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Gabriele Gilardi 2020-06-26 20:42:17 +09:00
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Code_Python/filters.py
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@ -4,68 +4,61 @@ Signal Filtering/Smoothing and Generation of Synthetic Time-Series.
Copyright (c) 2020 Gabriele Gilardi Copyright (c) 2020 Gabriele Gilardi
X (n_samples, n_series) Dataset to filter X (n_samples, ) Dataset to filter (input)
b (n_b, ) Numerator coefficients b (n_b, ) Transfer response coefficients (numerator)
a (n_a, ) Denominator coefficients a (n_a, ) Transfer response coefficients (denominator)
Y (n_samples, n_series) Filtered dataset Y (n_samples, ) Filtered dataset (output)
idx scalar First filtered element in Y idx scalar First filtered element in Y
n_samples Number of data to filter n_samples Number of samples in the input dataset
n_series Number of series to filter nb Number of coefficients in array <b>
nb Number of coefficients (numerator) na Number of coefficients in array <a>
na Number of coefficients (denominator)
Notes: Notes:
- the filter is applied starting from index. - the filter is applied starting from index idx = MAX(0, nb-1, na-1).
- non filtered data are set equal to the original input, i.e. - non filtered data are set equal to the input, i.e. Y[0:idx-1] = X[0:idx-1]
Y[0:idx-1,:] = X[0:idx-1,:] - X needs to be a 1D array.
- if n_series = 1 then must be ( ..., 1)
Filters:
Generic b,a Generic case Filter list:
SMA N Simple Moving Average -----------
EMA N/alpha Exponential Moving Average Generic b, Generic
SMA N Simple moving average
EMA N/alpha Exponential moving average
WMA N Weighted moving average WMA N Weighted moving average
MSMA N Modified Simple Moving Average MSMA N Modified simple moving average
MLSQ N Modified Least-Squares Quadratic (N=5,7,9,11) MLSQ N Modified least-squares quadratic (N = 5, 7, 9, 11)
ButterOrig P,N Butterworth original (N=2,3) ButterOrig P, N Butterworth original filter (N = 2, 3)
ButterMod P,N Butterworth modified (N=2,3) ButterMod P, N Butterworth modified filter (N = 2, 3)
SuperSmooth P,N Super smoother (N=2,3) SuperSmooth P, N Supersmoother filter (N = 2, 3)
GaussLow P,N Gauss low pass (P>=2) GaussLow P, N Gauss low pass filter (P > 1)
GaussHigh P,N Gauss high pass (P>=5) GaussHigh P, N Gauss high pass filter (P > 4)
BandPass P,delta Band-pass filter BandPass P, delta Band-pass filter
BandStop P,delta Band-stop filter BandStop P, delta Band-stop filter
ZEMA1 N/alpha,K,Vn Zero-lag EMA (type 1) ZEMA1 N/alpha, K, Vn Zero-lag EMA (type 1)
ZEMA2 N/alpha,K Zero-lag EMA (type 2) ZEMA2 N/alpha, K Zero-lag EMA (type 2)
InstTrend N/alpha Instantaneous trendline InstTrend N/alpha Instantaneous trendline
SincFunction N Sinc function SincFilter P, nel Sinc function filter (N > 1)
Decycler P Decycler, 1-GaussHigh (P>=5) Decycler P De-cycler filter (P >= 4)
DecyclerOsc P1,P2 Decycle oscillator, GH(P1) - GH(P2), (P1>=5) DecyclerOsc P1, P2 De-cycle oscillator (P >= 4)
ABG ABG alpha, beta, Alpha-beta-gamma filter (0 < alpha, beta < 1)
Kalman gamma, dt
Kalman sigma_x, dt One-dimensional steady-state Kalman filter
sigma_v
N Order/smoothing factor/number of previous samples N Order/smoothing factor/number of previous samples
alpha Damping term alpha Damping term
P, P1, P2 Cut-off/critical period (50% power loss, -3 dB) P, P1, P2 Cut-off/critical period (50% power loss, -3 dB)
delta Band centered in P and in fraction delta Semi-band centered in P
(30% of P => 0.3, = 0.3*P, if P = 12 => 0.3*12 = 4)
K Coefficient/gain K Coefficient/gain
Vn Look back sample (for the momentum) Vn Look-back sample
nel Number of frequencies in the sinc function
correction = update = measurement alpha Parameter(s) to correct the position in the ABG filter
prediction = motion beta Parameter(s) to correct the velocity in the ABG filter
gamma Parameter(s) to correct the acceleration in the ABG filter
X (n_states, 1) State estimate dt Sampling interval in the ABG and Kalman filters
P (n_states, n_states) Covariance estimate sigma_x Process variance in the Kalman filter
F (n_states, n_states) State transition model sigma_v Noise variance in the Kalman filter
Z (n_obs, 1) Observations
H (n_obs, n_states) Observation model
R (n_obs, n_obs) Covariance of the observation noise
S (n_obs, n_obs) Covariance of the observation residual
K (n_states, n_obs) Optimal Kalman gain
Q (n_states, n_states) Covariance of the process noise matrix
Y (n_obs, 1) Observation residual (innovation)
""" """
import numpy as np import numpy as np
@ -73,19 +66,32 @@ from scipy import signal
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
def plot_signals(signals, start=0): def plot_signals(signals, names=None, start=0):
""" """
signals must be a list Plot the signals specified in list <signals> with their names specified in
list <names>. Each signal is plotted in its full length.
""" """
# Identify the signals by index if their name is not specified
if (names is None):
legend = [] legend = []
count = 0 count = 0
else:
legend = names
# Loop over the signals
for signal in signals: for signal in signals:
signal = signal.flatten() signal = signal.flatten()
end = len(signal) end = len(signal)
t = np.arange(start, end) t = np.arange(start, end)
plt.plot(t, signal[start:end]) plt.plot(t, signal[start:end])
# If no name is given use the list index to identify the signals
if (names is None):
legend.append('Signal [' + str(count) + ']') legend.append('Signal [' + str(count) + ']')
count += 1 count += 1
# Plot and format
plt.xlabel('Index') plt.xlabel('Index')
plt.ylabel('Value') plt.ylabel('Value')
plt.grid(b=True) plt.grid(b=True)
@ -95,23 +101,27 @@ def plot_signals(signals, start=0):
def filter_data(data, b, a): def filter_data(data, b, a):
""" """
Applies a filter with transfer response coefficients <a> and <b>. Applies a filter with transfer response coefficients <b> (numerator) and
<a> (denominator).
""" """
n_samples = len(data) n_samples = len(data)
nb = len(b) nb = len(b)
na = len(a) na = len(a)
idx = np.amax([0, nb-1, na-1]) idx = np.amax([0, nb-1, na-1]) # Index of the 1st filtered sample
Y = data.copy() Y = data.copy()
# Loop over the samples
for i in range(idx, n_samples): for i in range(idx, n_samples):
tmp = 0 tmp = 0
# Contribution from the numerator term (input samples)
for j in range(nb): for j in range(nb):
tmp += b[j] * data[i-j] # Numerator term tmp += b[j] * data[i-j]
# Contribution from the denominator term (previous output samples)
for j in range(1, na): for j in range(1, na):
tmp -= a[j] * Y[i-j] # Denominator term tmp -= a[j] * Y[i-j]
Y[i] = tmp / a[0] Y[i] = tmp / a[0]
@ -122,13 +132,16 @@ class Filter:
def __init__(self, data): def __init__(self, data):
""" """
Initialize the filter object.
""" """
self.data = np.asarray(data) self.data = np.asarray(data).flatten()
self.n_samples = len(data) self.idx = 0
self.b = 0.0
self.a = 0.0
def Generic(self, b=1.0, a=1.0): def Generic(self, b=1.0, a=1.0):
""" """
Filter with generic transfer response coefficients <a> and <b>. Filter with generic transfer response coefficients <b> and <a>.
""" """
self.b = np.asarray(b) self.b = np.asarray(b)
self.a = np.asarray(a) self.a = np.asarray(a)
@ -136,9 +149,9 @@ class Filter:
return Y return Y
def SMA(self, N=10): def SMA(self, N=5):
""" """
Simple moving average (?? order, FIR, ?? band). Simple moving average.
""" """
self.b = np.ones(N) / N self.b = np.ones(N) / N
self.a = np.array([1.0]) self.a = np.array([1.0])
@ -146,9 +159,10 @@ class Filter:
return Y return Y
def EMA(self, N=10, alpha=None): def EMA(self, N=5, alpha=None):
""" """
Exponential moving average (?? order, IIR, pass ??). Exponential moving average.
If not given, <alpha> is determined as equivalent to a N-SMA. If not given, <alpha> is determined as equivalent to a N-SMA.
""" """
if (alpha is None): if (alpha is None):
@ -160,10 +174,11 @@ class Filter:
return Y return Y
def WMA(self, N=10): def WMA(self, N=5):
""" """
Weighted moving average (?? order, FIR, pass ??). Weighted moving average.
Example: N = 5 --> [5.0, 4.0, 3.0, 2.0, 1.0] / 15.0
Example: N = 5 --> [5, 4, 3, 2, 1] / 15.
""" """
w = np.arange(N, 0, -1) w = np.arange(N, 0, -1)
@ -173,10 +188,11 @@ class Filter:
return Y return Y
def MSMA(self, N=10): def MSMA(self, N=5):
""" """
Modified simple moving average (?? order, FIR, pass ??). Modified simple moving average.
Example: N = 4 --> [0.5, 1.0, 1.0, 1.0, 0.5] / 4.0
Example: N = 5 --> [1/2, 1, 1, 1, 1, 1/2] / 5.
""" """
w = np.ones(N+1) w = np.ones(N+1)
w[0] = 0.5 w[0] = 0.5
@ -190,26 +206,26 @@ class Filter:
def MLSQ(self, N=5): def MLSQ(self, N=5):
""" """
Modified simple moving average (?? order, FIR, pass ??). Modified least-squares quadratic.
Only N = 5, 7, 9, and 11 are implemented. If not returns the unfiltered
dataset. Must be N = 5, 7, 9, or 11. If wrong N, prints a warning and returns
the unfiltered dataset.
""" """
if (N == 5): if (N == 5):
w = np.array([7.0, 24.0, 34.0, 24.0, 7.0]) / 96.0 w = np.array([7, 24, 34, 24, 7]) / 96
elif (N == 7): elif (N == 7):
w = np.array([1.0, 6.0, 12.0, 14.0, 12.0, 6.0, 1.0]) / 52.0 w = np.array([1, 6, 12, 14, 12, 6, 1]) / 52
elif (N == 9): elif (N == 9):
w = np.array([-1.0, 28.0, 78.0, 108.0, 118.0, 108.0, 78.0, 28.0, w = np.array([-1, 28, 78, 108, 118, 108, 78, 28, -1]) / 544
-1.0]) / 544.0
elif (N == 11): elif (N == 11):
w = np.array([-11.0, 18.0, 88.0, 138.0, 168.0, 178.0, 168.0, 138.0, w = np.array([-11, 18, 88, 138, 168, 178, 168, 138, 88, 18,
88.0, 18.0, -11.0]) / 980.0 -11]) / 980
else: else:
print("Warning: data returned unfiltered (wrong N)") print("Warning: data returned unfiltered (MLSQ - Wrong N)")
self.idx = 0 self.idx = 0
return self.data return self.data
@ -221,8 +237,10 @@ class Filter:
def ButterOrig(self, N=2, P=10): def ButterOrig(self, N=2, P=10):
""" """
Butterworth original version (?? order, IIR, pass ??). Butterworth original filter.
Only N = 2 and 3 are implemented. If not returns the unfiltered dataset.
Must be N = 2 or 3. If wrong N, prints a warning and returns the
unfiltered dataset.
""" """
if (N == 2): if (N == 2):
beta = np.exp(-np.sqrt(2.0) * np.pi / P) beta = np.exp(-np.sqrt(2.0) * np.pi / P)
@ -239,7 +257,7 @@ class Filter:
(1.0 + alpha) * beta ** 2.0, - beta ** 4.0]) (1.0 + alpha) * beta ** 2.0, - beta ** 4.0])
else: else:
print("Warning: data returned unfiltered (wrong N)") print("Warning: data returned unfiltered (ButterOrig - Wrong N)")
self.idx = 0 self.idx = 0
return self.data return self.data
@ -251,8 +269,11 @@ class Filter:
def ButterMod(self, N=2, P=10): def ButterMod(self, N=2, P=10):
""" """
Butterworth modified version (?? order, IIR, pass ??). Butterworth modified filter. It is derived from the Butterworth original
Only N = 2 and 3 are implemented. If not returns the unfiltered dataset. filter deleting all but the constant term at the numerator.
Must be N = 2 or 3. If wrong N, prints a warning and returns the
unfiltered dataset.
""" """
if (N == 2): if (N == 2):
beta = np.exp(-np.sqrt(2.0) * np.pi / P) beta = np.exp(-np.sqrt(2.0) * np.pi / P)
@ -268,7 +289,7 @@ class Filter:
(1.0 + alpha) * beta ** 2.0, - beta ** 4.0]) (1.0 + alpha) * beta ** 2.0, - beta ** 4.0])
else: else:
print("Warning: data returned unfiltered (wrong N)") print("Warning: data returned unfiltered (ButterMod - Wrong N)")
self.idx = 0 self.idx = 0
return self.data return self.data
@ -280,8 +301,11 @@ class Filter:
def SuperSmooth(self, N=2, P=10): def SuperSmooth(self, N=2, P=10):
""" """
SuperSmooth (?? order, IIR, pass ??). Supersmoother filter. It is derived from the Butterworth modified
Only N = 2 and 3 are implemented. If not returns the unfiltered dataset. filter adding a two-element moving average at the numerator.
Must be N = 2 or 3. If wrong N, prints a warning and returns the
unfiltered dataset.
""" """
if (N == 2): if (N == 2):
beta = np.exp(-np.sqrt(2.0) * np.pi / P) beta = np.exp(-np.sqrt(2.0) * np.pi / P)
@ -298,7 +322,7 @@ class Filter:
(1.0 + alpha) * beta ** 2.0, - beta ** 4.0]) (1.0 + alpha) * beta ** 2.0, - beta ** 4.0])
else: else:
print("Warning: data returned unfiltered (wrong N)") print("Warning: data returned unfiltered (SuperSmooth - Wrong N)")
self.idx = 0 self.idx = 0
return self.data return self.data
@ -308,13 +332,15 @@ class Filter:
return Y return Y
def GaussLow(self, N=1, P=2): def GaussLow(self, N=1, P=10):
""" """
Gauss low pass (IIR, N-th order, low pass). Gauss low pass filter.
Must be P > 1. If not returns the unfiltered dataset.
Must be P > 1. If wrong P, prints a warning and returns the unfiltered
dataset.
""" """
if (P < 2): if (P <= 1):
print("Warning: data returned unfiltered (P < 2)") print("Warning: data returned unfiltered (GaussLow - Wrong P)")
self.idx = 0 self.idx = 0
return self.data return self.data
@ -331,13 +357,15 @@ class Filter:
return Y return Y
def GaussHigh(self, N=1, P=5): def GaussHigh(self, N=1, P=10):
""" """
Gauss high pass (IIR, Nth order, high pass). Gauss high pass filter.
Must be P > 4. If not returns the unfiltered dataset.
Must be P > 4. If wrong P, prints a warning and returns the unfiltered
dataset.
""" """
if (P < 5): if (P <= 4):
print("Warning: data returned unfiltered (P < 5)") print("Warning: data returned unfiltered (GaussHigh - Wrong P)")
self.idx = 0 self.idx = 0
return self.data return self.data
@ -354,11 +382,11 @@ class Filter:
return Y return Y
def BandPass(self, P=5, delta=0.3): def BandPass(self, P=10, delta=0.3):
""" """
Band-pass (type, order, IIR). Band-pass filter.
Example: delta = 0.3, P = 12
(30% of P => 0.3, = 0.3*P, if P = 12 => 0.3*12 = 4) Example: delta = 0.3, P = 10 --> 0.3 * 10 = 3 --> band is [7, 13]
""" """
beta = np.cos(2.0 * np.pi / P) beta = np.cos(2.0 * np.pi / P)
gamma = np.cos(4.0 * np.pi * delta / P) gamma = np.cos(4.0 * np.pi * delta / P)
@ -372,9 +400,9 @@ class Filter:
def BandStop(self, P=5, delta=0.3): def BandStop(self, P=5, delta=0.3):
""" """
Band-stop (type, order, IIR) Band-stop filter.
Example: delta = 0.3, P = 12
(30% of P => 0.3, = 0.3*P, if P = 12 => 0.3*12 = 4) Example: delta = 0.3, P = 10 --> 0.3 * 10 = 3 --> band is [7, 13]
""" """
beta = np.cos(2.0 * np.pi / P) beta = np.cos(2.0 * np.pi / P)
gamma = np.cos(4.0 * np.pi * delta / P) gamma = np.cos(4.0 * np.pi * delta / P)
@ -388,13 +416,15 @@ class Filter:
def ZEMA1(self, N=10, alpha=None, K=1.0, Vn=5): def ZEMA1(self, N=10, alpha=None, K=1.0, Vn=5):
""" """
Zero lag Exponential Moving Average (type 1). Zero-lag EMA (type 1). It is an alpha-beta type filter with sub-optimal
parameters.
If not given, <alpha> is determined as equivalent to a N-SMA. If not given, <alpha> is determined as equivalent to a N-SMA.
""" """
if (alpha is None): if (alpha is None):
alpha = 2.0 / (N + 1.0) alpha = 2.0 / (N + 1.0)
w = np.zeros(Vn+1) w = np.zeros(Vn + 1)
w[0] = alpha * (1.0 + K) w[0] = alpha * (1.0 + K)
w[Vn] = - alpha * K w[Vn] = - alpha * K
@ -406,7 +436,9 @@ class Filter:
def ZEMA2(self, N=10, alpha=None, K=1.0): def ZEMA2(self, N=10, alpha=None, K=1.0):
""" """
Zero lag Exponential Moving Average (type 2). Zero-lag EMA (type 2). It is derived from the type 1 ZEMA removing the
look-back term Vn.
If not given, <alpha> is determined as equivalent to a N-SMA. If not given, <alpha> is determined as equivalent to a N-SMA.
""" """
if (alpha is None): if (alpha is None):
@ -420,7 +452,9 @@ class Filter:
def InstTrend(self, N=10, alpha=None): def InstTrend(self, N=10, alpha=None):
""" """
Instantaneous Trendline (2nd order, IIR, low pass). Instantaneous Trendline. It is created by removing the dominant cycle
from the signal.
If not given, <alpha> is determined as equivalent to a N-SMA. If not given, <alpha> is determined as equivalent to a N-SMA.
""" """
if (alpha is None): if (alpha is None):
@ -433,15 +467,22 @@ class Filter:
return Y return Y
def SincFunction(self, N=10, nel=10): def SincFilter(self, P=10, nel=10):
""" """
Sinc function (order, FIR, pass). Sinc function filter. The cut off point is at 0.5/P.
(N > 1, cut off at 0.5/N)
Must be P > 1. If wrong P, prints a warning and returns the unfiltered
dataset.
""" """
if (P <= 1):
print("Warning: data returned unfiltered (SincFilter - Wrong P)")
self.idx = 0
return self.data
K = np.arange(1, nel) K = np.arange(1, nel)
w = np.zeros(nel) w = np.zeros(nel)
w[0] = 1.0 / N w[0] = 1.0 / N
w[1:] = np.sin(np.pi * K / N) / (np.pi * K) w[1:] = np.sin(np.pi * K / P) / (np.pi * K)
self.b = w self.b = w
self.a = np.array([1.0]) self.a = np.array([1.0])
@ -451,12 +492,14 @@ class Filter:
def Decycler(self, P=10): def Decycler(self, P=10):
""" """
Decycler (?? order, IIR, pass ??). Gauss,HP,1st,P De-cycler filter. It is derived subtracting a 1st order high pass Gauss
Built subtracting high pass Gauss filter from 1 (order 1) filter from 1.
Must be P > 4. If not returns the unfiltered dataset.
Must be P > 4. If wrong P, prints a warning and returns the unfiltered
dataset.
""" """
if (P < 5): if (P <= 4):
print("Warning: data returned unfiltered (P < 5)") print("Warning: data returned unfiltered (Decycler - Wrong P)")
self.idx = 0 self.idx = 0
return self.data return self.data
@ -465,17 +508,18 @@ class Filter:
def DecyclerOsc(self, P1=5, P2=10): def DecyclerOsc(self, P1=5, P2=10):
""" """
DecyclerOsc (?? order 2, IIR, pass ??). De-cycler oscillator. It is derived subtracting a 2nd order high pass
Gauss filter with higher cut-off period from a 2nd order high pass Gauss
filter with higher cut-off period.
(Gauss, HP, 2nd order, Pmax - Gauss, HP, 2nd order, Pmin) Must be P > 4. If wrong P, prints a warning and returns the unfiltered
P1 = 1st cut off period, P2 = 2nd cut off period. Automatically fixed. dataset.
Must be P1, P2 > 4. If not returns the unfiltered dataset.
""" """
P_low = np.amin([P1, P2]) P_low = np.amin([P1, P2])
P_high = np.amax([P1, P2]) P_high = np.amax([P1, P2])
if (P_low < 5): if (P_low <= 4):
print("Warning: data returned unfiltered (P_low < 5)") print("Warning: data returned unfiltered (DecyclerOsc - Wrong P)")
self.idx = 0 self.idx = 0
return self.data return self.data
@ -484,34 +528,38 @@ class Filter:
def ABG(self, alpha=0.0, beta=0.0, gamma=0.0, dt=1.0): def ABG(self, alpha=0.0, beta=0.0, gamma=0.0, dt=1.0):
""" """
alpha-beta-gamma Alpha-beta-gamma filter. It is a predictor-corrector type of filter.
For numerical stability: 0 < alpha, beta < 1
Arguments alpha, beta, and gamma can be a scalar (used for all samples)
or an array with one value for each sample. For numerical stability it
should be 0 < alpha, beta < 1.
""" """
# If necessary change scalars to arrays n_samples = len(data)
Y = np.zeros(n_samples)
# Change scalar arguments to arrays if necessary
if (np.ndim(alpha) == 0): if (np.ndim(alpha) == 0):
alpha = np.ones(self.n_samples) * alpha alpha = np.ones(n_samples) * alpha
if (np.ndim(beta) == 0): if (np.ndim(beta) == 0):
beta = np.ones(self.n_samples) * beta beta = np.ones(n_samples) * beta
if (np.ndim(gamma) == 0): if (np.ndim(gamma) == 0):
gamma = np.ones(self.n_samples) * gamma gamma = np.ones(n_samples) * gamma
# Initialize # Initialize
Y_corr = self.data.copy() x0 = self.data[0]
Y_pred = self.data.copy() v0 = 0.0
x0 = self.data[0, :] a0 = 0.0
v0 = np.zeros(self.n_series) Y[0] = x0
a0 = np.zeros(self.n_series)
for i in range(1, self.n_samples): for i in range(1, n_samples):
# Predictor (predicts state in <i>) # Predictor (predicts state in <i>)
x_pred = x0 + dt * v0 + 0.5 * a0 * dt ** 2.0 x_pred = x0 + dt * v0 + 0.5 * a0 * dt ** 2.0
v_pred = v0 + dt * a0 v_pred = v0 + dt * a0
a_pred = a0 a_pred = a0
Y_pred[i, :] = x_pred
# Residual (innovation) # Residual (innovation)
r = self.data[i, :] - x_pred r = self.data[i] - x_pred
# Corrector (corrects state in <i>) # Corrector (corrects state in <i>)
x_corr = x_pred + alpha[i] * r x_corr = x_pred + alpha[i] * r
@ -519,18 +567,23 @@ class Filter:
a_corr = a_pred + (2.0 * gamma[i] / dt ** 2.0) * r a_corr = a_pred + (2.0 * gamma[i] / dt ** 2.0) * r
# Save value and prepare next iteration # Save value and prepare next iteration
Y_corr[i, :] = x_corr
x0 = x_corr x0 = x_corr
v0 = v_corr v0 = v_corr
a0 = a_corr a0 = a_corr
Y[i] = x_corr
self.idx = 1 self.idx = 1
return Y_corr, Y_pred return Y
def Kalman(self, sigma_x, sigma_v, dt, abg_type="abg"): def Kalman(self, sigma_x, sigma_v, dt, abg_type="abg"):
""" """
Steady-state Kalman filter (also limited to one-dimension) One-dimensional steady-state Kalman filter. It is obtained from the
alpha-beta-gamma filter using the process variance, the noise variance
and optimizing the three parameters.
Arguments sigma_x and sigma_v can be a scalar (used for all samples) or
an array with one value for each sample.
""" """
L = (sigma_x / sigma_v) * dt ** 2.0 L = (sigma_x / sigma_v) * dt ** 2.0
@ -561,19 +614,23 @@ class Filter:
beta = 2.0 * (1 - s) ** 2.0 beta = 2.0 * (1 - s) ** 2.0
gamma = (beta ** 2.0) / (2.0 * alpha) gamma = (beta ** 2.0) / (2.0 * alpha)
# Apply filter # Apply the alpha-beta-gamma filter
Y = self.abg(alpha=alpha, beta=beta, gamma=gamma, dt=dt) Y = self.abg(alpha=alpha, beta=beta, gamma=gamma, dt=dt)
return Y return Y
def plot_frequency(self): def plot_frequency(self):
""" """
Plots the frequency response (in decibels) of the filter with transfer
response coefficients <b> and <a>.
""" """
w, h = signal.freqz(self.b, self.a) w, h = signal.freqz(self.b, self.a)
h_db = 20.0 * np.log10(np.abs(h)) h_db = 20.0 * np.log10(np.abs(h)) # Convert to decibels
wf = w / (2.0 * np.pi) wf = w / (2.0 * np.pi) # Scale to [0, 0.5]
# Plot and format
plt.plot(wf, h_db) plt.plot(wf, h_db)
plt.axhline(-3.0, lw=1.5, ls='--', C='r') plt.axhline(-3.0, lw=1.5, ls='--', C='r') # -3 dB (50% power loss)
plt.grid(b=True) plt.grid(b=True)
plt.xlim(np.amin(wf), np.amax(wf)) plt.xlim(np.amin(wf), np.amax(wf))
plt.xlabel(r'$\omega$ [rad/sample]') plt.xlabel(r'$\omega$ [rad/sample]')
@ -584,9 +641,13 @@ class Filter:
def plot_lag(self): def plot_lag(self):
""" """
Plots the lag (group delay) of the filter with transfer response
coefficients <b> and <a>.
""" """
w, gd = signal.group_delay((self.b, self.a)) w, gd = signal.group_delay((self.b, self.a))
wf = w / (2.0 * np.pi) wf = w / (2.0 * np.pi) # Scale to [0, 0.5]
# Plot and format
plt.plot(wf, gd) plt.plot(wf, gd)
plt.grid(b=True) plt.grid(b=True)
plt.xlim(np.amin(wf), np.amax(wf)) plt.xlim(np.amin(wf), np.amax(wf))

51
Code_Python/synthetic.py
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@ -49,14 +49,12 @@ def synthetic_wave(P, A=None, phi=None, num=1000):
def synthetic_FFT(X, n_reps=1): def synthetic_FFT(X, n_reps=1):
""" """
Generates surrogates of the time-serie X using the phase-randomized Generates surrogates of the time-serie X using the phase-randomized
Fourier-transform algorithm. Fourier-transform algorithm. Input X needs to be a 1D array.
X (n, ) Original time-series X (n, ) Original time-series
X_fft (n, ) FFT of the original time-series X_fft (n, ) FFT of the original time-series
X_synt_fft (n_reps, n) FFT of the synthetic time-series X_synt_fft (n_reps, n) FFT of the synthetic time-series
X_synt (n_reps, n) Synthetic time-series X_synt (n_reps, n) Synthetic time-series
Input array X needs to be a 1D array (of any shape).
""" """
X = X.flatten() # Reshape to (n, ) X = X.flatten() # Reshape to (n, )
n = len(X) n = len(X)
@ -95,13 +93,11 @@ def synthetic_FFT(X, n_reps=1):
def synthetic_sampling(X, n_reps=1, replace=True): def synthetic_sampling(X, n_reps=1, replace=True):
""" """
Generates surrogates of the time-serie X using randomized-sampling Generates surrogates of the time-serie X using randomized-sampling
(bootstrap) with or without replacement. (bootstrap) with or without replacement. Input X needs to be a 1D array.
X (n, ) Original time-series X (n, ) Original time-series
idx (n_reps, n) Random index of X idx (n_reps, n) Random index of X
X_synt (n_reps, n) Synthetic time-series X_synt (n_reps, n) Synthetic time-series
Input array X needs to be a 1D array (of any shape).
""" """
X = X.flatten() # Reshape to (n, ) X = X.flatten() # Reshape to (n, )
n = len(X) n = len(X)
@ -123,7 +119,7 @@ def synthetic_sampling(X, n_reps=1, replace=True):
def synthetic_MEboot(X, n_reps=1, alpha=0.1, bounds=False, scale=False): def synthetic_MEboot(X, n_reps=1, alpha=0.1, bounds=False, scale=False):
""" """
Generates surrogates of the time-serie X using the maximum entropy Generates surrogates of the time-serie X using the maximum entropy
bootstrap algorithm. bootstrap algorithm. Input X needs to be a 1D array.
X (n, ) Original time-series X (n, ) Original time-series
idx (n, ) Original order of X idx (n, ) Original order of X
@ -135,8 +131,6 @@ def synthetic_MEboot(X, n_reps=1, alpha=0.1, bounds=False, scale=False):
w_corr (n_reps, n) Interpolated new points with corrections for first w_corr (n_reps, n) Interpolated new points with corrections for first
and last interval and last interval
X_synt (n_reps, n) Synthetic time-series X_synt (n_reps, n) Synthetic time-series
Input array X needs to be a 1D array (of any shape).
""" """
X = X.flatten() # Reshape to (n, ) X = X.flatten() # Reshape to (n, )
n = len(X) n = len(X)
@ -259,13 +253,13 @@ def value2diff(X, percent=True):
""" """
Returns the 1st discrete difference of array X. Returns the 1st discrete difference of array X.
X (n, ) Original dataset X (n, ) Input dataset
dX (n-1, ) 1st discrete differences dX (n-1, ) 1st discrete differences
Notes: Notes:
- the discrete difference can be calculated in percent or in value. - the discrete difference can be calculated in percent or in value.
- array dX is one element shorter than array X. - dX is one element shorter than X.
- array X needs to be a 1D array (of any shape). - X needs to be a 1D array.
""" """
X = X.flatten() # Reshape to (n, ) X = X.flatten() # Reshape to (n, )
@ -282,36 +276,39 @@ def value2diff(X, percent=True):
def diff2value(dX, X0, percent=True): def diff2value(dX, X0, percent=True):
""" """
Returns array X from the 1st discrete difference. Returns array X from the 1st discrete difference using X0 as initial value.
dX (n, ) Discrete differences dX (n, ) Discrete differences
X0 scalar Initial value X0 scalar Initial value
X (n+1, ) Original dataset ????? X (n+1, ) Output dataset
Notes: Notes:
- the discrete difference can be in percent or in value. - the discrete difference can be in percent or in value.
- array X is one element longer than array dX. - X is one element longer than dX.
- array dX needs to be a 1D array (of any shape). - dX needs to be a 1D array.
If the discrete difference is in percent, the first column of X is set to If the discrete difference is in percent:
one. The original array is X[0] * X X[0] = X0
X[1] = X[0] * (1 + dX[0])
X[2] = X[1] * (1 + dX[1]) = X[0] * (1 + dX[0]) * (1 + dX[1])
....
If the discrete difference is in value, the first column of X is set to If the discrete difference is in value:
zero. X0+X X[0] = X0
- array X needs to be a 1D array (of any shape). X[1] = X[0] + dX[0]
X[2] = X[1] + dX[1] = X[0] + dX[0] + dX[1]
....
""" """
dX = dX.flatten() # Reshape to (n, ) dX = dX.flatten() # Reshape to (n, )
n = len(dX) X = np.zeros(len(dX) + 1)
X = np.zeros(n+1) X[0] = X0 # Initial value
# Discrete difference in percent # Discrete difference in percent
if (percent): if (percent):
X[0] = X0 X[1:] = X0 * np.cumprod(1.0 + dX)
X[1:] = np.cumprod(1.0 + dX) * X0 # !!!! check
# Discrete difference in value # Discrete difference in value
else: else:
X[0] = X0 X[1:] = X0 + np.cumsum(dX)
X[1:] = np.cumsum(dX) + X0 # !!!!!c check
return X return X

25
Code_Python/test.py
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@ -5,10 +5,10 @@ Copyright (c) 2020 Gabriele Gilardi
ToDo: ToDo:
- add comments to the code - do examples and check type (low pass, etc)
- in comments write what filters do - put type in description
- is necessary to copy X for Y untouched? - see paper ZEMA for tests
- decide default values in functions - check results with ML
""" """
import sys import sys
@ -31,18 +31,21 @@ data_file = sys.argv[1] + '.csv'
# Read data from a csv file (one time-series each column) # Read data from a csv file (one time-series each column)
data = np.loadtxt(data_file, delimiter=',') data = np.loadtxt(data_file, delimiter=',')
print(data.shape)
t, f = syn.synthetic_wave([1., 2., 3.], A=None, phi=None, num=1000) # t, f = syn.synthetic_wave([1., 2., 3.], A=None, phi=None, num=1000)
plt.plot(t,f) # plt.plot(t,f)
plt.show() # plt.show()
# spx = flt.Filter(data) spx = flt.Filter(data)
# res = spx.EMA(N=10) # res = spx.EMA(N=10)
# signals = [spx.data, res[0:400]] # signals = [spx.data, res[0:400]]
# flt.plot_signals(signals, start=100) # flt.plot_signals(signals, ['SPX', 'SMA'])
# spx.plot_frequency() res = spx.BandPass(P=10, delta=0.3)
# spx.plot_lag()
spx.plot_frequency()
spx.plot_lag()
# sigma_x = 0.1 # sigma_x = 0.1
# sigma_v = 0.1 * np.ones(n_samples) # sigma_v = 0.1 * np.ones(n_samples)

10
README.md
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@ -1,14 +1,12 @@
# Signal Filtering # Signal Smoothing / Filtering and Generation of Synthetic Time-Series
## Reference ## Reference
- John F. Ehlers, "[Cycle Analytics for Traders: Advanced Technical Trading Concepts](http://www.mesasoftware.com/ehlers_books.htm)." - John F. Ehlers, "[Cycle Analytics for Traders: Advanced Technical Trading Concepts](http://www.mesasoftware.com/ehlers_books.htm)".
- Wikipedia, "[Alpha beta filter](https://en.wikipedia.org/wiki/Alpha_beta_filter)." - D. Prichard and J. Theiler, "[Generating surrogate data for time series with several simultaneously measured variables](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.73.951)".
- D. Prichard, and J. Theiler, "[Generating surrogate data for time series with several simultaneously measured variables](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.73.951)." - H. Vinod and J. Lopez-de-Lacalle, "[Maximum entropy bootstrap for time series: the meboot R package](https://www.jstatsoft.org/article/view/v029i05)".
- H. Vinod, and J. Lopez-de-Lacalle, "[Maximum entropy bootstrap for time series: the meboot R package](https://www.jstatsoft.org/article/view/v029i05)."
## Characteristics ## Characteristics