Homework 2

Due: October 3

1. Given the following data sets, calculate the three scatter matrix criteria for each of the following data sets and analyze your findings. Assume that the classes are equiprobable.

   Class	X1	X2
   1	6.5	4.3
   1	5.4	5.5
   1	7.2	6.8
   1	4.8	5.2
   1	6.6	4.7
   1	7.4	6.7
   1	7.1	5.2
   1	5.7	6.6
   1	5.4	4.5
   1	6.0	6.6
   2	3.6	5.2
   2	4.2	3.5
   2	4.8	0.5
   2	3.2	-1.5
   2	2.6	3.2
   2	3.8	1.4
   2	5.1	1.9
   2	6.2	3.6
   2	4.5	2.4
   2	3.2	3.3
   3	1.5	5.0
   3	0.5	3.5
   3	-0.5	2.4
   3	6.8	7.4
   3	5.5	3.6
   3	4.6	3.8
   3	4.0	5.1
   3	3.9	4.2
   3	2.8	3.8
   3	3.3	4.4
   4	0.5	1.4
   4	-0.5	2.2
   4	0.6	0.1
   4	-0.4	0.4
   4	0.2	-0.1
   4	0.0	1.3
   4	-0.1	1.8
   4	-0.2	0.8
   4	0.3	1.1
   4	0.0	1.2

2. Can you do anything to improve the separability of this data? This is an open-ended type of question that pertains to our ability to transform data to get the maximum advantage from it.
3. We want to classify bolts of fabric as cotton or polyester, but can only sample the color. Let C₁ as cotton and C₂ as polyester with the a priori probabilities 1/3 and 2/3 respectively. The colors are yellow, white and red with the conditional probabilities:

   Class	Yellow	White	Red
   C1	3/5	1/5	1/5
   C2	2/5	1/5	2/5

   Construct the Bayes classifier for this problem.

4. For this problem, assume the actions {a₁ = choose Cotton a₂ = choose Polyester} and the following loss matrix:

   Class	a1	a2
   C1	-1	2.5
   C1	1.4	-1.5

   Determine the likelihood ratio and calculate the total (Bayes) risk.

5. A classifier problem has P(C₁) = 0.4, P(C₁) = 0.6, and x varying between zero and Pi. The conditional probability distributions are:

                      sin x                               2(Pi - x)
      p(x | C(1)) = -----------            p(x | C(2)) = -----------
                        2                                   Pi^2
   

   Construct the Bayes classifier. For what values of x is it classified as being in class 1?