Tutuorial 0b: Introduction to Jupyter notebooks¶

This tutorial was generated from a Jupyter notebook. You can download the notebook here.

The first thing we'll do, discussed later, is import all the modules we'll need. You should in general do this at the very beginning of each notebook, and in fact each .py file you write.

# Our numerical workhorses
import numpy as np
import pandas as pd
import scipy.integrate

# Import Bokeh modules for interactive plotting
import bokeh.io
import bokeh.plotting

# Import HoloViews for quick plotting of DataFrames
import holoviews as hv

# Set up Bokeh for inline viewing
bokeh.io.output_notebook()

# HoloViews uses Bokeh
hv.extension('bokeh')

# Global Holoviews settings
%opts Curve [show_grid=True, width=500, height=400, fontsize={'xlabel':14, 'ylabel':14}]

In this tutorial, you will learn the basics on how to use Jupyter notebooks. All of your homework will be submitted as Jupyter notebooks, so this is something you will need to master. It will be useful for you to go over Tutorial 0c to learn how to use $\LaTeX$ in your Jupyter notebooks. You should also look at Tutorial 0d, which gives an example homework problem an what we expect in terms of content and formatting in your homework.

You should, of course, read the official Jupyter documentation as well.

There are many sections to this tutorial, so I provide a table of contents.

What is Jupyter?¶

Jupyter is a way to combine text (with math!) and code (which runs and can display graphic output!) in an easy-to-read document that renders in a web browser. The notebook itself is stored as a text file in JSON format.

It is language agnostic as its name suggests. The name "Jupyter" is a combination of Julia (a new language for scientific computing), Python (which you know and love, or at least will when the course is over), and R (the dominant tool for statistical computation). However, you currently can run over 40 different languages in a Jupyter notebook, not just Julia, Python, and R.

Launching a Jupyter notebook¶

To launch a Jupyter notebook, you can do the following.

Mac: Use the Anaconda launcher and select Jupyter notebook.
Windows: Under "Search programs and files" from the Start menu, type jupyter notebook and select "Jupyter notebook."

A Jupyter notebook will then launch in your default web browser.

You can also launch Jupyter from the command line, which is what I prefer. To do this, simply enter

jupyter notebook

on the command line and hit enter. This also allows for greater flexibility, as you can launch Jupyter with command line flags. For example, I launch Jupyter using

jupyter notebook --browser=safari --NotebookApp.iopub_data_rate_limit=10000000

This fires up Jupyter with Safari as the browser. The last part allows Jupyter to send more data through your browser, which will be important when we look at larger data sets and do image processing later on in the course. If you launch Jupyter from the command line, your shell will be occupied with Jupyter and will occasionally print information to the screen. After you are finished with your Jupyter session (and have saved everything), you can kill Jupyter by hitting "ctrl + C" in the terminal/PowerShell window.

When you launch Jupyter, you will be presented with a menu of files in your current working directory to choose to edit. You can also navigate around the files on your computer to find a file you wish to edit by clicking the "Upload" button in the upper right corner. You can also click "New" in the upper right corner to get a new Jupyter notebook. After selecting the file you wish to edit, it will appear in a new window in your browser, beautifully formatted and ready to edit.

Cells¶

A Jupyter notebook consists of cells. The two main types of cells you will use are code cells and markdown cells, and we will go into their properties in depth momentarily. First, an overview.

A code cell contains actual code that you want to run. You can specify a cell as a code cell using the pulldown menu in the toolbar in your Jupyter notebook. Otherwise, you can can hit esc and then y (denoted "esc, y") while a cell is selected to specify that it is a code cell. Note that you will have to hit enter after doing this to start editing it.

If you want to execute the code in a code cell, hit "shift + enter." Note that code cells are executed in the order you execute them. That is to say, the ordering of the cells for which you hit "shift + enter" is the order in which the code is executed. If you did not explicitly execute a cell early in the document, its results are now known to the Python interpreter.

Markdown cells contain text. The text is written in markdown, a lightweight markup language. You can read about its syntax here. Note that you can also insert HTML into markdown cells, and this will be rendered properly. As you are typing the contents of these cells, the results appear as text. Hitting "shift + enter" renders the text in the formatting you specify.

You can specify a cell as being a markdown cell in the Jupyter toolbar, or by hitting "esc, m" in the cell. Again, you have to hit enter after using the quick keys to bring the cell into edit mode.

In general, when you want to add a new cell, you can use the "Insert" pulldown menu from the Jupyter toolbar. The shortcut to insert a cell below is "esc, b" and to insert a cell above is "esc, a." Alternatively, you can execute a cell and automatically add a new one below it by hitting "alt + enter."

Code cells¶

Below is an example of a code cell printing hello, world. Notice that the output of the print statement appears in the same cell, though separate from the code block.

# Say hello to the world.
print('hello, world.')

hello, world.

If you evaluate a Python expression that returns a value, that value is displayed as output of the code cell. This only happens, however, for the last line of the code cell.

# Would show 9 if this were the last line, but it is not, so shows nothing
4 + 5

# I hope we see 11.
5 + 6

11

Note, however, if the last line does not return a value, such as if we assigned a variable, there is no visible output from the code cell.

# Variable assignment, so no visible output.
a = 5 + 6

# However, now if we ask for a, its value will be displayed
a

11

Display of graphics¶

We will be using Bokeh almost exclusively during the course, except when we use some built-in plotting functions in PyMC3. To make sure the Bokeh plots get shown in the notebook, you should execute

bokeh.io.output_notebook()

in your notebook. In this notebook, it is executed in the first code cell.

Now, let's generate a plot of a pretty curve.

# Generate data to plot
x = np.linspace(0, 2 * np.pi, 200)
y = np.exp(np.sin(np.sin(x)))

# Make plot
p = bokeh.plotting.figure(plot_height=300,
                          plot_width=500,
                          x_axis_label='x',
                          y_axis_label='y')
p.line(x, y, line_width=2)

# Show the plot
bokeh.io.show(p)

We have a nice, interactive plot of a graceful curve!

We will get more into HoloViews in the second week of class or so, but for now, I point out that we import it as hv and run

hv.extension('bokeh')

to tell HoloViews that it will be giving us Bokeh plots. (See the first code cell in this notebook.) Note also that you can use IPython magic functions (these are functions that start with a % sign) to specify global options for the plots HoloViews creates.

Proper formatting of cells¶

Generally, it is a good idea to keep cells simple. You can define one function, or maybe two or three closely related functions, in a single cell, and that's about it. When you define a function, you should make sure it is properly commented with descriptive doc strings. Below is an example of how I might generate a plot of the Lorenz attractor (which I choose just because it is fun) with code cells and markdown cells with discussion of what I am doing.

We will use scipy.integrate.odeint() to numerically integrate the Lorenz attractor. We therefore first define a function that returns the right hand side of the system of ODEs that define the Lorentz attractor.

def lorenz_attractor(r, t, p):
    """
    Compute the right hand side of system of ODEs for Lorenz attractor.
    
    Parameters
    ----------
    r : array_like, shape (3,)
        (x, y, z) position of trajectory.
    t : dummy_argument
        Dummy argument, necessary to pass function into 
        scipy.integrate.odeint
    p : array_like, shape (3,)
        Parameters (s, k, b) for the attractor.
        
    Returns
    -------
    output : ndarray, shape (3,)
        Time derivatives of Lorenz attractor.
        
    Notes
    -----
    .. Returns the right hand side of the system of ODEs describing
       the Lorenz attractor.
        x' = s * (y - x)
        y' = x * (k - z) - y
        z' = x * y - b * z
    """
    # Unpack variables and parameters
    x, y, z = r
    s, p, b = p
    
    return np.array([s * (y - x), 
                     x * (p - z) - y, 
                     x * y - b * z])

With this function in hand, we just have to pick our initial conditions and time points and run the numerical integration.

# Parameters to use
p = np.array([10.0, 28.0, 8.0 / 3.0])

# Initial condition
r0 = np.array([0.1, 0.0, 0.0])

# Time points to sample
t = np.linspace(0.0, 80.0, 10000)

# Use scipy.integrate.odeint to integrate Lorentz attractor
r = scipy.integrate.odeint(lorenz_attractor, r0, t, args=(p,))

# Unpack results into x, y, z.
x, y, z = r.transpose()

To plot the trajectory, we will use HoloViews. As such, it is convenient to construct a Pandas DataFrame to hold our variables. (You'll learn about Pandas in the first week of class.)

df = pd.DataFrame(data=dict(x=x, y=y, z=z))

Now that we have the DataFrame, we can let HoloViews do the plotting.

%%opts Curve (line_width=1)
c1 = hv.Curve(df, kdims=['x', 'y'])
c2 = hv.Curve(df, kdims=['x', 'z'])
c3 = hv.Curve(df, kdims=['y', 'z'])

(c1 + c2 + c3).cols(1)

And we have three nice, interactive plots with projections onto the x-y, x-z, and y-z planes.

Best practices for code cells¶

Here is a summary of some general rules for composing and formatting your code cells.

Keep the width of code in cells below 80 characters.
Keep your code cells short. If you find yourself having one massive code cell, break it up.
Always properly comment your code. Provide complete doc strings for any functions you define.
Do all of your imports in the first code cell at the top of the notebook. With the exception of "from ... import ..." imports, import one module per line.
For submitting assignments, always display your graphics in the notebook.

Markdown cells¶

Markdown cells contain text. The text is written in markdown, a lightweight markup language. The list of syntactical constructions at this link are pretty much all you need to know for standard markdown. Note that you can also insert HTML into markdown cells, and this will be rendered properly. As you are typing the contents of these cells, the results appear as text. Hitting "shift + enter" renders the text in the formatting you specify.

You can specify a cell as being a markdown cell in the Jupyter tool bar, or by hitting "esc, m" in the cell. Again, you have to hit enter after using the quick keys to bring the cell into edit mode.

In addition to HTML, some $\LaTeX$ expressions may be inserted into markdown cells. $\LaTeX$ (pronounced "lay-tech") is a document markup language that uses the $\TeX$ typesetting software. It is particularly well-suited for beautiful typesetting of mathematical expressions. In Jupyter notebooks, the $\LaTeX$ mathematical input is rendered using software called MathJax. This is usually run off of a remote server, so if you are not connected to the internet, your equations may not be rendered. You will use $\LaTeX$ extensively in preparation of your assignments. There are plenty of resources on the internet for getting started with $\LaTeX$, but you will only need a tiny subset of its functionality in your assignments, and Tutorial 0c, plus cheat sheets you may find by Google (such as this one) are useful.