def demo_kalman_voltimeter():
"""
Examples
--------
>>> demo_kalman_voltimeter()
>>> plt.close()
"""
V0 = 12
h = np.atleast_2d(1) # voltimeter measure the voltage itself
q = 1e-9 # variance of process noise as the car operates
r = 0.05**2 # variance of measurement error
b = 0 # no system input
u = 0 # no system input
filt = Kalman(R=r, A=1, Q=q, H=h, B=b)
# Generate random voltages and watch the filter operate.
n = 50
truth = np.random.randn(n) * np.sqrt(q) + V0
z = truth + np.random.randn(n) * np.sqrt(r) # measurement
x = np.zeros(n)
for i, zi in enumerate(z):
x[i] = filt(zi, u) # perform a Kalman filter iteration
_hz = plt.plot(z, 'r.', label='observations')
# a-posteriori state estimates:
_hx = plt.plot(x, 'b-', label='Kalman output')
_ht = plt.plot(truth, 'g-', label='true voltage')
plt.legend()
plt.title('Automobile Voltimeter Example')
def kde_demo4(N=50):
"""Demonstrate that the improved Sheather-Jones plug-in (hisj) is superior
for 1D multimodal distributions
KDEDEMO4 shows that the improved Sheather-Jones plug-in smoothing is a
better compared to normal reference rules (in this case the hns)
Examples
--------
>>> kde_demo4()
"""
data = np.hstack((st.norm.rvs(loc=5, scale=1, size=(N,)),
st.norm.rvs(loc=-5, scale=1, size=(N,))))
# x = np.linspace(1.5e-3, 5, 55)
kde = KDE(data, kernel=Kernel('gauss', 'hns'))
f = kde(output='plot', title='Ordinary KDE', plotflag=1)
kde1 = KDE(data, kernel=Kernel('gauss', 'hisj'))
f1 = kde1(output='plot', label='Ordinary KDE', plotflag=1)
plt.figure(0)
f.plot('r', label='hns={0}'.format(kde.hs))
# plt.figure(2)
f1.plot('b', label='hisj={0}'.format(kde1.hs))
x = np.linspace(-9, 9)
plt.plot(x, (st.norm.pdf(x, loc=-5, scale=1) +
st.norm.pdf(x, loc=5, scale=1)) / 2, 'k:',
label='True density')
plt.legend()
def test_kalman():
V0 = 12
h = np.atleast_2d(1) # voltimeter measure the voltage itself
q = 1e-9 # variance of process noise as the car operates
r = 0.05 ** 2 # variance of measurement error
b = 0 # no system input
u = 0 # no system input
filt = Kalman(R=r, A=1, Q=q, H=h, B=b)
# Generate random voltages and watch the filter operate.
n = 50
truth = np.random.randn(n) * np.sqrt(q) + V0
z = truth + np.random.randn(n) * np.sqrt(r) # measurement
x = np.zeros(n)
for i, zi in enumerate(z):
x[i] = filt(zi, u) # perform a Kalman filter iteration
_hz = plt.plot(z, 'r.', label='observations')
# a-posteriori state estimates:
_hx = plt.plot(x, 'b-', label='Kalman output')
_ht = plt.plot(truth, 'g-', label='true voltage')
plt.legend()
plt.title('Automobile Voltimeter Example')
plt.show('hold')
def kreg_demo1(hs=None, fast=True, fun='hisj'):
"""Compare KRegression to KernelReg from statsmodels.nonparametric
Examples
--------
>>> kreg_demo1()
"""
N = 100
# ei = np.random.normal(loc=0, scale=0.075, size=(N,))
ei = np.array([
-0.08508516, 0.10462496, 0.07694448, -0.03080661, 0.05777525,
0.06096313, -0.16572389, 0.01838912, -0.06251845, -0.09186784,
-0.04304887, -0.13365788, -0.0185279, -0.07289167, 0.02319097,
0.06887854, -0.08938374, -0.15181813, 0.03307712, 0.08523183,
-0.0378058, -0.06312874, 0.01485772, 0.06307944, -0.0632959,
0.18963205, 0.0369126, -0.01485447, 0.04037722, 0.0085057,
-0.06912903, 0.02073998, 0.1174351, 0.17599277, -0.06842139,
0.12587608, 0.07698113, -0.0032394, -0.12045792, -0.03132877,
0.05047314, 0.02013453, 0.04080741, 0.00158392, 0.10237899,
-0.09069682, 0.09242174, -0.15445323, 0.09190278, 0.07138498,
0.03002497, 0.02495252, 0.01286942, 0.06449978, 0.03031802,
0.11754861, -0.02322272, 0.00455867, -0.02132251, 0.09119446,
-0.03210086, -0.06509545, 0.07306443, 0.04330647, 0.078111,
-0.04146907, 0.05705476, 0.02492201, -0.03200572, -0.02859788,
-0.05893749, 0.00089538, 0.0432551, 0.04001474, 0.04888828,
-0.17708392, 0.16478644, 0.1171006, 0.11664846, 0.01410477,
-0.12458953, -0.11692081, 0.0413047, -0.09292439, -0.07042327,
0.14119701, -0.05114335, 0.04994696, -0.09520663, 0.04829406,
-0.01603065, -0.1933216, 0.19352763, 0.11819496, 0.04567619,
-0.08348306, 0.00812816, -0.00908206, 0.14528945, 0.02901065])
x = np.linspace(0, 1, N)
va_1 = 0.3 ** 2
va_2 = 0.7 ** 2
y0 = np.exp(-x ** 2 / (2 * va_1)) + 1.3 * np.exp(-(x - 1) ** 2 / (2 * va_2))
y = y0 + ei
kernel = Kernel('gauss', fun=fun)
hopt = kernel.hisj(x)
kreg = KRegression(
x, y, p=0, hs=hs, kernel=kernel, xmin=-2 * hopt, xmax=1 + 2 * hopt)
if fast:
kreg.__call__ = kreg.eval_grid_fast
f = kreg(x, output='plot', title='Kernel regression', plotflag=1)
plt.figure(0)
f.plot(label='p=0')
kreg.p = 1
f1 = kreg(x, output='plot', title='Kernel regression', plotflag=1)
f1.plot(label='p=1')
# print(f1.data)
plt.plot(x, y, '.', label='data')
plt.plot(x, y0, 'k', label='True model')
from statsmodels.nonparametric.kernel_regression import KernelReg
kreg2 = KernelReg(y, x, ('c'))
y2 = kreg2.fit(x)
plt.plot(x, y2[0], 'm', label='statsmodel')
plt.legend()
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 demo_kalman_sine():
"""Kalman Filter demonstration with sine signal.
Examples
--------
>>> demo_kalman_sine()
>>> plt.close()
"""
sd = 0.5
dt = 0.1
w = 1
T = np.arange(0, 30 + dt / 2, dt)
n = len(T)
X = 3 * np.sin(w * T)
Y = X + sd * np.random.randn(n)
''' Initialize KF to values
x = 0
dx/dt = 0
with great uncertainty in derivative
'''
M = np.zeros((2, 1))
P = np.diag([0.1, 2])
R = sd**2
H = np.atleast_2d([1, 0])
q = 0.1
F = np.atleast_2d([[0, 1], [0, 0]])
A, Q = lti_disc(F, L=None, Q=np.diag([0, q]), dt=dt)
# Track and animate
m = M.shape[0]
_MM = np.zeros((m, n))
_PP = np.zeros((m, m, n))
'''In this demonstration we estimate a stationary sine signal from noisy
measurements by using the classical Kalman filter.'
'''
filt = Kalman(R=R, x=M, P=P, A=A, Q=Q, H=H, B=0)
# Generate random voltages and watch the filter operate.
# n = 50
# truth = np.random.randn(n) * np.sqrt(q) + V0
# z = truth + np.random.randn(n) * np.sqrt(r) # measurement
truth = X
z = Y
x = np.zeros((n, m))
for i, zi in enumerate(z):
x[i] = np.ravel(filt(zi, u=0))
_hz = plt.plot(z, 'r.', label='observations')
# a-posteriori state estimates:
_hx = plt.plot(x[:, 0], 'b-', label='Kalman output')
_ht = plt.plot(truth, 'g-', label='true voltage')
plt.legend()
plt.title('Automobile Voltimeter Example')
def test_kalman_sine():
"""Kalman Filter demonstration with sine signal."""
sd = 1.
dt = 0.1
w = 1
T = np.arange(0, 30 + dt / 2, dt)
n = len(T)
X = np.sin(w * T)
Y = X + sd * np.random.randn(n)
''' Initialize KF to values
x = 0
dx/dt = 0
with great uncertainty in derivative
'''
M = np.zeros((2, 1))
P = np.diag([0.1, 2])
R = sd ** 2
H = np.atleast_2d([1, 0])
q = 0.1
F = np.atleast_2d([[0, 1],
[0, 0]])
A, Q = lti_disc(F, L=None, Q=np.diag([0, q]), dt=dt)
# Track and animate
m = M.shape[0]
_MM = np.zeros((m, n))
_PP = np.zeros((m, m, n))
'''In this demonstration we estimate a stationary sine signal from noisy
measurements by using the classical Kalman filter.'
'''
filt = Kalman(R=R, x=M, P=P, A=A, Q=Q, H=H, B=0)
# Generate random voltages and watch the filter operate.
# n = 50
# truth = np.random.randn(n) * np.sqrt(q) + V0
# z = truth + np.random.randn(n) * np.sqrt(r) # measurement
truth = X
z = Y
x = np.zeros((n, m))
for i, zi in enumerate(z):
x[i] = filt(zi, u=0).ravel()
_hz = plt.plot(z, 'r.', label='observations')
# a-posteriori state estimates:
_hx = plt.plot(x[:, 0], 'b-', label='Kalman output')
_ht = plt.plot(truth, 'g-', label='true voltage')
plt.legend()
plt.title('Automobile Voltimeter Example')
plt.show()
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'))
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')
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')
