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.
- Select individual values and subsections from data.
- Perform operations on arrays of data.
This lesson presents an end-to-end scientific Python example, from analysing data (using a library), to visualisation (using a library).
We are studying inflammation in patients who have been given a new treatment for arthritis, and need to analyse the first dozen data sets of their daily inflammation. 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:
- load that data into memory,
- calculate the average inflammation per day across all patients, and
- plot the result.
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.
Let’s start by ensuring we are in the `swc-python-novice-websci/’ directory, e.g.:
$ pwd
And we should see:
/Users/nelle/swc-python-novice-websci
First, let’s go into the code subdirectory, and run the Python interpreter.
$ cd code
$ python
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 it’s done, we can ask the library to read our data file for us.
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:
data = numpy.loadtxt(fname='../data/inflammation-01.csv', delimiter=',')
This statement doesn’t produce any output because assignment doesn’t display anything.
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.
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.
Now that our data is in memory, we can start doing things with it.
If we want to check that our data has been loaded, we can print the variable’s value:
print(data)
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.]])
Let’s ask what type of thing data refers to:
print(type(data))
<type 'numpy.ndarray'>
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:
print(data.shape)
(60, 40)
This tells us that data has 60 rows and 40 columns, representing 60 patients over 40 days.
data.shape is a member of data,
i.e.,
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
The expression data[30, 20] may not surprise you,
but as with lists, data[0, 0] might.
So 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])
[[ 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.]]
The slice 0:4 means, numpy selects items between boundries [0,4] and [0,10].
See slide Slicing a List Example I.
Again, this 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 data:
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 doubledata[0,0] plus data[0,0],
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
print(data.mean())
6.14875
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.
This is why data.shape doesn’t need to be called
(it’s just a thing)
but data.mean() does
(it’s an action).
It is also why we need empty parentheses for data.mean():
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: 4.613833197118566
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 3:', data[2, :].max())
maximum inflammation for patient 3: 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 (representing the patients axis), we get:
print(data.mean(axis=0))
[ 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:
print(data.mean(axis=0).shape)
(40,)
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, we get:
print(data.mean(axis=1))
[ 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 for each patient across all days.
Thin slices
From our previous topic, the expression
element[3:3]produces an empty string, i.e., a string that contains no characters. Ifdataholds our array of patient data, what doesdata[3:3, 4:4]produce? What aboutdata[3:3, :]?