Analyzing Patient Data

Overview

Teaching: 40 min
Exercises: 20 min

Questions

• How can I process tabular data files in Python?

Objectives

• Explain what a library is and what libraries are used for.

• Import a Python library and use the functions it contains.

• Read tabular data from a file into a program.

• Select individual values and subsections from data.

• Perform operations on arrays of data.

Words are useful, but what’s more useful are the sentences and stories we build with them. Similarly, while a lot of powerful, general tools are built into Python, specialized tools built up from these basic units live in libraries that can be called upon when needed.

Loading data into Python

• load data using NumPy
  • numerical python
  • use this library for anything to do with lots of numbers, especially matrices/arrays
• import via:

import numpy

• import makes library available in our Python interpreter
• importing too many things can slow Python down
  • only import what you really need
• convention: imports at the top of the file, like a table of contents

Once we’ve imported the library, we can ask the library to read our data file for us:

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

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
  • asks Python to run the function loadtxt from the numpy library
• dot notation:
  • access properties and functions of an object
  • e.g. person.name or telephone.call()
• numpy.loadtxt has two parameters:
  • filename we want to read
  • delimiter that separates the values
  • both need to be strings -> put in quotes
• if we don’t assign the function’s output to a variable name, the notebook will just display it
• output is shortened and missing values indicated by ...
• trailing zeros removed -> 1.0 becomes 1.
• to keep the data in memory and work with it, assign to var

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

• no output, because we assigned to variable
• check data by printing:

print(data)

[[ 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.]]

• now that the data are in memory, we can manipulate them.
• first: find out data type:

print(type(data))

<class 'numpy.ndarray'>

• n-dimensional array, provided by numpy
• These data correspond to arthritis patients’ inflammation.
• rows are the individual patients
• columns are their daily inflammation measurements

Data Type

find out type of thing inside numpy array with:

print(data.dtype)

float64

This tells us that the NumPy array’s elements are floating-point numbers.

• data is a matrix, we can look at the shape with:

print(data.shape)

(60, 40)

• 60 rows, 40 columns
• shape is an attribute of our data, that was created when we loaded the data

• to access an attribute, same dot notation as before

• to get a single number, provide an index
• notation: square brackets
• two dimensional data, therefore two indices

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] accesses the element at row 30, column 20
• data[0, 0] might be surprising
  • Fortran, matlab, r start at 1 -> human way
  • Python starts counting at 0 -> closer to the machine way
• MxN array
  • indices 0 to M-1 in first axis
  • 0 to N-1 in second axis
• way to remember:
  • the index is how many steps we have to take from the start to get the item we want
• another surprising thing: 0,0 is in the top left
  • same as how mathematicians draw matrices
  • but different from cartesian coordinates

Slicing data

• previously: single element
• select whole sections
  • first ten days (columns)
  • first four patients (rows)

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.]]

• slice means: start at index and go up to but not including index
• slices don’t have to start 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.]]

• if we leave out upper or lower bound, Python will use 0, resp the end of the axis
• if we leave out both, i.e. only use :, slice includes everything

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

The above example selects rows 0 through 2 and columns 36 through to the end of the array.

small is:
[[ 2.  3.  0.  0.]
 [ 1.  1.  0.  1.]
 [ 2.  2.  1.  1.]]

• we can also slice strings
• this might be a bit easier to verify, so we will practise writing a slice with that, instead of the array

Slicing Strings

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

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 element[:4]? What about element[4:]? Or element[:]?

Solution

oxyg
en
oxygen

What is element[-1]? What is element[-2]?

Solution

n
e

Given those answers, explain what element[1:-1] does.

Solution

Creates a substring from index 1 up to (not including) the final index, effectively removing the first and last letters from ‘oxygen’

How can we rewrite the slice for getting the last three characters of element, so that it works even if we assign a different string to element? Test your solution with the following strings: carpentry, clone, hi.

Solution

element = 'oxygen'
print('last three characters:', element[-3:])
element = 'carpentry'
print('last three characters:', element[-3:])
element = 'clone'
print('last three characters:', element[-3:])
element = 'hi'
print('last three characters:', element[-3:])

last three characters: gen
last three characters: try
last three characters: one
last three characters: hi

Analyzing data

NumPy has several useful functions that take an array as input to perform operations on its values. If we want to find the average inflammation for all patients on all days, for example, we can ask NumPy to compute data’s mean value:

print(numpy.mean(data))

6.14875

• mean takes an array as an argument

Not All Functions Have Input

Generally, a function uses inputs to produce outputs. However, some functions produce outputs without needing any input. For example, checking the current time doesn’t require any input.

import time
print(time.ctime())

Sat Mar 26 13:07:33 2016

For functions that don’t take in any arguments, we still need parentheses (()) to tell Python to go and do something for us.

• other functions that are useful:
  • max, min, standard deviation
• We’ll also use multiple assignment, a convenient Python feature that will enable us to do this all in one line.

maxval, minval, stdval = numpy.max(data), numpy.min(data), numpy.std(data)

print('maximum inflammation:', maxval)
print('minimum inflammation:', minval)
print('standard deviation:', stdval)

Here we’ve assigned the return value from numpy.max(data) to the variable maxval, the value from numpy.min(data) to minval, and so on.

maximum inflammation: 20.0
minimum inflammation: 0.0
standard deviation: 4.61383319712

Mystery Functions in IPython

How did we know what functions NumPy has and how to use them? If you are working in IPython or in a Jupyter Notebook, there is an easy way to find out. If you type the name of something followed by a dot, then you can use tab completion (e.g. type numpy. and then press Tab) to see a list of all functions and attributes that you can use. After selecting one, you can also add a question mark (e.g. numpy.cumprod?), and IPython will return an explanation of the method! This is the same as doing help(numpy.cumprod). Similarly, if you are using the “plain vanilla” Python interpreter, you can type numpy. and press the Tab key twice for a listing of what is available. You can then use the help() function to see an explanation of the function you’re interested in, for example: help(numpy.cumprod).

When analyzing data, though, we often want to look at variations in statistical values, such as the maximum inflammation per patient or the average inflammation 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 (rows), everything on the second (columns)
print('maximum inflammation for patient 0:', numpy.max(patient_0))

maximum inflammation for patient 0: 18.0

Everything in a line of code following the ‘#’ symbol is a comment that is ignored by Python. Comments allow programmers to leave explanatory notes for other programmers or their future selves.

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

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

maximum inflammation for patient 2: 19.0

What if we need the maximum inflammation for each patient over all days (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 functionality, most array functions allow us to specify the axis we want to work on.

• the specified axis “disappears”

If we ask for the average across axis 0 (rows in our 2D example), we get:

print(numpy.mean(data, 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(numpy.mean(data, 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 (columns in our 2D example), we get:

print(numpy.mean(data, 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 per patient across all days.

INFO: 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, :]?

Solution

array([], shape=(0, 0), dtype=float64)
array([], shape=(0, 40), dtype=float64)

INFO: Stacking Arrays

Arrays can be concatenated and stacked on top of one another, using NumPy’s vstack and hstack functions for vertical and horizontal stacking, respectively.

import numpy

A = numpy.array([[1,2,3], [4,5,6], [7, 8, 9]])
print('A = ')
print(A)

B = numpy.hstack([A, A])
print('B = ')
print(B)

C = numpy.vstack([A, A])
print('C = ')
print(C)

A =
[[1 2 3]
 [4 5 6]
 [7 8 9]]
B =
[[1 2 3 1 2 3]
 [4 5 6 4 5 6]
 [7 8 9 7 8 9]]
C =
[[1 2 3]
 [4 5 6]
 [7 8 9]
 [1 2 3]
 [4 5 6]
 [7 8 9]]

Write some additional code that slices the first and last columns of A, and stacks them into a 3x2 array. Make sure to print the results to verify your solution.

Solution

A ‘gotcha’ with array indexing is that singleton dimensions are dropped by default. That means A[:, 0] is a one dimensional array, which won’t stack as desired. To preserve singleton dimensions, the index itself can be a slice or array. For example, A[:, :1] returns a two dimensional array with one singleton dimension (i.e. a column vector).

D = numpy.hstack((A[:, :1], A[:, -1:]))
print('D = ')
print(D)

D =
[[1 3]
 [4 6]
 [7 9]]

Solution

An alternative way to achieve the same result is to use Numpy’s delete function to remove the second column of A.

D = numpy.delete(A, 1, 1)
print('D = ')
print(D)

D =
[[1 3]
 [4 6]
 [7 9]]

INFO: Change In Inflammation

The patient data is longitudinal in the sense that each row represents a series of observations relating to one individual. This means that the change in inflammation over time is a meaningful concept. Let’s find out how to calculate changes in the data contained in an array with NumPy.

The numpy.diff() function takes an array and returns the differences between two successive values. Let’s use it to examine the changes each day across the first week of patient 3 from our inflammation dataset.

patient3_week1 = data[3, :7]
print(patient3_week1)

 [0. 0. 2. 0. 4. 2. 2.]

Calling numpy.diff(patient3_week1) would do the following calculations

[ 0 - 0, 2 - 0, 0 - 2, 4 - 0, 2 - 4, 2 - 2 ]

and return the 6 difference values in a new array.

numpy.diff(patient3_week1)

array([ 0.,  2., -2.,  4., -2.,  0.])

Note that the array of differences is shorter by one element (length 6).

When calling numpy.diff with a multi-dimensional array, an axis argument may be passed to the function to specify which axis to process. When applying numpy.diff to our 2D inflammation array data, which axis would we specify?

Solution

Since the row axis (0) is patients, it does not make sense to get the difference between two arbitrary patients. The column axis (1) is in days, so the difference is the change in inflammation – a meaningful concept.

numpy.diff(data, axis=1)

If the shape of an individual data file is (60, 40) (60 rows and 40 columns), what would the shape of the array be after you run the diff() function and why?

Solution

The shape will be (60, 39) because there is one fewer difference between columns than there are columns in the data.

How would you find the largest change in inflammation for each patient? Does it matter if the change in inflammation is an increase or a decrease?

Solution

By using the numpy.max() function after you apply the numpy.diff() function, you will get the largest difference between days.

numpy.max(numpy.diff(data, axis=1), axis=1)

array([  7.,  12.,  11.,  10.,  11.,  13.,  10.,   8.,  10.,  10.,   7.,
         7.,  13.,   7.,  10.,  10.,   8.,  10.,   9.,  10.,  13.,   7.,
        12.,   9.,  12.,  11.,  10.,  10.,   7.,  10.,  11.,  10.,   8.,
        11.,  12.,  10.,   9.,  10.,  13.,  10.,   7.,   7.,  10.,  13.,
        12.,   8.,   8.,  10.,  10.,   9.,   8.,  13.,  10.,   7.,  10.,
         8.,  12.,  10.,   7.,  12.])

If inflammation values decrease along an axis, then the difference from one element to the next will be negative. If you are interested in the magnitude of the change and not the direction, the numpy.absolute() function will provide that.

Notice the difference if you get the largest absolute difference between readings.

numpy.max(numpy.absolute(numpy.diff(data, axis=1)), axis=1)

array([ 12.,  14.,  11.,  13.,  11.,  13.,  10.,  12.,  10.,  10.,  10.,
        12.,  13.,  10.,  11.,  10.,  12.,  13.,   9.,  10.,  13.,   9.,
        12.,   9.,  12.,  11.,  10.,  13.,   9.,  13.,  11.,  11.,   8.,
        11.,  12.,  13.,   9.,  10.,  13.,  11.,  11.,  13.,  11.,  13.,
        13.,  10.,   9.,  10.,  10.,   9.,   9.,  13.,  10.,   9.,  10.,
        11.,  13.,  10.,  10.,  12.])

Key Points

• Import a library into a program using import libraryname.

• Use the numpy library to work with arrays in Python.

• The expression array.shape gives the shape of an array.

• Use array[x, y] to select a single element from a 2D array.

• Array indices start at 0, not 1.

• Use low:high to specify a slice that includes the indices from low to high-1.

• Use # some kind of explanation to add comments to programs.

• Use numpy.mean(array), numpy.max(array), and numpy.min(array) to calculate simple statistics.

• Use numpy.mean(array, axis=0) or numpy.mean(array, axis=1) to calculate statistics across the specified axis.