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564_sec2_hw1_updated

# 564_sec2_hw1_updated - unrecognizable If the cost for...

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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 | ω i ) α e -|x-a i | / b i for i= 1,2 and 0<b i. (a) Write an analytic expression for each density, that is, normalize each function for arbitrary a i and positive b i (b) Calculate the likelihood ratio as a function of your variables. (c) Sketch a graph of the likelihood ratio p(x | ω 1 ) / p(x | ω 2 ) for the case a 1 =0, b 1 =1, a 2 =1 and b 2 =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 |a 2 -a 1 |/b (c) What is the minimum value of P(error) and under which conditions can this occur? Explain.

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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
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Unformatted text preview: 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 = (x 1 ,…x d ) t be binary valued (0 or 1) and be the prior probability for the state of nature and j = 1,….c. Now define With the components of x i 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 ? Explain. (d) Show that approaches zero as Explain....
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