kopia lustrzana https://github.com/gabrielegilardi/SignalFilters
595 wiersze
19 KiB
Python
595 wiersze
19 KiB
Python
"""
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Class for filter/smooth data.
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Copyright (c) 2020 Gabriele Gilardi
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References (both from John F. Ehlers):
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[1] "Cycle Analytics for Traders: Advanced Technical Trading Concepts".
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[2] "Signal Analysis, Filters And Trading Strategies".
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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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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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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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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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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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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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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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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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"""
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import numpy as np
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from scipy import signal
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def plot_signals(signals, idx_start=0, idx_end=None):
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"""
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signals must be all same lenght
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"""
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if (idx_end is None):
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idx_end = len(signals[0])
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t = np.arange(idx_start, idx_end)
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names = []
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count = 0
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for signal in signals:
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plt.plot(t, signal[idx_start:idx_end])
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names.append(str(count))
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count += 1
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plt.grid(b=True)
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plt.legend(names)
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plt.show()
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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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"""
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n_samples, n_series = data.shape
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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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Y = data.copy()
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for i in range(idx, n_samples):
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tmp = np.zeros(n_series)
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for j in range(nb):
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tmp = tmp + b[j] * data[i-j, :] # Numerator term
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for j in range(1, na):
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tmp = tmp - a[j] * Y[i-j, :] # Denominator term
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Y[i, :] = tmp / a[0]
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return Y, idx
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class Filter:
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def __init__(self, data):
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"""
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"""
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self.data = np.asarray(data)
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self.n_samples, self.n_series = data.shape
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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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"""
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self.b = np.asarray(b)
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self.a = np.asarray(a)
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def SMA(self, N=10):
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"""
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Simple moving average (?? order, FIR, ?? band).
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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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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def EMA(self, N=10, alpha=None):
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"""
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Exponential moving average (?? order, IIR, pass ??).
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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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self.b = np.array([alpha])
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self.a = np.array([1.0, - (1.0 - alpha)])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def WMA(self, N=10):
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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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"""
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w = np.arange(N, 0, -1)
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self.b = w / np.sum(w)
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self.a = np.array([1.0])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def MSMA(self, N=10):
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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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"""
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w = np.ones(N+1)
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w[0] = 0.5
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w[N] = 0.5
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self.b = w / N
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self.a = np.array([1.0])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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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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"""
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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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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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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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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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else:
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print("Warning: data returned unfiltered (wrong N)")
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self.idx = 0
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return self.data
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self.b = w
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self.a = np.array([1.0])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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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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"""
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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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alpha = 2.0 * beta * np.cos(np.sqrt(2.0) * np.pi / P)
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wb = np.array([1.0, 2.0, 1.0]) * (1.0 - alpha + beta ** 2.0) / 4.0
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wa = np.array([1.0, - alpha, beta ** 2.0])
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elif (N == 3):
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beta = np.exp(-np.pi / P)
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alpha = 2.0 * beta * np.cos(np.sqrt(3.0) * np.pi / P)
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wb = np.array([1.0, 3.0, 3.0, 1.0]) * (1.0 - beta ** 2.0) \
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* (1.0 - alpha + beta ** 2.0) / 8.0
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wa = np.array([1.0, - (alpha + beta ** 2.0),
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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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self.idx = 0
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return self.data
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self.b = wb
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self.a = wa
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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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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"""
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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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alpha = 2.0 * beta * np.cos(np.sqrt(2.0) * np.pi / P)
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wb = np.array([1.0 - alpha + beta ** 2.0])
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wa = np.array([1.0, - alpha, beta ** 2.0])
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elif (N == 3):
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beta = np.exp(-np.pi / P)
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alpha = 2.0 * beta * np.cos(np.sqrt(3.0) * np.pi / P)
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wb = np.array([1.0 - alpha * (1.0 - beta ** 2.0) - beta ** 4.0])
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wa = np.array([1.0, - (alpha + beta ** 2.0),
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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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self.idx = 0
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return self.data
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self.b = wb
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self.a = wa
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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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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"""
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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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alpha = 2.0 * beta * np.cos(np.sqrt(2.0) * np.pi / P)
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wb = np.array([1.0, 1.0]) * (1.0 - alpha + beta ** 2.0) / 2.0
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wa = np.array([1.0, - alpha, beta ** 2.0])
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elif (N == 3):
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beta = np.exp(-np.pi / P)
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alpha = 2.0 * beta * np.cos(1.738 * np.pi / P)
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wb = np.array([1.0, 1.0]) \
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* (1.0 - alpha * (1.0 - beta ** 2.0) - beta ** 4.0) / 2.0
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wa = np.array([1.0, - (alpha + beta ** 2.0),
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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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self.idx = 0
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return self.data
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self.b = wb
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self.a = wa
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def GaussLow(self, N=1, P=2):
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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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"""
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if (P < 2):
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print("Warning: data returned unfiltered (P < 2)")
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self.idx = 0
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return self.data
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A = 2.0 ** (1.0 / N) - 1.0
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B = 4.0 * np.sin(np.pi / P) ** 2.0
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C = 2.0 * (np.cos(2.0 * np.pi / P) - 1.0)
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alpha = (-B + np.sqrt(B ** 2.0 - 4.0 * A * C)) / (2.0 * A)
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self.b = np.array([alpha])
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self.a = np.array([1.0, - (1.0 - alpha)])
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Y = self.data.copy()
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for i in range(N):
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Y, self.idx = filter_data(Y, self.b, self.a)
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return Y
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def GaussHigh(self, N=1, P=5):
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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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"""
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if (P < 5):
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print("Warning: data returned unfiltered (P < 5)")
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self.idx = 0
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return self.data
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A = 2.0 ** (1.0 / N) * np.sin(np.pi / P) ** 2.0 - 1.0
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B = 2.0 * (2.0 ** (1.0 / N) - 1.0) * (np.cos(2.0 * np.pi / P) - 1.0)
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C = - B
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alpha = (-B - np.sqrt(B ** 2.0 - 4.0 * A * C)) / (2.0 * A)
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self.b = np.array([1.0 - alpha / 2.0, -(1.0 - alpha / 2.0)])
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self.a = np.array([1.0, - (1.0 - alpha)])
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Y = self.data - self.data[0, :]
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for i in range(N):
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Y, self.idx = filter_data(Y, self.b, self.a)
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return Y
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def BandPass(self, P=5, 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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"""
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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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alpha = 1.0 / gamma - np.sqrt(1.0 / gamma ** 2 - 1.0)
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self.b = np.array([(1.0 - alpha) / 2.0, 0.0, - (1.0 - alpha) / 2.0])
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self.a = np.array([1.0, - beta * (1.0 + alpha), alpha])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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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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"""
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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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alpha = 1.0 / gamma - np.sqrt(1.0 / gamma ** 2 - 1.0)
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self.b = np.array([1.0, - 2.0 * beta, 1.0]) * (1.0 + alpha) / 2.0
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self.a = np.array([1.0, - beta * (1.0 + alpha), alpha])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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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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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[0] = alpha * (1.0 + K)
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w[Vn] = - alpha * K
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self.b = w
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self.a = np.array([1.0, - (1.0 - alpha)])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def ZEMA2(self, N=10, alpha=None, K=1.0):
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"""
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Zero lag Exponential Moving Average (type 2).
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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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self.b = np.array([alpha * (1.0 + K)])
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self.a = np.array([1.0, alpha * K - (1.0 - alpha)])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def InstTrend(self, N=10, alpha=None):
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"""
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Instantaneous Trendline (2nd order, IIR, low pass).
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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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self.b = np.array([alpha - alpha ** 2.0 / 4.0, alpha ** 2.0 / 2.0,
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- alpha + 3.0 * alpha ** 2.0 / 4.0])
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self.a = np.array([1.0, - 2.0 * (1.0 - alpha), (1.0 - alpha) ** 2.0])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def SincFunction(self, N=10, nel=10):
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"""
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Sinc function (order, FIR, pass).
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(N > 1, cut off at 0.5/N)
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"""
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K = np.arange(1, nel)
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w = np.zeros(nel)
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w[0] = 1.0 / N
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w[1:] = np.sin(np.pi * K / N) / (np.pi * K)
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self.b = w
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self.a = np.array([1.0])
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Y, self.idx = filter_data(self.data, self.b, self.a)
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return Y
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def Decycler(self, P=10):
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"""
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Decycler (?? order, IIR, pass ??). Gauss,HP,1st,P
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Built subtracting high pass Gauss filter from 1 (order 1)
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Must be P > 4. If not returns the unfiltered 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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self.idx = 0
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return self.data
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Y = self.data - self.GaussHigh(N=1, P=P)
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return Y
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def DecyclerOsc(self, P1=5, P2=10):
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"""
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DecyclerOsc (?? order 2, IIR, pass ??).
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(Gauss, HP, 2nd order, Pmax - Gauss, HP, 2nd order, Pmin)
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P1 = 1st cut off period, P2 = 2nd cut off period. Automatically fixed.
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Must be P1, P2 > 4. If not returns the unfiltered dataset.
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"""
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P_low = np.amin([P1, P2])
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P_high = np.amax([P1, P2])
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if (P_low < 5):
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print("Warning: data returned unfiltered (P_low < 5)")
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self.idx = 0
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return self.data
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Y = self.GaussHigh(N=2, P=P_low) - self.GaussHigh(N=2, P=P_high)
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return Y
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def ABG(self, alpha=0.0, beta=0.0, gamma=0.0, dt=1.0):
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"""
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alpha-beta-gamma
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For numerical stability: 0 < alpha, beta < 1
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"""
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# If necessary change scalars to arrays
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if (np.ndim(alpha) == 0):
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alpha = np.ones(self.n_samples) * alpha
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if (np.ndim(beta) == 0):
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beta = np.ones(self.n_samples) * beta
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if (np.ndim(gamma) == 0):
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gamma = np.ones(self.n_samples) * gamma
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|
|
# Initialize
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|
Y_corr = self.data.copy()
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Y_pred = self.data.copy()
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x0 = self.data[0, :]
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v0 = np.zeros(self.n_series)
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a0 = np.zeros(self.n_series)
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|
|
|
for i in range(1, self.n_samples):
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|
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|
# Predictor (predicts state in <i>)
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x_pred = x0 + dt * v0 + 0.5 * a0 * dt ** 2.0
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v_pred = v0 + dt * a0
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|
a_pred = a0
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Y_pred[i, :] = x_pred
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|
|
|
# Residual (innovation)
|
|
r = self.data[i, :] - x_pred
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|
|
|
# Corrector (corrects state in <i>)
|
|
x_corr = x_pred + alpha[i] * r
|
|
v_corr = v_pred + (beta[i] / dt) * r
|
|
a_corr = a_pred + (2.0 * gamma[i] / dt ** 2.0) * r
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|
|
|
# Save value and prepare next iteration
|
|
Y_corr[i, :] = x_corr
|
|
x0 = x_corr
|
|
v0 = v_corr
|
|
a0 = a_corr
|
|
|
|
self.idx = 1
|
|
|
|
return Y_corr, Y_pred
|
|
|
|
def Kalman(self, sigma_x, sigma_v, dt, abg_type="abg"):
|
|
"""
|
|
Steady-state Kalman filter (also limited to one-dimension)
|
|
"""
|
|
L = (sigma_x / sigma_v) * dt ** 2.0
|
|
|
|
# Alpha filter
|
|
if (abg_type == 'a'):
|
|
alpha = (-L ** 2.0 + np.sqrt(L ** 4.0 + 16.0 * L ** 2.0)) / 8.0
|
|
beta = 0.0
|
|
gamma = 0.0
|
|
|
|
# Alpha-Beta filter
|
|
elif(abg_type == 'ab'):
|
|
r = (4.0 + L - np.sqrt(8.0 * L + L ** 2.0)) / 4.0
|
|
alpha = 1.0 - r ** 2.0
|
|
beta = 2.0 * (2.0 - alpha) - 4.0 * np.sqrt(1.0 - alpha)
|
|
gamma = 0.0
|
|
|
|
# Alpha-Beta-Gamma filter
|
|
else:
|
|
b = (L / 2.0) - 3.0
|
|
c = (L / 2.0) + 3.0
|
|
d = - 1.0
|
|
p = c - b ** 2.0 / 3.0
|
|
q = (2.0 * b ** 3.0) / 27.0 - (b * c) / 3.0 + d
|
|
v = np.sqrt(q ** 2.0 + (4.0 * p ** 3.0) / 27.0)
|
|
z = - (q + v / 2.0) ** (1.0 / 3.0)
|
|
s = z - p / (3.0 * z) - b / 3.0
|
|
alpha = 1.0 - s ** 2.0
|
|
beta = 2.0 * (1 - s) ** 2.0
|
|
gamma = (beta ** 2.0) / (2.0 * alpha)
|
|
|
|
# Apply filter
|
|
Y = self.abg(alpha=alpha, beta=beta, gamma=gamma, dt=dt)
|
|
|
|
return Y
|
|
|
|
def plot_frequency(self):
|
|
"""
|
|
"""
|
|
w, h = signal.freqz(self.b, self.a)
|
|
h_db = 20.0 * np.log10(np.abs(h))
|
|
wf = w / (2.0 * np.pi)
|
|
plt.plot(wf, h_db)
|
|
plt.axhline(-3.0, lw=1.5, ls='--', C='r')
|
|
plt.grid(b=True)
|
|
plt.xlim(np.amin(wf), np.amax(wf))
|
|
plt.xlabel('$omega$ [rad/sample]')
|
|
plt.ylabel('$h$ [db]')
|
|
plt.show()
|
|
|
|
def plot_lag(self):
|
|
"""
|
|
"""
|
|
w, gd = signal.group_delay((self.b, self.a))
|
|
wf = w / (2.0 * np.pi)
|
|
plt.plot(wf, gd)
|
|
plt.grid(b=True)
|
|
plt.xlim(np.amin(wf), np.amax(wf))
|
|
plt.xlabel('$omega$ [rad/sample]')
|
|
plt.ylabel('$gd$ [samples]')
|
|
plt.show()
|