def kde_demo5(N=500):
"""Demonstrate that the improved Sheather-Jones plug-in (hisj) is superior
for 2D multimodal distributions
KDEDEMO5 shows that the improved Sheather-Jones plug-in smoothing is better
compared to normal reference rules (in this case the hns)
Examples
--------
>>> kde_demo5()
"""
data = np.hstack((st.norm.rvs(loc=5, scale=1, size=(2, N,)),
st.norm.rvs(loc=-5, scale=1, size=(2, N,))))
kde = KDE(data, kernel=Kernel('gauss', 'hns'))
f = kde(output='plot', plotflag=1,
title='Ordinary KDE, hns={0:s}'.format(str(list(kde.hs))))
kde1 = KDE(data, kernel=Kernel('gauss', 'hisj'))
f1 = kde1(output='plot', plotflag=1,
title='Ordinary KDE, hisj={0:s}'.format(str(list(kde1.hs))))
plt.figure(0)
plt.clf()
f.plot()
plt.plot(data[0], data[1], '.')
plt.figure(1)
plt.clf()
f1.plot()
plt.plot(data[0], data[1], '.')
lc.plot()
plt.show()
m_sea = ts.data.mean()
f0_sea = np.interp(m_sea, lc.args, lc.data)
extr_sea = len(tp.data) / (2 * T_sea)
alfa_sea = f0_sea / extr_sea
print('alfa = %g ' % alfa_sea)
#! Section 4.3.2 Extraction of rainflow cycles
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#! Min-max and rainflow cycle plots
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
mM_rfc = tp.cycle_pairs(h=0.3)
plt.clf()
plt.subplot(122),
mM.plot()
plt.title('min-max cycle pairs')
plt.subplot(121),
mM_rfc.plot()
plt.title('Rainflow filtered cycles')
plt.show()
#! Min-max and rainflow cycle distributions
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
import wafo.misc as wm
ampmM_sea = mM.amplitudes()
ampRFC_sea = mM_rfc.amplitudes()
plt.clf()
plt.subplot(121)
plt.show()
m_sea = ts.data.mean()
f0_sea = np.interp(m_sea, lc.args,lc.data)
extr_sea = len(tp.data)/(2*T_sea)
alfa_sea = f0_sea/extr_sea
print('alfa = %g ' % alfa_sea)
#! Section 4.3.2 Extraction of rainflow cycles
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#! Min-max and rainflow cycle plots
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
mM_rfc = tp.cycle_pairs(h=0.3)
plt.clf()
plt.subplot(122),
mM.plot()
plt.title('min-max cycle pairs')
plt.subplot(121),
mM_rfc.plot()
plt.title('Rainflow filtered cycles')
plt.show()
#! Min-max and rainflow cycle distributions
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
import wafo.misc as wm
ampmM_sea = mM.amplitudes()
ampRFC_sea = mM_rfc.amplitudes()
plt.clf()
plt.subplot(121)
import scipy.signal as ss
import wafo.data as wd
import wafo.misc as wm
import wafo.objects as wo
import wafo.stats as ws
import wafo.spectrum.models as wsm
xx = wd.sea()
xx[:, 1] = ss.detrend(xx[:, 1])
ts = wo.mat2timeseries(xx)
Tcrcr, ix = ts.wave_periods(vh=0, pdef='c2c', wdef='tw', rate=8)
Tc, ixc = ts.wave_periods(vh=0, pdef='u2d', wdef='tw', rate=8)
#! Histogram of crestperiod compared to the kernel density estimate
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
import wafo.kdetools as wk
plt.clf()
print(Tc.mean())
print(Tc.max())
t = np.linspace(0.01,8,200);
ftc = wk.TKDE(Tc, L2=0, inc=128)
plt.plot(t,ftc.eval_grid(t), t, ftc.eval_grid_fast(t),'-.')
wm.plot_histgrm(Tc, normed=True)
plt.title('Kernel Density Estimates')
plt.xlabel('Tc [s]')
plt.axis([0, 8, 0, 0.5])
plt.show()
#! Extreme waves - model check: the highest and steepest wave
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
def gaussq(fun, a, b, reltol=1e-3, abstol=1e-3, alpha=0, beta=0, wfun=1,
trace=False, args=None):
'''
Numerically evaluate integral, Gauss quadrature.
Parameters
----------
fun : callable
a,b : array-like
lower and upper integration limits, respectively.
reltol, abstol : real scalars, optional
relative and absolute tolerance, respectively. (default reltol=abstool=1e-3).
wfun : scalar integer, optional
defining the weight function, p(x). (default wfun = 1)
1 : p(x) = 1 a =-1, b = 1 Gauss-Legendre
2 : p(x) = exp(-x^2) a =-inf, b = inf Hermite
3 : p(x) = x^alpha*exp(-x) a = 0, b = inf Laguerre
4 : p(x) = (x-a)^alpha*(b-x)^beta a =-1, b = 1 Jacobi
5 : p(x) = 1/sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 1'st kind
6 : p(x) = sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 2'nd kind
7 : p(x) = sqrt((x-a)/(b-x)), a = 0, b = 1
8 : p(x) = 1/sqrt(b-x), a = 0, b = 1
9 : p(x) = sqrt(b-x), a = 0, b = 1
trace : bool, optional
If non-zero a point plot of the integrand (default False).
gn : scalar integer
number of base points to start the integration with (default 2).
alpha, beta : real scalars, optional
Shape parameters of Laguerre or Jacobi weight function
(alpha,beta>-1) (default alpha=beta=0)
Returns
-------
val : ndarray
evaluated integral
err : ndarray
error estimate, absolute tolerance abs(int-intold)
Notes
-----
GAUSSQ numerically evaluate integral using a Gauss quadrature.
The Quadrature integrates a (2m-1)th order polynomial exactly and the
integral is of the form
b
Int (p(x)* Fun(x)) dx
a
GAUSSQ is vectorized to accept integration limits A, B and
coefficients P1,P2,...Pn, as matrices or scalars and the
result is the common size of A, B and P1,P2,...,Pn.
Examples
---------
integration of x**2 from 0 to 2 and from 1 to 4
>>> from scitools import numpyutils as npt
>>> A = [0, 1]; B = [2,4]
>>> fun = npt.wrap2callable('x**2')
>>> [val1,err1] = gaussq(fun,A,B)
>>> val1
array([ 2.6666667, 21. ])
>>> err1
array([ 1.7763568e-15, 1.0658141e-14])
Integration of x^2*exp(-x) from zero to infinity:
>>> fun2 = npt.wrap2callable('1')
>>> val2, err2 = gaussq(fun2, 0, npt.inf, wfun=3, alpha=2)
>>> val3, err3 = gaussq(lambda x: x**2,0, npt.inf, wfun=3, alpha=0)
>>> val2, err2
(array([ 2.]), array([ 6.6613381e-15]))
>>> val3, err3
(array([ 2.]), array([ 1.7763568e-15]))
Integrate humps from 0 to 2 and from 1 to 4
>>> val4, err4 = gaussq(humps,A,B)
See also
--------
qrule
gaussq2d
'''
global _POINTS_AND_WEIGHTS
max_iter = 11
gn = 2
if not hasattr(fun, '__call__'):
raise ValueError('Function must be callable')
A, B = np.atleast_1d(a, b)
a_shape = np.atleast_1d(A.shape)
b_shape = np.atleast_1d(B.shape)
if np.prod(a_shape) == 1: # make sure the integration limits have correct size
A = A * ones(b_shape)
a_shape = b_shape
elif np.prod(b_shape) == 1:
B = B * ones(a_shape)
elif any(a_shape != b_shape):
raise ValueError('The integration limits must have equal size!')
if args is None:
num_parameters = 0
else:
num_parameters = len(args)
P0 = copy.deepcopy(args)
isvector1 = zeros(num_parameters)
nk = np.prod(a_shape) #% # of integrals we have to compute
for ix in xrange(num_parameters):
if is_numlike(P0[ix]):
p0_shape = np.shape(P0[ix])
Np0 = np.prod(p0_shape)
isvector1[ix] = (Np0 > 1)
if isvector1[ix]:
if nk == 1:
a_shape = p0_shape
nk = Np0
A = A * ones(a_shape)
B = B * ones(a_shape)
elif nk != Np0:
raise ValueError('The input must have equal size!')
P0[ix].shape = (-1, 1) # make sure it is a column
k = np.arange(nk)
val = zeros(nk)
val_old = zeros(nk)
abserr = zeros(nk)
#setup mapping parameters
A.shape = (-1, 1)
B.shape = (-1, 1)
jacob = (B - A) / 2
shift = 1
if wfun == 1:# Gauss-legendre
dx = jacob
elif wfun == 2 or wfun == 3:
shift = 0
jacob = ones((nk, 1))
A = zeros((nk, 1))
dx = jacob
elif wfun == 4:
dx = jacob ** (alpha + beta + 1)
elif wfun == 5:
dx = ones((nk, 1))
elif wfun == 6:
dx = jacob ** 2
elif wfun == 7:
shift = 0
jacob = jacob * 2
dx = jacob
elif wfun == 8:
shift = 0
jacob = jacob * 2
dx = sqrt(jacob)
elif wfun == 9:
shift = 0
jacob = jacob * 2
dx = sqrt(jacob) ** 3
else:
raise ValueError('unknown option')
dx = dx.ravel()
if trace:
x_trace = [0, ]*max_iter
y_trace = [0, ]*max_iter
if num_parameters > 0:
ix_vec, = np.where(isvector1)
if len(ix_vec):
P1 = copy.copy(P0)
#% Break out of the iteration loop for three reasons:
#% 1) the last update is very small (compared to int and compared to reltol)
#% 2) There are more than 11 iterations. This should NEVER happen.
for ix in xrange(max_iter):
x_and_w = 'wfun%d_%d_%g_%g' % (wfun, gn, alpha, beta)
if x_and_w in _POINTS_AND_WEIGHTS:
xn, w = _POINTS_AND_WEIGHTS[x_and_w]
else:
xn, w = qrule(gn, wfun, alpha, beta)
_POINTS_AND_WEIGHTS[x_and_w] = (xn, w)
# calculate the x values
x = (xn + shift) * jacob[k, :] + A[k, :]
# calculate function values y=fun(x,p1,p2,....,pn)
if num_parameters > 0:
if len(ix_vec):
#% Expand vector to the correct size
for iy in ix_vec:
P1[iy] = P0[iy][k, :]
y = fun(x, **P1)
else:
y = fun(x, **P0)
else:
y = fun(x)
val[k] = np.sum(w * y, axis=1) * dx[k] # do the integration sum(y.*w)
if trace:
x_trace.append(x.ravel())
y_trace.append(y.ravel())
hfig = plt.plot(x, y, 'r.')
#hold on
#drawnow,shg
#if trace>1:
# pause
plt.setp(hfig, 'color', 'b')
abserr[k] = abs(val_old[k] - val[k]) #absolute tolerance
if ix > 1:
k, = np.where(abserr > np.maximum(abs(reltol * val), abstol)) # abserr > abs(abstol))%indices to integrals which did not converge
nk = len(k)# of integrals we have to compute again
if nk :
val_old[k] = val[k]
else:
break
gn *= 2 #double the # of basepoints and weights
else:
if nk > 1:
if (nk == np.prod(a_shape)):
tmptxt = 'All integrals did not converge--singularities likely!'
else:
tmptxt = '%d integrals did not converge--singularities likely!' % (nk,)
else:
tmptxt = 'Integral did not converge--singularity likely!'
warnings.warn(tmptxt)
val.shape = a_shape # make sure int is the same size as the integration limits
abserr.shape = a_shape
if trace > 0:
plt.clf()
plt.plot(np.hstack(x_trace), np.hstack(y_trace), '+')
return val, abserr
import scipy.signal as ss
import wafo.data as wd
import wafo.misc as wm
import wafo.objects as wo
import wafo.stats as ws
import wafo.spectrum.models as wsm
xx = wd.sea()
xx[:, 1] = ss.detrend(xx[:, 1])
ts = wo.mat2timeseries(xx)
Tcrcr, ix = ts.wave_periods(vh=0, pdef='c2c', wdef='tw', rate=8)
Tc, ixc = ts.wave_periods(vh=0, pdef='u2d', wdef='tw', rate=8)
#! Histogram of crestperiod compared to the kernel density estimate
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
import wafo.kdetools as wk
plt.clf()
print(Tc.mean())
print(Tc.max())
t = np.linspace(0.01, 8, 200)
ftc = wk.TKDE(Tc, L2=0, inc=128)
plt.plot(t, ftc.eval_grid(t), t, ftc.eval_grid_fast(t), '-.')
wm.plot_histgrm(Tc, normed=True)
plt.title('Kernel Density Estimates')
plt.xlabel('Tc [s]')
plt.axis([0, 8, 0, 0.5])
plt.show()
#! Extreme waves - model check: the highest and steepest wave
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
def _plot_final_trace(self):
if self.trace > 0:
plt.clf()
plt.plot(np.hstack(self.x_trace), np.hstack(self.y_trace), '+')
