Analysing patient data

Learning Objectives

• Explain what a library is, and what libraries are used for.
• Load a Python library and use the things it contains.
• Read tabular data from a file into a program.
• Assign values to variables.
• Select individual values and subsections from data.
• Perform operations on arrays of data.
• Display simple graphs.

Words are useful, but what's more useful are the sentences and stories we build with them. Similarly, while a lot of powerful tools are built into languages like Python, even more live in the libraries they are used to build.

In order to load our inflammation data, we need to import a library called NumPy. In general you should use this library if you want to do fancy things with numbers, especially if you have matrices or arrays. We can load NumPy using:

import numpy

Importing a library is like getting a piece of lab equipment out of a storage locker and setting it up on the bench. Once you've loaded the library, we can ask the library to read our data file for us:

numpy.loadtxt(fname='data/inflammation-01.csv', delimiter=',')

The expression numpy.loadtxt(...) is a function call that asks Python to run the function loadtxt that belongs to the numpy library. This dotted notation is used everywhere in Python to refer to the parts of things as thing.component.

numpy.loadtxt has two parameters: the name of the file we want to read, and the delimiter that separates values on a line. These both need to be character strings (or strings for short), so we put them in quotes.

When we are finished typing and press Shift+Enter, the notebook runs our command. Since we haven't told it to do anything else with the function's output, the notebook displays it. In this case, that output is the data we just loaded. By default, only a few rows and columns are shown (with ... to omit elements when displaying big arrays). To save space, Python displays numbers as 1. instead of 1.0 when there's nothing interesting after the decimal point.

Our call to numpy.loadtxt read our file, but didn't save the data in memory. To do that, we need to assign the array to a variable. A variable is just a name for a value, such as x, current_temperature, or subject_id. Python's variables must begin with a letter and are case sensitive. We can create a new variable by assigning a value to it using =. As an illustration, let's step back and instead of considering a table of data, consider the simplest "collection" of data, a single value. The line below assigns the value 55 to a variable weight_kg:

weight_kg = 55

Once a variable has a value, we can print it to the screen:

print(weight_kg)

and do arithmetic with it:

print('weight in pounds:', 2.2 * weight_kg)

We can also change a variable's value by assigning it a new one:

weight_kg = 57.5
print('weight in kilograms is now:', weight_kg)

As the example above shows, we can print several things at once by separating them with commas.

If we imagine the variable as a sticky note with a name written on it, assignment is like putting the sticky note on a particular value:

This means that assigning a value to one variable does not change the values of other variables. For example, let's store the subject's weight in pounds in a variable:

weight_lb = 2.2 * weight_kg
print('weight in kilograms:', weight_kg, 'and in pounds:', weight_lb)

and then change weight_kg:

weight_kg = 100.0
print('weight in kilograms is now:', weight_kg, 'and weight in pounds is still:', weight_lb)

Since weight_lb doesn't "remember" where its value came from, it isn't automatically updated when weight_kg changes. This is different from the way spreadsheets work.

Just as we can assign a single value to a variable, we can also assign an array of values to a variable using the same syntax. Let's re-run numpy.loadtxt and save its result:

data = numpy.loadtxt(fname='data/inflammation-01.csv', delimiter=',')

This statement doesn't produce any output because assignment doesn't display anything. If we want to check that our data has been loaded, we can print the variable's value:

data

Now that our data is in memory, we can start doing things with it. First, let's ask what type of thing data refers to:

type(data)

The output tells us that data currently refers to an N-dimensional array created by the NumPy library. We can see what its shape is like this:

data.shape

This tells us that data has 60 rows and 40 columns. When we created the variable data to store our arthritis data, we didn't just create the array, we also created information about the array, called members or attributes. This extra information describes data in the same way an adjective describes a noun. data.shape is an attribute of data which described the dimensions of data. We use the same dotted notation for the attributes of variables that we use for the functions in libraries because they have the same part-and-whole relationship.

If we want to get a single number from the array, we must provide an index in square brackets, just as we do in math:

print('first value in data:', data[0, 0])

print('middle value in data:', data[30, 20])

The expression data[30, 20] may not surprise you, but data[0, 0] might. Programming languages like Fortran and MATLAB start counting at 1, because that's what human beings have done for thousands of years. Languages in the C family (including C++, Java, Perl, and Python) count from 0 because that's simpler for computers to do. As a result, if we have an M×N array in Python, its indices go from 0 to M-1 on the first axis and 0 to N-1 on the second. It takes a bit of getting used to, but one way to remember the rule is that the index is how many steps we have to take from the start to get the item we want.

In the Corner

What may also surprise you is that when Python displays an array, it shows the element with index [0, 0] in the upper left corner rather than the lower left. This is consistent with the way mathematicians draw matrices, but different from the Cartesian coordinates. The indices are (row, column) instead of (column, row) for the same reason, which can be confusing when plotting data.

An index like [30, 20] selects a single element of an array, but we can select whole sections as well. For example, we can select the first ten days (columns) of values for the first four (rows) patients like this:

print(data[0:4, 0:10])

The slice 0:4 means, "Start at index 0 and go up to, but not including, index 4." Again, the up-to-but-not-including takes a bit of getting used to, but the rule is that the difference between the upper and lower bounds is the number of values in the slice.

We don't have to start slices at 0:

print(data[5:10, 0:10])

We also don't have to include the upper and lower bound on the slice. If we don't include the lower bound, Python uses 0 by default; if we don't include the upper, the slice runs to the end of the axis, and if we don't include either (i.e., if we just use ':' on its own), the slice includes everything:

small = data[:3, 36:]
print('small is:')
print(small)

Arrays also know how to perform common mathematical operations on their values. The simplest operations with data are arithmetic: add, subtract, multiply, and divide. When you do such operations on arrays, the operation is done on each individual element of the array. Thus:

doubledata = data * 2.0

will create a new array doubledata whose elements have the value of two times the value of the corresponding elements in data:

print('original:')
print(data[:3, 36:])
print('doubledata:')
print(doubledata[:3, 36:])

If, instead of taking an array and doing arithmetic with a single value (as above) you did the arithmetic operation with another array of the same shape, the operation will be done on corresponding elements of the two arrays. Thus:

tripledata = doubledata + data

will give you an array where tripledata[0,0] will equal doubledata[0,0] plus data[0,0], and so on for all other elements of the arrays.

print('tripledata:')
print(tripledata[:3, 36:])

Often, we want to do more than add, subtract, multiply, and divide values of data. Arrays also know how to do more complex operations on their values. If we want to find the average inflammation for all patients on all days, for example, we can just ask the array for its mean value

print(data.mean())

mean is a method of the array, i.e., a function that belongs to it in the same way that the member shape does. If variables are nouns, methods are verbs: they are what the thing in question knows how to do. We need empty parentheses for data.mean(), even when we're not passing in any parameters, to tell Python to go and do something for us. data.shape doesn't need () because it is just a description but data.mean() requires the () because it is an action.

NumPy arrays have lots of useful methods:

print('maximum inflammation:', data.max())
print('minimum inflammation:', data.min())
print('standard deviation:', data.std())

When analyzing data, though, we often want to look at partial statistics, such as the maximum value per patient or the average value per day. One way to do this is to create a new temporary array of the data we want, then ask it to do the calculation:

patient_0 = data[0, :] # 0 on the first axis, everything on the second
print('maximum inflammation for patient 0:', patient_0.max())

We don't actually need to store the row in a variable of its own. Instead, we can combine the selection and the method call:

print('maximum inflammation for patient 2:', data[2, :].max())

What if we need the maximum inflammation for all patients (as in the next diagram on the left), or the average for each day (as in the diagram on the right)? As the diagram below shows, we want to perform the operation across an axis:

To support this, most array methods allow us to specify the axis we want to work on. If we ask for the average across axis 0 (rows in our 2D example), we get:

data.mean(axis=0)

As a quick check, we can ask this array what its shape is:

data.mean(axis=0).shape

The expression (40,) tells us we have an N×1 vector, so this is the average inflammation per day for all patients. If we average across axis 1 (columns in our 2D example), we get:

data.mean(axis=1)

which is the average inflammation per patient across all days.

The mathematician Richard Hamming once said, "The purpose of computing is insight, not numbers," and the best way to develop insight is often to visualize data. Visualization deserves an entire lecture (or course) of its own, but we can explore a few features of Python's matplotlib library here. While there is no "official" plotting library, this package is the de facto standard. First, we will import the pyplot module from matplotlib and use two of its functions to create and display a heat map of our data:

%matplotlib inline
import matplotlib.pyplot
matplotlib.pyplot.imshow(data)

Blue regions in this heat map are low values, while red shows high values. As we can see, inflammation rises and falls over a 40-day period.

Some IPython magic

If you're using an IPython / Jupyter notebook, you'll need to execute the following command in order for your matplotlib images to appear in the notebook when show() is called:

{.python}
% matplotlib inline

The % indicates an IPython magic function - a function that is only valid within the notebook environment. Note that you only have to execute this function once per notebook.

Let's take a look at the average inflammation over time:

ave_inflammation = data.mean(axis=0)
matplotlib.pyplot.plot(ave_inflammation)

Here, we have put the average per day across all patients in the variable ave_inflammation, then asked matplotlib.pyplot to create and display a line graph of those values. The result is roughly a linear rise and fall, which is suspicious: based on other studies, we expect a sharper rise and slower fall. Let's have a look at two other statistics:

matplotlib.pyplot.plot(data.max(axis=0))

matplotlib.pyplot.plot(data.min(axis=0))

The maximum value rises and falls perfectly smoothly, while the minimum seems to be a step function. Neither result seems particularly likely, so either there's a mistake in our calculations or something is wrong with our data.

You can group similar plots in a single figure using subplots. This script below uses a number of new commands. The function matplotlib.pyplot.figure() creates a space into which we will place all of our plots. The parameter figsize tells Python how big to make this space. Each subplot is placed into the figure using the subplot command. The subplot command takes 3 parameters. The first denotes how many total rows of subplots there are, the second parameter refers to the total number of subplot columns, and the final parameters denotes which subplot your variable is referencing. Each subplot is stored in a different variable (axes1, axes2, axes3). Once a subplot is created, the axes are can be titled using the set_xlabel() command (or set_ylabel()). Here are our three plots side by side:

import numpy
import matplotlib.pyplot

data = numpy.loadtxt(fname='data/inflammation-01.csv', delimiter=',')

fig = matplotlib.pyplot.figure(figsize=(10.0, 3.0))

axes1 = fig.add_subplot(1, 3, 1)
axes2 = fig.add_subplot(1, 3, 2)
axes3 = fig.add_subplot(1, 3, 3)

axes1.set_ylabel('average')
axes1.plot(data.mean(axis=0))

axes2.set_ylabel('max')
axes2.plot(data.max(axis=0))

axes3.set_ylabel('min')
axes3.plot(data.min(axis=0))

fig.tight_layout()

matplotlib.pyplot.show(fig)

The call to loadtxt reads our data, and the rest of the program tells the plotting library how large we want the figure to be, that we're creating three sub-plots, what to draw for each one, and that we want a tight layout. (Perversely, if we leave out that call to fig.tight_layout(), the graphs will actually be squeezed together more closely.)

Scientists dislike typing

We will always use the syntax import numpy to import NumPy. However, in order to save typing, it is often suggested to make a shortcut like so: import numpy as np. If you ever see Python code online using a NumPy function with np (for example, np.loadtxt(...)), it's because they've used this shortcut.

Check your understanding

Draw diagrams showing what variables refer to what values after each statement in the following program:

{.python}
mass = 47.5
age = 122
mass = mass * 2.0
age = age - 20

Sorting out references

What does the following program print out?

{.python}
first, second = 'Grace', 'Hopper'
third, fourth = second, first
print(third, fourth)

Slicing strings

A section of an array is called a slice. We can take slices of character strings as well:

{.python}
element = 'oxygen'
print 'first three characters:', element[0:3]
print 'last three characters:', element[3:6]

{.output}
first three characters: oxy
last three characters: gen

What is the value of element[:4]? What about element[4:]? Or element[:]?

What is element[-1]? What is element[-2]? Given those answers, explain what element[1:-1] does.

Thin slices

The expression element[3:3] produces an empty string, i.e., a string that contains no characters. If data holds our array of patient data, what does data[3:3, 4:4] produce? What about data[3:3, :]?

Check your understanding: plot scaling

Why do all of our plots stop just short of the upper end of our graph?

Check your understanding: drawing straight lines

Why are the vertical lines in our plot of the minimum inflammation per day not perfectly vertical?

Make your own plot

Create a plot showing the standard deviation (numpy.std) of the inflammation data for each day across all patients.

Moving plots around

Modify the program to display the three plots on top of one another instead of side by side.