test and format

Code_Python/filters.py

@@ -1,5 +1,5 @@
 """
-Signal Filtering/Smoothing and Generation of Synthetic Time-Series.
+Signal Filtering and Generation of Synthetic Time-Series.
 
 Copyright (c) 2020 Gabriele Gilardi
 
@@ -17,12 +17,12 @@ na              Number of coefficients in array <a>
 Notes:
 - the filter is applied starting from index idx = MAX(0, nb-1, na-1).
 - non filtered data are set equal to the input, i.e. Y[0:idx-1] = X[0:idx-1]
-- X needs to be a 1D array.
+- X must be a 1D array.
 
 
 Filter list:
 -----------
-Generic         b,              Generic
+Generic         b, a            Generic filter
 SMA             N               Simple moving average
 EMA             N/alpha         Exponential moving average
 WMA             N               Weighted moving average
@@ -66,10 +66,11 @@ from scipy import signal
 import matplotlib.pyplot as plt
 
 
-def plot_signals(signals, names=None, start=0):
+def plot_signals(signals, names=None, start=0, end=None):
     """
     Plot the signals specified in list <signals> with their names specified in
-    list <names>. Each signal is plotted in its full length.
+    list <names>. Each signal is plotted in its full length unless differently
+    specified.
     """
     # Identify the signals by index if their name is not specified
     if (names is None):
@@ -79,12 +80,12 @@ def plot_signals(signals, names=None, start=0):
         legend = names
 
     # Loop over the signals
+    t_max = 0
     for signal in signals:
 
         signal = signal.flatten()
-        end = len(signal)
-        t = np.arange(start, end)
-        plt.plot(t, signal[start:end])
+        t_max = np.amax([t_max, len(signal)])
+        plt.plot(signal)
 
         # If no name is given use the list index to identify the signals
         if (names is None):
@@ -96,6 +97,10 @@ def plot_signals(signals, names=None, start=0):
     plt.ylabel('Value')
     plt.grid(b=True)
     plt.legend(legend)
+    if (end is None):
+        plt.xlim(start, t_max)
+    else:
+        plt.xlim(start, end)
     plt.show()
 
 
@@ -315,7 +320,7 @@ class Filter:
 
         elif (N == 3):
             beta = np.exp(-np.pi / P)
-            alpha = 2.0 * beta * np.cos(1.738 * np.pi / P)
+            alpha = 2.0 * beta * np.cos(np.sqrt(3.0) * np.pi / P)
             wb = np.array([1.0, 1.0]) \
                  * (1.0 - alpha * (1.0 - beta ** 2.0) - beta ** 4.0) / 2.0
             wa = np.array([1.0, - (alpha + beta ** 2.0),
@@ -376,7 +381,7 @@ class Filter:
 
         self.b = np.array([1.0 - alpha / 2.0, -(1.0 - alpha / 2.0)])
         self.a = np.array([1.0, - (1.0 - alpha)])
-        Y = self.data - self.data[0, :]
+        Y = self.data - self.data[0]            # Shift to zero
         for i in range(N):
             Y, self.idx = filter_data(Y, self.b, self.a)
 
@@ -481,7 +486,7 @@ class Filter:
 
         K = np.arange(1, nel)
         w = np.zeros(nel)
-        w[0] = 1.0 / N
+        w[0] = 1.0 / P
         w[1:] = np.sin(np.pi * K / P) / (np.pi * K)
 
         self.b = w
@@ -534,7 +539,7 @@ class Filter:
         or an array with one value for each sample. For numerical stability it
         should be 0 < alpha, beta < 1.
         """
-        n_samples = len(data)
+        n_samples = len(self.data)
         Y = np.zeros(n_samples)
 
         # Change scalar arguments to arrays if necessary
@@ -615,20 +620,27 @@ class Filter:
             gamma = (beta ** 2.0) / (2.0 * alpha)
 
         # 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
 
-    def plot_frequency(self):
+    def plot_response(self):
         """
-        Plots the frequency response (in decibels) of the filter with transfer
-        response coefficients <b> and <a>.
+        Plots the frequency response (in decibels) and the lag (group delay)
+        of the filter with coefficients <b> and <a>.
         """
+        # Frequency response
         w, h = signal.freqz(self.b, self.a)
         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
+        # Lag / Group delay
+        w, gd = signal.group_delay((self.b, self.a))
+
+        # Scale frequency to [0, 0.5]
+        wf = w / (2.0 * np.pi)
+
+        # Plot and format frequency response
+        plt.subplot(1, 2, 1)
         plt.plot(wf, h_db)
         plt.axhline(-3.0, lw=1.5, ls='--', C='r')   # -3 dB (50% power loss)
         plt.grid(b=True)
@@ -637,20 +649,16 @@ class Filter:
         plt.ylabel('$h$ [db]')
         legend = ['Filter', '-3dB']
         plt.legend(legend)
-        plt.show()
+        plt.title('Frequency Response', fontweight="bold")
 
-    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)                  # Scale to [0, 0.5]
-
-        # Plot and format
+        # Plot and format lag/group delay
+        plt.subplot(1, 2, 2)
         plt.plot(wf, gd)
         plt.grid(b=True)
         plt.xlim(np.amin(wf), np.amax(wf))
         plt.xlabel(r'$\omega$ [rad/sample]')
         plt.ylabel('$gd$ [samples]')
+        plt.title('Lag / Group Delay', fontweight="bold")
+
+        # Show plots
         plt.show()

Code_Python/sine_fast.csv

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Code_Python/sine_fast_noise.csv

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Code_Python/sine_slow.csv

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Code_Python/sine_slow_noise.csv

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Code_Python/synthetic.py

@@ -1,5 +1,5 @@
 """
-Signal Filtering/Smoothing and Generation of Synthetic Time-Series.
+Signal Filtering and Generation of Synthetic Time-Series.
 
 Copyright (c) 2020 Gabriele Gilardi
 """
@@ -9,7 +9,7 @@ import numpy as np
 
 def synthetic_wave(P, A=None, phi=None, num=1000):
     """
-    Generates a multi-sine wave given a periods, amplitudes, and phases.
+    Generates a multi-sine wave given periods, amplitudes, and phases.
 
     P           (n, )               Periods
     A           (n, )               Amplitudes
@@ -46,10 +46,36 @@ def synthetic_wave(P, A=None, phi=None, num=1000):
     return t, f
 
 
+def synthetic_sampling(X, n_reps=1, replace=True):
+    """
+    Generates surrogates of the time-series X using randomized-sampling
+    (bootstrap) with or without replacement. Input X must 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
+    """
+    X = X.flatten()             # Reshape to (n, )
+    n = len(X)
+
+    # Sampling with replacement
+    if (replace):
+        idx = np.random.randint(0, n, size=(n_reps, n))
+
+    # Sampling without replacement
+    else:
+        idx = np.argsort(np.random.rand(n_reps, n), axis=1)
+
+    # Synthetic time-series
+    X_synt = X[idx]
+
+    return X_synt
+
+
 def synthetic_FFT(X, n_reps=1):
     """
-    Generates surrogates of the time-serie X using the phase-randomized
-    Fourier-transform algorithm. Input X needs to be a 1D array.
+    Generates surrogates of the time-series X using the phase-randomized
+    Fourier-transform algorithm. Input X must be a 1D array.
 
     X           (n, )               Original time-series
     X_fft       (n, )               FFT of the original time-series
@@ -90,36 +116,10 @@ def synthetic_FFT(X, n_reps=1):
     return X_synt
 
 
-def synthetic_sampling(X, n_reps=1, replace=True):
-    """
-    Generates surrogates of the time-serie X using randomized-sampling
-    (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
-    """
-    X = X.flatten()             # Reshape to (n, )
-    n = len(X)
-
-    # Sampling with replacement
-    if (replace):
-        idx = np.random.randint(0, n, size=(n_reps, n))
-
-    # Sampling without replacement
-    else:
-        idx = np.argsort(np.random.rand(n_reps, n), axis=1)
-
-    # Synthetic time-series
-    X_synt = X[idx]
-
-    return X_synt
-
-
 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. Input X needs to be a 1D array.
+    Generates surrogates of the time-series X using the maximum entropy
+    bootstrap algorithm. Input X must be a 1D array.
 
     X       (n, )           Original time-series
     idx     (n, )           Original order of X
@@ -259,7 +259,7 @@ def value2diff(X, percent=True):
     Notes:
     - the discrete difference can be calculated in percent or in value.
     - dX is one element shorter than X.
-    - X needs to be a 1D array.
+    - X must be a 1D array.
     """
     X = X.flatten()             # Reshape to (n, )
 
@@ -276,7 +276,7 @@ def value2diff(X, percent=True):
 
 def diff2value(dX, X0, percent=True):
     """
-    Returns array X from the 1st discrete difference using X0 as initial value.
+    Rebuilds array X from the 1st discrete difference using X0 as initial value.
 
     dX          (n, )           Discrete differences
     X0          scalar          Initial value
@@ -285,7 +285,7 @@ def diff2value(dX, X0, percent=True):
     Notes:
     - the discrete difference can be in percent or in value.
     - X is one element longer than dX.
-    - dX needs to be a 1D array.
+    - dX must be a 1D array.
 
     If the discrete difference is in percent:
         X[0] = X0

Code_Python/test.py

@@ -1,17 +1,12 @@
 """
-Signal Filtering/Smoothing and Generation of Synthetic Time-Series.
+Signal Filtering and Generation of Synthetic Time-Series.
 
 Copyright (c) 2020 Gabriele Gilardi
 
-
-ToDo:
-- do examples and check type (low pass, etc)
-- put type in description
-- see paper ZEMA for tests
-- check results with ML
 """
 
 import sys
+import warnings
 import numpy as np
 import matplotlib.pyplot as plt
 
@@ -20,83 +15,72 @@ import synthetic as syn
 
 # Added to avoid message warning "The group delay is singular at frequencies
 # [....], setting to 0" when plotting the lag for SMA filters. 
-np.seterr(all='ignore')
+# np.seterr(all='ignore')
+warnings.filterwarnings('ignore')
 np.random.seed(1294404794)
 
-# Read data to filter
+# Read example to run
 if len(sys.argv) != 2:
-    print("Usage: python test.py <data_file>")
+    print("Usage: python test.py <example>")
     sys.exit(1)
-data_file = sys.argv[1] + '.csv'
+example = sys.argv[1]
 
-# Read data from a csv file (one time-series each column)
-data = np.loadtxt(data_file, delimiter=',')
-print(data.shape)
+# Read data from a csv file
+data_file = 'spx'
+data = np.loadtxt(data_file + '.csv', delimiter=',')
 
-# t, f = syn.synthetic_wave([1., 2., 3.], A=None, phi=None, num=1000)
-# plt.plot(t,f)
-# plt.show()
+# Example with a few filters
+if (example == 'Filters'):
+    spx = flt.Filter(data)
+    ema = spx.EMA(N=10)
+    butter = spx.ButterMod(P=10, N=2)
+    zema = spx.ZEMA2(N=10, K=2.0)
+    signals = [spx.data, ema, butter, zema]
+    names = ['SPX', 'EMA', 'ButterMod', 'ZEMA2']
+    flt.plot_signals(signals, names=names, start=0, end=200)
 
-spx = flt.Filter(data)
-# res = spx.EMA(N=10)
-# signals = [spx.data, res[0:400]]
-# flt.plot_signals(signals, ['SPX', 'SMA'])
+# Example with the three types of Kalman filter
+elif (example == 'Kalman'):
+    spx = flt.Filter(data)
+    a_type = spx.Kalman(sigma_x=0.1, sigma_v=0.1, dt=1.0, abg_type="a")
+    ab_type = spx.Kalman(sigma_x=0.1, sigma_v=0.1, dt=1.0, abg_type="ab")
+    abg_type = spx.Kalman(sigma_x=0.1, sigma_v=0.1, dt=1.0, abg_type="abg")
+    signals = [spx.data, a_type, ab_type, abg_type]
+    names = ['SPX', 'a-type', 'ab-type', 'abg-type']
+    flt.plot_signals(signals, names=names, start=0, end=200)
 
-res = spx.BandPass(P=10, delta=0.3)
+# Example of filter frequency and lag
+elif (example == 'Response'):
+    spx = flt.Filter(data)
+    band = spx.BandPass(P=10, delta=0.3)
+    spx.plot_response()
 
-spx.plot_frequency()
-spx.plot_lag()
+# Example of surrogates time-series using the Fourier-transform algorithm
+elif (example == 'FFT_boot'):
+    pass
 
-# sigma_x = 0.1
-# sigma_v = 0.1 * np.ones(n_samples)
-# res = spx.Kalman(sigma_x=sigma_x, sigma_v=sigma_v, dt=1.0, abg_type="abg")
-# alpha = 0.5
-# beta = 0.005
-# gamma = 0.0
-# Yc, Yp = spx.ABG(alpha=alpha, beta=beta, gamma=gamma, dt=1.0)
-# signals = (spx.data[:, 0], Yc[:, 0], Yp[:, 0])
-# utl.plot_signals(signals, 0, 50)
+# Example of surrogates time-series using maximum entropy bootstrap algorithm
+elif (example == 'ME_boot'):
+    pass
 
-# t, f = syn.synthetic_wave([1., 2., 3.], A=None, phi=None, num=100)
-# plt.plot(t,f)
-# plt.show()
-# aa = np.array([
-#         [ 0.8252,  0.2820],
-#         [ 1.3790,  0.0335],
-#         [-1.0582, -1.3337],
-#         [-0.4686,  1.1275],
-#         [-0.2725,  0.3502],
-#         [ 1.0984, -0.2991],
-#         [-0.2779,  0.0229],
-#         [ 0.7015, -0.2620],
-#         [-2.0518, -1.7502],
-#         [-0.3538, -0.2857],
-#         [-0.8236, -0.8314],
-#         [-1.5771, -0.9792],
-#         [ 0.5080, -1.1564]])
-# synt_data = syn.synthetic_FFT(aa[0:5,0], n_reps=1)
-# print(synt_data)
+else:
+    print("Example not found")
+    sys.exit(1)
+    
 
-# plt.plot(synt_data1)
-# plt.plot(synt_data2)
+# dX = syn.value2diff(data, percent=True)
+# # X = syn.diff2value(dX, data[0], percent=True)
+# signals = [data, X]
+# flt.plot_signals(signals, start=0, end=200)
+
+# dX = syn.value2diff(data, percent=False)
+# dX_synt = syn.synthetic_sampling(dX, n_reps=1, replace=True)
+# dX_synt = syn.synthetic_FFT(dX, n_reps=2)
+# dX_synt = syn.synthetic_MEboot(dX, n_reps=4, alpha=0.1, bounds=False, scale=False)
+# for i in range(dX_synt.shape[0]):
+#     # aa = dX_synt[:, i]
+#     aa = syn.diff2value(dX_synt[i, :], data[0], percent=False)
+#     plt.plot(aa)
+# plt.xlim(0, 200)
 # plt.plot(data)
-# names = ['syn1', 'syn2', 'spx']
-# plt.legend(names)
 # plt.show()
-# percent = False
-# print(data[0:10, :])
-# bb = syn.value2diff(data, percent)
-# print(bb[0:10, :])
-# cc = syn.diff2value(bb, percent)
-# print(cc[0:10, :]+1399.48)
-
-# i = np.arange(aa.shape[1])
-# bb[:, i] = aa[idx[:, i], i]
-# aa = np.array([4, 12, 36, 20, 8])
-# bb = syn.synthetic_sampling(aa, n_reps=2, replace=True)
-# print(bb)
-# aa = np.array([4, 12, 36, 20, 8])
-# W = syn.synthetic_MEboot(aa, n_reps=1, alpha=0.1, bounds=False, scale=False)
-# print('W=')
-# print(W)
-

README.md

@@ -1,4 +1,4 @@
-# Signal Smoothing / Filtering and Generation of Synthetic Time-Series
+# Signal Filtering and Generation of Synthetic Time-Series
 
 ## Reference
 
@@ -10,6 +10,57 @@
 
 ## Characteristics
 
-## Parameters
+- The code has been written and tested in Python 3.7.7.
+- Implementation of several digital signal filters and functions for the generation of synthetic (surrogate) time-series.
+- Filter list (*filters.py*):
+  - **Generic** Generic filter.
+  - **SMA** Simple moving average.
+  - **EMA** Exponential moving average.
+  - **WMA** Weighted moving average.
+  - **MSMA** Modified simple moving average.
+  - **MLSQ** Modified least-squares quadratic.
+  - **ButterOrig** Butterworth original filter.
+  - **ButterMod** Butterworth modified filter.
+  - **SuperSmooth** Supersmoother filter.
+  - **GaussLow** Gauss low pass filter.
+  - **GaussHigh** Gauss high pass filter.
+  - **BandPass** Band-pass filter.
+  - **BandStop** Band-stop filter.
+  - **ZEMA1** Zero-lag EMA (type 1).
+  - **ZEMA2** Zero-lag EMA (type 2).
+  - **InstTrend** Instantaneous trendline.
+  - **SincFilter** Sinc function filter.
+  - **Decycler** De-cycler filter.
+  - **DecyclerOsc** De-cycle oscillator.
+  - **ABG** Alpha-beta-gamma filter.
+  - **Kalman** One-dimensional steady-state Kalman filter.
+- Synthetic time-series (*synthetic.py*):
+  - **synthetic_wave** Generates multi-sine wave given periods, amplitudes, and phases.
+  - **synthetic_sampling** Generates surrogates using randomized-sampling (bootstrap) with or without replacement.
+  - **synthetic_FFT** Generates surrogates using the phase-randomized Fourier-transform algorithm.
+  - **synthetic_MEboot** Generates surrogates using the maximum entropy bootstrap algorithm.
+- File *filters.py* includes also functions to plot the filter signal, frequency response, and group delay.
+- File *synthetic.py* includes also functions to differentiate, integrate, normalize, and scale the discrete time-series.
+- Usage: *python test.py example*.
+
+## Main Parameters
+
+`example` Name of the example to run.
+
+`data_file` File name with the dataset (csv format). The extension is added automatically.
+
+`X` Dataset to filter/time-series (input). It must be a 1D array, i.e. of shape `(:, )` or `(:, 1)` or `(1, :)`.
+
+`b` Transfer response coefficients (numerator).
+
+`a` Transfer response coefficients (denominator).
+
+`Y` Filtered dataset (output).
+
+`X_synt` Synthetic time-series (output)
+
+`n_reps` Number of surrogates/synthetic time-series to generate.
 
 ## Examples
+
+There are five examples: **Filters**, **Kalman**, **Response**, **FFT_boot**, **ME_boot**. For all, the dataset in *spx.csv* is used.