LUSEIS PRWTOU SET ASKHSEWN TOU MAJHMATOS STATISTIKH MONTELOPOIHSH KAI ANAGNWRISH PROTUPWN. Miqahl Maragkakhc, Hliac Iwshf

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1 LUSEIS PRWTOU SET ASKHSEWN TOU MAJHMATOS STATISTIKH MONTELOPOIHSH KAI ANAGNWRISH PROTUPWN Miqahl Maragkakhc, Hliac Iwshf Oktwbrioc 006

3 Askhsh.9 Jewreiste ton parakatw kanona apofashc gia kathgoriec monodiastatou problhmatoc kathgoriopoihshc : ˆω = ω ω an x > an x < a) Deixte oti h pijanothta lajouc gia auton ton kanona didetai apo ton tupo P (error) = P (ω ) P (x/ω )dx + P (ω ) P (x/ω )dx b) Pairnontac thn paragwgo, deixte oti mia aparaithth sunjhkh gia na elaqistopoihjei to P(error), einai P (/ω )P (ω ) = P (/ω )P (ω ) g) Auth h exiswsh kajorizei to monadika? d) Dwste ena paradeigma opou mia timh tou pou ikanopoiei thn sqesh tou erwthmatoc (b), sthn pragmatikothta megistopoiei thn pijanothta lajouc. LUSH a) Sumfwna me thn idiothta tou diamerismou, mporw na grayw P (error) = P (error, x)dx = P (error/x)p (x)dx = P (error/x)p (x)dx + P (error/x)p (x)dx ω ω = = P (ω /x)p (x)dx + P (ω /x)p (x)dx + P (ω /x)p (x)dx P (ω /x)p (x)dx Qrhsimopoiwntac se auto to shmeio ton kanona tou Bayes, mporw na grayw P (ω /x) = P (x/ω )P (ω ) P (x) P (ω /x) = P (x/ω )P (ω ) P (x) Antikajistwntac tic sthn exiswsh pou afora to P(error), eqoume P (error) = P (x/ω )P (ω )dx + P (x/ω )P (ω )dx Me dedomeno oti ta P (ω ), P (ω ) einai stajera, eqoume

P (error) = P (ω ) P (x/ω )dx + P (ω ) P (x/ω )dx b) Pairnontac thn paragwgo, deixte oti mia aparaithth sunjhkh gia na elaqistopoihjei to P(error), einai P (/ω )P (ω ) = P (/ω )P (ω ) g) Auth h

4 P (error) = P (ω ) P (x/ω )dx + P (ω ) P (x/ω )dx b) Gia na broume to elaqisto h to megisto miac sunarthshc, sthn periptwsh mac tou P(error), pairnoume thn paragwgo thc wc proc thn opoia kai sth suneqeia ja mhdenisoume. dp (error) d = P (ω ) P (x/ω )dx + P (ω ) P (x/ω )dx Se auto to shmeio, ja efarmosoume ton kanona tou Leibnitz, o opoioc dinei thn paragwgo enoc orismenou oloklhrwmatoc tou opoiou ta oria einai sunarthsh thc metablhthc wc proc thn opoia ginetai h paragwgish. Gia to prwto oloklhrwma thc pio panw sqeshc eqoume: = b() a() P (x/ω )dx = b() a() P (x/ω )dx P (x/ω ) dx + b() P (b()/ω ) a() P (a()/ω ) = P (a()/ω ) = 0 P (/ω ) Omoiwc P (x/ω )dx = P (/ω ) 0 Exiswnontac thn paragwgo me to mhden, eqoume th sunjhkh pou mac zhteitai: dp (error) d = 0 P (ω )P (/ω ) = P (ω )P (/ω ) g) H exiswsh tou erwthmatoc (b) den orizei monadika to shmeio, miac kai einai dunaton na uparqoun perissotera apo ena shmeia pou na ikanopoioun thn parapanw exiswsh, ta opoia apoteloun akrotata men, alla oqi aparaithta kai elaqista. Sto Sqhma ta shmeia,, 3 ikanopoioun thn exiswsh tou deuterou erwthmatoc. Gia na doume an ena akrotato apotelei elaqisto h megisto, pairnoume thn deuterh paragwgo, Upojetontac oti P (ω ) = P (ω ), tote d P (error) d = P (ω ) dp (/ω ) P (ω ) dp (/ω ) d d d ( P (error) dp (/ω ) d = P (ω ) d dp (/ω ) ) d O oroc mesa sthn parejensh kajorizei to an ja bgei h deuterh paragwgoc jetikh (elaqisto P (error) gia to ) h arnhtikh (megisto P (error) gia to ). d) An antistrefame to dojenta kanona apofashc tote h timh tou pou ja ikanopoiouse th sqesh tou erwthmatoc b) kai tautoqrona elaqistopoiouse thn pijanothta lajouc, ja megistopoiouse thn pijanothta lajouc.

dp (error) d = P (ω ) P (x/ω )dx + P (ω ) P (x/ω )dx Se auto to shmeio, ja efarmosoume ton kanona tou Leibnitz, o opoioc dinei thn paragwgo enoc orismenou oloklhrwmatoc tou opoiou ta oria einai

5 Askhsh.4 Sq ma : Ta shmeia,, 3 ikanopoioun thn exiswsh tou deuterou erwthmatoc. Jewreiste thn poludiastath kanonikh katanomh gia thn opoia o pinakac sundiakumanshc einai diagwnioc, σ ij = 0 kai σ ii = σ i i.e., Σ = diag(σ, σ,..., σ d ). a) Deixte oti P ( x) = d i= exp( πσi d (x i µ i ) /σi ) b) Sqediaste thn. g) Grayte thn ekfrash pou pairnei h Mahalanobis distance apo to x sto µ. LUSH O genikoc tupoc pou isquei gia thn poludiastath katanomh einai P ( x) = i= π d/ Σ / exp( ( x µ)t Σ ( x µ)) opou d einai h diastash twn qarakthristikwn. Gia pinaka diagwnio, isquei oti h orizousa tou einai ish me to ginomeno twn stoiqeiwn thc diagwniou, d Σ = σ σ...σ d = σi Σ / = i= enw gia ton antistrofo mporei eukola na deiqjei oti Σ = (/σ, /σ,..., /σ d ) Σ = /σ i I, i =,,..., d. miac kai ΣΣ = I. Me ton antistrofo tou pinaka sundiakumanshc se authn th morfh, mporoume na grayoume, d i= σ i ( x µ) T Σ ( x µ) = ( x µ) T σi I( x µ) = σi ( x µ) T ( x µ) = σi d i= ( x i µ i ) () 3

a) Deixte oti P ( x) = d i= exp( πσi d (x i µ i ) /σi ) b) Sqediaste thn. g) Grayte thn ekfrash pou pairnei h Mahalanobis distance apo to x sto µ.

6 afou ( x µ) T ( x µ) = d i= ( x i µ i ) einai h Eukleidia apostash. Twra, mporoume na xanagrayoume to P (x) lambanontac upoyin kai oti π d/ = ( π) d, P ( x) = d i= exp( πσi d (x i µ i ) /σi ) b) Gia na apanthsoume se auto to erwthma, mporoume na exetasoume orismenec periptwseic, lambanontac panta upoyin oti o Σ einai diagwnioc. Se periptwsh pou Σ = σ I, dhladh o Σ eqei ola ta stoiqeia thc diagwniou isa, tote gia d > 3 eqoume upersfairec, gia d = 3 eqoume sfairec, gia d = eqoume kuklouc. Se periptwsh pou o Σ einai apla diagwnioc, tote gia d > 3 eqoume uperelleiyeic, gia d = 3 eqoume elleiyoeideic sfairec, gia d = eqoume elleiyeic. Sto sqhma () fainetai ena tetoio paradeigma. i= Sq ma : Gia qarakthristika kai gia Σ diagwnio alla me diaforetika ta stoiqeia thc diagwniou tou, eqoume elleiyeic. g) Apo th jewria (Duda and Hart kef., sel.8 ) eqoume dei oti o genikoc tupoc orismou thc Mahalanobis distance einai d M ( x, µ) = ( x µ) T Σ ( x µ) Me ton metasqhmatismo pou eqoume apo th sqesh (), mporoume na xanagrayoume thn Mahalanobis distance wc exhc, h diaforetika, d M ( x, µ) = d (x i µ i ) /σi i= d M ( x, µ) = d ( xi µ i i= σ i ) 4

periptwseic, lambanontac panta upoyin oti o Σ einai diagwnioc.

7 Askhsh.7 Estw kanonikec katanomec me isouc pinakec sundiakumanshc Σ alla diaforetikec mesec timec, N(µ, Σ) kai N(µ, Σ). Sunarthsei twn a-priori pijanothtwn, breite th sunjhkh gia thn opoia to Bayesian decision boundary den dierqetai anamesa stic mesec timec. LUSH Anatreqontac sto biblio tou Duda and Hart sth selida 3, mporoume na doume th sqesh (63) gia to dianusma tou decision boundary: x o = ( µ + µ ) ln(p (ω )/P (ω )) ( µ µ ) T Σ ( µ µ ) ( µ µ ) () Sq ma 3: To decision boundary pernaei apo to meso twn meswn timwn. Parathrwntac to Sqhma 3, otan x o = ( µ + µ ) to x o dierqetai apo to meso thc apostashc µ µ. Epishc, me bash to Sqhma 3, otan x o = ( µ + µ ) ( µ µ ) to x o peftei eite panw sto µ eite panw sto µ. Antikajistwntac thn timh auth tou x o sth sqesh (): ln(p (ω )/P (ω )) ( µ µ ) T Σ ( µ µ ) = H posothta ( µ µ ) T Σ ( µ µ ) einai mia stajerh timh thn opoia sumbolizoume me a. Etsi: ln(p (ω )/P (ω )) = a Sunepwc otan P (ω ) timec. P (ω ) > exp a h P (ω ) P (ω ) < exp a to decision boundary den dierqetai anamesa stic mesec 5

LUSH Anatreqontac sto biblio tou Duda and Hart sth selida 3, mporoume na doume th sqesh (63) gia to dianusma tou decision boundary: x o = ( µ + µ ) ln(p (ω )/P (ω )) ( µ µ ) T Σ ( µ µ ) ( µ µ ) () Sq

8 Askhsh.30 Estw kathgoriec ω kai ω oi opoiec perigrafontai apo Gkaousianec katanomec N(µ, σ ) kai N(µ, σ ) kai me a priori pijanothtec P (ω ) = P (ω ) = 0.5. a) Deixte oti h minimum pijanothta lajouc didetai apo ton tupo: π opou α = µ µ σ. b) Qrhsimopoiwntac thn anisothta, π α α e u du e t dt na deixete oti to lajoc teinei sto mhden, otan to µ µ σ LUSH α π e α teinei sto apeiro. To sunoliko lajoc to opoio mporoume na eqoume se ena problhma kathgoriopoihshc qwrizetai sta parakatw pijana lajh: Bayes + model + estimation + (mismatch). H minimum pijanothta lajouc einai ish me to lajoc tou Bayes, afou jewroume oti kaname kalh douleia sthn epilogh tou montelou mac, ston upologismo twn parametrwn kai sthn exaleiyh tou mismatch. To lajoc tou Bayes, P (x/ω )P (ω )dx + Ω P (x/ω )P (ω )dx Ω To shmeio apofashc x 0 gia Gkaousianec me isec diasporec exei hdh upologisjei (Askhsh,ektoc set) oti einai iso me opote Me allagh metablhthc, x0 x0 x 0 = µ + µ P (x/ω )P (ω )dx + σ (x µ ) π e σ dx + x 0 P (x/ω )P (ω )dx x 0 σ π e (x µ ) σ dx enw ta nea oria oloklhrwshc einai, x µ σ x µ σ = u dx = σdu = z dx = σdz x = u = x = x 0 = µ + µ u = µ µ σ 6

b) Qrhsimopoiwntac thn anisothta, π α α e u du e t dt na deixete oti to lajoc teinei sto mhden, otan to µ µ σ LUSH α π e α teinei sto apeiro.

9 x = µ + µ z = µ µ σ x = z = Antikajistwntac tic neec metablhtec kai ta nea oria oloklhrwshc sta oloklhrwmata, kai µ µ σ e u π du + µ µ σ π e z dz µ µ σ µ µ σ π e z e u π du = Q(µ µ σ dz = ( Q(µ µ σ Ta duo oloklhrwmata antistoiqoun se isa embada, opote kai µ µ σ µ µ σ π e u du π e u du ) )) = Q(µ µ ) σ b) Me thn prwth matia, mporoume na doume oti o oroc µ µ / isoutai me thn Mahalanobis apostash, h opoia einai analogh tou bajmou diakritothtac twn katanomwn. Oso auth teinei sto apeiro, h diakritothta twn katanomwn auxanei, kai wc ek toutou o oroc P e elattwnetai kai teinei sto mhden. Qrhsimopoiwntac thn anisothta kai ekfrazomenh h Mahalanobis apostash wc, eqoume, O oroc r = µ µ σ π α e t = α α = r/ dt r e ( r ) π (3) r e ( r ) π = π r e ( r ) ston paranomasth tou eqei mia grammikh logarijmikh sunarthsh h opoia einai monotonika auxousa, dhl. oso megalwnei o oroc r, auth teinei sto apeiro. Auto shmainei oti oloc o oroc thc sqeshc () teinei sto mhden, kai ton akoloujei to P e afou h ekfwnhsh mac leei oti to P e einai panta mikrotero h estw iso me ton oro thc sqeshc (). 7

h opoia einai analogh tou bajmou diakritothtac twn katanomwn. Oso auth teinei sto apeiro, h diakritothta twn katanomwn auxanei, kai wc ek toutou o oroc P e elattwnetai kai teinei sto mhden.

Upojèseic - Jewr mata. Majhmatikˆ Plhroforik c. Upojèseic - EikasÐec. H qrus tom

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