Description
Problem 0: Getting Connected
Please join our class forum on Piazza,
http://piazza.com/uci/winter2017/cs178
Please post your questions and discussion points on Piazza, rather than by email to me or the TA,
since chances are that other students have the same or similar questions, and will be helped by
seeing the discussion.
Problem 1: Python & Data Exploration
In this problem, we will explore some basic statistics and visualizations of an example data set. First,
download the zip le for Homework 1, which contains some course code (the mltools directory)
and the Fisher iris data set, and load the latter into Python:
import numpy as np
import matplotlib.pyplot as plt
iris = np.genfromtxt(“data/iris.txt”,delimiter=None) # load the text file
Y = iris[:,-1] # target value is the last column
X = iris[:,0:-1] # features are the other columns
The Iris data consist of four real-valued features used to predict which of three types of iris ower
was measured (a three-class classication problem).
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(a) Use X.shape[1] to get the number of features, and X.shape[0] to get the number of data
points.
(b) For each feature, plot a histogram (plt.hist) of the data values
(c) Compute the mean & standard deviation of the data points for each feature (np.mean, np.std)
(d) For each pair of features (1,2), (1,3), and (1,4), 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, etc.)
Problem 2: kNN predictions
In this problem, you will continue to use the Iris data and explore a KNN classier using provided
knnClassify python class. While doing the problem, please explore the implementation to become
familiar with how it works.
First, we will shue and split the data into training and validation subsets:
iris = np.genfromtxt(“data/iris.txt”,delimiter=None) # load the data
Y = iris[:,-1]
X = iris[:,0:-1]
# Note: indexing with “:” indicates all values (in this case, all rows);
# indexing with a value (“0”, “1”, “-1”, etc.) extracts only that one value (here, columns);
# indexing rows/columns with a range (“1:-1”) extracts any row/column in that range.
import mltools as ml
# We’ll use some data manipulation routines in the provided class code
# Make sure the “mltools” directory is in a directory on your Python path, e.g.,
# export PYTHONPATH=${PYTHONPATH}:/path/to/parent/dir
# or add it to your path inside Python:
# import sys
# sys.path.append(‘/path/to/parent/dir/’);
X,Y = ml.shuffleData(X,Y); # shuffle data randomly
# (This is a good idea in case your data are ordered in some pathological way,
# as the Iris data are)
Xtr,Xva,Ytr,Yva = ml.splitData(X,Y, 0.75); # split data into 75/25 train/validation
Learner Objects Our learners (the parameterized functions that do the prediction) will be de-
ned as python objects, derived from either an abstract classier 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 a data set (i.e., set the parameters of the model to perform well on those
data), and make predictions on a data set.
So, you can build and train a kNN classier on Xtr,Ytr and make predictions on some data
Xva with it using e.g.,
knn = ml.knn.knnClassify() # create the object and train it
knn.train(Xtr, Ytr, K) # where K is an integer, e.g. 1 for nearest neighbor prediction
YvaHat = knn.predict(Xva) # get estimates of y for each data point in Xva
2
# Alternatively, the constructor provides a shortcut to “train”:
knn = ml.knn.knnClassify( Xtr, Ytr, K );
YvaHat = predict( knn, Xva );
If your data are 2D, you can visualize a data set and a classier’s decision regions using e.g.,
ml.plotClassify2D( knn, Xtr, Ytr ); # make 2D classification plot with data (Xtr,Ytr)
This function plots the training data, 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.
(a) Modify the code listed above to use only the rst two features of X (e.g., let X be only the
rst two columns of iris, instead of the rst four), and visualize (plot) the classication
boundary for varying values of K = [1, 5, 10, 50] using plotClassify2D.
(b) Again using only the rst two features, compute the error rate (number of misclassications)
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:
K=[1,2,5,10,50,100,200];
for i,k in enumerate(K):
learner = ml.knn.knnClassify(… # TODO: complete code to train model
Yhat = learner.predict(… # TODO: complete code to predict results on training data
errTrain[i] = … # TODO: ” ” to count what fraction of predictions are wrong
#TODO: repeat prediction / error evaluation for validation data
plt.semilogx(… #TODO: ” ” to 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?
Problem 3: Naïve Bayes Classiers
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 le 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 nal 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
In the case of any ties, we will prefer to predict class +1.
I decide to try a naïve Bayes classier to make my decisions and compute my uncertainty.
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(a) Compute all the probabilities necessary for a naïve Bayes classier, i.e., the class probability
p(y) and all the individual feature probabilities p(xi
|y), for each class y and feature xi
(b) Which class would be predicted for x = (0 0 0 0 0)? What about for x = (1 1 0 1 0)?
(c) Compute the posterior probability that y = +1 given the observation x = (1 1 0 1 0).
(d) Why should we probably not use a joint Bayes classier (using the joint probability of the
features x, as opposed to a naïve Bayes classier) for these data?
(e) 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. Should we re-train the model, and if so, how?
(e.g.: how does the model, and its parameters, change in this new situation?) Hint: what will
the naïve Bayes model over only features x2 . . . x5 look like, and what will its parameters be?
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