Problem 1:

1. Consider the following set of two dimensional vectors from three categories:

(a) Plot the decision boundary resulting from the nearest-neighbor rule just for categorizing ω₁ and ω₂. Find the sample means nil and m2 and on the same figure sketch the decision boundary corresponding to classifying x by assigning it to the category of the nearest sample mean.

(b) Repeat part (a) for categorizing only ω₁ and ω₃.

(c) Repeat part (a) for categorizing only ω₂ and ω₃,

(d) Repeat part (a) for a three-category classifier, classifying ω₁, ω₂ and ω₃.

Need code for plots as well in MATLAB

Problem 2:

Consider classifiers based on samples with priors P(ω₁) = P(ω₂) = 0.5 and the distributions

                  2x for 0 ≤ x ≤ 1

p(x|ω₁) =

                    0 otherwise,

                  2 - 2x for 0 ≤ x ≤ 1

p(x|ω₂) =

                    0 otherwise,

(a) What is the Bayes decision rule and the Bayes classification error?

(b) Suppose we randomly select a single point from col and a single point from 41)2, and create a nearest-neighbor classifier. Suppose too we select a test point from one of the categories (to] for definiteness). Integrate to find the expected error rate Pi (e).

(c) Repeat with two training samples from each category and a single test point in order to find P2(e).

(d) Generalize to show that in general.

Pn(e) = 1/3  + 1/((n+1)(n + 3) + 1/2(n + 2)(n + 3)

Confirm this formula makes sense in the n = 1 case.

(e) Compare lim_n→∞P_n(e)with the Bayes error.

Problem 3

Repeat Problem 2 but with

                3/2 for 0 ≤ x ≤ 2/3

p(x|ω₁) =

                    0 otherwise,

                  3/2  for 1/3 ≤ x ≤ 1

p(x|ω₂) =

                    0 otherwise,