\underline{Problem Set 2}

MAS 622J/1.126J: Pattern Recognition and Analysis

Due: 5:00 p.m. on September 30

[Note: All instructions to plot data or write a program should be carried out using Matlab. In order to maintain a reasonable level of consistency and simplicity we ask that you do not use other software tools.]

If you collaborated with other members of the class, please write their names at the end of the assignment. Moreover, you will need to write and sign the following statement: “In preparing my solutions, I did not look at any old homeworks, copy anybody’s answers or let them copy mine.”

Problem 1: [10 Points]

In many pattern classification problems we have the option to either assign a pattern to one of c classes, or reject it as unrecognizable - if the cost to reject is not too high. Let the cost of classification be defined as:

λ(ω_{i}|ω_{j}) = \begin{cases}0 & if ω _{i} = ω _{j}, (i.e. Correct Classification) \\ λ_{r} & if ω _{i} = ω _{0}, (i.e. Rejection) \\ λ_{s} & Otherwise, (i.e. Substitution Error) \end{cases}

Show that for the minimum risk classification, the decision rule should associate a test vector x with class ω _{i}, if P(ω _{i}|x ) ≥ P(ω _{j}|x ) for all j and and P(ω _{i} — x ) ≥ 1 - \frac{λ_{r}}{λ_{s}}, and reject otherwise.

Problem 2: [16 Points]

Use signal detection theory as well as the notation and basic Gaussian assumptions described in the text to address the following.