CENG 564 Spring 2006-2007
Homework 1
Due Date: 15.03.2007
1- Suppose two equally probable one-dimensional densities are of the
form p(x | ωi) α e -|x-ai| / bi for i= 1,2 and 0<bi.
(a) Write an analytic expression for each density, that is,
normalize each function for arbitrary ai and positive bi
(b) Calculate the likelihood ratio as a function of your
variables.
(c) Sketch a graph of the likelihood ratio p(x | ω1) / p(x | ω2) for
the case a1=0, b1=1, a2=1 and b2=2
Note: α : proportional
2- Use the conditional densities given by the formula;
Assume equal prior probabilities for the categories and for simplicity
a2>a1, the same ‘width’ b.
(a)
Show that the minimum probability of error is given by
(b)
Plot this as a function of |a2-a1|/b
(c) What is the minimum value of P(error) and under which
conditions can this occur? Explain.
3- In many pattern classification problems some has the opinion either
to assign the pattern to one of c classes, or to reject it as being
unrecognizable. If the cost for rejects is not too high, rejection may be
a desirable action. Let,
Where
is the loss incurred for choosing the (c+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
= 0? What happens if
?
4- Let the components of the vector x = (x1,…xd)t be binary valued (0
or 1) and
be the prior probability for the state of nature
j = 1,….c. Now define
and
With the components of xi being statistically independent for all x in
. Let x be distributed as described above, with c=2, d odd and
(a)
Show that the minimum-error-rate decision rule becomes:
(b) Show that the minimum probability of error is given by
(c) What is the limiting value of
(d) Show that
approaches zero as
? Explain.
Explain.