Homework 4¶
Problem 1¶
Construct the following numpy arrays. For full credit, you should not use the code pattern np.array(my_list) in any of your answers, nor should you use for-loops or any other solution that involves creating or modifying the array one entry at a time.
Please make sure to show your result so that the grader can evaluate it!
In [1]:
# run this block to import numpy
import numpy as np
(A).¶
array([[0, 1],
[2, 3],
[4, 5],
[6, 7],
[8, 9]])
Your Solution¶
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(B).¶
array([[0, 1, 2, 3, 4],
[5, 6, 7, 8, 9]])
Your Solution¶
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(C).¶
array([0. , 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1. ])
Your Solution¶
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(D).¶
array([ 1,1,1,1,1, 10, 10, 10, 10, 10])
Your Solution¶
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(E).¶
array([[30,1,2, 30,4],
[ 5, 30,7,8, 30]])
Your Solution¶
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(F).¶
array([[ 5,6,7,8,9],
[10, 11, 12, 13, 14]])
Your Solution¶
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(G).¶
array([[ 1,3],
[ 5,7],
[ 9, 11],
[13, 15],
[17, 19]])
Your Solution¶
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Problem 2¶
Consider the following array:
A = np.arange(12).reshape(4, 3)
A
array([[ 0,1,2],
[ 3,4,5],
[ 6,7,8],
[ 9, 10, 11]])
Construct the specified arrays by indexing A. For example, if asked for array([0, 1, 2]), a correct answer would be A[0,:]. Each of the parts below may be performed in a single line.
In [ ]:
# run this block to initialize A
A = np.arange(12).reshape(4, 3)
A
(A).¶
array([6, 7, 8])
Your Solution¶
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(B).¶
array([5, 8])
Your Solution¶
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(C).¶
array([ 6,7,8,9, 10, 11])
Your Solution¶
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(D).¶
array([ 0,2,4,6,8, 10])
Your Solution¶
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(E).¶
array([ 0,1,2,3,4,5, 11])
Your Solution¶
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(F).¶
array([ 4, 11])
Your Solution¶
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Problem 3¶
In this problem, we will use numpy array indexing to repair an image that has been artificially separated into multiple pieces. The following code will retrieve two images, one of which has been cutout from the other. Your job is to piece them back together again.
You’ve already seen urllib.request.urlopen() to retrieve online data. We’ll play with mpimg.imread() a bit more in the future. The short story is that it produces a representation of an image as a numpy array of RGB values; see below. You’ll see imshow() a lot more in the near future.
In [5]:
import matplotlib.image as mpimg
from matplotlib import pyplot as plt
import urllib
f= urllib.request.urlopen(“https://philchodrow.github.io/PIC16A/homework/main.jpg”)
main = mpimg.imread(f, format = “jpg”).copy()
f = urllib.request.urlopen(“https://philchodrow.github.io/PIC16A/homework/cutout.jpg”)
cutout= mpimg.imread(f, format = “jpg”).copy()
fig, ax = plt.subplots(1, 2)
ax[0].imshow(main)
ax[1].imshow(cutout)
Out[5]:

The images are stored as two np.arrays main and cutout. Inspect each one. You’ll observe that each is a 3-dimensional np.array of shape (height, width, 3). The 3 in this case indicates that the color of each pixel is encoded as an RGB (Red-Blue-Green) value. Each pixel has one RGB value, each of which has three elements.
Use array indexing to fix the image. The result should be that array main also contains the data for the face. This can be done just a few lines of carefully-crafted numpy. Once you’re done, visualize the result by running the indicated code block.
The black region in main starts at row 0, and column 50. You can learn more about its shape by inspecting the shape of cutout.
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# your solution
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# run this block to check your solution
plt.imshow(main)
Problem 4¶
(A).¶
Read these notes from the Python Data Science Handbook on array broadcasting. Broadcasting refers to the automatic expansion of one or more arrays to make a computation “make sense.” Here’s a simple example:
a = np.array([0, 1, 2])
b = np.ones((3, 3))
a.shape, b.shape
((3,), (3, 3))
a+b
array([[1., 2., 3.],
[1., 2., 3.],
[1., 2., 3.]])
What has happened here is that the first array a has been “broadcast” from a 1d array into a 2d array of size 3×3 in order to match the dimensions of b. The broadcast version of a looks like this:
array([[0., 1., 2.],
[0., 1., 2.],
[0., 1., 2.]])
Consult the notes above for many more examples of broadcasting. Pay special attention to the discussion of the rules of broadcasting.
(B).¶
Review, if needed, the unittest module for constructing automated unit tests in Python. You may wish to refer to the required reading from that week, the lecture notes, or the recorded lecture video.
(C).¶
Implement an automated test class called TestBroadcastingRules which tests the three rules of array broadcasting.
Rule 1: If the two arrays differ in their number of dimensions, the shape of the one with fewer dimensions is padded with ones on its leading (left) side.
To test this rule, write a method test_rule_1() that constructs the arrays:
a = np.ones(3)
b = np.arange(3).reshape(1, 3)
c = a + b
Then, within the method, check (a) that the shape of c has the value you would expect according to Rule 1 and (b) that the final entry of c has the value that you would expect. Note: you should use assertEqual() twice within this method.
In a docstring to this method, explain how this works. In particular, explain which of a or b is broadcasted, and what its new shape is according to Rule 1. You should also explain the value of the final entry of c.
Rule 2: If the shape of the two arrays does not match in any dimension, the array with shape equal to 1 in that dimension is stretched to match the other shape.
To test this rule, write a method test_rule_2() that constructs the following two arrays:
a = np.ones((1, 3))
b = np.arange(9).reshape(3, 3)
c = a + b
Then, within the method, check (a) that the shape of c has the value you would expect according to Rule 2 and (b) that the entry c[1,2] has the value that you would expect. You should again use assertEqual() twice within this method.
In a docstring to this method, explain how this works. In particular, explain which of a or b is broadcasted, and what its new shape is according to Rule 2. You should also explain the value of the entry c[1,2].
Rule 3: If in any dimension the sizes disagree and neither is equal to 1, an error is raised.
To test this rule, write a method test_rule_3 that constructs the arrays
a = np.ones((2, 3))
b = np.ones((3, 3))
It should then attempt to construct c = a + b. The test should pass if the Rule 3 error is raised, and fail otherwise. You will need to figure out what kind of error is raised by Rule 3 (is it a TypeError? ValueError? KeyError?). You will also need to handle the error using the assertRaises() method as demonstrated in the readings.
In a docstring to this method, explain why an error is raised according to Rule 3.
You should be able to perform the unit tests like this:
tester = TestBroadcastingRules()
tester.test_rule_1()
tester.test_rule_2()
tester.test_rule_3()
Your tests have passed if no output is printed when you run this code.
Your Solution¶
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# write your tester class here
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# run your tests
# your tests have passed if no output or errors are shown.
tester = TestBroadcastingRules()
tester.test_rule_1()
tester.test_rule_2()
tester.test_rule_3()
Problem 5¶
Recall the simple random walk. At each step, we flip a fair coin. If heads, we move “foward” one unit; if tails, we move “backward.”
(A).¶
Way back in Homework 1, you wrote some code to simulate a random walk in Python.
Start with this code, or use posted solutions for HW1. If you have since written random walk code that you prefer, you can use this instead. Regardless, take your code, modify it, and enclose it in a function rw(). This function should accept a single argument n, the length of the walk. The output should be a list giving the position of the random walker, starting with the position after the first step. For example,
rw(5)
[1, 2, 3, 2, 3]
Unlike in the HW1 problem, you should not use upper or lower bounds. The walk should always run for as long as the user-specified number of steps n.
Use your function to print out the positions of a random walk of length n = 10.
Don’t forget a helpful docstring!
In [1]:
# solution (with demonstration) here
(B).¶
Now create a function called rw2(n), where the argument n means the same thing that it did in Part A. Do so using numpy tools. Demonstrate your function as above, by creating a random walk of length 10. You can (and should) return your walk as a numpy array.
Requirements:
•No for-loops.
•This function is simple enough to be implemented as a one-liner of fewer than 80 characters, using lambda notation. Even if you choose not to use lambda notation, the body of your function definition should be no more than three lines long. Importing numpy does not count as a line.
•A docstring is required if and only if you take more than one line to define the function.
Hints:
•Check the documentation for np.random.choice().
•Discussion 9, and np.cumsum().
In [2]:
# solution (with demonstration) here
(C).¶
Use the %timeit magic macro to compare the runtime of rw() and rw2(). Test how each function does in computing a random walk of length n = 10000.
In [3]:
# solution (with demonstration) here
(D).¶
Write a few sentences in which you comment on (a) the performance of each function and (b) the ease of writing and reading each function.
Your discussion here
(E).¶
In this problem, we will perform a d-dimensional random walk. There are many ways to define such a walk. Here’s the definition we’ll use for this problem:
At each timestep, the walker takes one random step forward or backward in each of d directions.
For example, in a two-dimensional walk on a grid, in each timestep the walker would take a step either north or south, and then another step either east or west. Another way to think about is as the walker taking a single “diagonal” step either northeast, southeast, southwest, or northwest.
Write a function called rw_d(n,d) that implements a d-dimensional random walk. n is again the number of steps that the walker should take, and d is the dimension of the walk. The output should be given as a numpy array of shape (n,d), where the kth row of the array specifies the position of the walker after k steps. For example:
P = rw_d(5, 3)
P
array([[-1, -1, -1],
[ 0, -2, -2],
[-1, -3, -3],
[-2, -2, -2],
[-1, -3, -1]])
In this example, the third row P[2,:] = [-1, -3, -3] gives the position of the walk after 3 steps.
Demonstrate your function by generating a 3d walk with 5 steps, as shown in the example above.
All the same requirements and hints from Part B apply in this problem as well. It should be possible to solve this problem by making only a few small modifications to your solution from Part B. If you are finding that this is not possible, you may want to either (a) read the documentation for the relevant numpy functions more closely or (b) reconsider your Part B approach.
In [4]:
# solution (with demonstration) here
(F).¶
In a few sentences, describe how you would have solved Part E without numpy tools. Take a guess as to how many lines it would have taken you to define the appropriate function. Based on your findings in Parts C and D, how would you expect its performance to compare to your numpy-based function from Part E? Which approach would your recommend?
Note: while I obviously prefer the numpy approach, it is reasonable and valid to prefer the “vanilla” way instead. Either way, you should be ready to justify your preference on the basis of writeability, readability, and performance.
Your discussion here
(G).¶
Once you’ve implemented rw_d(), you can run the following code to generate a large random walk and visualize it.
from matplotlib import pyplot as plt
W = rw_d(20000, 2)
plt.plot(W[:,0], W[:,1])
You may be interested in looking at several other visualizations of multidimensional random walks on Wikipedia. Your result in this part will not look exactly the same, but should look qualitatively fairly similar.
You only need to show one plot. If you like, you might enjoy playing around with the plot settings. While ax.plot() is the normal method to use here, ax.scatter() with partially transparent points can also produce some intriguing images.
In [ ]:
# solution (with demonstration) here
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