def _plot_error(neval, err_dic, plot_error):
if plot_error:
plt.figure(0)
for name in err_dic:
plt.loglog(neval, err_dic[name], label=name)
plt.xlabel('number of function evaluations')
plt.ylabel('error')
plt.legend()
def plotecdf(self, symb1="r-", symb2="b."):
""" Plot Empirical and fitted Cumulative Distribution Function
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution.
If so the empirical CDF should resemble the model CDF.
Other distribution types will introduce deviations in the plot.
"""
n = len(self.data)
F = (arange(1, n + 1)) / n
plotbackend.plot(self.data, F, symb2, self.data, self.cdf(self.data), symb1)
plotbackend.xlabel("x")
plotbackend.ylabel("F(x) (%s)" % self.dist.name)
plotbackend.title("Empirical CDF plot")
def plotesf(self, symb1="r-", symb2="b."):
""" Plot Empirical and fitted Survival Function
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution.
If so the empirical CDF should resemble the model CDF.
Other distribution types will introduce deviations in the plot.
"""
n = len(self.data)
SF = (arange(n, 0, -1)) / n
plotbackend.semilogy(self.data, SF, symb2, self.data, self.sf(self.data), symb1)
# plotbackend.plot(self.data,SF,'b.',self.data,self.sf(self.data),'r-')
plotbackend.xlabel("x")
plotbackend.ylabel("F(x) (%s)" % self.dist.name)
plotbackend.title("Empirical SF plot")
def plotecdf(self, symb1='r-', symb2='b.'):
''' Plot Empirical and fitted Cumulative Distribution Function
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution.
If so the empirical CDF should resemble the model CDF.
Other distribution types will introduce deviations in the plot.
'''
n = len(self.data)
F = (arange(1, n + 1)) / n
plotbackend.plot(self.data, F, symb2,
self.data, self.cdf(self.data), symb1)
plotbackend.xlabel('x')
plotbackend.ylabel('F(x) (%s)' % self.dist.name)
plotbackend.title('Empirical CDF plot')
def plotesf(self, symb1='r-', symb2='b.'):
''' Plot Empirical and fitted Survival Function
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution.
If so the empirical CDF should resemble the model CDF.
Other distribution types will introduce deviations in the plot.
'''
n = len(self.data)
SF = (arange(n, 0, -1)) / n
plotbackend.semilogy(
self.data, SF, symb2, self.data, self.sf(self.data), symb1)
# plotbackend.plot(self.data,SF,'b.',self.data,self.sf(self.data),'r-')
plotbackend.xlabel('x')
plotbackend.ylabel('F(x) (%s)' % self.dist.name)
plotbackend.title('Empirical SF plot')
def plotresq(self, symb1='r-', symb2='b.'):
'''PLOTRESQ displays a residual quantile plot.
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution. If so the
plot will be linear. Other distribution types will introduce
curvature in the plot.
'''
n = len(self.data)
eprob = (arange(1, n + 1) - 0.5) / n
y = self.ppf(eprob)
y1 = self.data[[0, -1]]
plotbackend.plot(self.data, y, symb2, y1, y1, symb1)
plotbackend.xlabel('Empirical')
plotbackend.ylabel('Model (%s)' % self.dist.name)
plotbackend.title('Residual Quantile Plot')
plotbackend.axis('tight')
plotbackend.axis('equal')
def plotepdf(self, symb1="r-", symb2="b-"):
"""Plot Empirical and fitted Probability Density Function
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution.
If so the histogram should resemble the model density.
Other distribution types will introduce deviations in the plot.
"""
x, pdf = self._get_empirical_pdf()
ymax = pdf.max()
# plotbackend.hist(self.data,normed=True,fill=False)
plotbackend.plot(self.data, self.pdf(self.data), symb1, x, pdf, symb2)
ax = list(plotbackend.axis())
ax[3] = min(ymax * 1.3, ax[3])
plotbackend.axis(ax)
plotbackend.xlabel("x")
plotbackend.ylabel("f(x) (%s)" % self.dist.name)
plotbackend.title("Density plot")
def plotresq(self, symb1="r-", symb2="b."):
"""PLOTRESQ displays a residual quantile plot.
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution. If so the
plot will be linear. Other distribution types will introduce
curvature in the plot.
"""
n = len(self.data)
eprob = (arange(1, n + 1) - 0.5) / n
y = self.ppf(eprob)
y1 = self.data[[0, -1]]
plotbackend.plot(self.data, y, symb2, y1, y1, symb1)
plotbackend.xlabel("Empirical")
plotbackend.ylabel("Model (%s)" % self.dist.name)
plotbackend.title("Residual Quantile Plot")
plotbackend.axis("tight")
plotbackend.axis("equal")
def plotepdf(self, symb1='r-', symb2='b-'):
'''Plot Empirical and fitted Probability Density Function
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution.
If so the histogram should resemble the model density.
Other distribution types will introduce deviations in the plot.
'''
x, pdf = self._get_empirical_pdf()
ymax = pdf.max()
# plotbackend.hist(self.data,normed=True,fill=False)
plotbackend.plot(self.data, self.pdf(self.data), symb1,
x, pdf, symb2)
ax = list(plotbackend.axis())
ax[3] = min(ymax * 1.3, ax[3])
plotbackend.axis(ax)
plotbackend.xlabel('x')
plotbackend.ylabel('f(x) (%s)' % self.dist.name)
plotbackend.title('Density plot')
def plotresprb(self, symb1="r-", symb2="b."):
""" PLOTRESPRB displays a residual probability plot.
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution. If so the
plot will be linear. Other distribution types will introduce curvature
in the plot.
"""
n = len(self.data)
# ecdf = (0.5:n-0.5)/n;
ecdf = arange(1, n + 1) / (n + 1)
mcdf = self.cdf(self.data)
p1 = [0, 1]
plotbackend.plot(ecdf, mcdf, symb2, p1, p1, symb1)
plotbackend.xlabel("Empirical")
plotbackend.ylabel("Model (%s)" % self.dist.name)
plotbackend.title("Residual Probability Plot")
plotbackend.axis("equal")
plotbackend.axis([0, 1, 0, 1])
def plotresprb(self, symb1='r-', symb2='b.'):
''' PLOTRESPRB displays a residual probability plot.
The purpose of the plot is to graphically assess whether
the data could come from the fitted distribution. If so the
plot will be linear. Other distribution types will introduce curvature
in the plot.
'''
n = len(self.data)
# ecdf = (0.5:n-0.5)/n;
ecdf = arange(1, n + 1) / (n + 1)
mcdf = self.cdf(self.data)
p1 = [0, 1]
plotbackend.plot(ecdf, mcdf, symb2,
p1, p1, symb1)
plotbackend.xlabel('Empirical')
plotbackend.ylabel('Model (%s)' % self.dist.name)
plotbackend.title('Residual Probability Plot')
plotbackend.axis('equal')
plotbackend.axis([0, 1, 0, 1])
def demo_tide_filter():
"""
Examples
--------
>>> demo_tide_filter()
>>> plt.close()
"""
# import statsmodels.api as sa
import wafo.spectrum.models as sm
sd = 10
Sj = sm.Jonswap(Hm0=4. * sd)
S = Sj.tospecdata()
q = (0.1 * sd)**2 # variance of process noise s the car operates
r = (100 * sd)**2 # variance of measurement error
b = 0 # no system input
u = 0 # no system input
from scipy.signal import butter, filtfilt, lfilter_zi # lfilter,
freq_tide = 1. / (12 * 60 * 60)
freq_wave = 1. / 10
freq_filt = freq_wave / 10
dt = 1.
freq = 1. / dt
fn = (freq / 2)
P = 10 * np.diag([1, 0.01])
R = r
H = np.atleast_2d([1, 0])
F = np.atleast_2d([[0, 1], [0, 0]])
A, Q = lti_disc(F, L=None, Q=np.diag([0, q]), dt=dt)
t = np.arange(0, 60 * 12, 1. / freq)
w = 2 * np.pi * freq # 1 Hz
tide = 100 * np.sin(freq_tide * w * t + 2 * np.pi / 4) + 100
y = tide + S.sim(len(t), dt=1. / freq)[:, 1].ravel()
# lowess = sa.nonparametric.lowess
# y2 = lowess(y, t, frac=0.5)[:,1]
filt = Kalman(R=R, x=np.array([[tide[0]], [0]]), P=P, A=A, Q=Q, H=H, B=b)
filt2 = Kalman(R=R, x=np.array([[tide[0]], [0]]), P=P, A=A, Q=Q, H=H, B=b)
# y = tide + 0.5 * np.sin(freq_wave * w * t)
# Butterworth filter
b, a = butter(9, (freq_filt / fn), btype='low')
# y2 = [lowess(y[max(i-60,0):i + 1], t[max(i-60,0):i + 1], frac=.3)[-1,1]
# for i in range(len(y))]
# y2 = [lfilter(b, a, y[:i + 1])[i] for i in range(len(y))]
# y3 = filtfilt(b, a, y[:16]).tolist() + [filtfilt(b, a, y[:i + 1])[i]
# for i in range(16, len(y))]
# y0 = medfilt(y, 41)
_zi = lfilter_zi(b, a)
# y2 = lfilter(b, a, y)#, zi=y[0]*zi) # standard filter
y3 = filtfilt(b, a, y) # filter with phase shift correction
y4 = []
y5 = []
for _i, j in enumerate(y):
tmp = np.ravel(filt(j, u=u))
tmp = np.ravel(filt2(tmp[0], u=u))
# if i==0:
# print(filt.x)
# print(filt2.x)
y4.append(tmp[0])
y5.append(tmp[1])
_y0 = medfilt(y4, 41)
# print(filt.P)
# plot
plt.plot(t, y, 'r.-', linewidth=2, label='raw data')
# plt.plot(t, y2, 'b.-', linewidth=2, label='lowess @ %g Hz' % freq_filt)
# plt.plot(t, y2, 'b.-', linewidth=2, label='filter @ %g Hz' % freq_filt)
plt.plot(t, y3, 'g.-', linewidth=2, label='filtfilt @ %g Hz' % freq_filt)
plt.plot(t, y4, 'k.-', linewidth=2, label='kalman')
# plt.plot(t, y5, 'k.', linewidth=2, label='kalman2')
plt.plot(t, tide, 'y-', linewidth=2, label='True tide')
plt.legend(frameon=False, fontsize=14)
plt.xlabel("Time [s]")
plt.ylabel("Amplitude")
def qdemo(f, a, b, kmax=9, plot_error=False):
'''
Compares different quadrature rules.
Parameters
----------
f : callable
function
a,b : scalars
lower and upper integration limits
Details
-------
qdemo(f,a,b) computes and compares various approximations to
the integral of f from a to b. Three approximations are used,
the composite trapezoid, Simpson's, and Boole's rules, all with
equal length subintervals.
In a case like qdemo(exp,0,3) one can see the expected
convergence rates for each of the three methods.
In a case like qdemo(sqrt,0,3), the convergence rate is limited
not by the method, but by the singularity of the integrand.
Example
-------
>>> import numpy as np
>>> qdemo(np.exp,0,3)
true value = 19.08553692
ftn, Boole, Chebychev
evals approx error approx error
3, 19.4008539142, 0.3153169910, 19.5061466023, 0.4206096791
5, 19.0910191534, 0.0054822302, 19.0910191534, 0.0054822302
9, 19.0856414320, 0.0001045088, 19.0855374134, 0.0000004902
17, 19.0855386464, 0.0000017232, 19.0855369232, 0.0000000000
33, 19.0855369505, 0.0000000273, 19.0855369232, 0.0000000000
65, 19.0855369236, 0.0000000004, 19.0855369232, 0.0000000000
129, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
257, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
513, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
ftn, Clenshaw-Curtis, Gauss-Legendre
evals approx error approx error
3, 19.5061466023, 0.4206096791, 19.0803304585, 0.0052064647
5, 19.0834145766, 0.0021223465, 19.0855365951, 0.0000003281
9, 19.0855369150, 0.0000000082, 19.0855369232, 0.0000000000
17, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
33, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
65, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
129, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
257, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
513, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
ftn, Simps, Trapz
evals approx error approx error
3, 19.5061466023, 0.4206096791, 22.5366862979, 3.4511493747
5, 19.1169646189, 0.0314276957, 19.9718950387, 0.8863581155
9, 19.0875991312, 0.0020622080, 19.3086731081, 0.2231361849
17, 19.0856674267, 0.0001305035, 19.1414188470, 0.0558819239
33, 19.0855451052, 0.0000081821, 19.0995135407, 0.0139766175
65, 19.0855374350, 0.0000005118, 19.0890314614, 0.0034945382
129, 19.0855369552, 0.0000000320, 19.0864105817, 0.0008736585
257, 19.0855369252, 0.0000000020, 19.0857553393, 0.0002184161
513, 19.0855369233, 0.0000000001, 19.0855915273, 0.0000546041
'''
true_val, _tol = intg.quad(f, a, b)
print('true value = %12.8f' % (true_val,))
neval = zeros(kmax, dtype=int)
vals_dic = {}
err_dic = {}
# try various approximations
methods = [trapz, simps, boole, ]
for k in xrange(kmax):
n = 2 ** (k + 1) + 1
neval[k] = n
x = np.linspace(a, b, n)
y = f(x)
for method in methods:
name = method.__name__.title()
q = method(y, x)
vals_dic.setdefault(name, []).append(q)
err_dic.setdefault(name, []).append(abs(q - true_val))
name = 'Clenshaw-Curtis'
q, _ec3 = clencurt(f, a, b, (n - 1) / 2)
vals_dic.setdefault(name, []).append(q[0])
err_dic.setdefault(name, []).append(abs(q[0] - true_val))
name = 'Chebychev'
ck = np.polynomial.chebyshev.chebfit(x, y, deg=min(n-1, 36))
cki = np.polynomial.chebyshev.chebint(ck)
q = np.polynomial.chebyshev.chebval(x[-1], cki)
vals_dic.setdefault(name, []).append(q)
err_dic.setdefault(name, []).append(abs(q - true_val))
# ck = chebfit(f,n,a,b)
# q = chebval(b,chebint(ck,a,b),a,b)
# qc2[k] = q; ec2[k] = abs(q - true)
name = 'Gauss-Legendre' # quadrature
q = intg.fixed_quad(f, a, b, n=n)[0]
# [x, w]=qrule(n,1)
# x = (b-a)/2*x + (a+b)/2 % Transform base points X.
# w = (b-a)/2*w % Adjust weigths.
# q = sum(feval(f,x)*w)
vals_dic.setdefault(name, []).append(q)
err_dic.setdefault(name, []).append(abs(q - true_val))
# display results
names = sorted(vals_dic.keys())
num_cols = 2
formats = ['%4.0f, ', ] + ['%10.10f, ', ] * num_cols * 2
formats[-1] = formats[-1].split(',')[0]
formats_h = ['%4s, ', ] + ['%20s, ', ] * num_cols
formats_h[-1] = formats_h[-1].split(',')[0]
headers = ['evals'] + ['%12s %12s' % ('approx', 'error')] * num_cols
while len(names) > 0:
print(''.join(fi % t for fi, t in zip(formats_h,
['ftn'] + names[:num_cols])))
print(' '.join(headers))
data = [neval]
for name in names[:num_cols]:
data.append(vals_dic[name])
data.append(err_dic[name])
data = np.vstack(tuple(data)).T
for k in xrange(kmax):
tmp = data[k].tolist()
print(''.join(fi % t for fi, t in zip(formats, tmp)))
if plot_error:
plt.figure(0)
for name in names[:num_cols]:
plt.loglog(neval, err_dic[name], label=name)
names = names[num_cols:]
if plot_error:
plt.xlabel('number of function evaluations')
plt.ylabel('error')
plt.legend()
plt.show('hold')
##
# Return values in the Gumbel distribution
plt.clf()
T = np.r_[1:100000]
sT = gum[0] - gum[1] * np.log(-np.log1p(-1./T))
plt.semilogx(T, sT)
plt.hold(True)
# ws.edf(Hs).plot()
Nmax = len(Hs)
N = np.r_[1:Nmax+1]
plt.plot(Nmax/N, sorted(Hs, reverse=True), '.')
plt.title('Return values in the Gumbel model')
plt.xlabel('Return period')
plt.ylabel('Return value')
#wafostamp([],'(ER)')
plt.show()
#disp('Block = 4'),pause(pstate)
## Section 5.2 Generalized Pareto and Extreme Value distributions
## Section 5.2.1 Generalized Extreme Value distribution
# Empirical distribution of significant wave-height with estimated
# Generalized Extreme Value distribution,
gev = ws.genextreme.fit2(Hs)
gev.plotfitsummary()
# wafostamp([],'(ER)')
# disp('Block = 5a'),pause(pstate)
plt.clf()
#plot(rfc(:, 2), rfc(:, 1), '.')
#wafostamp('', '(ER)')
#hold off
#disp('Block = 9'), pause(pstate)
#! Section 1.4.5 Extreme value statistics
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Plot of yura87 data
plt.clf()
import wafo.data as wd
xn = wd.yura87()
#xn = load('yura87.dat');
plt.subplot(211)
plt.plot(xn[::30, 0] / 3600, xn[::30, 1], '.')
plt.title('Water level')
plt.ylabel('(m)')
#! Formation of 5 min maxima
yura = xn[:85500, 1]
yura = np.reshape(yura, (285, 300)).T
maxyura = yura.max(axis=0)
plt.subplot(212)
plt.plot(xn[299:85500:300, 0] / 3600, maxyura, '.')
plt.xlabel('Time (h)')
plt.ylabel('(m)')
plt.title('Maximum 5 min water level')
plt.show()
#! Estimation of GEV for yuramax
plt.clf()
import wafo.stats as ws
#plot(rfc(:, 2), rfc(:, 1), '.')
#wafostamp('', '(ER)')
#hold off
#disp('Block = 9'), pause(pstate)
#! Section 1.4.5 Extreme value statistics
#!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Plot of yura87 data
plt.clf()
import wafo.data as wd
xn = wd.yura87()
#xn = load('yura87.dat');
plt.subplot(211)
plt.plot(xn[::30, 0] / 3600, xn[::30, 1], '.')
plt.title('Water level')
plt.ylabel('(m)')
#! Formation of 5 min maxima
yura = xn[:85500, 1]
yura = np.reshape(yura, (285, 300)).T
maxyura = yura.max(axis=0)
plt.subplot(212)
plt.plot(xn[299:85500:300, 0] / 3600, maxyura, '.')
plt.xlabel('Time (h)')
plt.ylabel('(m)')
plt.title('Maximum 5 min water level')
plt.show()
#! Estimation of GEV for yuramax
plt.clf()
import wafo.stats as ws
def qdemo(f, a, b):
'''
Compares different quadrature rules.
Parameters
----------
f : callable
function
a,b : scalars
lower and upper integration limits
Details
-------
qdemo(f,a,b) computes and compares various approximations to
the integral of f from a to b. Three approximations are used,
the composite trapezoid, Simpson's, and Boole's rules, all with
equal length subintervals.
In a case like qdemo(exp,0,3) one can see the expected
convergence rates for each of the three methods.
In a case like qdemo(sqrt,0,3), the convergence rate is limited
not by the method, but by the singularity of the integrand.
Example
-------
>>> import numpy as np
>>> qdemo(np.exp,0,3)
true value = 19.08553692
ftn Trapezoid Simpsons Booles
evals approx error approx error approx error
3, 22.5366862979, 3.4511493747, 19.5061466023, 0.4206096791, 19.4008539142, 0.3153169910
5, 19.9718950387, 0.8863581155, 19.1169646189, 0.0314276957, 19.0910191534, 0.0054822302
9, 19.3086731081, 0.2231361849, 19.0875991312, 0.0020622080, 19.0856414320, 0.0001045088
17, 19.1414188470, 0.0558819239, 19.0856674267, 0.0001305035, 19.0855386464, 0.0000017232
33, 19.0995135407, 0.0139766175, 19.0855451052, 0.0000081821, 19.0855369505, 0.0000000273
65, 19.0890314614, 0.0034945382, 19.0855374350, 0.0000005118, 19.0855369236, 0.0000000004
129, 19.0864105817, 0.0008736585, 19.0855369552, 0.0000000320, 19.0855369232, 0.0000000000
257, 19.0857553393, 0.0002184161, 19.0855369252, 0.0000000020, 19.0855369232, 0.0000000000
513, 19.0855915273, 0.0000546041, 19.0855369233, 0.0000000001, 19.0855369232, 0.0000000000
ftn Clenshaw Chebychev Gauss-L
evals approx error approx error approx error
3, 19.5061466023, 0.4206096791, 0.0000000000, 1.0000000000, 19.0803304585, 0.0052064647
5, 19.0834145766, 0.0021223465, 0.0000000000, 1.0000000000, 19.0855365951, 0.0000003281
9, 19.0855369150, 0.0000000082, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
17, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
33, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
65, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
129, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
257, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
513, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
'''
# use quad8 with small tolerance to get "true" value
#true1 = quad8(f,a,b,1e-10)
#[true tol]= gaussq(f,a,b,1e-12)
#[true tol] = agakron(f,a,b,1e-13)
true_val, _tol = intg.quad(f, a, b)
print('true value = %12.8f' % (true_val,))
kmax = 9
neval = zeros(kmax, dtype=int)
qt = zeros(kmax)
qs = zeros(kmax)
qb = zeros(kmax)
qc = zeros(kmax)
qc2 = zeros(kmax)
qg = zeros(kmax)
et = ones(kmax)
es = ones(kmax)
eb = ones(kmax)
ec = ones(kmax)
ec2 = ones(kmax)
ec3 = ones(kmax)
eg = ones(kmax)
# try various approximations
for k in xrange(kmax):
n = 2 ** (k + 1) + 1
neval[k] = n
h = (b - a) / (n - 1)
x = np.linspace(a, b, n)
y = f(x)
# trapezoid approximation
q = np.trapz(y, x)
#h*( (y(1)+y(n))/2 + sum(y(2:n-1)) )
qt[k] = q
et[k] = abs(q - true_val)
# Simpson approximation
q = intg.simps(y, x)
#(h/3)*( y(1)+y(n) + 4*sum(y(2:2:n-1)) + 2*sum(y(3:2:n-2)) )
qs[k] = q
es[k] = abs(q - true_val)
# Boole's rule
#q = boole(x,y)
q = (2 * h / 45) * (7 * (y[0] + y[-1]) + 12 * np.sum(y[2:n - 1:4])
+ 32 * np.sum(y[1:n - 1:2]) + 14 * np.sum(y[4:n - 3:4]))
qb[k] = q
eb[k] = abs(q - true_val)
# Clenshaw-Curtis
[q, ec3[k]] = clencurt(f, a, b, (n - 1) / 2)
qc[k] = q
ec[k] = abs(q - true_val)
# Chebychev
#ck = chebfit(f,n,a,b)
#q = chebval(b,chebint(ck,a,b),a,b)
#qc2[k] = q; ec2[k] = abs(q - true)
# Gauss-Legendre quadrature
q = intg.fixed_quad(f, a, b, n=n)[0]
#[x, w]=qrule(n,1)
#x = (b-a)/2*x + (a+b)/2 % Transform base points X.
#w = (b-a)/2*w % Adjust weigths.
#q = sum(feval(f,x)*w)
qg[k] = q
eg[k] = abs(q - true_val)
#% display results
formats = ['%4.0f, ', ] + ['%10.10f, ', ]*6
formats[-1] = formats[-1].split(',')[0]
data = np.vstack((neval, qt, et, qs, es, qb, eb)).T
print(' ftn Trapezoid Simpson''s Boole''s')
print('evals approx error approx error approx error')
for k in xrange(kmax):
tmp = data[k].tolist()
print(''.join(fi % t for fi, t in zip(formats, tmp)))
# display results
data = np.vstack((neval, qc, ec, qc2, ec2, qg, eg)).T
print(' ftn Clenshaw Chebychev Gauss-L')
print('evals approx error approx error approx error')
for k in xrange(kmax):
tmp = data[k].tolist()
print(''.join(fi % t for fi, t in zip(formats, tmp)))
plt.loglog(neval, np.vstack((et, es, eb, ec, ec2, eg)).T)
plt.xlabel('number of function evaluations')
plt.ylabel('error')
plt.legend(('Trapezoid', 'Simpsons', 'Booles', 'Clenshaw', 'Chebychev', 'Gauss-L'))
##
# Return values in the Gumbel distribution
plt.clf()
T = np.r_[1:100000]
sT = gum[0] - gum[1] * np.log(-np.log1p(-1. / T))
plt.semilogx(T, sT)
plt.hold(True)
# ws.edf(Hs).plot()
Nmax = len(Hs)
N = np.r_[1:Nmax + 1]
plt.plot(Nmax / N, sorted(Hs, reverse=True), '.')
plt.title('Return values in the Gumbel model')
plt.xlabel('Return period')
plt.ylabel('Return value')
#wafostamp([],'(ER)')
plt.show()
#disp('Block = 4'),pause(pstate)
## Section 5.2 Generalized Pareto and Extreme Value distributions
## Section 5.2.1 Generalized Extreme Value distribution
# Empirical distribution of significant wave-height with estimated
# Generalized Extreme Value distribution,
gev = ws.genextreme.fit2(Hs)
gev.plotfitsummary()
# wafostamp([],'(ER)')
# disp('Block = 5a'),pause(pstate)
plt.clf()
def test_tide_filter():
# import statsmodels.api as sa
import wafo.spectrum.models as sm
sd = 10
Sj = sm.Jonswap(Hm0=4.*sd)
S = Sj.tospecdata()
q = (0.1 * sd) ** 2 # variance of process noise s the car operates
r = (100 * sd) ** 2 # variance of measurement error
b = 0 # no system input
u = 0 # no system input
from scipy.signal import butter, filtfilt, lfilter_zi # lfilter,
freq_tide = 1. / (12 * 60 * 60)
freq_wave = 1. / 10
freq_filt = freq_wave / 10
dt = 1.
freq = 1. / dt
fn = (freq / 2)
P = 10 * np.diag([1, 0.01])
R = r
H = np.atleast_2d([1, 0])
F = np.atleast_2d([[0, 1],
[0, 0]])
A, Q = lti_disc(F, L=None, Q=np.diag([0, q]), dt=dt)
t = np.arange(0, 60 * 12, 1. / freq)
w = 2 * np.pi * freq # 1 Hz
tide = 100 * np.sin(freq_tide * w * t + 2 * np.pi / 4) + 100
y = tide + S.sim(len(t), dt=1. / freq)[:, 1].ravel()
# lowess = sa.nonparametric.lowess
# y2 = lowess(y, t, frac=0.5)[:,1]
filt = Kalman(R=R, x=np.array([[tide[0]], [0]]), P=P, A=A, Q=Q, H=H, B=b)
filt2 = Kalman(R=R, x=np.array([[tide[0]], [0]]), P=P, A=A, Q=Q, H=H, B=b)
# y = tide + 0.5 * np.sin(freq_wave * w * t)
# Butterworth filter
b, a = butter(9, (freq_filt / fn), btype='low')
# y2 = [lowess(y[max(i-60,0):i + 1], t[max(i-60,0):i + 1], frac=.3)[-1,1]
# for i in range(len(y))]
# y2 = [lfilter(b, a, y[:i + 1])[i] for i in range(len(y))]
# y3 = filtfilt(b, a, y[:16]).tolist() + [filtfilt(b, a, y[:i + 1])[i]
# for i in range(16, len(y))]
# y0 = medfilt(y, 41)
_zi = lfilter_zi(b, a)
# y2 = lfilter(b, a, y)#, zi=y[0]*zi) # standard filter
y3 = filtfilt(b, a, y) # filter with phase shift correction
y4 = []
y5 = []
for _i, j in enumerate(y):
tmp = filt(j, u=u).ravel()
tmp = filt2(tmp[0], u=u).ravel()
# if i==0:
# print(filt.x)
# print(filt2.x)
y4.append(tmp[0])
y5.append(tmp[1])
_y0 = medfilt(y4, 41)
print(filt.P)
# plot
plt.plot(t, y, 'r.-', linewidth=2, label='raw data')
# plt.plot(t, y2, 'b.-', linewidth=2, label='lowess @ %g Hz' % freq_filt)
# plt.plot(t, y2, 'b.-', linewidth=2, label='filter @ %g Hz' % freq_filt)
plt.plot(t, y3, 'g.-', linewidth=2, label='filtfilt @ %g Hz' % freq_filt)
plt.plot(t, y4, 'k.-', linewidth=2, label='kalman')
# plt.plot(t, y5, 'k.', linewidth=2, label='kalman2')
plt.plot(t, tide, 'y-', linewidth=2, label='True tide')
plt.legend(frameon=False, fontsize=14)
plt.xlabel("Time [s]")
plt.ylabel("Amplitude")
plt.show('hold')