def inc(self, inc):
if inc is None:
_tau, tau = self.kernel.effective_support()
xyzrange = 8 * self.sigma
L1 = 10
inc = max(48, (L1 * xyzrange / (tau * self.hs)).max())
inc = 2 ** nextpow2(inc)
# pylint: disable=attribute-defined-outside-init
self._inc = inc
def test_nextpow2():
assert_equal(nextpow2(10), 4)
assert_equal(nextpow2(np.arange(5)), 3)
def tocovdata(self, timeseries):
'''
Return auto covariance function from data.
Return
-------
R : CovData1D object
with attributes:
data : ACF vector length L+1
args : time lags length L+1
sigma : 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 : bool
If false indicating that R is not normalized
Example:
--------
>>> import wafo.data
>>> import wafo.objects as wo
>>> x = wafo.data.sea()
>>> ts = wo.mat2timeseries(x)
>>> acf = ts.tocovdata(150)
>>> h = acf.plot()
'''
lag = self.lag
window = self.window
detrend = self.detrend
try:
x = timeseries.data.flatten('F')
dt = timeseries.sampling_period()
except Exception:
x = timeseries[:, 1:].flatten('F')
dt = sampling_period(timeseries[:, 0])
if not (self.dt is None):
dt = self.dt
if not (self.tr is None):
x = self.tr.dat2gauss(x)
n = len(x)
indnan = np.isnan(x)
if any(indnan):
x = x - x[1 - indnan].mean()
Ncens = n - indnan.sum()
x[indnan] = 0.
else:
Ncens = n
x = x - x.mean()
if hasattr(detrend, '__call__'):
x = detrend(x)
nfft = 2 ** nextpow2(n)
Rper = abs(fft(x, nfft)) ** 2 / Ncens # Raw periodogram
R = np.real(fft(Rper)) / nfft # ifft = fft/nfft since Rper is real!
if self.flag.startswith('unbiased'):
# unbiased result, i.e. divide by n-abs(lag)
R = R[:Ncens] * Ncens / np.arange(Ncens, 1, -1)
if self.norm:
R = R / R[0]
if lag is None:
lag = self._estimate_lag(R, Ncens)
lag = min(lag, n - 2)
if isinstance(window, str) or type(window) is tuple:
win = get_window(window, 2 * lag - 1)
else:
win = np.asarray(window)
R[:lag] = R[:lag] * win[lag - 1::]
R[lag] = 0
lags = slice(0, lag + 1)
t = np.linspace(0, lag * dt, lag + 1)
acf = CovData1D(R[lags], t)
acf.sigma = np.sqrt(np.r_[0, R[0] ** 2,
R[0] ** 2 + 2 * np.cumsum(R[1:] ** 2)] / Ncens)
acf.children = [PlotData(-2. * acf.sigma[lags], t),
PlotData(2. * acf.sigma[lags], t)]
acf.plot_args_children = ['r:']
acf.norm = self.norm
return acf
def test_nextpow2():
assert nextpow2(10) == 4
assert nextpow2(np.arange(5)) == 3
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.tocovdata()
>>> 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)
self._is_valid_acf()
acf = self.data.ravel()
n = acf.size
acf.shape = (n, 1)
dT = self.sampling_period()
x = 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 = 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 = r_[acf, zeros((nfft - m2, 1)), acf[-1:0:-1, :]]
# warnings,warn('I am now assuming that ACF(k)=0 for k>MAXLAG.')
else: # ACF(n)==0
m2 = 2 * n - 2
nfft = 2 ** nextpow2(max(m2, 2 * ns))
acf = r_[acf, 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 = flatnonzero(S < 0)
if k.size > 0:
_msg = '''
Not able to construct a nonnegative circulant vector from ACF.
Apply parzen windowfunction to the ACF in order to avoid this.
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 = 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
trunc = 1e-5
maxS = S[I]
k = 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 = int(cases / 2)
cases2 = int(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, axis=0)
x[:, 1:cases + 1] = hstack((y[2:ns + 2, 0:cases2].real,
y[2:ns + 2, 0:cases1].imag))
x[:, 0] = linspace(0, (ns - 1) * dT, ns) # (0:dT:(dT*(np-1)))'
if derivative:
Ssqr = Ssqr * \
r_[0:(nfft / 2 + 1), -(nfft / 2 - 1):0] * 2 * pi / nfft / dT
ephat = epsi * Ssqr # [:,newaxis]
y = fft(ephat, nfft, axis=0)
xder[:, 1:(cases + 1)] = 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:
print(' Transforming data.')
g = self.tr
if derivative:
for ix in range(cases):
tmp = g.gauss2dat(x[:, ix + 1], xder[:, ix + 1])
x[:, ix + 1] = tmp[0]
xder[:, ix + 1] = tmp[1]
else:
for ix in range(cases):
x[:, ix + 1] = g.gauss2dat(x[:, ix + 1])
if derivative:
return x, xder
else:
return x
def tospecdata(self, rate=None, method='fft', 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', 'fft'
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 as st
>>> import pylab
>>> L = 129
>>> t = np.linspace(0,75,L)
>>> R = np.zeros(L)
>>> win = st.parzen(41)
>>> R[0:21] = win[20:41]
>>> R0 = CovData1D(R,t)
>>> S0 = R0.tospecdata()
>>> Sj = sm.Jonswap()
>>> S = Sj.tospecdata()
>>> R2 = S.tocovdata()
>>> S1 = R2.tospecdata()
>>> abs(S1.data-S.data).max()
>>> S1.plot('r-')
>>> S.plot('b:')
>>> pylab.show()
>>> all(abs(S1.data-S.data)<1e-4)
See also
--------
spec2cov
datastructures
'''
dt = self.sampling_period()
# dt = time-step between data points.
acf, unused_ti = 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
nfft = 2 ** nextpow2(2 * n - 2) if fast else 2 * n - 2
if method == 'fft':
nfft *= rate
nf = nfft / 2 # number of frequencies
acf = r_[acf, zeros(nfft - 2 * n + 2), acf[n - 2:0:-1]]
Rper = (fft(acf, nfft).real).clip(0) # periodogram
RperMax = Rper.max()
Rper = where(Rper < trunc * RperMax, 0, Rper)
S = abs(Rper[0:(nf + 1)]) * dt / pi
w = linspace(0, pi / dt, nf + 1)
So = _wafospec.SpecData1D(S, w, type=spectype, freqtype=ftype)
So.tr = self.tr
So.h = self.h
So.norm = self.norm
if method != 'fft' and rate > 1:
So.args = 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 tocovdata(self, timeseries):
"""
Return auto covariance function from data.
Return
-------
acf : CovData1D object
with attributes:
data : ACF vector length L+1
args : time lags length L+1
sigma : estimated large lag standard deviation of the estimate
assuming x is a Gaussian process:
if acf[k]=0 for all lags k>q then an approximation
of the variance for large samples due to Bartlett
var(acf[k])=1/N*(acf[0]**2+2*acf[1]**2+2*acf[2]**2+ ..+2*acf[q]**2)
for k>q and where N=length(x). Special case is
white noise where it equals acf[0]**2/N for k>0
norm : bool
If false indicating that auto_cov is not normalized
Example:
--------
>>> import wafo.data
>>> import wafo.objects as wo
>>> x = wafo.data.sea()
>>> ts = wo.mat2timeseries(x)
>>> acf = ts.tocovdata(150)
h = acf.plot()
"""
lag = self.lag
window = self.window
detrend = self.detrend
try:
x = timeseries.data.flatten('F')
dt = timeseries.sampling_period()
except Exception:
x = timeseries[:, 1:].flatten('F')
dt = sampling_period(timeseries[:, 0])
if self.dt is not None:
dt = self.dt
if self.tr is not None:
x = self.tr.dat2gauss(x)
n = len(x)
indnan = np.isnan(x)
if any(indnan):
x = x - x[1 - indnan].mean()
Ncens = n - indnan.sum()
x[indnan] = 0.
else:
Ncens = n
x = x - x.mean()
if hasattr(detrend, '__call__'):
x = detrend(x)
nfft = 2 ** nextpow2(n)
raw_periodogram = abs(fft(x, nfft)) ** 2 / Ncens
# ifft = fft/nfft since raw_periodogram is real!
auto_cov = np.real(fft(raw_periodogram)) / nfft
if self.flag.startswith('unbiased'):
# unbiased result, i.e. divide by n-abs(lag)
auto_cov = auto_cov[:Ncens] * Ncens / np.arange(Ncens, 1, -1)
if self.norm:
auto_cov = auto_cov / auto_cov[0]
if lag is None:
lag = self._estimate_lag(auto_cov, Ncens)
lag = min(lag, n - 2)
if isinstance(window, str) or type(window) is tuple:
win = get_window(window, 2 * lag - 1)
else:
win = np.asarray(window)
auto_cov[:lag] = auto_cov[:lag] * win[lag - 1::]
auto_cov[lag] = 0
lags = slice(0, lag + 1)
t = np.linspace(0, lag * dt, lag + 1)
acf = CovData1D(auto_cov[lags], t)
acf.sigma = np.sqrt(np.r_[0, auto_cov[0] ** 2,
auto_cov[0] ** 2 + 2 * np.cumsum(auto_cov[1:] ** 2)] / Ncens)
acf.children = [PlotData(-2. * acf.sigma[lags], t),
PlotData(2. * acf.sigma[lags], t)]
acf.plot_args_children = ['r:']
acf.norm = self.norm
return acf
def test_nextpow2():
assert_equal(nextpow2(10), 4)
assert_equal(nextpow2(np.arange(5)), 3)
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.tocovdata()
>>> 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)
self._is_valid_acf()
acf = self.data.ravel()
n = acf.size
acf.shape = (n, 1)
dT = self.sampling_period()
x = 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 = 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 = r_[acf, zeros((nfft - m2, 1)), acf[-1:0:-1, :]]
# warnings,warn('I am now assuming that ACF(k)=0 for k>MAXLAG.')
else: # ACF(n)==0
m2 = 2 * n - 2
nfft = 2 ** nextpow2(max(m2, 2 * ns))
acf = r_[acf, 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 = flatnonzero(S < 0)
if k.size > 0:
_msg = '''
Not able to construct a nonnegative circulant vector from ACF.
Apply parzen windowfunction to the ACF in order to avoid this.
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 = 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
trunc = 1e-5
maxS = S[I]
k = 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 = int(cases / 2)
cases2 = int(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, axis=0)
x[:, 1:cases + 1] = hstack((y[2:ns + 2, 0:cases2].real,
y[2:ns + 2, 0:cases1].imag))
x[:, 0] = linspace(0, (ns - 1) * dT, ns) # (0:dT:(dT*(np-1)))'
if derivative:
Ssqr = Ssqr * \
r_[0:(nfft / 2 + 1), -(nfft / 2 - 1):0] * 2 * pi / nfft / dT
ephat = epsi * Ssqr # [:,newaxis]
y = fft(ephat, nfft, axis=0)
xder[:, 1:(cases + 1)] = 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:
print(' Transforming data.')
g = self.tr
if derivative:
for ix in range(cases):
tmp = g.gauss2dat(x[:, ix + 1], xder[:, ix + 1])
x[:, ix + 1] = tmp[0]
xder[:, ix + 1] = tmp[1]
else:
for ix in range(cases):
x[:, ix + 1] = g.gauss2dat(x[:, ix + 1])
if derivative:
return x, xder
else:
return x
def tospecdata(self, rate=None, method='fft', 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', 'fft'
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 as st
>>> import pylab
>>> L = 129
>>> t = np.linspace(0,75,L)
>>> R = np.zeros(L)
>>> win = st.parzen(41)
>>> R[0:21] = win[20:41]
>>> R0 = CovData1D(R,t)
>>> S0 = R0.tospecdata()
>>> Sj = sm.Jonswap()
>>> S = Sj.tospecdata()
>>> R2 = S.tocovdata()
>>> S1 = R2.tospecdata()
>>> abs(S1.data-S.data).max()
>>> S1.plot('r-')
>>> S.plot('b:')
>>> pylab.show()
>>> all(abs(S1.data-S.data)<1e-4)
See also
--------
spec2cov
datastructures
'''
dt = self.sampling_period()
# dt = time-step between data points.
acf, unused_ti = 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
nfft = 2 ** nextpow2(2 * n - 2) if fast else 2 * n - 2
if method == 'fft':
nfft *= rate
nf = nfft / 2 # number of frequencies
acf = r_[acf, zeros(nfft - 2 * n + 2), acf[n - 2:0:-1]]
Rper = (fft(acf, nfft).real).clip(0) # periodogram
RperMax = Rper.max()
Rper = where(Rper < trunc * RperMax, 0, Rper)
S = abs(Rper[0:(nf + 1)]) * dt / pi
w = linspace(0, pi / dt, nf + 1)
So = _wafospec.SpecData1D(S, w, type=spectype, freqtype=ftype)
So.tr = self.tr
So.h = self.h
So.norm = self.norm
if method != 'fft' and rate > 1:
So.args = 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
