Filters

The filters module wraps some filters to use directly with ktk’s TimeSeries objects. These filters are convenience wrappers for numpy’s and scipy’s filters.

import kineticstoolkit.lab as ktk
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

In this tutorial, we will see the effect of filtering on a TimeSeries that includes different forms of signal:

ts_source = ktk.TimeSeries(time=np.linspace(0, 5, 1000))

# Generate some fancy functions
ts_source.data['square'] = np.array([int(t) % 2 == 0 for t in ts_source.time]) + 4.5
ts_source.data['triangle'] = np.block([
    0.01 * (sp.integrate.cumtrapz(ts_source.data['square'] - 5)) + 3, 4.])
ts_source.data['sine'] = 0.5 * np.sin(ts_source.time * np.pi) + 2
ts_source.data['pulse'] = np.array([float((i + 50) % 100 == 0) for i in range(1000)])

ts_source.plot()
plt.tight_layout()

Let’s add some white and quantification noise to these nice functions.

# White noise
np.random.seed(0)
noise = (np.random.rand(1000) - 0.5) * 1E-1
ts = ts_source.copy()
ts.data['square'] += noise
ts.data['triangle'] += noise
ts.data['sine'] += noise
ts.data['pulse'] += noise

# Quantification noise: resolution of 1E-2
for key in ts.data:
    ts.data[key] = np.floor(ts.data[key] * 1E2) * 1E-2

ts.plot()
plt.tight_layout()

Filtering a TimeSeries using a Butterworth filter

The ktk.filters.butter() function method applies a butterworth filter on a TimeSeries, using scipy.signal’s functions.

Here is how to filter using a no-lag Butterworth filter of order 2 at different cut-off frequencies:

plt.subplot(1, 2, 1)
ts.plot()
plt.title('Before filtering')

plt.subplot(1, 2, 2)
temp = ktk.filters.butter(ts, 20)
temp.plot()
plt.title('Low-pass 2nd order Butterworth at 20 Hz')
plt.tight_layout()

plt.subplot(1, 2, 1)
ts.plot()
plt.title('Before filtering')

plt.subplot(1, 2, 2)
temp = ktk.filters.butter(ts, 6)  # 4 Hz
temp.plot()
plt.title('Low-pass 2nd order Butterworth at 6 Hz')
plt.tight_layout()

Smoothing a TimeSeries with a moving average

The ktk.filters.smooth() function applies a moving average on TimeSeries data.

Here is how to smooth a signal using a moving average filter using different window lengths.

plt.subplot(1, 2, 1)
ts.plot()
plt.title('Before filtering')

plt.subplot(1, 2, 2)
temp = ktk.filters.smooth(ts, window_length=5)
temp.plot()
plt.title('Moving average of length 5')
plt.tight_layout()

plt.subplot(1, 2, 1)
ts.plot()
plt.title('Before filtering')

plt.subplot(1, 2, 2)
temp = ktk.filters.smooth(ts, window_length=25)
temp.plot()
plt.title('Moving average of length 25')
plt.tight_layout()

Smoothing a TimeSeries using a Savitzky-Golay filter

The ktk.filters.savgol() function applies the scipy.signal.savgol_filter filter to a TimeSeries. This is an extension of the moving average using a polynom of higher degrees than 0.

Here is how to smooth a signal using a Savitsky-Golay filter of order 2 using different window lengths.

plt.subplot(1, 2, 1)
ts.plot()
plt.title('Before filtering')

plt.subplot(1, 2, 2)
temp = ktk.filters.savgol(ts, window_length=5, poly_order=2)
temp.plot()
plt.title('2nd order smoothing Savitsky Golay of length 5')
plt.tight_layout()

plt.subplot(1, 2, 1)
ts.plot()
plt.title('Before filtering')

plt.subplot(1, 2, 2)
temp = ktk.filters.savgol(ts, window_length=25, poly_order=2)
temp.plot()
plt.title('2nd order smoothing Savitsky Golay of length 25')
plt.tight_layout()

Removing artefacts using a median filter

The ktk.filters.median() function filters a TimeSeries using a median filter. This type of filter is pretty powerful for removing artefacts that rarely happen twice in a row. Here is how to use this filter with a window length of 3.

plt.subplot(1, 2, 1)
ts.plot()
plt.title('Before filtering')

plt.subplot(1, 2, 2)
temp = ktk.filters.median(ts, window_length=3)
temp.plot()
plt.title('Median filter with a window length of 3')
plt.tight_layout()

Derivating using a differential filter

The ktk.filters.deriv() function derivates a signal in time using the centered difference formula. Since the derivative is calculated between the original data points, the resulting TimeSeries has a modified time vector that is also between the original time points.

Here is how to use this function to obtain the first derivative of a signal without preliminary filtering.

plt.subplot(1, 2, 1)
temp = ktk.filters.deriv(ts_source)
temp.plot()
plt.axis([0, 5, -5, 5])
plt.title('First derivative of the reference signal')

plt.subplot(1, 2, 2)
temp = ktk.filters.deriv(ts)
temp.plot()
plt.axis([0, 5, -5, 5])
plt.title('First derivative without filtering')

plt.tight_layout()

Obviously there is a lot of noise in this signal, which gets it very difficult to derivate.

Here is the same after filtering the data using Butterworth filters of different cut-off frequencies prior to derivating.

plt.subplot(1, 2, 1)
temp = ktk.filters.deriv(ts_source)
temp.plot()
plt.axis([0, 5, -5, 5])
plt.title('First derivative of the reference signal')

plt.subplot(1, 2, 2)
temp = ktk.filters.deriv(
    ktk.filters.butter(ts, fc=20))
temp.plot()
plt.axis([0, 5, -5, 5])
plt.title('First derivative after filtering at 20 Hz')

plt.tight_layout()

plt.subplot(1, 2, 1)
temp = ktk.filters.deriv(ts_source)
temp.plot()
plt.axis([0, 5, -5, 5])
plt.title('First derivative of the reference signal')

plt.subplot(1, 2, 2)
temp = ktk.filters.deriv(
    ktk.filters.butter(ts, fc=6))
temp.plot()
plt.axis([0, 5, -5, 5])
plt.title('First derivative after filtering at 6 Hz')

plt.tight_layout()

Derivating using a Savitsky-Golay filter

The ktk.filters.savgol() can smooth but also derivate with a very good performance. Here is how to derivate a signal using this type of filter.

plt.subplot(1, 2, 1)
temp = ktk.filters.deriv(ts_source)
temp.plot()
plt.axis([0, 5, -5, 5])
plt.title('First derivative of the reference signal')

plt.subplot(1, 2, 2)
temp = ktk.filters.savgol(ts, window_length=25, poly_order=2, deriv=1)
temp.plot()
plt.axis([0, 5, -5, 5])
plt.title('First derivative using a Savitzky-Golay filter of length 25')

plt.tight_layout()

For more information, please check the API Reference for the filters module.