Description
Problem 0: Get Connected (0 points, but it will make the course easier!)
Please visit our class forum on Piazza: piazza.com/uci/fall2020/cs178. Piazza will be the place to post your
questions and discussions, rather than by email to the instructor or TA. Often, other students have the same or
similar questions, and will be helped by seeing the online discussion.
Problem 1: Python & Data Exploration (20 points)
In this problem, we will compute some basic statistics and create visualizations of an example data set. First,
download the zip file for Homework 1, which contains some course code (the mltools directory) and a dataset of
New York area real estate sales, “nyc_housing”. Load the data into Python:
1 import numpy as np
2 import matplotlib.pyplot as plt
3
4 nych = np.genfromtxt(“data/nyc_housing.txt”,delimiter=None) # load the text file
5 Y = nych[:,-1] # target value (NYC borough) is the last column
6 X = nych[:,0:-1] # features are the other columns
These data are from the “NYC Open Data” initiative, and consist of three real-valued features and a class value Y
representing in which of three boroughs the house or apartment was located (Manhattan, the Bronx, or Staten
Island).
1. Use X.shape to get the number of features and the number of data points. Report both numbers, mentioning
which number is which. (5 points)
2. For each feature, plot a histogram ( plt.hist ) of the data values. (5 points)
3. Compute the mean & standard deviation of the data points for each feature ( np.mean , np.std ). (5 points)
4. For each pair of features (1,2), (1,3), and (2,3), plot a scatterplot (see plt.plot or plt.scatter ) of the
feature values, colored according to their target value (class). (For example, plot all data points with y = 0
as blue, y = 1 as green, and y = 2 as red.) (5 points)
1For example, by doing a Print Preview in Chrome and printing it to a PDF. Please also remember to check the resulting PDF before submitting.
Homework 1
CS 178: Machine Learning & Data Mining
Problem 2: k-nearest-neighbor predictions (25 points)
In this problem, you will continue to use the NYC Housing data and create a k-nearest-neighbor (kNN) classifier
using the provided knnClassify python class. While completing this problem, please explore the implementation
to become familiar with how it works.
First, we will shuffle and split the data into training and validation subsets:
1 nych = np.genfromtxt(“data/nyc_housing.txt”,delimiter=None) # load the data
2 Y = nych[:,-1]
3 X = nych[:,0:-1]
4 # Note: indexing with “:” indicates all values (in this case, all rows);
5 # indexing with a value (“0”, “1”, “-1”, etc.) extracts only that value (here, columns);
6 # indexing rows/columns with a range (“1:-1”) extracts any row/column in that range.
7
8 import mltools as ml
9 # We’ll use some data manipulation routines in the provided class code
10 # Make sure the “mltools” directory is in a directory on your Python path, e.g.,
11 # export PYTHONPATH=$\$${PYTHONPATH}:/path/to/parent/dir
12 # or add it to your path inside Python:
13 # import sys
14 # sys.path.append(‘/path/to/parent/dir/’);
15
16 np.random.seed(0) # set the random number seed
17 X,Y = ml.shuffleData(X,Y); # shuffle data randomly
18 # (This is a good idea in case your data are ordered in some systematic way.)
19
20 Xtr,Xva,Ytr,Yva = ml.splitData(X,Y, 0.75); # split data into 75/25 train/validation
Make sure to set the random number seed to 0 before calling shuffleData as in the example above (and in general,
for every assignment). This ensures consistent behavior each time the code is run.
Learner Objects Our learners (the parameterized functions that do the prediction) will be defined as python
objects, derived from either an abstract classifier or abstract regressor class. The abstract base classes have a few
useful functions, such as computing error rates or other measures of quality. More importantly, the learners will all
follow a generic behavioral pattern, allowing us to train the function on one data set (i.e., set the parameters of
the model to perform well on those data), and then make predictions on another data set.
You can now build and train a kNN classifier on Xtr,Ytr and make predictions on some data Xva with it:
1 knn = ml.knn.knnClassify() # create the object and train it
2 knn.train(Xtr, Ytr, K) # where K is an integer, e.g. 1 for nearest neighbor prediction
3 YvaHat = knn.predict(Xva) # get estimates of y for each data point in Xva
4
5 # Alternatively, the constructor provides a shortcut to “train”:
6 knn = ml.knn.knnClassify( Xtr, Ytr, K );
7 YvaHat = predict( knn, Xva );
If your data are 2D, you can visualize the data set and a classifier’s decision regions using the function
1 ml.plotClassify2D( knn, Xtr, Ytr ); # make 2D classification plot with data (Xtr,Ytr)
This function plots the training data and colored points as per their labels, then calls knn ’s predict function on a
densely spaced grid of points in the 2D space, and uses this to produce the background color. Calling the function
with knn=None will plot only the data.
1. Modify the code listed above to use only the first two features of X (e.g., let X be only the first two columns
of nych , instead of the first three), and visualize (plot) the classification boundary for varying values of
K = [1, 5, 10, 50] using plotClassify2D . (10 points)
2. Again using only the first two features, compute the error rate (number of misclassifications) on both the
training and validation data as a function of K = [1, 2, 5, 10, 50, 100, 200]. You can do this most easily with
a for-loop:
Homework 1
CS 178: Machine Learning & Data Mining
1 K=[1,2,5,10,50,100,200];
2 errTrain = [None]*len(K) # (preallocate storage for training error)
3 for i,k in enumerate(K):
4 learner = ml.knn.knnClassify(… # TODO: complete code to train model
5 Yhat = learner.predict(… # TODO: predict results on training data
6 errTrain[i] = … # TODO: count what fraction of predictions are wrong
7 #TODO: repeat prediction / error evaluation for validation data
8
9 plt.semilogx(… #TODO: average and plot results on semi-log scale
Plot the resulting error rate functions using a semi-log plot ( semilogx ), with training error in red and
validation error in green. Based on these plots, what value of K would you recommend? (10 points)
3. Create the same error rate plots as the previous part, but with all the features in the dataset. Are the plots
very different? Is your recommendation for the best K different? (5 points)
Problem 3: Naïve Bayes Classifiers (35 points)
In order to reduce my email load, I decide to implement a machine learning algorithm to decide whether or
not I should read an email, or simply file it away instead. To train my model, I obtain the following data set of
binary-valued features about each email, including whether I know the author or not, whether the email is long or
short, and whether it has any of several key words, along with my final decision about whether to read it ( y = +1
for “read”, y = −1 for “discard”).
x1 x2 x3 x4 x5 y
know author? is long? has ‘research’ has ‘grade’ has ‘lottery’ ⇒ read?
0 0 1 1 0 -1
1 1 0 1 0 -1
0 1 1 1 1 -1
1 1 1 1 0 -1
0 1 0 0 0 -1
1 0 1 1 1 1
0 0 1 0 0 1
1 0 0 0 0 1
1 0 1 1 0 1
1 1 1 1 1 -1
I decide to try a naïve Bayes classifier to make my decisions and compute my uncertainty. In the case of any ties
where both classes have equal probability, we will prefer to predict class +1.
1. Compute all the probabilities necessary for a naïve Bayes classifier, i.e., the class probability p( y) and all the
individual feature probabilities p(xi
| y), for each class y and feature xi
. (7 points)
2. Which class would be predicted for x = (0 0 0 0 0)? What about for x = (1 1 0 1 0)? (7 points)
3. Compute the posterior probability that y = +1 given the observation x = (0 0 0 0 0). Also compute the
posterior probability that y = +1 given the observation x = (1 1 0 1 0). (7 points)
4. Why should we probably not use a “joint” Bayes classifier (using the joint probability of the features x, as
opposed to the conditional independencies assumed by naïve Bayes) for these data? (7 points)
5. Suppose that before we make our predictions, we lose access to my address book, so that we cannot tell
whether the email author is known. Do we need to re-train the model to classify based solely on the other
four features? If so, how? If not, what changes about how our trained parameters are used? Hint: what
parameters do I need for a naïve Bayes model over only features x2
, . . . , x5? Do I need to re-calculate any
new parameter values in our new setting? What, if anything, changes about the parameters or the way they
are used? (7 points)
Homework 1
CS 178: Machine Learning & Data Mining
Problem 4: Gaussian Bayes Classifiers (15 points)
Now, using the NYC Housing data, we will explore a classifier based on Bayes rule. Again, we’ll use only the first
two features of NYC Housing, shuffled and split in to training and validation sets as before.
1. Splitting your training data by class, compute the empirical mean vector and covariance matrix of the data
in each class. (You can use mean and cov for this.) (5 points)
2. Plot a scatterplot of the data, coloring each data point by its class, and use plotGauss2D to plot contours
on your scatterplot for each class, i.e., plot a Gaussian contour for each class using its empirical parameters,
in the same color you used for those data points. 5 points
3. Visualize the classifier and its boundaries that result from applying Bayes rule, using
1 bc = ml.bayes.gaussClassify( Xtr, Ytr );
2 ml.plotClassify2D(bc, Xtr, Ytr);
Also compute the empirical error rate (number of misclassified points) on the training and validation data.
(5 points)
Problem 5: Statement of Collaboration (5 points)
It is mandatory to include a Statement of Collaboration in each submission, that follows the guidelines below.
Include the names of everyone involved in the discussions (especially in-person ones), and what was discussed.
All students are required to follow the academic honesty guidelines posted on the course website. For
programming assignments in particular, I encourage students to organize (perhaps using Piazza) to discuss the
task descriptions, requirements, possible bugs in the support code, and the relevant technical content before they
start working on it. However, you should not discuss the specific solutions, and as a guiding principle, you are
not allowed to take anything written or drawn away from these discussions (no photographs of the blackboard,
written notes, referring to Piazza, etc.). Especially after you have started working on the assignment, try to restrict
the discussion to Piazza as much as possible, so that there is no doubt as to the extent of your collaboration.
Homework 1
