Homework 2
CS 591Q/791V - Pattern Recognition
Instructor: Dr. Arun Ross
Due Date: Feb 27, 2007
Note: You are permitted to discuss the following questions with others in the class.
However, you must write up your own solutions to these questions. Any indication to the
contrary will be considered an act of academic dishonesty.
1. Solve the following problems in DHS.
(a) [5 points] 2.27, p. 72.
(b) [15 points] 2.48 (a), p. 78.
(c) [20 points] 2.52, p. 79.
2. [15 points] Consider the following class-conditional density function for feature vector x=
x1
x2:
p(x|ω)∼N(µ, Σ)
where,
µ=µ1
µ2;Σ=σ2
1σ12
σ21 σ2
2;σ12 =σ21.
(a) Write down the expression for the Euclidean distance between point xand mean vector
µ.
(b) Write down the expression for the Mahalanobis distance between point xand mean
vector µ. Simplify this expression by expanding the quadratic term.
(c) Compare the two expressions. How does the Mahalanobis distance differ from the
Euclidean distance? When are the two distances equal? When is it more appropriate
to use the Mahalanobis distance?
3. [20 points] Consider a two-category classification problem with two-dimensional feature vec-
tor x= (x1, x2). The two categories are ω1and ω2
p(x|ω1)∼N(0, I),
p(x|ω2)∼N([1,1]T, I), and
P(ω1) = P(ω2) = 1
2.
