def showResults(challenger_data, model):
''' Show the original data, and the resulting logit-fit'''
temperature = challenger_data[:,0]
failures = challenger_data[:,1]
# First plot the original data
plt.figure()
setFonts()
sns.set_style('darkgrid')
np.set_printoptions(precision=3, suppress=True)
plt.scatter(temperature, failures, s=200, color="k", alpha=0.5)
plt.yticks([0, 1])
plt.ylabel("Damage Incident?")
plt.xlabel("Outside Temperature [F]")
plt.title("Defects of the Space Shuttle O-Rings vs temperature")
plt.tight_layout
# Plot the fit
x = np.arange(50, 85)
alpha = model.params[0]
beta = model.params[1]
y = logistic(x, beta, alpha)
plt.hold(True)
plt.plot(x,y,'r')
plt.xlim([50, 85])
outFile = 'ChallengerPlain.png'
showData(outFile)
def showAndSave(temperature: np.ndarray, failures: np.ndarray) -> None:
"""Shows the input data, and saves the resulting figure
Parameters
----------
temperature : temperature data
failureData : corresponding failure status
"""
# Plot it, as a function of tempature
plt.figure()
setFonts()
sns.set_style('darkgrid')
np.set_printoptions(precision=3, suppress=True)
plt.scatter(temperature, failures, s=200, color="k", alpha=0.5)
plt.yticks([0, 1])
plt.ylabel("Damage Incident?")
plt.xlabel("Outside Temperature [F]")
plt.title("Defects of the Space Shuttle O-Rings vs temperature")
plt.tight_layout
outFile = 'Challenger_ORings.png'
showData(outFile)
def show_poisson_views():
"""Show different views of a Poisson distribution"""
sns.set_palette(sns.color_palette('muted'))
fig, ax = plt.subplots(3,1)
k = np.arange(25)
pd = stats.poisson(10)
setFonts(12)
ax[0].plot(k, pd.pmf(k),'x-')
ax[0].set_title('Poisson distribution', fontsize=24)
ax[0].set_xticklabels([])
ax[0].set_ylabel('PMF (X)')
ax[1].plot(k, pd.cdf(k))
ax[1].set_xlabel('X')
ax[1].set_ylabel('CDF (X)')
y = np.linspace(0,1,100)
ax[2].plot(y, pd.ppf(y))
ax[2].set_xlabel('X')
ax[2].set_ylabel('PPF (X)')
plt.tight_layout()
plt.show()
def showSimResults(alpha_samples, beta_samples):
'''Show the results of the simulations, and save them to an outFile'''
plt.figure(figsize=(12.5, 6))
sns.set_style('darkgrid')
setFonts(18)
# Histogram of the samples:
plt.subplot(211)
plt.title(r"Posterior distributions of the variables $\alpha, \beta$")
plt.hist(beta_samples,
histtype='stepfilled',
bins=35,
alpha=0.85,
label=r"posterior of $\beta$",
color="#7A68A6",
normed=True)
plt.legend()
plt.subplot(212)
plt.hist(alpha_samples,
histtype='stepfilled',
bins=35,
alpha=0.85,
label=r"posterior of $\alpha$",
color="#A60628",
normed=True)
plt.legend()
outFile = 'Challenger_Parameters.png'
showData(outFile)
def KS_principle(inData):
'''Show the principle of the Kolmogorov-Smirnov test.'''
# CDF of normally distributed data
nd = stats.norm()
nd_x = np.linspace(-4, 4, 101)
nd_y = nd.cdf(nd_x)
# Empirical CDF of the sample data, which range for approximately 0 to 10
numPts = 50
lowerLim = 0
upperLim = 10
ecdf_x = np.linspace(lowerLim, upperLim, numPts)
ecdf_y = stats.cumfreq(data, numPts, (lowerLim, upperLim))[0] / len(inData)
#Add zero-point by hand
ecdf_x = np.hstack((0., ecdf_x))
ecdf_y = np.hstack((0., ecdf_y))
# Plot the data
sns.set_style('ticks')
sns.set_context('poster')
setFonts(36)
plt.plot(nd_x, nd_y, 'k--')
plt.hold(True)
plt.plot(ecdf_x, ecdf_y, color='k')
plt.xlabel('X')
plt.ylabel('Cumulative Probability')
# For the arrow, find the start
ecdf_startIndex = np.min(np.where(ecdf_x >= 2))
arrowStart = np.array([ecdf_x[ecdf_startIndex], ecdf_y[ecdf_startIndex]])
nd_startIndex = np.min(np.where(nd_x >= 2))
arrowEnd = np.array([nd_x[nd_startIndex], nd_y[nd_startIndex]])
arrowDelta = arrowEnd - arrowStart
plt.arrow(arrowStart[0],
arrowStart[1],
0,
arrowDelta[1],
width=0.05,
length_includes_head=True,
head_length=0.04,
head_width=0.4,
color='k')
plt.arrow(arrowStart[0],
arrowStart[1] + arrowDelta[1],
0,
-arrowDelta[1],
width=0.05,
length_includes_head=True,
head_length=0.04,
head_width=0.4,
color='k')
outFile = 'KS_Example.png'
showData(outFile)
def showResults(challenger_data, model):
''' Show the original data, and the resulting logit-fit'''
temperature = challenger_data[:,0]
failures = challenger_data[:,1]
# First plot the original data
plt.figure()
setFonts()
sns.set_style('darkgrid')
np.set_printoptions(precision=3, suppress=True)
plt.scatter(temperature, failures, s=200, color="k", alpha=0.5)
plt.yticks([0, 1])
plt.ylabel("Damage Incident?")
plt.xlabel("Outside Temperature [F]")
plt.title("Defects of the Space Shuttle O-Rings vs temperature")
plt.tight_layout
# Plot the fit
x = np.arange(50, 85)
alpha = model.params[0]
beta = model.params[1]
y = logistic(x, beta, alpha)
plt.hold(True)
plt.plot(x,y,'r')
plt.xlim([50, 85])
outFile = 'ChallengerPlain.png'
showData(outFile)
def show_poisson_views():
"""Show different views of a Poisson distribution"""
sns.set_palette(sns.color_palette('muted'))
fig, ax = plt.subplots(3, 1)
k = np.arange(25)
pd = stats.poisson(10)
setFonts(12)
ax[0].plot(k, pd.pmf(k), 'x-')
ax[0].set_title('Poisson distribution', fontsize=24)
ax[0].set_xticklabels([])
ax[0].set_ylabel('PMF (X)')
ax[1].plot(k, pd.cdf(k))
ax[1].set_xlabel('X')
ax[1].set_ylabel('CDF (X)')
y = np.linspace(0, 1, 100)
ax[2].plot(y, pd.ppf(y))
ax[2].set_xlabel('X')
ax[2].set_ylabel('PPF (X)')
plt.tight_layout()
plt.show()
def showProbabilities(linearTemperature, temperature, failures,
mean_prob_t, p_t, quantiles) -> None:
"""Show the posterior probabilities, and save the resulting figures
Parameters
----------
linearTemperature :
temperature :
failures :
mean_prob_t :
p_t :
quantiles :
"""
# --- Show the probability curve ----
plt.figure(figsize=(12.5, 4))
setFonts(18)
plt.plot(linearTemperature, mean_prob_t, lw=3, label="Average posterior\n \
probability of defect")
plt.plot(linearTemperature, p_t[0, :], ls="--", label="Realization from posterior")
plt.plot(linearTemperature, p_t[-2, :], ls="--", label="Realization from posterior")
plt.scatter(temperature, failures, color="k", s=50, alpha=0.5)
plt.title("Posterior expected value of probability of defect, plus realizations")
plt.legend(loc="lower left")
plt.ylim(-0.1, 1.1)
plt.xlim(linearTemperature.min(), linearTemperature.max())
plt.ylabel("Probability")
plt.xlabel("Temperature [F]")
outFile = 'Challenger_Probability.png'
showData(outFile)
# --- Draw CIs ---
setFonts()
sns.set_style('darkgrid')
plt.fill_between(linearTemperature[:, 0], *quantiles, alpha=0.7,
color="#7A68A6")
plt.plot(linearTemperature[:, 0], quantiles[0], label="95% CI", color="#7A68A6", alpha=0.7)
plt.plot(linearTemperature, mean_prob_t, lw=1, ls="--", color="k",
label="average posterior \nprobability of defect")
plt.xlim(linearTemperature.min(), linearTemperature.max())
plt.ylim(-0.02, 1.02)
plt.legend(loc="lower left")
plt.scatter(temperature, failures, color="k", s=50, alpha=0.5)
plt.xlabel("Temperature [F]")
plt.ylabel("Posterior Probability Estimate")
outFile = 'Challenger_CIs.png'
showData(outFile)
def show3D():
'''Generation of 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))
setFonts(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)
showData('3dGraph.png')
def KS_principle(inData):
'''Show the principle of the Kolmogorov-Smirnov test.'''
# CDF of normally distributed data
nd = stats.norm()
nd_x = np.linspace(-4, 4, 101)
nd_y = nd.cdf(nd_x)
# Empirical CDF of the sample data, which range for approximately 0 to 10
numPts = 50
lowerLim = 0
upperLim = 10
ecdf_x = np.linspace(lowerLim, upperLim, numPts)
ecdf_y = stats.cumfreq(data, numPts, (lowerLim, upperLim))[0]/len(inData)
#Add zero-point by hand
ecdf_x = np.hstack((0., ecdf_x))
ecdf_y = np.hstack((0., ecdf_y))
# Plot the data
sns.set_style('ticks')
sns.set_context('poster')
setFonts(36)
plt.plot(nd_x, nd_y, 'k--')
plt.hold(True)
plt.plot(ecdf_x, ecdf_y, color='k')
plt.xlabel('X')
plt.ylabel('Cumulative Probability')
# For the arrow, find the start
ecdf_startIndex = np.min(np.where(ecdf_x >= 2))
arrowStart = np.array([ecdf_x[ecdf_startIndex], ecdf_y[ecdf_startIndex]])
nd_startIndex = np.min(np.where(nd_x >= 2))
arrowEnd = np.array([nd_x[nd_startIndex], nd_y[nd_startIndex]])
arrowDelta = arrowEnd - arrowStart
plt.arrow(arrowStart[0], arrowStart[1], 0, arrowDelta[1],
width=0.05, length_includes_head=True, head_length=0.04, head_width=0.4, color='k')
plt.arrow(arrowStart[0], arrowStart[1]+arrowDelta[1], 0, -arrowDelta[1],
width=0.05, length_includes_head=True, head_length=0.04, head_width=0.4, color='k')
outFile = 'KS_Example.png'
showData(outFile)
def main():
'''Demonstrate central limit theorem.'''
setFonts(24)
# Generate data
data = np.random.random(ndata)
# Show three histograms, side-by-side
fig, axs = plt.subplots(1,3)
showAsHistogram(axs[0], data, 'Random data')
showAsHistogram(axs[1], np.mean(data.reshape((ndata//2, 2 )), axis=1), 'Average over 2')
showAsHistogram(axs[2], np.mean(data.reshape((ndata//10,10)), axis=1), 'Average over 10')
# Format them and show them
axs[0].set_ylabel('Counts')
plt.tight_layout()
showData('CentralLimitTheorem.png')
def showAndSave(temperature, failures):
'''Shows the input data, and saves the resulting figure'''
# Plot it, as a function of tempature
plt.figure()
setFonts()
sns.set_style('darkgrid')
np.set_printoptions(precision=3, suppress=True)
plt.scatter(temperature, failures, s=200, color="k", alpha=0.5)
plt.yticks([0, 1])
plt.ylabel("Damage Incident?")
plt.xlabel("Outside Temperature [F]")
plt.title("Defects of the Space Shuttle O-Rings vs temperature")
plt.tight_layout
outFile = 'Challenger_ORings.png'
showData(outFile)
def generate_probplot():
'''Generate a prob-plot for a chi2-distribution of sample data'''
# Define the skewed distribution
chi2 = stats.chi2(3)
# Generate the data
x = np.linspace(0,10, 100)
y = chi2.pdf(x)
np.random.seed(12345)
numData = 100
data = chi2.rvs(numData)
# Arrange subplots
sns.set_context('paper')
sns.set_style('white')
setFonts(11)
fig, axs = plt.subplots(1,2)
# Plot distribution
axs[0].plot(x,y)
axs[0].set_xlabel('X')
axs[0].set_ylabel('PDF(X)')
axs[0].set_title('chi2(x), k=3')
sns.set_style('white')
x0, x1 = axs[0].get_xlim()
y0, y1 = axs[0].get_ylim()
axs[0].set_aspect((x1-x0)/(y1-y0))
# Plot probplot
plt.axes(axs[1])
stats.probplot(data, plot=plt)
x0, x1 = axs[1].get_xlim()
y0, y1 = axs[1].get_ylim()
axs[1].axhline(0, lw=0.5, ls='--')
axs[1].axvline(0, lw=0.5, ls='--')
axs[1].set_aspect((x1-x0)/(y1-y0))
showData('chi2pp.png')
return(data)
'''
def generate_probplot():
'''Generate a prob-plot for a chi2-distribution of sample data'''
# Define the skewed distribution
chi2 = stats.chi2(3)
# Generate the data
x = np.linspace(0,10, 100)
y = chi2.pdf(x)
np.random.seed(12345)
numData = 100
data = chi2.rvs(numData)
# Arrange subplots
sns.set_context('paper')
sns.set_style('white')
setFonts(11)
fig, axs = plt.subplots(1,2)
# Plot distribution
axs[0].plot(x,y)
axs[0].set_xlabel('X')
axs[0].set_ylabel('PDF(X)')
axs[0].set_title('chi2(x), k=3')
sns.set_style('white')
x0, x1 = axs[0].get_xlim()
y0, y1 = axs[0].get_ylim()
axs[0].set_aspect((x1-x0)/(y1-y0))
# Plot probplot
plt.axes(axs[1])
stats.probplot(data, plot=plt)
x0, x1 = axs[1].get_xlim()
y0, y1 = axs[1].get_ylim()
axs[1].axhline(0, lw=0.5, ls='--')
axs[1].axvline(0, lw=0.5, ls='--')
axs[1].set_aspect((x1-x0)/(y1-y0))
showData('chi2pp.png')
return(data)
'''
def showProbabilities(linearTemperature, temperature, failures, mean_prob_t, p_t, quantiles):
'''Show the posterior probabilities, and save the resulting figures'''
# --- Show the probability curve ----
plt.figure(figsize=(12.5, 4))
setFonts(18)
plt.plot(linearTemperature, mean_prob_t, lw=3, label="Average posterior\n \
probability of defect")
plt.plot(linearTemperature, p_t[0, :], ls="--", label="Realization from posterior")
plt.plot(linearTemperature, p_t[-2, :], ls="--", label="Realization from posterior")
plt.scatter(temperature, failures, color="k", s=50, alpha=0.5)
plt.title("Posterior expected value of probability of defect, plus realizations")
plt.legend(loc="lower left")
plt.ylim(-0.1, 1.1)
plt.xlim(linearTemperature.min(), linearTemperature.max())
plt.ylabel("Probability")
plt.xlabel("Temperature [F]")
outFile = 'Challenger_Probability.png'
showData(outFile)
# --- Draw CIs ---
setFonts()
sns.set_style('darkgrid')
plt.fill_between(linearTemperature[:, 0], *quantiles, alpha=0.7,
color="#7A68A6")
plt.plot(linearTemperature[:, 0], quantiles[0], label="95% CI", color="#7A68A6", alpha=0.7)
plt.plot(linearTemperature, mean_prob_t, lw=1, ls="--", color="k",
label="average posterior \nprobability of defect")
plt.xlim(linearTemperature.min(), linearTemperature.max())
plt.ylim(-0.02, 1.02)
plt.legend(loc="lower left")
plt.scatter(temperature, failures, color="k", s=50, alpha=0.5)
plt.xlabel("Temperature [F]")
plt.ylabel("Posterior Probability Estimate")
outFile = 'Challenger_CIs.png'
showData(outFile)
def main():
'''Demonstrate central limit theorem.'''
setFonts(24)
# Generate data
# Show three histograms, side-by-side
fig, axs = plt.subplots(1,4)
showAsHistogram(axs[0], data, 'Random data')
showAsHistogram(axs[1], np.mean(data.reshape((ndata//2, 2 )), axis=1), 'Average over 2')
showAsHistogram(axs[2], np.mean(data.reshape((ndata//10,10)), axis=1), 'Average over 10')
showAsHistogram(axs[3], np.mean(data.reshape((ndata//100,100)), axis=1), 'Average over 100')
# Format them and show them
axs[0].set_ylabel('Counts')
plt.tight_layout()
showData('CentralLimitTheorem.png')
def show3D():
'''Generation of 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))
setFonts(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)
showData('3dGraph.png')
def showSimResults(alpha_samples, beta_samples):
'''Show the results of the simulations, and save them to an outFile'''
plt.figure(figsize=(12.5, 6))
sns.set_style('darkgrid')
setFonts(18)
# Histogram of the samples:
plt.subplot(211)
plt.title(r"Posterior distributions of the variables $\alpha, \beta$")
plt.hist(beta_samples, histtype='stepfilled', bins=35, alpha=0.85,
label=r"posterior of $\beta$", color="#7A68A6", normed=True)
plt.legend()
plt.subplot(212)
plt.hist(alpha_samples, histtype='stepfilled', bins=35, alpha=0.85,
label=r"posterior of $\alpha$", color="#A60628", normed=True)
plt.legend()
outFile = 'Challenger_Parameters.png'
showData(outFile)
def main():
# Generate some dummy data
np.set_printoptions(precision=2)
N = 20
study_duration = 12
# Note: a constant dropout rate is equivalent to an exponential distribution!
subsciption_list = [ [exponential(18), exponential(3)][uniform()<0.5] \
for i in range(N) ]
actual_subscriptiontimes = np.array(subsciption_list)
observed_subscriptiontimes = np.minimum(actual_subscriptiontimes,study_duration)
observed= actual_subscriptiontimes < study_duration
# Show the data
setFonts(18)
plt.xlim(0,24)
plt.vlines(12, 0, 30, lw=2, linestyles="--")
plt.xlabel('time')
plt.title('Subscription Times, at $t=12$ months')
plot_lifetimes(observed_subscriptiontimes, event_observed=observed)
plt.show()
print(f'Observed subscription time at time {study_duration:d}', \
observed_subscriptiontimes)
def simplePlots():
'''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
sns.set(context='poster', style='ticks', palette=sns.color_palette('muted'))
# Set the fonts the way I like them
setFonts(32)
# Scatter plot
plt.scatter(np.arange(len(x)), x)
plt.xlim([0, len(x)])
# Save and show the data, in a systematic format
printout('scatterPlot.png', xlabel='Datapoints', ylabel='Values', title='Scatter')
# Histogram
plt.hist(x)
printout('histogram_plain.png', xlabel='Data Values',
ylabel='Frequency', title='Histogram, default settings')
plt.hist(x,25)
printout('histogram.png', xlabel='Data Values', ylabel='Frequency',
title='Histogram, 25 bins')
# Cumulative probability density
numbins = 20
plt.plot(stats.cumfreq(x,numbins)[0])
printout('CumulativeFrequencyFunction.png', xlabel='Data Values',
ylabel='CumFreq', title='Cumulative Frequency')
# KDE-plot
sns.kdeplot(x)
printout('kde.png', xlabel='Data Values', ylabel='Density',
title='KDE_plot')
# Boxplot
# The ox consists of the first, second (middle) and third quartile
plt.boxplot(x, sym='*')
printout('boxplot.png', 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.png', xlabel='Data Values', ylabel='Measurements', title='Errorbars')
# Violinplot
nd = stats.norm
data = nd.rvs(size=(100))
nd2 = stats.norm(loc = 3, scale = 1.5)
data2 = nd2.rvs(size=(100))
# Use pandas and the seaborn package for the violin plot
df = pd.DataFrame({'Girls':data, 'Boys':data2})
sns.violinplot(df)
printout('violinplot.png', title='Violinplot')
# Barplot
# The font-size is set such that the legend does not overlap with the data
np.random.seed(1234)
setFonts(20)
df = pd.DataFrame(np.random.rand(10, 4), columns=['a', 'b', 'c', 'd'])
df.plot(kind='bar', grid=False, color=sns.color_palette('muted'))
showData('barplot.png')
setFonts(28)
# 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.png')
# Grouped Boxplot
sns.set_style('whitegrid')
sns.boxplot(df)
setFonts(28)
printout('groupedBoxplot.png', title='sns.boxplot')
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.png', title=' ')
def show_fig(std, ax, title):
"""Create a plot of normally distributed data in a given axis"""
for ii in range(3):
data = stats.norm(centers[ii], std).rvs(numData)
offset = ii * numData
ax.plot(offset + np.arange(numData), data, '.', ms=10)
ax.xaxis.set_ticks([50, 150, 250])
ax.set_xticklabels(['Group1', 'Group2', 'Group3'])
ax.set_title(title)
sns.despine()
if __name__ == '__main__':
# Set up the figure
sns.set_context('paper')
sns.set_style('whitegrid')
setFonts(14)
# Create 2 plots of 3 different, normally distributed data groups, with different SDs
fig, axs = plt.subplots(1, 2)
centers = [5, 5.3, 4.7]
stds = [0.1, 2]
numData = 100
show_fig(0.1, axs[0], 'SD=0.1')
show_fig(2, axs[1], 'SD=2.0')
showData('anova_oneway.png')
except ImportError:
# Ensure correct performance otherwise
def setFonts(*options):
return
# Generate some dummy data
np.set_printoptions(precision=2)
N = 20
study_duration = 12
# Note: a constant dropout rate is equivalent to an exponential distribution!
actual_subscriptiontimes = np.array([[exponential(18),
exponential(3)][uniform() < 0.5]
for i in range(N)])
observed_subscriptiontimes = np.minimum(actual_subscriptiontimes,
study_duration)
observed = actual_subscriptiontimes < study_duration
# Show the data
setFonts(18)
plt.xlim(0, 24)
plt.vlines(12, 0, 30, lw=2, linestyles="--")
plt.xlabel('time')
plt.title('Subscription Times, at $t=12$ months')
plot_lifetimes(observed_subscriptiontimes, event_observed=observed)
print("Observed subscription time at time %d:\n" % (study_duration),
observed_subscriptiontimes)
return
# Calculate the values
nd = stats.norm()
x = np.linspace(-3, 3, 100)
yp = nd.pdf(x)
y = nd.cdf(x)
x1 = np.linspace(-3, 1)
y1 = nd.pdf(x1)
# Make the plot
sns.set_context('paper')
sns.set_style('white')
setFonts(12)
figs, axs = plt.subplots(1, 2)
axs[0].plot(x, yp, 'k')
axs[0].fill_between(x1, y1, facecolor='#CCCCCC')
axs[0].text(0,
0.1,
'CDF(x)',
family='cursive',
fontsize=14,
horizontalalignment='center',
style='italic')
axs[0].set_xlabel('x')
axs[0].set_ylabel('PDF(x)')
sns.despine()
def show_fig(std, ax, title):
'''Create a plot of normally distributed data in a given axis'''
for ii in range(3):
data = stats.norm(centers[ii], std).rvs(numData)
offset = ii*numData
ax.plot( offset+np.arange(numData), data, '.', ms=10)
ax.xaxis.set_ticks([50,150,250])
ax.set_xticklabels(['Group1', 'Group2', 'Group3'])
ax.set_title(title)
sns.despine()
if __name__ == '__main__':
# Set up the figure
sns.set_context('paper')
sns.set_style('whitegrid')
setFonts(14)
# Create 2 plots of 3 different, normally distributed data groups, with different SDs
fig, axs = plt.subplots(1, 2)
centers = [5, 5.3, 4.7]
stds = [0.1, 2]
numData = 100
show_fig(0.1, axs[0], 'SD=0.1')
show_fig(2, axs[1], 'SD=2.0')
showData('anova_oneway.png')
ax.xaxis.set_ticks([50,150,250])
ax.set_xticklabels(['Group1', 'Group2', 'Group3'])
ax.yaxis.set_ticks([])
ax.set_title(title)
grandMean = np.mean(groupMean)
ax.axhline(grandMean, color='#999999')
ax.plot([80, 220], [groupMean[1], groupMean[1]], '#999999')
ax.plot([80, 120], [groupMean[1]+0.2, groupMean[1]+0.2], '#999999')
ax.annotate('', xy=(210, grandMean), xytext=(210,groupMean[1]),
arrowprops=dict(arrowstyle='<->, head_width=0.1', facecolor='black'))
ax.annotate('', xy=(90, groupMean[1]), xytext=(90,groupMean[1]+0.2),
arrowprops=dict(arrowstyle='<->, head_width=0.1', facecolor='black'))
ax.text(210, (grandMean + groupMean[1])/2., '$SS_{Treatment}$', fontsize=36)
ax.text(90, groupMean[1]+0.1, '$SS_{Error}$', ha='right', fontsize=36)
if __name__ == '__main__':
centers = [5, 5.3, 4.7]
np.random.seed(123)
setFonts(30)
fig = plt.figure()
ax = fig.add_subplot(111)
std = 0.1
numData = 100
show_fig(0.1, ax, 'Sum-Squares')
# Save and show
showData('anova_annotated.png')
try:
# Import formatting commands if directory "Utilities" is available
from ISP_mystyle import setFonts, showData
except ImportError:
# Ensure correct performance otherwise
def setFonts(*options):
return
def showData(*options):
plt.show()
return
# General formatting options
sns.set(context='poster', style='ticks')
sns.set_palette(sns.color_palette('hls', 3))
setFonts(24)
#----------------------------------------------------------------------
def show_binomial():
"""Show an example of binomial distributions"""
# Arbitrarily select 3 total numbers, and 3 probabilities
ns = [20,20,40]
ps = [0.5, 0.7, 0.5]
# For each (p,n)-pair, plot the corresponding binomial PMFs
for (p,n) in zip(ps, ns):
bd = stats.binom(n,p) # generate the "frozen function"
x = np.arange(n+1) # generate the x-values
plt.plot(x, bd.pmf(x), 'o--', label='p={0:3.1f}, n={1}'.format(p,n))
def setFonts(*options):
return
def showData(*options):
plt.show()
return
# Generate the data
x = np.arange(-20, 80)
y = 10 + 0.2 * x + 4 * np.random.randn(len(x))
# Make the plot
sns.set_style('ticks')
sns.set_context('poster')
setFonts()
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(x, y, '.')
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')
sns.despine()
# Draw the fitted line
p = np.polyfit(x, y, 1)
yFit = np.polyval(p, x)
ax.plot(x, yFit, 'r')
def simplePlots():
"""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
sns.set(context="poster", style="ticks", palette=sns.color_palette("muted"))
# Set the fonts the way I like them
setFonts(32)
# Scatter plot
plt.scatter(np.arange(len(x)), x)
plt.xlim([0, len(x)])
# Save and show the data, in a systematic format
printout("scatterPlot.png", xlabel="Datapoints", ylabel="Values", title="Scatter")
# Histogram
plt.hist(x)
printout("histogram_plain.png", xlabel="Data Values", ylabel="Frequency", title="Histogram, default settings")
plt.hist(x, 25)
printout("histogram.png", xlabel="Data Values", ylabel="Frequency", title="Histogram, 25 bins")
# Cumulative probability density
numbins = 20
plt.plot(stats.cumfreq(x, numbins)[0])
printout("CumulativeFrequencyFunction.png", xlabel="Data Values", ylabel="CumFreq", title="Cumulative Frequency")
# KDE-plot
sns.kdeplot(x)
printout("kde.png", xlabel="Data Values", ylabel="Density", title="KDE_plot")
# Boxplot
# The ox consists of the first, second (middle) and third quartile
plt.boxplot(x, sym="*")
printout("boxplot.png", 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.png", xlabel="Data Values", ylabel="Measurements", title="Errorbars")
# Violinplot
nd = stats.norm
data = nd.rvs(size=(100))
nd2 = stats.norm(loc=3, scale=1.5)
data2 = nd2.rvs(size=(100))
# Use pandas and the seaborn package for the violin plot
df = pd.DataFrame({"Girls": data, "Boys": data2})
sns.violinplot(df)
printout("violinplot.png", title="Violinplot")
# Barplot
# The font-size is set such that the legend does not overlap with the data
np.random.seed(1234)
setFonts(20)
df = pd.DataFrame(np.random.rand(10, 4), columns=["a", "b", "c", "d"])
df.plot(kind="bar", grid=False, color=sns.color_palette("muted"))
showData("barplot.png")
setFonts(28)
# 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.png")
# Grouped Boxplot
sns.set_style("whitegrid")
sns.boxplot(df)
setFonts(28)
printout("groupedBoxplot.png", title="sns.boxplot")
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.png", title=" ")
def simplePlots():
'''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
sns.set(context='poster', style='ticks', palette=sns.color_palette('muted'))
# Set the fonts the way I like them
setFonts(32)
# Scatter plot
plt.scatter(np.arange(len(x)), x)
plt.xlim([0, len(x)])
# Save and show the data, in a systematic format
printout('scatterPlot.png', xlabel='Datapoints', ylabel='Values', title='Scatter')
# Histogram
plt.hist(x)
printout('histogram_plain.png', xlabel='Data Values',
ylabel='Frequency', title='Histogram, default settings')
plt.hist(x,25)
printout('histogram.png', xlabel='Data Values', ylabel='Frequency',
title='Histogram, 25 bins')
# Cumulative probability density
numbins = 20
plt.plot(stats.cumfreq(x,numbins)[0])
printout('CumulativeFrequencyFunction.png', xlabel='Data Values',
ylabel='CumFreq', title='Cumulative Frequency')
# KDE-plot
sns.kdeplot(x)
printout('kde.png', xlabel='Data Values', ylabel='Density',
title='KDE_plot')
# Boxplot
# The ox consists of the first, second (middle) and third quartile
plt.boxplot(x, sym='*')
printout('boxplot.png', 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.png', xlabel='Data Values', ylabel='Measurements', title='Errorbars')
# Violinplot
nd = stats.norm
data = nd.rvs(size=(100))
nd2 = stats.norm(loc = 3, scale = 1.5)
data2 = nd2.rvs(size=(100))
# Use pandas and the seaborn package for the violin plot
df = pd.DataFrame({'Girls':data, 'Boys':data2})
sns.violinplot(df)
printout('violinplot.png', title='Violinplot')
# Barplot
# The font-size is set such that the legend does not overlap with the data
np.random.seed(1234)
setFonts(20)
df = pd.DataFrame(np.random.rand(10, 4), columns=['a', 'b', 'c', 'd'])
df.plot(kind='bar', grid=False, color=sns.color_palette('muted'))
showData('barplot.png')
setFonts(28)
# 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.png')
# Grouped Boxplot
sns.set_style('whitegrid')
sns.boxplot(df)
setFonts(28)
printout('groupedBoxplot.png', title='sns.boxplot')
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=False, startangle=90,
colors=sns.color_palette('muted') )
plt.axis('equal')
printout('piePlot.png', title=' ')
def setFonts(*options):
return
def showData(*options):
plt.show()
return
# Generate the data
x = np.r_[3, 1.5, 4, 6, 3, 2]
dx = np.r_[0.1, 0.3, 0.2, 0.2, 0.3, 0.25]
xs = x - dx
index = range(len(x))
# plot the data
setFonts(20)
plt.plot(x, 'o', ms=10, label='pre')
plt.plot(xs, 'r*', ms=12, label='post')
plt.bar(index,
dx,
width=0.5,
align='center',
color=0.75 * np.ones(3),
label='pre-post')
# Format the plot
plt.legend(loc='upper left')
plt.axhline(0, ls='--')
plt.xlim(-0.3, 5.3)
plt.ylim(-0.2, 6.2)
plt.xlabel('Subject Nr')
import sys
sys.path.append(os.path.join('..', '..', 'Utilities'))
try:
from ISP_mystyle import setFonts, showData
except ImportError:
# Ensure correct performance otherwise
def setFonts(*options):
return
def showData(*options):
plt.show()
return
sns.set_context('poster')
sns.set_style('ticks')
setFonts()
# Generate the data
np.random.seed(1234)
nd = stats.norm(100, 20)
scores = nd.rvs(10)
# Make the plot
plt.plot(scores, 'o')
plt.axhline(110, ls='--')
plt.axhline(np.mean(scores), ls='-.')
plt.xlim(-0.2, 9.2)
plt.ylim(50, 130)
plt.xlabel('Student-Nr')
plt.ylabel('Score')
"""Subroutine showing a histogram and formatting it"""
axis.hist(data, bins=nbins)
axis.set_xticks([0, 0.5, 1])
axis.set_title(title)
if __name__ == '__main__':
# Formatting options
sns.set(context='poster', style='ticks', palette='muted')
# Input data
ndata = 100000
nbins = 50
setFonts(24)
# Generate data
data = np.random.random(ndata)
# Show three histograms, side-by-side
fig, axs = plt.subplots(1, 3)
showAsHistogram(axs[0], data, 'Random data')
showAsHistogram(axs[1], np.mean(data.reshape((int(ndata / 2), 2)), axis=1),
'Average over 2')
showAsHistogram(axs[2], np.mean(data.reshape((int(ndata / 10), 10)),
axis=1), 'Average over 10')
# Format them and show them
axs[0].set_ylabel('Counts')
plt.tight_layout()
ax.annotate('',
xy=(210, grandMean),
xytext=(210, groupMean[1]),
arrowprops=dict(arrowstyle='<->, head_width=0.1',
facecolor='black'))
ax.annotate('',
xy=(90, groupMean[1]),
xytext=(90, groupMean[1] + 0.2),
arrowprops=dict(arrowstyle='<->, head_width=0.1',
facecolor='black'))
ax.text(210, (grandMean + groupMean[1]) / 2.,
'$SS_{Treatment}$',
fontsize=36)
ax.text(90, groupMean[1] + 0.1, '$SS_{Error}$', ha='right', fontsize=36)
if __name__ == '__main__':
centers = [5, 5.3, 4.7]
np.random.seed(123)
setFonts(30)
fig = plt.figure()
ax = fig.add_subplot(111)
std = 0.1
numData = 100
show_fig(0.1, ax, 'Sum-Squares')
# Save and show
showData('anova_annotated.png')
plt.show()
return
# Calculate the values
nd = stats.norm()
x = np.linspace(-3,3,100)
yp = nd.pdf(x)
y = nd.cdf(x)
x1 = np.linspace(-3, 1)
y1 = nd.pdf(x1)
# Make the plot
sns.set_context('paper')
sns.set_style('white')
setFonts(12)
figs, axs = plt.subplots(1,2)
axs[0].plot(x,yp, 'k')
axs[0].fill_between(x1, y1, facecolor='#CCCCCC')
axs[0].text(0, 0.1, 'CDF(x)', family='cursive', fontsize=14, horizontalalignment='center', style='italic')
axs[0].set_xlabel('x')
axs[0].set_ylabel('PDF(x)')
sns.despine()
axs[1].plot(x, y, '#999999', lw=3)
axs[1].set_xlabel('x')
axs[1].set_ylabel('CDF(x)')
plt.vlines(0, 0, 1, linestyles='--')
sns.despine()
def setFonts(*options):
return
def showData(*options):
plt.show()
return
# Generate the data
x = np.r_[3, 1.5, 4, 6, 3, 2]
dx = np.r_[0.1, 0.3, 0.2, 0.2, 0.3, 0.25]
xs = x - dx
index = range(len(x))
# plot the data
setFonts(20)
plt.plot(x, "o", ms=10, label="pre")
plt.plot(xs, "r*", ms=12, label="post")
plt.bar(index, dx, width=0.5, align="center", color=0.75 * np.ones(3), label="pre-post")
# Format the plot
plt.legend(loc="upper left")
plt.axhline(0, ls="--")
plt.xlim(-0.3, 5.3)
plt.ylim(-0.2, 6.2)
plt.xlabel("Subject Nr")
plt.ylabel("Value")
plt.tight_layout()
# P-values for paired and unpaired T-tests
_, p_paired = stats.ttest_rel(x, xs)
sys.path.append(os.path.join('..', '..', 'Utilities'))
try:
# Import formatting commands if directory "Utilities" is available
from ISP_mystyle import setFonts
except ImportError:
# Ensure correct performance otherwise
def setFonts(*options):
return
# Generate some dummy data
np.set_printoptions(precision=2)
N = 20
study_duration = 12
# Note: a constant dropout rate is equivalent to an exponential distribution!
actual_subscriptiontimes = np.array([[exponential(18), exponential(3)][uniform()<0.5] for i in range(N)])
observed_subscriptiontimes = np.minimum(actual_subscriptiontimes,study_duration)
observed= actual_subscriptiontimes < study_duration
# Show the data
setFonts(18)
plt.xlim(0,24)
plt.vlines(12, 0, 30, lw=2, linestyles="--")
plt.xlabel('time')
plt.title('Subscription Times, at $t=12$ months')
plot_lifetimes(observed_subscriptiontimes, event_observed=observed)
print("Observed subscription time at time %d:\n"%(study_duration), observed_subscriptiontimes)
