## Description

1) A binary classifier decides whether data is from one of two classes, labeled 1 and -1.

In this problem the data is described by two features x1 and x2 and the classification

decision is made as follows. The class with label 1 is decided if x1a1 + x2a2 > b while

the class with label -1 is decided when x1a1 + x2a2 < b. Here a1, a2, and b are given

real numbers.

a) The decision boundary is the set of {x1, x2} that satisfy x1a1 + x2a2 = b. Thus,

we may assign the label to the data using the sign of y = x1a1 + x2a2 − b since

label 1 is decided if y > 0 and label -1 is decided if y < 0. That is, the label may

be obtained as sign{y}. Express y as an inner product of a vector x containing

the features and w containing weights, that is, write y = x

T w.

b) Let x2 be the vertical axis and x1 be the horizontal axis. Show that the decision

boundary y = 0 is a straight line. Find the slope and intercept with the vertical

axis as a function of a1, a2, b.

c) You classify n data samples using sign{y} where y =

y1

y2

.

.

.

yn

= Xw. Suppose

n = 4 and the features for the 4 data samples are 1 : (0, 0.4), 2 : (0.2, 0.1), 3 :

(0.5, 0.6), 4 : (0.9, 0.8). Write out the matrix X.

d) Suppose a1 = 1, a2 = 2, and b = 1. Sketch the decision boundary in the x1-x2

plane assuming x2 is the vertical axis and x1 is the horizontal axis. Graph the

four data samples from the previous part and classify them.

e) Download and run the linear classifier script. This script classifies 5000 examples

of (randomly generated) data consisting of two features using the linear classifier.

Save the figure and include it in your submission. Describe the decision boundary

you observe using a sentence.

f) Change the classifier weights to w = [1.6 2 − 1.6]T

. Rerun the scrip. Include

the figure in your pdf file. Briefly describe how the change in the weights changed

the decision boundary.