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Universidade de São Paulo
Escola Superior de Agricultura “Luiz de Queiroz”
Um estudo sobre reconhecimento de padrões:
um aprendizado supervisionado com classificador bayesiano
Pedro Henrique Ramos Cerqueira
Dissertação apresentada para obtenção do título de Mestre em Ciências. Área de concentração: Estatística e Experimentação Agronômica
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Pedro Henrique Ramos Cerqueira Estatístico
Um estudo sobre reconhecimento de padrões:
um aprendizado supervisionado com classificador bayesiano
Orientadora:
Profa. Dra. ROSELI APARECIDA LEANDRO
Dissertação apresentada para obtenção do título de Mestre em Ciências. Área de concentração: Estatística e Experimentação Agronômica
Dados Internacionais de Catalogação na Publicação DIVISÃO DE BIBLIOTECA E DOCUMENTAÇÃO - ESALQ/USP
Cerqueira, Pedro Henrique Ramos
Um estudo sobre reconhecimento de padrões: um aprendizado supervisionado com classificador bayesiano / Pedro Henrique Ramos Cerqueira. - - Piracicaba, 2010.
108 p. : il.
Dissertação (Mestrado) - - Escola Superior de Agricultura “Luiz de Queiroz”, 2010. Bibliografia.
1. Aprendizagem 2. Classificação 3. Estatística computacional 4. Inferência bayesiana 5. Marcador molecular 6. Mineração de dados 7. Probabilidade 8. Reconhecimento de padrões 9. Softwares I. Título
CDD 519.50285 C416u
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❉❡❞✐❝❛tór✐❛
❊♠ ♠❡♠ór✐❛ ❞❡ ♠✐♥❤❛ ♠ã❡✱ ❏✉r❛❝✐ ❘❛♠♦s ❈❡rq✉❡✐r❛✳
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✺
❆●❘❆❉❊❈■▼❊◆❚❖❙
Pr✐♠❡✐r❛♠❡♥t❡ ❛♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✿ ♠✐♥❤❛ ❛✈ó✱ ❆❞❡❧❛✐❞❡ ❘❛♠♦s ❈❡rq✉❡✐r❛✱ ♠✐✲ ♥❤❛s t✐❛s✱ ❘♦s❡❧✐ ❘❛♠♦s ❈❡rq✉❡✐r❛ ❡ ❈❛r♠❡♠✱ ♠❡✉ t✐♦ ❈■❉ ❘❛♠♦s ❈❡rq✉❡✐r❛ ❡ ♠❡✉ ♣r✐♠♦ ❘♦❞r✐❣♦ ❙❛✛✉❛♥✱ ♣♦r ❡st❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦✱ ♠❡s♠♦ ❛ ❛ q✉✐❧ô♠❡tr♦s ❞❡ ❞✐stâ♥❝✐❛ ❞✉r❛♥t❡ t♦❞❛ ❡st❛ ❥♦r♥❛❞❛✱ ♣♦r ❜r✐❣❛r❡♠ ❝♦♠✐❣♦ q✉❛♥❞♦ ❡st❛✈❛ ❡rr❛❞♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r s❡r❡♠ ♠✐♥❤❛ ❢❛♠í❧✐❛✳
➚ ♠✐♥❤❛ ♥❛♠♦r❛❞❛✱ ❈❛♠✐❧❛✱ ♣♦r t♦❞♦ ♦ s❡✉ ❛♣♦✐♦✱ ❛❥✉❞❛ ❡ ♣❡♥s❛♠❡♥t♦ ♣♦s✐t✐✈♦✳ ❙❡♠♣r❡ ♠❡ ❢❛③❡♥❞♦ ♦❧❤❛r ♣❛r❛ ❢r❡♥t❡✱ ♥✉♥❝❛ ♠❡ ❞❡✐①❛♥❞♦ ❛❜❛❧❛r ❝♦♠ ❛❞✈❡rs✐❞❛❞❡s ❡ ◆❯◆❈❆ ❞❡s✐st✐r✳
❆♦s ♠❡✉s ♣❛r❡♥t❡s✿ ♠❡✉ ♣r✐♠♦ ❋❡❧✐♣❡ ❡ s✉❛ ❡s♣♦s❛ ❈❛♠✐❧❛✱ ♠❡✉s ♣r✐♠♦s ●✉st❛✈s♦♥ ❡ ❇r✉♥♦✱ ♠✐♥❤❛s t✐❛s ❉✐r❝❡ ❡ ❆❧✐❝❡✱ ♠✐♥❤❛ ♠❛❞r✐♥❤❛ ■r❡♥❡✱ ♣♦r t♦❞♦ ❛♣♦✐♦ ❞✉✲ r❛♥t❡ t♦❞❛s ❛s ❡t❛♣❛s ❞❡ ♠✐♥❤❛ ✈✐❞❛✳
❆♦s ♠❡✉s ❛♠✐❣♦s✿ ❉✐❡❣♦ ✏❈❛✈❛❧♦✑✱ ❆❧✐♥❡✱ ❘✐❝❤❛r❞ ✏▲❡ô♥❝✐♦✑✱ ❇❡t♦✱ ❈❛♠✐❧❛✱ ❉❛♥✐❡❧ ❡ ❏✉❧✐❛♥❛✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❛❧❡❣r✐❛ ❞❡s❝♦♥tr❛çã♦ ❡ ❞✐s❝✉ssõ❡s s♦❜r❡ ✈❛❧♦r❡s q✉❡ s❡♠♣r❡ ✜③❡r❛♠ ❛❜r✐r ♠❡✉s ♦❧❤♦s ♣❛r❛ ♦✉tr♦s ♣♦♥t♦s ❞❡ ✈✐st❛✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❞❡ ♣ós✲❣r❛❞✉❛çã♦✿ ❆❞r✐❛♥❛✱ ▼❛r❝❡❧✐♥♦ ✏P♦♣ó✑✱ ❈áss✐♦✱ ●✉✐❧❤❡r♠❡ ❇✐③✱ ▲✉✐③ ❘✐❝❛r❞♦✱ ❆♥❛ ❏✉❧✐❛✱ ❡ ❊✈❡rt♦♥✱ ❙✐♠♦♥❡✱ ❘❡♥❛t❛✱ ❮t❛❧♦✱ ●❛❜r✐❡❧✱ ❏♦ã♦✱ ❉❛♥✐❡❧✱ ❘✐❝❛r❞♦ ❡ ❚❤✐❛❣♦✳
❊♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ❛♠✐❣♦s ❞❡ ♣ós✲❣r❛❞✉❛çã♦✿ ❘♦❞r✐❣♦ ✏❉✐❣ã♦✑✱ ❘❛♣❤❛❡❧ ✏❑❝♦♥❞❡✑ ♣♦r s❡♠♣r❡ ❡st❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✱ ♥❛s ❞✐s❝✉ssõ❡s s♦❜r❡ ❡♥s✐♥♦ ❡ ❝♦♠♦ s❡r ♣r♦✜ss✐♦♥❛✐s ♠❡❧❤♦r❡s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r ❡st❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❛♣❡rt♦ s❡♠♣r❡ ♦❢❡r❡❝❡♥❞♦ ❛❥✉❞❛✳
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➚s s❡❝r❡tár✐❛s ❞♦ ♣r♦❣r❛♠❛ ❞❡ ♣ós ❣r❛❞✉❛çã♦✿ ▲✉❝✐❛♥❡ ❡ ❙♦❧❛♥❣❡ ♣♦r t♦❞❛ ❛ ❛❥✉❞❛ ♥♦s ♠♦♠❡♥t♦s ❝r✉❝✐❛✐s ❡ ♥❛ ❝♦rr❡r✐❛ ♥❛ r❡t❛ ✜♥❛❧✳
❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ♣r♦❣r❛♠❛ ❞❡ ♣ós ❣r❛❞✉❛çã♦✿ ❊❞✉❛r❞♦✱ ❏♦r❣❡ ❡ ❘♦s♥✐ ♣♦r ♠❡ ❛❥✉❞❛r q✉❛♥❞♦ ❡✉ ♣r❡❝✐s❛✈❛ ❞❡ ❛❧❣✉♠❛ ❛❥✉❞❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ♣r♦❣r❛♠❛ ❞❡ ♣ós ❣r❛❞✉❛çã♦ ❡♠ ❊st❛tíst✐❝❛ ❡ ❊①♣❡r✐♠❡♥t❛çã♦ ❆❣r♦♥ô♠✐❝❛✿ ❈❛r❧♦s ❚❛❞❡✉ ❞♦s ❙❛♥t♦s ❉✐❛s✱ ❊❞✇✐♥ ▼♦✐sés ▼❛r❝♦s ❖rt❡❣❛✱ ❙✐❧✈✐♦ ❙❛♥❞♦✈❛❧ ❩♦❝❝❤✐✱ ❙ô♥✐❛ ▼❛r✐❛ ❉❡ ❙t❡❢❛♥♦ P✐❡❞❛❞❡ ❡ ❚❛❝✐❛♥❛ ❱✐❧❧❡❧❛ ❙❛✈✐❛♥ ♣❡❧♦s ❝♦♥s❡❧❤♦s ❡ ❛❥✉❞❛ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❞ú✈✐❞❛s✳
➚ ♣r♦❢❡ss♦r❛ ❞♦✉t♦r❛ ❏✉❧✐❛♥❛ ●❛r❝✐❛ ❈❡s♣❡❞❡s✱ ♣♦r t♦❞❛ ❛❥✉❞❛ ❡ ❝♦❧❛❜♦r❛çã♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❛ ❞✐ss❡rt❛çã♦✳
❆❣r❛❞❡ç♦ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ♣r♦❢❡ss♦r❛ ❘♦s❡❧✐ ❆♣❛r❡❝✐❞❛ ▲❡❛♥❞r♦✱ ♣♦r ♥✉♥❝❛ t❡r ❞❡s✐st✐❞♦ ❞❡ ♠✐♠ ♠❡s♠♦ ❝♦♠ t♦❞❛ ❛ ♠✐♥❤❛ ❛♣❛r❡♥t❡ ❢❛❧t❛ ❞❡ ❞❡❞✐❝❛çã♦✱ ♣♦r s❡♠♣r❡ ❡st❛r ♣r❡s❡♥t❡ ♣❛r❛ ♠❡ ❛✉①✐❧✐❛r ❡ ❞✐r❡❝✐♦♥❛r ♦ ♠❡✉ ❝❛♠✐♥❤♦✳
❆❣r❛❞❡ç♦ ❡♠ ❡s♣❡❝✐❛❧ à ♣r♦❢❡ss♦r❛ ❈❧❛r✐❝❡ ●❛r❝✐❛ ❇♦r❣❡s ❉❡♠étr✐♦✱ ♦s ♣✉①õ❡s ❞❡ ♦r❡❧❤❛ ❡ ❛s ❝♦♥✈❡rs❛s ❡♠ s✉❛ s❛❧❛✱ s❡♠♣r❡ ♠❡ ❝❤❛♠❛♥❞♦ ❛ ❛t❡♥çã♦ q✉❛♥❞♦ ❡✉ ❡st❛✈❛ ❡rr❛❞♦✱ ❢❛③❡♥❞♦ ❡✉ ♠♦❞✐✜❝❛r ❛ ♠✐♥❤❛ ♣♦st✉r❛✱ t♦r❛♥❞♦✲♠❡ ✉♠ ♣r♦✜ss✐♦♥❛❧ ❡ ♣❡ss♦❛ ♠❡❧❤♦r✱ ❝♦♠ s❡✉s ❝♦♥s❡❧❤♦s✳
❆♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✲ ❈◆Pq ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❡♠ ❢♦r♠❛ ❞❡ ❜♦❧s❛ ❞❡ ❡st✉❞♦s✳
P♦r ✜♠ ❛ t♦❞♦s q✉❡ s❡♠♣r❡ ❛❝r❡❞✐t❛r❛♠ ❡♠ ♠✐♠✳
✼
❙❯▼➪❘■❖
❘❊❙❯▼❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ❆❇❙❚❘❆❈❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ▲■❙❚❆ ❉❊ ❚❆❇❊▲❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶ ■◆❚❘❖❉❯➬➹❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷ ❉❊❙❊◆❱❖▲❱■▼❊◆❚❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✶ ❘❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ P❛❞rõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✷ ❆♣r❡♥❞✐③❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✷✳✶ ❆♣r❡♥❞✐③❛❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✳✷✳✷ ❆♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✳✸ ❈❧❛ss✐✜❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✳✸✳✶ ❈❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✶✳✸✳✷ ❋✉♥çã♦ ❞✐s❝r✐♠✐♥❛♥t❡ ♣❛r❛ ❞✉❛s ❝❧❛ss❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✶✳✸✳✸ ❋✉♥çã♦ ❞✐s❝r✐♠✐♥❛♥t❡ ♣❛r❛ ❞❛❞♦s ❜✐♥ár✐♦s ❡ ♠✉❧t✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✶✳✸✳✹ ❋✉♥çã♦ ❞✐s❝r✐♠✐♥❛♥t❡ ♣❛r❛ ❞✐str✐❜✉✐çã♦ ◆♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✶✳✸✳✺ ❋✉♥çã♦ ❞✐s❝r✐♠✐♥❛♥t❡ ♣❛r❛ ♥♦r♠❛❧ ♠✉❧t✐✈❛r✐❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✶✳✸✳✺✳✶ ❈❛s♦ ✶✿ Σi =σ2■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✷✳✶✳✸✳✺✳✷ ❈❛s♦ ✷✿ Σi = Σ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✷✳✶✳✸✳✺✳✸ ❈❛s♦ ✸✿ Σi❂❛r❜✐trár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✽
✾
❘❊❙❯▼❖
❯♠ ❡st✉❞♦ s♦❜r❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s✿
✉♠ ❛♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ❝♦♠ ❝❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦
❆ ❢❛❝✐❧✐❞❛❞❡ q✉❡ t❡♠♦s ♣❛r❛ r❡❝♦♥❤❡❝❡r ✉♠ r♦st♦✱ ❝♦♠♣r❡❡♥❞❡r ♣❛❧❛✈r❛s ❢❛❧❛❞❛s✱ ❧❡r ♠❛♥✉s❝r✐t♦s✱ ✐❞❡♥t✐✜❝❛r ❝❤❛✈❡s ❞♦ ❝❛rr♦ ♥♦ ❜♦❧s♦ ❡ ❞❡❝✐❞✐r s❡ ✉♠❛ ♠❛çã ❡stá ♠❛❞✉r❛ ♣❡❧♦ s❡✉ ❝❤❡✐r♦✱ ❞❡s♠❡♥t❡♠ ♦s ♣r♦❝❡ss♦s ❝♦♠♣❧❡①♦s q✉❡ ❡stã♦ ♣♦r trás ❞❡ss❡s ❛t♦s ❞❡ r❡❝♦♥❤❡❝❡r ❡st❡s ♣❛❞rõ❡s✳ ❊st❡s r❡❝♦♥❤❡❝✐♠❡♥t♦s tê♠ s✐❞♦ ❝r✉❝✐❛✐s ♣❛r❛ ❛ ♥♦ss❛ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❡ ❛♦ ❧♦♥❣♦ ❞❛s ú❧t✐♠❛s ❞❡③❡♥❛s ❞❡ ♠✐❧❤õ❡s ❞❡ ❛♥♦s ❞❡s❡♥✈♦❧✈❡♠♦s s✐st❡♠❛s s♦✜st✐❝❛❞♦s ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❛s t❛r❡❢❛s✳ ❖ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s t❡♠ ♣♦r ♦❜❥❡t✐✈♦ r❡❛❧✐③❛r ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❞❡t❡r✲ ♠✐♥❛❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❡♠ ❞❡t❡r♠✐♥❛❞❛s ❝❧❛ss❡s ♦✉ ❣r✉♣♦s✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦s s❡✉s ♣❛❞rõ❡s ❡ ♦s ❞❛s ❝❧❛ss❡s✱ ♣❡r♠✐t✐♥❞♦ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ❞♦❝✉✲ ♠❡♥t♦s✱ ❧❡✐t♦r❡s ❞❡ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛❀ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ♣❡ss♦❛s✱ ❧❡✐t♦r❡s ót✐❝♦s ♦✉ ❞❡ ✐♠♣r❡ssã♦ ❞✐❣✐t❛❧❀ ❛✉t♦♠❛çã♦ ✐♥❞✉str✐❛❧✱ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♠❛❣❡♥s ❡ ❛♣❧✐❝❛çõ❡s ❛❣r♦♥ô♠✐❝❛s✱ ❛♥á❧✐s❡ ❞❡ ♠❛r❝❛❞♦r❡s ♠♦❧❡❝✉❧❛r❡s ❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ♣❧❛♥t❛s✱ t♦r♥❛♥❞♦✲s❡ ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ✉♠❛ té❝♥✐❝❛ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛✳ P❛r❛ ✉♠❛ ♠❡❧❤♦r ❝❧❛ss✐✜❝❛çã♦ é ♥❡❝❡ssár✐♦ r❡❛❧✐③❛r ❛♣r❡♥❞✐③❛❞♦s✱ q✉❡ ♣♦❞❡♠ s❡r ❡❧❛❜♦r❛❞♦s ♣❡❧♦ ♠ét♦❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ♦✉ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ❛ ✜♠ ❞❡ ❞❡s❡♥✲ ✈♦❧✈❡r ♦s ❝❧❛ss✐✜❝❛❞♦r❡s✱ t❛✐s ❝♦♠♦ ♦ ❝❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦ ❡ ❛s r❡❞❡s ♥❡✉r❛✐s✱ q✉❡ ♣❡r♠✐t❡♠ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✳ P❛r❛ ✈❡r✐✜❝❛r ❛ q✉❛❧✐❞❛❞❡ ❞❛s ❝❧❛ss✐✜❝❛çõ❡s ❞❡✈❡♠ s❡r ✉t✐❧✐③❛❞❛s ♠❡❞✐çõ❡s ❡s♣❡❝í✜❝❛s✱ ❝♦♠♦ ♦ í♥❞✐❝❡ ❦❛♣♣❛ ♦✉ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡rr♦ ❣❡r❛❧✳ ❉❡st❡ ♠♦❞♦ é ❡ss❡♥❝✐❛❧ ❛ ✉t✐❧✐③❛çã♦ ❞❡ s♦❢t✇❛r❡ ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✱ ❡♥tr❡ ❡❧❡s ♦ ❘ ❡ ♦ ❲❊❑❆✱ ❞❡s❡♥✈♦❧✈✐❞♦ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s✳ ❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ s♦❧✉❝✐♦♥❛r ♦s ♣r♦❜❧❡♠❛s ❡s♣❡❝í✜❝♦s ❞❛s ár❡❛s ❞❡ ❛✉t♦♠❛çã♦ ❡ ❛❣r♦♥♦♠✐❛✱ ❢♦✐ ✉t✐❧✐③❛❞♦ ♦ ♠ét♦❞♦ ❞❡ ❛♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ❝♦♠ ❝❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦ ❡ ♣❛r❛ ✈❡r✐✜❝❛r ❛ q✉❛❧✐❞❛❞❡ ❞❛s ❝❧❛ss✐✜❝❛çõ❡s ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ♦ í♥❞✐❝❡ ❞❡ ❦❛♣♣❛ ❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡rr♦ ❣❡r❛❧ ♣♦r ♠❡✐♦ ❞♦s s♦❢t✇❛r❡ ❘ ❡ ❲❊❑❆✱ ♣❛r❛ ❛s ❝❧❛ss✐✜❝❛çõ❡s ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ❞❛❞♦s ❞❡ ♠❛r❝❛❞♦r❡s ♠♦❧❡❝✉❧❛r❡s✱ ❞❛❞♦s ❞❡ s♦❥❛ ❡ t✐♣♦s ❞❡ ❡♠❜❛❧❛❣❡♥s ❞❡ ♣❡ç❛s ❞❡ ❛✉t♦♠ó✈❡❧✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❆♣r❡♥❞✐③❛❣❡♠❀ ❈❧❛ss✐✜❝❛çã♦❀ ❊st❛tíst✐❝❛ ❝♦♠♣✉t❛❝✐♦♥❛❧❀ ▼❛r❝❛❞♦r ♠♦❧❡❝✉❧❛r❀
✶✵
✶✶
❆❇❙❚❘❆❈❚
❆ st✉❞② ♦♥ ♣❛tt❡r♥ r❡❝♦❣♥✐t✐♦♥✿
s✉♣❡r✈✐s❡❞ ❧❡❛r♥✐♥❣ ✇✐t❤ ❛ ❇❛②❡s✐❛♥ ❝❧❛ss✐✜❡r
❚❤❡ ❢❛❝✐❧✐t② ✇❡ ❤❛✈❡ t♦ r❡❝♦❣♥✐③❡ ❛ ❢❛❝❡✱ t♦ ✉♥❞❡rst❛♥❞ s♣♦❦❡♥ ✇♦r❞s✱ r❡❛❞✐♥❣ ♠❛♥✉s❝r✐♣ts✱ ✐❞❡♥t✐❢②✐♥❣ ❝❛r ❦❡②s ✐♥ ♦✉r ♣♦❝❦❡t ❛♥❞ ❞❡❝✐❞✐♥❣ ✇❤❡t❤❡r ❛♥ ❛♣♣❧❡ ✐s r✐♣❡ ❜② ✐ts s♠❡❧❧✱ ❜❡❧✐❡ t❤❡ ❝♦♠♣❧❡① ♣r♦❝❡ss❡s t❤❛t ❛r❡ ❜❡❤✐♥❞ t❤❡ ❛❝t t♦ r❡❝♦❣♥✐③❡ t❤❡s❡ ♣❛tt❡r♥s✳ ❚❤❡s❡ r❡❝♦❣♥✐t✐♦♥s ❤❛✈❡ ❜❡❡♥ ❝r✉❝✐❛❧ t♦ ♦✉r s✉r✈✐✈❛❧✱ ❛♥❞ ♦✈❡r t❤❡ ♣❛st t❡♥s ♦❢ ♠✐❧❧✐♦♥s ♦❢ ②❡❛rs s♦♣❤✐st✐❝❛t❡❞ s②st❡♠s ✇❡r❡ ❞❡✈❡❧♦♣❡❞ t♦ ❛❝❝♦♠♣❧✐s❤ t❤❡s❡ t❛s❦s✳ ❚❤❡ ♣❛tt❡r♥ r❡❝♦❣♥✐t✐♦♥ ❤❛s ❛✐♠❡❞ t♦ ❝❛rr② ♦✉t t❤❡ r❛t✐♥❣ ♦❢ ❛ ❣✐✈❡♥ s❡t ♦❢ ❞❛t❛ ✐♥ ❝❡rt❛✐♥ ❝❧❛ss❡s ♦r ❣r♦✉♣s✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡✐r st❛♥❞❛r❞s ❛♥❞ t❤♦s❡ ♦❢ t❤❡✐r ❝❧❛ss❡s✱ ❛❧❧♦✇✐♥❣ s❡✈❡r❛❧ ❛♣♣❧✐❝❛t✐♦♥s✱ s✉❝❤ ❛s✿ ❞♦❝✉♠❡♥t ♣r♦❝❡ss✐♥❣✱ ❜❛r ❝♦❞❡ r❡❛❞❡rs✱ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ♣❡♦♣❧❡✱ ♦♣t✐❝❛❧ ❛♥❞ ✜♥❣❡r♣r✐♥t ❞r✐✈❡rs✱ ✐♥❞✉str✐❛❧ ❛✉t♦♠❛t✐♦♥✱ ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣ ❛♥❞ ❛❣r♦♥♦♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s✱ ♠♦❧❡❝✉❧❛r ♠❛r❦❡rs ❛♥❛❧②s✐s ❛♥❞ ♣❧❛♥t✬s ❝❧❛ss✐✜❝❛t✐♦♥✱ ✇❤✐❝❤ ❜❡❝❛♠❡ ❛ t❡❝❤♥✐q✉❡ ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡ ✐♥ r❡❝❡♥t ②❡❛rs✳ ❋♦r ❛ ❜❡tt❡r r❛t✐♥❣ ✐s ♥❡❝❡ss❛r② t♦ ♣❡r❢♦r♠ s♦♠❡ st✉❞✐❡s✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❜② s✉♣❡r✈✐s❡❞ ❛♥❞ ✉♥s✉♣❡r✈✐s❡❞ ♠❡t❤♦❞s✱ ✐♥ ♦r❞❡r t♦ ❞❡✈❡❧♦♣ ❝❧❛ss✐✜❡rs✱ s✉❝❤ ❛s t❤❡ ❇❛②❡s✐❛♥ ❝❧❛ss✐✜❡r ❛♥❞ ♥❡✉r❛❧ ♥❡t✇♦r❦s✱ ✇❤✐❝❤ ❡♥❛❜❧❡ t❤❡ ❞❡❝✐s✐♦♥✲♠❛❦✐♥❣✳ ❚♦ ❝❤❡❝❦ t❤❡ q✉❛❧✐t② ♦❢ r❛t✐♥❣s✱ s♣❡❝✐✜❝ ♠❡❛s✉r❡♠❡♥ts ♠✉st ❜❡ ✉s❡❞✱ s✉❝❤ ❛s t❤❡ ❦❛♣♣❛ ✐♥❞❡① ♦r t❤❡ ❣❡♥❡r❛❧ ❡rr♦r ♣r♦❜❛❜✐❧✐t②✳ ❚❤✉s ✐t ✐s ❡ss❡♥t✐❛❧ t♦ ✉s❡ s♦❢t✇❛r❡ t♦ ♠❛❦❡ ❛ ❞❡❝✐s✐♦♥✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❘ ❛♥❞ ❲❊❑❆✱ ❞❡✈❡❧♦♣❡❞ s♣❡❝✐✜❝❛❧❧② t♦ s♦❧✈❡ ♣❛tt❡r♥ r❡❝♦❣♥✐t✐♦♥ ♣r♦❜❧❡♠s✳ ■♥ ♦r❞❡r t♦ s♦❧✈❡ ♣r♦❜❧❡♠s ♦❢ s♣❡❝✐✜❝ ❛r❡❛s✱ ❛s ❛✉t♦♠❛t✐♦♥ ❛♥❞ ❛❣r♦♥♦♠②✱ t❤❡ s✉♣❡r✈✐s❡❞ ❧❡❛r♥✐♥❣ ♠❡t❤♦❞ ✇✐t❤ ❇❛②❡s✐❛♥ ❝❧❛ss✐✜❡r ✇❛s ✉s❡❞✱ ❛♥❞ t♦ ✈❡r✐❢② t❤❡ r❛t✐♥❣s q✉❛❧✐t②✱ t❤❡ ❦❛♣♣❛ ✐♥❞❡① ❛♥❞ t❤❡ ❣❡♥❡r❛❧ ❡rr♦r ♣r♦❜❛❜✐❧✐t② ✇❡r❡ ✉s❡❞✳ ❙♦❢t✇❛r❡ ❘ ❛♥❞ ❲❊❑❆ ✇❡r❡ ✉s❡❞✱ ✐♥ ♦r❞❡r t♦ ♣❡r❢♦r♠ r❛t✐♥❣s ❢♦r ♠♦❧❡❝✉❧❛r ♠❛r❦❡rs ❞❛t❛✱ s♦②❜❡❛♥ ❞❛t❛ ❛♥❞ t②♣❡s ♦❢ ♣❛❝❦❛❣✐♥❣ ❢♦r ❛✉t♦ ♣❛rts✳
❑❡②✇♦r❞s✿ ❈❧❛ss✐✜❝❛t✐♦♥❀ ▲❡❛r♥✐♥❣❀ ▼♦❧❡❝✉❧❛r ♠❛r❦❡r❀ P❛tt❡r♥ ❘❡❝♦❣♥✐t✐♦♥❀ ❙t❛t✐st✐❝❛❧
✶✷
✶✸
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙
❋✐❣✉r❛ ✶ ✲ ❊str✉t✉r❛ ♣❛r❛ ❝r✐❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ❋✐❣✉r❛ ✷ ✲ ❊str✉t✉r❛ ❞❡ ✉♠ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ❋✐❣✉r❛ ✸ ✲ ❯♥✐✈❡rs♦ ❞❛s ❝❧❛ss❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ❋✐❣✉r❛ ✹ ✲ ❊str✉t✉r❛ ❞♦ ❝❧❛ss✐✜❝❛❞♦r ❞❡ ▼❆P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❋✐❣✉r❛ ✺ ✲ ❱❛❧♦r❡s ♣❛r❛P(ωi|①)❡♠ r❡❧❛çã♦ ❛s ♦❜s❡r✈❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✶✺
▲■❙❚❆ ❉❊ ❚❆❇❊▲❆❙
✶✼
✶ ■◆❚❘❖❉❯➬➹❖
❖ ❝♦♥❝❡✐t♦ ❞❡ r❡❝♦♥❤❡❝❡r ♣❛❞rõ❡s ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♥♦ss❛s ❛çõ❡s ❝♦t✐❞✐❛♥❛s✱ ❞❡s❞❡ ❛ s✐♠♣❧❡s ❞✐❢❡r❡♥❝✐❛çã♦ ❞❡ ❝❤❛✈❡s ❡♠ ♥♦ss♦ ❜♦❧s♦ ♣❡❧♦ t❛t♦ ❝♦♠♦ ❛ ❝♦♥s✉❧t❛ ❞♦ ✈❛❧♦r ❞❡ ✉♠ ♣r♦❞✉t♦ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❧❡✐t♦r ❞❡ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛s ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽✮✳
❖ ❝❛♠♣♦ ❞❛ ❡st❛tíst✐❝❛ é ❝♦♥st❛♥t❡♠❡♥t❡ ❞❡s❛✜❛❞♦ ♣♦r ♣r♦❜❧❡♠❛s q✉❡ ❛ ❝✐ê♥❝✐❛ ❡ ❛ ✐♥❞ústr✐❛ ♣r♦♣õ❡♠✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❡ss❡s ♣r♦❜❧❡♠❛s s✉r❣✐❛♠ ♠✉✐t❛s ✈❡③❡s ❞❡ ❡①♣❡r✐ê♥❝✐❛s ❛❣rí❝♦❧❛s ❡ ✐♥❞✉str✐❛✐s ❡ ❡r❛♠ r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s✳ ❈♦♠ ♦ ❛✈❛♥ç♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡ ♠❛✐♦r ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çõ❡s✱ ♦s ❞❛❞♦s ❛✉♠❡♥t❛r❛♠ ❡♠ r❡❧❛çã♦ à q✉❛♥t✐❞❛❞❡ ❡ ❝♦♠♣❧❡①✐❞❛❞❡✳ ❉✐✜❝✉❧❞❛❞❡s ♥♦ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❧❡✈❛r❛♠ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❞❛t❛ ♠✐♥✐♥❣✳ ❈♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❡st✉❞❛r ♦s ♣r♦❜❧❡♠❛s ♥❛ ❜✐♦❧♦❣✐❛ ❡ ♥❛ ♠❡❞✐❝✐♥❛✱ q✉❡ ♥❡❝❡ss✐t❛♠ ❞♦ ✉s♦ ❞❡ ❡st❛tís✲ t✐❝❛ ❡ ✐♥❢♦r♠át✐❝❛✱ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ❛ ❜✐♦✐♥❢♦r♠át✐❝❛ ✭❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ▼■❚❈❍❊▲▲✱ ✶✾✾✼✮✳
❈♦♠ ♦ ❛♣r✐♠♦r❛♠❡♥t♦ ❞❛s té❝♥✐❝❛s ♣❛r❛ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❡ ❛ ❢❛❝✐❧✐❞❛❞❡ ❞❡ s❡ ♦❜t❡r ❛❝❡ss♦ à ✐♥❢♦r♠❛çã♦✱ ♠✉✐t♦s ❞❛❞♦s tê♠ s✐❞♦ ❝♦❧❡t❛❞♦s ❡♠ ❞✐✈❡rs❛s ár❡❛s✳ ❖ tr❛❜❛❧❤♦ ❡st❛tíst✐❝♦ ❝♦♥s✐st❡ ❡♠ ✉t✐❧✐③❛r ❛s ❢❡rr❛♠❡♥t❛s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ❝♦♥s❡❣✉✐r ❡①tr❛✐r ♣❛❞rõ❡s ✐♠✲ ♣♦rt❛♥t❡s ❡ t❡♥❞ê♥❝✐❛s✱ ❞❡st❡ ♠♦❞♦✱ é ♣♦ssí✈❡❧ ❡♥t❡♥❞❡r ♦ q✉❡ ♦s ✏❞❛❞♦s ❞✐③❡♠✑ ✭❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽✮✳
❖ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ❞❛❞♦s ♣♦ss✐❜✐❧✐t❛✱ r❡❛❧✐③❛r ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❛♥á❧✐s❡s ♣♦r ♠❡✐♦ ❞❡ ♠ét♦❞♦s ❞✐❢❡r❡♥t❡s✳ ❈♦♠ ♦s ❞❛❞♦s ❞✐s♣♦♥í✈❡✐s ❤á ♦ ✐♥t❡r❡ss❡ ❡♠ s❡ ♦❜t❡r ✐♥❢♦r♠❛çõ❡s q✉❡ ♣♦ss❛♠ ❛❥✉❞❛r ❡ ❢❛❝✐❧✐t❛r ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✱ ♦✉ s❡❥❛✱ ❞❡s❝♦❜r✐r ♣❛❞rõ❡s ❡♠ r❡❧❛çã♦ ❛♦s ❞❛❞♦s✳ ❖ ♣r♦❜❧❡♠❛ ❞❛ ♣r♦❝✉r❛ ❞❡ ♣❛❞rõ❡s t❡♠ ✉♠❛ ❧♦♥❣❛ ❤✐stór✐❛ ❞❡ s✉❝❡ss♦✱ ❛s ♦❜s❡r✈❛çõ❡s ❛str♦♥ô♠✐❝❛s ❞❡ ❚②❝❤♦ ❇r❛❤❡✱ ♥♦ sé❝✉❧♦ ❳❱■✱ ♣❡r♠✐t✐r❛♠ ❏♦❤❛♥♥❡s ❑❡♣❧❡r ❞❡s❝♦❜r✐r ❛s ❧❡✐s ❡♠♣ír✐❝❛s ❞♦ ♠♦✈✐♠❡♥t♦ ♣❧❛♥❡tár✐♦✱ q✉❡ ♣♦r s✉❛ ✈❡③✱ s❡r✈✐r❛♠ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ r❡❣✉❧❛r✐❞❛❞❡s ♥♦s ❡s♣❡❝tr♦s ❛tô♠✐❝♦s ❞❡s❡♠♣❡♥❤♦✉ ✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ✈❡r✐✜❝❛çã♦ ❞❛ ❢ís✐❝❛ q✉â♥t✐❝❛ ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳ ✭❇■❙❍❖P✱ ✷✵✵✻✮✳
✶✽
❝❛t❡❣♦r✐❛s✳ ❆ss✐♠ s❡♥❞♦✱ ♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❛t♦ ❞❡ ✉t✐❧✐③❛r ♦s ❞❛❞♦s ❜r✉t♦s ❡ ❝♦♠ ❜❛s❡ ♥♦s ♣❛❞rõ❡s ❞❛ ✏❝❛t❡❣♦r✐❛✑ ♣❛r❛ t♦♠❛r ❞❡❝✐sõ❡s✳ ❖ ❛t♦ ❞❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s t❡♠ s✐❞♦ ❝r✉❝✐❛❧ ♣❛r❛ ❛ ♥♦ss❛ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❛♦ ❧♦♥❣♦ ❞♦s ú❧t✐♠♦s ❛♥♦s ✭❇■❙❍❖P✱ ✷✵✵✻❀ ❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❏❆■◆ ❡t ❛❧✱ ✷✵✵✵✮✳
❆ ár❡❛ ❞❡ s❡❣✉r❛♥ç❛✱ t❡♠ ✉t✐❧✐③❛❞♦ ♠✉✐t♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s✱ ♣♦r ♠❡✐♦ ❞❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ♣❡ss♦❛s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✿ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ✈♦③❡s✱ ❞✐❣✐t❛✐s✱ ❢❛❝❡ ❡♥tr❡ ♦✉tr❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ✐♥❞✐✈í❞✉♦s✳ ❆t✉❛❧♠❡♥t❡ ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ t❡♠ s✐❞♦ ✉t✐❧✐③❛❞♦ ❡♠ ❞✐✈❡rs❛s ♦❝❛s✐õ❡s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ❡❧❡✐çõ❡s✱ ❛❝❡ss♦ ❛ ❝♦♠♣✉t❛❞♦r❡s✱ s✉❜st✐✲ t✉✐çã♦ ❞❡ s❡♥❤❛s ❡♠ ❛❣ê♥❝✐❛s ❜❛♥❝ár✐❛s✱ ❞❡♥tr❡ ♦✉tr❛s s✐t✉❛çõ❡s ❡♠ q✉❡ é ✐♠♣r❡s❝✐♥❞í✈❡❧ t❡r ❝❡rt❡③❛ q✉❡ ♦ ❛❝❡ss♦ s❡rá ❞❛❞♦ à ♣❡ss♦❛ ❝♦rr❡t❛ ✭❈❆▼P❖❙✱ ✷✵✵✶❀ ❏❆■◆ ❡t ❛❧✱ ✷✵✵✵❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
❖s ❞❡s❛✜♦s ♣❛r❛ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ❞❛❞♦s r❡❛❧✐③❛r❛♠ ✉♠❛ r❡✈♦❧✉çã♦ ♥❛s ❝✐ê♥❝✐❛s ❡st❛tíst✐❝❛s✱ ❧❡✈❛♥❞♦ ❛ ❡st✉❞♦s ❡s♣❡❝í✜❝♦s ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✱ s❡♠♣r❡ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♠❡❧❤♦r❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✱ ♦✉ s❡❥❛✱ t♦♠❛r ❛ ❞❡❝✐sã♦ ❝♦rr❡t❛ ✭❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺✮✳
❱✐s❛♥❞♦ ❛♣❡r❢❡✐ç♦❛r ❡ss❡s r❡s✉❧t❛❞♦s✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ♦s ❛♣r❡♥❞✐③❛❞♦s✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ✉t✐❧✐③❛r ♦s ❞❛❞♦s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ❡❧❛❜♦r❛r ❝r✐tér✐♦s ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✱ ♣♦✐s ❞❡st❡ ♠♦❞♦ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ♠❡❧❤♦r❡s r❡s✉❧t❛❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦✳ ❖ ❛♣r❡♥❞✐③❛❞♦ ♣♦❞❡ s❡r s✉♣❡r✈✐s✐♦♥❛❞♦ ♦✉ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦✳ ◆♦ ❛♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ♦ ♦❜❥❡t✐✈♦ é ♣r❡❞✐③❡r ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ✉♠❛ ♦❜s❡r✈❛çã♦ ❝♦♠ ❜❛s❡ ❡♠ ✉♠❛ sér✐❡ ❞❡ ♠❡❞✐❞❛s ❞❡ ❡♥tr❛❞❛✱ s❡♥❞♦ q✉❡ ❛s q✉❛♥t✐❞❛❞❡s ❞❡ ❝❛t❡❣♦r✐❛s sã♦ ❝♦♥❤❡❝✐❞❛s✱ ♥♦ ❛♣r❡♥❞✐③❛❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ♦ ♦❜❥❡t✐✈♦ é ❞❡s❝r❡✈❡r ❛s ❛ss♦❝✐❛çõ❡s ❡ ♦s ♣❛❞rõ❡s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛s ❞❡ ❡♥tr❛❞❛✱ ❡ ♥❡ss❡ ❝❛s♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛t❡❣♦r✐❛s ♥ã♦ é ❝♦♥❤❡❝✐❞❛ ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
❉❡♥tr❡ ❛s té❝♥✐❝❛s ❞✐s♣♦♥í✈❡✐s ♣❛r❛r r❡❛❧✐③❛r ❛s ❝❧❛ss✐✜❝❛çõ❡s ❞♦s ❞❛❞♦s✱ ❛ té❝♥✐❝❛ ❞❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s ♠❡r❡❝❡ ✉♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦✳ ❊st❛ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛✉①✐❧✐❛r ❛s ♠❛✐s ❞✐✈❡rs❛s ár❡❛s✱ t❛✐s ❝♦♠♦✿
✶✾
✷✮ ❆✉t♦♠❛çã♦ ✐♥❞✉str✐❛❧✿ ■♥s♣❡❝çã♦ ❡ ♠♦♥t❛❣❡♠ ❞❡ ♦❜❥❡t♦s ❝♦♠♣❧❡①♦s ✭✐♥s♣❡❝çã♦ ❞❡ ❝✐r✲ ❝✉✐t♦s ✐♠♣r❡ss♦s✱ ✐♥s♣❡❝çã♦ ❞❡ ♣❛rt❡s ❞❡ ♠áq✉✐♥❛s ❡ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♠❛❣❡♠✮❀
✸✮ ❉❡t❡❝çã♦ r❡♠♦t❛ ✭❘❡♠♦t❡ s❡♥s✐♥❣✮✿ ❖❜s❡r✈❛çã♦ ❞♦ ♣❧❛♥❡t❛ ♣♦r ♠❡✐♦ ❞❡ s❡♥s♦r❡s ❡♠ s❛té❧✐t❡s ♦✉ ❛✈✐õ❡s ✭♣r❡✈✐sã♦ ❞❛ ❡✈♦❧✉çã♦ ❞❡ ❝✉❧t✉r❛s✱ ♣❧❛♥❡❥❛♠❡♥t♦ ❞❡ ✉s♦ ❞❡ t❡rr❛s✱ ♠♦♥✐t♦r✐③❛çã♦ ❛♠❜✐❡♥t❛❧✱ ♠❡t❡♦r♦❧♦❣✐❛✱ ❡①♣❧♦r❛çã♦ ❞❡ ♠✐♥❛s ❡ ♠❛♣❛s t♦♣♦❣rá✜❝♦s✮❀
✹✮ ▼❡❞✐❝✐♥❛ ❡ ❜✐♦❧♦❣✐❛✿ Pr♦❝❡ss❛♠❡♥t♦ ❞❡ ❞✐✈❡rs♦s s✐♥❛✐s ❡ ✐♠❛❣❡♥s ♠é❞✐❝❛s ✭❝♦♥t❛❣❡♠ ❞❡ ❝é❧✉❧❛s ♥♦ s❛♥❣✉❡✱ ❞❡t❡❝çã♦ ❞❡ t✉♠♦r❡s ❡♠ ✐♠❛❣❡♥s ❞❡ ❘❛✐♦s✲❳✱ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ t❡❝✐❞♦s ✉s❛♥❞♦ ✉❧tr❛✲s♦♥s✱ ❛♥á❧✐s❡ ❞❡ ✐♠❛❣❡♥s ❞❡ ❝r♦♠♦ss♦♠♦s✱ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ❡❧❡tr♦❝❛r❞✐♦❣r❛✲ ♠❛s ❡ ❞✐❛❣♥óst✐❝♦ ♠é❞✐❝♦❀
✺✮ ■❞❡♥t✐✜❝❛çã♦ ❞❡ ♣❡ss♦❛s✿ ❘❡str✐çã♦ ❞❡ ❛❝❡ss♦ ❡♠ ✐♥st❛❧❛çõ❡s ❞❡ s❡❣✉r❛♥ç❛ ✭r❡❝♦♥❤❡❝✐✲ ♠❡♥t♦ ❞❡ ✈♦③✱ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ✐♠♣r❡ssõ❡s ❞✐❣✐t❛✐s ❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❝❛r❛s✮❀
✻✮ ❆♣❧✐❝❛çõ❡s ❝✐❡♥tí✜❝❛s ✭✐♥t❡r♣r❡t❛çã♦ ❞❡ ♦♥❞❛s sís♠✐❝❛s ♣❛r❛ ♣r❡✈✐sã♦ ❞❡ t❡rr❡♠♦t♦s ❡ ❛♥á❧✐s❡ ❞❛ ❝♦♠♣♦s✐çã♦ ♠♦❧❡❝✉❧❛r ♣♦r ♠❡✐♦ ❞❡ ✐♠❛❣❡♥s ❞❡ ♠✐❝r♦s❝ó♣✐♦ ❡❧❡❝tró♥✐❝♦✮❀
✼✮ ❆♣❧✐❝❛çõ❡s ♥❛ ❛❣r✐❝✉❧t✉r❛ ✭❞✐r❡❝✐♦♥❛♠❡♥t♦ ❞❡ ❡q✉✐♣❛♠❡♥t♦✱ ✐♥s♣❡❝çã♦ ❞❡ ♣r♦❞✉t♦s✱ ♦r✲ ❞❡♥❛çã♦ ❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ♣r♦❞✉t♦s✮✳
P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ♣❛❞rõ❡s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✱ é ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ✉♠❛ ❛❜♦r❞❛❣❡♠ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✿ ♠♦r❢♦❧♦❣✐❛ ♠❛t❡♠át✐❝❛ ❝♦♠ ❛♣r❡♥❞✐✲ ③❛❞♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡ ❡st❛tíst✐❝❛✳ ❖✉tr♦s ♣r♦❝❡❞✐♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s sã♦✿ ✶✮ ♦s ❛♣r❡♥❞✐③❛❞♦s ✷✮ ❛ ❡s❝♦❧❤❛ ❞♦ ❝❧❛ss✐✜❝❛❞♦r✱ q✉❡ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❡❧❛❜♦r❛r ♦s ❛❧❣♦r✐t♠♦s ❞❡ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✱ ❡♥tr❡ ❡❧❡s tê♠✲s❡✿ ♦ ❝❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦✱ r❡❞❡s ♥❡✉r❛✐s✱ ár✈♦r❡ ❞❡ ❞❡❝✐sã♦ ♦✉ ❛ r❡❣r❛ ❞♦s k✲✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s ✭●❆❉■✱ ✷✵✵✽❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
✷✵
✷✶
✷ ❉❊❙❊◆❱❖▲❱■▼❊◆❚❖
P❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ s❡rá ❢❡✐t❛ ✉♠❛ r❡✈✐sã♦ s♦❜r❡ r❡❝♦♥❤❡❝✐✲ ♠❡♥t♦ ❞❡ ♣❛❞rõ❡s ❡ s❡❧❡çã♦ ❞❡ ✈❛r✐á✈❡✐s✳
✷✳✶ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛
✷✳✶✳✶ ❘❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ P❛❞rõ❡s
P❛r❛ ❡♥t❡♥❞❡r ♠❡❧❤♦r ♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s é ✐♥t❡r❡ss❛♥t❡ ❡♥t❡♥❞❡r ❛ ♠♦r❢♦❧♦❣✐❛ ❞❡st❛s ♣❛❧❛✈r❛s✱ ♣♦r ❞❡✜♥✐çã♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ é ❝♦♥s✐❞❡r❛❞♦ ♦ ❛t♦ ❞❡ r❡❝♦♥❤❡❝❡r✱ q✉❡ ♣♦r s✉❛ ✈❡③ é ❞❡✜♥✐❞♦ ❝♦♠♦ ❞✐st✐♥❣✉✐r ♣♦r ♠❡✐♦ ❞❡ ❝❡rt♦s ❝❛r❛❝t❡r❡s✱ ❡ ♣❛❞rõ❡s sã♦ ♠♦❞❡❧♦s q✉❡ s❡r✈❡♠ ❞❡ r❡❢❡rê♥❝✐❛✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ r❡❧❛çã♦ ❞❡ ✉♠❛s ❝♦✐s❛s ❝♦♠ ♦✉tr❛s✳ ❉❡st❡ ♠♦❞♦ ♣♦❞❡✲s❡ ♣❡♥s❛r ❡♠ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s ❝♦♠♦ ✉♠❛ té❝♥✐❝❛ q✉❡ ♣♦r ♠❡✐♦ ❞❡ ❡str✉t✉r❛s ❞❡ r❡❢❡rê♥❝✐❛ r❡❛❧✐③❛ ❛ ❞✐st✐♥çã♦ ❞❡st❛s✳ P❛r❛ ❉✉❞❛ ❡t ❛❧ ✭✷✵✵✵✮ ♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❛t♦ ❞❡ ✉t✐❧✐③❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦❜s❡r✈❛çõ❡s ✭❞❛❞♦s✮ ♣❛r❛ t♦♠❛r ✉♠❛ ❛çã♦✱ ❜❛s❡❛❞❛ ♥♦ ♣❛❞rã♦ ❞❡ s✉❛ ✏❝❛t❡❣♦r✐❛✑✳
❖ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❞✐✈❡rs❛s ár❡❛s ❞❡ ♣❡sq✉✐s❛ ❡ ❛❜r❛♥❣❡ ❞❡s❞❡ ❛ ❞❡t❡❝çã♦ ❞❡ ♣❛❞rõ❡s ❞❛ ❡s❝♦❧❤❛ ♠❛✐s s✐♠♣❧❡s ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s✱ ❝♦♠♦ ❛ ❝♦♠♣❧❡①❛ r❡❛❧✐③❛çã♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠✱ ❡st❡ ❡stá ♣r❡s❡♥t❡ ❡♠ ♥♦ss♦ ❝♦t✐❞✐❛♥♦ ❡♠ ❞✐✈❡rs❛s ♦❝❛s✐õ❡s✱ t❛✐s q✉❛✐s ❛ ❡s❝♦❧❤❛ ❞❡ ♦❜❥❡t♦s ♣♦r ♠❡✐♦ ❞♦ t❛t♦ ♦✉ ♦ ❢❛t♦ ❞❡ ✉t✐❧✐③❛r ❧❡✐t♦r❡s ❞❡ ❝ó❞✐❣♦s ❞❡ ❜❛rr❛ ♣❛r❛ ♦ ♣❛❣❛♠❡♥t♦ ❞❡ ❝♦♥t❛s ✭❈❆▼P❖❙✱ ✷✵✵✶❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ❏❆■◆ ❡t ❛❧✱ ✷✵✵✵❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
❈♦♠ ❛ t❡❝♥♦❧♦❣✐❛ ❞✐s♣♦♥í✈❡❧ ❛t✉❛❧♠❡♥t❡ é ♣♦ssí✈❡❧ ❛r♠❛③❡♥❛r ✉♠❛ ❣r❛♥❞❡ q✉❛♥✲ t✐❞❛❞❡ ❞❡ ❞❛❞♦s✱ ❞❡st❡ ♠♦❞♦ ❤á ✉♠❛ ♠❛✐♦r ♣♦ss✐❜✐❧✐❞❛❞❡ q✉❡ s❡❥❛♠ ❡♥❝♦♥tr❛❞♦s ♣❛❞rõ❡s✱ t❡♥❞ê♥❝✐❛s ❡ ❛♥♦♠❛❧✐❛s ❡ ❛ss✐♠ tr❛♥s❢♦r♠❛r ❡st❡s ❞❛❞♦s ❡♠ ✐♥❢♦r♠❛çõ❡s✱ ❡ ❛s ✐♥❢♦r♠❛çõ❡s✱ ❡♠ ❜❛s❡ ♣❛r❛ ❛s ❝❧❛ss✐✜❝❛çõ❡s ✭❲■❚❚❊◆❀ ❋❘❆◆❑✱ ✷✵✵✺✮✳
P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ♣❛❞rõ❡s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✱ é ♥❡❝❡ssár✐♦ ❞❡✜♥✐r q✉❛❧ ❛❜♦r❞❛❣❡♠ s❡rá ✉t✐❧✐③❛❞❛✱ ❞❡♥tr❡ ❡❧❛s tê♠✲s❡✿
✶✮ ❈❛s❛♠❡♥t♦ ♦✉ ❛ss♦❝✐❛çã♦ ✭t❡♠♣❧❛t❡ ♠❛t❝❤✐♥❣✮❀
✷✷
✸✮ ▲ó❣✐❝❛ ♥❡❜✉❧♦s❛❀
✹✮ ▼♦r❢♦❧♦❣✐❛ ♠❛t❡♠át✐❝❛ ❝♦♠ ❛♣r❡♥❞✐③❛❞♦ ❝♦♠♣✉t❛❝✐♦♥❛❧❀
✺✮ ❊st❛tíst✐❝❛✳
❉❡♣♦✐s ❞❛ ❞❡✜♥✐çã♦ ❞❛ ❛❜♦r❞❛❣❡♠ ❛ s❡r ✉t✐❧✐③❛❞❛✱ é ♥❡❝❡ssár✐♦ ❞❡✜♥✐r ♦ ❛♣r❡♥❞✐✲ ③❛❞♦ ❞❡ ✐♥t❡r❡ss❡✱ ♦ ❛♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ❡♠ q✉❡✱ sã♦ ❝♦♥❤❡❝✐❞❛s ❛s ❝❧❛ss❡s ♣❛r❛ ❛ ❝❧❛ss✐✜❝❛çã♦✱ ♦✉✱ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ❡♠ q✉❡ ♥ã♦ sã♦ ❝♦♥❤❡❝✐❞❛s ❛s ❝❧❛ss❡s✱ ❞❡st❡ ♠♦❞♦ sã♦ ❝r✐❛❞♦s ❣r✉♣♦s ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛s s❡♠❡❧❤❛♥t❡s✳
P❛r❛ s❡r❡♠ t♦♠❛❞❛s ❛s ❞❡❝✐sõ❡s ♦s ❛♣r❡♥❞✐③❛❞♦s sã♦ ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛✱ ♣♦✐s✱ ♣♦r ♠❡✐♦ ❞❡❧❡s sã♦ ❞❡✜♥✐❞♦s ♦s ❛❧❣♦r✐t♠♦s ❞❡ ❞❡❝✐sã♦✱ q✉❡ sã♦ ❢❡rr❛♠❡♥t❛s q✉❡ r❡❛❧✐③❛♠ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✳ ❊st❡s ❛♣r❡♥❞✐③❛❞♦s sã♦ r❡❛❧✐③❛❞♦s ❛ ♣❛rt✐r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s r❡t✐r❛❞♦s ❞♦ ♦r✐❣✐♥❛❧✱ ❡ sã♦ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞❡ tr❡✐♥♦✱ ❡st❡ ❝♦♥❥✉♥t♦ ♣♦ss✉✐ ❡①❡♠♣❧♦s ❞❡ ♣❛❞rõ❡s ❞❡ t♦❞❛s ❛s ❝❧❛ss❡s ❡①✐st❡♥t❡s ♥♦ s✐st❡♠❛✱ ♣❛r❛ ✈❡r✐✜❝❛r ❛ ❡✜❝✐ê♥❝✐❛ ❞❛ ❞❡❝✐sã♦ é ✉t✐❧✐③❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❛ ✈❛❧✐❞❛çã♦✱ ♣r❡❢❡r❡♥❝✐❛❧♠❡♥t❡ ❝♦♠ ❞❛❞♦s q✉❡ ♥ã♦ ❡st❡❥❛♠ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ tr❡✐♥♦✱ ♦✉ s❡❥❛✱ ❛s ♦❜s❡r✈❛çõ❡s r❡st❛♥t❡s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ♦r✐❣✐♥❛❧ ✭❈❆▼P❖❙✱ ✷✵✵✶❀ ❉❯❉❆ ❡t ❛❧✱ ✷✵✵✽❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
❊①✐st❡♠ ❞✐❢❡r❡♥t❡s ♠ét♦❞♦s ♣❛r❛ s❡❧❡❝✐♦♥❛r ♦ ❝♦♥❥✉♥t♦ ❞❡ tr❡✐♥♦ ❡ r❡❛❧✐③❛r ❛ ✈❛❧✐❞❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❞❡❝✐sã♦ ❞❡♥tr❡ ❡❧❡s tê♠✲s❡✿
✶✮ P♦r❝❡♥t❛❣❡♠ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✿ ❈♦♥s✐st❡ ❡♠ ✉t✐❧✐③❛r ✉♠❛ ❛♠♦str❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ♣❛r❛ ❡❧❛❜♦r❛r ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❡ ❝♦♠ ❛ ♠♦str❛ ♥ã♦ ✉t✐❧✐③❛❞❛ r❡❛❧✐③❛r ❛ ✈❛❧✐❞❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ✉t✐❧✐③❛❞♦❀
✷✮ ❱❛❧✐❞❛çã♦ ❝r✉③❛❞❛✿ ❈♦♥s✐st❡ ❡♠ ❞✐✈✐❞✐r ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❡♠ p s✉❜❝♦♥❥✉♥t♦s✱ ❡♠
s❡q✉ê♥❝✐❛ ✉t✐❧✐③❛r p−1 s✉❜❝♦♥❥✉♥t♦s ♣❛r❛ ❝r✐❛r ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦✱ ❡♠ s❡❣✉✐❞❛
✉t✐❧✐③❛✲s❡ ♦ s✉❜❝♦♥❥✉♥t♦ q✉❡ ♥ã♦ ❢♦✐ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ ✈❛❧✐❞❛çã♦✳ ❊st❡ ♣r♦❝❡ss♦ é r❡❛❧✐③❛❞♦ ♣❛r❛ t♦❞♦s ♦s p s✉❜❝♦♥❥✉♥t♦s✱ ♦✉ s❡❥❛✱ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛
✈❛❧✐❞❛çã♦ ❡ ♣❛r❛ ❝r✐❛r ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦✳
✷✸
❋✐❣✉r❛ ✶ ✲ ❊str✉t✉r❛ ♣❛r❛ ❝r✐❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦
✶✮ ❆q✉✐s✐çã♦ ❞❡ ❞❛❞♦s✿ ❖❜t❡♥çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ♣❛r❛ ❛♥á❧✐s❡❀
✷✮ Pré✲♣r♦❝❡ss❛♠❡♥t♦✿ ❙❡r✈❡ ♣❛r❛ ❡❧✐♠✐♥❛r r✉í❞♦s ♦✉ ❛s ❞✐st♦rçõ❡s❀
✸✮ ❊①tr❛çã♦ ❡ s❡❧❡çã♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s✿ ❆ ❡①tr❛çã♦ ❝r✐❛ ✉♠ ✈❡t♦r ❝♦♠ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ❞❛❞♦s✱ tr❛♥s❢♦r♠❛♥❞♦✲♦s ❡♠ ❛tr✐❜✉t♦s✱ ♣r♦♣r✐❡❞❛❞❡s ♦✉ ❝❛r❛❝t❡ríst✐❝❛s✱ q✉❛♥❞♦ ♥❡❝❡ssár✐♦ ♦ s❡❧❡t♦r ❡❧✐♠✐♥❛ ❛s ❝❛r❛❝t❡ríst✐❝❛s ♠❛✐s r❡❞✉♥❞❛♥t❡s❀
✹✮ ❈♦♠ ❜❛s❡ ♥❛ ❞❡❝✐sã♦ ❞❡ ✐♥t❡r❡ss❡ é ❡❧❛❜♦r❛❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❞❡❝✐sã♦❀
❖ ♣r✐♠❡✐r♦ ♣❛ss♦ é ❛ ♦❜t❡♥çã♦ ❞♦s ❞❛❞♦s✱ ❡ ❛ss✐♠ ❞❡✜♥✐r✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ s✉❛s ❡str✉t✉r❛ q✉❛❧ ♦ t✐♣♦ ❞❡ ❛♣r❡♥❞✐③❛❞♦ s❡rá ❡❧❛❜♦r❛❞♦✱ q✉❛♥t♦s ❡ ❞❡ q✉❛ ♠❛♥❡✐r❛ ♦s ❞❛❞♦s s❡rã♦ ❞❡✜♥✐❞♦s ❝♦♠♦ ♣❡rt❡♥❝❡♥t❡ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ tr❡✐♥♦ ♦✉ ❞❡ ✈❛❧✐❞❛çã♦ ❡ q✉❛❧ t✐♣♦ ❞❡ ✈❛❧✐❞❛çã♦ q✉❡ s❡rá ✉t✐❧✐③❛❞❛✳
❖ ♣ré✲♣r♦❝❡ss❛♠❡♥t♦ ❡ ❛ s❡❧❡çã♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s sã♦ r❡❛❧✐③❛❞♦s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛❝❡❧❡r❛r ♦ ♣r♦❝❡ss♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ ♦ ♦❜❥❡t✐✈♦ ❞❡ ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❢♦r ❞❡t❡❝çã♦ ❞❡ ✉♠❛ ❢❛❝❡ ❡♠ t❡♠♣♦ r❡❛❧✱ ♦ ❝♦♠♣✉t❛❞♦r ♥❡❝❡ss✐t❛ ❧✐❞❛r ❝♦♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ♣✐①❡❧s ♣♦r s❡❣✉♥❞♦✱ ❡ ♥❡❝❡ss✐t❛ ❛♣r❡s❡♥t❛r ❡st❡s ❞✐r❡t❛♠❡♥t❡ ❛ ✉♠ ❝♦♠♣❧❡①♦ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s✱ q✉❡ ♣♦❞❡ s❡ t♦r♥❛r ❡♠ ❛❧❣✉♥s ❝❛s♦s ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♥✈✐á✈❡❧✱ ❞❡st❡ ♠♦❞♦ ❝♦♠ ❛ s❡❧❡çã♦ ❞❡ ❝❛r❛t❡ríst✐❝❛s ❡ ❝♦♠ ❛ ❡❧✐♠✐♥❛çã♦ ❞❡ r✉í❞♦s ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣♦❞❡ s❡r s✐♠♣❧✐✜❝❛❞♦ ✭❇■❙❍❖P✱ ✷✵✵✻❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ❏■❆◆● ❡t ❛❧✱ ✷✵✵✺❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
✷✹
✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ ♣❡sq✉✐s❛❞♦r✱ q✉❡ ♥ã♦ ♣♦❞❡ ❞❡✐①❛r ❞❡ ❛♥á❧✐s❛✲❧❛✱ ❡ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦✲ ❜❧❡♠❛✱ ♦✉ s❡❥❛✱ ♣♦❞❡ ✐♥✢✉❡♥❝✐❛r ♥❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦ s✐st❡♠❛ ♣♦❞❡♥❞♦ ♣r❡❥✉❞✐❝❛r ❛ s✉❛ ❡①❛t✐❞ã♦ ✭❈❆▼P❖❙✱ ✷✵✵✶❀ ❏❆■◆ ❡t ❛❧✱ ✷✵✵✵❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
❆♣ós ✉t✐❧✐③❛r ♦s ❞❛❞♦s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ tr❡✐♥♦ ♣❛r❛ ❡❧❛❜♦r❛r ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❞❡❝✐sã♦ é ✉t✐❧✐③❛❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❞❡ ✈❛❧✐❞❛çã♦ ♣❛r❛ r❡❛❧✐③❛r ❛s ❞❡❝✐sõ❡s ❞❡ ✐♥t❡r❡ss❡✳
P❡❧❛ ❋✐❣✉r❛ ✷ ♣♦❞❡✲s❡ ✈❡r✐✜❝❛r ❝♦♠♦ é ❛ ❡str✉t✉r❛ ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦ ♥♦s r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s✳
❋✐❣✉r❛ ✷ ✲ ❊str✉t✉r❛ ❞❡ ✉♠ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s
P❛r❛ ❛s ❡t❛♣❛ ❞❡ ❈❧❛ss✐✜❝❛çã♦ tê♠✲s❡✿
✶✮ Pré✲Pr♦❝❡ss❛♠❡♥t♦✿ ❙❡r✈❡ ♣❛r❛ ❡❧✐♠✐♥❛r r✉í❞♦s ♦✉ ❛s ❞✐st♦rçõ❡s❀
✷✮ ▼❡❞✐çã♦ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s✿ ❙ã♦ r❡❛❧✐③❛❞❛s ❛s ♠❡❞✐çõ❡s ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ✐♥t❡r❡ss❡ ❡ ❛ss✐♠ ❝r✐❛✲s❡ ♦ ✈❡t♦r ❞❡ ❝❛r❛❝t❡ríst✐❝❛s❀
✸✮ ❘❡s✉❧t❛❞♦ ❞♦ ❛❧❣♦r✐t♠♦✿ ❖s ❞❛❞♦s ♠❡❞✐❞♦s sã♦ ✉t✐❧✐③❛❞♦s ♥♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ ✈❛❧♦r❡s❀
✹✮ ❉❡❝✐sã♦✿ ❈♦♠ ♦s ✈❛❧♦r❡s ♦❜t✐❞♦s ♣♦r ♠❡✐♦ ❞♦s ❛❧❣♦r✐t♠♦s é r❡❛❧✐③❛❞❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦✳
✷✺
❆s ❝❛r❛❝t❡ríst✐❝❛s ♠❡❞✐❞❛s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❣❡r❛♠ ✈❛❧♦r❡s q✉❡ sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦✳
◆❛ ❛❜♦r❞❛❣❡♠ ❞❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s✱ ❝❛❞❛ ♣❛❞rã♦ é r❡♣r❡s❡♥t❛❞♦ ❡♠ t❡r♠♦s ❞❡ d ❝❛r❛❝t❡ríst✐❝❛s ♦✉ ❛tr✐❜✉t♦s ❡ é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ✈❡t♦r ① = [x1, . . . , xd]t✱ ❡♠
q✉❡ ❝❛❞❛xi é ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛✳ ❖ ❡s♣❛ç♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡s ❞❡ ❝❛r❛❝t❡ríst✐❝❛s é ❝❤❛♠❛❞♦
❞❡ ❡s♣❛ç♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❏❆■◆ ❡t ❛❧✱ ✷✵✵✵✱ ▼❆❘◗❯❊❙✱ ✷✵✵✺✮✳
❖ ❝♦♥❥✉♥t♦ ❞❡ ♣❛❞rõ❡s ✏s✐♠✐❧❛r❡s✑✱ ♦✉ s❡❥❛✱ ♦s ✈❛❧♦r❡s ❛ss✉♠✐❞♦s ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s d ❝❛r❛❝t❡ríst✐❝❛s✱ q✉❡ ♣♦❞❡♠ s❡r ❝❛t❡❣ór✐❝❛s✱ ❞✐s❝r❡t❛s ♦✉ ❝♦♥tí♥✉❛s ❣❡r❛♠ ✉♠❛ ❝❧❛ss❡ωi✱
q✉❡ r❡♣r❡s❡♥t❛ i✲és✐♠❛ ❝❧❛ss❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝❧❛ss❡s Ω✭❋✐❣✉r❛ ✸✮✳
❋✐❣✉r❛ ✸ ✲ ❯♥✐✈❡rs♦ ❞❛s ❝❧❛ss❡s
❈❛❞❛ ❝❧❛ss❡ ❝♦♥tê♠ ♣❛❞rõ❡s q✉❡ ♣♦ss✉❡♠ ❛❧❣✉♠❛ r❡❧❛çã♦ ♦✉ ♣❡❝✉❧✐❛r✐❞❛❞❡ ❡♠ ❝♦♠✉♠✳ P❛r❛ ❡①❡♠♣❧✐✜❝❛r ❝♦♠♦ sã♦ ❡ss❡s ♣❛❞rõ❡s ♣♦❞❡✲s❡ ♣❡♥s❛r ❡♠ ✉♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s ❞❡ ❜✐♦♠❡tr✐❛✿ ❙❡❥❛ ♦ ❡s♣❛ç♦ ❞❡ ♣❛❞rõ❡s ①= (x1, x2, x3)❡♠ q✉❡x1 r❡♣r❡s❡♥t❛ ❛❧t✉r❛✱x2 r❡♣r❡s❡♥t❛
♣❡s♦ ❡ x3 r❡♣r❡s❡♥t❛ ♦ t❛♠❛♥❤♦ ❞♦s ♣és✳ ◆❡ss❡ ❡s♣❛ç♦ ❞❡ três ❞✐♠❡♥sõ❡s✱ ❝❛❞❛ ♦❜s❡r✈❛çã♦ ①
r❡♣r❡s❡♥t❛ ❛s ♠❡❞✐çõ❡s ❞❡ ✉♠❛ ♣❡ss♦❛ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥st❛♥t❡ ❡ ❝❛❞❛ ❝❧❛ss❡ r❡♣r❡s❡♥t❛ ✉♠❛ ❢❛♠✐❧✐❛✳ ◆❡ss❡ ❝❛s♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡✜♥❡✲s❡ ♣♦r✿ ❞❛❞❛ ✉♠❛ ♣❡ss♦❛ ❞❡s❝♦♥❤❡✲ ❝✐❞❛✱ ❡①tr❛✐r s✉❛s ❝❛r❛❝t❡ríst✐❝❛s ♣❛r❛ ♦❜t❡r s❡✉ ✈❡t♦r ❞❡ ❝❛r❛❝t❡ríst✐❝❛s ① ❡ ❞❡t❡r♠✐♥❛r ❛ q✉❛❧ ❢❛♠✐❧✐❛ ❡ss❛ ♣❡ss♦❛ ♣❡rt❡♥❝❡ ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼❖❘❆■❙✱ ✷✵✶✵❀ ❲■❚❚❊◆❀ ❋❘❆◆❑✱ ✷✵✵✺✮✳
✷✳✶✳✷ ❆♣r❡♥❞✐③❛❣❡♠
✷✻
③❛❞♦ s❡rá ✉t✐❧✐③❛❞♦✳ ❆ ❛♣r❡♥❞✐③❛❣❡♠ ❝♦♥s✐st❡ ❡♠ ✉t✐❧✐③❛r ♦s ❞❛❞♦s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ♣♦❞❡r ❝r✐❛r ♦s ❛❧❣♦r✐t♠♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❡ ❡♠ ❛❧❣✉♥s ❝❛s♦s ❣r✉♣♦s ❝♦♠ ♣❛❞rã♦ s❡♠❡❧❤❛♥t❡ ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
❖ ❛♣r❡♥❞✐③❛❞♦ ♣♦❞❡✲s❡ ❞❡s❡♥✈♦❧✈✐❞♦ ❞❡ ❞✉❛s ♠❛♥❡✐r❛s✿
✶✮ ❆♣r❡♥❞✐③❛❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ♦s ♣❛❞rõ❡s ❞❡ tr❡✐♥♦ ♥ã♦ s❡ ❡♥❝♦♥tr❛♠ ❝❧❛ss✐✜❝❛❞♦s✱ ♦✉ s❡❥❛✱ ♥ã♦ s❡ ❝♦♥❤❡❝❡♠ ❛s ❝❧❛ss❡s q✉❡ ❣❡r❛r❛♠ ♦s ♣❛❞rõ❡s❀
✷✮ ❆♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ❡♠ q✉❡ ❛s ❝❧❛ss❡s q✉❡ ❣❡r❛r❛♠ ♦s ♣❛❞rõ❡s sã♦ ❝♦♥❤❡❝✐❞❛s✳
✷✳✶✳✷✳✶ ❆♣r❡♥❞✐③❛❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦
P❛r❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ♦ ❝♦♥❥✉♥t♦ ♣♦ss✉✐ s♦♠❡♥t❡ ♣❛❞rõ❡s ❞❡ ❡♥tr❛❞❛✱ ♦✉ s❡❥❛✱ ♥ã♦ ❤á ♦ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ❝♦♠♦ sã♦ ♦s ❞❛❞♦s ❡ r❡❧❛çã♦ às ❝❧❛ss❡s✱ ♥❡♠ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ q✉❛♥t✐❞❛❞❡✱ ❞♦s ♥♦♠❡s ❡ ♦✉tr❛s ✐♥❢♦r♠❛çõ❡s✱ ❛♦ ❝♦♥trár✐♦ ❞♦ ❛♣r❡♥❞✐✲ ③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ q✉❡ ♦ ❝♦♥❥✉♥t♦ ♣♦ss✉✐ ❡♥tr❛❞❛ ❡ s❛í❞❛ ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ●❆❉■✱ ✷✵✵✽❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼❖❘❆■❙✱ ✷✵✶✵✮✳
P❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❛♣r❡♥❞✐③❛❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦✱ ♣♦❞❡✲s❡ ✉t✐❧✐③❛r✿
✶✮ ❚é❝♥✐❝❛s ❞❡ ❛♥á❧✐s❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦s
✷✮ ❚é❝♥✐❝❛s ❞❡ ❛♥á❧✐s❡ ❞❡ ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s
❊ss❛s té❝♥✐❝❛s ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ♦ ♦❜❥❡t✐✈♦ ♥❡st❡ t✐♣♦ ❞❡ ❛♣r❡♥❞✐③❛❞♦ ❡stá ❡♠ ❝r✐❛r ❣r✉♣♦s q✉❡ ♣♦ss✉❛♠ ❝❛r❛❝t❡ríst✐❝❛s s❡♠❡❧❤❛♥t❡s ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ❏❆■◆ ❡t ❛❧✱ ✷✵✵✵❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼❖❘❆■❙✱ ✷✵✶✵❀ ❙❯◆✱ ✷✵✵✾✮✳
✷✳✶✳✷✳✷ ❆♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦
✷✼
✉t✐❧✐③❛♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❝♦♠♦ tr❡✐♥♦✱ ♣❛r❛ ❡❧❛❜♦r❛r ♦s ❝r✐tér✐♦s ❞❡ ❞❡❝✐sã♦ ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ▼❖❘❆■❙✱ ✷✵✶✵❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ❲■❚❚❊◆❀ ❋❘❆◆❑✱ ✷✵✵✺✮✳
❊♠❜♦r❛ ♥ã♦ s❡❥❛ ❞❡ ✐♥t❡r❡ss❡✱ ♥❡st❡ ❝❛s♦✱ r❡❛❧✐③❛r ❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s t✐♣♦s ❞❡ ❛♣r❡♥❞✐③❛❞♦✱ ❛❧❣✉♥s ❛✉t♦r❡s ❞❡❢❡♥❞❡♠ q✉❡ ❛s té❝♥✐❝❛s ❞❡ ❛♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ❣❡r❛♠ ♠❡❧❤♦r❡s r❡s✉❧t❛❞♦s q✉❡ ♦s ❞♦ ❛♣r❡♥❞✐③❛❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ✭●❆❉■✱ ✷✵✵✽❀ ❏❆■◆ ❡t ❛❧✱ ✷✵✵✵✱ ▼❆❘◗❯❊❙✱ ✷✵✵✺✮✳
P❛r❛ r❡❛❧✐③❛r ❛ ❝❧❛ss✐✜❝❛çã♦ ✉t✐❧✐③❛♥❞♦ ♦ ❛♣r❡♥❞✐③❛❞♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ❞♦✐s ♠ét♦❞♦s ✭▼❆❘◗❯❊❙✱ ✷✵✵✺✮✳
✶✮ ❊st✐♠❛r ♦ ♠♦❞❡❧♦ ♣r♦❜❛❜✐❧íst✐❝♦ ❛ ♣❛rt✐r ❞♦s ❞❛❞♦s ❞✐s♣♦♥í✈❡✐s✱ s✉❜st✐t✉✐♥❞♦ ♦ ♠♦❞❡❧♦ ❡st✐♠❛❞♦ ♣❡❧❛ ❝❧❛ss✐✜❝❛çã♦ ✉t✐❧✐③❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❇❛②❡s❀
✷✮ ❊st✐♠❛r ❞✐r❡t❛♠❡♥t❡ ❛s ❢✉♥çõ❡s ❞✐s❝r✐♠✐♥❛♥t❡s q✉❡ ❞❡✜♥❡♠ ❛s ❢r♦♥t❡✐r❛s ❞❡ ❞❡❝✐sã♦ ❞♦ ❝❧❛ss✐✜❝❛❞♦r ❛ ♣❛rt✐r ❞♦s ❞❛❞♦s✳
❊ss❡ ❛♣r❡♥❞✐③❛❞♦ é ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡①❡❝✉çã♦ ❞♦s ❛❧❣♦r✐t♠♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♣❛r❛ ❡❧❛❜♦r❛r ❛s ❢✉♥çõ❡s ❞✐s❝r✐♠✐♥❛t❡s q✉❡ s❡r✈❡♠ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛s ❢r♦♥t❡✐r❛s ❞❡ ❞❡❝✐sã♦✱ q✉❡ sã♦ ❛s r❡❣✐õ❡s q✉❡ ❞❡❧✐♠✐t❛♠ ❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ♦✉ s❡❥❛✱ ❞✐✈✐❞❡♠ ♦ ❡s♣❛ç♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s ❡♠cr❡❣✐õ❡s✱ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡c❝❧❛ss❡s✱ s❡♥❞♦ q✉❡ ❝❛❞❛ r❡❣✐ã♦ ❝♦rr❡s♣♦♥❞❡ ❛
✉♠❛ ❝❧❛ss❡✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♣❡❧❛ ❋✐❣✉r❛ ✸ ✭❇■❙❍❖P✱ ✷✵✵✻❀ ❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❋❘■❊❉▼❆◆ ❡t ❛❧✱ ✷✵✵✽✮✳ ❊st❡ ❢♦✐ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞❡✜♥✐❞♦ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦✱ ❞❡st❡ ♠♦❞♦ é ❡ss❡♥❝✐❛❧ r❡❛❧✐③❛r ✉♠❛ r❡✈✐sã♦ s♦❜r❡ ♦s ❝❧❛ss✐✜❝❛❞♦r❡s ❡ ♦s ♠ét♦❞♦s ❞❡ ❝♦♠♦ ❡❧❛❜♦r❛r ♦s s❡✉s ❛❧❣♦r✐t♠♦s ❞❡ ❞❡❝✐sõ❡s✱ s✉❛s ❢✉♥çõ❡s ❞✐s❝r✐♠✐♥❛♥t❡s✱ ❛ss✐♠ ❝♦♠♦ ♦s ❝r✐tér✐♦s ♣❛r❛ ❛ t♦♠❛❞❛ ❞❛ ♠❡s♠❛✳
✷✳✶✳✸ ❈❧❛ss✐✜❝❛❞♦r❡s
❙❡❥❛ xi ✉♠ ♣❛❞rã♦ ❞❡s❝♦♥❤❡❝✐❞♦✱ ♣❡rt❡♥❝❡♥t❡ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛❞rõ❡s ❳ ❡♠ ✉♠
❡s♣❛ç♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s ①✱ ❡ ♦ ❝♦♥❥✉♥t♦ Ω ❝♦♠ t♦❞❛s ❛s ❝❧❛ss❡s ❡①✐st❡♥t❡s✱ ♦ ❝❧❛ss✐✜❝❛❞♦r é
✉♠❛ ❢✉♥çã♦ ❨:X −→Ω t❛❧ q✉❡ ❨(①) = ωi✱ ❡♠ q✉❡ωi r❡♣r❡s❡♥t❛ ❛ i✲és✐♠❛ ❝❧❛ss❡ ❞❡ Ω✳
✷✽
❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❝❧❛ss✐✜❝❛❞♦r❡s ❡stá ♥♦ ♠ét♦❞♦ ❡♠ q✉❡ s❡ ❝r✐❛♠ ❛s ❢r♦♥t❡✐r❛s ❞❡ ❞❡❝✐sõ❡s✱ ♣♦ré♠ ❛s ❞✐❢❡r❡♥t❡s té❝♥✐❝❛s ♣❛r❛ ❡st✐♠❛r ❛s ❢r♦♥t❡✐r❛s ❞❡ ❞❡❝✐sõ❡s tê♠ ❡♠ ❝♦♠✉♠ ♦s s❡❣✉✐♥t❡s ♦❜❥❡t✐✈♦s✿
✶✮ ▼✐♥✐♠✐③❛r ♦ ❡rr♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦❀
✷✮ P❡r♠✐t✐r q✉❡ ❛ ❝❧❛ss✐✜❝❛çã♦ s❡❥❛ ❡✜❝✐❡♥t❡ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡✱ ♦t✐♠✐③❛♥❞♦ ♦ ♣r♦❝❡ss♦✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ❧❡✈❡ ♦ ♠❡♥♦r t❡♠♣♦ ♣♦ssí✈❡❧✳
❆ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❡s ♦❜❥❡t✐✈♦s ✈❛r✐❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❝❧❛ss✐✜❝❛❞♦r✱ s❡♥❞♦ q✉❡ ♦ ♠❡❧❤♦r r❡s✉❧t❛❞♦ ♣♦ssí✈❡❧ ♦❝♦rr❡ q✉❛♥❞♦ ✉♠ ❝❧❛ss✐✜❝❛❞♦r é rá♣✐❞♦ ❡ ♣r❡❝✐s♦✱ ♠❛s ❡st❛ s✐t✉❛çã♦ ❞✐✜❝✐❧♠❡♥t❡ ♦❝♦rr❡ ❡♠ ♣r♦❜❧❡♠❛s ❝♦♠♣❧❡①♦s✳
❖✉tr♦ ❢❛t♦r q✉❡ ♣♦❞❡ ✐♥✢✉❡♥❝✐❛r ♥❛s ❝❧❛ss✐✜❝❛çõ❡s sã♦ ❛s ❝♦♥s❡q✉ê♥❝✐❛s q✉❡ ❝❛❞❛ ❞❡❝✐sã♦ ♣♦❞❡ ♦❝❛s✐♦♥❛r✳ P♦r ❡①❡♠♣❧♦✱ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❞❡t❡❝çã♦ ❞❡ ✐♥❝ê♥❞✐♦ ❡♠ q✉❡ ❛ ❝❛❞❛ ♠♦♠❡♥t♦ ❞❡✈❡✲s❡ ✈❡r✐✜❝❛r s❡ ❤á ♦✉ ♥ã♦ ❛❧❣✉♠ ❢♦❝♦ ❞❡ ✐♥❝ê♥❞✐♦✱ s❡♥t❡ ❝❛s♦✱ ♣♦❞❡♠ ♦❝♦rr❡r ❞♦✐s t✐♣♦s ❞❡ ❡rr♦✿ ❞❡t❡❝t❛r ✐♥❝ê♥❞✐♦ q✉❛♥❞♦ ♥ã♦ ❡①✐st❡ ✭❛❧❛r♠❡ ❢❛❧s♦✮ ❡ ♥ã♦ ❞❡t❡❝t❛r q✉❛♥❞♦ ❡①✐st❡ ✭❢❛❧❤❛ ♥♦ s✐st❡♠❛✮✳ ❆♠❜♦s ❡rr♦s sã♦ ♣r❡❥✉❞✐❝✐❛✐s✱ ♣♦ré♠✱ q✉❛♥❞♦ ♦❝♦rr❡ ❛ ❢❛❧❤❛ ♥♦ s✐st❡♠❛✱ ❡st❛ ❧❡✈❛ à ❝♦♥s❡q✉ê♥❝✐❛s ♠❛✐s ❣r❛✈❡s✳ ❖✉tr♦ ❡①❡♠♣❧♦✿ é ♦ ❞❡ s✐st❡♠❛ ❞❡ ❞❡t❡❝çã♦ ❞❡ ❛✈✐ã♦ ❡♠ r❛❞❛r✱ ✜❝❛ ❡✈✐❞❡♥t❡ q✉❡ é ♠❛✐s ♣r❡❥✉❞✐❝✐❛❧ ♥ã♦ ❞❡t❡❝t❛r ✉♠ ❛✈✐ã♦ ❞♦ q✉❡ ❞❡t❡❝t❛r ✉♠ ❢❛❧s♦ ❛✈✐ã♦✳
❊♠ ❛♠❜♦s ♦s ❝❛s♦s é ♣r❡❢❡rí✈❡❧ q✉❡ ♦ ❡rr♦ ❞❡ ♦❝♦rr❡r ❢❛❧❤❛ ♥♦ s✐st❡♠❛ s❡❥❛ ♠❡♥♦r✱ ♠❡s♠♦ q✉❡ ♣❛r❛ ✐ss♦ s❡❥❛ ♥❡❝❡ssár✐♦ ❛✉♠❡♥t❛r ♦ ❡rr♦ ♣❛r❛ ❛❧❛r♠❡s ❢❛❧s♦s✱ ❞❡st❡ ♠♦❞♦ ♣❛r❛ r❡s♦❧✈❡r ❡st❡ ♣r♦❜❧❡♠❛ ♣♦❞❡✲s❡ ❛tr✐❜✉✐r ✉♠ ❝✉st♦ ♣❛r❛ ❝❛❞❛ ❞❡❝✐sã♦ ❡ ❞❡st❛ ♠❛♥❡✐r❛ ✉t✐❧✐③❛r ♦ ❝❧❛ss✐✜❝❛❞♦r ♣❛r❛ ♠✐♥✐♠✐③❛r ♦ ❝✉st♦ ♠é❞✐♦ ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺✮✳
P❛r❛ r❡❛❧✐③❛r ❛s ❝❧❛ss✐✜❝❛çõ❡s é ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ✉♠ ❝❧❛ss✐✜❝❛❞♦r ❞❡♥tr❡ ❡❧❡s ♣♦❞❡✲s❡ ❞❡st❛❝❛r✿
✶✮ ➪r✈♦r❡s ❞❡ ❉❡❝✐sã♦
✷✮ ❘❡❞❡s ◆❡✉r❛✐s
✸✮ ❈❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦ ♦✉ ◆❛✐✈❡ ❇❛②❡s
✷✾
✺✮ ❆❧❣♦r✐t♠♦ ●❡♥ét✐❝♦
✻✮ ❘❡❣r❛ ❞♦s ❑✲✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s
P❛r❛ ♦ s❡❣✉✐♠❡♥t♦ ❞❡st❛ ❢♦✐ ❡s❝♦❧❤✐❞♦ ♦ ❝❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦✱ ♣♦r ✐ss♦ ❛ s❡❣✉✐r s❡rá r❡❛❧✐③❛❞♦ ✉♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ s♦❜r❡ ♦ ♠❡s♠♦✳
✷✳✶✳✸✳✶ ❈❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦
❖ ❝❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♥❛✐✈❡ ❇❛②❡s✱ é ✉♠❛ ❛❜♦r✲ ❞❛❣❡♠ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦s ♣r♦❜❧❡♠❛s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ♣❛❞rõ❡s✳ ❊st❛ ❛❜♦r❞❛❣❡♠ é ❜❛s❡❛❞❛ ♥❛ q✉❛♥t✐✜❝❛çã♦ ❞❛s ❝♦♠♣❛r❛çõ❡s ❡♥tr❡ ❛s ✈ár✐❛s ❞❡❝✐sõ❡s ✉t✐❧✐③❛♥❞♦ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡ ❝✉st♦s ❞❡ t❛✐s ❞❡❝✐sõ❡s✱ ❛ss✉♠✐♥❞♦ q✉❡ ♦s ♣r♦❜❧❡♠❛s ❞❡ ❞❡❝✐sã♦ sã♦ ❝♦❧♦❝❛❞♦s ❡♠ t❡r♠♦s ♣r♦❜❛❜✐❧íst✐❝♦s ❡ q✉❡ ❡st❡s ✈❛❧♦r❡s sã♦ ❝♦♥❤❡❝✐❞♦s ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❏■❆◆● ❡t ❛❧✱ ✷✵✵✺❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼■❚❈❍❊▲▲✱ ✶✾✾✼✮✳
❊st❡ ❝❧❛ss✐✜❝❛❞♦r é ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ❞❡✈✐❞♦ ❛ s✉❛ s✐♠♣❧✐❝✐❞❛❞❡ ❡ s✉❛ ❡✜❝✐ê♥❝✐❛✱ ♦✉ s❡❥❛✱ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ s❡✉ ❛❧❣♦r✐t♠♦ ❞❡ ❡st✐♠❛çã♦ é ❢á❝✐❧ ❡ ❝♦♥s❡❣✉❡♠✲s❡ ❜♦♥s r❡s✉❧t❛❞♦s ❞❡ ♠♦❞♦ rá♣✐❞♦✳ P♦ré♠ é ❧✐♠✐t❛❞♦ à ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞❡ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s ✭●❆❉■✱ ✷✵✵✽✱ ❘❊◆◆■❊ ❡t ❛❧✱ ✷✵✵✸❀ ❲❊❇❇ ❡t ❛❧✱ ✷✵✶✵✮✳
❆♦ ✉t✐❧✐③❛r ❝❧❛ss✐✜❝❛❞♦r ❜❛②❡s✐❛♥♦ ❤á ♦ ✐♥t❡r❡ss❡ ❡♠ ❝❧❛ss✐✜❝❛r ✉♠❛ ♦❜s❡r✈❛çã♦ ❡♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝❧❛ss❡✱ ♣❛r❛ t❛❧ é ✉t✐❧✐③❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣❛r❛ s❡r❡♠ ❞❡s❡♥✈♦❧✈✐❞❛s ❛s s✉❛s ❢✉♥çõ❡s ❞✐s❝r✐♠✐♥❛♥t❡s é ✉t✐❧✐③❛❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❇❛②❡s ❞❛❞♦ ♣♦r✿
P(ωi|①) =
p(①|ωi)P(ωi)
p(①) i= 1, . . . , c
❡♠ q✉❡ P(ωi|①) é r❡♣r❡s❡♥t❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ ❞♦ ✐t❡♠ s❡r ❞❛ ❝❧❛ss❡ ωi ❞❛❞♦ q✉❡
❢♦✐ ♦❜s❡r✈❛❞♦ ♦ ♣❛❞rã♦ ①✱p(①|ωi)r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧✱
♣❛r❛ ❞❛❞♦s ❝♦♥tí♥✉♦s ❡ ❢✉♥çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧✱ ♣❛r❛ ❞❛❞♦s ❞✐s❝r❡t♦s✱ P(ωi) é
❝♦♥s✐❞❡r❛❞❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣r✐♦r✐ ♣❛r❛ ❝❛❞❛ ❝❧❛ss❡ ❡ ♦ ✈❛❧♦r ❞❡ P(①) é ❞❛❞♦ ♣♦r✿ c
X
i=1
p(①|ωi)P(ωi)
✸✵
✶✾✾✼❀ P➱❘❊❩✱ ✷✵✵✼❀ ❘❊◆◆■❊ ❡t ❛❧✱ ✷✵✵✸❀ ❲❊❇❇ ❡t ❛❧✱ ✷✵✶✵✮✳ P♦❞❡✲s❡ r❡♣r❡s❡♥t❛✲❧♦ ❞❡ ♠♦❞♦ ✐♥❢♦r♠❛❧ ❝♦♠♦✿
♣♦st❡r✐♦r✐= ✈❡r♦ss✐♠✐❧❤❛♥ç❛ × ♣r✐♦r✐
❡✈✐❞ê♥❝✐❛
❊st❡ t❡♦r❡♠❛ ♠♦str❛ q✉❡✱ ♦❜s❡r✈❛❞♦ ♦ ✈❡t♦r ① ♣♦❞❡✲s❡ ♠♦❞✐✜❝❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣r✐♦r✐ P(ωi) ♣❛r❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ P(ωi|①)✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❝❧❛ss❡ ωi
♦❝♦rr❡r✱ s❛❜❡♥❞♦ q✉❡ ❢♦r❛♠ ♦❜s❡r✈❛❞♦s ✈❛❧♦r❡s ❞❡ ①✳ ❆ ❢✉♥çã♦p(①|ωi)é ❝❤❛♠❛❞❛ ❞❡ ✈❡r♦ss✐♠✲
✐❧❤❛♥ç❛ ❞❡ ω ❡♠ r❡❧❛çã♦ ❛ ①✳ ❊st❡ ♣r♦❞✉t♦ ❡♥tr❡ ❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣r✐♦r✐
é ♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ ✭❉❯❉❆✱ ✷✵✵❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺✮✳
❆ ❢✉♥çã♦ ❞✐s❝r✐♠✐♥❛♥t❡ ♣❛r❛ ❝❛❞❛ ❝❧❛ss❡ r❡♣r❡s❡♥t❛ ♦ ❧♦❣❛r✐t♠♦ ❞❡ s✉❛ ♣r♦❜❛❜✐✲ ❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐✱ ❞❡st❡ ♠♦❞♦ ❛ ❢✉♥çã♦ ❞✐s❝r✐♠✐♥❛t❡ é ❞❛❞❛ ♣♦r✿
lnP(ωi|①) = gi(①) = lnp(①|ωi) + lnP(ωi)−lnP(①)
◆❡ss❡ t✐♣♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❤á ♦ ✐♥t❡r❡ss❡ ❡♠ ❡s❝♦❧❤❡r ❛ ❝❧❛ss❡ q✉❡ ♠❛①✐♠✐③❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐✱ ♦✉ s❡❥❛✱ q✉❡ ♠✐♥✐♠✐③❡ ♦ ❡rr♦ ❞❡ ✉♠❛ ❡s❝♦❧❤❛✱ ❞❡st❡ ♠♦❞♦ é ✉t✐❧✐③❛❞♦ ♦ ❝❧❛ss✐✜❝❛❞♦r ❞❡ ♠á①✐♠♦ ❛ ♣♦st❡r✐♦r✐ ✭▼❆P✮ ❝♦♠♦ ❝r✐tér✐♦ ♣❛r❛ ♦ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦ ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ❏■❆◆ ❡t ❛❧✱ ✷✵✵✺❀ ▼❆❘◗❯❊❙✱ ✷✵✵✺❀ ▼■❚❈❍❊▲▲✱ ✶✾✾✼❀ P➱❘❊❩✱ ✷✵✵✼❀ ❘❊◆◆■❊ ❡t ❛❧✱ ✷✵✵✸❀ ❲❊❇❇ ❡t ❛❧✱ ✷✵✶✵✮✳ ❊st❡ ❝r✐tér✐♦ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦✿
ˆ
ω =❛r❣ ♠❛①ω∈ΩP(ω|①)
s❡♥❞♦ q✉❡ ❛r❣ ♠❛① r❡♣r❡s❡♥t❛ ♦ ❛r❣✉♠❡♥t♦ q✉❡ ♠❛①✐♠✐③❛ ❛ ❡①♣r❡ssã♦
❆♦ ✉t✐❧✐③❛r ❛ ❢✉♥çã♦ ❞✐s❝r✐♠✐♥❛♥t❡ ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦ ❝♦♠ ♦ ▼❆P ♦ ✐♥t❡r❡ss❡ ♣❡r♠❛♥❡❝❡ ❡♠ t♦♠❛r ✉♠❛ ❞❡❝✐sã♦ q✉❡ ♠❛①✐♠✐③❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ ✭❉❯❉❆ ❡t ❛❧✱ ✷✵✵✵❀ ▼❖❘❆■❙✱ ✷✵✶✵❀ ❲❊❇❇ ❡t ❛❧✱ ✷✵✶✵✮✳ ❙❡❥❛ (g1(①), . . . , gc(①)) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡
❢✉♥çõ❡s ❞✐s❝r✐♠✐♥❛t❡s ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s c ❝❧❛ss❡✱ ❡♥tã♦ ❛ ❞❡❝✐sã♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♣♦❞❡ s❡r
✈✐st❛ ❝♦♠♦✿
ˆ
ωi =ωk :k=❛r❣ ♠❛①i=1,...,cgi(①)
