We require plots, charts and other statistical graphics for the written communication of quantitative ideas.
They allow us to more easily convey relationships and reveal deviations from patterns.
Gelman and Unwin 2011:
A well-designed graph can display more information than a table of the same size, and more information than numbers embedded in text. Graphical displays allow and encourage direct visual comparisons.
Matplotlib is an excellent 2D and 3D graphics library for generating scientific figures in Python. Some of the many advantages of this library includes:
One of the of the key features of matplotlib that I would like to emphasize, and that I think makes matplotlib highly suitable for generating figures for scientific publications is that all aspects of the figure can be controlled programmatically. This is important for reproducibility, convenient when one need to regenerate the figure with updated data or changes its appearance.
The convention for loading Matplotlib is in its own shortened namespace:
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%matplotlib inline
import matplotlib.pyplot as plt
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import numpy as np
rain = np.genfromtxt('../data/nashville_precip.txt', delimiter='\t',
names=True, missing_values='NA')
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rain[:3]
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x = rain['Year']
y = rain['Jan']
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plt.figure()
plt.plot(x, y, 'r')
plt.xlabel('Year')
plt.ylabel('Rainfall')
plt.title('January rainfall in Nashville')
It is straightforward to customize plotting symbols and create subplots.
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plt.figure(figsize=(14,6))
plt.subplot(1,2,2)
plt.plot(x, y, 'r--')
plt.subplot(1,2,1)
plt.plot(x, rain['Feb'], 'g*-')
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# Write your answer here
While the MATLAB-like API is easy and convenient, it is worth learning matplotlib's object-oriented plotting API. It is remarkably powerful and for advanced figures, with subplots, insets and other components it is very nice to work with.
The main idea with object-oriented programming is to have objects with associated methods and functions that operate on them, and no object or program states should be global.
To use the object-oriented API we start out very much like in the previous example, but instead of creating a new global figure instance we store a reference to the newly created figure instance in the fig variable, and from it we create a new axis instance axes using the add_axes method in the Figure class instance fig.
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fig = plt.figure()
# left, bottom, width, height (range 0 to 1)
# as fractions of figure size
axes = fig.add_axes([0.1, 0.1, 0.8, 0.8])
axes.plot(x, y, 'r', label='rainfall')
axes.set_xlabel('Year')
axes.set_ylabel('Rainfall')
Although a little bit more code is involved, the advantage is that we now have full control of where the plot axes are place, and we can easily add more than one axis to the figure.
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fig = plt.figure()
axes1 = fig.add_axes([0.1, 0.1, 0.9, 0.9]) # main axes
axes2 = fig.add_axes([0.65, 0.65, 0.3, 0.3]) # inset axes
# main figure
axes1.plot(x, y, 'r')
axes1.set_xlabel('Year')
axes1.set_ylabel('Rainfall')
axes1.set_title('January rainfall in Nashville');
# insert
axes2.plot(x, np.log(y), 'g')
axes2.set_title('Log rainfall');
If we don't care to be explicit about where our plot axes are placed in the figure canvas, then we can use one of the many axis layout managers in matplotlib, such as subplots.
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fig, axes = plt.subplots(nrows=4, ncols=1)
months = rain.dtype.names[1:]
for i,ax in enumerate(axes):
ax.plot(x, rain[months[i]], 'r')
ax.set_xlabel('Year')
ax.set_ylabel('Rainfall')
ax.set_title(months[i])
That was easy, but it's not so pretty with overlapping figure axes and labels, right?
We can deal with that by using the fig.tight_layout method, which automatically adjusts the positions of the axes on the figure canvas so that there is no overlapping content:
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fig, axes = plt.subplots(nrows=4, ncols=1, figsize=(10,10))
months = rain.dtype.names[1:]
for i,ax in enumerate(axes):
ax.plot(rain['Year'], rain[months[i]], 'r')
ax.set_xlabel('Year')
ax.set_ylabel('Rainfall')
ax.set_title(months[i])
fig.tight_layout()
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# Write your answer here
Matplotlib allows the aspect ratio, DPI and figure size to be specified when the Figure object is created, using the figsize and dpi keyword arguments. figsize is a tuple with width and height of the figure in inches, and dpi is the dot-per-inch (pixel per inch). To create a figure with size 800 by 400 pixels we can do:
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fig = plt.figure(figsize=(8,4), dpi=100)
The same arguments can also be passed to layout managers, such as the subplots function.
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fig, axes = plt.subplots(figsize=(12,3))
axes.plot(x, y, 'r')
axes.set_xlabel('Year')
axes.set_ylabel('Rainfall')
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fig.savefig("jan_rainfall.png", format='jpg')
Notice that the graphic file format is inferred from the extension of the file name (though there is an optional format argument).
Here we can also optionally specify the DPI, and chose between different output formats.
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fig.savefig("jan_rainfall.png", dpi=200)
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fig.savefig("jan_rainfall.svg")
Matplotlib can generate high-quality output in a number formats, including:
Figure titles
A title can be added to each axis instance in a figure. To set the title use the set_title method in the axes instance:
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ax.set_title("January rainfall")
Axis labels
Similarly, using the methods set_xlabel and set_ylabel we can set the labels of the X and Y axes:
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ax.set_xlabel("Year")
ax.set_ylabel("Rainfall")
Legends
Legends to curves in a figure can be added in two ways. First method is to use the legend method of the axis object and pass a list/tuple of legend texts for the curves that have previously been added:
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ax.legend(["Jan", "Feb"]);
A more robust method for associating labels with plots is to use the label keyword argument when plots a other objects are added to the figure, and then using the legend method without arguments to add the legend.
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ax.plot(x, rain['Jan'], label="Jan")
ax.plot(x, rain['Feb'], label="Feb")
ax.legend()
The advantage with this method is that if curves are added or removed from the figure, the legend is automatically updated accordingly.
The legend function takes and optional keywork argument loc that can be used to specify where in the figure the legend is to be drawn. The allowed values of loc are numerical codes for the various places the legend can be drawn.
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ax.legend(loc=0) # let matplotlib decide the optimal location
ax.legend(loc=1) # upper right corner
ax.legend(loc=2) # upper left corner
ax.legend(loc=3) # lower left corner
ax.legend(loc=4) # lower right corner
The following figure show how to use the figure title, axis labels and legends described above:
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fig, ax = plt.subplots()
ax.plot(x, rain['Jan'], label="Jan")
ax.plot(x, rain['Aug'], label="Aug")
ax.set_xlabel('Year')
ax.set_ylabel('Rainfall')
ax.legend(loc=1); # upper left corner
Matplotlib has great support for $\LaTeX$. All we need to do is to use dollar signs encapsulate $\LaTeX$ in the text, just as with Markdown and MathJax.
In order to properly process $\LaTeX$ commands, which begin with a backslash, we must specify the $\LaTeX$ code as a raw string, by prepending the string with an "r". This is because the backslash is interpreted by Python as the escape code character. For example, rather than "\alpha" we use r"\alpha".
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import scipy.stats as stats
fig, ax = plt.subplots()
x = np.linspace(-3, 3, 100)
y1 = stats.distributions.norm.pdf(x)
ax.plot(x, y1, label=r"$\sigma=1$")
y2 = stats.distributions.norm.pdf(x, scale=0.5)
ax.plot(x, y2, label=r"$\sigma=0.5$")
ax.set_xlabel('x', fontsize=18)
ax.set_ylabel('f(x)', fontsize=18)
ax.set_title('Normal distributions')
ax.legend(loc=2); # upper left corner
We can also change the global font size and font family, which applies to all text elements in a figure (tick labels, axis labels and titles, legends, etc.):
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import matplotlib as mpl
# Update the matplotlib configuration parameters:
mpl.rcParams.update({'font.size': 18,
'font.family': 'serif'})
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fig, ax = plt.subplots()
x = np.linspace(-3, 3, 100)
y1 = stats.distributions.norm.pdf(x)
ax.plot(x, y1, label=r"$\sigma=1$")
y2 = stats.distributions.norm.pdf(x, scale=0.5)
ax.plot(x, y2, label=r"$\sigma=0.5$")
ax.set_xlabel('x', fontsize=18)
ax.set_ylabel('f(x)', fontsize=18)
ax.set_title('Normal distributions')
ax.legend(loc=2); # upper left corner
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# restore
mpl.rcParams.update({'font.size': 12, 'font.family': 'sans'})
A covenient approach for temporarily changing plotting options is to use a context manager:
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with mpl.rc_context(rc={'font.family': 'serif', 'font.weight': 'bold', 'font.size': 8}):
fig = plt.figure(figsize=(6,3))
ax1 = fig.add_subplot(121)
ax1.set_xlabel('some random numbers')
ax1.set_ylabel('more random numbers')
ax1.set_title("Random scatterplot")
plt.plot(np.random.normal(size=100), np.random.normal(size=100), 'r.')
ax2 = fig.add_subplot(122)
plt.hist(np.random.normal(size=100), bins=15)
ax2.set_xlabel('sample')
ax2.set_ylabel('cumulative sum')
ax2.set_title("Normal distrubution")
plt.tight_layout()
plt.savefig("normalvars.png", dpi=150)
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fig, ax = plt.subplots()
ax.plot(x, y1, 'b:') # blue dotted line
ax.plot(x, y2, 'g--') # green dashed line
If we want a broader range of colors, they can be specified by their name or RGB hex code using the color keyword. An alpha value can also be specified.
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fig, ax = plt.subplots()
ax.plot(x, y1, color="red", alpha=0.1) # faint red
ax.plot(x, y2, color="#15cc55") # RGB hex code for a greenish color
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fig, ax = plt.subplots(figsize=(12,6))
x = np.linspace(0, 10, 10)
ax.plot(x, x+1, color="blue", linewidth=0.25)
ax.plot(x, x+2, color="blue", linewidth=0.50)
ax.plot(x, x+3, color="blue", linewidth=1.00)
ax.plot(x, x+4, color="blue", linewidth=2.00)
# possible linestype options ‘-‘, ‘–’, ‘-.’, ‘:’, ‘steps’
ax.plot(x, x+5, color="red", lw=2, linestyle='-')
ax.plot(x, x+6, color="red", lw=2, ls='-.')
ax.plot(x, x+7, color="red", lw=2, ls=':')
# custom dash
line, = ax.plot(x, x+8, color="black", lw=1.50)
line.set_dashes([5, 10, 15, 10]) # format: line length, space length, ...
# possible marker symbols: marker = '+', 'o', '*', 's', ',', '.', '1', '2', '3', '4', ...
ax.plot(x, x+ 9, color="green", lw=2, ls=':', marker='+')
ax.plot(x, x+10, color="green", lw=2, ls=':', marker='o')
ax.plot(x, x+11, color="green", lw=2, ls=':', marker='s')
ax.plot(x, x+12, color="green", lw=2, ls=':', marker='1')
# marker size and color
ax.plot(x, x+13, color="purple", lw=1, ls='-', marker='o', markersize=2)
ax.plot(x, x+14, color="purple", lw=1, ls='-', marker='o', markersize=4)
ax.plot(x, x+15, color="purple", lw=1, ls='-', marker='o', markersize=8,
markerfacecolor="red")
ax.plot(x, x+16, color="purple", lw=1, ls='-', marker='s', markersize=8,
markerfacecolor="yellow", markeredgewidth=2, markeredgecolor="blue");
The appearance of the axes is an important aspect of a figure that we often need to modify to make a publication quality graphics. We need to be able to control where the ticks and labels are placed, modify the font size and possibly the labels used on the axes.
We can specify the axis range using the set_ylim and set_xlim methods of the axis object, or axis('tight') for automatrically setting axes ranges.
In the figure below, give the second set of axes a tight axis range, and constrain the x- and y-ranges of the third subplot to a subset of their respective ranges:
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fig, axes = plt.subplots(3, 1, figsize=(8, 12))
axes[0].plot(rain['Year'], rain['Jan'], rain['Year'], rain['Feb'])
axes[0].set_title("default axis ranges")
axes[1].plot(rain['Year'], rain['Jan'], rain['Year'], rain['Feb'])
axes[1].set_title("tight axes")
axes[2].plot(rain['Year'], rain['Jan'], rain['Year'], rain['Feb'])
axes[2].set_title("custom axis ranges");
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fig, ax = plt.subplots(figsize=(10, 4))
ax.plot(rain['Year'], rain['Jan'], rain['Year'], rain['Feb'])
ax.set_xlim(1880, 1970)
ax.set_xticks([1900, 1920, 1940, 1960])
ax.set_xticklabels(['Turn of\ncentury', 'Twenties', 'Forties', 'Sixties'],
fontsize=12)
yticks = [2, 6, 10, 14]
ax.set_yticks(yticks)
ax.set_yticklabels(['dry', 'moist', 'wet', 'soggy'], fontsize=18)
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fig, axes = plt.subplots(1, 2, figsize=(10,3))
# default grid appearance
axes[0].plot(rain['Year'], rain['Jan'], linewidth=2)
axes[0].grid(True)
# custom grid appearance
axes[1].plot(rain['Year'], rain['Jan'], linewidth=2)
axes[1].grid(True)
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fig, ax = plt.subplots(figsize=(6,2))
ax.plot(rain['Year'], rain['Jan'])
ax.spines['top'].set_color('none')
ax.spines['left'].set_color('none')
ax.spines['left'].set_linewidth(2)
ax.spines['right'].set_color("none")
ax.xaxis.tick_top()
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fig, ax1 = plt.subplots()
ax1.plot(rain['Year'], rain['Jan'], color='blue')
ax1.set_ylabel("rain (mm)", fontsize=18, color="blue")
for label in ax1.get_yticklabels():
label.set_color("blue")
ax2 = ax1.twinx()
ax2.plot(rain['Year'], np.log(rain['Jan']), color='red')
ax2.set_ylabel("rain (log mm)", fontsize=18, color="red")
for label in ax2.get_yticklabels():
label.set_color("red")
Axes can be placed at arbitrary locations, with the commonest choice being $x=0$ and $y=0$.
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fig, ax = plt.subplots()
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
# Set location of x-axis
ax.xaxis.set_ticks_position('bottom')
ax.spines['bottom'].set_position(('data',0))
# Set location of y-axis
ax.yaxis.set_ticks_position('left')
ax.spines['left'].set_position(('data',0))
x = np.linspace(-1, 1., 100)
ax.plot(x, x**3);
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fig, axes = plt.subplots(2, 2, figsize=(12,6))
axes[0,0].scatter(rain['Jan'], rain['Jul'])
axes[0,1].step(rain['Year'][:10], np.cumsum(rain['Jan'][:10]), lw=2)
axes[1,0].bar(rain['Year'][:10], rain['Apr'][:10], align="center", width=0.5, alpha=0.5)
minrain = [np.nanmin(rain[m]) for m in months]
maxrain = [np.nanmax(rain[m]) for m in months]
axes[1,1].fill_between(range(12), minrain, maxrain, color="green", alpha=0.2)
_ = axes[1,1].set_xticklabels(months[::2])
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# Write your answer here
Error bars
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x = np.linspace(1, 10, 30)
y = [np.random.normal(0, xi, 20) for xi in x]
fig, ax = plt.subplots()
plt.errorbar(x, np.mean(y, 1), np.std(y, 1), fmt='.k')
Polar plot
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# polar plot using add_axes and polar projection
fig = plt.figure()
ax = fig.add_axes([0.0, 0.0, .6, .6], polar=True)
t = np.linspace(0, 2 * np.pi, 12)
ax.plot(t, list(rain[1])[1:], color='blue', lw=3)
_ = ax.set_xticklabels(months)
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fig, ax = plt.subplots()
xx = np.linspace(-0.75, 1., 100)
ax.plot(xx, xx**2)
ax.plot(xx, xx**3)
ax.text(0.15, 0.2, r"$y=x^2$", fontsize=20, color="blue")
ax.text(0.65, 0.1, r"$y=x^3$", fontsize=20, color="green");
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fig, ax = plt.subplots(2, 3)
for i,m in enumerate(months[:6]):
(ax.ravel()[i]).plot(rain['Year'], rain[m])
(ax.ravel()[i]).set_title(m)
fig.tight_layout()
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ax = plt.subplot2grid((2,2),(0, 0))
Here, the grid is specified by shape=(2,2), at location of loc=(0,0).
This is identical to:
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ax = plt.subplot(2,2,1)
It is handy when we are assembling a figure composed of heterogeneously-sized subplots.
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fig = plt.figure()
ax1 = plt.subplot2grid((3,3), (0,0), colspan=3)
ax2 = plt.subplot2grid((3,3), (1,0), colspan=2)
ax3 = plt.subplot2grid((3,3), (1,2), rowspan=2)
ax4 = plt.subplot2grid((3,3), (2,0))
ax5 = plt.subplot2grid((3,3), (2,1))
fig.tight_layout()
gridspec is a module which specifies the location of the subplot in the figure.
GridSpec specifies the geometry of the grid that a subplot will be placed. The number of rows and number of columns of the grid need to be set. Optionally, the subplot layout parameters (e.g., left, right, etc.) can be tuned.SubplotSpec specifies the location of the subplot in the given GridSpec.A gridspec.GridSpec instance provides array-like (2d or 1d) indexing that returns a SubplotSpec instance. For, SubplotSpec that spans multiple cells, use slice.
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import matplotlib.gridspec as gridspec
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fig = plt.figure()
gs = gridspec.GridSpec(3, 3)
# identical to ax1 = plt.subplot(gs.new_subplotspec((0,0), colspan=3))
ax1 = plt.subplot(gs[0, :], axisbg='red')
ax2 = plt.subplot(gs[1,:-1], axisbg='blue')
ax3 = plt.subplot(gs[1:,-1], axisbg='magenta')
ax4 = plt.subplot(gs[-1,0], axisbg='yellow')
ax5 = plt.subplot(gs[-1,-2], axisbg='cyan')
plt.tight_layout()
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fig, ax = plt.subplots()
ax.plot(rain['Year'], rain['Jan'])
# inset
inset_ax = fig.add_axes([0.6, 0.6, 0.35, 0.3], axisbg='white') # X, Y, width, height
inset_ax.plot(rain['Year'], rain['Jan'])
inset_ax.set_title('zoom to 1950\'s')
# set inset axis range
inset_ax.set_xlim(1949, 1961)
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#from pymc.distributions import mv_normal_cov_like as mvnorm
from scipy.stats import multivariate_normal as mvnorm
x1 = x2 = np.linspace(-4, 4, 100)
mu = np.array([0,0])
Sigma = [[1, 0.5],[0.5, 1]]
z = np.array([[mvnorm(mu, Sigma).pdf([xi,yi]) for xi in x1] for yi in x2])
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fig, ax = plt.subplots()
x, y = np.meshgrid(x1, x2)
p = ax.pcolor(x, y, z, cmap=mpl.cm.Reds)
cb = fig.colorbar(p)
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fig, ax = plt.subplots()
im = plt.imshow(z, cmap=mpl.cm.RdBu)
im.set_interpolation('bilinear')
cb = fig.colorbar(im)
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fig, ax = plt.subplots()
cnt = plt.contour(z, cmap=mpl.cm.Blues, vmin=abs(z).min(), vmax=abs(z).max(), extent=[0, 1, 0, 1])
Matplotlib has a number of "backends", which are responsible for rendering graphs. The different backends are able to generate graphics with different formats or using different display technologies. There is a distinction between noninteractive backends (such as 'agg', 'svg', 'pdf', etc.) that are only used to generate images files (with for example the savefig function), and interactive backends (such as Qt4Agg, GTK, MaxOSX) that can display a GUI window for interactively exploring figures.
A list of available backends are:
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mpl.rcsetup.all_backends
The standard backend is called agg, and is based on a library for raster graphics and is great for generating raster formats such as PNG.
Normally we don't need to bother with changing the default backend, but sometimes it can be useful to switch to for example the PDF or GTKCairo (if you are using Linux) to produce high-quality vector graphics instead of raster based graphics.
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# %load http://matplotlib.org/mpl_examples/showcase/xkcd.py
import matplotlib.pyplot as plt
import numpy as np
with plt.xkcd():
# Based on "Stove Ownership" from XKCD by Randall Monroe
# http://xkcd.com/418/
fig = plt.figure()
ax = fig.add_axes((0.1, 0.2, 0.8, 0.7))
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
plt.xticks([])
plt.yticks([])
ax.set_ylim([-30, 10])
data = np.ones(100)
data[70:] -= np.arange(30)
plt.annotate(
'THE DAY I REALIZED\nI COULD COOK BACON\nWHENEVER I WANTED',
xy=(70, 1), arrowprops=dict(arrowstyle='->'), xytext=(15, -10))
plt.plot(data)
plt.xlabel('time')
plt.ylabel('my overall health')
fig.text(
0.5, 0.05,
'"Stove Ownership" from xkcd by Randall Monroe',
ha='center')
# Based on "The Data So Far" from XKCD by Randall Monroe
# http://xkcd.com/373/
fig = plt.figure()
ax = fig.add_axes((0.1, 0.2, 0.8, 0.7))
ax.bar([-0.125, 1.0 - 0.125], [0, 100], 0.25)
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.set_xticks([0, 1])
ax.set_xlim([-0.5, 1.5])
ax.set_ylim([0, 110])
ax.set_xticklabels(['CONFIRMED BY\nEXPERIMENT', 'REFUTED BY\nEXPERIMENT'])
plt.yticks([])
plt.title("CLAIMS OF SUPERNATURAL POWERS")
fig.text(
0.5, 0.05,
'"The Data So Far" from xkcd by Randall Monroe',
ha='center')
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random_dists = ['Normal(1,1)',' Lognormal(1,1)', 'Exp(1)', 'Gumbel(6,4)',
'Triangular(2,9,11)']
num_dists = len(random_dists)
N = 500
norm = np.random.normal(1,1, N)
logn = np.random.lognormal(1,1, N)
expon = np.random.exponential(1, N)
gumb = np.random.gumbel(6, 4, N)
tri = np.random.triangular(2, 9, 11, N)
Resampling is easy: we can simply generate random integers (with replacement) and use them to index the values of the original sample.
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resample_indices = np.random.randint(0, N-1, N)
norm_resample = norm[resample_indices]
expon_resample = expon[resample_indices]
gumb_resample = gumb[resample_indices]
logn_resample = logn[resample_indices]
tri_resample = tri[resample_indices]
As closely as possible, try to replicate this plot:
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# Write your answer here