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(COMP327)midterm96F.pdf
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HongKongUniversityofScienceandTechnology
COMP327/527:PatternRecognition
Fall1996
MidtermExamination
15October1996,12:00{1:20pm
StudentName:
StudentNumber:
Instructions 1.Thisisanopen-book,open-notesexamination. 2.Checkthatyouhaveall8pages(excludingthiscoverpage). 3.Writeyournameandstudentnumberonthispage. 4.Answerallquestionsinthespaceprovided.Roughworkshouldbedoneontheback
pages. 5.Makeyouranswersasconciseaspossible.
Question1(20%):
Question2(35%):
Question3(20%):
Question4(25%):
TOTAL(100%): 1.(20%)
(a)[10%]Densityestimationanddiscriminantfunctionsrepresenttwodi.erentap-proachestothedesignofclassi.ers.Brie.ydiscussthemajordi.erencesbetween thesetwoapproaches.
(b)[10%]Anyclassi.erbasedonparametricdensityestimationcanbeconvertedinto anequivalentclassi.erbasedondiscriminantfunctions.Ontheotherhand,not everyclassi.erbasedondiscriminantfunctionshasanequivalentparametricclas-si.er.Explainwhy.
2.(35%)
Considertheone-dimensionaldistributionsfortwoclasses,whereclass1isuniformly
distributedin[0.2]andclass2isuniformlydistributedin[1.5].AssumethatPr(!1).
Pr(!2).0:5.
(a)[5%]Sketchtheprobabilitydensityfunctionsforthetwoclasses.Labeltheaxes andmarkthedensityfunctionvaluesappropriately.
(b)[5%]UsingBayesian(optimal)decision,whatrangeofthexvaluewillbeclassi.ed toclass1andwhatrangewillbeclassi.edtoclass2.
(c)[5%]UsingBayesian(optimal)decision,whatistheprobabilitythatarandom examplexgeneratedfromclass1willbemisclassi.edtoclass2.
(d)[5%]UsingBayesian(optimal)decision,whatistheprobabilitythatarandom examplexgeneratedfromclass2willbemisclassi.edtoclass1.
(e)[5%]ComputetheBayeserror(i.e.minimumaverageprobabilityoferror).
(f)[8%]Usingthe1-NNclassi.cationruleunderthelarge-sampleassumption,com-putetheaverageprobabilityoferror.
(g)[2%]Does1-NNachieveoptimaldecisionforthisproblem.

3.(20%) Considerthefollowingdistribution
p(xj.). .x ;.
e
x! wherex.0.1.2.:::and..0istheparameterofthisdistribution.Supposeweare
(1)(n)
givenasampleSofnexamples,x.:::.x,drawnindependentlyfromtheunderlying distribution.Usingthesenexamples,.ndthemaximumlikelihoodestimatefor..
4.(25%) Considerthefollowingfourtrainingexamples:
x1 x2 x3 t
1 ;1 3 1
1 0 3 ;1
2 ;1 2 ;1
2 0 2 1

(a)[10%]Showthattheredoesnotexistasimpleperceptronwiththreeinputunits thatcanlinearlyseparatethefourexamplesintotwoclassescorrectly.

(b)[15%]Supposex3ofthesecondtrainingexampleaboveisequalto2insteadof3. Showthatthefourexamplesbecomelinearlyseparable.Giveasetofperceptron weightsthatcorrespondstoonesolution.
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