kopia lustrzana https://github.com/gabrielegilardi/SignalFilters
complete comments
Gabriele Gilardi
rodzic
753245f285
commit
87ef343228
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@ -4,68 +4,61 @@ Signal Filtering/Smoothing and Generation of Synthetic Time-Series.
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Copyright (c) 2020 Gabriele Gilardi
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X (n_samples, n_series) Dataset to filter
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b (n_b, ) Numerator coefficients
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a (n_a, ) Denominator coefficients
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Y (n_samples, n_series) Filtered dataset
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idx scalar First filtered element in Y
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X (n_samples, ) Dataset to filter (input)
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b (n_b, ) Transfer response coefficients (numerator)
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a (n_a, ) Transfer response coefficients (denominator)
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Y (n_samples, ) Filtered dataset (output)
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idx scalar First filtered element in Y
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n_samples Number of data to filter
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n_series Number of series to filter
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nb Number of coefficients (numerator)
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na Number of coefficients (denominator)
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n_samples Number of samples in the input dataset
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nb Number of coefficients in array <b>
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na Number of coefficients in array <a>
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Notes:
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- the filter is applied starting from index.
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- non filtered data are set equal to the original input, i.e.
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Y[0:idx-1,:] = X[0:idx-1,:]
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- if n_series = 1 then must be ( ..., 1)
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- the filter is applied starting from index idx = MAX(0, nb-1, na-1).
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- non filtered data are set equal to the input, i.e. Y[0:idx-1] = X[0:idx-1]
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- X needs to be a 1D array.
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Filters:
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Generic b,a Generic case
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SMA N Simple Moving Average
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EMA N/alpha Exponential Moving Average
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Filter list:
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-----------
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Generic b, Generic
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SMA N Simple moving average
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EMA N/alpha Exponential moving average
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WMA N Weighted moving average
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MSMA N Modified Simple Moving Average
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MLSQ N Modified Least-Squares Quadratic (N=5,7,9,11)
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ButterOrig P,N Butterworth original (N=2,3)
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ButterMod P,N Butterworth modified (N=2,3)
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SuperSmooth P,N Super smoother (N=2,3)
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GaussLow P,N Gauss low pass (P>=2)
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GaussHigh P,N Gauss high pass (P>=5)
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BandPass P,delta Band-pass filter
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BandStop P,delta Band-stop filter
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ZEMA1 N/alpha,K,Vn Zero-lag EMA (type 1)
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ZEMA2 N/alpha,K Zero-lag EMA (type 2)
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MSMA N Modified simple moving average
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MLSQ N Modified least-squares quadratic (N = 5, 7, 9, 11)
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ButterOrig P, N Butterworth original filter (N = 2, 3)
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ButterMod P, N Butterworth modified filter (N = 2, 3)
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SuperSmooth P, N Supersmoother filter (N = 2, 3)
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GaussLow P, N Gauss low pass filter (P > 1)
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GaussHigh P, N Gauss high pass filter (P > 4)
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BandPass P, delta Band-pass filter
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BandStop P, delta Band-stop filter
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ZEMA1 N/alpha, K, Vn Zero-lag EMA (type 1)
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ZEMA2 N/alpha, K Zero-lag EMA (type 2)
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InstTrend N/alpha Instantaneous trendline
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SincFunction N Sinc function
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Decycler P Decycler, 1-GaussHigh (P>=5)
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DecyclerOsc P1,P2 Decycle oscillator, GH(P1) - GH(P2), (P1>=5)
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ABG
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Kalman
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SincFilter P, nel Sinc function filter (N > 1)
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Decycler P De-cycler filter (P >= 4)
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DecyclerOsc P1, P2 De-cycle oscillator (P >= 4)
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ABG alpha, beta, Alpha-beta-gamma filter (0 < alpha, beta < 1)
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gamma, dt
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Kalman sigma_x, dt One-dimensional steady-state Kalman filter
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sigma_v
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N Order/smoothing factor/number of previous samples
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alpha Damping term
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P, P1, P2 Cut-off/critical period (50% power loss, -3 dB)
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delta Band centered in P and in fraction
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(30% of P => 0.3, = 0.3*P, if P = 12 => 0.3*12 = 4)
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delta Semi-band centered in P
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K Coefficient/gain
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Vn Look back sample (for the momentum)
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correction = update = measurement
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prediction = motion
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X (n_states, 1) State estimate
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P (n_states, n_states) Covariance estimate
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F (n_states, n_states) State transition model
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Z (n_obs, 1) Observations
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H (n_obs, n_states) Observation model
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R (n_obs, n_obs) Covariance of the observation noise
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S (n_obs, n_obs) Covariance of the observation residual
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K (n_states, n_obs) Optimal Kalman gain
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Q (n_states, n_states) Covariance of the process noise matrix
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Y (n_obs, 1) Observation residual (innovation)
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Vn Look-back sample
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nel Number of frequencies in the sinc function
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alpha Parameter(s) to correct the position in the ABG filter
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beta Parameter(s) to correct the velocity in the ABG filter
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gamma Parameter(s) to correct the acceleration in the ABG filter
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dt Sampling interval in the ABG and Kalman filters
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sigma_x Process variance in the Kalman filter
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sigma_v Noise variance in the Kalman filter
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"""
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import numpy as np
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@ -73,19 +66,32 @@ from scipy import signal
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import matplotlib.pyplot as plt
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def plot_signals(signals, start=0):
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def plot_signals(signals, names=None, start=0):
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"""
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signals must be a list
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Plot the signals specified in list <signals> with their names specified in
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list <names>. Each signal is plotted in its full length.
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"""
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legend = []
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count = 0
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# Identify the signals by index if their name is not specified
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if (names is None):
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legend = []
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count = 0
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else:
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legend = names
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# Loop over the signals
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for signal in signals:
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signal = signal.flatten()
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end = len(signal)
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t = np.arange(start, end)
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plt.plot(t, signal[start:end])
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legend.append('Signal [' + str(count) + ']')
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count += 1
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# If no name is given use the list index to identify the signals
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if (names is None):
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legend.append('Signal [' + str(count) + ']')
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count += 1
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# Plot and format
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plt.xlabel('Index')
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plt.ylabel('Value')
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plt.grid(b=True)
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@ -95,23 +101,27 @@ def plot_signals(signals, start=0):
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def filter_data(data, b, a):
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"""
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Applies a filter with transfer response coefficients <a> and <b>.
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Applies a filter with transfer response coefficients <b> (numerator) and
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<a> (denominator).
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"""
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n_samples = len(data)
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nb = len(b)
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na = len(a)
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idx = np.amax([0, nb-1, na-1])
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idx = np.amax([0, nb-1, na-1]) # Index of the 1st filtered sample
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Y = data.copy()
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# Loop over the samples
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for i in range(idx, n_samples):
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tmp = 0
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# Contribution from the numerator term (input samples)
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for j in range(nb):
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tmp += b[j] * data[i-j] # Numerator term
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tmp += b[j] * data[i-j]
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# Contribution from the denominator term (previous output samples)
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for j in range(1, na):
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tmp -= a[j] * Y[i-j] # Denominator term
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tmp -= a[j] * Y[i-j]
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Y[i] = tmp / a[0]
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@ -122,13 +132,16 @@ class Filter:
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def __init__(self, data):
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"""
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Initialize the filter object.
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"""
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self.data = np.asarray(data)
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self.n_samples = len(data)
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self.data = np.asarray(data).flatten()
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self.idx = 0
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self.b = 0.0
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self.a = 0.0
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def Generic(self, b=1.0, a=1.0):
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"""
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Filter with generic transfer response coefficients <a> and <b>.
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Filter with generic transfer response coefficients <b> and <a>.
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"""
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self.b = np.asarray(b)
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self.a = np.asarray(a)
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@ -136,9 +149,9 @@ class Filter:
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return Y
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def SMA(self, N=10):
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def SMA(self, N=5):
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"""
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Simple moving average (?? order, FIR, ?? band).
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Simple moving average.
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"""
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self.b = np.ones(N) / N
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self.a = np.array([1.0])
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@ -146,9 +159,10 @@ class Filter:
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return Y
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def EMA(self, N=10, alpha=None):
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def EMA(self, N=5, alpha=None):
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"""
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Exponential moving average (?? order, IIR, pass ??).
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Exponential moving average.
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If not given, <alpha> is determined as equivalent to a N-SMA.
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"""
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if (alpha is None):
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@ -160,10 +174,11 @@ class Filter:
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return Y
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def WMA(self, N=10):
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def WMA(self, N=5):
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"""
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Weighted moving average (?? order, FIR, pass ??).
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Example: N = 5 --> [5.0, 4.0, 3.0, 2.0, 1.0] / 15.0
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Weighted moving average.
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Example: N = 5 --> [5, 4, 3, 2, 1] / 15.
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"""
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w = np.arange(N, 0, -1)
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return Y
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def MSMA(self, N=10):
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def MSMA(self, N=5):
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"""
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Modified simple moving average (?? order, FIR, pass ??).
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Example: N = 4 --> [0.5, 1.0, 1.0, 1.0, 0.5] / 4.0
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Modified simple moving average.
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Example: N = 5 --> [1/2, 1, 1, 1, 1, 1/2] / 5.
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"""
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w = np.ones(N+1)
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w[0] = 0.5
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@ -190,26 +206,26 @@ class Filter:
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def MLSQ(self, N=5):
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"""
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Modified simple moving average (?? order, FIR, pass ??).
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Only N = 5, 7, 9, and 11 are implemented. If not returns the unfiltered
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dataset.
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Modified least-squares quadratic.
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Must be N = 5, 7, 9, or 11. If wrong N, prints a warning and returns
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the unfiltered dataset.
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"""
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if (N == 5):
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w = np.array([7.0, 24.0, 34.0, 24.0, 7.0]) / 96.0
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w = np.array([7, 24, 34, 24, 7]) / 96
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elif (N == 7):
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w = np.array([1.0, 6.0, 12.0, 14.0, 12.0, 6.0, 1.0]) / 52.0
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w = np.array([1, 6, 12, 14, 12, 6, 1]) / 52
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elif (N == 9):
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w = np.array([-1.0, 28.0, 78.0, 108.0, 118.0, 108.0, 78.0, 28.0,
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-1.0]) / 544.0
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w = np.array([-1, 28, 78, 108, 118, 108, 78, 28, -1]) / 544
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elif (N == 11):
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w = np.array([-11.0, 18.0, 88.0, 138.0, 168.0, 178.0, 168.0, 138.0,
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88.0, 18.0, -11.0]) / 980.0
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w = np.array([-11, 18, 88, 138, 168, 178, 168, 138, 88, 18,
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-11]) / 980
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else:
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print("Warning: data returned unfiltered (wrong N)")
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print("Warning: data returned unfiltered (MLSQ - Wrong N)")
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self.idx = 0
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return self.data
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@ -221,8 +237,10 @@ class Filter:
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def ButterOrig(self, N=2, P=10):
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"""
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Butterworth original version (?? order, IIR, pass ??).
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Only N = 2 and 3 are implemented. If not returns the unfiltered dataset.
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Butterworth original filter.
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Must be N = 2 or 3. If wrong N, prints a warning and returns the
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unfiltered dataset.
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"""
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if (N == 2):
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beta = np.exp(-np.sqrt(2.0) * np.pi / P)
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@ -239,7 +257,7 @@ class Filter:
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(1.0 + alpha) * beta ** 2.0, - beta ** 4.0])
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else:
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print("Warning: data returned unfiltered (wrong N)")
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print("Warning: data returned unfiltered (ButterOrig - Wrong N)")
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self.idx = 0
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return self.data
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@ -251,8 +269,11 @@ class Filter:
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def ButterMod(self, N=2, P=10):
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"""
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Butterworth modified version (?? order, IIR, pass ??).
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Only N = 2 and 3 are implemented. If not returns the unfiltered dataset.
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Butterworth modified filter. It is derived from the Butterworth original
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filter deleting all but the constant term at the numerator.
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Must be N = 2 or 3. If wrong N, prints a warning and returns the
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unfiltered dataset.
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"""
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if (N == 2):
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beta = np.exp(-np.sqrt(2.0) * np.pi / P)
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@ -268,7 +289,7 @@ class Filter:
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(1.0 + alpha) * beta ** 2.0, - beta ** 4.0])
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else:
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print("Warning: data returned unfiltered (wrong N)")
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print("Warning: data returned unfiltered (ButterMod - Wrong N)")
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self.idx = 0
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return self.data
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@ -280,8 +301,11 @@ class Filter:
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def SuperSmooth(self, N=2, P=10):
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"""
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SuperSmooth (?? order, IIR, pass ??).
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Only N = 2 and 3 are implemented. If not returns the unfiltered dataset.
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Supersmoother filter. It is derived from the Butterworth modified
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filter adding a two-element moving average at the numerator.
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Must be N = 2 or 3. If wrong N, prints a warning and returns the
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unfiltered dataset.
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"""
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if (N == 2):
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beta = np.exp(-np.sqrt(2.0) * np.pi / P)
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@ -298,7 +322,7 @@ class Filter:
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(1.0 + alpha) * beta ** 2.0, - beta ** 4.0])
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else:
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print("Warning: data returned unfiltered (wrong N)")
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print("Warning: data returned unfiltered (SuperSmooth - Wrong N)")
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self.idx = 0
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return self.data
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@ -308,13 +332,15 @@ class Filter:
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return Y
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def GaussLow(self, N=1, P=2):
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def GaussLow(self, N=1, P=10):
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"""
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Gauss low pass (IIR, N-th order, low pass).
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Must be P > 1. If not returns the unfiltered dataset.
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Gauss low pass filter.
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Must be P > 1. If wrong P, prints a warning and returns the unfiltered
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dataset.
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"""
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if (P < 2):
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print("Warning: data returned unfiltered (P < 2)")
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if (P <= 1):
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print("Warning: data returned unfiltered (GaussLow - Wrong P)")
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self.idx = 0
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return self.data
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@ -331,13 +357,15 @@ class Filter:
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return Y
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def GaussHigh(self, N=1, P=5):
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def GaussHigh(self, N=1, P=10):
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"""
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Gauss high pass (IIR, Nth order, high pass).
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Must be P > 4. If not returns the unfiltered dataset.
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Gauss high pass filter.
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Must be P > 4. If wrong P, prints a warning and returns the unfiltered
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dataset.
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"""
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if (P < 5):
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print("Warning: data returned unfiltered (P < 5)")
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if (P <= 4):
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print("Warning: data returned unfiltered (GaussHigh - Wrong P)")
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self.idx = 0
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return self.data
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@ -354,11 +382,11 @@ class Filter:
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return Y
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def BandPass(self, P=5, delta=0.3):
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def BandPass(self, P=10, delta=0.3):
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"""
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Band-pass (type, order, IIR).
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Example: delta = 0.3, P = 12
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(30% of P => 0.3, = 0.3*P, if P = 12 => 0.3*12 = 4)
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Band-pass filter.
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Example: delta = 0.3, P = 10 --> 0.3 * 10 = 3 --> band is [7, 13]
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"""
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beta = np.cos(2.0 * np.pi / P)
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gamma = np.cos(4.0 * np.pi * delta / P)
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@ -372,9 +400,9 @@ class Filter:
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def BandStop(self, P=5, delta=0.3):
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"""
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Band-stop (type, order, IIR)
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Example: delta = 0.3, P = 12
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(30% of P => 0.3, = 0.3*P, if P = 12 => 0.3*12 = 4)
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Band-stop filter.
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Example: delta = 0.3, P = 10 --> 0.3 * 10 = 3 --> band is [7, 13]
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"""
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beta = np.cos(2.0 * np.pi / P)
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gamma = np.cos(4.0 * np.pi * delta / P)
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@ -388,13 +416,15 @@ class Filter:
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def ZEMA1(self, N=10, alpha=None, K=1.0, Vn=5):
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"""
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Zero lag Exponential Moving Average (type 1).
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Zero-lag EMA (type 1). It is an alpha-beta type filter with sub-optimal
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parameters.
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If not given, <alpha> is determined as equivalent to a N-SMA.
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"""
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if (alpha is None):
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alpha = 2.0 / (N + 1.0)
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w = np.zeros(Vn+1)
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w = np.zeros(Vn + 1)
|
||||
w[0] = alpha * (1.0 + K)
|
||||
w[Vn] = - alpha * K
|
||||
|
||||
|
|
@ -406,7 +436,9 @@ class Filter:
|
|||
|
||||
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 (alpha is None):
|
||||
|
|
@ -420,7 +452,9 @@ class Filter:
|
|||
|
||||
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 (alpha is None):
|
||||
|
|
@ -433,15 +467,22 @@ class Filter:
|
|||
|
||||
return Y
|
||||
|
||||
def SincFunction(self, N=10, nel=10):
|
||||
def SincFilter(self, P=10, nel=10):
|
||||
"""
|
||||
Sinc function (order, FIR, pass).
|
||||
(N > 1, cut off at 0.5/N)
|
||||
Sinc function filter. The cut off point is at 0.5/P.
|
||||
|
||||
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)
|
||||
w = np.zeros(nel)
|
||||
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.a = np.array([1.0])
|
||||
|
|
@ -451,12 +492,14 @@ class Filter:
|
|||
|
||||
def Decycler(self, P=10):
|
||||
"""
|
||||
Decycler (?? order, IIR, pass ??). Gauss,HP,1st,P
|
||||
Built subtracting high pass Gauss filter from 1 (order 1)
|
||||
Must be P > 4. If not returns the unfiltered dataset.
|
||||
De-cycler filter. It is derived subtracting a 1st order high pass Gauss
|
||||
filter from 1.
|
||||
|
||||
Must be P > 4. If wrong P, prints a warning and returns the unfiltered
|
||||
dataset.
|
||||
"""
|
||||
if (P < 5):
|
||||
print("Warning: data returned unfiltered (P < 5)")
|
||||
if (P <= 4):
|
||||
print("Warning: data returned unfiltered (Decycler - Wrong P)")
|
||||
self.idx = 0
|
||||
return self.data
|
||||
|
||||
|
|
@ -465,17 +508,18 @@ class Filter:
|
|||
|
||||
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)
|
||||
P1 = 1st cut off period, P2 = 2nd cut off period. Automatically fixed.
|
||||
Must be P1, P2 > 4. If not returns the unfiltered dataset.
|
||||
Must be P > 4. If wrong P, prints a warning and returns the unfiltered
|
||||
dataset.
|
||||
"""
|
||||
P_low = np.amin([P1, P2])
|
||||
P_high = np.amax([P1, P2])
|
||||
|
||||
if (P_low < 5):
|
||||
print("Warning: data returned unfiltered (P_low < 5)")
|
||||
if (P_low <= 4):
|
||||
print("Warning: data returned unfiltered (DecyclerOsc - Wrong P)")
|
||||
self.idx = 0
|
||||
return self.data
|
||||
|
||||
|
|
@ -484,34 +528,38 @@ class Filter:
|
|||
|
||||
def ABG(self, alpha=0.0, beta=0.0, gamma=0.0, dt=1.0):
|
||||
"""
|
||||
alpha-beta-gamma
|
||||
For numerical stability: 0 < alpha, beta < 1
|
||||
Alpha-beta-gamma filter. It is a predictor-corrector type of filter.
|
||||
|
||||
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):
|
||||
alpha = np.ones(self.n_samples) * alpha
|
||||
alpha = np.ones(n_samples) * alpha
|
||||
if (np.ndim(beta) == 0):
|
||||
beta = np.ones(self.n_samples) * beta
|
||||
beta = np.ones(n_samples) * beta
|
||||
if (np.ndim(gamma) == 0):
|
||||
gamma = np.ones(self.n_samples) * gamma
|
||||
gamma = np.ones(n_samples) * gamma
|
||||
|
||||
# Initialize
|
||||
Y_corr = self.data.copy()
|
||||
Y_pred = self.data.copy()
|
||||
x0 = self.data[0, :]
|
||||
v0 = np.zeros(self.n_series)
|
||||
a0 = np.zeros(self.n_series)
|
||||
x0 = self.data[0]
|
||||
v0 = 0.0
|
||||
a0 = 0.0
|
||||
Y[0] = x0
|
||||
|
||||
for i in range(1, self.n_samples):
|
||||
for i in range(1, n_samples):
|
||||
|
||||
# Predictor (predicts state in <i>)
|
||||
x_pred = x0 + dt * v0 + 0.5 * a0 * dt ** 2.0
|
||||
v_pred = v0 + dt * a0
|
||||
a_pred = a0
|
||||
Y_pred[i, :] = x_pred
|
||||
|
||||
# Residual (innovation)
|
||||
r = self.data[i, :] - x_pred
|
||||
r = self.data[i] - x_pred
|
||||
|
||||
# Corrector (corrects state in <i>)
|
||||
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
|
||||
|
||||
# Save value and prepare next iteration
|
||||
Y_corr[i, :] = x_corr
|
||||
x0 = x_corr
|
||||
v0 = v_corr
|
||||
a0 = a_corr
|
||||
Y[i] = x_corr
|
||||
|
||||
self.idx = 1
|
||||
|
||||
return Y_corr, Y_pred
|
||||
return Y
|
||||
|
||||
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
|
||||
|
||||
|
|
@ -561,19 +614,23 @@ class Filter:
|
|||
beta = 2.0 * (1 - s) ** 2.0
|
||||
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)
|
||||
|
||||
return Y
|
||||
|
||||
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)
|
||||
h_db = 20.0 * np.log10(np.abs(h))
|
||||
wf = w / (2.0 * np.pi)
|
||||
h_db = 20.0 * np.log10(np.abs(h)) # Convert to decibels
|
||||
wf = w / (2.0 * np.pi) # Scale to [0, 0.5]
|
||||
|
||||
# Plot and format
|
||||
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.xlim(np.amin(wf), np.amax(wf))
|
||||
plt.xlabel(r'$\omega$ [rad/sample]')
|
||||
|
|
@ -584,9 +641,13 @@ class Filter:
|
|||
|
||||
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))
|
||||
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.grid(b=True)
|
||||
plt.xlim(np.amin(wf), np.amax(wf))
|
||||
|
|
|
|||
|
|
@ -49,14 +49,12 @@ def synthetic_wave(P, A=None, phi=None, num=1000):
|
|||
def synthetic_FFT(X, n_reps=1):
|
||||
"""
|
||||
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_fft (n, ) FFT of the original time-series
|
||||
X_synt_fft (n_reps, n) FFT of the 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, )
|
||||
n = len(X)
|
||||
|
|
@ -82,7 +80,7 @@ def synthetic_FFT(X, n_reps=1):
|
|||
|
||||
# FFT of the synthetic time-series (1st sample is unchanged)
|
||||
X_synt_fft = np.zeros((n_reps, n), dtype=complex)
|
||||
X_synt_fft[:, 0] = X_fft[0]
|
||||
X_synt_fft[:, 0] = X_fft[0]
|
||||
X_synt_fft[:, idx1] = X_fft[idx1] * phases1 # 1st half
|
||||
X_synt_fft[:, idx2] = X_fft[idx2] * phases2 # 2nd half
|
||||
|
||||
|
|
@ -95,13 +93,11 @@ def synthetic_FFT(X, n_reps=1):
|
|||
def synthetic_sampling(X, n_reps=1, replace=True):
|
||||
"""
|
||||
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
|
||||
idx (n_reps, n) Random index of X
|
||||
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, )
|
||||
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):
|
||||
"""
|
||||
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
|
||||
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
|
||||
and last interval
|
||||
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, )
|
||||
n = len(X)
|
||||
|
|
@ -259,16 +253,16 @@ def value2diff(X, percent=True):
|
|||
"""
|
||||
Returns the 1st discrete difference of array X.
|
||||
|
||||
X (n, ) Original dataset
|
||||
X (n, ) Input dataset
|
||||
dX (n-1, ) 1st discrete differences
|
||||
|
||||
|
||||
Notes:
|
||||
- the discrete difference can be calculated in percent or in value.
|
||||
- array dX is one element shorter than array X.
|
||||
- array X needs to be a 1D array (of any shape).
|
||||
- dX is one element shorter than X.
|
||||
- X needs to be a 1D array.
|
||||
"""
|
||||
X = X.flatten() # Reshape to (n, )
|
||||
|
||||
|
||||
# Discrete difference in percent
|
||||
if (percent):
|
||||
dX = X[1:] / X[:-1] - 1.0
|
||||
|
|
@ -282,36 +276,39 @@ def value2diff(X, 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
|
||||
X0 scalar Initial value
|
||||
X (n+1, ) Original dataset ?????
|
||||
|
||||
X (n+1, ) Output dataset
|
||||
|
||||
Notes:
|
||||
- the discrete difference can be in percent or in value.
|
||||
- array X is one element longer than array dX.
|
||||
- array dX needs to be a 1D array (of any shape).
|
||||
- X is one element longer than dX.
|
||||
- dX needs to be a 1D array.
|
||||
|
||||
If the discrete difference is in percent, the first column of X is set to
|
||||
one. The original array is X[0] * X
|
||||
If the discrete difference is in percent:
|
||||
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
|
||||
zero. X0+X
|
||||
- array X needs to be a 1D array (of any shape).
|
||||
If the discrete difference is in value:
|
||||
X[0] = X0
|
||||
X[1] = X[0] + dX[0]
|
||||
X[2] = X[1] + dX[1] = X[0] + dX[0] + dX[1]
|
||||
....
|
||||
"""
|
||||
dX = dX.flatten() # Reshape to (n, )
|
||||
n = len(dX)
|
||||
X = np.zeros(n+1)
|
||||
dX = dX.flatten() # Reshape to (n, )
|
||||
X = np.zeros(len(dX) + 1)
|
||||
X[0] = X0 # Initial value
|
||||
|
||||
# Discrete difference in percent
|
||||
if (percent):
|
||||
X[0] = X0
|
||||
X[1:] = np.cumprod(1.0 + dX) * X0 # !!!! check
|
||||
X[1:] = X0 * np.cumprod(1.0 + dX)
|
||||
|
||||
# Discrete difference in value
|
||||
else:
|
||||
X[0] = X0
|
||||
X[1:] = np.cumsum(dX) + X0 # !!!!!c check
|
||||
X[1:] = X0 + np.cumsum(dX)
|
||||
|
||||
return X
|
||||
|
|
|
|||
|
|
@ -5,10 +5,10 @@ Copyright (c) 2020 Gabriele Gilardi
|
|||
|
||||
|
||||
ToDo:
|
||||
- add comments to the code
|
||||
- in comments write what filters do
|
||||
- is necessary to copy X for Y untouched?
|
||||
- decide default values in functions
|
||||
- do examples and check type (low pass, etc)
|
||||
- put type in description
|
||||
- see paper ZEMA for tests
|
||||
- check results with ML
|
||||
"""
|
||||
|
||||
import sys
|
||||
|
|
@ -31,18 +31,21 @@ data_file = sys.argv[1] + '.csv'
|
|||
|
||||
# Read data from a csv file (one time-series each column)
|
||||
data = np.loadtxt(data_file, delimiter=',')
|
||||
print(data.shape)
|
||||
|
||||
t, f = syn.synthetic_wave([1., 2., 3.], A=None, phi=None, num=1000)
|
||||
plt.plot(t,f)
|
||||
plt.show()
|
||||
# t, f = syn.synthetic_wave([1., 2., 3.], A=None, phi=None, num=1000)
|
||||
# plt.plot(t,f)
|
||||
# plt.show()
|
||||
|
||||
# spx = flt.Filter(data)
|
||||
spx = flt.Filter(data)
|
||||
# res = spx.EMA(N=10)
|
||||
# signals = [spx.data, res[0:400]]
|
||||
# flt.plot_signals(signals, start=100)
|
||||
# flt.plot_signals(signals, ['SPX', 'SMA'])
|
||||
|
||||
# spx.plot_frequency()
|
||||
# spx.plot_lag()
|
||||
res = spx.BandPass(P=10, delta=0.3)
|
||||
|
||||
spx.plot_frequency()
|
||||
spx.plot_lag()
|
||||
|
||||
# sigma_x = 0.1
|
||||
# sigma_v = 0.1 * np.ones(n_samples)
|
||||
|
|
|
|||
10
README.md
10
README.md
|
|
@ -1,14 +1,12 @@
|
|||
# Signal Filtering
|
||||
# Signal Smoothing / Filtering and Generation of Synthetic Time-Series
|
||||
|
||||
## 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
|
||||
|
||||
|
|
|
|||
Ładowanie…
Reference in New Issue