DataFrames - Part 1

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Data Frames - Part 2

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Image Handling

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Figure 1: Example image, rat cerebellum
Figure 1: Example image, rat cerebellum

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Time Series

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Fourier Transform of EEG data

We import the Fourier Transform function fft from the library scipy.fftpack where it can be used to transform all columns at the same time.

PYTHON 

from scipy.fftpack import fft

data_back_fft = fft(data_back, axis=0)
To plot the results of the Fourier Transform, the following steps must be taken.

Firstly, we must obtain a Fourier spectrum for every data column. Thus, we need to define how many plots we want to have. If we take only the columns in our data, we should be able to display them all, simultaneously.

Secondly, the Fourier Transform results in twice the number of complex coefficients; it produces both positive and negative frequency components, of which we only need the first (positive) half.

Lastly, the Fourier Transform outputs complex numbers. To display the ‘amplitude’ of each frequency, we take the absolute value of the complex numbers, using the abs() function.

PYTHON 

no_win = 2

rows = data_back.shape[0]

freqs = (sr/2)*linspace(0, 1, int(rows/2))

amplitudes_back = (2.0 / rows) * abs(data_back_fft[:rows//2, :2])


fig, axes = subplots(figsize=(6, 5), ncols=1, nrows=no_win, sharex=False)

names = df_back.columns[:2]

for index, ax in enumerate(axes.flat):
    axes[index].plot(freqs, amplitudes_back[:, index])
    axes[index].set_xlim(0, 8)
    axes[index].set(ylabel=f'Amplitude {names[index]}')

axes[index].set(xlabel='Frequency (Hz)');

show()

In these two channels, we can clearly see that the main amplitude contributions lie in the low frequencies, below 2 Hz.

Let us compare the corresponding figure for the case of seizure activity:

PYTHON 

data_epil_fft = fft(data_epil, axis=0)

PYTHON 

fig, axes = subplots(figsize=(6, 5), ncols=1, nrows=no_win, sharex=False)

names = df_epil.columns[:2]

amplitudes_epil = (2.0 / rows) * abs(data_epil_fft[:rows//2, :2])

for index, ax in enumerate(axes.flat):
    axes[index].plot(freqs, amplitudes_epil[:, index])
    axes[index].set_xlim(0, 12)
    axes[index].set(ylabel=f'Amplitude {names[index]}')

axes[index].set(xlabel='Frequency (Hz)');

show()
During the seizure, it is clear that the main frequency of the epileptic rhythm is between 2 and 3 Hz.

As we can see from the Fourier spectra generated above, the amplitudes are high for low frequencies, and tend to decrease as the frequency increases. Thus, it can sometimes be useful to see the high frequencies enhanced. This can be achieved with a logarithmic plot of the powers.

PYTHON 

fig, axes = subplots(figsize=(6, 6), ncols=1, nrows=no_win, sharex=False)

for index, ax in enumerate(axes.flat):

    axes[index].plot(freqs, amplitudes_back[:, index])
    axes[index].set_xlim(0, 30)
    axes[index].set(ylabel=f'Amplitude {names[index]}')
    axes[index].set_yscale('log')

axes[no_win-1].set(xlabel='Frequency (Hz)');
fig.suptitle('Logarithmic Fourier Spectra of Background EEG', fontsize=16);

show()

And for the seizure data:

PYTHON 

fig, axes = subplots(figsize=(6, 10), ncols=1, nrows=no_win, sharex=False)

for index, ax in enumerate(axes.flat):

    axes[index].plot(freqs, amplitudes_epil[:, index])
    axes[index].set_xlim(0, 30)
    axes[index].set(ylabel=f'Power {names[index]}')
    axes[index].set_yscale('log')

axes[no_win-1].set(xlabel='Frequency (Hz)');
fig.suptitle('Logarithmic Fourier Spectra of Seizure EEG', fontsize=16);

show()

In the spectrum of the absence data, it is now more obvious that there are further maxima at 6, 9, 12 and perhaps 15Hz. These are integer multiples or ‘harmonics’ of the basic frequency at around 3Hz, which we term as the fundamental frequency.

A feature that can be used as a summary statistic, is to caclulate the band power for each channel. Band power is the total power of a signal within a specific frequency range. The band power can be obtained by calculating the sum of all powers within a specified range of frequencies; this range is also referred to as the ‘band’. The band power, thus, is given as a single number.

Practice Exercise 2

Fourier spectra of filtered data

Calculate and display the Fourier spectra of the first two channels filtered between 4 and 12 Hz for the absence seizure data. Can you find any harmonics?

PYTHON 

data_epil_filt = data_filter(data_epil, sr, 4, 12)

data_epil_fft = fft(data_epil_filt, axis=0)

rows = data_epil.shape[0]

freqs = (sr/2)*linspace(0, 1, int(rows/2))

amplitudes_epil = (2.0 / rows) * abs(data_epil_fft[:rows//2, :no_win])

fig, axes = subplots(figsize=(6, 10), ncols=1, nrows=no_win, sharex=False)

for index, ax in enumerate(axes.flat):
    axes[index].plot(freqs, amplitudes_epil[:, index])
    axes[index].set_xlim(0, 12)
    axes[index].set(ylabel=f'Amplitudes {names[index]}')
axes[no_win-1].set(xlabel='Frequency (Hz)');

fig.suptitle('Fourier Spectra of Seizure EEG', fontsize=16);

show()

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