Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών Εθνικό Μετσόβιο Πολυτεχνείο Αναγνώριση Προτύπων και Νευρωνικά Δίκτυα η Σειρά Ασκήσεων Θεόδωρος Αλεξόπουλος
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Jeìdwroc Alexìpouloc Anaplhrw c Kajhgh c Theodoros Alexopoulos Associae Professor EJNIKO METSOBIO POLUTEQNEIO NATIONAL TECHNICAL UNIVERSITY SQOLH EFARMOSMENWN MAJHMATIKWN KAI DEPARTMENT OF PHYSICS FUSIKWN EPISTHMWN - TOMEAS FUSIKHS ZOGRAFOU CAMPUS HRWWN POLUTEQNEIOU 9 57 8 ATHENS - GREECE AJHNA 57 8 Phone : +3 77-39 Fax: +3 77-35 Thl: 77-39 Fax: 77-35 e-mail: Theodoros.Alexopoulos@cern.ch e-mail: heoalex@cenral.nua.gr hml://www.physics.nua.gr/faculy/heoalex Anagn rish ProÔpwn & Neurwnikˆ DÐkua Probl maa (Kefˆlaio: Saisik JewrÐa ou Bayes (Episrof : 4 NoembrÐou 3. O akìloujoc pðnakac ou sq maoc ( mac dðnei ic upì sunj kh pijanìhec miac uqaðac meablh c X gia reic klˆseic ω ω kai ω 3. 'Esw ìi gnwrðzoume ic a priori pijanìhec P (ω = 3 kai P (ω = 3. UpologÐse hn pijanìha lˆjouc hc axinìmhshc qrhsimopoi nac on kanìna apìfashc Bayes. ParahreÐse ìi x p(x/ω i = kai h uqaða meablh X paðrnei imèc so diˆshma [ 6]. LÔsh: Gia ic a priori pijanìhec èqoume: P (ω = P (ω = 3 P (ω 3 = P (ω P (ω = 4. Oi imèc P (ω i p(x/ω i i = 3 pou ja qrhsimopoi soume gia hn axinìmhsh wn klˆsewn faðnonai son pðnaka. ParahroÔme ìi x P (ω ip(x/ω i = P (ω i. O axinomh c Bayes ja apofasðsei gia hn klˆsh ω i ìan: P (ω i p(x/ω i > P (ω j p(x/ω j j i.
i x p(x=x/ i X= X= X=3 X=4 X=5 X=6 3 4 5 5 3 3 5 5 3 Sq ma : ParahreÐse ìi x p(x/ω i = kai h uqaða meablh X paðrnei imèc so diˆshma [ 6]. P (ω i p(x/ω i x = x = x = 3 x = 4 x = 5 x = 6 ω 9 6 3 3 6 3 ω 6 6 5 3 5 ω 3 4 6 4 PÐnakac : [ Sq ma :
Epomènwc apì on pðnaka h apìfash so q ro wn qarakhrisik n shmeðwn eðnai au pou faðneai so sq ma (. H pijanìha lˆjouc axinìmhshc eðnai: P e = P c = [ 9 + + + 3 + + 4] = 48 ìpou P c eðnai h pijanìha hc orj c axinìmhshc.. Na upologðsee hn pijanìha ou lˆjouc axinìmhshc kaˆ Bayes gia duo klˆseic ω ω ìpou jewroôme deðgmaa se -diasˆseic apì gkaousianèc kaanomèc pou perigrˆfonai apì ic akìloujec puknìhec pijanìhac: me kai me µ = µ = p( x/ω N( µ Σ ( ( 4 Σ = 9 p( x/ω N( µ Σ ( Jewr se ìi h a priori pijanìha P (ω = 5. LÔsh: ( 4 Σ = 9. ParahroÔme ìi gia ouc pðnakec diasporˆc Σ = Σ èqoume: Σ = Σ = Σ. O anðsrofoc pðnakac diasporˆc eðnai: Σ = 35 Oi sunar seic diˆkrishc g ( x g ( x eðnai: ( 9 4. me g i ( x = (Σ i µ i x µ i Σ i µ i + ln P (ω i P (ω = 5 P (ω = 75 kai h sunˆrhsh apìfashc eðnai: d( x = g ( x g ( x 3
d( ( x = Σ µ x µ Σ µ ( Σ µ x + µ Σ µ + ln ( P (ω P (ω d( x = 4x + 9x 3 ìpou x = ( x x. H epifˆneia apìfashc eðnai: d( x = 4x + 9x 3 = w x + w = ìpou w = ( 4/35 9/35 w = 3/35 = 885. OrÐzoume mia nèa meablh y = w x. Tìe oi upì sunj kh pijanìhec eðnai: p( ( x /ω i N µ i Σ i p( w x /ω i N ( w µ i w Σ i w p( w x /ω N ( 89 p( w x /ω N ( 63 89. Dhlad o prìblhmˆ mac ègine prìblhma mðac diˆsashc ìpwc faðneai so sq ma (3. H pijanìha orj c axinìmhshc eðnai: P c = P (ω p( w x /ω d( w x + P (ω R p( w x /ω d( ω x R P c = 4 885 N ( 89 dy + 3 4 885 N ( 68 89 dy P c = 4 ( 885 4 Φ + 3 ( 885 + 68 83 4 Φ 83 4
5 5 \ Sq ma 3: P c = 4 4 Φ( 87 + 3 Φ( 494 4 P c = 4 4 ( 45 + 3 ( 933 4 P c = 84..3 JewreÐse mia mèrhsh se reic diasˆseic x = (x x x 3. 'Esw ìi èqoume 4 deðgmaa apì hn klˆsh ω kai 4 deðgmaa apì hn klˆsh ω : ω : { ( ( ( ( } ω : { ( ( ( ( }. Na upojèsee ìi h meablh x eðnai gkaousian ìpou mporeðe na qrhsimopoi see ic akìloujec sqèseic gia on prosdiorismì hc mèshc im c kai ou pðnaka diasporˆc: µ = N N x k k= ìpou N eðnai o pl joc wn deigmˆwn. Σ = N N x k x k µ µ k= 5
Na qrhsimopoi see on kanìna apìfashc Bayes gia on prosdiorismì hc epifˆneiac apìfashc. ProsdiorÐse hn olik pijanìha gia hn orj axinìmhsh ou kanìna apìfashc Bayes. LÔsh: Oi mèsec imèc eðnai: kai oi pðnakec diasporˆc: µ = 4 3 Σ = Σ = 6 µ = 3 3 3 3 3. Efìson Σ = Σ o kanìnac ou Bayes ja mac d sei èna axinomh elˆqishc apìsashc Mahalanobis me sunar seic diˆkrishc: ìpou Epomènwc: H epifˆneia apìfashc eðnai: Σ = g i ( ( x = Σ µ i x µ i Σ µ i 8 4 4 4 8 4 4 4 8 kai x = x x x 3. g ( x = 4x 3 g ( x = 4x + 8x + 8x 3. d( x = g ( x g ( x = 8x 8x 8x 3 + 4 = ( x x x 3 w x + w = + = 6
ìpou w = kai w = /. Efìson Σ = Σ kai p( x /ω i N( µ i Σ i ìe: p( w x /ω i N( w µ i w Σ i w. Epomènwc oi nèec monodiˆsaec puknìhec pijanìhac eðnai: p( w x /ω N( 5 875 p( w x /ω N( 5 875. \ Z W [ 5 5 H pijanìha ou lˆjouc axinìmhshc eðnai: [ Φ P e = Sq ma 4: ( 5 + 5 875 + Φ ( 5 875 ] = 4..4 'Esw a deðgmaa: ( ( (3 (3 ( 3 7
an koun se mia klˆsh ω. Epiplèon èsw a deðgmaa: (7 9 (8 9 (9 8 (9 9 (8 an koun se mia klˆsh ω. 'Esw ìi a deðgmaa wn duo klˆsewn proèrqonai apì gkaousianèc puknìhec pijanìhac. Na breðe hn epifˆneia apìfashc efarmìzonac on kanìna Bayes. JewreÐse isopðjanec klˆseic. LÔsh: 'Oi mèsec imèc wn dôo kaanom n eðnai: µ = ( /5 µ = ( 4/5 9 ìpou qrhsimopoi same h sqèsh: µ i = 5 x k i = (gia ic dôo klˆseic 5 Gia on prosdiorismì wn mèswn im n kai gia ouc pðnakec diasporˆc: k= Epomènwc ja èqoume: Σ i = 5 5 x k x k µ i µ i k= Σ = {( ( ( ( 4 4 9 3 9 6 + + + 5 4 4 4 3 6 4 ( } ( 4 6 (/5 (/5 + 6 9 (/5 ( /5 /5 Σ = /5 /5 ( Σ 7 9 =. 9 3 4 + DeÐxe ìi Σ = Σ. Oi sunar seic diˆkrishc g ( x g ( x kai h sunˆrhsh apìfashc eðnai: d( x = g ( x g ( x d( x = { ( Σ µ x } { ( µ Σ µ Σ } µ x µ Σ µ 8
d( x = ( µ µ Σ ( x µ Σ µ Σ µ ìpou x = ( x x. Epomènwc h epifˆneia apìfashc eðnai: d( x = x + 35x 6 = x + 35x 6 =..5 (a H epifˆneia apìfashc gia duo klˆseic ω kai ω pou eðnai kaanemhmènec kaˆ Gauss me pðnakec diasporˆc: Σ = Σ σ I kai mèsec imèc µ µ kai a priori pijanìhec: P (ω P (ω eðnai mia uperepifˆneia pou perigrˆfeai apì hn akìloujh grammik exðswsh: A ( x A =. Na ekfrˆsee ic paramèrouc A kai A wc sunˆrhsh wn µ µ Σ P (ω kai P (ω. (b O Ðdioc kanìnac apìfashc Bayes gia duo kahgorðec me pðnakec diasporˆc: Σ Σ P (ω P (ω ja d sei mia epifˆneia apìfashc pou perigrˆfeai apì hn exðswsh: x B x + A x + C =. Na kajorðsee ic paramèrouc B A kai C. LÔsh: (a H sunˆrhsh diˆkrishc g i ( x eðnai: g i ( x = x Σ x + ( µ i Σ x µ i Σ µ i + ln P (ω i 9
me epifˆneia apìfashc d( x = g ( x g ( x = g ( x = g( x 'Esw ìi: Tìe h sqèsh ( ja gðnei: ìpou C = ( µ µ Σ x ( µ Σ µ µ Σ µ + ln P (ω P (ω ( µ Σ µ µ Σ µ ln P (ω P (ω. ( µ µ Σ ( x C( µ + µ ( µ µ Σ ( µ + µ A ( x A = A = Σ ( µ µ =. ( = ParahroÔme ìi: A = / ( µ Σ µ µ Σ µ ln P (ω P (ω ( µ µ Σ ( µ + ( µ + µ. µ ( µ µ Σ ( µ + µ = µ Σ µ µ Σ µ. Epomènwc A = ( µ + ln (P (ω /P (ω µ ( µ µ Σ ( µ + ( µ + µ. µ (b Gia Σ Σ kai P (ω P (ω h sunˆrhsh diˆkrishc eðnai:
H epifˆneia apìfashc d ( x eðnai: g i ( x = x Σ ( i x + µ i Σ x i µ i Σ i ln Σ i + ln P (ω i. g ( x = g ( x x (Σ Σ x + ( µ Σ µ Σ x ( µ Σ µ µ Σ µ + + ln Σ Σ + ln P (ω P (ω = x B x + A x + C = ìpou profan c èqoume orðsei ic akìloujec paramèrouc: B = (Σ Σ kai C = A = Σ µ Σ µ ( µ Σ µ µ Σ µ + + ln Σ Σ + ln P (ω P (ω.
.6 (a JewreÐse èna prìblhma anagn rishc proôpwn me M-kahgorÐec miac diˆsashc ìpou kˆje kahgorða qarakhrðzeai apì mia kaanom puknìhac pijanìhac Rayleigh: p(x/ω i = { (x/σ i e x /σi gia x gia x <. BreÐe h sunˆrhsh apìfashc Bayes ou probl maoc upojèonac ìi oi a priori pijanìhec eðnai P (ω i = /M. (b Epanalˆbae o Ðdio prìblhma shn perðpwsh gia hn kaanom puknìhac pijanìhac: p(x/ω i = { (x/ti e x/t i gia x gia x <. LÔsh: (a H sunˆrhsh diˆkrishc eðnai: g i (x = p(x/ω i P (ω i mporoôme na qrhsimopoi soume o fusikì logˆrijmo g i (x = ln p(x/ω i + ln P (ω i g i (x = ln x ln σ i x σ i + ln M g i (x = ln x ln σ i x σ i ln M. Gia dôo klˆseic i j o axinomh c ja mac d sei mia epifˆneia apìfashc: ln x = d ij (x = g i (x g j (x = ( σi σ j x 4 ln ( σj σ i ( σ i / ( = σj σ i σ j. (b
H sunˆrhsh diˆkrishc eðnai: ìpou kai h epifˆneia apìfashc eðnai: g i (x = ln p(x/ω i ln P (ω i p(x/ω i = x e x/t i P (ω Ti i = d ij (x = g i (x g j (x ( ln T i xti ( ln T j xtj = x T = ln(t j/t i. T i T j S[ 3 S[ 3 [ 7 [ Sq ma 5: H pijanìha ou lˆjouc axinìmhshc eðnai: P e = [ xt x ] e x/t j x dx + e x/t i dx Tj x T Tj P e = ( [x T e x T /T j + e xt /Ti + ( e x T /T j + e x T /T i ] T j T i P e = [ ( ( ] e x T /T xt i + e x T /T xt j + T i T j 3
ìpou x T eðnai o shmeðo apìfashc: x T = ln(t j/t i. T i T j.7 'Esw ìi duo klˆseic perigrˆfonai apì ic sunar seic puknìhac pijanìhac ou sq - maoc (6. BreÐe o sônoro apìfashc Bayes. Na upojèsee ìi èqoume isopðjanec klˆseic ω kai ω. S[ S[ [ Sq ma 6: LÔsh: Apì o sq ma (6 èqoume: p(x/ω = x + p(x/ω = x. P (ω = P (ω =. H sunˆrhsh diˆkrishc eðnai: g i (x = p(x/ω i P (ω i kai h epifˆneia apìfashc eðnai: 4
d (x = g (x g (x = ( x + (x = H pijanìha ou lˆjouc hc axinìmhshc eðnai: P e = P (ω 4/3 x = 4 3. (x dx + P (ω P e = 4/3 ( = 3. ( x + dx.8 Upojèoume ìi oi akìloujec duo kahgorðec (ω ω perigrˆfonai apì gkaousianèc sunar - seic puknìhac pijanìhac: kai ω : { ( ( ( ( } ω : { (4 4 (6 4 (6 6 (4 6 }. 'Esw ìi oi a priori pijanìhec eðnai Ðsec P (ω = P (ω. Na breðe hn exðswsh pou perigrˆfei o sônoro apìfashc Bayes meaxô wn duo au n kahgori n. LÔsh: 'Eqoume P (ω = P (ω = ìpou x j eðnai a prìupa hc klˆshc ω µ = 4 µ = 4 4 j= 4 j= x j = ( x j = ( 5 5 5
[ Z Z [ Sq ma 7: ìpou x j eðnai a prìupa hc klˆshc ω. Oi pðnakec diasporˆc eðnai: Σ = 4 [( Σ = 4 ( + ( 4 x j x j µ µ j= ( ( = kai omoðwc breðe oi o pðnakac diasporˆc Σ eðnai: ( ( + ( = I ( ( + ] Σ = 4 4 x j x j = I. j= Efìson Σ = Σ = Σ = I oi sunar seic diˆkrishc eðnai: g ( x = x Σ µ µ Σ µ g ( x = x µ ( µ µ = ( x x ( ( g ( x = x + x. 'Omoia gia h g ( x èqoume: 6
g ( x = x Σ µ µ Σ µ Epomènwc h epifˆneia apìfashc eðnai: g ( x = 5x + 5x 5. d ( x = g ( x g ( x = ìpwc faðneai so sq ma (7. x + x 6 =.9 Na epanalˆbee o prìblhma (.8 gia ic akìloujec klˆseic: ω : { ( ( ( ( } kai ω : { ( ( ( ( }. ParahreÐse ìi oi duo auèc klˆseic den eðnai grammikˆ diaqwrðsimec. LÔsh: [ Z Z [ Sq ma 8: Oi mèsec imèc wn dôo klˆsewn eðnai: 7
ìpou a deðgmaa x j kai x j eðnai: x j kai x j Oi pðnakec diasporˆc eðnai: µ = 4 µ = 4 {( {( 4 j= 4 j= ( ( x j = ( x j = ( ( ( ( ( } }. 4 kai Σ = 4 [( Σ = 4 j= ( + + x j x j µ µ ( ] ( ( Σ = ( = I ( Σ = [( 4 ( + ( + ( ( + ( ( Σ = ] ( ( ( + ( = I. ( + ( ( ( + ParahroÔme ìi Σ Σ epomènwc oi dôo auèc klˆseic ω ω den eðnai grammikˆ diaqwrðsimec. Oi sunar seic diˆkrishc eðnai: g ( x = ln Σ ( x µ Σ ( x µ 8
g ( x = ln(/4 ( x µ ( x µ ìpou g ( x = ln 4 ( ( x + x = ln x + x x = ( x x. 'Omoia: g ( x = ln Σ ( x µ Σ ( x µ g ( x = ln 4 4 x x Epomènwc h epifˆneia apìfashc eðnai: g ( x = ln 4 ( x 4 + x ( = ln x 4 + x. d ( x = g ( x g ( x = ln 3 4 (x + x = x + x = 8 ln. 3 9
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