2.11 Feature Engineering#

1. Introduction#

The process of feature engineering involves manipulating, transforming, and selecting raw data into features that can be used in statistical analysis and prediction.

Key types of features include:

Statistical features#

Statistical features summarize the distribution of data values, providing insights into the central tendency, dispersion, and shape of the dataset.

Mean: Average value of the data points.
Variance/Standard Deviation: Measure of the spread of the data.
Skewness: Asymmetry in the distribution of values.
Kurtosis: “Tailedness” of the distribution.
Percentiles : Threshold values for different segments of the distribution.

Temporal Features#

Temporal features focus on the time-dependent patterns within the series.

Autocorrelation: Correlation of a time series with a lagged version of itself.
Trend: Long-term increase or decrease in the data.
Seasonality: Regular patterns that repeat over a specific period.
Change Points: Locations where statistical properties of the series change significantly.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# Create a time series with trend and seasonality
t = np.arange(0, 365*3)
seasonal_series = np.random.randn(3*365) + np.linspace(0, 10, 3*365) + \
        2* np.sin(2 * np.pi * np.linspace(0, 3, 3*365))

# fit a trend using least squares regression
t = np.arange(len(seasonal_series))
trend = np.polyfit(t, seasonal_series, 1)[0] * t + np.polyfit(t, seasonal_series, 1)[1]
# calculate the seasonal component by subtracting the trend
seasonal = seasonal_series - trend
# calculate the autocorrelation of the seasonal component
autocorr = np.corrcoef(seasonal[:-1], seasonal[1:])[0, 1]
print(f'The autocorrelation of the seasonal component is {autocorr:.2f}')

# Plot the original time series, trend and seasonal components
plt.figure(figsize=(10, 6))
plt.plot(seasonal_series, label='Original Time Series')
plt.plot(trend, label='Trend')
plt.plot(seasonal, label='Seasonal')
plt.legend()

The autocorrelation of the seasonal component is 0.63

<matplotlib.legend.Legend at 0x319ac7fd0>

../_images/2.11_feature_engineering_1_2.png

crap = np.random.randn(3*365) 

autocorr = np.corrcoef(crap[:-1], crap[1:])[0, 1]
print(f'The autocorrelation of the seasonal component is {autocorr:.2f}')

The autocorrelation of the seasonal component is 0.01

Spatial features#

For geospatial data, spatial features capture the relationships between different locations in an image or dataset.

Texture: Patterns in the local intensity variations (e.g., smooth, rough).
Spatial Correlation: Measures how similar nearby locations are in terms of intensity values.

import numpy as np
import matplotlib.pyplot as plt
from scipy.ndimage import gaussian_filter
from scipy.fft import fft2, ifft2, fftshift

# Generate a 2D grid (e.g., geospatial data, such as a topographic map)
n = 100  # Size of the grid
x = np.linspace(0, 10, n)
y = np.linspace(0, 10, n)
X, Y = np.meshgrid(x, y)

# Generate 2D white noise
white_noise_2d = np.random.normal(0, 1, (n, n))

# Generate spatially correlated noise using a Gaussian filter
spatially_correlated_noise = gaussian_filter(white_noise_2d, sigma=3)

# Plot the synthetic 2D white noise, correlated noise, and autocorrelation function
plt.figure(figsize=(18, 6))

plt.subplot(1, 2, 1)
plt.imshow(white_noise_2d, extent=[0, 10, 0, 10], cmap='viridis')
plt.xlabel('x')
plt.ylabel('y')
plt.colorbar(label='Amplitude')
plt.title('2D White Noise')

# Spatially correlated noise
plt.subplot(1, 2, 2)
plt.imshow(spatially_correlated_noise, extent=[0, 10, 0, 10], cmap='viridis')
plt.xlabel('x')
plt.ylabel('y')
plt.colorbar(label='Amplitude')
plt.title('2D Spatially Correlated Noise')

Text(0.5, 1.0, '2D Spatially Correlated Noise')

../_images/2.11_feature_engineering_4_1.png

spatial correlation of the white noise

# Compute the 2D Fourier Transform of the spatially correlated noise
fft_white_noise = fft2(white_noise_2d)

# Compute the power spectrum
power_spectrum = np.abs(fft_white_noise) ** 2

# Compute the inverse 2D Fourier Transform of the power spectrum to get the autocorrelation function
autocorrelation_white = np.real(ifft2(power_spectrum))

# Shift the zero-frequency component to the center of the spectrum
autocorrelation_white = fftshift(autocorrelation_white)

# Normalize the autocorrelation function
autocorrelation_white /= autocorrelation_white.max()

spatial correlation of the correlated noise

# Compute the 2D Fourier Transform of the spatially correlated noise
fft_noise = fft2(spatially_correlated_noise)

# Compute the power spectrum
power_spectrum = np.abs(fft_noise) ** 2

# Compute the inverse 2D Fourier Transform of the power spectrum to get the autocorrelation function
autocorrelation = np.real(ifft2(power_spectrum))

# Shift the zero-frequency component to the center of the spectrum
autocorrelation = fftshift(autocorrelation)

# Normalize the autocorrelation function
autocorrelation /= autocorrelation.max()

# Plot the synthetic 2D white noise, correlated noise, and autocorrelation function
plt.figure(figsize=(11, 6))

# White noise (unfiltered)
plt.subplot(2, 2, 1)
plt.imshow(white_noise_2d, extent=[0, 10, 0, 10], cmap='viridis')
plt.xlabel('x')
plt.ylabel('y')
plt.colorbar(label='Amplitude')
plt.title('2D White Noise')


plt.subplot(2, 2, 2)
plt.imshow(autocorrelation_white, extent=[-5, 5, -5, 5], cmap='viridis')
plt.xlabel('x')
plt.ylabel('y')
plt.colorbar(label='Amplitude')
plt.title('2D White Noise Autocorrelation')


# Spatially correlated noise
plt.subplot(2, 2, 3)
plt.imshow(spatially_correlated_noise, extent=[0, 10, 0, 10], cmap='viridis')
plt.xlabel('x')
plt.ylabel('y')
plt.colorbar(label='Amplitude')
plt.title('2D Spatially Correlated Noise')

# Autocorrelation function
plt.subplot(2, 2, 4)
plt.imshow(autocorrelation, extent=[-5, 5, -5, 5], cmap='viridis')
plt.xlabel('x')
plt.ylabel('y')
plt.colorbar(label='Normalized Amplitude')
plt.title('Autocorrelation Function')

plt.tight_layout()
plt.show()

../_images/2.11_feature_engineering_9_0.png

Estimate the feature “correlation length” of the two images. The correlation length is often measured as the distance at which the autocorelation function decays to 1/e of its maximum value.

center = n // 2
# calculate correlation length for the white noise
autocorr_center = autocorrelation_white[center, center:]
distances = np.linspace(0, 5, center)
correlation_length_white = np.interp(1/np.e, autocorr_center[::-1], distances[::-1])

# calculate correlation length for the spatially correlated noise 
autocorr_center = autocorrelation[center, center:]
distances = np.linspace(0, 5, center)
correlation_length = np.interp(1/np.e, autocorr_center[::-1], distances[::-1])

print(f'Estimated correlation length for white noise: {correlation_length_white:.2f} and for spatially correlated noise: {correlation_length:.2f}')

Estimated correlation length for white noise: 0.06 and for spatially correlated noise: 0.66

Fractal Features#

Fractal features describe the self-similarity or complexity of the data at different scales. A common fractal feature is the Fractal Dimension, which quantifies the complexity of a dataset, especially useful in understanding geospatial structures and irregular patterns.

2. Seismic Time Series Data Example#

This lecture will demonstrate how to automatically extract features using a popular and simple Python package, tsfel, to extract common time series features. We will apply this to the example of seismic waveforms recorded in the Pacific Northwest.

An example will follow. The Pacific Northwest data includes labeled seismic waveforms for events of various origins:

Earthquakes
Explosions (mostly quarry blasts)
Surface events (such as avalanches and landslides)
Seismic noise (ambient Earth vibrations between events)

We will explore how these features vary across the four categories, or classes, of seismic events.

# Import modules for seismic data and feature extraction

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import scipy
import scipy.stats as st
import os
import h5py # for reading .h5 files

We download 2 files from the class storage: a CSV file with the waveforms themselves as an HDF5 file and their associated metadata as a CSV file:

#make sure to install wget
!pip install wget
import wget

Requirement already satisfied: wget in /Users/marinedenolle/opt/miniconda3/envs/mlgeo/lib/python3.9/site-packages (3.2)

[notice] A new release of pip is available: 23.3.1 -> 24.2
[notice] To update, run: pip install --upgrade pip

wget.download("https://dasway.ess.washington.edu/shared/niyiyu/PNW-ML/miniPNW_metadata.csv")
wget.download("https://dasway.ess.washington.edu/shared/niyiyu/PNW-ML/miniPNW_waveforms.hdf5")

'miniPNW_waveforms.hdf5'

Metadata#

We first read the metadata and arange them into a Pandas Data frame

import pandas as pd
# Handle irregular rows by skipping them
df = pd.read_csv("miniPNW_metadata.csv")
# Display the first few rows of the DataFrame
df.head()

	Unnamed: 0	event_id	source_origin_time	source_latitude_deg	source_longitude_deg	source_type	source_depth_km	preferred_source_magnitude	preferred_source_magnitude_type	preferred_source_magnitude_uncertainty	...	trace_S_onset	trace_P_onset	trace_snr_db	year	source_type_pnsn_label	source_local_magnitude	source_local_magnitude_uncertainty	source_duration_magnitude	source_duration_magnitude_uncertainty	source_hand_magnitude
0	0	uw61669232	2020-09-07T03:44:14.690000Z	46.560	-119.797	earthquake	23.300	1.30	ml	0.241000	...	impulsive	impulsive	-1.444\|2.612\|9.921	2020.0	eq	1.30	0.241273	1.17	0.187840	NaN
1	1	uw60888282	2014-10-08T15:39:31.330000Z	45.371	-121.708	earthquake	-0.947	1.67	ml	0.128000	...	impulsive	impulsive	0.368\|3.526\|5.981	2014.0	eq	1.67	0.128000	1.63	0.099000	NaN
2	2	uw61361706	2017-12-30T04:37:46.870000Z	46.165	-120.543	earthquake	13.520	2.46	ml	0.158000	...	impulsive	emergent	11.274\|13.32\|15.828	2017.0	eq	2.46	0.158000	3.37	0.389000	NaN
3	3	uw61639436	2020-06-09T23:37:10.420000Z	46.542	-119.589	earthquake	16.370	1.59	ml	0.157000	...	impulsive	emergent	27.007\|20.797\|19.252	2020.0	eq	1.59	0.156750	1.69	0.354773	NaN
4	4	uw61735446	2021-05-24T10:42:37.810000Z	46.857	-121.941	earthquake	12.380	0.83	ml	0.082983	...	impulsive	emergent	18.579\|18.912\|7.609	2021.0	eq	0.83	0.082983	0.50	0.381880	NaN

5 rows × 37 columns

The nature of the event source is located in one of the metadata attributes

df['source_type'].unique()

array(['earthquake', 'explosion', 'sonic_boom', 'thunder',
       'surface_event'], dtype=object)

Let’s assume that we are exploring features to classify the waveforms into the categories of the event types. We will attribute the *labels as the source_type attribute.

labels=df['source_type']
print(labels.value_counts())

source_type
earthquake       500
explosion        500
surface_event    500
sonic_boom       126
thunder           94
Name: count, dtype: int64

How many seismic waveforms are there in each of the category?

##

Would you say that this is a balanced data set with respect to the four classes of interest?

Now we will look at the seismic data, taking a random waveform from each of the categories

plt.hist(labels)

(array([500.,   0., 500.,   0.,   0., 126.,   0.,  94.,   0., 500.]),
 array([0. , 0.4, 0.8, 1.2, 1.6, 2. , 2.4, 2.8, 3.2, 3.6, 4. ]),
 <BarContainer object of 10 artists>)

../_images/2.11_feature_engineering_29_1.png

Now are read the data. It is stored in an HDF5 files under a finite number of groups. Each groups has an array of datasets that correspond to the waveforms. To link the metadata to the waveform files, the key trace_name has the dataset ID. The address is labeled as follows:

bucketX$i,:3,:n

where X is the HDF5 group number, i is the index. The file has typically 3 waveforms from each direction of ground motions N, E, Z. In the following exercise, we will focus on the vertical waveforms.

f = h5py.File("miniPNW_waveforms.hdf5", "r")

chan_list = ['N','E','Z']

Below a function to read the file in the data

def read_data(tn,f):
    """
    Function to read the waveform data from the .h5 file
    tn: trace_name of the waveform
    f: file object of the .h5 file
    """
    bucket, narray = tn.split('$')  # split the string of trace_name into bucket and narray
    x, y, z = iter([int(i) for i in narray.split(',:')]) # split the narray into x, y, z
    data = f['/data/%s' % bucket][x, :y, :z] # read the data as a 3D array
    return data

The trace name is stored as data attriobute in the metadata.

ldata=list(df['trace_name'])

ldata[0]

'bucket1$0,:3,:15001'

crap=read_data(ldata[530],f)
#print the dimension of a givan data
print(f'The first dimension of the data is {crap.shape[0]}')
print(f'The second dimension of the data is {crap.shape[1]}')

# the time vector of the data goes from 0 to 150 seconds with a sampling rate of 100Hz
t = np.linspace(-50, 100, crap.shape[1])

# plot an example of the data in a 3 row subplot
fig, ax = plt.subplots(3, 1, figsize=(10, 6))
for i in range(3):
    ax[i].plot(t,crap[i, :])
    ax[i].set_title(f'Channel {chan_list[i]}')
    ax[i].set_xlabel('Time (s)')
    ax[i].set_ylabel('Amplitude')
    ax[i].grid(True)
    ax[i].set_xlim(-50, 100)
plt.tight_layout()

The first dimension of the data is 3
The second dimension of the data is 15001

../_images/2.11_feature_engineering_38_1.png

We will just extract the Z component and reshape them into a single array.

nt=crap.shape[-1]
ndata=len(labels)
Z=np.zeros(shape=(ndata,nt))
for i in range(ndata-1):
    Z[i,:]=read_data(df.iloc[i]["trace_name"],f)[2,:nt]

Now we have data and its attributes, in particular the label as source type.

We are going to extract features automatically from tsfel and explore how varied the

!pip install tsfel
import tsfel

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[notice] A new release of pip is available: 23.3.1 -> 24.2
[notice] To update, run: pip install --upgrade pip

We need to format for input into tsfresh. It needs 1 column with the id (or label), one column for the time stamps (sort)

cfg = tsfel.get_features_by_domain()    
print(cfg)

{'spectral': {'FFT mean coefficient': {'complexity': 'constant', 'description': 'Computes the mean value of each spectrogram frequency.', 'function': 'tsfel.fft_mean_coeff', 'parameters': {'fs': 100, 'nfreq': 256}, 'n_features': 'nfreq', 'use': 'yes'}, 'Fundamental frequency': {'complexity': 'log', 'description': 'Computes the fundamental frequency.', 'function': 'tsfel.fundamental_frequency', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Human range energy': {'complexity': 'log', 'description': 'Computes the human range energy ratio given by the ratio between the energy in frequency 0.6-2.5Hz and the whole energy band.', 'function': 'tsfel.human_range_energy', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes', 'tag': 'inertial'}, 'LPCC': {'complexity': 'log', 'description': 'Computes the linear prediction cepstral coefficients.', 'function': 'tsfel.lpcc', 'parameters': {'n_coeff': 12}, 'n_features': 'n_coeff', 'use': 'yes', 'tag': 'audio'}, 'MFCC': {'complexity': 'constant', 'description': 'Computes the MEL cepstral coefficients.', 'function': 'tsfel.mfcc', 'parameters': {'cep_lifter': 22, 'fs': 100, 'nfft': 512, 'nfilt': 40, 'num_ceps': 12, 'pre_emphasis': 0.97}, 'n_features': 'num_ceps', 'use': 'yes', 'tag': ['audio', 'emg']}, 'Max power spectrum': {'complexity': 'log', 'description': 'Computes the maximum power spectrum density.', 'function': 'tsfel.max_power_spectrum', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Maximum frequency': {'complexity': 'log', 'description': 'Computes the maximum frequency.', 'function': 'tsfel.max_frequency', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Median frequency': {'complexity': 'log', 'description': 'Computes the median frequency.', 'function': 'tsfel.median_frequency', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Power bandwidth': {'complexity': 'log', 'description': 'Computes power spectrum density bandwidth of the signal.', 'function': 'tsfel.power_bandwidth', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral centroid': {'complexity': 'linear', 'description': 'Computes the barycenter of the spectrum.', 'function': 'tsfel.spectral_centroid', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes', 'tag': 'audio'}, 'Spectral decrease': {'complexity': 'log', 'description': 'Computes the amount of decreasing of the spectra amplitude.', 'function': 'tsfel.spectral_decrease', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral distance': {'complexity': 'log', 'description': 'Computes the signal spectral distance.', 'function': 'tsfel.spectral_distance', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral entropy': {'complexity': 'log', 'description': 'Computes the spectral entropy of the signal based on Fourier transform.', 'function': 'tsfel.spectral_entropy', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes', 'tag': 'eeg'}, 'Spectral kurtosis': {'complexity': 'linear', 'description': 'Computes the flatness of a distribution around its mean value.', 'function': 'tsfel.spectral_kurtosis', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral positive turning points': {'complexity': 'log', 'description': 'Computes number of positive turning points of the fft magnitude signal', 'function': 'tsfel.spectral_positive_turning', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral roll-off': {'complexity': 'log', 'description': 'Computes the frequency where 95% of the signal magnitude is contained below of this value.', 'function': 'tsfel.spectral_roll_off', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes', 'tag': 'audio'}, 'Spectral roll-on': {'complexity': 'log', 'description': 'Computes the frequency where 5% of the signal magnitude is contained below of this value.', 'function': 'tsfel.spectral_roll_on', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral skewness': {'complexity': 'linear', 'description': 'Computes the asymmetry of a distribution around its mean value.', 'function': 'tsfel.spectral_skewness', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral slope': {'complexity': 'log', 'description': 'Computes the spectral slope, obtained by linear regression of the spectral amplitude.', 'function': 'tsfel.spectral_slope', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral spread': {'complexity': 'linear', 'description': 'Computes the spread of the spectrum around its mean value.', 'function': 'tsfel.spectral_spread', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Spectral variation': {'complexity': 'log', 'description': 'Computes the amount of variation of the spectrum along time.', 'function': 'tsfel.spectral_variation', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Wavelet absolute mean': {'complexity': 'linear', 'description': 'Computes CWT absolute mean value of each wavelet scale.', 'function': 'tsfel.wavelet_abs_mean', 'parameters': {'function': 'scipy.signal.ricker', 'widths': 'np.arange(1,10)'}, 'n_features': 'widths', 'use': 'yes', 'tag': ['eeg', 'ecg']}, 'Wavelet energy': {'complexity': 'linear', 'description': 'Computes CWT energy of each wavelet scale.', 'function': 'tsfel.wavelet_energy', 'parameters': {'function': 'scipy.signal.ricker', 'widths': 'np.arange(1,10)'}, 'n_features': 'widths', 'use': 'yes', 'tag': 'eeg'}, 'Wavelet entropy': {'complexity': 'linear', 'description': 'Computes CWT entropy of the signal.', 'function': 'tsfel.wavelet_entropy', 'parameters': {'function': 'scipy.signal.ricker', 'widths': 'np.arange(1,10)'}, 'n_features': 1, 'use': 'yes', 'tag': 'eeg'}, 'Wavelet standard deviation': {'complexity': 'linear', 'description': 'Computes CWT std value of each wavelet scale.', 'function': 'tsfel.wavelet_std', 'parameters': {'function': 'scipy.signal.ricker', 'widths': 'np.arange(1,10)'}, 'n_features': 'widths', 'use': 'yes', 'tag': 'eeg'}, 'Wavelet variance': {'complexity': 'linear', 'description': 'Computes CWT variance value of each wavelet scale.', 'function': 'tsfel.wavelet_var', 'parameters': {'function': 'scipy.signal.ricker', 'widths': 'np.arange(1,10)'}, 'n_features': 'widths', 'use': 'yes', 'tag': 'eeg'}}, 'statistical': {'Absolute energy': {'complexity': 'log', 'description': 'Computes the absolute energy of the signal.', 'function': 'tsfel.abs_energy', 'parameters': '', 'n_features': 1, 'use': 'yes', 'tag': 'audio'}, 'Average power': {'complexity': 'constant', 'description': 'Computes the average power of the signal.', 'function': 'tsfel.average_power', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes', 'tag': 'audio'}, 'ECDF': {'complexity': 'log', 'description': 'Computes the values of ECDF (empirical cumulative distribution function) along the time axis.', 'function': 'tsfel.ecdf', 'parameters': {'d': 10}, 'n_features': 'd', 'use': 'yes'}, 'ECDF Percentile': {'complexity': 'log', 'description': 'Determines the percentile value of the ECDF.', 'function': 'tsfel.ecdf_percentile', 'parameters': {'percentile': '[0.2, 0.8]'}, 'n_features': 'percentile', 'use': 'yes'}, 'ECDF Percentile Count': {'complexity': 'log', 'description': 'Determines the cumulative sum of samples that are less than the percentile.', 'function': 'tsfel.ecdf_percentile_count', 'parameters': {'percentile': '[0.2, 0.8]'}, 'n_features': 'percentile', 'use': 'yes'}, 'Entropy': {'complexity': 'log', 'description': 'Computes the entropy of the signal using the Shannon Entropy.', 'function': 'tsfel.entropy', 'parameters': {'prob': 'standard'}, 'n_features': 1, 'use': 'yes', 'tag': 'eeg'}, 'Histogram': {'complexity': 'log', 'description': 'Computes histogram of the signal.', 'function': 'tsfel.hist', 'parameters': {'nbins': 10, 'r': 1}, 'n_features': 'nbins', 'use': 'yes'}, 'Interquartile range': {'complexity': 'constant', 'description': 'Computes interquartile range of the signal.', 'function': 'tsfel.interq_range', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Kurtosis': {'complexity': 'constant', 'description': 'Computes kurtosis of the signal.', 'function': 'tsfel.kurtosis', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Max': {'complexity': 'constant', 'description': 'Computes the maximum value of the signal.', 'function': 'tsfel.calc_max', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Mean': {'complexity': 'constant', 'description': 'Computes the mean value of the signal.', 'function': 'tsfel.calc_mean', 'parameters': '', 'n_features': 1, 'use': 'yes', 'tag': 'inertial'}, 'Mean absolute deviation': {'complexity': 'log', 'description': 'Computes mean absolute deviation of the signal.', 'function': 'tsfel.mean_abs_deviation', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Median': {'complexity': 'constant', 'description': 'Computes median of the signal.', 'function': 'tsfel.calc_median', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Median absolute deviation': {'complexity': 'constant', 'description': 'Computes median absolute deviation of the signal.', 'function': 'tsfel.median_abs_deviation', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Min': {'complexity': 'constant', 'description': 'Computes the minimum value of the signal.', 'function': 'tsfel.calc_min', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Peak to peak distance': {'complexity': 'constant', 'description': 'Computes the peak to peak distance.', 'function': 'tsfel.pk_pk_distance', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Root mean square': {'complexity': 'constant', 'description': 'Computes root mean square of the signal.', 'function': 'tsfel.rms', 'parameters': '', 'n_features': 1, 'use': 'yes', 'tag': ['emg', 'inertial']}, 'Skewness': {'complexity': 'constant', 'description': 'Computes skewness of the signal.', 'function': 'tsfel.skewness', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Standard deviation': {'complexity': 'constant', 'description': 'Computes standard deviation of the signal.', 'function': 'tsfel.calc_std', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Variance': {'complexity': 'constant', 'description': 'Computes variance of the signal.', 'function': 'tsfel.calc_var', 'parameters': '', 'n_features': 1, 'use': 'yes'}}, 'temporal': {'Area under the curve': {'complexity': 'log', 'description': 'Computes the area under the curve of the signal computed with trapezoid rule.', 'function': 'tsfel.auc', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Autocorrelation': {'complexity': 'constant', 'description': 'Computes autocorrelation of the signal.', 'function': 'tsfel.autocorr', 'parameters': '', 'n_features': 1, 'use': 'yes', 'tag': 'inertial'}, 'Centroid': {'complexity': 'constant', 'description': 'Computes the centroid along the time axis.', 'function': 'tsfel.calc_centroid', 'parameters': {'fs': 100}, 'n_features': 1, 'use': 'yes'}, 'Mean absolute diff': {'complexity': 'constant', 'description': 'Computes mean absolute differences of the signal.', 'function': 'tsfel.mean_abs_diff', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Mean diff': {'complexity': 'constant', 'description': 'Computes mean of differences of the signal.', 'function': 'tsfel.mean_diff', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Median absolute diff': {'complexity': 'constant', 'description': 'Computes median absolute differences of the signal.', 'function': 'tsfel.median_abs_diff', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Median diff': {'complexity': 'constant', 'description': 'Computes median of differences of the signal.', 'function': 'tsfel.median_diff', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Negative turning points': {'complexity': 'constant', 'description': 'Computes number of negative turning points of the signal.', 'function': 'tsfel.negative_turning', 'parameters': '', 'n_features': 1, 'use': 'yes', 'tag': 'emg'}, 'Neighbourhood peaks': {'complexity': 'constant', 'description': 'Computes the number of peaks from a defined neighbourhood of the signal.', 'function': 'tsfel.neighbourhood_peaks', 'parameters': {'n': 10}, 'n_features': 1, 'use': 'yes'}, 'Positive turning points': {'complexity': 'constant', 'description': 'Computes number of positive turning points of the signal.', 'function': 'tsfel.positive_turning', 'parameters': '', 'n_features': 1, 'use': 'yes', 'tag': 'emg'}, 'Signal distance': {'complexity': 'constant', 'description': 'Computes signal traveled distance.', 'function': 'tsfel.distance', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Slope': {'complexity': 'log', 'description': 'Computes the slope of the signal by fitting a linear equation to the observed data.', 'function': 'tsfel.slope', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Sum absolute diff': {'complexity': 'constant', 'description': 'Computes sum of absolute differences of the signal.', 'function': 'tsfel.sum_abs_diff', 'parameters': '', 'n_features': 1, 'use': 'yes'}, 'Zero crossing rate': {'complexity': 'constant', 'description': 'Computes Zero-crossing rate of the signal.', 'function': 'tsfel.zero_cross', 'parameters': '', 'n_features': 1, 'use': 'yes', 'tag': ['audio', 'emg']}}}

print(f'We have a total of {Z.shape[0]} data sample and each has {Z.shape[1]} data poins')

We have a total of 1720 data sample and each has 15001 data poins

for i in range(10): #Z.shape[0]):
    print(f'Extracting features from sample {i}/{Z.shape[0]}')
    if i==0:
        X=tsfel.time_series_features_extractor(cfg, Z[i,:],fs=100., verbose = 0)
        X['source_type']=df.iloc[i]['source_type']
    else:
        XX = tsfel.time_series_features_extractor(cfg, Z[i,:],fs=100., verbose = 0)
        XX['source_type']=df.iloc[i]['source_type']
        X=pd.concat([XX,X],axis=0,ignore_index=True)


# change the column names to remove the prefix 0_
X.columns = X.columns.str.replace('0_', '')

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X.head()

	Absolute energy	Area under the curve	Autocorrelation	Average power	Centroid	ECDF Percentile Count_0	ECDF Percentile Count_1	ECDF Percentile_0	ECDF Percentile_1	ECDF_0	...	Wavelet variance_1	Wavelet variance_2	Wavelet variance_3	Wavelet variance_4	Wavelet variance_5	Wavelet variance_6	Wavelet variance_7	Wavelet variance_8	Zero crossing rate	source_type
0	9.742864e+08	31207.637387	9.742864e+08	6.495243e+06	78.466491	3000.0	12000.0	-298.208729	151.301746	0.000067	...	1395.348318	2495.550398	3674.667437	5530.106304	8072.751642	11149.616998	14670.487177	18719.058340	235.0	earthquake
1	8.402399e+07	9063.005871	8.402399e+07	5.601600e+05	58.830080	3000.0	12000.0	-77.708776	48.153323	0.000067	...	921.217046	1395.363737	1567.561286	1256.932809	880.666612	682.995650	637.912389	674.893518	272.0	earthquake
2	9.015331e+07	6910.552647	9.015331e+07	6.010221e+05	63.002674	3000.0	12000.0	-29.206673	47.147688	0.000067	...	12754.759956	23536.460101	26033.368648	22937.003326	18408.577823	14822.361811	12640.580469	11471.174193	1085.0	earthquake
3	7.096144e+08	24553.426193	7.096144e+08	4.730763e+06	66.389598	3000.0	12000.0	-114.501055	213.176390	0.000067	...	27267.403246	55930.185792	66419.654741	64470.555161	60589.434231	58693.473343	60371.152906	66502.811011	803.0	earthquake
4	9.082897e+08	28869.941529	9.082897e+08	6.055265e+06	50.864641	3000.0	12000.0	-294.543856	38.136901	0.000067	...	5822.304317	10699.777184	15313.786784	17756.953569	18258.559868	18086.534963	17987.856997	18139.370933	1361.0	earthquake

5 rows × 390 columns

In a general code, you may package this into a function.

def calculate_features(Z):
    """
    Function to calculate features from the data
    Z: 2D array of seismic data data

    Returns:
    X: DataFrame of the features
    """
    for i in range(Z.shape[0]):
        print(f'Extracting features from sample {i}/{Z.shape[0]}')
        if i==0:
            X=tsfel.time_series_features_extractor(cfg, Z[i,:],fs=100., verbose = 0)
            X['source_type']=df.iloc[i]['source_type']
        else:
            XX = tsfel.time_series_features_extractor(cfg, Z[i,:],fs=100., verbose = 0)
            XX['source_type']=df.iloc[i]['source_type']
            X=pd.concat([XX,X],axis=0,ignore_index=True)

    # change the column names to remove the prefix 0_
    X.columns = X.columns.str.replace('0_', '')

    return X

# Calculate features from the seismic data and time it
import time
start = time.time()
X = calculate_features(Z)
end = time.time()
print(f'Time taken to calculate features: {end - start:.2f} seconds')

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Extracting features from sample 1695/1720
Extracting features from sample 1696/1720
Extracting features from sample 1697/1720
Extracting features from sample 1698/1720
Extracting features from sample 1699/1720
Extracting features from sample 1700/1720
Extracting features from sample 1701/1720
Extracting features from sample 1702/1720
Extracting features from sample 1703/1720
Extracting features from sample 1704/1720
Extracting features from sample 1705/1720
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Extracting features from sample 1709/1720
Extracting features from sample 1710/1720
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Extracting features from sample 1716/1720
Extracting features from sample 1717/1720
Extracting features from sample 1718/1720
Extracting features from sample 1719/1720
Time taken to calculate features: 1115.64 seconds

X.head()

	Absolute energy	Area under the curve	Autocorrelation	Average power	Centroid	ECDF Percentile Count_0	ECDF Percentile Count_1	ECDF Percentile_0	ECDF Percentile_1	ECDF_0	...	Wavelet variance_1	Wavelet variance_2	Wavelet variance_3	Wavelet variance_4	Wavelet variance_5	Wavelet variance_6	Wavelet variance_7	Wavelet variance_8	Zero crossing rate	source_type
0	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000	0.0	0.0	0.000000	0.000000	0.000067	...	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.0	surface_event
1	2.325603e+09	5.627850e+04	2.325603e+09	1.550402e+07	76.418530	3000.0	12000.0	292.999790	450.250905	0.000067	...	6.678689e+02	1.055538e+03	2.530908e+03	5.790352e+03	1.165462e+04	2.118641e+04	3.548474e+04	5.549514e+04	28.0	surface_event
2	8.705101e+11	7.477590e+05	8.705101e+11	5.803400e+09	74.967691	3000.0	12000.0	2434.988468	5493.021391	0.000067	...	9.263918e+07	2.386667e+08	2.936940e+08	2.466776e+08	1.717919e+08	1.109657e+08	6.954319e+07	4.344068e+07	258.0	surface_event
3	9.606099e+11	1.168706e+06	9.606099e+11	6.404066e+09	74.660640	3000.0	12000.0	-9228.169002	-6414.699254	0.000067	...	2.431281e+06	7.334504e+06	1.268369e+07	1.649068e+07	1.778231e+07	1.683463e+07	1.469724e+07	1.235497e+07	16.0	surface_event
4	3.595338e+11	7.343233e+05	3.595338e+11	2.396892e+09	74.976185	3000.0	12000.0	-4927.503218	-4863.251218	0.000067	...	4.896885e+03	1.023186e+04	1.774939e+04	2.750474e+04	3.936635e+04	5.330725e+04	6.934682e+04	8.750847e+04	0.0	surface_event

5 rows × 390 columns

New DataFrame Clean up#

Removing the samples with NaN or Infinity

print(f'no of samples in the dataframe:{X.shape[0]}')
# Replace infinities with NaN
new_X = X.replace([np.inf, -np.inf], np.nan, inplace= False)

# Drop rows with NaN values
new_X = new_X.dropna(inplace= False)
print(f'no of samples in the new dataframe:{new_X.shape[0]}')

no of samples in the dataframe:1720
no of samples in the new dataframe:1717

X.head()

	Absolute energy	Area under the curve	Autocorrelation	Average power	Centroid	ECDF Percentile Count_0	ECDF Percentile Count_1	ECDF Percentile_0	ECDF Percentile_1	ECDF_0	...	Wavelet variance_1	Wavelet variance_2	Wavelet variance_3	Wavelet variance_4	Wavelet variance_5	Wavelet variance_6	Wavelet variance_7	Wavelet variance_8	Zero crossing rate	source_type
0	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000	0.0	0.0	0.000000	0.000000	0.000067	...	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.000000e+00	0.0	surface_event
1	2.325603e+09	5.627850e+04	2.325603e+09	1.550402e+07	76.418530	3000.0	12000.0	292.999790	450.250905	0.000067	...	6.678689e+02	1.055538e+03	2.530908e+03	5.790352e+03	1.165462e+04	2.118641e+04	3.548474e+04	5.549514e+04	28.0	surface_event
2	8.705101e+11	7.477590e+05	8.705101e+11	5.803400e+09	74.967691	3000.0	12000.0	2434.988468	5493.021391	0.000067	...	9.263918e+07	2.386667e+08	2.936940e+08	2.466776e+08	1.717919e+08	1.109657e+08	6.954319e+07	4.344068e+07	258.0	surface_event
3	9.606099e+11	1.168706e+06	9.606099e+11	6.404066e+09	74.660640	3000.0	12000.0	-9228.169002	-6414.699254	0.000067	...	2.431281e+06	7.334504e+06	1.268369e+07	1.649068e+07	1.778231e+07	1.683463e+07	1.469724e+07	1.235497e+07	16.0	surface_event
4	3.595338e+11	7.343233e+05	3.595338e+11	2.396892e+09	74.976185	3000.0	12000.0	-4927.503218	-4863.251218	0.000067	...	4.896885e+03	1.023186e+04	1.774939e+04	2.750474e+04	3.936635e+04	5.330725e+04	6.934682e+04	8.750847e+04	0.0	surface_event

5 rows × 390 columns

Explore correlation among features#

import seaborn as sns
import matplotlib.pyplot as plt

# Compute pairwise correlation of columns
corr_matrix = new_X.drop('source_type',axis=1).corr().abs()

# Create a mask to only show the lower triangle of the matrix (since it's mirrored around its 
# top-left to bottom-right diagonal)
mask = np.triu(np.ones_like(corr_matrix, dtype=bool))

# Draw the heatmap with the mask
plt.figure(figsize=(15, 10))
sns.heatmap(corr_matrix, mask=mask, cmap='coolwarm', vmax=1, vmin=-1, center=0, square=True, linewidths=.5, 
            #cbar_kws={"shrink": .5}, 
            annot= False)

plt.show()

../_images/2.11_feature_engineering_56_0.png

Exploring the feature space for classification purpose#

Here we will plot distributions of certain features among the four classes. And we will explore what features are most correlated with each others in each of the categories.

def plot_feature_histograms(new_X, feature):
    # Get unique source types
    source_types = new_X['source_type'].unique()

    # Define colorblind-friendly palette
    colors = sns.color_palette('colorblind')

    for i, source_type in enumerate(source_types):
        # Select data for this source type
        data = new_X[new_X['source_type'] == source_type][feature]
        
        # Plot histogram for this source type
        plt.hist(np.log10(data), color=colors[i % len(colors)], alpha=0.5, label=source_type)

    # Add title
    plt.title(f'np.log10({feature})')
    
    # Add x-axis label
    plt.xlabel(f'np.log10({feature})')
    
    # Add y-axis label
    plt.ylabel('Count')
    
    # Add legend
    plt.legend()

    # Show plot
    plt.show()

feature = 'Area under the curve'
plot_feature_histograms(new_X, feature)

feature = 'Kurtosis'
plot_feature_histograms(new_X, feature)

feature = 'Spectral variation'
plot_feature_histograms(new_X, feature)

../_images/2.11_feature_engineering_59_0.png

/Users/marinedenolle/opt/miniconda3/envs/mlgeo/lib/python3.9/site-packages/pandas/core/arraylike.py:396: RuntimeWarning: invalid value encountered in log10
  result = getattr(ufunc, method)(*inputs, **kwargs)

../_images/2.11_feature_engineering_59_2.png

../_images/2.11_feature_engineering_59_3.png

Student exercise#

Which of the features will be best to classify among the event type classes?

ML Geo Curriculum

2.11 Feature Engineering

Contents

2.11 Feature Engineering#

1. Introduction#

Statistical features#

Temporal Features#

Spatial features#

Fractal Features#

2. Seismic Time Series Data Example#

Metadata#

New DataFrame Clean up#

Explore correlation among features#

Exploring the feature space for classification purpose#

Student exercise#