EEL6825: HW#2

Due Friday, September 25, 1998 in class. Do not be late to class. Late homework will lose $e^{\char93 ~of~days~late} -1$ percentage points. Do not hand in any matlab code or plots-instructions for turning in matlab code will be discussed in class.

PART A: Textbook Problems

A1

3.1 in DH&S

A2

3.2 in DH&S

A3

3.6 in DH&S

A4

3.7 in DH&S

A5

5.4 in DH&S

PART B: Short-Answer Problems Answer each of the following with a short statement, derivation and/or sketch.

B1

If x is a 1D random variable given by a normal distribution with mean $\mu$ and variance $\sigma^2$, what is $E\{x^2\}$?

B2

It is well known that if two normal distributions have the same covariance matrix, the Bayes discrimination function is linear. However, given two non-normal probability distributions are identical, except for their means, is the Bayes classifier necessarily linear? Why or why not?

B3

You are given data drawn from two Normal distributions. It turns out that the data points are linearly separable. Is the Bayes Classifier you design guaranteed to correctly classify all of the data points?

B4

A certain classifier $g(\underline{x})$ gives an error of 55% on a given dataset for a two-category classification problem. Explain a simple way of improving the performance of this classifier on the same data.

B5

Write an equation for a linear classifier that successfully classifies points (1,1) and (2,0) as class $\omega_1$ and points (2,1) and (3,0) as class $\omega_2$.

turn over

PART C: Computer Experiment: Bayes Classifier with 3 classes

You are given the heights and weights of all of the players in the WNBA (Women's National Basketball Association) at the end of last season. The file can be found in
http://www.cnel.ufl.edu/analog/courses/EEL6825/wnba.asc Each line of the file consists of four numbers: (1) Player's position. 1=guard 2=forward 3=center. (Guards are usually smaller than forwards who are usually smaller than centers.) (2) Player's height: truncated value in feet (3) Player's height - truncated value: in inches (4) Player's weight: pounds.

C1

Read in the data and print a scatter plot showing the three different classes with height on the x₁ axis and weight on the x₂axis. Are any two of the classes linearly separable?

C2

Compute the sampled mean and covariance matrix for each position.

C3

Assume that all three classes are generated from normal distributions with equal a priori probabilities. Design a Bayes classifier using the sampled mean and covariance matrix. Draw the discriminant boundaries on the scatter plots. How many players are classified incorrectly?

C4

Can you draw two linear boundaries by hand that classify better than the sampled Bayes Classifier?

C5

What position would you play if you played in the WNBA?