def test_skew(self):
for n in self.get_n():
x, y, xm, ym = self.generate_xy_sample(n)
r = stats.skew(x)
rm = stats.mstats.skew(xm)
assert_almost_equal(r, rm, 10)
r = stats.skew(y)
rm = stats.mstats.skew(ym)
assert_almost_equal(r, rm, 10)
def trdata(self, timeseries):
'''
Returns
-------
tr, tr_emp : TrData objects
with the smoothed and empirical transformation, respectively.
TRDATA estimates the transformation in a transformed Gaussian model.
Assumption: a Gaussian process, Y, is related to the
non-Gaussian process, X, by Y = g(X).
The empirical crossing intensity is usually very irregular.
More than one local maximum of the empirical crossing intensity may
cause poor fit of the transformation. In such case one should use a
smaller value of CSM. In order to check the effect of smoothing it is
recomended to also plot g and g2 in the same plot or plot the smoothed
g against an interpolated version of g (when CSM=GSM=1).
Example
-------
>>> import wafo.spectrum.models as sm
>>> import wafo.transform.models as tm
>>> from wafo.objects import mat2timeseries
>>> Hs = 7.0
>>> Sj = sm.Jonswap(Hm0=Hs)
>>> S = Sj.tospecdata() #Make spectrum object from numerical values
>>> S.tr = tm.TrOchi(mean=0, skew=0.16, kurt=0,
... sigma=Hs/4, ysigma=Hs/4)
>>> xs = S.sim(ns=2**16, iseed=10)
>>> ts = mat2timeseries(xs)
>>> g0, g0emp = ts.trdata(monitor=True)
>>> g1, g1emp = ts.trdata(method='m', gvar=0.5 )
>>> g2, g2emp = ts.trdata(method='n', gvar=[3.5, 0.5, 3.5])
>>> int(S.tr.dist2gauss()*100)
141
>>> int(g0emp.dist2gauss()*100)
217949
>>> int(g0.dist2gauss()*100)
93
>>> int(g1.dist2gauss()*100)
66
>>> int(g2.dist2gauss()*100)
84
See also
--------
LevelCrossings.trdata
wafo.transform.models
References
----------
Rychlik, I. , Johannesson, P and Leadbetter, M. R. (1997)
"Modelling and statistical analysis of ocean wavedata using
transformed Gaussian process."
Marine structures, Design, Construction and Safety, Vol. 10, No. 1,
pp 13--47
Brodtkorb, P, Myrhaug, D, and Rue, H (1999)
"Joint distribution of wave height and crest velocity from
reconstructed data"
in Proceedings of 9th ISOPE Conference, Vol III, pp 66-73
'''
data = np.atleast_1d(timeseries.data)
ma = data.mean()
sa = data.std()
method = self.method[0]
if method == 'l':
return TrLinear(mean=ma, sigma=sa), TrLinear(mean=ma, sigma=sa)
if method == 'n':
tp = timeseries.turning_points()
mM = tp.cycle_pairs()
lc = mM.level_crossings(self.crossdef)
return self._trdata_lc(lc)
elif method == 'm':
return self._trdata_cdf(data)
elif method == 'h':
ga1 = skew(data)
ga2 = kurtosis(data, fisher=True) # kurt(xx(n+1:end))-3;
up = min(4 * (4 * ga1 / 3)**2, 13)
lo = (ga1**2) * 3 / 2
kurt1 = min(up, max(ga2, lo)) + 3
return TrHermite(mean=ma, var=sa**2, skew=ga1, kurt=kurt1)
elif method[0] == 'o':
ga1 = skew(data)
return TrOchi(mean=ma, var=sa**2, skew=ga1)
def trdata(self, timeseries):
'''
Returns
-------
tr, tr_emp : TrData objects
with the smoothed and empirical transformation, respectively.
TRDATA estimates the transformation in a transformed Gaussian model.
Assumption: a Gaussian process, Y, is related to the
non-Gaussian process, X, by Y = g(X).
The empirical crossing intensity is usually very irregular.
More than one local maximum of the empirical crossing intensity may
cause poor fit of the transformation. In such case one should use a
smaller value of CSM. In order to check the effect of smoothing it is
recomended to also plot g and g2 in the same plot or plot the smoothed
g against an interpolated version of g (when CSM=GSM=1).
Example
-------
>>> import wafo.spectrum.models as sm
>>> import wafo.transform.models as tm
>>> from wafo.objects import mat2timeseries
>>> Hs = 7.0
>>> Sj = sm.Jonswap(Hm0=Hs)
>>> S = Sj.tospecdata() #Make spectrum object from numerical values
>>> S.tr = tm.TrOchi(mean=0, skew=0.16, kurt=0,
... sigma=Hs/4, ysigma=Hs/4)
>>> xs = S.sim(ns=2**16, iseed=10)
>>> ts = mat2timeseries(xs)
>>> g0, g0emp = ts.trdata(monitor=True)
>>> g1, g1emp = ts.trdata(method='m', gvar=0.5 )
>>> g2, g2emp = ts.trdata(method='n', gvar=[3.5, 0.5, 3.5])
>>> int(S.tr.dist2gauss()*100)
141
>>> int(g0emp.dist2gauss()*100)
217949
>>> int(g0.dist2gauss()*100)
93
>>> int(g1.dist2gauss()*100)
66
>>> int(g2.dist2gauss()*100)
84
See also
--------
LevelCrossings.trdata
wafo.transform.models
References
----------
Rychlik, I. , Johannesson, P and Leadbetter, M. R. (1997)
"Modelling and statistical analysis of ocean wavedata using
transformed Gaussian process."
Marine structures, Design, Construction and Safety, Vol. 10, No. 1,
pp 13--47
Brodtkorb, P, Myrhaug, D, and Rue, H (1999)
"Joint distribution of wave height and crest velocity from
reconstructed data"
in Proceedings of 9th ISOPE Conference, Vol III, pp 66-73
'''
data = np.atleast_1d(timeseries.data)
ma = data.mean()
sa = data.std()
method = self.method[0]
if method == 'l':
return TrLinear(mean=ma, sigma=sa), TrLinear(mean=ma, sigma=sa)
if method == 'n':
tp = timeseries.turning_points()
mM = tp.cycle_pairs()
lc = mM.level_crossings(self.crossdef)
return self._trdata_lc(lc)
elif method == 'm':
return self._trdata_cdf(data)
elif method == 'h':
ga1 = skew(data)
ga2 = kurtosis(data, fisher=True) # kurt(xx(n+1:end))-3;
up = min(4 * (4 * ga1 / 3) ** 2, 13)
lo = (ga1 ** 2) * 3 / 2
kurt1 = min(up, max(ga2, lo)) + 3
return TrHermite(mean=ma, var=sa ** 2, skew=ga1, kurt=kurt1)
elif method[0] == 'o':
ga1 = skew(data)
return TrOchi(mean=ma, var=sa ** 2, skew=ga1)
#! It can be seen in Figure above that the covariance corresponding to S1
#! agrees much better with the estimated covariance function
clf()
R2 = S2.tocovdata(nr=1)
R2.plot('.')
Rest.plot()
show()
#! Section 2.2.2 Transformed Gaussian models
#!-------------------------------------------
#! We begin with computing skewness and kurtosis
#! for the data set xx and compare it with the second order wave approximation
#! proposed by Winterstein:
import wafo.stats as ws
rho3 = ws.skew(xx[:, 1])
rho4 = ws.kurtosis(xx[:, 1])
sk, ku = S1.stats_nl(moments='sk')
#! Comparisons of 3 transformations
clf()
import wafo.transform.models as wtm
gh = wtm.TrHermite(mean=me, sigma=sa, skew=sk, kurt=ku).trdata()
g = wtm.TrLinear(mean=me, sigma=sa).trdata() # Linear transformation
glc, gemp = lc.trdata(mean=me, sigma=sa)
glc.plot('b-') #! Transf. estimated from level-crossings
gh.plot('b-.') #! Hermite Transf. estimated from moments
g.plot('r')
grid('on')
#! It can be seen in Figure above that the covariance corresponding to S1
#! agrees much better with the estimated covariance function
clf()
R2 = S2.tocovdata(nr=1)
R2.plot('.')
Rest.plot()
show()
#! Section 2.2.2 Transformed Gaussian models
#!-------------------------------------------
#! We begin with computing skewness and kurtosis
#! for the data set xx and compare it with the second order wave approximation
#! proposed by Winterstein:
import wafo.stats as ws
rho3 = ws.skew(xx[:, 1])
rho4 = ws.kurtosis(xx[:, 1])
sk, ku = S1.stats_nl(moments='sk')
#! Comparisons of 3 transformations
clf()
import wafo.transform.models as wtm
gh = wtm.TrHermite(mean=me, sigma=sa, skew=sk, kurt=ku).trdata()
g = wtm.TrLinear(mean=me, sigma=sa).trdata() # Linear transformation
glc, gemp = lc.trdata(mean=me, sigma=sa)
glc.plot('b-') #! Transf. estimated from level-crossings
gh.plot('b-.') #! Hermite Transf. estimated from moments
g.plot('r')
grid('on')
