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Python Data Science Handbook by Jake VanderPlas

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Chapter 4. Visualization with Matplotlib

We’ll now take an in-depth look at the Matplotlib tool for visualization in Python. Matplotlib is a multiplatform data visualization library built on NumPy arrays, and designed to work with the broader SciPy stack. It was conceived by John Hunter in 2002, originally as a patch to IPython for enabling interactive MATLAB-style plotting via gnuplot from the IPython command line. IPython’s creator, Fernando Perez, was at the time scrambling to finish his PhD, and let John know he wouldn’t have time to review the patch for several months. John took this as a cue to set out on his own, and the Matplotlib package was born, with version 0.1 released in 2003. It received an early boost when it was adopted as the plotting package of choice of the Space Telescope Science Institute (the folks behind the Hubble Telescope), which financially supported Matplotlib’s development and greatly expanded its capabilities.

One of Matplotlib’s most important features is its ability to play well with many operating systems and graphics backends. Matplotlib supports dozens of backends and output types, which means you can count on it to work regardless of which operating system you are using or which output format you wish. This cross-platform, everything-to-everyone approach has been one of the great strengths of Matplotlib. It has led to a large userbase, which in turn has led to an active developer base and Matplotlib’s powerful tools and ubiquity within the scientific Python world.

In recent years, however, the interface and style of Matplotlib have begun to show their age. Newer tools like ggplot and ggvis in the R language, along with web visualization toolkits based on D3js and HTML5 canvas, often make Matplotlib feel clunky and old-fashioned. Still, I’m of the opinion that we cannot ignore Matplotlib’s strength as a well-tested, cross-platform graphics engine. Recent Matplotlib versions make it relatively easy to set new global plotting styles (see “Customizing Matplotlib: Configurations and Stylesheets”), and people have been developing new packages that build on its powerful internals to drive Matplotlib via cleaner, more modern APIs—for example, Seaborn (discussed in “Visualization with Seaborn”), ggplot, HoloViews, Altair, and even Pandas itself can be used as wrappers around Matplotlib’s API. Even with wrappers like these, it is still often useful to dive into Matplotlib’s syntax to adjust the final plot output. For this reason, I believe that Matplotlib itself will remain a vital piece of the data visualization stack, even if new tools mean the community gradually moves away from using the Matplotlib API directly.

General Matplotlib Tips

Before we dive into the details of creating visualizations with Matplotlib, there are a few useful things you should know about using the package.

Importing matplotlib

Just as we use the np shorthand for NumPy and the pd shorthand for Pandas, we will use some standard shorthands for Matplotlib imports:

In[1]: import matplotlib as mpl
       import matplotlib.pyplot as plt

The plt interface is what we will use most often, as we’ll see throughout this chapter.

Setting Styles

We will use the plt.style directive to choose appropriate aesthetic styles for our figures. Here we will set the classic style, which ensures that the plots we create use the classic Matplotlib style:

In[2]: plt.style.use('classic')

Throughout this section, we will adjust this style as needed. Note that the stylesheets used here are supported as of Matplotlib version 1.5; if you are using an earlier version of Matplotlib, only the default style is available. For more information on stylesheets, see “Customizing Matplotlib: Configurations and Stylesheets”.

show() or No show()? How to Display Your Plots

A visualization you can’t see won’t be of much use, but just how you view your Matplotlib plots depends on the context. The best use of Matplotlib differs depending on how you are using it; roughly, the three applicable contexts are using Matplotlib in a script, in an IPython terminal, or in an IPython notebook.

Plotting from a script

If you are using Matplotlib from within a script, the function plt.show() is your friend. plt.show() starts an event loop, looks for all currently active figure objects, and opens one or more interactive windows that display your figure or figures.

So, for example, you may have a file called myplot.py containing the following:

# ------- file: myplot.py ------
import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(0, 10, 100)

plt.plot(x, np.sin(x))
plt.plot(x, np.cos(x))

plt.show()

You can then run this script from the command-line prompt, which will result in a window opening with your figure displayed:

$ python myplot.py

The plt.show() command does a lot under the hood, as it must interact with your system’s interactive graphical backend. The details of this operation can vary greatly from system to system and even installation to installation, but Matplotlib does its best to hide all these details from you.

One thing to be aware of: the plt.show() command should be used only once per Python session, and is most often seen at the very end of the script. Multiple show() commands can lead to unpredictable backend-dependent behavior, and should mostly be avoided.

Plotting from an IPython shell

It can be very convenient to use Matplotlib interactively within an IPython shell (see Chapter 1). IPython is built to work well with Matplotlib if you specify Matplotlib mode. To enable this mode, you can use the %matplotlib magic command after starting ipython:

In [1]: %matplotlib
Using matplotlib backend: TkAgg

In [2]: import matplotlib.pyplot as plt

At this point, any plt plot command will cause a figure window to open, and further commands can be run to update the plot. Some changes (such as modifying properties of lines that are already drawn) will not draw automatically; to force an update, use plt.draw(). Using plt.show() in Matplotlib mode is not required.

Plotting from an IPython notebook

The IPython notebook is a browser-based interactive data analysis tool that can combine narrative, code, graphics, HTML elements, and much more into a single executable document (see Chapter 1).

Plotting interactively within an IPython notebook can be done with the %matplotlib command, and works in a similar way to the IPython shell. In the IPython notebook, you also have the option of embedding graphics directly in the notebook, with two possible options:

  • %matplotlib notebook will lead to interactive plots embedded within the notebook

  • %matplotlib inline will lead to static images of your plot embedded in the notebook

For this book, we will generally opt for %matplotlib inline:

In[3]: %matplotlib inline

After you run this command (it needs to be done only once per kernel/session), any cell within the notebook that creates a plot will embed a PNG image of the resulting graphic (Figure 4-1):

In[4]: import numpy as np
       x = np.linspace(0, 10, 100)

       fig = plt.figure()
       plt.plot(x, np.sin(x), '-')
       plt.plot(x, np.cos(x), '--');
png
Figure 4-1. Basic plotting example

Saving Figures to File

One nice feature of Matplotlib is the ability to save figures in a wide variety of formats. You can save a figure using the savefig() command. For example, to save the previous figure as a PNG file, you can run this:

In[5]: fig.savefig('my_figure.png')

We now have a file called my_figure.png in the current working directory:

In[6]: !ls -lh my_figure.png
-rw-r--r--  1 jakevdp  staff    16K Aug 11 10:59 my_figure.png

To confirm that it contains what we think it contains, let’s use the IPython Image object to display the contents of this file (Figure 4-2):

In[7]: from IPython.display import Image
       Image('my_figure.png')
png
Figure 4-2. PNG rendering of the basic plot

In savefig(), the file format is inferred from the extension of the given filename. Depending on what backends you have installed, many different file formats are available. You can find the list of supported file types for your system by using the following method of the figure canvas object:

In[8]: fig.canvas.get_supported_filetypes()
Out[8]: {'eps': 'Encapsulated Postscript',
         'jpeg': 'Joint Photographic Experts Group',
         'jpg': 'Joint Photographic Experts Group',
         'pdf': 'Portable Document Format',
         'pgf': 'PGF code for LaTeX',
         'png': 'Portable Network Graphics',
         'ps': 'Postscript',
         'raw': 'Raw RGBA bitmap',
         'rgba': 'Raw RGBA bitmap',
         'svg': 'Scalable Vector Graphics',
         'svgz': 'Scalable Vector Graphics',
         'tif': 'Tagged Image File Format',
         'tiff': 'Tagged Image File Format'}

Note that when saving your figure, it’s not necessary to use plt.show() or related commands discussed earlier.

Two Interfaces for the Price of One

A potentially confusing feature of Matplotlib is its dual interfaces: a convenient MATLAB-style state-based interface, and a more powerful object-oriented interface. We’ll quickly highlight the differences between the two here.

MATLAB-style interface

Matplotlib was originally written as a Python alternative for MATLAB users, and much of its syntax reflects that fact. The MATLAB-style tools are contained in the pyplot (plt) interface. For example, the following code will probably look quite familiar to MATLAB users (Figure 4-3):

In[9]: plt.figure()  # create a plot figure

       # create the first of two panels and set current axis
       plt.subplot(2, 1, 1) # (rows, columns, panel number)
       plt.plot(x, np.sin(x))

       # create the second panel and set current axis
       plt.subplot(2, 1, 2)
       plt.plot(x, np.cos(x));
png
Figure 4-3. Subplots using the MATLAB-style interface

It’s important to note that this interface is stateful: it keeps track of the “current” figure and axes, which are where all plt commands are applied. You can get a reference to these using the plt.gcf() (get current figure) and plt.gca() (get current axes) routines.

While this stateful interface is fast and convenient for simple plots, it is easy to run into problems. For example, once the second panel is created, how can we go back and add something to the first? This is possible within the MATLAB-style interface, but a bit clunky. Fortunately, there is a better way.

Object-oriented interface

The object-oriented interface is available for these more complicated situations, and for when you want more control over your figure. Rather than depending on some notion of an “active” figure or axes, in the object-oriented interface the plotting functions are methods of explicit Figure and Axes objects. To re-create the previous plot using this style of plotting, you might do the following (Figure 4-4):

In[10]: # First create a grid of plots
        # ax will be an array of two Axes objects
        fig, ax = plt.subplots(2)

        # Call plot() method on the appropriate object
        ax[0].plot(x, np.sin(x))
        ax[1].plot(x, np.cos(x));
png
Figure 4-4. Subplots using the object-oriented interface

For more simple plots, the choice of which style to use is largely a matter of preference, but the object-oriented approach can become a necessity as plots become more complicated. Throughout this chapter, we will switch between the MATLAB-style and object-oriented interfaces, depending on what is most convenient. In most cases, the difference is as small as switching plt.plot() to ax.plot(), but there are a few gotchas that we will highlight as they come up in the following sections.

Simple Line Plots

Perhaps the simplest of all plots is the visualization of a single function y = f ( x ) . Here we will take a first look at creating a simple plot of this type. As with all the following sections, we’ll start by setting up the notebook for plotting and importing the functions we will use:

In[1]: %matplotlib inline
       import matplotlib.pyplot as plt
       plt.style.use('seaborn-whitegrid')
       import numpy as np

For all Matplotlib plots, we start by creating a figure and an axes. In their simplest form, a figure and axes can be created as follows (Figure 4-5):

In[2]: fig = plt.figure()
       ax = plt.axes()
png
Figure 4-5. An empty gridded axes

In Matplotlib, the figure (an instance of the class plt.Figure) can be thought of as a single container that contains all the objects representing axes, graphics, text, and labels. The axes (an instance of the class plt.Axes) is what we see above: a bounding box with ticks and labels, which will eventually contain the plot elements that make up our visualization. Throughout this book, we’ll commonly use the variable name fig to refer to a figure instance, and ax to refer to an axes instance or group of axes instances.

Once we have created an axes, we can use the ax.plot function to plot some data. Let’s start with a simple sinusoid (Figure 4-6):

In[3]: fig = plt.figure()
       ax = plt.axes()

       x = np.linspace(0, 10, 1000)
       ax.plot(x, np.sin(x));
png
Figure 4-6. A simple sinusoid

Alternatively, we can use the pylab interface and let the figure and axes be created for us in the background (Figure 4-7; see “Two Interfaces for the Price of One” for a discussion of these two interfaces):

In[4]: plt.plot(x, np.sin(x));
png
Figure 4-7. A simple sinusoid via the object-oriented interface

If we want to create a single figure with multiple lines, we can simply call the plot function multiple times (Figure 4-8):

In[5]: plt.plot(x, np.sin(x))
       plt.plot(x, np.cos(x));
png
Figure 4-8. Over-plotting multiple lines

That’s all there is to plotting simple functions in Matplotlib! We’ll now dive into some more details about how to control the appearance of the axes and lines.

Adjusting the Plot: Line Colors and Styles

The first adjustment you might wish to make to a plot is to control the line colors and styles. The plt.plot() function takes additional arguments that can be used to specify these. To adjust the color, you can use the color keyword, which accepts a string argument representing virtually any imaginable color. The color can be specified in a variety of ways (Figure 4-9):

In[6]:
plt.plot(x, np.sin(x - 0), color='blue')        # specify color by name
plt.plot(x, np.sin(x - 1), color='g')           # short color code (rgbcmyk)
plt.plot(x, np.sin(x - 2), color='0.75')        # Grayscale between 0 and 1
plt.plot(x, np.sin(x - 3), color='#FFDD44')     # Hex code (RRGGBB from 00 to FF)
plt.plot(x, np.sin(x - 4), color=(1.0,0.2,0.3)) # RGB tuple, values 0 and 1
plt.plot(x, np.sin(x - 5), color='chartreuse'); # all HTML color names supported
png
Figure 4-9. Controlling the color of plot elements

If no color is specified, Matplotlib will automatically cycle through a set of default colors for multiple lines.

Similarly, you can adjust the line style using the linestyle keyword (Figure 4-10):

In[7]: plt.plot(x, x + 0, linestyle='solid')
       plt.plot(x, x + 1, linestyle='dashed')
       plt.plot(x, x + 2, linestyle='dashdot')
       plt.plot(x, x + 3, linestyle='dotted');

       # For short, you can use the following codes:
       plt.plot(x, x + 4, linestyle='-')  # solid
       plt.plot(x, x + 5, linestyle='--') # dashed
       plt.plot(x, x + 6, linestyle='-.') # dashdot
       plt.plot(x, x + 7, linestyle=':');  # dotted
png
Figure 4-10. Example of various line styles

If you would like to be extremely terse, these linestyle and color codes can be combined into a single nonkeyword argument to the plt.plot() function (Figure 4-11):

In[8]: plt.plot(x, x + 0, '-g')  # solid green
       plt.plot(x, x + 1, '--c') # dashed cyan
       plt.plot(x, x + 2, '-.k') # dashdot black
       plt.plot(x, x + 3, ':r');  # dotted red
png
Figure 4-11. Controlling colors and styles with the shorthand syntax

These single-character color codes reflect the standard abbreviations in the RGB (Red/Green/Blue) and CMYK (Cyan/Magenta/Yellow/blacK) color systems, commonly used for digital color graphics.

There are many other keyword arguments that can be used to fine-tune the appearance of the plot; for more details, I’d suggest viewing the docstring of the plt.plot() function using IPython’s help tools (see “Help and Documentation in IPython”).

Adjusting the Plot: Axes Limits

Matplotlib does a decent job of choosing default axes limits for your plot, but sometimes it’s nice to have finer control. The most basic way to adjust axis limits is to use the plt.xlim() and plt.ylim() methods (Figure 4-12):

In[9]: plt.plot(x, np.sin(x))

       plt.xlim(-1, 11)
       plt.ylim(-1.5, 1.5);
png
Figure 4-12. Example of setting axis limits

If for some reason you’d like either axis to be displayed in reverse, you can simply reverse the order of the arguments (Figure 4-13):

In[10]: plt.plot(x, np.sin(x))

        plt.xlim(10, 0)
        plt.ylim(1.2, -1.2);
png
Figure 4-13. Example of reversing the y-axis

A useful related method is plt.axis() (note here the potential confusion between axes with an e, and axis with an i). The plt.axis() method allows you to set the x and y limits with a single call, by passing a list that specifies [xmin, xmax, ymin, ymax] (Figure 4-14):

In[11]: plt.plot(x, np.sin(x))
        plt.axis([-1, 11, -1.5, 1.5]);
png
Figure 4-14. Setting the axis limits with plt.axis

The plt.axis() method goes even beyond this, allowing you to do things like automatically tighten the bounds around the current plot (Figure 4-15):

In[12]: plt.plot(x, np.sin(x))
        plt.axis('tight');
png
Figure 4-15. Example of a “tight” layout

It allows even higher-level specifications, such as ensuring an equal aspect ratio so that on your screen, one unit in x is equal to one unit in y (Figure 4-16):

In[13]: plt.plot(x, np.sin(x))
        plt.axis('equal');
png
Figure 4-16. Example of an “equal” layout, with units matched to the output resolution

For more information on axis limits and the other capabilities of the plt.axis() method, refer to the plt.axis() docstring.

Labeling Plots

As the last piece of this section, we’ll briefly look at the labeling of plots: titles, axis labels, and simple legends.

Titles and axis labels are the simplest such labels—there are methods that can be used to quickly set them (Figure 4-17):

In[14]: plt.plot(x, np.sin(x))
        plt.title("A Sine Curve")
        plt.xlabel("x")
        plt.ylabel("sin(x)");
png
Figure 4-17. Examples of axis labels and title

You can adjust the position, size, and style of these labels using optional arguments to the function. For more information, see the Matplotlib documentation and the docstrings of each of these functions.

When multiple lines are being shown within a single axes, it can be useful to create a plot legend that labels each line type. Again, Matplotlib has a built-in way of quickly creating such a legend. It is done via the (you guessed it) plt.legend() method. Though there are several valid ways of using this, I find it easiest to specify the label of each line using the label keyword of the plot function (Figure 4-18):

In[15]: plt.plot(x, np.sin(x), '-g', label='sin(x)')
        plt.plot(x, np.cos(x), ':b', label='cos(x)')
        plt.axis('equal')

        plt.legend();
png
Figure 4-18. Plot legend example

As you can see, the plt.legend() function keeps track of the line style and color, and matches these with the correct label. More information on specifying and formatting plot legends can be found in the plt.legend() docstring; additionally, we will cover some more advanced legend options in “Customizing Plot Legends”.

Simple Scatter Plots

Another commonly used plot type is the simple scatter plot, a close cousin of the line plot. Instead of points being joined by line segments, here the points are represented individually with a dot, circle, or other shape. We’ll start by setting up the notebook for plotting and importing the functions we will use:

In[1]: %matplotlib inline
       import matplotlib.pyplot as plt
       plt.style.use('seaborn-whitegrid')
       import numpy as np

Scatter Plots with plt.plot

In the previous section, we looked at plt.plot/ax.plot to produce line plots. It turns out that this same function can produce scatter plots as well (Figure 4-20):

In[2]: x = np.linspace(0, 10, 30)
       y = np.sin(x)

       plt.plot(x, y, 'o', color='black');
png
Figure 4-20. Scatter plot example

The third argument in the function call is a character that represents the type of symbol used for the plotting. Just as you can specify options such as '-' and '--' to control the line style, the marker style has its own set of short string codes. The full list of available symbols can be seen in the documentation of plt.plot, or in Matplotlib’s online documentation. Most of the possibilities are fairly intuitive, and we’ll show a number of the more common ones here (Figure 4-21):

In[3]: rng = np.random.RandomState(0)
       for marker in ['o', '.', ',', 'x', '+', 'v', '^', '<', '>', 's', 'd']:
           plt.plot(rng.rand(5), rng.rand(5), marker,
                    label="marker='{0}'".format(marker))
       plt.legend(numpoints=1)
       plt.xlim(0, 1.8);
png
Figure 4-21. Demonstration of point numbers

For even more possibilities, these character codes can be used together with line and color codes to plot points along with a line connecting them (Figure 4-22):

In[4]: plt.plot(x, y, '-ok');   # line (-), circle marker (o), black (k)
png
Figure 4-22. Combining line and point markers

Additional keyword arguments to plt.plot specify a wide range of properties of the lines and markers (Figure 4-23):

In[5]: plt.plot(x, y, '-p', color='gray',
                markersize=15, linewidth=4,
                markerfacecolor='white',
                markeredgecolor='gray',
                markeredgewidth=2)
       plt.ylim(-1.2, 1.2);
png
Figure 4-23. Customizing line and point numbers

This type of flexibility in the plt.plot function allows for a wide variety of possible visualization options. For a full description of the options available, refer to the plt.plot documentation.

Scatter Plots with plt.scatter

A second, more powerful method of creating scatter plots is the plt.scatter function, which can be used very similarly to the plt.plot function (Figure 4-24):

In[6]: plt.scatter(x, y, marker='o');
png
Figure 4-24. A simple scatter plot

The primary difference of plt.scatter from plt.plot is that it can be used to create scatter plots where the properties of each individual point (size, face color, edge color, etc.) can be individually controlled or mapped to data.

Let’s show this by creating a random scatter plot with points of many colors and sizes. In order to better see the overlapping results, we’ll also use the alpha keyword to adjust the transparency level (Figure 4-25):

In[7]: rng = np.random.RandomState(0)
       x = rng.randn(100)
       y = rng.randn(100)
       colors = rng.rand(100)
       sizes = 1000 * rng.rand(100)

       plt.scatter(x, y, c=colors, s=sizes, alpha=0.3,
                   cmap='viridis')
       plt.colorbar();  # show color scale
png
Figure 4-25. Changing size, color, and transparency in scatter points

Notice that the color argument is automatically mapped to a color scale (shown here by the colorbar() command), and the size argument is given in pixels. In this way, the color and size of points can be used to convey information in the visualization, in order to illustrate multidimensional data.

For example, we might use the Iris data from Scikit-Learn, where each sample is one of three types of flowers that has had the size of its petals and sepals carefully measured (Figure 4-26):

In[8]: from sklearn.datasets import load_iris
       iris = load_iris()
       features = iris.data.T

       plt.scatter(features[0], features[1], alpha=0.2,
                   s=100*features[3], c=iris.target, cmap='viridis')
       plt.xlabel(iris.feature_names[0])
       plt.ylabel(iris.feature_names[1]);
png
Figure 4-26. Using point properties to encode features of the Iris data

We can see that this scatter plot has given us the ability to simultaneously explore four different dimensions of the data: the (x, y) location of each point corresponds to the sepal length and width, the size of the point is related to the petal width, and the color is related to the particular species of flower. Multicolor and multifeature scatter plots like this can be useful for both exploration and presentation of data.

plot Versus scatter: A Note on Efficiency

Aside from the different features available in plt.plot and plt.scatter, why might you choose to use one over the other? While it doesn’t matter as much for small amounts of data, as datasets get larger than a few thousand points, plt.plot can be noticeably more efficient than plt.scatter. The reason is that plt.scatter has the capability to render a different size and/or color for each point, so the renderer must do the extra work of constructing each point individually. In plt.plot, on the other hand, the points are always essentially clones of each other, so the work of determining the appearance of the points is done only once for the entire set of data. For large datasets, the difference between these two can lead to vastly different performance, and for this reason, plt.plot should be preferred over plt.scatter for large datasets.

Visualizing Errors

For any scientific measurement, accurate accounting for errors is nearly as important, if not more important, than accurate reporting of the number itself. For example, imagine that I am using some astrophysical observations to estimate the Hubble Constant, the local measurement of the expansion rate of the universe. I know that the current literature suggests a value of around 71 (km/s)/Mpc, and I measure a value of 74 (km/s)/Mpc with my method. Are the values consistent? The only correct answer, given this information, is this: there is no way to know.

Suppose I augment this information with reported uncertainties: the current literature suggests a value of around 71 ± 2.5 (km/s)/Mpc, and my method has measured a value of 74 ± 5 (km/s)/Mpc. Now are the values consistent? That is a question that can be quantitatively answered.

In visualization of data and results, showing these errors effectively can make a plot convey much more complete information.

Basic Errorbars

A basic errorbar can be created with a single Matplotlib function call (Figure 4-27):

In[1]: %matplotlib inline
       import matplotlib.pyplot as plt
       plt.style.use('seaborn-whitegrid')
       import numpy as np
In[2]: x = np.linspace(0, 10, 50)
       dy = 0.8
       y = np.sin(x) + dy * np.random.randn(50)

       plt.errorbar(x, y, yerr=dy, fmt='.k');
png
Figure 4-27. An errorbar example

Here the fmt is a format code controlling the appearance of lines and points, and has the same syntax as the shorthand used in plt.plot, outlined in “Simple Line Plots” and “Simple Scatter Plots”.

In addition to these basic options, the errorbar function has many options to fine-tune the outputs. Using these additional options you can easily customize the aesthetics of your errorbar plot. I often find it helpful, especially in crowded plots, to make the errorbars lighter than the points themselves (Figure 4-28):

In[3]: plt.errorbar(x, y, yerr=dy, fmt='o', color='black',
                    ecolor='lightgray', elinewidth=3, capsize=0);
png
Figure 4-28. Customizing errorbars

In addition to these options, you can also specify horizontal errorbars (xerr), one-sided errorbars, and many other variants. For more information on the options available, refer to the docstring of plt.errorbar.

Continuous Errors

In some situations it is desirable to show errorbars on continuous quantities. Though Matplotlib does not have a built-in convenience routine for this type of application, it’s relatively easy to combine primitives like plt.plot and plt.fill_between for a useful result.

Here we’ll perform a simple Gaussian process regression (GPR), using the Scikit-Learn API (see “Introducing Scikit-Learn” for details). This is a method of fitting a very flexible nonparametric function to data with a continuous measure of the uncertainty. We won’t delve into the details of Gaussian process regression at this point, but will focus instead on how you might visualize such a continuous error measurement:

In[4]: from sklearn.gaussian_process import GaussianProcess

       # define the model and draw some data
       model = lambda x: x * np.sin(x)
       xdata = np.array([1, 3, 5, 6, 8])
       ydata = model(xdata)

       # Compute the Gaussian process fit
       gp = GaussianProcess(corr='cubic', theta0=1e-2, thetaL=1e-4, thetaU=1E-1,
                            random_start=100)
       gp.fit(xdata[:, np.newaxis], ydata)

       xfit = np.linspace(0, 10, 1000)
       yfit, MSE = gp.predict(xfit[:, np.newaxis], eval_MSE=True)
       dyfit = 2 * np.sqrt(MSE)  # 2*sigma ~ 95% confidence region

We now have xfit, yfit, and dyfit, which sample the continuous fit to our data. We could pass these to the plt.errorbar function as above, but we don’t really want to plot 1,000 points with 1,000 errorbars. Instead, we can use the plt.fill_between function with a light color to visualize this continuous error (Figure 4-29):

In[5]: # Visualize the result
       plt.plot(xdata, ydata, 'or')
       plt.plot(xfit, yfit, '-', color='gray')

       plt.fill_between(xfit, yfit - dyfit, yfit + dyfit,
                        color='gray', alpha=0.2)
       plt.xlim(0, 10);
png
Figure 4-29. Representing continuous uncertainty with filled regions

Note what we’ve done here with the fill_between function: we pass an x value, then the lower y-bound, then the upper y-bound, and the result is that the area between these regions is filled.

The resulting figure gives a very intuitive view into what the Gaussian process regression algorithm is doing: in regions near a measured data point, the model is strongly constrained and this is reflected in the small model errors. In regions far from a measured data point, the model is not strongly constrained, and the model errors increase.

For more information on the options available in plt.fill_between() (and the closely related plt.fill() function), see the function docstring or the Matplotlib documentation.

Finally, if this seems a bit too low level for your taste, refer to “Visualization with Seaborn”, where we discuss the Seaborn package, which has a more streamlined API for visualizing this type of continuous errorbar.

Density and Contour Plots

Sometimes it is useful to display three-dimensional data in two dimensions using contours or color-coded regions. There are three Matplotlib functions that can be helpful for this task: plt.contour for contour plots, plt.contourf for filled contour plots, and plt.imshow for showing images. This section looks at several examples of using these. We’ll start by setting up the notebook for plotting and importing the functions we will use:

In[1]: %matplotlib inline
       import matplotlib.pyplot as plt
       plt.style.use('seaborn-white')
       import numpy as np

Visualizing a Three-Dimensional Function

We’ll start by demonstrating a contour plot using a function z = f ( x , y ) , using the following particular choice for f (we’ve seen this before in “Computation on Arrays: Broadcasting”, when we used it as a motivating example for array broadcasting):

In[2]: def f(x, y):
           return np.sin(x) ** 10 + np.cos(10 + y * x) * np.cos(x)

A contour plot can be created with the plt.contour function. It takes three arguments: a grid of x values, a grid of y values, and a grid of z values. The x and y values represent positions on the plot, and the z values will be represented by the contour levels. Perhaps the most straightforward way to prepare such data is to use the np.meshgrid function, which builds two-dimensional grids from one-dimensional arrays:

In[3]: x = np.linspace(0, 5, 50)
       y = np.linspace(0, 5, 40)

       X, Y = np.meshgrid(x, y)
       Z = f(X, Y)

Now let’s look at this with a standard line-only contour plot (Figure 4-30):

In[4]: plt.contour(X, Y, Z, colors='black');
png
Figure 4-30. Visualizing three-dimensional data with contours

Notice that by default when a single color is used, negative values are represented by dashed lines, and positive values by solid lines. Alternatively, you can color-code the lines by specifying a colormap with the cmap argument. Here, we’ll also specify that we want more lines to be drawn—20 equally spaced intervals within the data range (Figure 4-31):

In[5]: plt.contour(X, Y, Z, 20, cmap='RdGy');
png
Figure 4-31. Visualizing three-dimensional data with colored contours

Here we chose the RdGy (short for Red-Gray) colormap, which is a good choice for centered data. Matplotlib has a wide range of colormaps available, which you can easily browse in IPython by doing a tab completion on the plt.cm module:

plt.cm.<TAB>

Our plot is looking nicer, but the spaces between the lines may be a bit distracting. We can change this by switching to a filled contour plot using the plt.contourf() function (notice the f at the end), which uses largely the same syntax as plt.contour().

Additionally, we’ll add a plt.colorbar() command, which automatically creates an additional axis with labeled color information for the plot (Figure 4-32):

In[6]: plt.contourf(X, Y, Z, 20, cmap='RdGy')
       plt.colorbar();
png
Figure 4-32. Visualizing three-dimensional data with filled contours

The colorbar makes it clear that the black regions are “peaks,” while the red regions are “valleys.”

One potential issue with this plot is that it is a bit “splotchy.” That is, the color steps are discrete rather than continuous, which is not always what is desired. You could remedy this by setting the number of contours to a very high number, but this results in a rather inefficient plot: Matplotlib must render a new polygon for each step in the level. A better way to handle this is to use the plt.imshow() function, which interprets a two-dimensional grid of data as an image.

Figure 4-33 shows the result of the following code:

In[7]: plt.imshow(Z, extent=[0, 5, 0, 5], origin='lower',
                  cmap='RdGy')
       plt.colorbar()
       plt.axis(aspect='image');

There are a few potential gotchas with imshow(), however:

  • plt.imshow() doesn’t accept an x and y grid, so you must manually specify the extent [xmin, xmax, ymin, ymax] of the image on the plot.

  • plt.imshow() by default follows the standard image array definition where the origin is in the upper left, not in the lower left as in most contour plots. This must be changed when showing gridded data.

  • plt.imshow() will automatically adjust the axis aspect ratio to match the input data; you can change this by setting, for example, plt.axis(aspect='image') to make x and y units match.

png
Figure 4-33. Representing three-dimensional data as an image

Finally, it can sometimes be useful to combine contour plots and image plots. For example, to create the effect shown in Figure 4-34, we’ll use a partially transparent background image (with transparency set via the alpha parameter) and over-plot contours with labels on the contours themselves (using the plt.clabel() function):

In[8]: contours = plt.contour(X, Y, Z, 3, colors='black')
       plt.clabel(contours, inline=True, fontsize=8)

       plt.imshow(Z, extent=[0, 5, 0, 5], origin='lower',
                  cmap='RdGy', alpha=0.5)
       plt.colorbar();
png
Figure 4-34. Labeled contours on top of an image

The combination of these three functions—plt.contour, plt.contourf, and plt.imshow—gives nearly limitless possibilities for displaying this sort of three-dimensional data within a two-dimensional plot. For more information on the options available in these functions, refer to their docstrings. If you are interested in three-dimensional visualizations of this type of data, see “Three-Dimensional Plotting in Matplotlib”.

Histograms, Binnings, and Density

A simple histogram can be a great first step in understanding a dataset. Earlier, we saw a preview of Matplotlib’s histogram function (see “Comparisons, Masks, and Boolean Logic”), which creates a basic histogram in one line, once the normal boilerplate imports are done (Figure 4-35):

In[1]: %matplotlib inline
       import numpy as np
       import matplotlib.pyplot as plt
       plt.style.use('seaborn-white')

       data = np.random.randn(1000)
In[2]: plt.hist(data);
png
Figure 4-35. A simple histogram

The hist() function has many options to tune both the calculation and the display; here’s an example of a more customized histogram (Figure 4-36):

In[3]: plt.hist(data, bins=30, normed=True, alpha=0.5,
                histtype='stepfilled', color='steelblue',
                edgecolor='none');
png
Figure 4-36. A customized histogram

The plt.hist docstring has more information on other customization options available. I find this combination of histtype='stepfilled' along with some transparency alpha to be very useful when comparing histograms of several distributions (Figure 4-37):

In[4]: x1 = np.random.normal(0, 0.8, 1000)
       x2 = np.random.normal(-2, 1, 1000)
       x3 = np.random.normal(3, 2, 1000)

       kwargs = dict(histtype='stepfilled', alpha=0.3, normed=True, bins=40)

       plt.hist(x1, **kwargs)
       plt.hist(x2, **kwargs)
       plt.hist(x3, **kwargs);
png
Figure 4-37. Over-plotting multiple histograms

If you would like to simply compute the histogram (that is, count the number of points in a given bin) and not display it, the np.histogram() function is available:

In[5]: counts, bin_edges = np.histogram(data, bins=5)
       print(counts)
[ 12 190 468 301  29]

Two-Dimensional Histograms and Binnings

Just as we create histograms in one dimension by dividing the number line into bins, we can also create histograms in two dimensions by dividing points among two-dimensional bins. We’ll take a brief look at several ways to do this here. We’ll start by defining some data—an x and y array drawn from a multivariate Gaussian distribution:

In[6]: mean = [0, 0]
       cov = [[1, 1], [1, 2]]
       x, y = np.random.multivariate_normal(mean, cov, 10000).T

plt.hist2d: Two-dimensional histogram

One straightforward way to plot a two-dimensional histogram is to use Matplotlib’s plt.hist2d function (Figure 4-38):

In[12]: plt.hist2d(x, y, bins=30, cmap='Blues')
        cb = plt.colorbar()
        cb.set_label('counts in bin')
png
Figure 4-38. A two-dimensional histogram with plt.hist2d

Just as with plt.hist, plt.hist2d has a number of extra options to fine-tune the plot and the binning, which are nicely outlined in the function docstring. Further, just as plt.hist has a counterpart in np.histogram, plt.hist2d has a counterpart in np.histogram2d, which can be used as follows:

In[8]: counts, xedges, yedges = np.histogram2d(x, y, bins=30)

For the generalization of this histogram binning in dimensions higher than two, see the np.histogramdd function.

plt.hexbin: Hexagonal binnings

The two-dimensional histogram creates a tessellation of squares across the axes. Another natural shape for such a tessellation is the regular hexagon. For this purpose, Matplotlib provides the plt.hexbin routine, which represents a two-dimensional dataset binned within a grid of hexagons (Figure 4-39):

In[9]: plt.hexbin(x, y, gridsize=30, cmap='Blues')
       cb = plt.colorbar(label='count in bin')
png
Figure 4-39. A two-dimensional histogram with plt.hexbin

plt.hexbin has a number of interesting options, including the ability to specify weights for each point, and to change the output in each bin to any NumPy aggregate (mean of weights, standard deviation of weights, etc.).

Kernel density estimation

Another common method of evaluating densities in multiple dimensions is kernel density estimation (KDE). This will be discussed more fully in “In-Depth: Kernel Density Estimation”, but for now we’ll simply mention that KDE can be thought of as a way to “smear out” the points in space and add up the result to obtain a smooth function. One extremely quick and simple KDE implementation exists in the scipy.stats package. Here is a quick example of using the KDE on this data (Figure 4-40):

In[10]: from scipy.stats import gaussian_kde

        # fit an array of size [Ndim, Nsamples]
        data = np.vstack([x, y])
        kde = gaussian_kde(data)

        # evaluate on a regular grid
        xgrid = np.linspace(-3.5, 3.5, 40)
        ygrid = np.linspace(-6, 6, 40)
        Xgrid, Ygrid = np.meshgrid(xgrid, ygrid)
        Z = kde.evaluate(np.vstack([Xgrid.ravel(), Ygrid.ravel()]))

        # Plot the result as an image
        plt.imshow(Z.reshape(Xgrid.shape),
                   origin='lower', aspect='auto',
                   extent=[-3.5, 3.5, -6, 6],
                   cmap='Blues')
        cb = plt.colorbar()
        cb.set_label("density")
png
Figure 4-40. A kernel density representation of a distribution

KDE has a smoothing length that effectively slides the knob between detail and smoothness (one example of the ubiquitous bias–variance trade-off). The literature on choosing an appropriate smoothing length is vast: gaussian_kde uses a rule of thumb to attempt to find a nearly optimal smoothing length for the input data.

Other KDE implementations are available within the SciPy ecosystem, each with its own various strengths and weaknesses; see, for example, sklearn.neighbors.KernelDensity and statsmodels.nonparametric.kernel_density.KDEMultivariate. For visualizations based on KDE, using Matplotlib tends to be overly verbose. The Seaborn library, discussed in “Visualization with Seaborn”, provides a much more terse API for creating KDE-based visualizations.

Customizing Plot Legends

Plot legends give meaning to a visualization, assigning labels to the various plot elements. We previously saw how to create a simple legend; here we’ll take a look at customizing the placement and aesthetics of the legend in Matplotlib.

The simplest legend can be created with the plt.legend() command, which automatically creates a legend for any labeled plot elements (Figure 4-41):

In[1]: import matplotlib.pyplot as plt
       plt.style.use('classic')
In[2]: %matplotlib inline
       import numpy as np
In[3]: x = np.linspace(0, 10, 1000)
       fig, ax = plt.subplots()
       ax.plot(x, np.sin(x), '-b', label='Sine')
       ax.plot(x, np.cos(x), '--r', label='Cosine')
       ax.axis('equal')
       leg = ax.legend();
png
Figure 4-41. A default plot legend

But there are many ways we might want to customize such a legend. For example, we can specify the location and turn off the frame (Figure 4-42):

In[4]: ax.legend(loc='upper left', frameon=False)
       fig
png
Figure 4-42. A customized plot legend

We can use the ncol command to specify the number of columns in the legend (Figure 4-43):

In[5]: ax.legend(frameon=False, loc='lower center', ncol=2)
       fig
png
Figure 4-43. A two-column plot legend

We can use a rounded box (fancybox) or add a shadow, change the transparency (alpha value) of the frame, or change the padding around the text (Figure 4-44):

In[6]: ax.legend(fancybox=True, framealpha=1, shadow=True, borderpad=1)
       fig
png
Figure 4-44. A fancybox plot legend

For more information on available legend options, see the plt.legend docstring.

Choosing Elements for the Legend

As we’ve already seen, the legend includes all labeled elements by default. If this is not what is desired, we can fine-tune which elements and labels appear in the legend by using the objects returned by plot commands. The plt.plot() command is able to create multiple lines at once, and returns a list of created line instances. Passing any of these to plt.legend() will tell it which to identify, along with the labels we’d like to specify (Figure 4-45):

In[7]: y = np.sin(x[:, np.newaxis] + np.pi * np.arange(0, 2, 0.5))
       lines = plt.plot(x, y)

       # lines is a list of plt.Line2D instances
       plt.legend(lines[:2], ['first', 'second']);
png
Figure 4-45. Customization of legend elements

I generally find in practice that it is clearer to use the first method, applying labels to the plot elements you’d like to show on the legend (Figure 4-46):

In[8]: plt.plot(x, y[:, 0], label='first')
       plt.plot(x, y[:, 1], label='second')
       plt.plot(x, y[:, 2:])
       plt.legend(framealpha=1, frameon=True);
png
Figure 4-46. Alternative method of customizing legend elements

Notice that by default, the legend ignores all elements without a label attribute set.

Legend for Size of Points

Sometimes the legend defaults are not sufficient for the given visualization. For example, perhaps you’re using the size of points to mark certain features of the data, and want to create a legend reflecting this. Here is an example where we’ll use the size of points to indicate populations of California cities. We’d like a legend that specifies the scale of the sizes of the points, and we’ll accomplish this by plotting some labeled data with no entries (Figure 4-47):

In[9]: import pandas as pd
       cities = pd.read_csv('data/california_cities.csv')

       # Extract the data we're interested in
       lat, lon = cities['latd'], cities['longd']
       population, area = cities['population_total'], cities['area_total_km2']

       # Scatter the points, using size and color but no label
       plt.scatter(lon, lat, label=None,
                   c=np.log10(population), cmap='viridis',
                   s=area, linewidth=0, alpha=0.5)
       plt.axis(aspect='equal')
       plt.xlabel('longitude')
       plt.ylabel('latitude')
       plt.colorbar(label='log$_{10}$(population)')
       plt.clim(3, 7)

       # Here we create a legend:
       # we'll plot empty lists with the desired size and label
       for area in [100, 300, 500]:
           plt.scatter([], [], c='k', alpha=0.3, s=area,
                       label=str(area) + ' km$^2$')
       plt.legend(scatterpoints=1, frameon=False,
                  labelspacing=1, title='City Area')

       plt.title('California Cities: Area and Population');
png
Figure 4-47. Location, geographic size, and population of California cities

The legend will always reference some object that is on the plot, so if we’d like to display a particular shape we need to plot it. In this case, the objects we want (gray circles) are not on the plot, so we fake them by plotting empty lists. Notice too that the legend only lists plot elements that have a label specified.

By plotting empty lists, we create labeled plot objects that are picked up by the legend, and now our legend tells us some useful information. This strategy can be useful for creating more sophisticated visualizations.

Finally, note that for geographic data like this, it would be clearer if we could show state boundaries or other map-specific elements. For this, an excellent choice of tool is Matplotlib’s Basemap add-on toolkit, which we’ll explore in “Geographic Data with Basemap”.

Multiple Legends

Sometimes when designing a plot you’d like to add multiple legends to the same axes. Unfortunately, Matplotlib does not make this easy: via the standard legend interface, it is only possible to create a single legend for the entire plot. If you try to create a second legend using plt.legend() or ax.legend(), it will simply override the first one. We can work around this by creating a new legend artist from scratch, and then using the lower-level ax.add_artist() method to manually add the second artist to the plot (Figure 4-48):

In[10]: fig, ax = plt.subplots()

       lines = []
       styles = ['-', '--', '-.', ':']
       x = np.linspace(0, 10, 1000)

       for i in range(4):
           lines += ax.plot(x, np.sin(x - i * np.pi / 2),
                            styles[i], color='black')
       ax.axis('equal')

       # specify the lines and labels of the first legend
       ax.legend(lines[:2], ['line A', 'line B'],
                 loc='upper right', frameon=False)

       # Create the second legend and add the artist manually.
       from matplotlib.legend import Legend
       leg = Legend(ax, lines[2:], ['line C', 'line D'],
                    loc='lower right', frameon=False)
       ax.add_artist(leg);
png
Figure 4-48. A split plot legend

This is a peek into the low-level artist objects that compose any Matplotlib plot. If you examine the source code of ax.legend() (recall that you can do this within the IPython notebook using ax.legend??) you’ll see that the function simply consists of some logic to create a suitable Legend artist, which is then saved in the legend_ attribute and added to the figure when the plot is drawn.

Customizing Colorbars

Plot legends identify discrete labels of discrete points. For continuous labels based on the color of points, lines, or regions, a labeled colorbar can be a great tool. In Matplotlib, a colorbar is a separate axes that can provide a key for the meaning of colors in a plot. Because the book is printed in black and white, this section has an accompanying online appendix where you can view the figures in full color (https://github.com/jakevdp/PythonDataScienceHandbook). We’ll start by setting up the notebook for plotting and importing the functions we will use:

In[1]: import matplotlib.pyplot as plt
       plt.style.use('classic')
In[2]: %matplotlib inline
       import numpy as np

As we have seen several times throughout this section, the simplest colorbar can be created with the plt.colorbar function (Figure 4-49):

In[3]: x = np.linspace(0, 10, 1000)
       I = np.sin(x) * np.cos(x[:, np.newaxis])

       plt.imshow(I)
       plt.colorbar();
png
Figure 4-49. A simple colorbar legend

We’ll now discuss a few ideas for customizing these colorbars and using them effectively in various situations.

Customizing Colorbars

We can specify the colormap using the cmap argument to the plotting function that is creating the visualization (Figure 4-50):

In[4]: plt.imshow(I, cmap='gray');
png
Figure 4-50. A grayscale colormap

All the available colormaps are in the plt.cm namespace; using IPython’s tab-completion feature will give you a full list of built-in possibilities:

plt.cm.<TAB>

But being able to choose a colormap is just the first step: more important is how to decide among the possibilities! The choice turns out to be much more subtle than you might initially expect.

Choosing the colormap

A full treatment of color choice within visualization is beyond the scope of this book, but for entertaining reading on this subject and others, see the article “Ten Simple Rules for Better Figures”. Matplotlib’s online documentation also has an interesting discussion of colormap choice.

Broadly, you should be aware of three different categories of colormaps:

Sequential colormaps

These consist of one continuous sequence of colors (e.g., binary or viridis).

Divergent colormaps

These usually contain two distinct colors, which show positive and negative deviations from a mean (e.g., RdBu or PuOr).

Qualitative colormaps

These mix colors with no particular sequence (e.g., rainbow or jet).

The jet colormap, which was the default in Matplotlib prior to version 2.0, is an example of a qualitative colormap. Its status as the default was quite unfortunate, because qualitative maps are often a poor choice for representing quantitative data. Among the problems is the fact that qualitative maps usually do not display any uniform progression in brightness as the scale increases.

We can see this by converting the jet colorbar into black and white (Figure 4-51):

In[5]:
from matplotlib.colors import LinearSegmentedColormap

def grayscale_cmap(cmap):
    """Return a grayscale version of the given colormap"""
    cmap = plt.cm.get_cmap(cmap)
    colors = cmap(np.arange(cmap.N))

    # convert RGBA to perceived grayscale luminance
    # cf. http://alienryderflex.com/hsp.html
    RGB_weight = [0.299, 0.587, 0.114]
    luminance = np.sqrt(np.dot(colors[:, :3] ** 2, RGB_weight))
    colors[:, :3] = luminance[:, np.newaxis]

    return LinearSegmentedColormap.from_list(cmap.name + "_gray", colors, cmap.N)


def view_colormap(cmap):
    """Plot a colormap with its grayscale equivalent"""
    cmap = plt.cm.get_cmap(cmap)
    colors = cmap(np.arange(cmap.N))

    cmap = grayscale_cmap(cmap)
    grayscale = cmap(np.arange(cmap.N))

    fig, ax = plt.subplots(2, figsize=(6, 2),
                           subplot_kw=dict(xticks=[], yticks=[]))
    ax[0].imshow([colors], extent=[0, 10, 0, 1])
    ax[1].imshow([grayscale], extent=[0, 10, 0, 1])
In[6]: view_colormap('jet')
png
Figure 4-51. The jet colormap and its uneven luminance scale

Notice the bright stripes in the grayscale image. Even in full color, this uneven brightness means that the eye will be drawn to certain portions of the color range, which will potentially emphasize unimportant parts of the dataset. It’s better to use a colormap such as viridis (the default as of Matplotlib 2.0), which is specifically constructed to have an even brightness variation across the range. Thus, it not only plays well with our color perception, but also will translate well to grayscale printing (Figure 4-52):

In[7]: view_colormap('viridis')
png
Figure 4-52. The viridis colormap and its even luminance scale

If you favor rainbow schemes, another good option for continuous data is the cubehelix colormap (Figure 4-53):

In[8]: view_colormap('cubehelix')
png
Figure 4-53. The cubehelix colormap and its luminance

For other situations, such as showing positive and negative deviations from some mean, dual-color colorbars such as RdBu (short for Red-Blue) can be useful. However, as you can see in Figure 4-54, it’s important to note that the positive-negative information will be lost upon translation to grayscale!

In[9]: view_colormap('RdBu')
png
Figure 4-54. The RdBu (Red-Blue) colormap and its luminance

We’ll see examples of using some of these color maps as we continue.

There are a large number of colormaps available in Matplotlib; to see a list of them, you can use IPython to explore the plt.cm submodule. For a more principled approach to colors in Python, you can refer to the tools and documentation within the Seaborn library (see “Visualization with Seaborn”).

Color limits and extensions

Matplotlib allows for a large range of colorbar customization. The colorbar itself is simply an instance of plt.Axes, so all of the axes and tick formatting tricks we’ve learned are applicable. The colorbar has some interesting flexibility; for example, we can narrow the color limits and indicate the out-of-bounds values with a triangular arrow at the top and bottom by setting the extend property. This might come in handy, for example, if you’re displaying an image that is subject to noise (Figure 4-55):

In[10]: # make noise in 1% of the image pixels
        speckles = (np.random.random(I.shape) < 0.01)
        I[speckles] = np.random.normal(0, 3, np.count_nonzero(speckles))

        plt.figure(figsize=(10, 3.5))

        plt.subplot(1, 2, 1)
        plt.imshow(I, cmap='RdBu')
        plt.colorbar()

        plt.subplot(1, 2, 2)
        plt.imshow(I, cmap='RdBu')
        plt.colorbar(extend='both')
        plt.clim(-1, 1);
png
Figure 4-55. Specifying colormap extensions

Notice that in the left panel, the default color limits respond to the noisy pixels, and the range of the noise completely washes out the pattern we are interested in. In the right panel, we manually set the color limits, and add extensions to indicate values that are above or below those limits. The result is a much more useful visualization of our data.

Discrete colorbars

Colormaps are by default continuous, but sometimes you’d like to represent discrete values. The easiest way to do this is to use the plt.cm.get_cmap() function, and pass the name of a suitable colormap along with the number of desired bins (Figure 4-56):

In[11]: plt.imshow(I, cmap=plt.cm.get_cmap('Blues', 6))
        plt.colorbar()
        plt.clim(-1, 1);
png
Figure 4-56. A discretized colormap

The discrete version of a colormap can be used just like any other colormap.

Example: Handwritten Digits

For an example of where this might be useful, let’s look at an interesting visualization of some handwritten digits data. This data is included in Scikit-Learn, and consists of nearly 2,000 8×8 thumbnails showing various handwritten digits.

For now, let’s start by downloading the digits data and visualizing several of the example images with plt.imshow() (Figure 4-57):

In[12]: # load images of the digits 0 through 5 and visualize several of them
        from sklearn.datasets import load_digits
        digits = load_digits(n_class=6)

        fig, ax = plt.subplots(8, 8, figsize=(6, 6))
        for i, axi in enumerate(ax.flat):
            axi.imshow(digits.images[i], cmap='binary')
            axi.set(xticks=[], yticks=[])
png
Figure 4-57. Sample of handwritten digit data

Because each digit is defined by the hue of its 64 pixels, we can consider each digit to be a point lying in 64-dimensional space: each dimension represents the brightness of one pixel. But visualizing relationships in such high-dimensional spaces can be extremely difficult. One way to approach this is to use a dimensionality reduction technique such as manifold learning to reduce the dimensionality of the data while maintaining the relationships of interest. Dimensionality reduction is an example of unsupervised machine learning, and we will discuss it in more detail in “What Is Machine Learning?”.

Deferring the discussion of these details, let’s take a look at a two-dimensional manifold learning projection of this digits data (see “In-Depth: Manifold Learning” for details):

In[13]: # project the digits into 2 dimensions using IsoMap
        from sklearn.manifold import Isomap
        iso = Isomap(n_components=2)
        projection = iso.fit_transform(digits.data)

We’ll use our discrete colormap to view the results, setting the ticks and clim to improve the aesthetics of the resulting colorbar (Figure 4-58):

In[14]: # plot the results
        plt.scatter(projection[:, 0], projection[:, 1], lw=0.1,
                    c=digits.target, cmap=plt.cm.get_cmap('cubehelix', 6))
        plt.colorbar(ticks=range(6), label='digit value')
        plt.clim(-0.5, 5.5)
png
Figure 4-58. Manifold embedding of handwritten digit pixels

The projection also gives us some interesting insights on the relationships within the dataset: for example, the ranges of 5 and 3 nearly overlap in this projection, indicating that some handwritten fives and threes are difficult to distinguish, and therefore more likely to be confused by an automated classification algorithm. Other values, like 0 and 1, are more distantly separated, and therefore much less likely to be confused. This observation agrees with our intuition, because 5 and 3 look much more similar than do 0 and 1.

We’ll return to manifold learning and digit classification in Chapter 5.

Multiple Subplots

Sometimes it is helpful to compare different views of data side by side. To this end, Matplotlib has the concept of subplots: groups of smaller axes that can exist together within a single figure. These subplots might be insets, grids of plots, or other more complicated layouts. In this section, we’ll explore four routines for creating subplots in Matplotlib. We’ll start by setting up the notebook for plotting and importing the functions we will use:

In[1]: %matplotlib inline
       import matplotlib.pyplot as plt
       plt.style.use('seaborn-white')
       import numpy as np

plt.axes: Subplots by Hand

The most basic method of creating an axes is to use the plt.axes function. As we’ve seen previously, by default this creates a standard axes object that fills the entire figure. plt.axes also takes an optional argument that is a list of four numbers in the figure coordinate system. These numbers represent [bottom, left, width, height] in the figure coordinate system, which ranges from 0 at the bottom left of the figure to 1 at the top right of the figure.

For example, we might create an inset axes at the top-right corner of another axes by setting the x and y position to 0.65 (that is, starting at 65% of the width and 65% of the height of the figure) and the x and y extents to 0.2 (that is, the size of the axes is 20% of the width and 20% of the height of the figure). Figure 4-59 shows the result of this code:

In[2]: ax1 = plt.axes()  # standard axes
       ax2 = plt.axes([0.65, 0.65, 0.2, 0.2])
png
Figure 4-59. Example of an inset axes

The equivalent of this command within the object-oriented interface is fig.add_axes(). Let’s use this to create two vertically stacked axes (Figure 4-60):

In[3]: fig = plt.figure()
       ax1 = fig.add_axes([0.1, 0.5, 0.8, 0.4],
                          xticklabels=[], ylim=(-1.2, 1.2))
       ax2 = fig.add_axes([0.1, 0.1, 0.8, 0.4],
                          ylim=(-1.2, 1.2))

       x = np.linspace(0, 10)
       ax1.plot(np.sin(x))
       ax2.plot(np.cos(x));
png
Figure 4-60. Vertically stacked axes example

We now have two axes (the top with no tick labels) that are just touching: the bottom of the upper panel (at position 0.5) matches the top of the lower panel (at position 0.1 + 0.4).

plt.subplot: Simple Grids of Subplots

Aligned columns or rows of subplots are a common enough need that Matplotlib has several convenience routines that make them easy to create. The lowest level of these is plt.subplot(), which creates a single subplot within a grid. As you can see, this command takes three integer arguments—the number of rows, the number of columns, and the index of the plot to be created in this scheme, which runs from the upper left to the bottom right (Figure 4-61):

In[4]: for i in range(1, 7):
           plt.subplot(2, 3, i)
           plt.text(0.5, 0.5, str((2, 3, i)),
                    fontsize=18, ha='center')
png
Figure 4-61. A plt.subplot() example

The command plt.subplots_adjust can be used to adjust the spacing between these plots. The following code (the result of which is shown in Figure 4-62) uses the equivalent object-oriented command, fig.add_subplot():

In[5]: fig = plt.figure()
       fig.subplots_adjust(hspace=0.4, wspace=0.4)
       for i in range(1, 7):
           ax = fig.add_subplot(2, 3, i)
           ax.text(0.5, 0.5, str((2, 3, i)),
                  fontsize=18, ha='center')
png
Figure 4-62. plt.subplot() with adjusted margins

We’ve used the hspace and wspace arguments of plt.subplots_adjust, which specify the spacing along the height and width of the figure, in units of the subplot size (in this case, the space is 40% of the subplot width and height).

plt.subplots: The Whole Grid in One Go

The approach just described can become quite tedious when you’re creating a large grid of subplots, especially if you’d like to hide the x- and y-axis labels on the inner plots. For this purpose, plt.subplots() is the easier tool to use (note the s at the end of subplots). Rather than creating a single subplot, this function creates a full grid of subplots in a single line, returning them in a NumPy array. The arguments are the number of rows and number of columns, along with optional keywords sharex and sharey, which allow you to specify the relationships between different axes.

Here we’ll create a 2×3 grid of subplots, where all axes in the same row share their y-axis scale, and all axes in the same column share their x-axis scale (Figure 4-63):

In[6]: fig, ax = plt.subplots(2, 3, sharex='col', sharey='row')
png
Figure 4-63. Shared x and y axis in plt.subplots()

Note that by specifying sharex and sharey, we’ve automatically removed inner labels on the grid to make the plot cleaner. The resulting grid of axes instances is returned within a NumPy array, allowing for convenient specification of the desired axes using standard array indexing notation (Figure 4-64):

In[7]: # axes are in a two-dimensional array, indexed by [row, col]
       for i in range(2):
           for j in range(3):
               ax[i, j].text(0.5, 0.5, str((i, j)),
                             fontsize=18, ha='center')
       fig
png
Figure 4-64. Identifying plots in a subplot grid

In comparison to plt.subplot(), plt.subplots() is more consistent with Python’s conventional 0-based indexing.

plt.GridSpec: More Complicated Arrangements

To go beyond a regular grid to subplots that span multiple rows and columns, plt.GridSpec() is the best tool. The plt.GridSpec() object does not create a plot by itself; it is simply a convenient interface that is recognized by the plt.subplot() command. For example, a gridspec for a grid of two rows and three columns with some specified width and height space looks like this:

In[8]: grid = plt.GridSpec(2, 3, wspace=0.4, hspace=0.3)

From this we can specify subplot locations and extents using the familiar Python slicing syntax (Figure 4-65):

In[9]: plt.subplot(grid[0, 0])
       plt.subplot(grid[0, 1:])
       plt.subplot(grid[1, :2])
       plt.subplot(grid[1, 2]);
png
Figure 4-65. Irregular subplots with plt.GridSpec

This type of flexible grid alignment has a wide range of uses. I most often use it when creating multi-axes histogram plots like the one shown here (Figure 4-66):

In[10]: # Create some normally distributed data
        mean = [0, 0]
        cov = [[1, 1], [1, 2]]
        x, y = np.random.multivariate_normal(mean, cov, 3000).T

        # Set up the axes with gridspec
        fig = plt.figure(figsize=(6, 6))
        grid = plt.GridSpec(4, 4, hspace=0.2, wspace=0.2)
        main_ax = fig.add_subplot(grid[:-1, 1:])
        y_hist = fig.add_subplot(grid[:-1, 0], xticklabels=[], sharey=main_ax)
        x_hist = fig.add_subplot(grid[-1, 1:], yticklabels=[], sharex=main_ax)

        # scatter points on the main axes
        main_ax.plot(x, y, 'ok', markersize=3, alpha=0.2)

        # histogram on the attached axes
        x_hist.hist(x, 40, histtype='stepfilled',
                    orientation='vertical', color='gray')
        x_hist.invert_yaxis()

        y_hist.hist(y, 40, histtype='stepfilled',
                    orientation='horizontal', color='gray')
        y_hist.invert_xaxis()
png
Figure 4-66. Visualizing multidimensional distributions with plt.GridSpec

This type of distribution plotted alongside its margins is common enough that it has its own plotting API in the Seaborn package; see “Visualization with Seaborn” for more details.

Text and Annotation

Creating a good visualization involves guiding the reader so that the figure tells a story. In some cases, this story can be told in an entirely visual manner, without the need for added text, but in others, small textual cues and labels are necessary. Perhaps the most basic types of annotations you will use are axes labels and titles, but the options go beyond this. Let’s take a look at some data and how we might visualize and annotate it to help convey interesting information. We’ll start by setting up the notebook for plotting and importing the functions we will use:

In[1]: %matplotlib inline
       import matplotlib.pyplot as plt
       import matplotlib as mpl
       plt.style.use('seaborn-whitegrid')
       import numpy as np
       import pandas as pd

Example: Effect of Holidays on US Births

Let’s return to some data we worked with earlier in “Example: Birthrate Data”, where we generated a plot of average births over the course of the calendar year; as already mentioned, this data can be downloaded at https://raw.githubusercontent.com/jakevdp/data-CDCbirths/master/births.csv.

We’ll start with the same cleaning procedure we used there, and plot the results (Figure 4-67):

In[2]:
births = pd.read_csv('births.csv')

quartiles = np.percentile(births['births'], [25, 50, 75])
mu, sig = quartiles[1], 0.74 * (quartiles[2] - quartiles[0])
births = births.query('(births > @mu - 5 * @sig) & (births < @mu + 5 * @sig)')

births['day'] = births['day'].astype(int)

births.index = pd.to_datetime(10000 * births.year +
                              100 * births.month +
                              births.day, format='%Y%m%d')
births_by_date = births.pivot_table('births',
                                    [births.index.month, births.index.day])
births_by_date.index = [pd.datetime(2012, month, day)
                        for (month, day) in births_by_date.index]
In[3]: fig, ax = plt.subplots(figsize=(12, 4))
       births_by_date.plot(ax=ax);
png
Figure 4-67. Average daily births by date

When we’re communicating data like this, it is often useful to annotate certain features of the plot to draw the reader’s attention. This can be done manually with the plt.text/ax.text command, which will place text at a particular x/y value (Figure 4-68):

In[4]: fig, ax = plt.subplots(figsize=(12, 4))
       births_by_date.plot(ax=ax)

       # Add labels to the plot
       style = dict(size=10, color='gray')

       ax.text('2012-1-1', 3950, "New Year's Day", **style)
       ax.text('2012-7-4', 4250, "Independence Day", ha='center', **style)
       ax.text('2012-9-4', 4850, "Labor Day", ha='center', **style)
       ax.text('2012-10-31', 4600, "Halloween", ha='right', **style)
       ax.text('2012-11-25', 4450, "Thanksgiving", ha='center', **style)
       ax.text('2012-12-25', 3850, "Christmas ", ha='right', **style)

       # Label the axes
       ax.set(title='USA births by day of year (1969-1988)',
              ylabel='average daily births')

       # Format the x axis with centered month labels
       ax.xaxis.set_major_locator(mpl.dates.MonthLocator())
       ax.xaxis.set_minor_locator(mpl.dates.MonthLocator(bymonthday=15))
       ax.xaxis.set_major_formatter(plt.NullFormatter())
       ax.xaxis.set_minor_formatter(mpl.dates.DateFormatter('%h'));
png
Figure 4-68. Annotated average daily births by date

The ax.text method takes an x position, a y position, a string, and then optional keywords specifying the color, size, style, alignment, and other properties of the text. Here we used ha='right' and ha='center', where ha is short for horizonal alignment. See the docstring of plt.text() and of mpl.text.Text() for more information on available options.

Transforms and Text Position

In the previous example, we anchored our text annotations to data locations. Sometimes it’s preferable to anchor the text to a position on the axes or figure, independent of the data. In Matplotlib, we do this by modifying the transform.

Any graphics display framework needs some scheme for translating between coordinate systems. For example, a data point at ( x , y ) = ( 1 , 1 ) needs to somehow be represented at a certain location on the figure, which in turn needs to be represented in pixels on the screen. Mathematically, such coordinate transformations are relatively straightforward, and Matplotlib has a well-developed set of tools that it uses internally to perform them (the tools can be explored in the matplotlib.transforms submodule).

The average user rarely needs to worry about the details of these transforms, but it is helpful knowledge to have when considering the placement of text on a figure. There are three predefined transforms that can be useful in this situation:

ax.transData

Transform associated with data coordinates

ax.transAxes

Transform associated with the axes (in units of axes dimensions)

fig.transFigure

Transform associated with the figure (in units of figure dimensions)

Here let’s look at an example of drawing text at various locations using these transforms (Figure 4-69):

In[5]: fig, ax = plt.subplots(facecolor='lightgray')
       ax.axis([0, 10, 0, 10])

       # transform=ax.transData is the default, but we'll specify it anyway
       ax.text(1, 5, ". Data: (1, 5)", transform=ax.transData)
       ax.text(0.5, 0.1, ". Axes: (0.5, 0.1)", transform=ax.transAxes)
       ax.text(0.2, 0.2, ". Figure: (0.2, 0.2)", transform=fig.transFigure);
png
Figure 4-69. Comparing Matplotlib’s coordinate systems

Note that by default, the text is aligned above and to the left of the specified coordinates; here the “.” at the beginning of each string will approximately mark the given coordinate location.

The transData coordinates give the usual data coordinates associated with the x- and y-axis labels. The transAxes coordinates give the location from the bottom-left corner of the axes (here the white box) as a fraction of the axes size. The transFigure coordinates are similar, but specify the position from the bottom left of the figure (here the gray box) as a fraction of the figure size.

Notice now that if we change the axes limits, it is only the transData coordinates that will be affected, while the others remain stationary (Figure 4-70):

In[6]: ax.set_xlim(0, 2)
       ax.set_ylim(-6, 6)
       fig
png
Figure 4-70. Comparing Matplotlib’s coordinate systems

You can see this behavior more clearly by changing the axes limits interactively; if you are executing this code in a notebook, you can make that happen by changing %matplotlib inline to %matplotlib notebook and using each plot’s menu to interact with the plot.

Arrows and Annotation

Along with tick marks and text, another useful annotation mark is the simple arrow.

Drawing arrows in Matplotlib is often much harder than you might hope. While there is a plt.arrow() function available, I wouldn’t suggest using it; the arrows it creates are SVG objects that will be subject to the varying aspect ratio of your plots, and the result is rarely what the user intended. Instead, I’d suggest using the plt.annotate() function. This function creates some text and an arrow, and the arrows can be very flexibly specified.

Here we’ll use annotate with several of its options (Figure 4-71):

In[7]: %matplotlib inline

       fig, ax = plt.subplots()

       x = np.linspace(0, 20, 1000)
       ax.plot(x, np.cos(x))
       ax.axis('equal')

       ax.annotate('local maximum', xy=(6.28, 1), xytext=(10, 4),
                   arrowprops=dict(facecolor='black', shrink=0.05))

       ax.annotate('local minimum', xy=(5 * np.pi, -1), xytext=(2, -6),
                   arrowprops=dict(arrowstyle="->",
                                   connectionstyle="angle3,angleA=0,angleB=-90"));
png
Figure 4-71. Annotation examples

The arrow style is controlled through the arrowprops dictionary, which has numerous options available. These options are fairly well documented in Matplotlib’s online documentation, so rather than repeating them here I’ll quickly show some of the possibilities. Let’s demonstrate several of the possible options using the birthrate plot from before (Figure 4-72):

In[8]:
fig, ax = plt.subplots(figsize=(12, 4))
births_by_date.plot(ax=ax)

# Add labels to the plot
ax.annotate("New Year's Day", xy=('2012-1-1', 4100),  xycoords='data',
            xytext=(50, -30), textcoords='offset points',
            arrowprops=dict(arrowstyle="->",
                            connectionstyle="arc3,rad=-0.2"))


ax.annotate("Independence Day", xy=('2012-7-4', 4250),  xycoords='data',
            bbox=dict(boxstyle="round", fc="none", ec="gray"),
            xytext=(10, -40), textcoords='offset points', ha='center',
            arrowprops=dict(arrowstyle="->"))

ax.annotate('Labor Day', xy=('2012-9-4', 4850), xycoords='data', ha='center',
            xytext=(0, -20), textcoords='offset points')
ax.annotate('', xy=('2012-9-1', 4850), xytext=('2012-9-7', 4850),
            xycoords='data', textcoords='data',
            arrowprops={'arrowstyle': '|-|,widthA=0.2,widthB=0.2', })

ax.annotate('Halloween', xy=('2012-10-31', 4600),  xycoords='data',
            xytext=(-80, -40), textcoords='offset points',
            arrowprops=dict(arrowstyle="fancy",
                            fc="0.6", ec="none",
                            connectionstyle="angle3,angleA=0,angleB=-90"))

ax.annotate('Thanksgiving', xy=('2012-11-25', 4500),  xycoords='data',
            xytext=(-120, -60), textcoords='offset points',
            bbox=dict(boxstyle="round4,pad=.5", fc="0.9"),
            arrowprops=dict(arrowstyle="->",
                            connectionstyle="angle,angleA=0,angleB=80,rad=20"))


ax.annotate('Christmas', xy=('2012-12-25', 3850),  xycoords='data',
             xytext=(-30, 0), textcoords='offset points',
             size=13, ha='right', va="center",
             bbox=dict(boxstyle="round", alpha=0.1),
             arrowprops=dict(arrowstyle="wedge,tail_width=0.5", alpha=0.1));

# Label the axes
ax.set(title='USA births by day of year (1969-1988)',
       ylabel='average daily births')

# Format the x axis with centered month labels
ax.xaxis.set_major_locator(mpl.dates.MonthLocator())
ax.xaxis.set_minor_locator(mpl.dates.MonthLocator(bymonthday=15))
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.xaxis.set_minor_formatter(mpl.dates.DateFormatter('%h'));

ax.set_ylim(3600, 5400);
png
Figure 4-72. Annotated average birth rates by day

You’ll notice that the specifications of the arrows and text boxes are very detailed: this gives you the power to create nearly any arrow style you wish. Unfortunately, it also means that these sorts of features often must be manually tweaked, a process that can be very time-consuming when one is producing publication-quality graphics! Finally, I’ll note that the preceding mix of styles is by no means best practice for presenting data, but rather included as a demonstration of some of the available options.

More discussion and examples of available arrow and annotation styles can be found in the Matplotlib gallery, in particular http://matplotlib.org/examples/pylab_examples/annotation_demo2.html.

Customizing Ticks

Matplotlib’s default tick locators and formatters are designed to be generally sufficient in many common situations, but are in no way optimal for every plot. This section will give several examples of adjusting the tick locations and formatting for the particular plot type you’re interested in.

Before we go into examples, it will be best for us to understand further the object hierarchy of Matplotlib plots. Matplotlib aims to have a Python object representing everything that appears on the plot: for example, recall that the figure is the bounding box within which plot elements appear. Each Matplotlib object can also act as a container of sub-objects; for example, each figure can contain one or more axes objects, each of which in turn contain other objects representing plot contents.

The tick marks are no exception. Each axes has attributes xaxis and yaxis, which in turn have attributes that contain all the properties of the lines, ticks, and labels that make up the axes.

Major and Minor Ticks

Within each axis, there is the concept of a major tick mark and a minor tick mark. As the names would imply, major ticks are usually bigger or more pronounced, while minor ticks are usually smaller. By default, Matplotlib rarely makes use of minor ticks, but one place you can see them is within logarithmic plots (Figure 4-73):

In[1]: %matplotlib inline
       import matplotlib.pyplot as plt
       plt.style.use('seaborn-whitegrid')
       import numpy as np
In[2]: ax = plt.axes(xscale='log', yscale='log')
png
Figure 4-73. Example of logarithmic scales and labels

We see here that each major tick shows a large tick mark and a label, while each minor tick shows a smaller tick mark with no label.

We can customize these tick properties—that is, locations and labels—by setting the formatter and locator objects of each axis. Let’s examine these for the x axis of the plot just shown:

In[3]: print(ax.xaxis.get_major_locator())
       print(ax.xaxis.get_minor_locator())
<matplotlib.ticker.LogLocator object at 0x107530cc0>
<matplotlib.ticker.LogLocator object at 0x107530198>
In[4]: print(ax.xaxis.get_major_formatter())
       print(ax.xaxis.get_minor_formatter())
<matplotlib.ticker.LogFormatterMathtext object at 0x107512780>
<matplotlib.ticker.NullFormatter object at 0x10752dc18>

We see that both major and minor tick labels have their locations specified by a LogLocator (which makes sense for a logarithmic plot). Minor ticks, though, have their labels formatted by a NullFormatter; this says that no labels will be shown.

We’ll now show a few examples of setting these locators and formatters for various plots.

Hiding Ticks or Labels

Perhaps the most common tick/label formatting operation is the act of hiding ticks or labels. We can do this using plt.NullLocator() and plt.NullFormatter(), as shown here (Figure 4-74):

In[5]: ax = plt.axes()
       ax.plot(np.random.rand(50))

       ax.yaxis.set_major_locator(plt.NullLocator())
       ax.xaxis.set_major_formatter(plt.NullFormatter())
png
Figure 4-74. Plot with hidden tick labels (x-axis) and hidden ticks (y-axis)

Notice that we’ve removed the labels (but kept the ticks/gridlines) from the x axis, and removed the ticks (and thus the labels as well) from the y axis. Having no ticks at all can be useful in many situations—for example, when you want to show a grid of images. For instance, consider Figure 4-75, which includes images of different faces, an example often used in supervised machine learning problems (for more information, see “In-Depth: Support Vector Machines”):

In[6]: fig, ax = plt.subplots(5, 5, figsize=(5, 5))
       fig.subplots_adjust(hspace=0, wspace=0)

       # Get some face data from scikit-learn
       from sklearn.datasets import fetch_olivetti_faces
       faces = fetch_olivetti_faces().images

       for i in range(5):
           for j in range(5):
               ax[i, j].xaxis.set_major_locator(plt.NullLocator())
               ax[i, j].yaxis.set_major_locator(plt.NullLocator())
               ax[i, j].imshow(faces[10 * i + j], cmap="bone")
png
Figure 4-75. Hiding ticks within image plots

Notice that each image has its own axes, and we’ve set the locators to null because the tick values (pixel number in this case) do not convey relevant information for this particular visualization.

Reducing or Increasing the Number of Ticks

One common problem with the default settings is that smaller subplots can end up with crowded labels. We can see this in the plot grid shown in Figure 4-76:

In[7]: fig, ax = plt.subplots(4, 4, sharex=True, sharey=True)
png
Figure 4-76. A default plot with crowded ticks

Particularly for the x ticks, the numbers nearly overlap, making them quite difficult to decipher. We can fix this with the plt.MaxNLocator(), which allows us to specify the maximum number of ticks that will be displayed. Given this maximum number, Matplotlib will use internal logic to choose the particular tick locations (Figure 4-77):

In[8]: # For every axis, set the x and y major locator
       for axi in ax.flat:
           axi.xaxis.set_major_locator(plt.MaxNLocator(3))
           axi.yaxis.set_major_locator(plt.MaxNLocator(3))
       fig
png
Figure 4-77. Customizing the number of ticks

This makes things much cleaner. If you want even more control over the locations of regularly spaced ticks, you might also use plt.MultipleLocator, which we’ll discuss in the following section.

Fancy Tick Formats

Matplotlib’s default tick formatting can leave a lot to be desired; it works well as a broad default, but sometimes you’d like to do something more. Consider the plot shown in Figure 4-78, a sine and a cosine:

In[9]: # Plot a sine and cosine curve
       fig, ax = plt.subplots()
       x = np.linspace(0, 3 * np.pi, 1000)
       ax.plot(x, np.sin(x), lw=3, label='Sine')
       ax.plot(x, np.cos(x), lw=3, label='Cosine')

       # Set up grid, legend, and limits
       ax.grid(True)
       ax.legend(frameon=False)
       ax.axis('equal')
       ax.set_xlim(0, 3 * np.pi);
png
Figure 4-78. A default plot with integer ticks

There are a couple changes we might like to make. First, it’s more natural for this data to space the ticks and grid lines in multiples of π . We can do this by setting a MultipleLocator, which locates ticks at a multiple of the number you provide. For good measure, we’ll add both major and minor ticks in multiples of π / 4 (Figure 4-79):

In[10]: ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))
        ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 4))
        fig
png
Figure 4-79. Ticks at multiples of pi/2

But now these tick labels look a little bit silly: we can see that they are multiples of π , but the decimal representation does not immediately convey this. To fix this, we can change the tick formatter. There’s no built-in formatter for what we want to do, so we’ll instead use plt.FuncFormatter, which accepts a user-defined function giving fine-grained control over the tick outputs (Figure 4-80):

In[11]: def format_func(value, tick_number):
            # find number of multiples of pi/2
            N = int(np.round(2 * value / np.pi))
            if N == 0:
                return "0"
            elif N == 1:
                return r"$\pi/2$"
            elif N == 2:
                return r"$\pi$"
            elif N % 2 > 0:
                return r"${0}\pi/2$".format(N)
            else:
                return r"${0}\pi$".format(N // 2)

        ax.xaxis.set_major_formatter(plt.FuncFormatter(format_func))
        fig
png
Figure 4-80. Ticks with custom labels

This is much better! Notice that we’ve made use of Matplotlib’s LaTeX support, specified by enclosing the string within dollar signs. This is very convenient for display of mathematical symbols and formulae; in this case, "$\pi$" is rendered as the Greek character π .

The plt.FuncFormatter() offers extremely fine-grained control over the appearance of your plot ticks, and comes in very handy when you’re preparing plots for presentation or publication.

Summary of Formatters and Locators

We’ve mentioned a couple of the available formatters and locators. We’ll conclude this section by briefly listing all the built-in locator and formatter options. For more information on any of these, refer to the docstrings or to the Matplotlib online documentation. Each of the following is available in the plt namespace:

Locator class Description

NullLocator

No ticks

FixedLocator

Tick locations are fixed

IndexLocator

Locator for index plots (e.g., where x = range(len(y)))

LinearLocator

Evenly spaced ticks from min to max

LogLocator

Logarithmically ticks from min to max

MultipleLocator

Ticks and range are a multiple of base

MaxNLocator

Finds up to a max number of ticks at nice locations

AutoLocator

(Default) MaxNLocator with simple defaults

AutoMinorLocator

Locator for minor ticks

Formatter class Description

NullFormatter

No labels on the ticks

IndexFormatter

Set the strings from a list of labels

FixedFormatter

Set the strings manually for the labels

FuncFormatter

User-defined function sets the labels

FormatStrFormatter

Use a format string for each value

ScalarFormatter

(Default) Formatter for scalar values

LogFormatter

Default formatter for log axes

We’ll see additional examples of these throughout the remainder of the book.

Customizing Matplotlib: Configurations and Stylesheets

Matplotlib’s default plot settings are often the subject of complaint among its users. While much is slated to change in the 2.0 Matplotlib release, the ability to customize default settings helps bring the package in line with your own aesthetic preferences.

Here we’ll walk through some of Matplotlib’s runtime configuration (rc) options, and take a look at the newer stylesheets feature, which contains some nice sets of default configurations.

Plot Customization by Hand

Throughout this chapter, we’ve seen how it is possible to tweak individual plot settings to end up with something that looks a little bit nicer than the default. It’s possible to do these customizations for each individual plot. For example, here is a fairly drab default histogram (Figure 4-81):

In[1]: import matplotlib.pyplot as plt
       plt.style.use('classic')
       import numpy as np

       %matplotlib inline
In[2]: x = np.random.randn(1000)
       plt.hist(x);
png
Figure 4-81. A histogram in Matplotlib’s default style

We can adjust this by hand to make it a much more visually pleasing plot, shown in Figure 4-82:

In[3]: # use a gray background
       ax = plt.axes(axisbg='#E6E6E6')
       ax.set_axisbelow(True)

       # draw solid white grid lines
       plt.grid(color='w', linestyle='solid')

       # hide axis spines
       for spine in ax.spines.values():
           spine.set_visible(False)

       # hide top and right ticks
       ax.xaxis.tick_bottom()
       ax.yaxis.tick_left()

       # lighten ticks and labels
       ax.tick_params(colors='gray', direction='out')
       for tick in ax.get_xticklabels():
           tick.set_color('gray')
       for tick in ax.get_yticklabels():
           tick.set_color('gray')

       # control face and edge color of histogram
       ax.hist(x, edgecolor='#E6E6E6', color='#EE6666');
png
Figure 4-82. A histogram with manual customizations

This looks better, and you may recognize the look as inspired by the look of the R language’s ggplot visualization package. But this took a whole lot of effort! We definitely do not want to have to do all that tweaking each time we create a plot. Fortunately, there is a way to adjust these defaults once in a way that will work for all plots.

Changing the Defaults: rcParams

Each time Matplotlib loads, it defines a runtime configuration (rc) containing the default styles for every plot element you create. You can adjust this configuration at any time using the plt.rc convenience routine. Let’s see what it looks like to modify the rc parameters so that our default plot will look similar to what we did before.

We’ll start by saving a copy of the current rcParams dictionary, so we can easily reset these changes in the current session:

In[4]: IPython_default = plt.rcParams.copy()

Now we can use the plt.rc function to change some of these settings:

In[5]: from matplotlib import cycler
       colors = cycler('color',
                       ['#EE6666', '#3388BB', '#9988DD',
                        '#EECC55', '#88BB44', '#FFBBBB'])
       plt.rc('axes', facecolor='#E6E6E6', edgecolor='none',
              axisbelow=True, grid=True, prop_cycle=colors)
       plt.rc('grid', color='w', linestyle='solid')
       plt.rc('xtick', direction='out', color='gray')
       plt.rc('ytick', direction='out', color='gray')
       plt.rc('patch', edgecolor='#E6E6E6')
       plt.rc('lines', linewidth=2)

With these settings defined, we can now create a plot and see our settings in action (Figure 4-83):

In[6]: plt.hist(x);
png
Figure 4-83. A customized histogram using rc settings

Let’s see what simple line plots look like with these rc parameters (Figure 4-84):

In[7]: for i in range(4):
           plt.plot(np.random.rand(10))
png
Figure 4-84. A line plot with customized styles

I find this much more aesthetically pleasing than the default styling. If you disagree with my aesthetic sense, the good news is that you can adjust the rc parameters to suit your own tastes! These settings can be saved in a .matplotlibrc file, which you can read about in the Matplotlib documentation. That said, I prefer to customize Matplotlib using its stylesheets instead.

Stylesheets

The version 1.4 release of Matplotlib in August 2014 added a very convenient style module, which includes a number of new default stylesheets, as well as the ability to create and package your own styles. These stylesheets are formatted similarly to the .matplotlibrc files mentioned earlier, but must be named with a .mplstyle extension.

Even if you don’t create your own style, the stylesheets included by default are extremely useful. The available styles are listed in plt.style.available—here I’ll list only the first five for brevity:

In[8]: plt.style.available[:5]
Out[8]: ['fivethirtyeight',
         'seaborn-pastel',
         'seaborn-whitegrid',
         'ggplot',
         'grayscale']

The basic way to switch to a stylesheet is to call:

plt.style.use('stylename')

But keep in mind that this will change the style for the rest of the session! Alternatively, you can use the style context manager, which sets a style temporarily:

with plt.style.context('stylename'):
    make_a_plot()

Let’s create a function that will make two basic types of plot:

In[9]: def hist_and_lines():
           np.random.seed(0)
           fig, ax = plt.subplots(1, 2, figsize=(11, 4))
           ax[0].hist(np.random.randn(1000))
           for i in range(3):
               ax[1].plot(np.random.rand(10))
           ax[1].legend(['a', 'b', 'c'], loc='lower left')

We’ll use this to explore how these plots look using the various built-in styles.

Default style

The default style is what we’ve been seeing so far throughout the book; we’ll start with that. First, let’s reset our runtime configuration to the notebook default:

In[10]: # reset rcParams
        plt.rcParams.update(IPython_default);

Now let’s see how it looks (Figure 4-85):

In[11]: hist_and_lines()
png
Figure 4-85. Matplotlib’s default style

FiveThirtyEight style

The FiveThirtyEight style mimics the graphics found on the popular FiveThirtyEight website. As you can see in Figure 4-86, it is typified by bold colors, thick lines, and transparent axes.

In[12]: with plt.style.context('fivethirtyeight'):
            hist_and_lines()
png
Figure 4-86. The FiveThirtyEight style

ggplot

The ggplot package in the R language is a very popular visualization tool. Matplotlib’s ggplot style mimics the default styles from that package (Figure 4-87):

In[13]: with plt.style.context('ggplot'):
            hist_and_lines()
png
Figure 4-87. The ggplot style

Bayesian Methods for Hackers style

There is a very nice short online book called Probabilistic Programming and Bayesian Methods for Hackers; it features figures created with Matplotlib, and uses a nice set of rc parameters to create a consistent and visually appealing style throughout the book. This style is reproduced in the bmh stylesheet (Figure 4-88):

In[14]: with plt.style.context('bmh'):
            hist_and_lines()
png
Figure 4-88. The bmh style

Dark background

For figures used within presentations, it is often useful to have a dark rather than light background. The dark_background style provides this (Figure 4-89):

In[15]: with plt.style.context('dark_background'):
            hist_and_lines()
png
Figure 4-89. The dark_background style

Grayscale

Sometimes you might find yourself preparing figures for a print publication that does not accept color figures. For this, the grayscale style, shown in Figure 4-90, can be very useful:

In[16]: with plt.style.context('grayscale'):
            hist_and_lines()
png
Figure 4-90. The grayscale style

Seaborn style

Matplotlib also has stylesheets inspired by the Seaborn library (discussed more fully in “Visualization with Seaborn”). As we will see, these styles are loaded automatically when Seaborn is imported into a notebook. I’ve found these settings to be very nice, and tend to use them as defaults in my own data exploration (see Figure 4-91):

In[17]: import seaborn
        hist_and_lines()
png
Figure 4-91. Seaborn’s plotting style

With all of these built-in options for various plot styles, Matplotlib becomes much more useful for both interactive visualization and creation of figures for publication. Throughout this book, I will generally use one or more of these style conventions when creating plots.

Three-Dimensional Plotting in Matplotlib

Matplotlib was initially designed with only two-dimensional plotting in mind. Around the time of the 1.0 release, some three-dimensional plotting utilities were built on top of Matplotlib’s two-dimensional display, and the result is a convenient (if somewhat limited) set of tools for three-dimensional data visualization. We enable three-dimensional plots by importing the mplot3d toolkit, included with the main Matplotlib installation (Figure 4-92):

In[1]: from mpl_toolkits import mplot3d

Once this submodule is imported, we can create a three-dimensional axes by passing the keyword projection='3d' to any of the normal axes creation routines:

In[2]: %matplotlib inline
       import numpy as np
       import matplotlib.pyplot as plt
In[3]: fig = plt.figure()
       ax = plt.axes(projection='3d')
png
Figure 4-92. An empty three-dimensional axes

With this 3D axes enabled, we can now plot a variety of three-dimensional plot types. Three-dimensional plotting is one of the functionalities that benefits immensely from viewing figures interactively rather than statically in the notebook; recall that to use interactive figures, you can use %matplotlib notebook rather than %matplotlib inline when running this code.

Three-Dimensional Points and Lines

The most basic three-dimensional plot is a line or scatter plot created from sets of (x, y, z) triples. In analogy with the more common two-dimensional plots discussed earlier, we can create these using the ax.plot3D and ax.scatter3D functions. The call signature for these is nearly identical to that of their two-dimensional counterparts, so you can refer to “Simple Line Plots” and “Simple Scatter Plots” for more information on controlling the output. Here we’ll plot a trigonometric spiral, along with some points drawn randomly near the line (Figure 4-93):

In[4]: ax = plt.axes(projection='3d')

       # Data for a three-dimensional line
       zline = np.linspace(0, 15, 1000)
       xline = np.sin(zline)
       yline = np.cos(zline)
       ax.plot3D(xline, yline, zline, 'gray')

       # Data for three-dimensional scattered points
       zdata = 15 * np.random.random(100)
       xdata = np.sin(zdata) + 0.1 * np.random.randn(100)
       ydata = np.cos(zdata) + 0.1 * np.random.randn(100)
       ax.scatter3D(xdata, ydata, zdata, c=zdata, cmap='Greens');
png
Figure 4-93. Points and lines in three dimensions

Notice that by default, the scatter points have their transparency adjusted to give a sense of depth on the page. While the three-dimensional effect is sometimes difficult to see within a static image, an interactive view can lead to some nice intuition about the layout of the points.

Three-Dimensional Contour Plots

Analogous to the contour plots we explored in “Density and Contour Plots”, mplot3d contains tools to create three-dimensional relief plots using the same inputs. Like two-dimensional ax.contour plots, ax.contour3D requires all the input data to be in the form of two-dimensional regular grids, with the Z data evaluated at each point. Here we’ll show a three-dimensional contour diagram of a three-dimensional sinusoidal function (Figure 4-94):

In[5]: def f(x, y):
           return np.sin(np.sqrt(x ** 2 + y ** 2))

       x = np.linspace(-6, 6, 30)
       y = np.linspace(-6, 6, 30)

       X, Y = np.meshgrid(x, y)
       Z = f(X, Y)
In[6]: fig = plt.figure()
       ax = plt.axes(projection='3d')
       ax.contour3D(X, Y, Z, 50, cmap='binary')
       ax.set_xlabel('x')
       ax.set_ylabel('y')
       ax.set_zlabel('z');
png
Figure 4-94. A three-dimensional contour plot

Sometimes the default viewing angle is not optimal, in which case we can use the view_init method to set the elevation and azimuthal angles. In this example (the result of which is shown in Figure 4-95), we’ll use an elevation of 60 degrees (that is, 60 degrees above the x-y plane) and an azimuth of 35 degrees (that is, rotated 35 degrees counter-clockwise about the z-axis):

In[7]: ax.view_init(60, 35)
       fig
png
Figure 4-95. Adjusting the view angle for a three-dimensional plot

Again, note that we can accomplish this type of rotation interactively by clicking and dragging when using one of Matplotlib’s interactive backends.

Wireframes and Surface Plots

Two other types of three-dimensional plots that work on gridded data are wireframes and surface plots. These take a grid of values and project it onto the specified three-dimensional surface, and can make the resulting three-dimensional forms quite easy to visualize. Here’s an example using a wireframe (Figure 4-96):

In[8]: fig = plt.figure()
       ax = plt.axes(projection='3d')
       ax.plot_wireframe(X, Y, Z, color='black')
       ax.set_title('wireframe');
png
Figure 4-96. A wireframe plot

A surface plot is like a wireframe plot, but each face of the wireframe is a filled polygon. Adding a colormap to the filled polygons can aid perception of the topology of the surface being visualized (Figure 4-97):

In[9]: ax = plt.axes(projection='3d')
       ax.plot_surface(X, Y, Z, rstride=1, cstride=1,
                       cmap='viridis', edgecolor='none')
       ax.set_title('surface');
png
Figure 4-97. A three-dimensional surface plot

Note that though the grid of values for a surface plot needs to be two-dimensional, it need not be rectilinear. Here is an example of creating a partial polar grid, which when used with the surface3D plot can give us a slice into the function we’re visualizing (Figure 4-98):

In[10]: r = np.linspace(0, 6, 20)
        theta = np.linspace(-0.9 * np.pi, 0.8 * np.pi, 40)
        r, theta = np.meshgrid(r, theta)

        X = r * np.sin(theta)
        Y = r * np.cos(theta)
        Z = f(X, Y)

        ax = plt.axes(projection='3d')
        ax.plot_surface(X, Y, Z, rstride=1, cstride=1,
                        cmap='viridis', edgecolor='none');
png
Figure 4-98. A polar surface plot

Surface Triangulations

For some applications, the evenly sampled grids required by the preceding routines are overly restrictive and inconvenient. In these situations, the triangulation-based plots can be very useful. What if rather than an even draw from a Cartesian or a polar grid, we instead have a set of random draws?

In[11]: theta = 2 * np.pi * np.random.random(1000)
        r = 6 * np.random.random(1000)
        x = np.ravel(r * np.sin(theta))
        y = np.ravel(r * np.cos(theta))
        z = f(x, y)

We could create a scatter plot of the points to get an idea of the surface we’re sampling from (Figure 4-99):

In[12]: ax = plt.axes(projection='3d')
        ax.scatter(x, y, z, c=z, cmap='viridis', linewidth=0.5);
png
Figure 4-99. A three-dimensional sampled surface

This leaves a lot to be desired. The function that will help us in this case is ax.plot_trisurf, which creates a surface by first finding a set of triangles formed between adjacent points (the result is shown in Figure 4-100; remember that x, y, and z here are one-dimensional arrays):

In[13]: ax = plt.axes(projection='3d')
        ax.plot_trisurf(x, y, z,
                        cmap='viridis', edgecolor='none');
png
Figure 4-100. A triangulated surface plot

The result is certainly not as clean as when it is plotted with a grid, but the flexibility of such a triangulation allows for some really interesting three-dimensional plots. For example, it is actually possible to plot a three-dimensional Möbius strip using this, as we’ll see next.

Example: Visualizing a Möbius strip

A Möbius strip is similar to a strip of paper glued into a loop with a half-twist. Topologically, it’s quite interesting because despite appearances it has only a single side! Here we will visualize such an object using Matplotlib’s three-dimensional tools. The key to creating the Möbius strip is to think about its parameterization: it’s a two-dimensional strip, so we need two intrinsic dimensions. Let’s call them θ , which ranges from 0 to 2 π around the loop, and w which ranges from –1 to 1 across the width of the strip:

In[14]: theta = np.linspace(0, 2 * np.pi, 30)
        w = np.linspace(-0.25, 0.25, 8)
        w, theta = np.meshgrid(w, theta)

Now from this parameterization, we must determine the (x, y, z) positions of the embedded strip.

Thinking about it, we might realize that there are two rotations happening: one is the position of the loop about its center (what we’ve called θ ), while the other is the twisting of the strip about its axis (we’ll call this ϕ ). For a Möbius strip, we must have the strip make half a twist during a full loop, or Δ ϕ = Δ θ / 2 .

In[15]: phi = 0.5 * theta

Now we use our recollection of trigonometry to derive the three-dimensional embedding. We’ll define r , the distance of each point from the center, and use this to find the embedded ( x , y , z ) coordinates:

In[16]: # radius in x-y plane
        r = 1 + w * np.cos(phi)

        x = np.ravel(r * np.cos(theta))
        y = np.ravel(r * np.sin(theta))
        z = np.ravel(w * np.sin(phi))

Finally, to plot the object, we must make sure the triangulation is correct. The best way to do this is to define the triangulation within the underlying parameterization, and then let Matplotlib project this triangulation into the three-dimensional space of the Möbius strip. This can be accomplished as follows (Figure 4-101):

In[17]: # triangulate in the underlying parameterization
        from matplotlib.tri import Triangulation
        tri = Triangulation(np.ravel(w), np.ravel(theta))

        ax = plt.axes(projection='3d')
        ax.plot_trisurf(x, y, z, triangles=tri.triangles,
                        cmap='viridis', linewidths=0.2);

        ax.set_xlim(-1, 1); ax.set_ylim(-1, 1); ax.set_zlim(-1, 1);
png
Figure 4-101. Visualizing a Möbius strip

Combining all of these techniques, it is possible to create and display a wide variety of three-dimensional objects and patterns in Matplotlib.

Geographic Data with Basemap

One common type of visualization in data science is that of geographic data. Matplotlib’s main tool for this type of visualization is the Basemap toolkit, which is one of several Matplotlib toolkits that live under the mpl_toolkits namespace. Admittedly, Basemap feels a bit clunky to use, and often even simple visualizations take much longer to render than you might hope. More modern solutions, such as leaflet or the Google Maps API, may be a better choice for more intensive map visualizations. Still, Basemap is a useful tool for Python users to have in their virtual toolbelts. In this section, we’ll show several examples of the type of map visualization that is possible with this toolkit.

Installation of Basemap is straightforward; if you’re using conda you can type this and the package will be downloaded:

$ conda install basemap

We add just a single new import to our standard boilerplate:

In[1]: %matplotlib inline
       import numpy as np
       import matplotlib.pyplot as plt
       from mpl_toolkits.basemap import Basemap

Once you have the Basemap toolkit installed and imported, geographic plots are just a few lines away (the graphics in Figure 4-102 also require the PIL package in Python 2, or the pillow package in Python 3):

In[2]: plt.figure(figsize=(8, 8))
       m = Basemap(projection='ortho', resolution=None, lat_0=50, lon_0=-100)
       m.bluemarble(scale=0.5);
png
Figure 4-102. A “bluemarble” projection of the Earth

The meaning of the arguments to Basemap will be discussed momentarily.

The useful thing is that the globe shown here is not a mere image; it is a fully functioning Matplotlib axes that understands spherical coordinates and allows us to easily over-plot data on the map! For example, we can use a different map projection, zoom in to North America, and plot the location of Seattle. We’ll use an etopo image (which shows topographical features both on land and under the ocean) as the map background (Figure 4-103):

In[3]: fig = plt.figure(figsize=(8, 8))
       m = Basemap(projection='lcc', resolution=None,
                   width=8E6, height=8E6,
                   lat_0=45, lon_0=-100,)
       m.etopo(scale=0.5, alpha=0.5)

       # Map (long, lat) to (x, y) for plotting
       x, y = m(-122.3, 47.6)
       plt.plot(x, y, 'ok', markersize=5)
       plt.text(x, y, ' Seattle', fontsize=12);
png
Figure 4-103. Plotting data and labels on the map

This gives you a brief glimpse into the sort of geographic visualizations that are possible with just a few lines of Python. We’ll now discuss the features of Basemap in more depth, and provide several examples of visualizing map data. Using these brief examples as building blocks, you should be able to create nearly any map visualization that you desire.

Map Projections

The first thing to decide when you are using maps is which projection to use. You’re probably familiar with the fact that it is impossible to project a spherical map, such as that of the Earth, onto a flat surface without somehow distorting it or breaking its continuity. These projections have been developed over the course of human history, and there are a lot of choices! Depending on the intended use of the map projection, there are certain map features (e.g., direction, area, distance, shape, or other considerations) that are useful to maintain.

The Basemap package implements several dozen such projections, all referenced by a short format code. Here we’ll briefly demonstrate some of the more common ones.

We’ll start by defining a convenience routine to draw our world map along with the longitude and latitude lines:

In[4]: from itertools import chain

       def draw_map(m, scale=0.2):
           # draw a shaded-relief image
           m.shadedrelief(scale=scale)

           # lats and longs are returned as a dictionary
           lats = m.drawparallels(np.linspace(-90, 90, 13))
           lons = m.drawmeridians(np.linspace(-180, 180, 13))

           # keys contain the plt.Line2D instances
           lat_lines = chain(*(tup[1][0] for tup in lats.items()))
           lon_lines = chain(*(tup[1][0] for tup in lons.items()))
           all_lines = chain(lat_lines, lon_lines)

           # cycle through these lines and set the desired style
           for line in all_lines:
               line.set(linestyle='-', alpha=0.3, color='w')

Cylindrical projections

The simplest of map projections are cylindrical projections, in which lines of constant latitude and longitude are mapped to horizontal and vertical lines, respectively. This type of mapping represents equatorial regions quite well, but results in extreme distortions near the poles. The spacing of latitude lines varies between different cylindrical projections, leading to different conservation properties, and different distortion near the poles. In Figure 4-104, we show an example of the equidistant cylindrical projection, which chooses a latitude scaling that preserves distances along meridians. Other cylindrical projections are the Mercator (projection='merc') and the cylindrical equal-area (projection='cea') projections.

In[5]: fig = plt.figure(figsize=(8, 6), edgecolor='w')
       m = Basemap(projection='cyl', resolution=None,
                   llcrnrlat=-90, urcrnrlat=90,
                   llcrnrlon=-180, urcrnrlon=180, )
       draw_map(m)
png
Figure 4-104. Cylindrical equal-area projection

The additional arguments to Basemap for this view specify the latitude (lat) and longitude (lon) of the lower-left corner (llcrnr) and upper-right corner (urcrnr) for the desired map, in units of degrees.

Pseudo-cylindrical projections

Pseudo-cylindrical projections relax the requirement that meridians (lines of constant longitude) remain vertical; this can give better properties near the poles of the projection. The Mollweide projection (projection='moll') is one common example of this, in which all meridians are elliptical arcs (Figure 4-105). It is constructed so as to preserve area across the map: though there are distortions near the poles, the area of small patches reflects the true area. Other pseudo-cylindrical projections are the sinusoidal (projection='sinu') and Robinson (projection='robin') projections.

In[6]: fig = plt.figure(figsize=(8, 6), edgecolor='w')
       m = Basemap(projection='moll', resolution=None,
                   lat_0=0, lon_0=0)
       draw_map(m)
png
Figure 4-105. The Molleweide projection

The extra arguments to Basemap here refer to the central latitude (lat_0) and longitude (lon_0) for the desired map.

Perspective projections

Perspective projections are constructed using a particular choice of perspective point, similar to if you photographed the Earth from a particular point in space (a point which, for some projections, technically lies within the Earth!). One common example is the orthographic projection (projection='ortho'), which shows one side of the globe as seen from a viewer at a very long distance. Thus, it can show only half the globe at a time. Other perspective-based projections include the gnomonic projection (projection='gnom') and stereographic projection (projection='stere'). These are often the most useful for showing small portions of the map.

Here is an example of the orthographic projection (Figure 4-106):

In[7]: fig = plt.figure(figsize=(8, 8))
       m = Basemap(projection='ortho', resolution=None,
                   lat_0=50, lon_0=0)
       draw_map(m);
png
Figure 4-106. The orthographic projection

Conic projections

A conic projection projects the map onto a single cone, which is then unrolled. This can lead to very good local properties, but regions far from the focus point of the cone may become very distorted. One example of this is the Lambert conformal conic projection (projection='lcc'), which we saw earlier in the map of North America. It projects the map onto a cone arranged in such a way that two standard parallels (specified in Basemap by lat_1 and lat_2) have well-represented distances, with scale decreasing between them and increasing outside of them. Other useful conic projections are the equidistant conic (projection='eqdc') and the Albers equal-area (projection='aea') projection (Figure 4-107). Conic projections, like perspective projections, tend to be good choices for representing small to medium patches of the globe.

In[8]: fig = plt.figure(figsize=(8, 8))
       m = Basemap(projection='lcc', resolution=None,
                   lon_0=0, lat_0=50, lat_1=45, lat_2=55,
                   width=1.6E7, height=1.2E7)
       draw_map(m)
png
Figure 4-107. The Albers equal-area projection

Other projections

If you’re going to do much with map-based visualizations, I encourage you to read up on other available projections, along with their properties, advantages, and disadvantages. Most likely, they are available in the Basemap package. If you dig deep enough into this topic, you’ll find an incredible subculture of geo-viz geeks who will be ready to argue fervently in support of their favorite projection for any given application!

Drawing a Map Background

Earlier we saw the bluemarble() and shadedrelief() methods for projecting global images on the map, as well as the drawparallels() and drawmeridians() methods for drawing lines of constant latitude and longitude. The Basemap package contains a range of useful functions for drawing borders of physical features like continents, oceans, lakes, and rivers, as well as political boundaries such as countries and US states and counties. The following are some of the available drawing functions that you may wish to explore using IPython’s help features:

  • Physical boundaries and bodies of water

    drawcoastlines()

    Draw continental coast lines

    drawlsmask()

    Draw a mask between the land and sea, for use with projecting images on one or the other

    drawmapboundary()

    Draw the map boundary, including the fill color for oceans

    drawrivers()

    Draw rivers on the map

    fillcontinents()

    Fill the continents with a given color; optionally fill lakes with another color

  • Political boundaries

    drawcountries()

    Draw country boundaries

    drawstates()

    Draw US state boundaries

    drawcounties()

    Draw US county boundaries

  • Map features

    drawgreatcircle()

    Draw a great circle between two points

    drawparallels()

    Draw lines of constant latitude

    drawmeridians()

    Draw lines of constant longitude

    drawmapscale()

    Draw a linear scale on the map

  • Whole-globe images

    bluemarble()

    Project NASA’s blue marble image onto the map

    shadedrelief()

    Project a shaded relief image onto the map

    etopo()

    Draw an etopo relief image onto the map

    warpimage()

    Project a user-provided image onto the map

For the boundary-based features, you must set the desired resolution when creating a Basemap image. The resolution argument of the Basemap class sets the level of detail in boundaries, either 'c' (crude), 'l' (low), 'i' (intermediate), 'h' (high), 'f' (full), or None if no boundaries will be used. This choice is important: setting high-resolution boundaries on a global map, for example, can be very slow.

Here’s an example of drawing land/sea boundaries, and the effect of the resolution parameter. We’ll create both a low- and high-resolution map of Scotland’s beautiful Isle of Skye. It’s located at 57.3°N, 6.2°W, and a map of 90,000×120,000 kilometers shows it well (Figure 4-108):

In[9]: fig, ax = plt.subplots(1, 2, figsize=(12, 8))

       for i, res in enumerate(['l', 'h']):
           m = Basemap(projection='gnom', lat_0=57.3, lon_0=-6.2,
                       width=90000, height=120000, resolution=res, ax=ax[i])
           m.fillcontinents(color="#FFDDCC", lake_color='#DDEEFF')
           m.drawmapboundary(fill_color="#DDEEFF")
           m.drawcoastlines()
           ax[i].set_title("resolution='{0}'".format(res));
png
Figure 4-108. Map boundaries at low and high resolution

Notice that the low-resolution coastlines are not suitable for this level of zoom, while high-resolution works just fine. The low level would work just fine for a global view, however, and would be much faster than loading the high-resolution border data for the entire globe! It might require some experimentation to find the correct resolution parameter for a given view; the best route is to start with a fast, low-resolution plot and increase the resolution as needed.

Plotting Data on Maps

Perhaps the most useful piece of the Basemap toolkit is the ability to over-plot a variety of data onto a map background. For simple plotting and text, any plt function works on the map; you can use the Basemap instance to project latitude and longitude coordinates to (x, y) coordinates for plotting with plt, as we saw earlier in the Seattle example.

In addition to this, there are many map-specific functions available as methods of the Basemap instance. These work very similarly to their standard Matplotlib counterparts, but have an additional Boolean argument latlon, which if set to True allows you to pass raw latitudes and longitudes to the method, rather than projected (x, y) coordinates.

Some of these map-specific methods are:

contour()/contourf()

Draw contour lines or filled contours

imshow()

Draw an image

pcolor()/pcolormesh()

Draw a pseudocolor plot for irregular/regular meshes

plot()

Draw lines and/or markers

scatter()

Draw points with markers

quiver()

Draw vectors

barbs()

Draw wind barbs

drawgreatcircle()

Draw a great circle

We’ll see examples of a few of these as we continue. For more information on these functions, including several example plots, see the online Basemap documentation.

Example: California Cities

Recall that in “Customizing Plot Legends”, we demonstrated the use of size and color in a scatter plot to convey information about the location, size, and population of California cities. Here, we’ll create this plot again, but using Basemap to put the data in context.

We start with loading the data, as we did before:

In[10]: import pandas as pd
        cities = pd.read_csv('data/california_cities.csv')

        # Extract the data we're interested in
        lat = cities['latd'].values
        lon = cities['longd'].values
        population = cities['population_total'].values
        area = cities['area_total_km2'].values

Next, we set up the map projection, scatter the data, and then create a colorbar and legend (Figure 4-109):

In[11]: # 1. Draw the map background
        fig = plt.figure(figsize=(8, 8))
        m = Basemap(projection='lcc', resolution='h',
                    lat_0=37.5, lon_0=-119,
                    width=1E6, height=1.2E6)
        m.shadedrelief()
        m.drawcoastlines(color='gray')
        m.drawcountries(color='gray')
        m.drawstates(color='gray')

        # 2. scatter city data, with color reflecting population
        # and size reflecting area
        m.scatter(lon, lat, latlon=True,
                  c=np.log10(population), s=area,
                  cmap='Reds', alpha=0.5)

        # 3. create colorbar and legend
        plt.colorbar(label=r'$\log_{10}({\rm population})$')
        plt.clim(3, 7)

        # make legend with dummy points
        for a in [100, 300, 500]:
            plt.scatter([], [], c='k', alpha=0.5, s=a,
                        label=str(a) + ' km$^2$')
        plt.legend(scatterpoints=1, frameon=False,
                   labelspacing=1, loc='lower left');
png
Figure 4-109. Scatter plot over a map background

This shows us roughly where larger populations of people have settled in California: they are clustered near the coast in the Los Angeles and San Francisco areas, stretched along the highways in the flat central valley, and avoiding almost completely the mountainous regions along the borders of the state.

Example: Surface Temperature Data

As an example of visualizing some more continuous geographic data, let’s consider the “polar vortex” that hit the eastern half of the United States in January 2014. A great source for any sort of climatic data is NASA’s Goddard Institute for Space Studies. Here we’ll use the GIS 250 temperature data, which we can download using shell commands (these commands may have to be modified on Windows machines). The data used here was downloaded on 6/12/2016, and the file size is approximately 9 MB:

In[12]: # !curl -O http://data.giss.nasa.gov/pub/gistemp/gistemp250.nc.gz
        # !gunzip gistemp250.nc.gz

The data comes in NetCDF format, which can be read in Python by the netCDF4 library. You can install this library as shown here:

$ conda install netcdf4

We read the data as follows:

In[13]: from netCDF4 import Dataset
        data = Dataset('gistemp250.nc')

The file contains many global temperature readings on a variety of dates; we need to select the index of the date we’re interested in—in this case, January 15, 2014:

In[14]: from netCDF4 import date2index
        from datetime import datetime
        timeindex = date2index(datetime(2014, 1, 15),
                               data.variables['time'])

Now we can load the latitude and longitude data, as well as the temperature anomaly for this index:

In[15]: lat = data.variables['lat'][:]
        lon = data.variables['lon'][:]
        lon, lat = np.meshgrid(lon, lat)
        temp_anomaly = data.variables['tempanomaly'][timeindex]

Finally, we’ll use the pcolormesh() method to draw a color mesh of the data. We’ll look at North America, and use a shaded relief map in the background. Note that for this data we specifically chose a divergent colormap, which has a neutral color at zero and two contrasting colors at negative and positive values (Figure 4-110). We’ll also lightly draw the coastlines over the colors for reference:

In[16]: fig = plt.figure(figsize=(10, 8))
        m = Basemap(projection='lcc', resolution='c',
                    width=8E6, height=8E6,
                    lat_0=45, lon_0=-100,)
        m.shadedrelief(scale=0.5)
        m.pcolormesh(lon, lat, temp_anomaly,
                     latlon=True, cmap='RdBu_r')
        plt.clim(-8, 8)
        m.drawcoastlines(color='lightgray')

        plt.title('January 2014 Temperature Anomaly')
        plt.colorbar(label='temperature anomaly (°C)');

The data paints a picture of the localized, extreme temperature anomalies that happened during that month. The eastern half of the United States was much colder than normal, while the western half and Alaska were much warmer. Regions with no recorded temperature show the map background.

png
Figure 4-110. The temperature anomaly in January 2014

Visualization with Seaborn

Matplotlib has proven to be an incredibly useful and popular visualization tool, but even avid users will admit it often leaves much to be desired. There are several valid complaints about Matplotlib that often come up:

  • Prior to version 2.0, Matplotlib’s defaults are not exactly the best choices. It was based off of MATLAB circa 1999, and this often shows.

  • Matplotlib’s API is relatively low level. Doing sophisticated statistical visualization is possible, but often requires a lot of boilerplate code.

  • Matplotlib predated Pandas by more than a decade, and thus is not designed for use with Pandas DataFrames. In order to visualize data from a Pandas DataFrame, you must extract each Series and often concatenate them together into the right format. It would be nicer to have a plotting library that can intelligently use the DataFrame labels in a plot.

An answer to these problems is Seaborn. Seaborn provides an API on top of Matplotlib that offers sane choices for plot style and color defaults, defines simple high-level functions for common statistical plot types, and integrates with the functionality provided by Pandas DataFrames.

To be fair, the Matplotlib team is addressing this: it has recently added the plt.style tools (discussed in “Customizing Matplotlib: Configurations and Stylesheets”), and is starting to handle Pandas data more seamlessly. The 2.0 release of the library will include a new default stylesheet that will improve on the current status quo. But for all the reasons just discussed, Seaborn remains an extremely useful add-on.

Seaborn Versus Matplotlib

Here is an example of a simple random-walk plot in Matplotlib, using its classic plot formatting and colors. We start with the typical imports:

In[1]: import matplotlib.pyplot as plt
       plt.style.use('classic')
       %matplotlib inline
       import numpy as np
       import pandas as pd

Now we create some random walk data:

In[2]: # Create some data
       rng = np.random.RandomState(0)
       x = np.linspace(0, 10, 500)
       y = np.cumsum(rng.randn(500, 6), 0)

And do a simple plot (Figure 4-111):

In[3]: # Plot the data with Matplotlib defaults
       plt.plot(x, y)
       plt.legend('ABCDEF', ncol=2, loc='upper left');
png
Figure 4-111. Data in Matplotlib’s default style

Although the result contains all the information we’d like it to convey, it does so in a way that is not all that aesthetically pleasing, and even looks a bit old-fashioned in the context of 21st-century data visualization.

Now let’s take a look at how it works with Seaborn. As we will see, Seaborn has many of its own high-level plotting routines, but it can also overwrite Matplotlib’s default parameters and in turn get even simple Matplotlib scripts to produce vastly superior output. We can set the style by calling Seaborn’s set() method. By convention, Seaborn is imported as sns:

In[4]: import seaborn as sns
       sns.set()

Now let’s rerun the same two lines as before (Figure 4-112):

In[5]: # same plotting code as above!
       plt.plot(x, y)
       plt.legend('ABCDEF', ncol=2, loc='upper left');
png
Figure 4-112. Data in Seaborn’s default style

Ah, much better!

Exploring Seaborn Plots

The main idea of Seaborn is that it provides high-level commands to create a variety of plot types useful for statistical data exploration, and even some statistical model fitting.

Let’s take a look at a few of the datasets and plot types available in Seaborn. Note that all of the following could be done using raw Matplotlib commands (this is, in fact, what Seaborn does under the hood), but the Seaborn API is much more convenient.

Histograms, KDE, and densities

Often in statistical data visualization, all you want is to plot histograms and joint distributions of variables. We have seen that this is relatively straightforward in Matplotlib (Figure 4-113):

In[6]: data = np.random.multivariate_normal([0, 0], [[5, 2], [2, 2]], size=2000)
       data = pd.DataFrame(data, columns=['x', 'y'])

       for col in 'xy':
           plt.hist(data[col], normed=True, alpha=0.5)
png
Figure 4-113. Histograms for visualizing distributions

Rather than a histogram, we can get a smooth estimate of the distribution using a kernel density estimation, which Seaborn does with sns.kdeplot (Figure 4-114):

In[7]: for col in 'xy':
           sns.kdeplot(data[col], shade=True)
png
Figure 4-114. Kernel density estimates for visualizing distributions

Histograms and KDE can be combined using distplot (Figure 4-115):

In[8]: sns.distplot(data['x'])
       sns.distplot(data['y']);
png
Figure 4-115. Kernel density and histograms plotted together

If we pass the full two-dimensional dataset to kdeplot, we will get a two-dimensional visualization of the data (Figure 4-116):

In[9]: sns.kdeplot(data);
png
Figure 4-116. A two-dimensional kernel density plot

We can see the joint distribution and the marginal distributions together using sns.jointplot. For this plot, we’ll set the style to a white background (Figure 4-117):

In[10]: with sns.axes_style('white'):
            sns.jointplot("x", "y", data, kind='kde');
png
Figure 4-117. A joint distribution plot with a two-dimensional kernel density estimate

There are other parameters that can be passed to jointplot—for example, we can use a hexagonally based histogram instead (Figure 4-118):

In[11]: with sns.axes_style('white'):
            sns.jointplot("x", "y", data, kind='hex')
png
Figure 4-118. A joint distribution plot with a hexagonal bin representation

Pair plots

When you generalize joint plots to datasets of larger dimensions, you end up with pair plots. This is very useful for exploring correlations between multidimensional data, when you’d like to plot all pairs of values against each other.

We’ll demo this with the well-known Iris dataset, which lists measurements of petals and sepals of three iris species:

In[12]: iris = sns.load_dataset("iris")
        iris.head()
Out[12]:    sepal_length  sepal_width  petal_length  petal_width species
         0           5.1          3.5           1.4          0.2  setosa
         1           4.9          3.0           1.4          0.2  setosa
         2           4.7          3.2           1.3          0.2  setosa
         3           4.6          3.1           1.5          0.2  setosa
         4           5.0          3.6           1.4          0.2  setosa

Visualizing the multidimensional relationships among the samples is as easy as calling sns.pairplot (Figure 4-119):

In[13]: sns.pairplot(iris, hue='species', size=2.5);
png
Figure 4-119. A pair plot showing the relationships between four variables

Faceted histograms

Sometimes the best way to view data is via histograms of subsets. Seaborn’s FacetGrid makes this extremely simple. We’ll take a look at some data that shows the amount that restaurant staff receive in tips based on various indicator data (Figure 4-120):

In[14]: tips = sns.load_dataset('tips')
        tips.head()
Out[14]:    total_bill   tip     sex smoker  day    time  size
         0       16.99  1.01  Female     No  Sun  Dinner     2
         1       10.34  1.66    Male     No  Sun  Dinner     3
         2       21.01  3.50    Male     No  Sun  Dinner     3
         3       23.68  3.31    Male     No  Sun  Dinner     2
         4       24.59  3.61  Female     No  Sun  Dinner     4
In[15]: tips['tip_pct'] = 100 * tips['tip'] / tips['total_bill']

        grid = sns.FacetGrid(tips, row="sex", col="time", margin_titles=True)
        grid.map(plt.hist, "tip_pct", bins=np.linspace(0, 40, 15));
png
Figure 4-120. An example of a faceted histogram

Factor plots

Factor plots can be useful for this kind of visualization as well. This allows you to view the distribution of a parameter within bins defined by any other parameter (Figure 4-121):

In[16]: with sns.axes_style(style='ticks'):
            g = sns.factorplot("day", "total_bill", "sex", data=tips, kind="box")
            g.set_axis_labels("Day", "Total Bill");
png
Figure 4-121. An example of a factor plot, comparing distributions given various discrete factors

Joint distributions

Similar to the pair plot we saw earlier, we can use sns.jointplot to show the joint distribution between different datasets, along with the associated marginal distributions (Figure 4-122):

In[17]: with sns.axes_style('white'):
            sns.jointplot("total_bill", "tip", data=tips, kind='hex')
png
Figure 4-122. A joint distribution plot

The joint plot can even do some automatic kernel density estimation and regression (Figure 4-123):

In[18]: sns.jointplot("total_bill", "tip", data=tips, kind='reg');
png
Figure 4-123. A joint distribution plot with a regression fit

Bar plots

Time series can be plotted with sns.factorplot. In the following example (visualized in Figure 4-124), we’ll use the Planets data that we first saw in “Aggregation and Grouping”:

In[19]: planets = sns.load_dataset('planets')
        planets.head()
Out[19]:             method  number  orbital_period   mass  distance  year
         0  Radial Velocity       1         269.300   7.10     77.40  2006
         1  Radial Velocity       1         874.774   2.21     56.95  2008
         2  Radial Velocity       1         763.000   2.60     19.84  2011
         3  Radial Velocity       1         326.030  19.40    110.62  2007
         4  Radial Velocity       1         516.220  10.50    119.47  2009
In[20]: with sns.axes_style('white'):
            g = sns.factorplot("year", data=planets, aspect=2,
                               kind="count", color='steelblue')
            g.set_xticklabels(step=5)
png
Figure 4-124. A histogram as a special case of a factor plot

We can learn more by looking at the method of discovery of each of these planets, as illustrated in Figure 4-125:

In[21]: with sns.axes_style('white'):
            g = sns.factorplot("year", data=planets, aspect=4.0, kind='count',
                               hue='method', order=range(2001, 2015))
            g.set_ylabels('Number of Planets Discovered')
png
Figure 4-125. Number of planets discovered by year and type (see the online appendix for a full-scale figure)

For more information on plotting with Seaborn, see the Seaborn documentation, a tutorial, and the Seaborn gallery.

Example: Exploring Marathon Finishing Times

Here we’ll look at using Seaborn to help visualize and understand finishing results from a marathon. I’ve scraped the data from sources on the Web, aggregated it and removed any identifying information, and put it on GitHub where it can be downloaded (if you are interested in using Python for web scraping, I would recommend Web Scraping with Python by Ryan Mitchell). We will start by downloading the data from the Web, and loading it into Pandas:

In[22]:
# !curl -O https://raw.githubusercontent.com/jakevdp/marathon-data/
# master/marathon-data.csv
In[23]: data = pd.read_csv('marathon-data.csv')
        data.head()
Out[23]:    age gender     split     final
         0   33      M  01:05:38  02:08:51
         1   32      M  01:06:26  02:09:28
         2   31      M  01:06:49  02:10:42
         3   38      M  01:06:16  02:13:45
         4   31      M  01:06:32  02:13:59

By default, Pandas loaded the time columns as Python strings (type object); we can see this by looking at the dtypes attribute of the DataFrame:

In[24]: data.dtypes
Out[24]: age        int64
         gender    object
         split     object
         final     object
         dtype: object

Let’s fix this by providing a converter for the times:

In[25]: def convert_time(s):
            h, m, s = map(int, s.split(':'))
            return pd.datetools.timedelta(hours=h, minutes=m, seconds=s)

        data = pd.read_csv('marathon-data.csv',
                           converters={'split':convert_time, 'final':convert_time})
        data.head()
Out[25]:    age gender    split    final
         0   33      M 01:05:38 02:08:51
         1   32      M 01:06:26 02:09:28
         2   31      M 01:06:49 02:10:42
         3   38      M 01:06:16 02:13:45
         4   31      M 01:06:32 02:13:59
In[26]: data.dtypes
Out[26]: age                 int64
         gender             object
         split     timedelta64[ns]
         final     timedelta64[ns]
         dtype: object

That looks much better. For the purpose of our Seaborn plotting utilities, let’s next add columns that give the times in seconds:

In[27]: data['split_sec'] = data['split'].astype(int) / 1E9
        data['final_sec'] = data['final'].astype(int) / 1E9
        data.head()
Out[27]:    age gender    split    final  split_sec  final_sec
         0   33      M 01:05:38 02:08:51     3938.0     7731.0
         1   32      M 01:06:26 02:09:28     3986.0     7768.0
         2   31      M 01:06:49 02:10:42     4009.0     7842.0
         3   38      M 01:06:16 02:13:45     3976.0     8025.0
         4   31      M 01:06:32 02:13:59     3992.0     8039.0

To get an idea of what the data looks like, we can plot a jointplot over the data (Figure 4-126):

In[28]: with sns.axes_style('white'):
            g = sns.jointplot("split_sec", "final_sec", data, kind='hex')
            g.ax_joint.plot(np.linspace(4000, 16000),
                            np.linspace(8000, 32000), ':k')
png
Figure 4-126. The relationship between the split for the first half-marathon and the finishing time for the full marathon

The dotted line shows where someone’s time would lie if they ran the marathon at a perfectly steady pace. The fact that the distribution lies above this indicates (as you might expect) that most people slow down over the course of the marathon. If you have run competitively, you’ll know that those who do the opposite—run faster during the second half of the race—are said to have “negative-split” the race.

Let’s create another column in the data, the split fraction, which measures the degree to which each runner negative-splits or positive-splits the race:

In[29]: data['split_frac'] = 1 - 2 * data['split_sec'] / data['final_sec']
        data.head()
Out[29]:    age gender    split    final  split_sec  final_sec  split_frac
         0   33      M 01:05:38 02:08:51     3938.0     7731.0   -0.018756
         1   32      M 01:06:26 02:09:28     3986.0     7768.0   -0.026262
         2   31      M 01:06:49 02:10:42     4009.0     7842.0   -0.022443
         3   38      M 01:06:16 02:13:45     3976.0     8025.0    0.009097
         4   31      M 01:06:32 02:13:59     3992.0     8039.0    0.006842

Where this split difference is less than zero, the person negative-split the race by that fraction. Let’s do a distribution plot of this split fraction (Figure 4-127):

In[30]: sns.distplot(data['split_frac'], kde=False);
        plt.axvline(0, color="k", linestyle="--");
png
Figure 4-127. The distribution of split fractions; 0.0 indicates a runner who completed the first and second halves in identical times
In[31]: sum(data.split_frac < 0)
Out[31]: 251

Out of nearly 40,000 participants, there were only 250 people who negative-split their marathon.

Let’s see whether there is any correlation between this split fraction and other variables. We’ll do this using a pairgrid, which draws plots of all these correlations (Figure 4-128):

In[32]:
g = sns.PairGrid(data, vars=['age', 'split_sec', 'final_sec', 'split_frac'],
                 hue='gender', palette='RdBu_r')
g.map(plt.scatter, alpha=0.8)
g.add_legend();
png
Figure 4-128. The relationship between quantities within the marathon dataset

It looks like the split fraction does not correlate particularly with age, but does correlate with the final time: faster runners tend to have closer to even splits on their marathon time. (We see here that Seaborn is no panacea for Matplotlib’s ills when it comes to plot styles: in particular, the x-axis labels overlap. Because the output is a simple Matplotlib plot, however, the methods in “Customizing Ticks” can be used to adjust such things if desired.)

The difference between men and women here is interesting. Let’s look at the histogram of split fractions for these two groups (Figure 4-129):

In[33]: sns.kdeplot(data.split_frac[data.gender=='M'], label='men', shade=True)
        sns.kdeplot(data.split_frac[data.gender=='W'], label='women', shade=True)
        plt.xlabel('split_frac');
png
Figure 4-129. The distribution of split fractions by gender

The interesting thing here is that there are many more men than women who are running close to an even split! This almost looks like some kind of bimodal distribution among the men and women. Let’s see if we can suss out what’s going on by looking at the distributions as a function of age.

A nice way to compare distributions is to use a violin plot (Figure 4-130):

In[34]:
sns.violinplot("gender", "split_frac", data=data,
               palette=["lightblue", "lightpink"]);
png
Figure 4-130. A violin plot showing the split fraction by gender

This is yet another way to compare the distributions between men and women.

Let’s look a little deeper, and compare these violin plots as a function of age. We’ll start by creating a new column in the array that specifies the decade of age that each person is in (Figure 4-131):

In[35]: data['age_dec'] = data.age.map(lambda age: 10 * (age // 10))
data.head()
Out[35]:
   age gender    split    final  split_sec  final_sec  split_frac  age_dec
0   33      M 01:05:38 02:08:51     3938.0     7731.0   -0.018756       30
1   32      M 01:06:26 02:09:28     3986.0     7768.0   -0.026262       30
2   31      M 01:06:49 02:10:42     4009.0     7842.0   -0.022443       30
3   38      M 01:06:16 02:13:45     3976.0     8025.0    0.009097       30
4   31      M 01:06:32 02:13:59     3992.0     8039.0    0.006842       30
In[36]:
men = (data.gender == 'M')
women = (data.gender == 'W')

with sns.axes_style(style=None):
    sns.violinplot("age_dec", "split_frac", hue="gender", data=data,
                  split=True, inner="quartile",
                  palette=["lightblue", "lightpink"]);
png
Figure 4-131. A violin plot showing the split fraction by gender and age

Looking at this, we can see where the distributions of men and women differ: the split distributions of men in their 20s to 50s show a pronounced over-density toward lower splits when compared to women of the same age (or of any age, for that matter).

Also surprisingly, the 80-year-old women seem to outperform everyone in terms of their split time. This is probably due to the fact that we’re estimating the distribution from small numbers, as there are only a handful of runners in that range:

In[38]: (data.age > 80).sum()
Out[38]: 7

Back to the men with negative splits: who are these runners? Does this split fraction correlate with finishing quickly? We can plot this very easily. We’ll use regplot, which will automatically fit a linear regression to the data (Figure 4-132):

In[37]: g = sns.lmplot('final_sec', 'split_frac', col='gender', data=data,
                       markers=".", scatter_kws=dict(color='c'))
        g.map(plt.axhline, y=0.1, color="k", ls=":");
png
Figure 4-132. Split fraction versus finishing time by gender

Apparently the people with fast splits are the elite runners who are finishing within ~15,000 seconds, or about 4 hours. People slower than that are much less likely to have a fast second split.

Further Resources

Matplotlib Resources

A single chapter in a book can never hope to cover all the available features and plot types available in Matplotlib. As with other packages we’ve seen, liberal use of IPython’s tab-completion and help functions (see “Help and Documentation in IPython”) can be very helpful when you’re exploring Matplotlib’s API. In addition, Matplotlib’s online documentation can be a helpful reference. See in particular the Matplotlib gallery linked on that page: it shows thumbnails of hundreds of different plot types, each one linked to a page with the Python code snippet used to generate it. In this way, you can visually inspect and learn about a wide range of different plotting styles and visualization techniques.

For a book-length treatment of Matplotlib, I would recommend Interactive Applications Using Matplotlib, written by Matplotlib core developer Ben Root.

Other Python Graphics Libraries

Although Matplotlib is the most prominent Python visualization library, there are other more modern tools that are worth exploring as well. I’ll mention a few of them briefly here:

  • Bokeh is a JavaScript visualization library with a Python frontend that creates highly interactive visualizations capable of handling very large and/or streaming datasets. The Python frontend outputs a JSON data structure that can be interpreted by the Bokeh JS engine.

  • Plotly is the eponymous open source product of the Plotly company, and is similar in spirit to Bokeh. Because Plotly is the main product of a startup, it is receiving a high level of development effort. Use of the library is entirely free.

  • Vispy is an actively developed project focused on dynamic visualizations of very large datasets. Because it is built to target OpenGL and make use of efficient graphics processors in your computer, it is able to render some quite large and stunning visualizations.

  • Vega and Vega-Lite are declarative graphics representations, and are the product of years of research into the fundamental language of data visualization. The reference rendering implementation is JavaScript, but the API is language agnostic. There is a Python API under development in the Altair package. Though it’s not mature yet, I’m quite excited for the possibilities of this project to provide a common reference point for visualization in Python and other languages.

The visualization space in the Python community is very dynamic, and I fully expect this list to be out of date as soon as it is published. Keep an eye out for what’s coming in the future!

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