particle acceleration in y-direction
14, 'W_t' : Water particle acceleration in z-direction
15, 'X_p' : Water particle displacement in x-direction from its mean position
16, 'Y_p' : Water particle displacement in y-direction from its mean position
17, 'Z_p' : Water particle displacement in z-direction from its mean position
Example:
>>> sensortypeid('W','v')
[11, 10]
>>> sensortypeid('rubbish')
[-1.#IND]
See also sensortype, id
'''
sensorid_table = dict(n=0,n_t=1,n_tt=2,n_x=3,n_y=4,n_xx=5,
n_yy=6,n_xy=7,p=8,u=9,v=10,w=11,u_t=12,
v_t=13,w_t=14,x_p=15,y_p=16,z_p=17)
try:
return [sensorid_table.get(name.lower(),np.NAN) for name in sensortypes]
except:
raise ValueError('Input must be a string!')
def sensortype(*sensorids):
''' Return sensortype name
Parameter
---------
sensorids : vector or list of integers defining the sensortype
Returns
-------
sensornames : list of strings defining the sensortype
Valid senor-ids and -types for time series are as follows:
0, 'n' : Surface elevation (n=Eta)
1, 'n_t' : Vertical surface velocity
2, 'n_tt' : Vertical surface acceleration
3, 'n_x' : Surface slope in x-direction
4, 'n_y' : Surface slope in y-direction
5, 'n_xx' : Surface curvature in x-direction
6, 'n_yy' : Surface curvature in y-direction
7, 'n_xy' : Surface curvature in xy-direction
8, 'P' : Pressure fluctuation about static MWL pressure
9, 'U' : Water particle velocity in x-direction
10, 'V' : Water particle velocity in y-direction
11, 'W' : Water particle velocity in z-direction
12, 'U_t' : Water particle acceleration in x-direction
13, 'V_t' : Water particle acceleration in y-direction
14, 'W_t' : Water particle acceleration in z-direction
15, 'X_p' : Water particle displacement in x-direction from its mean position
16, 'Y_p' : Water particle displacement in y-direction from its mean position
17, 'Z_p' : Water particle displacement in z-direction from its mean position
Example:
>>> sensortype(range(3))
['n', 'n_t', 'n_tt']
See also sensortypeid, tran
'''
validNames = ('n','n_t','n_tt','n_x','n_y','n_xx',
'n_yy','n_xy','p','u','v','w','u_t',
'v_t','w_t','x_p','y_p','z_p',np.NAN)
ids = np.atleast_1d(*sensorids)
if isinstance(ids,list):
ids = np.hstack(ids)
N = len(validNames)-1
try:
return [validNames[id] for id in ids.clip(0,N)]
except:
raise ValueError('Input must be an integer!')
def main():
from wafo.spectrum import models as sm
sensortype(range(21))
w = np.linspace(0,3,100)
Sj = sm.Jonswap()
S = Sj.toSpecData()
#S = SpecData1D(Sj(w),w)
R = S.toacf(nr=1)
S1 = S.copy()
Si = R.tospec()
ns =5000;
dt = .2;
x1 = S.sim_nl(ns= ns,dt=dt);
x2 = TimeSeries(x1[:,1],x1[:,0])
R = x2.acf(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')
np.disp('done')
if __name__ == '__main__':
if False: #True: #
import doctest
doctest.testmod()
else:
#main()
import wafo.spectrum.models as sm
Sj = sm.Jonswap();
S = Sj.toSpecData();
R = S.toacf()
x = R.sim(ns=1000,dt=0.2)
S.characteristic(['hm0','tm02'])
ns =1000; dt = .2;
x1 = S.sim_nl(ns,dt=dt);
x = np.arange(-2,2,0.2)
# Plot 2 objects in one call
d2 = WafoData(np.sin(x),x,xlab='x',ylab='sin',title='sinus')
d0 = d2.copy()
d0.data = d0.data*0.9
d1 = d2.copy()
d1.data = d1.data*1.2
d1.children = [d0]
d2.children = [d1]
d2.plot()
print 'Done'
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