- a)
- [2 P]
In many pattern classification problems one has the option either to assign the pattern
to one of
classes, or to
*reject*it as being unrecognizable. If the cost for rejects is not too high, rejection may be a desirable action. Suppose that, for a new value of , the true class is and that we assign to class . Assume, In doing so, we incur the loss(2)

where is the loss incurred for choosing the ( +1)th action, rejection, and is the loss incurred for making a substitution error. Show that the minimum risk is obtained if we decide if for all and if , and reject otherwise. What happens if ? What happens if ? - b)
- [2* P]
Consider the classification problem with rejection option of a).
- Use the results of a) to show that the following discriminant functions are optimal for such problems:
(3)

- Plot these discriminant functions and the decision regions for the two-class one-dimensional case having
- ,
- ,
- , and
- ,

- Describe qualitatively what happens as is increased from 0 to 1.
- Repeat for the case having
- ,
- ,
- , and

- Use the results of a) to show that the following discriminant functions are optimal for such problems: