We are studying inflammation in patients who have been given a new treatment for arthritis, and need to analyze the first dozen data sets. The data sets are stored in comma-separated values (CSV) format: each row holds information for a single patient, and the columns represent successive days. The first few rows of our first file look like this:
0,0,1,3,1,2,4,7,8,3,3,3,10,5,7,4,7,7,12,18,6,13,11,11,7,7,4,6,8,8,4,4,5,7,3,4,2,3,0,0 0,1,2,1,2,1,3,2,2,6,10,11,5,9,4,4,7,16,8,6,18,4,12,5,12,7,11,5,11,3,3,5,4,4,5,5,1,1,0,1 0,1,1,3,3,2,6,2,5,9,5,7,4,5,4,15,5,11,9,10,19,14,12,17,7,12,11,7,4,2,10,5,4,2,2,3,2,2,1,1 0,0,2,0,4,2,2,1,6,7,10,7,9,13,8,8,15,10,10,7,17,4,4,7,6,15,6,4,9,11,3,5,6,3,3,4,2,3,2,1 0,1,1,3,3,1,3,5,2,4,4,7,6,5,3,10,8,10,6,17,9,14,9,7,13,9,12,6,7,7,9,6,3,2,2,4,2,0,1,1
We want to:
To do all that, we'll have to learn a little bit about programming.
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. We can load NumPy using:
Importing a library is like getting a piece of lab equipment out of a storage locker and setting it up on the bench. Once it's done, we can ask the library to read our data file for us:
array([[ 0., 0., 1., ..., 3., 0., 0.], [ 0., 1., 2., ..., 1., 0., 1.], [ 0., 1., 1., ..., 2., 1., 1.], ..., [ 0., 1., 1., ..., 1., 1., 1.], [ 0., 0., 0., ..., 0., 2., 0.], [ 0., 0., 1., ..., 1., 1., 0.]])
numpy.loadtxt(...) is a function call
that asks Python to run the function
loadtxt that belongs to the
This dotted notation is used everywhere in Python
to refer to the parts of things as
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.
only a few rows and columns are shown
... to omit elements when displaying big arrays).
To save space,
Python displays numbers as
1. instead of
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,
subject_id. Python's variables must begin with a letter.
We can create a new variable simply 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 a value to a variable:
weight_kg = 55
Once a variable has a value, we can print it:
and do arithmetic with it:
print 'weight in pounds:', 2.2 * weight_kg
weight in pounds: 121.0
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
weight in kilograms is now: 57.5
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
weight in kilograms: 57.5 and in pounds: 126.5
and then change
weight_kg = 100.0 print 'weight in kilograms is now:', weight_kg, 'and weight in pounds is still:', weight_lb
weight in kilograms is now: 100.0 and weight in pounds is still: 126.5
weight_lb doesn't "remember" where its value came from,
it isn't automatically updated when
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='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:
[[ 0. 0. 1. ..., 3. 0. 0.] [ 0. 1. 2. ..., 1. 0. 1.] [ 0. 1. 1. ..., 2. 1. 1.] ..., [ 0. 1. 1. ..., 1. 1. 1.] [ 0. 0. 0. ..., 0. 2. 0.] [ 0. 0. 1. ..., 1. 1. 0.]]
Draw diagrams showing what variables refer to what values after each statement in the following program:
mass = 47.5 age = 122 mass = mass * 2.0 age = age - 20
What does the following program print out?
first, second = 'Grace', 'Hopper' third, fourth = second, first print third, fourth
Now that our data is in memory,
we can start doing things with it.
let's ask what type of thing
data refers to:
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:
This tells us that
data has 60 rows and 40 columns.
data.shape is a member of
a value that is stored as part of a larger value.
We use the same dotted notation for the members of values
that we use for the functions in libraries
because they have the same part-and-whole relationship.
If we want to get a single value from the matrix, we must provide an index in square brackets, just as we do in math:
print 'first value in data:', data[0, 0]
first value in data: 0.0
print 'middle value in data:', data[30, 20]
middle value in data: 13.0
data[30, 20] may not surprise you,
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.
we can select the first ten days (columns) of values
for the first four (rows) patients like this:
print data[0:4, 0:10]
[[ 0. 0. 1. 3. 1. 2. 4. 7. 8. 3.] [ 0. 1. 2. 1. 2. 1. 3. 2. 2. 6.] [ 0. 1. 1. 3. 3. 2. 6. 2. 5. 9.] [ 0. 0. 2. 0. 4. 2. 2. 1. 6. 7.]]
"Start at index 0 and go up to, but not including, index 4."
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]
[[ 0. 0. 1. 2. 2. 4. 2. 1. 6. 4.] [ 0. 0. 2. 2. 4. 2. 2. 5. 5. 8.] [ 0. 0. 1. 2. 3. 1. 2. 3. 5. 3.] [ 0. 0. 0. 3. 1. 5. 6. 5. 5. 8.] [ 0. 1. 1. 2. 1. 3. 5. 3. 5. 8.]]
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
small is: [[ 2. 3. 0. 0.] [ 1. 1. 0. 1.] [ 2. 2. 1. 1.]]
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
print 'original:' print data[:3, 36:] print 'doubledata:' print doubledata[:3, 36:]
original: [[ 2. 3. 0. 0.] [ 1. 1. 0. 1.] [ 2. 2. 1. 1.]] doubledata: [[ 4. 6. 0. 0.] [ 2. 2. 0. 2.] [ 4. 4. 2. 2.]]
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 size and 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
and so on for all other elements of the arrays.
print 'tripledata:' print tripledata[:3, 36:]
tripledata: [[ 6. 9. 0. 0.] [ 3. 3. 0. 3.] [ 6. 6. 3. 3.]]
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
mean is a method of the array,
a function that belongs to it
in the same way that the member
If variables are nouns, methods are verbs:
they are what the thing in question knows how to do.
This is why
data.shape doesn't need to be called
(it's just a thing)
(it's an action).
It is also why we need empty parentheses for
even when we're not passing in any parameters,
parentheses are how we tell Python to go and do something for us.
NumPy arrays have lots of useful methods:
print 'maximum inflammation:', data.max() print 'minimum inflammation:', data.min() print 'standard deviation:', data.std()
maximum inflammation: 20.0 minimum inflammation: 0.0 standard deviation: 4.61383319712
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 select the data we want to create a new temporary array, 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()
maximum inflammation for patient 0: 18.0
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()
maximum inflammation for patient 2: 19.0
What if we need the maximum inflammation for all patients, or the average for each day? 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, we get:
[ 0. 0.45 1.11666667 1.75 2.43333333 3.15 3.8 3.88333333 5.23333333 5.51666667 5.95 5.9 8.35 7.73333333 8.36666667 9.5 9.58333333 10.63333333 11.56666667 12.35 13.25 11.96666667 11.03333333 10.16666667 10. 8.66666667 9.15 7.25 7.33333333 6.58333333 6.06666667 5.95 5.11666667 3.6 3.3 3.56666667 2.48333333 1.5 1.13333333 0.56666667]
As a quick check, we can ask this array what its shape is:
(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, we get:
[ 5.45 5.425 6.1 5.9 5.55 6.225 5.975 6.65 6.625 6.525 6.775 5.8 6.225 5.75 5.225 6.3 6.55 5.7 5.85 6.55 5.775 5.825 6.175 6.1 5.8 6.425 6.05 6.025 6.175 6.55 6.175 6.35 6.725 6.125 7.075 5.725 5.925 6.15 6.075 5.75 5.975 5.725 6.3 5.9 6.75 5.925 7.225 6.15 5.95 6.275 5.7 6.1 6.825 5.975 6.725 5.7 6.25 6.4 7.05 5.9 ]
which is the average inflammation per patient across all days.
element = 'oxygen' print 'first three characters:', element[0:3] print 'last three characters:', element[3:6]
first three characters: oxy last three characters: gen
What is the value of
Given those answers,
element[3:3] produces an empty string,
i.e., a string that contains no characters.
data holds our array of patient data,
data[3:3, 4:4] produce?
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 here. While there is no "official" plotting library, this package is the de facto standard.
let's tell the IPython Notebook that we want our plots displayed inline,
rather than in a separate viewing window:
% at the start of the line signals that this is a command for the notebook,
rather than a statement in Python.
we will import the
pyplot module from
and use two of its functions to create and display a heat map of our data:
from matplotlib import pyplot pyplot.imshow(data) pyplot.show()
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. Let's take a look at the average inflammation over time:
ave_inflammation = data.mean(axis=0) pyplot.plot(ave_inflammation) pyplot.show()
we have put the average per day across all patients in the variable
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:
print 'maximum inflammation per day' pyplot.plot(data.max(axis=0)) pyplot.show() print 'minimum inflammation per day' pyplot.plot(data.min(axis=0)) pyplot.show()
maximum inflammation per day
minimum inflammation per day
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.
Why do all of our plots stop just short of the upper end of our graph? Why are the vertical lines in our plot of the minimum inflammation per day not perfectly vertical?
Create a plot showing the standard deviation of the inflammation data for each day across all patients.
It's very common to create an alias for a library when importing it
in order to reduce the amount of typing we have to do.
Here are our three plots side by side using aliases for
import numpy as np from matplotlib import pyplot as plt data = np.loadtxt(fname='inflammation-01.csv', delimiter=',') plt.figure(figsize=(10.0, 3.0)) plt.subplot(1, 3, 1) plt.ylabel('average') plt.plot(data.mean(0)) plt.subplot(1, 3, 2) plt.ylabel('max') plt.plot(data.max(0)) plt.subplot(1, 3, 3) plt.ylabel('min') plt.plot(data.min(0)) plt.tight_layout() plt.show()
The first two lines re-load our libraries as
which are the aliases most Python programmers use.
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.
if we leave out that call to
the graphs will actually be squeezed together more closely.)
numpylibrary to work with arrays in Python.
variable = valueto assign a value to a variable in order to record it in memory.
print somethingto display the value of
array.shapegives the shape of an array.
array[x, y]to select a single element from an array.
low:highto specify a slice that includes the indices from
# some kind of explanationto add comments to programs.
array.min()to calculate simple statistics.
array.mean(axis=1)to calculate statistics across the specified axis.
matplotlibfor creating simple visualizations.
Our work so far has convinced us that something's wrong with our first data file. We would like to check the other 11 the same way, but typing in the same commands repeatedly is tedious and error-prone. Since computers don't get bored (that we know of), we should create a way to do a complete analysis with a single command, and then figure out how to repeat that step once for each file. These operations are the subjects of the next two lessons.