def fourier_intro(time, data, freq, Pxx, time_slice, out_file):
"""Time- and frequency presentation of sound.
Corresponds to Fig. 9.1 in the book"""
fig, axs = plt.subplots(1, 3)
axs[0].plot(time, data, lw=0.5)
axs[0].set_xlabel('Time (s)')
axs[0].set_ylabel('Sound-pressure ()')
axs[0].margins(x=0)
axs[1].plot(time, data)
axs[1].set_xlim(time_slice)
axs[1].set_xlabel('Time (s)')
axs[1].set_yticklabels([])
axs[2].plot(freq, np.sqrt(Pxx))
axs[2].set_xlim([0, 4000])
axs[2].set_xlabel('Frequency (Hz)')
axs[2].set_ylabel('|FFT| ()')
axs[2].ticklabel_format(style='sci', scilimits=(0, 4))
# Position the plots
axs[0].set_position([0.125, 0.11, 0.2, 0.775])
axs[1].set_position([0.35, 0.11, 0.2, 0.775])
show_data(out_file)
def noise_effects(time, sound, Pxx, time_slice, duration, out_file):
"""Effect of adding noise to a signal on the powerspectrum
Corresponds to Fig. 9.8 in the book
"""
# Note that a1 has been normalized to have a range of 1
sound_noisy = sound + 1 / 100 * np.random.randn(len(sound))
fft_noisy = np.fft.fft(sound_noisy)
Pxx_noisy = np.abs(fft_noisy)**2
# Normalize to 'density'
Pxx_noisy = Pxx_noisy * 2 / (np.sum(Pxx_noisy) / duration)
fig, axs = plt.subplots(1, 2)
axs[0].plot(time, sound, label='original')
axs[0].plot(time, sound_noisy, label='noise added')
axs[0].set_xlabel('Time (s)')
axs[0].set_ylabel('Sound-pressure ()')
axs[0].set_xlim(time_slice)
axs[0].legend()
axs[1].semilogy(freq, Pxx, label='original', lw=0.5)
axs[1].semilogy(freq, Pxx_noisy, label='noise added', lw=0.5)
axs[1].set_xlabel('Frequency (Hz)')
axs[1].set_ylabel('Powerspectrum (P**2/Hz)')
axs[1].set_xlim([1200, 1700])
plt.tight_layout()
show_data(out_file)
def main():
# Generate dummy data
x = np.array([-2.1, -1.3, -0.4, 1.9, 5.1, 6.2])
# Define the two plots
fig, axs = plt.subplots(1, 2, sharey=True)
# Generate the left plot
plot_histogram(axs[0], x)
# Generate the right plot
explain_KDE(axs[1], x)
# Save and show
show_data('KDEexplained.jpg')
def show3D() -> None:
""" Generate 3D plots. """
# imports specific to the plots in this example
from matplotlib import cm # colormaps
# This module is required for 3D plots!
from mpl_toolkits.mplot3d import Axes3D
# Twice as wide as it is tall.
fig = plt.figure(figsize=plt.figaspect(0.5))
set_fonts(16)
#---- First subplot
# Generate the data
X = np.arange(-5, 5, 0.1)
Y = np.arange(-5, 5, 0.1)
X, Y = np.meshgrid(X, Y)
R = np.sqrt(X**2 + Y**2)
Z = np.sin(R)
# Note the definition of "projection", required for 3D plots
#plt.style.use('ggplot')
ax = fig.add_subplot(1, 2, 1, projection='3d')
surf = ax.plot_surface(X,
Y,
Z,
rstride=1,
cstride=1,
cmap=cm.GnBu,
linewidth=0,
antialiased=False)
#surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.viridis_r,
#linewidth=0, antialiased=False)
ax.set_zlim3d(-1.01, 1.01)
fig.colorbar(surf, shrink=0.5, aspect=10)
#---- Second subplot
# Get some 3d test-data
from mpl_toolkits.mplot3d.axes3d import get_test_data
ax = fig.add_subplot(1, 2, 2, projection='3d')
X, Y, Z = get_test_data(0.05)
ax.plot_wireframe(X, Y, Z, rstride=10, cstride=10)
show_data('3dGraph.jpg')
def bode_log(out_file):
"""Logarithmic presentation of transfer function """
fig, axs = plt.subplots(2, 1, sharex=True)
# Indicate the approximate gains
axs[0].plot([0, 1 / tau], [0, 0], ls='dashed', lw=3, color='C1')
axs[0].plot([1 / tau, 10 / tau], [0, -20], ls='dashed', lw=3, color='C1')
axs[0].set_xlim(1e-2, 1)
# Bode magnitude plot
axs[0].semilogx(w, mag, color='C0')
axs[0].plot(w[index], mag[index], 'ro') # Indicate cut-off frequency
# Bode phase plot
axs[1].semilogx(w, phase)
axs[1].plot(w[index], phase[index], 'ro')
# Format the plots
axs[0].set_ylabel('Gain [dB]')
axs[0].grid(True, ls='dotted')
axs[0].margins(x=0)
axs[0].annotate(r'$\omega = \frac{1}{\tau}$',
xy=(w[index], mag[index]),
xycoords='data',
xytext=(-10, -50),
textcoords='offset points',
fontsize=16,
arrowprops=dict(arrowstyle="->", connectionstyle="arc3"))
axs[1].set_xlabel('Frequency [Hz]')
axs[1].set_ylabel('Phase [deg]')
axs[1].set_yticks([0, -45, -90], minor=False)
axs[1].margins(x=0)
axs[1].grid(True, ls='dotted')
axs[1].annotate(r'$\omega = \frac{1}{\tau}$',
xy=(w[index], phase[index]),
xycoords='data',
xytext=(-10, 30),
textcoords='offset points',
fontsize=16,
arrowprops=dict(arrowstyle="->", connectionstyle="arc3"))
show_data(out_file)
def linear_and_log(freq, Pxx, out_file):
"""Comparison of linear and log representatino of powerspectrum
Corresponds to Fig. 9.7 in the book.
"""
fig, axs = plt.subplots(1, 2)
upper_limit = 4000
axs[0].plot(freq, Pxx)
axs[0].set_xlim([0, upper_limit])
axs[0].set_xlabel('Frequency (Hz)')
axs[0].set_ylabel('Powerspectrum ()')
axs[0].ticklabel_format(style='sci', scilimits=(0, 4))
axs[1].semilogy(freq, Pxx, lw=0.5)
axs[1].set_xlim([0, upper_limit])
axs[1].set_xlabel('Frequency (Hz)')
axs[1].set_ylabel('Powerspectrum (dB)')
plt.tight_layout()
show_data(out_file)
def bode_linear(out_file):
"""Linear presentation of transfer function """
fig, axs = plt.subplots(2, 1, sharex=True)
# Bode magnitude plot
axs[0].plot(w, 10**(mag / 20))
axs[0].plot(w[index], 10**(mag[index] / 20), 'ro')
# Bode phase plot
axs[1].plot(w, phase)
axs[1].plot(w[index], phase[index], 'ro')
# Format the plots
axs[0].set_ylabel('Gain [linear]')
axs[0].set_ylim([-0.10, 1.1])
axs[0].margins(x=0)
axs[0].grid(True, ls='dotted')
axs[0].annotate(r'$\omega = \frac{1}{\tau}$',
xy=(w[index], 10**(mag[index] / 20)),
xycoords='data',
xytext=(10, 20),
textcoords='offset points',
fontsize=16,
arrowprops=dict(arrowstyle="->", connectionstyle="arc3"))
axs[1].set_xlabel('Frequency [Hz]')
axs[1].set_ylabel('Phase [deg]')
axs[1].margins(x=0)
axs[1].set_yticks([0, -45, -90], minor=False)
axs[1].grid(True, ls='dotted')
axs[1].annotate(r'$\omega = \frac{1}{\tau}$',
xy=(w[index], phase[index]),
xycoords='data',
xytext=(10, 30),
textcoords='offset points',
fontsize=16,
arrowprops=dict(arrowstyle="->", connectionstyle="arc3"))
show_data(out_file)
def feedback(tau: float, fb_gain: float) -> None:
"""Simulation of a negative feedback system, using the package 'control'.
'time', 'in_signal', 'num' and 'den' are taken from the global workspace
Parameters
----------
tau : time constant of a first order lag [sec]
fb_gain : feedback gain
"""
# First, define the feedforward transfer function
sys = control.TransferFunction(num, den)
print(sys) # 1 / (tau*s + 1)
# Simulate the response of the feedforward system
t_ff, out_ff = control.forced_response(sys, T=time, U=in_signal)
# Then define a feedback-loop, with gain fb_gain
sys_total = control.feedback(sys, fb_gain)
print(sys_total) # 1 / (tau*s + (1+k))
# Simulate the response of the feedback system
t_fb, out_fb = control.forced_response(sys_total, T=time, U=in_signal)
# Show the signals
plt.plot(time, in_signal, '-', label='Input')
plt.plot(t_ff, out_ff, '--', label='Feedforward')
plt.plot(t_fb, out_fb, '-.', label='Feedback')
# Format the plot
plt.xlabel('Time [sec]')
plt.title(f"First order lag (tau={tau} sec)")
print('Simulated with "control"')
plt.legend()
out_file = 'feedback.jpg'
show_data(out_file)
def welch_periodogram(data, rate, freq, Pxx, out_file):
"""Comparison of simple powerspectrum to Welch-periodogram
Corresponds to Fig. 9.9 in the book.
"""
fig, axs = plt.subplots(1, 2)
f, welch = signal.welch(data, fs=rate, nperseg=len(data) / 8)
df = np.diff(f)[0]
# "normalize" welch
welch = welch / (np.sum(welch) * df)
axs[0].semilogy(freq, Pxx, label='Periodogram', lw=0.5)
axs[0].semilogy(f, welch, label='Welch', lw=0.8)
axs[0].set_xlim([0, rate / 2])
axs[0].set_ylabel('Powerspectrum (P**2/Hz)')
axs[0].set_xlabel('Frequency (Hz)')
axs[1].semilogy(freq, Pxx, label='Pxx')
axs[1].semilogy(f, welch, label='Welch')
axs[1].set_xlim([800, 1100])
axs[1].set_xlabel('Frequency (Hz)')
axs[1].legend()
show_data(out_file)
# Add the text
fs = 18
plt.text(11, 5.5, '$\pm$ 1SD', fontsize=fs,
fontstyle='italic', color='C0')
plt.text(62, 5.2, '$\pm$ 1SEM', fontsize=fs,
fontstyle='italic', color='C1')
# plt.annotate('mean', (110,np.mean(x)),xycoords='data',
# fontsize=fs, xytext=(110, 5.5), textcoords='data',
# arrowprops={
# 'facecolor':'black',
# 'shrink':1,
# 'headwidth':6,
# 'headlength':6,
# 'width': 1})
plt.annotate('mean', (-5,np.mean(x)),xycoords='data',
fontsize=fs, xytext=(-40, 4.5), textcoords='data',
color = [0.6, 0.6, 0.6],
arrowprops={
'color':[0.6, 0.6, 0.6],
'shrink':1,
'headwidth':6,
'headlength':6,
'width': 1})
# Save and show
plt.tight_layout()
outFile = 'standardError.jpg'
show_data(outFile, out_dir='.')
# Import the required libraries import numpy as np import matplotlib.pyplot as plt from skimage import data from utilities.my_style import set_fonts, show_data # Get a color-image img = data.astronaut() nrows, ncols = img.shape[:2] # Make vectors from 1 to 0, with lengths matching the image alpha_row = np.linspace(1, 0, ncols) alpha_col = np.linspace(1, 0, nrows) # Make coordinate-grids X, Y = np.meshgrid(alpha_row, alpha_col) # Scale the vector from 0 to 255, and # let the image fade from top-right to bottom-left X_Y = np.uint8(X * Y * 255) X_Y = np.atleast_3d(X_Y) #make sure the dimensions matches the image # Add the alpha-layer img_alpha = np.concatenate((img, X_Y), axis=2) plt.imshow(img_alpha) out_file = 'fading_astronout.png' show_data(out_file)
# Generate the base image
data = np.zeros((99, 99))
data[34:66, 33:67] = 1
data[85:87, 85:87] = 1
data[49:51, 49:51] = 0
# Show the original image
fig, ax = plt.subplots()
plt.gray()
ax.imshow(data)
ax.set_title('Original Image')
ax.get_xaxis().set_visible(False)
ax.get_yaxis().set_visible(False)
out_file = 'Square.jpg'
show_data(out_file)
# Perform the morphological operations
selem = morphology.square(5)
fig, axs = plt.subplots(2, 2, figsize=(8, 8))
show_modImage(data, 'binary_erosion', axs[0][0], 'Eroded')
show_modImage(data, 'binary_dilation', axs[0][1], 'Dilated')
show_modImage(data, 'binary_opening', axs[1][0],
'Opened (Dilation after Erosion)')
show_modImage(data, 'binary_closing', axs[1][1],
'Closed (Erosion after Dilation)')
out_file = 'Square_Morphological.jpg'
show_data(out_file, out_dir='.')
def simplePlots() -> None:
""" Demonstrate the generation of different statistical standard plots. """
# Univariate data -------------------------
# Make sure that always the same random numbers are generated
np.random.seed(1234)
# Generate data that are normally distributed
x = np.random.randn(500)
# Other graphics settings
# Set " context='poster' " for printouts, and "set_fonts(32)"
sns.set(context='notebook', style='ticks', palette='muted')
# Set the fonts the way I like them
set_fonts(16)
# Scatter plot
plt.plot(x, '.', markersize=7)
plt.xlim([0, len(x)])
# Save and show the data, in a systematic format
printout('scatterPlot.jpg',
xlabel='Datapoints',
ylabel='Values',
title='Scatter')
# Histogram
plt.hist(x)
printout('histogram_plain.jpg',
xlabel='Data Values',
ylabel='Frequency',
title='Histogram, default settings')
plt.hist(x, 25, density=True)
printout('density_histogram.jpg',
xlabel='Data Values',
ylabel='Probability',
title='Density Histogram, 25 bins')
# Boxplot
# The ox consists of the first, second (middle) and third quartile
set_fonts(18)
plt.boxplot(x, sym='*')
printout('boxplot.jpg', xlabel='Values', title='Boxplot')
plt.boxplot(x, sym='*', vert=False)
plt.title('Boxplot, horizontal')
plt.xlabel('Values')
plt.show()
# Errorbars
x = np.arange(5)
y = x**2
errorBar = x / 2
plt.errorbar(x, y, yerr=errorBar, fmt='o', capsize=5, capthick=3)
plt.xlim([-0.2, 4.2])
plt.ylim([-0.2, 19])
printout('Errorbars.jpg',
xlabel='Data Values',
ylabel='Measurements',
title='Errorbars')
# SD for two groups
weight = {'USA': 89, 'Austria': 74}
weight_SD_male = 12
plt.errorbar([1, 2],
weight.values(),
yerr=weight_SD_male * np.r_[1, 1],
capsize=5,
LineStyle='',
marker='o')
plt.xlim([0.5, 2.5])
plt.xticks([1, 2], weight.keys())
plt.ylabel('Weight [kg]')
plt.title('Adult male, mean +/- SD')
show_data('SD_groups.jpg', out_dir='.')
# Barplot
# The font-size is set such that the legend does not overlap with the data
np.random.seed(1234)
set_fonts(16)
df = pd.DataFrame(np.random.rand(7, 3), columns=['one', 'two', 'three'])
df.plot(kind='bar', grid=False, color=sns.color_palette('muted'))
show_data('barplot.jpg')
# Bivariate Plots
df2 = pd.DataFrame(np.random.rand(50, 3), columns=['a', 'b', 'c'])
df2.plot(kind='scatter', x='a', y='b', s=df2['c'] * 500)
plt.axhline(0, ls='--', color='#999999')
plt.axvline(0, ls='--', color='#999999')
printout('bivariate.jpg')
sns.set_style('ticks')
# Pieplot
txtLabels = 'Cats', 'Dogs', 'Frogs', 'Others'
fractions = [45, 30, 15, 10]
offsets = (0, 0.05, 0, 0)
plt.pie(fractions,
explode=offsets,
labels=txtLabels,
autopct='%1.1f%%',
shadow=True,
startangle=90,
colors=sns.color_palette('muted'))
plt.axis('equal')
printout('piePlot.jpg', title=' ')
# Formatting
set_fonts(14)
# Noisy sine
t = np.arange(0, 20, 0.05)
x = np.sin(t)
x_noisy = x + 0.06 * t + 0.3 * np.random.randn(len(x))
#plt.plot(t, x)
#show_data('sine.jpg')
fig, axs = plt.subplots(1, 2)
axs[0].plot(t, x_noisy)
axs[1].plot(t, x)
show_data('noisy_sine.svg')
# BW-Image
img_file = '../../data/HagenbergRocks.jpg'
img = color.rgb2gray(plt.imread(img_file))
# Edges
edges = feature.canny(img, sigma=2.5)
fig, ax = plt.subplots(1, 1)
#axs[0].imshow(img, cmap='gray')
#axs[0].axis('off')
ax.imshow(edges, cmap='gray')
ax.axis('off')
fs = 18
plt.text(11, 5.5, '$\pm$ 1SD', fontsize=fs)
plt.text(62, 5.2, '$\pm$ 1SEM', fontsize=fs)
# plt.annotate('mean', (110,np.mean(x)),xycoords='data',
# fontsize=fs, xytext=(110, 5.5), textcoords='data',
# arrowprops={
# 'facecolor':'black',
# 'shrink':1,
# 'headwidth':6,
# 'headlength':6,
# 'width': 1})
plt.annotate('mean', (-5, np.mean(x)),
xycoords='data',
fontsize=fs,
xytext=(-40, 4.5),
textcoords='data',
color=[0.6, 0.6, 0.6],
arrowprops={
'color': [0.6, 0.6, 0.6],
'shrink': 1,
'headwidth': 6,
'headlength': 6,
'width': 1
})
# Save and show
outFile = ('standardError.jpg')
show_data(outFile)
def power_spectrum(t: np.ndarray, dt: float, sig: np.ndarray,
sig_ideal: np.ndarray) -> None:
""" Demonstrate three different ways to calculate the power-spectrum
Parameters
----------
t : time [sec]
dt : sample period [sec]
sig : sample signal to be analyzed
sig_ideal : signal without noise
"""
set_fonts(16)
fig, axs = plt.subplots(1,2, figsize=(10, 5))
if latex_installed:
txt ='$\displaystyle signal=offset + \sum_{i=0}^{2} a_i*sin(\omega_i*t) + noise$'
label = '$|FFT|\; ()$'
label2 = '$|FFT|^2 ()$'
else:
txt = 'signal = offset + sum(i=0:2) a_i*sin(omega_i*t)'
label = '|FFT| ()'
label2 = '|FFT|^2 ()'
# From a quick look we learn - nothing
axs[0].plot(t, sig, lw=0.7, label='noisy')
axs[0].plot(t, sig_ideal, ls='dashed', lw=2, label='ideal')
axs[0].set_xlim(0, 0.4)
axs[0].set_ylim(-15, 25)
axs[0].set_xlabel('Time (s)')
axs[0].set_ylabel('Signal ()')
axs[0].legend()
axs[0].text(0.2, 24, s=txt, fontsize=16)
# Calculate the spectrum by hand
fft = np.fft.fft(sig)
print(fft[:3])
fft_abs = np.abs(fft)
# The easiest way to calculate the frequencies
freq = np.fft.fftfreq(len(sig), dt)
axs[1].plot(freq, fft_abs)
axs[1].set_xlim(0, 35)
axs[1].set_xlabel('Frequency (Hz)')
axs[1].set_ylabel(label)
axs[1].set_yticklabels([])
show_data('FFT_sines.jpg')
# Also show the double-sided spectrum
fig, ax = plt.subplots(1,1)
ax.plot(fft_abs)
#ax.set_xlim(0, 35)
ax.set_xlabel('Points')
ax.set_ylabel(label)
plt.tight_layout()
show_data('FFT_doublesided.jpg')
# With real input, the power spectrum is symmetrical and we only one half
fft_abs = fft_abs[:int(len(fft_abs)/2)]
freq = freq[:int(len(freq)/2)]
# The power is the norm of the amplitude squared
Pxx = fft_abs**2
# Showing the same data on a linear and a log scale
fig, axs = plt.subplots(2,1, sharex=True)
axs[0].plot(freq, Pxx)
axs[0].set_ylabel('Power (linear)')
axs[1].semilogy(freq, Pxx)
axs[1].set_xlabel('Frequency (Hz)')
axs[1].set_ylabel('Power (dB)')
show_data('FFT_sines_power_lin_log.jpg')
# Periodogram and Welch-Periodogram
f_pgram, P_pgram = signal.periodogram(sig, fs = 1/dt)
f_welch, P_welch = signal.welch(sig, fs = 100, nperseg=2**8)
fig, axs = plt.subplots(2, 1, sharex=True)
axs[0].semilogy(f_pgram, P_pgram, label='periodogram')
axs[1].semilogy(f_welch, P_welch, label='welch')
axs[0].set_ylabel('Spectral Density (dB)')
axs[0].legend()
axs[0].set_ylim(1e-4, 1e3)
axs[1].set_xlabel('Frequency (Hz)')
axs[1].set_ylabel('Spectral Density (dB)')
axs[1].legend()
show_data('FFT_sines_periodogram.jpg')
if __name__ == '__main__':
# Generate the data
signal = np.zeros(20)
signal[7:10] = 1
signal[14:17] = 1
feature = np.zeros(7)
feature[2:5] = 1
# Plot the auto-correlation
set_fonts(14)
fig, axs = plt.subplots(2, 1)
axs[0].plot(signal, 'o-')
axs[0].hlines(0, -0.5, 20, ls='dotted')
axs[0].set_xticks(np.arange(0, 21, 2))
axs[0].set_xlim(-0.5, 20)
axs[0].set_ylabel('Signal')
axs[0].set_yticks([0, 0.5, 1])
axs[1].plot(feature, 'o-')
axs[1].hlines(0, -0.5, 20, ls='dotted')
axs[1].set_xticks(np.arange(0, 20, 2))
axs[1].set_xticks(np.arange(0, 21, 2))
axs[1].set_xlim(-0.5, 20)
axs[1].set_xlabel('Shift')
axs[1].set_ylabel('Feature')
axs[1].set_yticks([0, 0.5, 1])
sig_file = 'CorrDemo.jpg'
show_data(sig_file, out_dir='.')
