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pywafo/wafo/demo_sg.py

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import numpy as np
from scipy.sparse.linalg import expm
from scipy.signal import medfilt
from wafo.plotbackend import plotbackend as plt
from wafo.sg_filter import (SavitzkyGolay, smoothn, Kalman, HodrickPrescott,
HampelFilter)
def demo_savitzky_on_noisy_chirp():
"""
Example
-------
>>> demo_savitzky_on_noisy_chirp()
>>> plt.close()
"""
plt.figure(figsize=(7, 12))
# generate chirp signal
tvec = np.arange(0, 6.28, .02)
true_signal = np.sin(tvec * (2.0 + tvec))
true_d_signal = (2+tvec) * np.cos(tvec * (2.0 + tvec))
# add noise to signal
noise = np.random.normal(size=true_signal.shape)
signal = true_signal + .15 * noise
# plot signal
plt.subplot(311)
plt.plot(signal)
plt.title('signal')
# smooth and plot signal
plt.subplot(312)
savgol = SavitzkyGolay(n=8, degree=4)
s_signal = savgol.smooth(signal)
s2 = smoothn(signal, robust=True)
plt.plot(s_signal)
plt.plot(s2)
plt.plot(true_signal, 'r--')
plt.title('smoothed signal')
# smooth derivative of signal and plot it
plt.subplot(313)
savgol1 = SavitzkyGolay(n=8, degree=1, diff_order=1)
dt = tvec[1]-tvec[0]
d_signal = savgol1.smooth(signal) / dt
plt.plot(d_signal)
plt.plot(true_d_signal, 'r--')
plt.title('smoothed derivative of signal')
def demo_kalman_voltimeter():
"""
Example
-------
>>> 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 lti_disc(F, L=None, Q=None, dt=1):
"""LTI_DISC Discretize LTI ODE with Gaussian Noise.
Syntax:
[A,Q] = lti_disc(F,L,Qc,dt)
In:
F - NxN Feedback matrix
L - NxL Noise effect matrix (optional, default identity)
Qc - LxL Diagonal Spectral Density (optional, default zeros)
dt - Time Step (optional, default 1)
Out:
A - Transition matrix
Q - Discrete Process Covariance
Description:
Discretize LTI ODE with Gaussian Noise. The original
ODE model is in form
dx/dt = F x + L w, w ~ N(0,Qc)
Result of discretization is the model
x[k] = A x[k-1] + q, q ~ N(0,Q)
Which can be used for integrating the model
exactly over time steps, which are multiples
of dt.
"""
n = np.shape(F)[0]
if L is None:
L = np.eye(n)
if Q is None:
Q = np.zeros((n, n))
# Closed form integration of transition matrix
A = expm(F * dt)
# Closed form integration of covariance
# by matrix fraction decomposition
Phi = np.vstack((np.hstack((F, np.dot(np.dot(L, Q), L.T))),
np.hstack((np.zeros((n, n)), -F.T))))
AB = np.dot(expm(Phi * dt), np.vstack((np.zeros((n, n)), np.eye(n))))
# Q = AB[:n, :] / AB[n:(2 * n), :]
Q = np.linalg.solve(AB[n:(2 * n), :].T, AB[:n, :].T)
return A, Q
def demo_kalman_sine():
"""Kalman Filter demonstration with sine signal.
Example
-------
>>> 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')
plt.show('hold')
# for k in range(m):
# [M,P] = kf_predict(M,P,A,Q);
# [M,P] = kf_update(M,P,Y(k),H,R);
#
# MM(:,k) = M;
# PP(:,:,k) = P;
#
# %
# % Animate
# %
# if rem(k,10)==1
# plot(T,X,'b--',...
# T,Y,'ro',...
# T(k),M(1),'k*',...
# T(1:k),MM(1,1:k),'k-');
# legend('Real signal','Measurements','Latest estimate',
# 'Filtered estimate')
# title('Estimating a noisy sine signal with Kalman filter.');
# drawnow;
#
# pause;
# end
# end
#
# clc;
# disp('In this demonstration we estimate a stationary sine signal '
# 'from noisy measurements by using the classical Kalman filter.');
# disp(' ');
# disp('The filtering results are now displayed sequantially for 10 time '
# 'step at a time.');
# disp(' ');
# disp('<push any key to see the filtered and smoothed results together>')
# pause;
# %
# % Apply Kalman smoother
# %
# SM = rts_smooth(MM,PP,A,Q);
# plot(T,X,'b--',...
# T,MM(1,:),'k-',...
# T,SM(1,:),'r-');
# legend('Real signal','Filtered estimate','Smoothed estimate')
# title('Filtered and smoothed estimate of the original signal');
#
# clc;
# disp('The filtered and smoothed estimates of the signal are now '
# 'displayed.')
# disp(' ');
# disp('RMS errors:');
# %
# % Errors
# %
# fprintf('KF = %.3f\nRTS = %.3f\n',...
# sqrt(mean((MM(1,:)-X(1,:)).^2)),...
# sqrt(mean((SM(1,:)-X(1,:)).^2)));
def demo_hampel():
"""
Example
-------
>>> demo_hampel()
>>> plt.close()
"""
randint = np.random.randint
Y = 5000 + np.random.randn(1000)
outliers = randint(0, 1000, size=(10,))
Y[outliers] = Y[outliers] + randint(1000, size=(10,))
YY, res = HampelFilter(dx=3, t=3, fulloutput=True)(Y)
YY1, res1 = HampelFilter(dx=1, t=3, adaptive=0.1, fulloutput=True)(Y)
YY2, res2 = HampelFilter(dx=3, t=0, fulloutput=True)(Y) # median
plt.figure(1)
plot_hampel(Y, YY, res)
plt.title('Standard HampelFilter')
plt.figure(2)
plot_hampel(Y, YY1, res1)
plt.title('Adaptive HampelFilter')
plt.figure(3)
plot_hampel(Y, YY2, res2)
plt.title('Median filter')
def plot_hampel(Y, YY, res):
X = np.arange(len(YY))
plt.plot(X, Y, 'b.') # Original Data
plt.plot(X, YY, 'r') # Hampel Filtered Data
plt.plot(X, res['Y0'], 'b--') # Nominal Data
plt.plot(X, res['LB'], 'r--') # Lower Bounds on Hampel Filter
plt.plot(X, res['UB'], 'r--') # Upper Bounds on Hampel Filter
i = res['outliers']
plt.plot(X[i], Y[i], 'ks') # Identified Outliers
def demo_tide_filter():
"""
Example
-------
>>> 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 demo_savitzky_on_exponential():
"""
Example
-------
>>> demo_savitzky_on_exponential()
>>> plt.close()
"""
t = np.linspace(-4, 4, 500)
y = np.exp(-t ** 2) + np.random.normal(0, 0.05, np.shape(t))
n = 11
ysg = SavitzkyGolay(n, degree=1, diff_order=0)(y)
plt.plot(t, y, t, ysg, '--')
def demo_smoothn_on_1d_cos():
"""
Example
-------
>>> demo_smoothn_on_1d_cos()
>>> plt.close()
"""
x = np.linspace(0, 100, 2 ** 8)
y = np.cos(x / 10) + (x / 50) ** 2 + np.random.randn(np.size(x)) / 10
y[np.r_[70, 75, 80]] = np.array([5.5, 5, 6])
z = smoothn(y) # Regular smoothing
zr = smoothn(y, robust=True) # Robust smoothing
_h0 = plt.subplot(121),
_h = plt.plot(x, y, 'r.', x, z, 'k', linewidth=2)
plt.title('Regular smoothing')
plt.subplot(122)
plt.plot(x, y, 'r.', x, zr, 'k', linewidth=2)
plt.title('Robust smoothing')
def demo_smoothn_on_2d_exp_sin():
"""
Example
-------
>>> demo_smoothn_on_2d_exp_sin()
>>> plt.close()
"""
xp = np.arange(0, 1, 0.02) # np.r_[0:1:0.02]
[x, y] = np.meshgrid(xp, xp)
f = np.exp(x + y) + np.sin((x - 2 * y) * 3)
fn = f + np.random.randn(*f.shape) * 0.5
_fs, s = smoothn(fn, fulloutput=True)
fs2 = smoothn(fn, s=2 * s)
_h = plt.subplot(131),
_h = plt.contourf(xp, xp, fn)
_h = plt.subplot(132),
_h = plt.contourf(xp, xp, fs2)
_h = plt.subplot(133),
_h = plt.contourf(xp, xp, f)
def _cardioid(n=1000):
t = np.linspace(0, 2 * np.pi, n)
x0 = 2 * np.cos(t) * (1 - np.cos(t))
y0 = 2 * np.sin(t) * (1 - np.cos(t))
x = x0 + np.random.randn(x0.size) * 0.1
y = y0 + np.random.randn(y0.size) * 0.1
return x, y, x0, y0
def demo_smoothn_on_cardioid():
"""
Example
-------
>>> demo_smoothn_cardoid()
>>> plt.close()
"""
x, y, x0, y0 = _cardioid()
z = smoothn(x + 1j * y, robust=False)
plt.plot(x0, y0, 'y',
x, y, 'r.',
np.real(z), np.imag(z), 'k', linewidth=2)
def demo_hodrick_on_cardioid():
"""
Example
-------
>>> demo_hodrick_on_cardioid()
>>> plt.close()
"""
x, y, x0, y0 = _cardioid()
smooth = HodrickPrescott(w=20000)
# smooth = HampelFilter(adaptive=50)
xs, ys = smooth(x), smooth(y)
plt.plot(x0, y0, 'y',
x, y, 'r.',
xs, ys, 'k', linewidth=2)
if __name__ == '__main__':
from wafo.testing import test_docstrings
test_docstrings(__file__)
# demo_savitzky_on_noisy_chirp()
# plt.show('hold') # show plot
# demo_kalman_sine()
# demo_tide_filter()
# demo_hampel()
# demo_kalman_voltimeter()
# demo_savitzky_on_exponential()
# plt.figure(1)
# demo_hodrick_on_cardioid()
# plt.figure(2)
# # demo_smoothn_on_1d_cos()
# demo_smoothn_on_cardioid()
# plt.show('hold')
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