Redes Probabilísticas de K-dependência para problemas de classi cação binária

❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ❞❡ ❑✲❞❡♣❡♥❞ê♥❝✐❛

♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❜✐♥ár✐❛

❆♥❞❡rs♦♥ ▲✉✐③ ❞❡ ❙♦✉③❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❋r❛♥❝✐s❝♦ ▲♦✉③❛❞❛ ◆❡t♦

❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▲✉✐s ❆♣❛r❡❝✐❞♦ ▼✐❧❛♥

❙ã♦ ❈❛r❧♦s

❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ❞❡ ❑✲❞❡♣❡♥❞ê♥❝✐❛

♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❜✐♥ár✐❛

❆♥❞❡rs♦♥ ▲✉✐③ ❞❡ ❙♦✉③❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❋r❛♥❝✐s❝♦ ▲♦✉③❛❞❛ ◆❡t♦

❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▲✉✐s ❆♣❛r❡❝✐❞♦ ▼✐❧❛♥

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r✲ ❧♦s ✲ ❉❊s✴❯❋❙❈❛r✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❊st❛tís✲ t✐❝❛✳

❙ã♦ ❈❛r❧♦s

Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar

S729rp

Souza, Anderson Luiz de.

Redes probabilísticas de K-dependência para problemas de classificação binária / Anderson Luiz de Souza. -- São Carlos : UFSCar, 2012.

128 f.

Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2011.

1. Estatística. 2. Classificadores. 3. Redes probabilísticas. 4. Combinação de classificadores. 5. Redes Bayesianas. I. Título.

❆❣r❛❞❡❝✐♠❡♥t♦s

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♠❡✉s ♣❛✐s✱ ❈❛r♠❡♥ ❆♣❛r❡❝✐❞❛ ❆r❛ ❞❡ ❙♦✉③❛ ❡ ❱❛❧❞❡❝✐ ❋❡❧✐s❜❡rt♦ ❞❡ ❙♦✉③❛✱ ♣♦r t♦❞♦ ❡s❢♦rç♦✱ ❝♦♠♣r❡❡♥sã♦ ❡ ❛❧✐❝❡r❝❡ ❢♦r♥❡❝✐❞♦ ♣❛r❛ ♠❡✉ ❛✈❛♥ç♦ ❡❞✉❝❛❝✐♦♥❛❧ ❡ ♣r♦✜ss✐♦♥❛❧✳

➚ ♠✐♥❤❛ ❛✈ó ❆♣❛r❡❝✐❞❛ ❇❛♣t✐st❛ ❆r❛✱ ♣♦r t♦❞♦ ③❡❧♦✱ ✐♥t❡r❡ss❡ ❡ s♦❧✐❞❛r✐❡❞❛❞❡ ❡♠ t♦❞♦s ♦s ❛s♣❡❝t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳

➚ ♠✐♥❤❛ ✐r♠ã ❈r②st✐❛♥❡ ❋❡r♥❛♥❞❛ ❞❡ ❙♦✉③❛ ♣❡❧❛ t♦❧❡râ♥❝✐❛ ❡ ❧❛③❡r✳ ❆ ♠❡✉ t✐♦ ❏♦sé ▲✉✐s ❆r❛ ❙♦❜r✐♥❤♦ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ✐♥s♣✐r❛çã♦✳

❆ ❈❧❡②t♦♥ ❩❛♥❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❡ ❋❡❧✐♣❡ ◆❛rt✐s ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ✐♠❡♥s♦ ❛♣♦✐♦✱ ❛♦s ♥♦ss♦s ♣❛ss❛t❡♠♣♦s ❡ ❧♦♥❣❛s ❝♦♥✈❡rs❛s s♦❜r❡ ♦s ♠❛✐s ✈❛r✐❛❞♦s ❛ss✉♥t♦s✳

❆ ♠❡✉ ♦r✐❡♥t❛❞♦r ❋r❛♥❝✐s❝♦ ▲♦✉③❛❞❛ ◆❡t♦ ♣❡❧❛ ❛♠✐③❛❞❡✱ ♦♣♦rt✉♥✐❞❛❞❡s ❡ ♣❡❧❛ ❛ ❡①♣❡r✐ê♥❝✐❛ q✉❡ t❡♠ ♠❡ ♣❛ss❛❞♦ ❡♠ t♦❞♦s ❡ss❡s ❛♥♦s ❞❡ tr❛❜❛❧❤♦✳

❆♦s ♠❡✉s ♠❡❧❤♦r❡s ♣r♦❢❡ss♦r❡s ❞❡s❞❡ ♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ ♣♦✐s s❡♠ ❜♦♥s ♣r♦❢❡ss♦r❡s ♥✉♥❝❛ ❝❤❡❣❛rí❛♠♦s ❛ tr✐❧❤❛r ❛s ❧✉③❡s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳

❖ ❛rt✐st❛ q✉❡ ✜❝❛ s❛t✐s❢❡✐t♦ ❝♦♠ s✉❛ ♦❜r❛ ❢❛❧t♦✉ à ✈♦❝❛çã♦✳

❘❡s✉♠♦

❆ ❝❧❛ss✐✜❝❛çã♦ ❝♦♥s✐st❡ ♥❛ ❞❡s❝♦❜❡rt❛ ❞❡ r❡❣r❛s ❞❡ ♣r❡✈✐sã♦ ♣❛r❛ ❛✉①í❧✐♦ ♥♦ ♣❧❛♥❡❥❛✲ ♠❡♥t♦ ❡ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✱ s❡♥❞♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ✐♥❞✐s♣❡♥sá✈❡❧ ❡ ✉♠ t❡♠❛ ❜❛st❛♥t❡ ❞✐s❝✉t✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛✳ ❈♦♠♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❡ ❝❧❛ss✐✜❝❛çã♦✱ t❡♠♦s ♦ ♣r♦❝❡ss♦ ❞❡ ❛✈❛✲ ❧✐❛çã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♥♦ q✉❛❧ t❡♠♦s ♦ ✐♥t❡r❡ss❡ ❞❡ ✐❞❡♥t✐✜❝❛r ❝❧✐❡♥t❡s ❜♦♥s ❡ ♠❛✉s ♣❛❣❛❞♦r❡s ❛tr❛✈és ❞❡ ♠ét♦❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❜✐♥ár✐❛✳ ❆ss✐♠✱ ❡♠ ❞✐✈❡rs♦s ❡♥r❡❞♦s ❞❡ ❛♣❧✐❝❛çã♦✱ ❝♦♠♦ ♥❛s ✜♥❛♥❝❡✐r❛s✱ ❞✐✈❡rs❛s té❝♥✐❝❛s ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s✱ t❛✐s ❝♦♠♦ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐♥❛♥t❡✱ ❛♥á❧✐s❡ ♣r♦❜✐t♦✱ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ ❡ r❡❞❡s ♥❡✉r❛✐s✳ P♦ré♠✱ ❛ té❝♥✐❝❛ ❞❡ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❘❡❞❡s ❇❛②❡s✐❛♥❛s✱ t❡♠ s❡ ♠♦str❛❞♦ ✉♠ ♠ét♦❞♦ ♣rát✐❝♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❡ ❝♦♠ ❛♣❧✐❝❛çõ❡s ❜❡♠ s✉❝❡❞✐❞❛s ❡♠ ❞✐✈❡rs♦s ❝❛♠♣♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ✈✐s❛♠♦s ❡①✐❜✐r ❛ ❛♣❧✐❝❛çã♦ ❞❛s ❘❡❞❡s Pr♦✲ ❜❛❜✐❧íst✐❝❛s ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦✱ ❡♠ ❡s♣❡❝í✜❝♦✱ ❛ té❝♥✐❝❛ ❞❡♥♦♠✐♥❛❞❛ ❘❡❞❡s Pr♦❜✐❜✐❧íst✐❝❛s ❝♦♠ ❑✲❞❡♣❡♥❞ê♥❝✐❛✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ r❡❞❡s ❑❉❇✱ ❜❡♠ ❝♦♠♦ ❝♦♠♣❛r❛r s❡✉ ❞❡s❡♠♣❡♥❤♦ ❝♦♠ ❛s té❝♥✐❝❛s ❝♦♥✈❡♥❝✐♦♥❛✐s ❛♣❧✐❝❛❞❛s ♥♦ ❝♦♥t❡①t♦ ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❡ ❉✐❛❣♥♦s❡ ▼é❞✐❝❛✳ ❊①✐❜✐r❡♠♦s ❝♦♠♦ r❡s✉❧t❛❞♦ ❛♣❧✐❝❛çõ❡s ❞❛ té❝♥✐❝❛ ❜❛s❡❛❞❛s ❡♠ ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s r❡❛✐s ❡ ❛rt✐✜❝✐❛✐s ❡ s❡✉ ❞❡s❡♠♣❡♥❤♦ ❛✉①✐❧✐❛❞♦ ♣❡❧♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❜❛❣❣✐♥❣✳

❆❜str❛❝t

❈❧❛ss✐✜❝❛t✐♦♥ ❝♦♥s✐sts ✐♥ t❤❡ ❞✐s❝♦✈❡r② ♦❢ r✉❧❡s ♦❢ ♣r❡❞✐❝t✐♦♥ t♦ ❛ss✐st ✇✐t❤ ♣❧❛♥♥✐♥❣ ❛♥❞ ❞❡❝✐s✐♦♥✲♠❛❦✐♥❣✱ ❜❡✐♥❣ ❛ ❝♦♥t✐♥✉♦✉s❧② ✐♥❞✐s♣❡♥s❛❜❧❡ t♦♦❧ ❛♥❞ ❛ ❤✐❣❤❧② ❞✐s❝✉s✲ s❡❞ s✉❜❥❡❝t ✐♥ ❧✐t❡r❛t✉r❡✳ ❆s ❛ s♣❡❝✐❛❧ ❝❛s❡ ✐♥ ❝❧❛ss✐✜❝❛t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ♣r♦❝❡ss ♦❢ ❝r❡❞✐t r✐s❦ r❛t✐♥❣✱ ✇✐t❤✐♥ ✇❤✐❝❤ t❤❡r❡ ✐s ✐♥t❡r❡st ✐♥ ✐❞❡♥t✐❢②✐♥❣ ❣♦♦❞ ❛♥❞ ❜❛❞ ♣❛②✐♥❣ ❝✉st♦♠❡rs t❤r♦✉❣❤ ❜✐♥❛r② ❝❧❛ss✐✜❝❛t✐♦♥ ♠❡t❤♦❞s✳ ❚❤❡r❡❢♦r❡✱ ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥ ❜❛❝❦❣r♦✉♥❞s✱ ❛s ✐♥ ✜♥❛♥❝✐❛❧✱ s❡✈❡r❛❧ t❡❝❤♥✐q✉❡s ❝❛♥ ❜❡ ✉t✐❧✐③❡❞✱ s✉❝❤ ❛s ❞✐s❝r✐♠✐✲ ♥❛t✐♥❣ ❛♥❛❧②s✐s✱ ♣r♦❜✐t ❛♥❛❧②s✐s✱ ❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥ ❛♥❞ ♥❡✉r❛❧ ♥❡ts✳ ❍♦✇❡✈❡r✱ t❤❡ Pr♦❜❛❜✐❧✐st✐❝ ◆❡ts t❡❝❤♥✐q✉❡✱ ❛❧s♦ ❦♥♦✇♥ ❛s ❇❛②❡s✐❛♥ ◆❡t✇♦r❦s✱ ❤❛✈❡ s❤♦✇❡❞ ✐ts❡❧❢ ❛s ❛ ♣r❛❝t✐❝❛❧ ❝♦♥✈❡♥✐❡♥t ❝❧❛ss✐✜❝❛t✐♦♥ ♠❡t❤♦❞ ✇✐t❤ s✉❝❝❡ss❢✉❧ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ s❡✈❡✲ r❛❧ ❛r❡❛s✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❛✐♠ t♦ ❞✐s♣❧❛② t❤❡ ❛♣♣❧✐❛♥❝❡ ♦❢ Pr♦❜❛❜✐❧✐st✐❝ ◆❡ts ✐♥ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ s❝❡♥❛r✐♦✱ s♣❡❝✐✜❝❛❧❧②✱ t❤❡ t❡❝❤♥✐q✉❡ ♥❛♠❡❞ ❑✲❞❡♣❡♥❞❡♥❝❡ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦s ❛❧s♦ ❦♥♦✇♥ ❛s ❑❉❇ ♥❡ts✱❛s ✇❡❧❧ ❛s ❝♦♠♣❛r❡❞ ✐ts ♣❡r❢♦r♠❛♥❝❡ ✇✐t❤ ❝♦♥✈❡♥✲ t✐♦♥❛❧ t❡❝❤♥✐q✉❡s ❛♣♣❧✐❡❞ ✇✐t❤✐♥ ❝♦♥t❡①t ♦❢ t❤❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❛♥❞ ▼❡❞✐❝❛❧ ❞✐❛❣♥♦s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❡❝❤♥✐q✉❡ ❜❛s❡❞ ✐♥ r❡❛❧ ❛♥❞ ❛rt✐✜❝✐❛❧ ❞❛t❛s❡ts ❛♥❞ ✐ts ♣❡r❢♦r♠❛♥❝❡ ❛ss✐st❡❞ ❜② t❤❡ ❜❛❣❣✐♥❣ ♣r♦❝❡❞✉r❡ ✇✐❧❧ ❜❡ ❞✐s♣❧❛②❡❞ ❛s r❡s✉❧ts✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❈r❡❞✐t ❙❝♦r✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❉✐❛❣♥óst✐❝♦ ❞❡ ❉♦❡♥ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ Pr♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸✳✶ ❚❤♦♠❛s ❇❛②❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸✳✷ ❈♦♥❝❡✐t♦s Pr♦❜❛❜✐❧íst✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸✳✷✳✶ Pr♦❜❛❜✐❧✐❞❛❞❡ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳✷✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡ ❈♦♥❞✐❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸✳✷✳✸ ■♥❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥❞✐❝✐♦♥❛❧ ♣r♦❜❛❜✐❧íst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸✳✷✳✹ ❚❡♦r❡♠❛ ❞❡ ❇❛②❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸✳✷✳✺ ❆s ❞✐str✐❜✉✐çõ❡s ▼✉❧t✐♥♦♠✐❛❧ ❡ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸✳✷✳✻ ❉✐str✐❜✉✐çã♦ ◆♦r♠❛❧ ❡ ◆♦r♠❛❧ ▼✉❧t✐✈❛r✐❛❞❛ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✸ ❆s ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ♣♦❞❡♠ s❡r ❝❤❛♠❛❞❛s ❞❡ ❘❡❞❡s ❇❛②❡✲

s✐❛♥❛s❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✹ ▼étr✐❝❛s ❡ ❞❡✜♥✐çõ❡s ❞❛ ❚❡♦r✐❛ ❞❛ ■♥❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹✳✶ ❊♥tr♦♣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹✳✷ ❉✐stâ♥❝✐❛ ❞❡ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✹✳✸ ■♥❢♦r♠❛çã♦ ▼út✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✺ ❖ ❙♦❢t✇❛r❡ ❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✶✳✻ ❈♦♠❡♥tár✐♦s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ✷✻

✷✳✶ ❊str✉t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✳✶ ❊❧❡♠❡♥t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✳✷ ❊str✉t✉r❛s ❞❡ t❡♦r✐❛ ❞❡ ❣r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✳✸ ❍✐❡r❛rq✉✐❛ ❡♥tr❡ ♥ós ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✳✹ ❋♦r♠❛❧✐③❛çã♦ ❡st❛tíst✐❝❛ ❞❛ ❡str✉t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✶✳✺ ❚❛❜❡❧❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❝♦♥❞✐❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✶✳✻ ❊①❡♠♣❧♦ ❇ás✐❝♦ ❞❡ ✉♠❛ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷ ❊✈✐❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸ Pr♦♣r✐❡❞❛❞❡s ▼❛r❦♦✈✐❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✹ ❆ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❞✲s❡♣❛r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✺ ❊q✉✐✈❛❧ê♥❝✐❛ ❞❡ ▼❛r❦♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✻ ▼ét♦❞♦ ❣❡r❛❧ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✼ ❈♦♠❡♥tár✐♦s ✜♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✸ ❊st✐♠❛çã♦ ❡♠ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ✹✷

✸✳✶ ❊st✐♠❛çã♦ ❞❡ ❡str✉t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✶✳✶ ❆❧❣♦r✐t♠♦ ❑✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✶✳✷ ❆❧❣♦r✐t♠♦ P❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✷ ❊st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✷✳✶ ❊st✐♠❛çã♦ ❋r❡q✉❡♥t✐st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✷✳✷ ❊st✐♠❛çã♦ ❇❛②❡s✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✸ ❈♦♠❡♥tár✐♦s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

✹ ❈❧❛ss✐✜❝❛çã♦ ✻✺

✹✳✷ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ❙✐♠♣❧❡s ❝♦♠ ❑✲❞❡♣❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✸ ❖✉tr♦s ♠ét♦❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✸✳✶ ❆♥á❧✐s❡ ❉✐s❝r✐♠✐♥❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✸✳✷ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✸✳✸ ❘❡❣r❡ssã♦ Pr♦❜✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✸✳✹ ❘❡❞❡s ◆❡✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✹ ▼❡❞✐❞❛s ❞❡ ❞❡s❡♠♣❡♥❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✹✳✺ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ❇❛❣❣✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✹✳✻ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✹✳✼ ❊st✉❞♦ ❞❡ ❙✐♠✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼

✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✾✺

✺✳✶ P❡rs♣❡❝t✐✈❛s ❋✉t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻

❇✐❜❧✐♦❣r❛✜❛ ✾✽

❆ ❈Ó❉■●❖ ❘ ✲ ●❡r❛r ❜❛s❡ P❈ ✶✵✻

❇ ❈Ó❉■●❖ ❘ ✲ ❆❧❣♦r✐t♠♦ P❈ ✶✵✼

❈ ❈Ó❉■●❖ ❘ ✲ ❚P❈ ✶✶✵

❉ ❈Ó❉■●❖ ❘ ✲ ❑❉❇ ❉✐s❝r❡t♦ ✶✶✶

❊ ❈Ó❉■●❖ ❘ ✲ ❑❉❇ ❈♦♥tí♥✉♦ ✶✶✹

❋ ❈Ó❉■●❖ ❘ ✲ ❉❛❞♦s ❙✐♠✉❧❛❞♦s ✶✶✽

● ❈Ó❉■●❖ ❘ ✲ ●rá✜❝♦ ✶✶✾

❍ ❈Ó❉■●❖ ❘ ✲ ❋✉♥çõ❡s ✶✷✵

■ ❈❖◆❏❯◆❚❖ ❉❊ ❉❆❉❖❙ ✶✷✷

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❈♦♥❡①õ❡s ❡♥tr❡ ♦s ♦❜❥❡t✐✈♦s ❡ t❛r❡❢❛s ❡♠ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s✳ ❆❞❛♣✲ t❛❞♦ ❞❡ ❱❡❧✐❝❦♦✈ ❡ ❙♦❧♦♠❛t✐♥❡ ✭✷✵✵✵✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ Ú♥✐❝❛ ■❧✉str❛çã♦ ❝♦♥❤❡❝✐❞❛ ❞❡ ❚❤♦♠❛s ❇❛②❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❉✐❛❣r❛♠❛s ❞❡ ❊ü❧❧❡r✲❱❡♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✶ ❊❧❡♠❡♥t♦s ❜ás✐❝♦s ❞❛ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷ ❊str✉t✉r❛s ❜ás✐❝❛s ❡①✐st❡♥t❡s ❞❡♥tr♦ ❞❛ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸ ❊①❡♠♣❧♦ ❞❡ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ♣❛r❛ ❞❛❞♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ t❡♥❞♦ ❝♦♠♦ ❡✈✐❞ê♥❝✐❛ ❛ ✈❛r✐á✈❡❧ ■❞❛❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✷✳✺ ❈♦❜❡rt✉r❛ ❞❡ ▼❛r❦♦✈ ❞❡ ❆ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛s ✈❛r✐á✈❡✐s✲♥ó ❡♠ ❝✐♥③❛✳ ✸✼

✷✳✻ ❚✐♣♦s ❞❡ ❞✲s❡♣❛r❛çã♦✱ ❯ ❡ ❲ ❞✲s❡♣❛r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✼ ❊①❡♠♣❧♦ ❞❡ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ▼❛r❦♦✈ ❡q✉✐✈❛❧❡♥t❡s✳ ✹✵

✸✳✶ ❆❧❣♦r✐t♠♦ P❈✲ P❛ss♦ ✶✿ ■♥✐❝✐❛✲s❡ ❝♦♠ t♦❞❛s ❛s ❝♦♥❡①õ❡s ❡♥tr❡ ❛s ✈❛✲ r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✷ ❆❧❣♦r✐t♠♦ P❈✲ P❛ss♦ ✷✿ ❱❡r✐✜❝❛♥❞♦ ✐♥❞❡♣❡♥❞ê♥❝✐❛s ❝♦♥❞✐❝✐♦♥❛✐s✳ ❆

✈❛r✐á✈❡❧ ❙❡①♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ✈❛r✐á✈❡❧ ■❞❛❞❡ ❞❛❞♦ ❈r❡❞✐t ❘❛t✐♥❣✳ ✳ ✳ ✺✵ ✸✳✸ P❛ss♦ ✷ ❞♦ ❆❧❣♦r✐t♠♦ P❈✿ ❱❡r✐✜❝❛♥❞♦ ✐♥❞❡♣❡♥❞ê♥❝✐❛s ❝♦♥❞✐❝✐♦♥❛✐s✳ ❆

✈❛r✐á✈❡❧ ■❞❛❞❡ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ✈❛r✐á✈❡❧ ❈r❡❞✐t ❘❛t✐♥❣ ❞❛❞❛ ❛ ✈❛r✐á✈❡❧ ❈ré❞✐t♦s ❆♥t❡r✐♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✸✳✹ P❛ss♦ ✷ ❞♦ ❆❧❣♦r✐t♠♦ P❈✿ ❱❡r✐✜❝❛♥❞♦ ✐♥❞❡♣❡♥❞ê♥❝✐❛s ❝♦♥❞✐❝✐♦♥❛✐s✳ ❆ ✈❛r✐á✈❡❧ ❙❡①♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ✈❛r✐á✈❡❧ ❈r❡❞✐t ❘❛t✐♥❣ ❞❛❞❛ ❛ ■❞❛❞❡ ❡ ❈ré❞✐t♦s ❆♥t❡r✐♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✺ ❆❧❣♦r✐t♠♦ P❈✲ P❛ss♦ ✸✿ ❉❛❞❛ ❛ ❚r✐♣❧❛ ❢♦r♠❛❞❛ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s

❙❡①♦✱ ❈ré❞✐t♦s ❆♥t❡r✐♦r❡s ❡ ■❞❛❞❡✱ é ❞❡✜♥✐❞❛ ❛ ❝♦♥❡①ã♦ ❤❡❛❞✲t♦✲❤❡❛❞✳ ✺✶ ✸✳✻ ❆❧❣♦r✐t♠♦ P❈✲ P❛ss♦ ✹✿ ♦r✐❡♥t❛çã♦ ❣❡r❛♥❞♦ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ▼❛r❦♦✈✳

❊st❛s r❡❞❡s sã♦ ▼❛r❦♦✈ ❡q✉✐✈❛❧❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✼ ❊str✉t✉r❛ ❡st✐♠❛❞❛ ✉t✐❧✐③❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ P❈ ✐♠♣❧❡♠❡♥t❛❞♦ ♥♦ ❙♦❢t✲

✇❛r❡ ❘✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✽ ✳ ❆❥✉st❡ ❞♦ ❛❧❣♦r✐t♠♦ P❈ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s r❡❛✐s ❏❛♣❛♥❡s❡ ❈r❡❞✐t

❙❝r❡❡♥✐♥❣ ❉❛t❛ ❙❡t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✾ P♦ssí✈❡❧ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ♣❛r❛ ❞❛❞♦s ❛♣❧✐❝❛❞♦s ❛ ❝r❡❞✐t s❝♦r✐♥❣✳ ✳ ✳ ✺✻ ✸✳✶✵ P♦ssí✈❡❧ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ❝♦♠ ❚P❈ ♣❛r❛ ❞❛❞♦s ❞❡ ❝r❡❞✐t s❝♦r✐♥❣✳ ✳ ✻✵ ✸✳✶✶ ❊st✐♠❛çã♦ ❇❛②❡s✐❛♥❛ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❛ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛✳ ✳ ✳ ✳ ✻✸

✹✳✶ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✷ ❊①❡♠♣❧✐✜❝❛çã♦ ❞❡ ✉♠❛ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ❙✐♠♣❧❡s ❝♦♠ ✵✲ ❞❡♣❡♥❞ê♥❝✐❛✳ ✻✾ ✹✳✸ ❊①❡♠♣❧✐✜❝❛çã♦ ❞❡ ✉♠❛ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ❙✐♠♣❧❡s ❝♦♠ ✶✲ ❞❡♣❡♥❞ê♥❝✐❛✳ ✼✵ ✹✳✹ ❊①❡♠♣❧✐✜❝❛çã♦ ❞❡ ✉♠❛ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ❙✐♠♣❧❡s ❝♦♠ ✷✲ ❞❡♣❡♥❞ê♥❝✐❛✳ ✼✵ ✹✳✺ ❊①❡♠♣❧✐✜❝❛çã♦ ❞❡ ✉♠❛ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ❙✐♠♣❧❡s ❝♦♠ ✸✲❞❡♣❡♥❞ê♥❝✐❛✳ ✼✶ ✹✳✻ ❊①❡♠♣❧♦ ❞❡ ❘❡❞❡ ◆❡✉r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✼ ❊①❡♠♣❧♦ ❞❡ ❈✉r✈❛ ❘❖❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✽ ❊sq✉❡♠❛t✐③❛çã♦ ❞♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❇❛❣❣✐♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✹✳✾ ❊str✉t✉r❛s ❞❡ ❘❡❞❡ Pr♦❜❛❜✐❧íst✐❝❛ ♣❛r❛ ♦s ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ❝♦♠

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✷✳✶ ❚❛❜❡❧❛ ❞❡ Pr♦❜❛❜✐❧✐❞❛❞❡ ❈♦♥❞✐❝✐♦♥❛❧ P✭❈⑤❆✱❇✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✸✳✶ ❈♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s r❡❢❡r❡♥t❡s ❛ ❝r❡❞✐t s❝♦r✐♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❥✉♥t❛ P(CA, S) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✸✳✸ Pr♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❥✉♥t❛ P(CR, CA) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✸✳✹ Pr♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧P(CA,|S) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✸✳✺ Pr♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧P(CR_|CA) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✸✳✻ ❋r❡qüê♥❝✐❛ ❆❜s♦❧✉t❛ ❞❡ (CR, CA) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✸✳✼ Pr♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧P(CR_|CA) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✹✳✶ ▼❛tr✐③ ❞❡ ❝♦♥❢✉sã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✷ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❛tr❛✈és ❞❡ ❞❛❞♦s r❡❛✐s

❞✐s❝r❡t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✹✳✸ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❛tr❛✈és ❞❡ ❞❛❞♦s r❡❛✐s

❝♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✹✳✹ ❆♣❧✐❝❛çã♦ ❞♦ ♣r♦❝❡❞✐♠❡♥t♦ ❇❛❣❣✐♥❣✲✺ ♣❛r❛ ♦s ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ❝♦♠

✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❞✐s❝r❡t❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✹✳✺ ❆♣❧✐❝❛çã♦ ❞♦ ♣r♦❝❡❞✐♠❡♥t♦ ❇❛❣❣✐♥❣✲✺ ♣❛r❛ ♦s ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ❝♦♠

✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❝♦♥tí♥✉❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼

✹✳✻ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❛tr❛✈és ❞❡ s✐♠✉❧❛çã♦ ❡♠ ❞❛❞♦s ❞✐s❝r❡t♦s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✹✳✼ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❛tr❛✈és ❞❡ s✐♠✉❧❛çã♦ ❡♠ ❞❛❞♦s ❞✐s❝r❡t♦s

❡ ❞❡♣❡♥❞❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✹✳✽ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❛tr❛✈és ❞❡ s✐♠✉❧❛çã♦ ❡♠ ❞❛❞♦s ❝♦♥tí✲

♥✉♦s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✹✳✾ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❛tr❛✈és ❞❡ s✐♠✉❧❛çã♦ ❡♠ ❞❛❞♦s ❝♦♥tí✲

♥✉♦s ❡ ❞❡♣❡♥❞❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❆ q✉❛♥t✐❞❛❞❡ ❞❡ ❞❛❞♦s ❞✐s♣♦♥í✈❡❧ ♥♦ ♠✉♥❞♦ t❡♠ ❛✉♠❡♥t❛❞♦ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ❛ ❝❛❞❛ ❞✐❛✳ ❆ ♥❡❝❡ss✐❞❛❞❡ ♣♦r ❢❡rr❛♠❡♥t❛s ❝❛♣❛③❡s ❞❡ ❛♥❛❧✐s❛r ❡ss❡s ❞❛❞♦s ♠♦t✐✈♦✉ ♦ s✉r❣✐♠❡♥t♦ ❞❛ ár❡❛ ❞❡ ♣❡sq✉✐s❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s✱ s❡♥❞♦ ❡st❛ ✉♠❛ ár❡❛ ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ ❛♦s ♠ét♦❞♦s ❡st❛tíst✐❝♦s✳

◆❡st❡ ❡♥r❡❞♦✱ ❞❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s ❡stá ✐♥s❡r✐❞♦ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❑♥♦✇❧❡❞❣❡ ❉✐s❝♦✈❡r② ✐♥ ❉❛t❛❜❛s❡s ✲ ❑❉❉✱ ♦✉ ❞❡s❝♦❜❡rt❛ ❞❡ ❝♦♥❤❡✲ ❝✐♠❡♥t♦s ❡♠ ❜❛♥❝♦s ❞❡ ❞❛❞♦s✱ ♦ q✉❛❧ é r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❡①tr❛çã♦ ❞❡ ✐♥❢♦r♠❛çõ❡s s❡♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❞❡ ✉♠ ❣r❛♥❞❡ ❜❛♥❝♦ ❞❡ ❞❛❞♦s ❡ s❡✉ ✉s♦ ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s ✭❉■◆■❩❀ ▲❖❯❩❆❉❆✲◆❊❚❖✱ ✷✵✵✵✮✳

❇❛s✐❝❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ❡st❡s ♣r♦❝❡❞✐♠❡♥t♦s ♣❡r♠✐t❡♠ ❛ tr❛♥s❢♦r✲ ♠❛çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❜r✉t♦s ❡♠ ✐♥❢♦r♠❛çã♦ ❡ ❝♦♥❤❡❝✐♠❡♥t♦ út❡✐s ❡♠ ❞✐✈❡rs❛s ár❡❛s✳

❆ss✐♠✱ ♥♦t♦r✐❛♠❡♥t❡✱ ❡①✐st❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❝♦♥tí♥✉❛ ❞❡ t❡♦r✐❛s ❡ ❢❡rr❛♠❡♥t❛s ❡s✲ t❛tíst✐❝❛s ❡ ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ❛✉①✐❧✐❛r ♦s s❡r❡s ❤✉♠❛♥♦s ❛ ❡①tr❛✐r ❝♦♥❤❡❝✐♠❡♥t♦✱ ✐♥❢♦r♠❛çã♦ út✐❧ ❡ t❛♥❣í✈❡❧✱ ❞❡ ❝r❡s❝❡♥t❡s ✈♦❧✉♠❡s ❞❡ ❞❛❞♦s✳

❆❧é♠ ❞✐ss♦✱ ♦s ♣r♦❝❡❞✐♠❡♥t♦s ❞❡ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s sã♦ ❝♦♥s✐❞❡r❛❞♦s ✐♥t❡r❛t✐✈♦s ❡ ✐t❡r❛t✐✈♦s✳ ❆ ✐♥t❡r❛t✐✈✐❞❛❞❡ é ❞❡✈✐❞❛ ❛♦ ❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❝♦♦♣❡r❛çã♦ ❞❡ ✉♠ ❣r✉♣♦

✷

❋✐❣✉r❛ ✶✳✶✿ ❈♦♥❡①õ❡s ❡♥tr❡ ♦s ♦❜❥❡t✐✈♦s ❡ t❛r❡❢❛s ❡♠ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s✳ ❆❞❛♣t❛❞♦ ❞❡ ❱❡❧✐❝❦♦✈ ❡ ❙♦❧♦♠❛t✐♥❡ ✭✷✵✵✵✮✳

r❡s♣♦♥sá✈❡❧✱ ❝✉❥♦ ❝♦♥❤❡❝✐♠❡♥t♦ r❡❢❡r❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛ ❛♥❛❧✐s❛❞♦ ❛✉①✐❧✐❛rá ♥❛ ❡①❡❝✉çã♦ ❞❡ t♦❞♦ ♦ ♣r♦❝❡ss♦✳ P♦r s✉❛ ✈❡③✱ ❛ ✐t❡r❛t✐✈✐❞❛❞❡ ♣r♦✈é♠ ❞❡ q✉❡✱ ❢r❡q✉❡♥t❡♠❡♥t❡✱ ❡st❡ ♣r♦❝❡ss♦ ❡♥✈♦❧✈❡ r❡♣❡t✐❞❛s s❡❧❡çõ❡s ❞❡ ❛♠♦str❛s ❡ ❛♣❧✐❝❛çõ❡s ❞❛s té❝♥✐❝❛s ❞❡ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s ❡ ♣♦st❡r✐♦r ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❛ ✜♠ ❞❡ r❡✜♥❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❡①tr❛í❞♦s ✭❇❘❆❈❍▼❆◆❀ ❆◆❆◆❉✱✶✾✾✻✮✳

❖s ♣r♦❜❧❡♠❛s tr❛t❛❞♦s ❡♠ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s sã♦ r❡s♦❧✈✐❞♦s ♣♦r ❞♦✐s ❣r❛♥❞❡s ❣r✉♣♦s ❞❡ ♦❜❥❡t✐✈♦s ✭❱❊▲■❈❑❖❱❀ ❙❖▲❖▼❆❚■◆❊✱ ✷✵✵✵✮✳

❼ ❉❡s❝r✐çã♦✿ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❡♥❝♦♥tr❛r ♣❛❞rõ❡s✱ ❛ss♦❝✐❛çõ❡s ♦✉ ❝♦rr❡❧❛çõ❡s ✐♥t❡r♣r❡tá✈❡✐s ❛tr❛✈és ❞❛ ❞❡s❝r✐çã♦ ❞♦s ❞❛❞♦s✳

❼ Pr❡❞✐çã♦✿ r❡❛❧✐③❛r ✐♥❢❡rê♥❝✐❛s s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❡①✐st❡♥t❡✱ ❛ ✜♠ ❞❡ ♣r❡✈❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ♥♦✈❛s ♦❜s❡r✈❛çõ❡s✳ ■ss♦ ♣♦❞❡ s❡r ❢❡✐t♦ ❛tr❛✈és ❞❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♦✉ ♠❛✐s ♠♦❞❡❧♦s✳

✸

❆ ❝❧❛ss✐✜❝❛çã♦ é ❛ t❛r❡❢❛ ♠❛✐s ❝♦♠✉♠ ❞❡♥tr❡ ❛s ❞✐✈❡rs❛s t❛r❡❢❛s ❞❡ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s ✭❇❊❘❘❨❀ ▲■◆❖❋❋✱ ✶✾✾✼✮✳ ❊❧❛ ❝♦♥s✐st❡ ♥❛ ❞❡s❝♦❜❡rt❛ ❞❡ r❡❣r❛s ❞❡ ♣r❡✈✐sã♦ ♣❛r❛ ❛✉①í❧✐♦ ♥♦ ♣❧❛♥❡❥❛♠❡♥t♦ ❡ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✳

❉❡st❛ ❢♦r♠❛✱ ❣❡r❛❧♠❡♥t❡ tr❛❞✉③✐❞♦ ❡♠ ✉♠ ❛❧❣♦r✐t♠♦✱ ✉♠ ♠ét♦❞♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❝♦♥s✐st❡ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ♣r❡❞✐çã♦ ♣❛r❛ ✉♠❛ ✈❛r✐á✈❡❧ ❝❛t❡❣ór✐❝❛ ❜❛s❡❛❞♦ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ♣ré✲❞❡✜♥✐❞❛s✱ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✳

❖s ♠ét♦❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ tê♠ s✐❞♦ ❧❛r❣❛♠❡♥t❡ ✉t✐❧✐③❛❞♦s ❡ s❡ ♠♦str❛♠ ♥❡❝❡s✲ sár✐♦s ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ■❞❡♥t✐✜❝❛♥❞♦ ❛♣❡♥❛s ❛❧❣✉♠❛s ár❡❛s ♣❛r❛ ❡①❡♠♣❧✐✜❝❛çã♦✿ ♥❛ ár❡❛ ❞❡ ❜✐♦♠❡❞✐❝✐♥❛✱ ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❡ ❞✐❛❣♥óst✐❝♦s✱ ✈❡r✐✜❝❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s ❣❡♥ét✐❝❛s✱ ❡♥tr❡ ♦✉tr❛s ❛♣❧✐❝❛çõ❡s❀ ♥❛ ár❡❛ ✜♥❛♥❝❡✐r❛ ❡ ❞❡ ♥❡❣ó❝✐♦s✱ ♣❛r❛ ❝❧❛ss✐✜❝❛r ❡ q✉❛♥t✐✜❝❛r r✐s❝♦ ❞❡ ❡♠♣rést✐♠♦ ❛ ❝❧✐❡♥t❡s✱ ♠❛r❦❡t✐♥❣✱ ❢❛❧ê♥❝✐❛ ❞❡ ❡♠♣r❡❡♥❞✐♠❡♥t♦s✱ ♦♣❡r❛çõ❡s ❢r❛✉❞✉❧❡♥t❛s✱ ❡♥tr❡ ♦✉tr♦s❀ ♥❛ ■♥❞ústr✐❛✱ ♣❛r❛ ♣r❡❞✐③❡r ❡ q✉❛♥t✐✜❝❛r ❝❤❛♥❝❡s ❞❡ ♣r♦❞✉çã♦ ❞❡ ✐t❡♥s ❞❡❢❡✐t✉♦s♦s ❡✱ ❛té ♠❡s♠♦✱ ♥❛ ■♥t❡r♥❡t ♣❛r❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ s♣❛♠✱ ♠ét♦❞♦s ❞❡ ❜✉s❝❛ t❡①t✉❛✐s❀ ❡♥tr❡ ♦✉tr♦s ❝❛s♦s ❝♦♠♦ ✈✐❣✐❧â♥❝✐❛ ❡ ❞❡s❝♦❜❡rt❛s ❝✐❡♥tí✜❝❛s✳

❊♠❜♦r❛ ❛s s♦❧✉çõ❡s ❡♠ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s ♣♦ss❛♠ s❡r ❞✐✈✐❞✐❞❛s ❡♠ ❞♦✐s ❣r❛♥❞❡s ❣r✉♣♦s✱ ❝♦♠♦ ❝✐t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❡①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♠ét♦❞♦s r❡❧❛t✐✈♦s ❛ ❝❛❞❛ ✉♠❛ ❞❛s t❛r❡❢❛s✱ s❡♥❞♦ q✉❡✱ ❣❡r❛❧♠❡♥t❡✱ ✉♠ ♠ét♦❞♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ♠❛✐s ❞❡ ✉♠❛ t❛r❡❢❛✱ ♦✉ s❡❥❛✱ ✉♠ ♠❡s♠♦ ♠ét♦❞♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❡ r❡❣r❡ssã♦✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡ ❡♠ ❝❧❛ss✐✜❝❛çã♦✱ ♣♦❞❡♠♦s ❝✐t❛r ❝♦♠♦ ♠❛✐s ❝♦♠✉♥s ♦s ♠ét♦❞♦s✿ ❆♥á❧✐s❡ ❉✐s❝r✐♠✐♥❛♥t❡✱ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛ ❡ Pr♦❜✐t♦✱ ❘❡❞❡s ◆❡✉r❛✐s ❡ ❈❧❛ss✐✜❝❛çã♦ ♣♦r ➪r✈♦r❡s ✭❆❇❉❖❯ ❡t ❛❧✳✱ ✷✵✵✽✮✳

✹

♠❡♥t❡ ❡ ❞❡ ❢♦r♠❛ ❜❡♠✲s✉❝❡❞✐❞❛ ❡♠ ❞✐✈❡rs❛s ár❡❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❡st✐♠❛çã♦ ❞❡ r✐s❝♦ ♦♣❡r❛❝✐♦♥❛❧✱ ❞✐❛❣♥óst✐❝♦ ♠é❞✐❝♦✱ ❝r❡❞✐t s❝♦r✐♥❣✱ ♣r♦❥❡t♦ ❞❡ ❥♦❣♦s ❝♦♠♣✉t❛❝✐♦✲ ♥❛✐s✱ ✐♠♣✉t❛çã♦ ❞❡ ❞❛❞♦s✱ ❡♥tr❡ ♦✉tr❛s✳

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❡♥tr❡ s❡✉s ♣r✐♥❝✐♣❛✐s ♦❜❥❡t✐✈♦s ✐♥✈❡st✐❣❛r ❛ ❛♣❧✐❝❛çã♦ ❞❛ té❝♥✐❝❛ ❞❡ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❜✐♥ár✐❛✱ ❝♦♠♣❛r❛♥❞♦✲ ❛ ❡♥tr❡ s❡✉s ❞✐✈❡rs♦s t✐♣♦s ❞❡ ❛❥✉st❡ ❡✱ t❛♠❜é♠✱ ❝♦♠ ❛s ♣r✐♥❝✐♣❛✐s té❝♥✐❝❛s ❛t✉❛✐s ❞❡st❡ ❡♥r❡❞♦✳ ❆ ✜♠ ❞❡ ❝♦♥tr✐❜✉✐r ❝♦♠ ❛ ❡st❛tíst✐❝❛ ♥❛❝✐♦♥❛❧✱ ♥♦ s❡♥t✐❞♦ ❞❛ ❡s❝❛ss❡③ ❞❡ ❧✐t❡r❛t✉r❛ r❡❢❡r❡♥t❡ ❛ ❡st❛ té❝♥✐❝❛ ❡♠ ♥♦ss♦ ♣❛ís✳ ❇❡♠ ❝♦♠♦ ❛ ❝♦♥str✉çã♦ ❞❡ r♦t✐♥❛s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❡s♣❡❝í✜❝❛s q✉❡ ♣❡rt✐♠❡♠ ❛ ✉t✐❧✐③❛çã♦ ❣❡r❛❧ ❞❡st❛ t❡♦r✐❛✳

P♦r s✐♠♣❧✐❝✐❞❛❞❡ ❡ ❡①❡♠♣❧✐✜❝❛çã♦✱ ❛❜♦r❞❛♠♦s ❛ t❛r❡❢❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❛♣❧✐❝❛❞❛ ❛♦ ❡♥r❡❞♦ ❞❡ ♥❡❣ó❝✐♦s✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♦ ❝♦♥t❡①t♦ ❞❡ ❝r❡❞✐t s❝♦r✐♥❣✱ ❡ ❛♦ ❡♥r❡❞♦ ❞❛ s❛ú❞❡✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ à ♣r♦❜❧é♠❛t✐❝❛ ❞❡ ❞✐❛❣♥óst✐❝♦ ❡ ❞❡t❡❝çã♦ ❞❡ ❞♦❡♥ç❛s✱ ♦s q✉❛✐s s❡rã♦ ❡①♣♦st♦s ❛ s❡❣✉✐r✳ ❆ss✐♠✱ ❛ ♠❛✐♦r✐❛ ❞❛s ❛♣❧✐❝❛çõ❡s ❡♠ ❞❛❞♦s r❡❛✐s ❡ ❡①❡♠♣❧♦s t❡ór✐❝♦s ❡stã♦ ❜❛s❡❛❞♦s ♥❡st❛s ♣r♦❜❧❡♠át✐❝❛s ❡ ❝♦♥s✐❞❡r❛♠ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❜✐♥ár✐❛✳

✺

✜♠✱ ♦ ❈❛♣ít✉❧♦ ✺ ❡①✐❜❡ ❝♦♠❡♥tár✐♦s ✜♥❛✐s s♦❜r❡ ♦ tr❛❜❛❧❤♦✳

✶✳✶ ❈r❡❞✐t ❙❝♦r✐♥❣

❆ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦ ♥❛s❝❡✉ ♥♦s ♣r✐♠ór❞✐♦s ❞♦ ❝♦♠ér❝✐♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❝♦♥❝❡ssã♦ ❞❡ ❡♠♣rést✐♠♦s ❞❡ ❞✐♥❤❡✐r♦ ♦✉ ❝♦♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❡ ❝♦♠♣r❛s ❛ ♣❛❣❛r ❢✉t✉r❛♠❡♥t❡✱ ♣♦✐s✱ ❞❡s❞❡ ❛q✉❡❧❛ é♣♦❝❛✱ q✉❛♥❞♦ ✉♠ ❝♦♠❡r❝✐❛♥t❡ ♦❢❡r❡❝✐❛ ❞❡♠❛s✐❛❞♦ ❝ré❞✐t♦ à ♣❡ss♦❛ ❡rr❛❞❛✱ ❝♦rr✐❛ ♦ r✐s❝♦ ❞❡ ♣❡r❞❡r ❞✐♥❤❡✐r♦ ❡ t❡r ❢✉t✉r♦s ♣r♦❜❧❡♠❛s ✜♥❛♥❝❡✐r♦s✳ ❈♦♠ ♦ ♣❛ss❛r ❞♦s ❛♥♦s✱ ♦s ❝♦♠❡r❝✐❛♥t❡s ❝♦♠❡ç❛r❛♠ ❛ ❧❡✈❛♥t❛r ✐♥❢♦r♠❛✲ çõ❡s s♦❜r❡ ♦s s♦❧✐❝✐t❛♥t❡s ❞❡ ❝ré❞✐t♦ ❡ ❝❛t❛❧♦❣á✲❧♦s ♣❛r❛ ❞❡❝✐❞✐r s❡ ❡♠♣r❡st❛r✐❛♠ ♦✉ ♥ã♦ ❞❡t❡r♠✐♥❛❞❛ q✉❛♥t✐❛ ❡♠ ❞✐♥❤❡✐r♦✳

❈♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❝✐ê♥❝✐❛ ❡♠ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s r❡✢❡t✐❞❛ ❡♠ ♠ét♦❞♦s ♣r❡❝✐s♦s✱ ❤♦❥❡ ❝r❡❞✐t s❝♦r✐♥❣ é ✉♠ ♠ét♦❞♦ ❞❡ ❛✈❛❧✐❛çã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣❛r❛ ❛♣❧✐❝❛çã♦ ❞❡ ❡♠♣rést✐♠♦s ✭▼❊❙❚❊❘✱ ✶✾✾✼✮✳ ❇❛s❡❛❞♦ ❡♠ ♠ét♦❞♦s ❡st❛tíst✐❝♦s ♣❛r❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s✱ t❛❧ ♠ét♦❞♦ ♣r♦❞✉③ ✉♠ s❝♦r❡ ♣❛r❛ ❝❛❞❛ ❝❧✐❡♥t❡✱ q✉❛♥t✐✜❝❛♥❞♦ ♦ r✐s❝♦ ❞❡st❡ ❝❧✐❡♥t❡ s❡r ❜♦♠ ♦✉ ♠❛✉ ♣❛❣❛❞♦r✱ ❛ ✜♠ ❞❡ ♠✐♥✐♠✐③❛r ❛s ♣❡r❞❛s ♦✉ ♠❛①✐♠✐③❛r ♦s ❣❛♥❤♦s ❞❡ ✉♠❛ ❡♠♣r❡s❛✱ ❣❡r❛❧♠❡♥t❡ ✜♥❛♥❝❡✐r❛✳

P♦r t❡r ❝♦♠♦ ♦❜❥❡t✐✈♦ ✜♥❛❧ ❛ ❝❧❛ss✐✜❝❛çã♦ ❜✐♥ár✐❛ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝❛r❛❝✲ t❡ríst✐❝❛✱ sã♦ ❛♣❧✐❝❛❞♦s ❞✐✈❡rs♦s ♠ét♦❞♦s ❞❡ tr❛t❛♠❡♥t♦ ❞❡ ❞❛❞♦s ♥❛ ár❡❛ ❞❡ ❝r❡❞✐t s❝♦r✐♥❣✳

✻

✶✳✷ ❉✐❛❣♥óst✐❝♦ ❞❡ ❉♦❡♥ç❛s

❯♠ ❛♠♣❧♦ s✐st❡♠❛ ❞❡ ✐♥❢♦r♠❛çõ❡s ❤♦s♣✐t❛❧❛r ♣❛r❛ ❛t❡♥❞❡r às ♥❡❝❡ss✐❞❛❞❡s ❡s♣❡❝í✜❝❛s ❞❡ ✉♠ ❤♦s♣✐t❛❧ ❝♦♥té♠ ♠ó❞✉❧♦s ❞❡ ✐♥t❡r♥❛çã♦✱ r❡❣✐str♦ ❞❡ ❛♠❜✉❧❛tór✐♦✱ ❛ss✐stê♥❝✐❛ ❛♦ ♣❛❝✐❡♥t❡✱ r❡❣✐str♦ ❞❡ ❢❛r♠á❝✐❛✱ ♣❧❛♥❡❥❛♠❡♥t♦ ❞❡ ❞✐❡t❛✱ ❡♥tr❡ ♦✉tr♦s✳ ❊♠ s✉♠❛✱ ✉♠ ❡q✉✐♣❛♠❡♥t♦ s♦✜st✐❝❛❞♦ ✉t✐❧✐③❛❞♦ ♥❛ ♣rát✐❝❛ ❞❛ ♠❡❞✐❝✐♥❛ ♠♦❞❡r♥❛ ❡ ❣❡r❛❞♦r ❞❡ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❞❛❞♦s✱ ✉♠ ❧♦❝❛❧ ✐❞❡❛❧ ♣❛r❛ ♣r♦❝✉r❛ ❞❡ ♥♦✈❛s ❛♥á❧✐s❡s ❡ ♣❛❞rõ❡s✱ ♦✉ ♣❛r❛ ✈❛❧✐❞❛çã♦ ❞❡ ❤✐♣ót❡s❡s ♣r♦♣♦st❛s ✭❲❆❙❆◆ ❡t ❛❧✳✱ ✷✵✵✻✮✳ P❛r❛ ❡①♣❧♦r❛r ❡st❡s ❞❛❞♦s ♠é❞✐❝♦s✱ ✐♥ú♠❡r❛s té❝♥✐❝❛s ❞❡ ❛♥á❧✐s❡ ❡st❛tíst✐❝❛✱ ♣r♦✈❡♥✐❡♥t❡s ❞♦ ❡♥r❡❞♦ ❞❡ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s✱ sã♦ ❛♣❧✐❝❛❞❛s ❝♦♠ s✉❝❡ss♦ ♣❛r❛ ❞❡s❝♦❜r✐r ❝♦♥❤❡❝✐♠❡♥t♦ út✐❧ ❡ ♥♦✈♦✱ ♦ q✉❛❧ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ rá♣✐❞❛ ❡ ♠❡❧❤♦r t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s ❝❧í♥✐❝❛s ✭❇❆❘◆❊❙✱ ✷✵✵✸✮✭▲❆❇■❇ ❡ ▼❆▲❊❑✱ ✷✵✵✺✮✳ ❆ss✐♠✱ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✐♥❢♦r♠❛✲ çõ❡s r❡❧❛t✐✈❛s ❛ ✉♠❛ ❞♦❡♥ç❛✱ ❞❡s❡❥❛♠♦s ✈❡r✐✜❝❛r ❛ ❝❤❛♥❝❡ ❞❡ ✉♠ ♣❛❝✐❡♥t❡ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❞♦❡♥ç❛ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✐♥❢❛rt♦ ❞♦ ♠✐♦❝ár❞✐♦✱ ❝â♥❝❡r ❞❡ ♠❛♠❛✱ ❞✐❛❜❡t❡s✱ ❞♦r ❛❜❞♦♠✐♥❛❧✱ ❡♥tr❡ ♦✉tr❛s✱ ❛❧é♠ ❞❡❝✐❞✐r s♦❜r❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s✉❛ ✐♥t❡r✲ ♥❛çã♦ ♦✉ ✈❡r✐✜❝❛r ❢❛t♦r❡s q✉❡ ♣♦❞❡♠ ❧❡✈❛r à ❝❛✉s❛ ❞❡ s✉❛ ❡♥❢❡r♠✐❞❛❞❡✳

❉❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ ❛s té❝♥✐❝❛s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♣r♦♣✐❝✐❛♠ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❞✐❛❣✲ ♥♦s❡ ❞✐❢❡r❡♥❝✐❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ s❡ ❜❛s❡✐❛♠ ♥♦ ❡st✉❞♦ ❞❡ ❞♦❡♥ç❛s q✉❛♥t✐✜❝❛❞❛s ❛tr❛✈és ❞❡ t❡st❡s ♠é❞✐❝♦s ♦✉ ❤✐stór✐❝♦ ❞♦ ♣❛❝✐❡♥t❡✱ ❛ ✜♠ ❞❡ ❞❡t❡r♠✐♥❛r s❡ ❡st❡ é ♣♦rt❛❞♦r ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝❛r❛❝t❡ríst✐❝❛ ♦✉ s❡ ♥❡❝❡ss✐t❛ ❞❡ ✉♠ tr❛t❛♠❡♥t♦ ❞✐❢❡r❡♥❝✐❛❞♦✳

✼

✶✳✸ Pr♦❜❛❜✐❧✐❞❛❞❡s

❖ ❝á❧❝✉❧♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s t❡✈❡ ♦r✐❣❡♠ ❡♠ ❡st✉❞♦s ❞❡ ❥♦❣♦s ❞❡ ❛③❛r ♥❛ ■❞❛❞❡ ▼é❞✐❛✳ ❆ss✐♠✱ ❡♠ ✶✻✺✹✱ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ❝✐ê♥❝✐❛ é ❞❡✈✐❞♦ ❛ ✉♠❛ sér✐❡ ❞❡ ❝❛rt❛s tr♦❝❛❞❛s ❡♥tr❡ ❞♦✐s ♠❛t❡♠át✐❝♦s ❡ ♣❡♥s❛❞♦r❡s ♥♦tá✈❡✐s✱ ❇❧❛✐s❡ P❛s❝❛❧ ✭✶✻✷✸✲ ✶✻✻✷✮ ❡ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✭✶✻✵✶✲✶✻✻✺✮✱ s♦❜r❡ ♣r♦❜❧❡♠❛s ❝♦♠ ❛♣♦st❛s ❡♠ ✉♠ ❥♦❣♦ ❝♦♠♣♦st♦ ♣♦r ♠♦❡❞❛s ❡ ❞❛❞♦s✳

❉❡s❞❡ ❡♥tã♦✱ ❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❢♦✐ ❛♠♣❧❛♠❡♥t❡ ❡st✉❞❛❞❛✱ ✐♥❝❧✉s✐✈❡ ♣❡❧♦ t❛♠❜é♠ r❡♥♦♠❛❞♦ ❚❤♦♠❛s ❇❛②❡s✱ s❡♥❞♦ ❤♦❥❡ ✉t✐❧✐③❛❞❛ ❡♠ ❞✐✈❡rs♦s ♣r♦❝❡❞✐♠❡♥t♦s ❞❛s ❈✐ê♥❝✐❛s ❊①❛t❛s✳

◆❡st❛ s❡çã♦ ✐♥tr♦❞✉③✐♠♦s ✉♠❛ ❜r❡✈❡ ❤✐stór✐❛ s♦❜r❡ ❚❤♦♠❛s ❇❛②❡s ❡ ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❡♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ q✉❡ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞❡ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s✳

✶✳✸✳✶ ❚❤♦♠❛s ❇❛②❡s

◆❛s❝✐❞♦ ❡♠ ▲♦♥❞r❡s ♥♦ ❛♥♦ ❞❡ ✶✼✵✷ ❡ ❢❛❧❡❝✐❞♦ ❡♠ ❑❡♥t✱ ❛ ✺✽ ❦♠ ❞❡ ▲♦♥❞r❡s✱ ❡♠ ✶✼✻✶✱ ♦ ✐♥❣❧ês ❚❤♦♠❛s ❇❛②❡s ✭❋✐❣✉r❛ ✶✳✷✮ ❢♦✐ ♠❛t❡♠át✐❝♦ ❡ r❡✈❡r❡♥❞♦ ❞❛ ✐❣r❡❥❛ ♣r❡s✲ ❜✐t❡r✐❛♥❛ ❡ ✐♠♦rt❛❧✐③❛❞♦ ♣♦r ❢♦r♠✉❧❛r ✉♠ ✐♠♣♦rt❛♥t❡ t❡♦r❡♠❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ♦ q✉❛❧ é ✐♥t✐t✉❧❛❞♦ ♣❡❧♦ s❡✉ ♥♦♠❡ ❡ ❞❡✉ ♦r✐❣❡♠✱ ❛♥♦s ❞❡♣♦✐s✱ ❛ ✉♠ ♥♦✈♦ r❛♠♦ ❞❛ ❝✐ê♥❝✐❛ ❡st❛tíst✐❝❛✱ ❞❡♥♦♠✐♥❛❞❛ ❊st❛tíst✐❝❛ ❇❛②❡s✐❛♥❛✳

✽

❋✐❣✉r❛ ✶✳✷✿ Ú♥✐❝❛ ■❧✉str❛çã♦ ❝♦♥❤❡❝✐❞❛ ❞❡ ❚❤♦♠❛s ❇❛②❡s

❛♣❡♥❛s ❞♦✐s tr❛❜❛❧❤♦s ❡♠ ✈✐❞❛✱ ♦ ♣r✐♠❡✐r♦ ✐♥t✐t✉❧❛❞♦ ✧❇❡♥❡✈♦❧ê♥❝✐❛ ❞✐✈✐♥❛✧ ✭✶✼✸✶✮ ❡ ♦ s❡❣✉♥❞♦ ✧❯♠❛ ■♥tr♦❞✉çã♦ ❛ ❞♦✉tr✐♥❛ ❞♦s ✢✉①✐♦♥s✧ ♥♦ q✉❛❧ ❡❧❡ ❞❡❢❡♥❞✐❛ ■s❛❛❝ ◆❡✇t♦♥ ❝♦♥tr❛ ❛ ❝rít✐❝❛ ❞❡ ●❡♦r❣❡ ❇❡r❦❧❡②✱ ❝♦♥❤❡❝✐❞♦ ✜❧♦s♦❢♦ ✐r❧❛♥❞ês ❞❛ é♣♦❝❛✳ ❆♣ós s✉❛ ♠♦rt❡✱ ♦✉tr♦ tr❛❜❛❧❤♦ ❞❡ s✉❛ ❛✉t♦r✐❛ ❢♦✐ r❡✈❡❧❛❞♦ ✧❊♥s❛✐♦ ❜✉s❝❛♥❞♦ r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛ ♥❛ ❞♦✉tr✐♥❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✧✱ ♥♦ q✉❛❧ ❤❛✈✐❛ ❢♦r♠✉❧❛❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❇❛②❡s✳

P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s s♦❜r❡ ❛ ✈✐❞❛ ❞❡ ❚❤♦♠❛s ❇❛②❡s ❝♦♥s✉❧t❛r ❇❡❧❧❤♦✉s❡ ✭✷✵✵✹✮✱ ✉♠❛ ❝♦♠♣❧❡t❛ ❜✐♦❣r❛✜❛ r❡❛❧✐③❛❞❛ ❡♠ ❝♦♠❡♠♦r❛çã♦ ❛♦ s❡✉ ✸✵✵➸ ❛♥✐✈❡rsár✐♦ ❞❡ ♥❛s❝✐✲ ♠❡♥t♦✳

✶✳✸✳✷ ❈♦♥❝❡✐t♦s Pr♦❜❛❜✐❧íst✐❝♦s

✾

✶✳✸✳✷✳✶ Pr♦❜❛❜✐❧✐❞❛❞❡ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s

❊♠ ♣♦✉❝❛s ♣❛❧❛✈r❛s✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣♦❞❡ s❡r ✐♥tr♦❞✉③✐❞❛✱ s❡❣✉♥❞♦ ❈♦st❛ ◆❡t♦ ❡ ❈②♠❜❛❧✐st❛ ✭✷✵✵✻✮✱ ❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ q✉❡ ♠❡❞❡ ❛ ♠❛✐♦r ♦✉ ♠❡♥♦r ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ❞✐✈❡rs♦s ❡✈❡♥t♦s✳

P♦ré♠✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ é✱ ❤✐st♦r✐❝❛♠❡♥t❡✱ ❝❡♥ár✐♦ ❞❡ ❛♠♣❧❛ ❞✐s❝✉ssã♦ ❡ t❡♠ s✐❞♦ ❞❡✜♥✐❞♦ ❞❡ ❞✐❢❡r❡♥t❡s ♠❛♥❡✐r❛s✱ s❡♥❞♦ q✉❡ ❛❧❣✉♠❛s sã♦ ❛s ❞❡✜♥✐çõ❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❢r❡qü❡♥t✐st❛✱ ❝❧áss✐❝❛ ❡ s✉❜❥❡t✐✈❛✳

❍♦❥❡ ❡♠ ❞✐❛✱ ❛ ❞❡✜♥✐çã♦ ❛①✐♦♠át✐❝❛✱ ❞❛❞❛ ♣♦r ❑♦♠♦❧❣♦r♦✈ ❡♠ ✶✾✸✸✱ é ❝♦♠✉♠❡♥t❡ ❛❞♦t❛❞❛ ❡ ❝♦♥s✐❞❡r❛ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ é ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ❝❧❛ss❡ℑ ❞❡

❡✈❡♥t♦s ❞❡ Ω✱ s❡♥❞♦ _ℑ ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ Ω ❛ q✉❛❧ é ❢❡❝❤❛❞❛ s♦❜r❡

♦♣❡r❛çõ❡s ❡♥✉♠❛rá✈❡✐s ❞❡ ✉♥✐ã♦✱ ✐♥t❡rs❡çã♦ ❡ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ ❝♦♥❥✉♥t♦s✳ ❉❡st❡ ♠♦❞♦✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

✭❛✮ P(A)>0♣❛r❛ t♦❞♦ A ǫℑ;

✭❜✮ ❙❡(An)n≥1é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡✈❡♥t♦s ❞❡ℑ✱ t❛❧ q✉❡(An)n≥1 sã♦ ♠✉t✉❛♠❡♥t❡

❡①❝❧✉s✐✈♦s✱ ❡♥tã♦✿

P ∞

[

n=1 An

!

=

∞

X

n=1

P(An) ✭✶✳✶✮

✭❝✮ P(Ω) = 1

♦♥❞❡ A é ✉♠ ❡✈❡♥t♦ ♥♦ ❡s♣❛ç♦ _ℑ ❡Ω é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡✈❡♥t♦s ❞❡ ✐♥t❡r❡ss❡

❞❡♥♦♠✐♥❛❞♦ ❡s♣❛ç♦ ❛♠♦str❛❧✳

❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♦r✐❣✐♥❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❧✐st❛❞❛s ❛❜❛✐①♦✱ s❡♥❞♦E✱ F ❡K q✉❛✐s✲

q✉❡r ❝♦♥❥✉♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛Ω ❡E ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ❡❧❡♠❡♥t♦s ♥ã♦ ♣❡rt❡♥✲

✶✵

✭❞✮ P(_∅) = 0

✭❡✮ P E = 1₋P(E)

✭❢✮ P (ESF) =P(E) +P(F)−P(ETF)

✭❣✮ ❙❡E, F, . . . , K sã♦ ❡✈❡♥t♦s q✉❡ ♥ã♦ ♣♦ss✉❡♠ ✐♥t❡rs❡❝çã♦ ❞♦✐s ❛ ❞♦✐s✱ ❞✐t♦s

♠✉t✉❛♠❡♥t❡ ❡①❝❧✉s✐✈♦s✿

P E[F [. . .[K=P(E) +P(F) +. . .+P(K) ✭✶✳✷✮

❡♥tr❡ ♦✉tr❛s✳

❆ss✐♠✱ ✉♠❛ ❢♦r♠❛ ♦❜❥❡t✐✈❛ ❞❡ ❛tr✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛♦ ❡✈❡♥t♦ F✱ q✉❛♥❞♦

Ωé ✜♥✐t♦ ❡ ❡♥✉♠❡rá✈❛❧✱ é ❞❛❞❛ ♣♦r✿

P (F) = ♯(F)

♯(Ω) ✭✶✳✸✮

♦♥❞❡ ★✭❋✮ é ♦ ♥ú♠❡r♦ ❞❡ r❡s✉❧t❛❞♦s ❢❛✈♦rá✈❡✐s ❛♦ ❡✈❡♥t♦ ❋ ❡ ★(Ω) é ♦ ♥ú♠❡r♦ ❞❡

r❡s✉❧t❛❞♦s t♦t❛✐s✱ ♦✉ s❡❥❛✱ ♦ ♥ú♠❡r♦ ❞❡ r❡s✉❧t❛❞♦s ♥♦ ❡s♣❛ç♦ ❛♠♦str❛❧ ✳

P❛r❛ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s t❡r♠♦s ♣r♦❜❛❜✐❧íst✐❝♦s✱ ❝♦♥s✐❞❡r❡ ♦s ✐t❡♥s ✶✱ ✷✱ ✸ ❡ ✹ ❞❛ ❋✐❣✉r❛ ✶✳✸✱ ♦s q✉❛✐s ❡①✐❜❡♠ ✉♠❛ ✈✐s✉❛❧✐③❛çã♦ ❢r❡q✉❡♥t❡ ♥❛ ❧✐t❡r❛t✉r❛ ❞❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❜❛s❡❛❞❛ ♥❛ ❞✐❛❣r❛♠❛çã♦ ❞❡ ❊ü❧❧❡r✲❱❡♥♥ ♣❛r❛ ♦s ❡✈❡♥t♦s ❡ ♦ s❡✉ ❡s♣❛ç♦ ❛♠♦str❛❧✳

◆❛ ❋✐❣✉r❛ ✶✳✸✱ ♦ ✐t❡♠ ✭✶✮ ❡①✐❜❡ t♦❞♦ ♦ ❡s♣❛ç♦ ❛♠♦str❛❧ ✱ ♦ ✐t❡♠ ✭✷✮ ❡①✐❜❡ ♦ ❡✈❡♥t♦

❊ s♦❜ ♦ ❡s♣❛ç♦ ❛♠♦str❛❧✱ ♦ ✐t❡♠ ✭✸✮ ❡①✐❜❡ ♦s ❡✈❡♥t♦s E ❡ F s❡♥❞♦ ♠✉t✉❛♠❡♥t❡

❡①❝❧✉s✐✈♦s✱ ♦✉ s❡❥❛✱ P(ETF) = 0 ❡✱ ✜♥❛❧♠❡♥t❡✱ ♦ ✐t❡♠ ✭✹✮ ❡①✐❜❡ ♦s ❡✈❡♥t♦s E ❡ F

✶✶

❋✐❣✉r❛ ✶✳✸✿ ❉✐❛❣r❛♠❛s ❞❡ ❊ü❧❧❡r✲❱❡♥♥

✶✳✸✳✷✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡ ❈♦♥❞✐❝✐♦♥❛❧

❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧ tr❛t❛ ❞♦ ❢❛t♦ ❞❡ q✉❡ ♠✉✐t❛s ✈❡③❡s t❡♠♦s ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡✈❡♥t♦✱ s❡♥❞♦ s✉❛ ♦❝♦rrê♥❝✐❛ ♦✉ ✉♠❛ ✐♥❢♦r♠❛çã♦ t♦♠❛❞❛ ❛ ♣r✐♦r✐✳ ❉❡st❛ ❢♦r♠❛✱ s✉r❣❡ ♦ ✐♥t❡r❡ss❡ ❞❡ ❝❛❧❝✉❧❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦✉tr♦ ❡✈❡♥t♦ ♣♦ss✐✈❡❧♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❛♦ ❛♥t❡r✐♦r✳

❉❡♥♦t❛♠♦s ❝♦♠♦ P(E|F) ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞♦ ❡✈❡♥t♦ E✱ s❛❜❡♥❞♦

q✉❡ ♦ ❡✈❡♥t♦F ♦❝♦rr❡✉✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❊ ❞❛❞♦ ❋✳

❉❡st❛ ❢♦r♠❛✱ t❡♠♦s✿

P(E|F) = P(E

T

F)

P(F) ✭✶✳✹✮

❆♥❛❧♦❣❛♠❡♥t❡✱

✶✷

❚❡♠♦s t❛♠❜é♠✱ ❣❡♥❡r❛❧✐③❛♥❞♦ ✶✳✺ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♥♦t❛çã♦P(ETF) = P(E, F)✱

P(E1,E2,. . . , En) = P(E1)P(E2|E1)P(E3|E2, E1), . . . , P(En|E1, E2, . . . , En−1)

❆❧é♠ ❞✐ss♦✱ ❝♦♥s✐❞❡r❛♥❞♦ E1, E2, . . . , En ❡✈❡♥t♦s ❡①❝❧✉s✐✈♦s ❡ ❡①❛✉st✐✈♦s✱ ♦✉ s❡❥❛✱ ❡✈❡♥t♦s q✉❡ ♥ã♦ ♣♦ss✉❡♠ ✐♥t❡rs❡❝çã♦ ❡ s✉❛ ✉♥✐ã♦ é ✐❣✉❛❧ ❛♦ ❡s♣❛ç♦ ❛♠♦str❛❧✱ t❡♠♦s

♣❛r❛ ✉♠ ❡✈❡♥t♦F

P(F) =

n

X

i=1

P(F_|Ei)P(Ei)

❆ ♣r♦♣r✐❡❞❛❞❡ ❛❝✐♠❛ é ❝♦♠✉♠❡♥t❡ ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❢ór♠✉❧❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s t♦t❛✐s✳ ◆♦t❡ q✉❡ ❡st❛ ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ❡✈❡♥t♦ F q✉❛♥❞♦ s❡

❝♦♥❤❡❝❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡✈❡♥t♦s ❞✐st✐♥t♦s✱ s❡♥❞♦ q✉❡ s✉❛ ✉♥✐ã♦ ❢♦r♠❛ ♦ ❡s♣❛ç♦ ❛♠♦str❛❧✳

✶✳✸✳✷✳✸ ■♥❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥❞✐❝✐♦♥❛❧ ♣r♦❜❛❜✐❧íst✐❝❛

❆ss✐♠ ❝♦♠♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧✱ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ♣r♦❜❛❜✐❧íst✐❝❛ é ✉♠❛ ❞❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s ✉t✐❧✐③❛❞❛s ♥❛ t❡♦r✐❛ ❞❡ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s✳ ❇❛s✐❝❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ♦s ❡✈❡♥t♦sE ❡F sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s q✉❛♥❞♦ ❡①✐st❡ ❛ r❡❧❛çã♦✿

✶✸

❆ r❡❧❛çã♦ ✶✳✻ ❛❞✈é♠ ❞❡ ♦✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ❜ás✐❝❛ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥❞✐❝✐♦♥❛❧ ♣r♦❜❛❜✐❧íst✐❝❛ ❡♥tr❡ ❞♦✐s ❡✈❡♥t♦s✱ s❡♥❞♦P(E, F) =P(E)P(F)✳

✶✳✸✳✷✳✹ ❚❡♦r❡♠❛ ❞❡ ❇❛②❡s

❈♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♥s✐❞❡r❡ ♦ ❡✈❡♥t♦ F ❡ ♦s ❡✈❡♥t♦s E1, E2, . . . , En ❡①❝❧✉s✐✈♦s ❡ ❡①❛✉st✐✈♦s✱ ♦✉ s❡❥❛✱ q✉❡ ♥ã♦ ♣♦ss✉❡♠ ✐♥t❡rs❡❝çã♦ ❞♦✐s ❛ ❞♦✐s ❡ s✉❛ ❛ ✉♥✐ã♦ ❢♦r♠❛ ♦ ❡s♣❛ç♦ ❛♠♦str❛❧✳ ❆ss✐♠✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❇❛②❡s é ❞❡✜♥✐❞♦ ❝♦♠♦✿

P(Ei|F) =

P(F_|Ei)P(Ei)

Pn

i=1P(F|Ei)P(Ei) ✭✶✳✼✮

❖ t❡♦r❡♠❛ ❞❡ ❇❛②❡s é ✉♠❛ ❥✉♥çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧ ❡ ❞❛ ❢ór♠✉❧❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s t♦t❛✐s✳ ❆ss✐♠✱ P(Ei) ♣♦❞❡ s❡r ❞❡♥♦♠✐♥❛❞❛ ❝♦♠♦ ♣r♦❜❛✲ ❜✐❧✐❞❛❞❡ ❛ ♣r✐♦r✐✱ P(F_|Ei) ❝♦♠♦ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡ P(Ei|F) ❝♦♠♦ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐✱ ♦✉ s❡❥❛✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣♦st❡r✐♦r à ♦❜s❡r✈❛çã♦ ❞♦ ❡✈❡♥t♦ ❋✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❞❡♥♦♠✐♥❛❞♦r é ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡P(F)✱ ♦✉ s❡❥❛✱ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ❝♦♥s✲

t❛♥t❡ ♥♦r♠❛❧✐③❛❞♦r❛❀ ❞❡st❛ ❢♦r♠❛✱ ✶✳✼ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ♥❛ ❢♦r♠❛ ✶✳✽✳

P(Ei|F)∝P(F|Ei)P(Ei) ✭✶✳✽✮

s❡♥❞♦ ∝✐♥❞✐❝❛❞♦r ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✳

✶✹

✶✳✸✳✷✳✺ ❆s ❞✐str✐❜✉✐çõ❡s ▼✉❧t✐♥♦♠✐❛❧ ❡ ❉✐r✐❝❤❧❡t

❊st❛s ❞✉❛s ❞✐str✐❜✉✐çõ❡s✱ ❛q✉✐ ✐♥tr♦❞✉③✐❞❛s✱ sã♦ ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞❛s ♥♦ ❝♦♥t❡①t♦ ❞❡ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s q✉❛♥❞♦ ♠ét♦❞♦s ❞❡ ❡st✐♠❛çã♦ ❜❛②❡s✐❛♥❛ sã♦ r❡q✉❡r✐❞♦s✳

❈♦♥s✐❞❡r❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ X ❞✐s❝r❡t❛ q✉❡ r❡♣r❡s❡♥t❡ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❝♦♠ r ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s✱ s❡♥❞♦ q✉❡ ❝❛❞❛ t✐♣♦ ❞❡ r❡s✉❧t❛❞♦ ♣♦ss✉✐ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡

❡s♣❡❝í✜❝❛ P(X = xr) = pr ❡ Pr_i=1pi = 1✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❡①♣❡r✐♠❡♥t♦ é r❡♣❡t✐❞♦ ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡N ✈❡③❡s✱ ❞❡ ❢♦r♠❛ q✉❡ ❛ ✈❛r✐á✈❡❧Xi s❡❥❛ ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡ ♦ r❡s✉❧t❛❞♦xi ❡stá ♣r❡s❡♥t❡ ♥❛ ❛♠♦str❛ ❝♦♠i= 1, ..., r✳ ❚❡♠♦s q✉❡ ❛ ✈❛r✐á✈❡❧X s❡❣✉❡ ❞✐str✐❜✉✐çã♦ ▼✉❧t✐♥♦♠✐❛❧✱ s❡♥❞♦ s✉❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡①♣r❡ss❛ ♣❡❧❛ ❢ór♠✉❧❛ ✶✳✾✳

P(X1 =x1, . . . , Xr=xr|N, p1, . . . , pr) =

N!

x1!x2!. . . xr!

px1

1 px

2

2 . . . pxrr ✭✶✳✾✮

s❡♥❞♦ q✉❡Pn

i=1Xi =N✳

❈♦♥s✐❞❡r❛♥❞♦ ♦ t❡r♠♦ N!

x1!x2!...xr! ❝♦♠♦ ♥♦r♠❛❧✐③❛❞♦r✱ t❡♠♦s ✶✳✶✵

P(X1 =x1, . . . , Xr =xr|N, p1, . . . , pr)∝p x1

1 px

2

2 . . . pxrr ✭✶✳✶✵✮

❚❡♠♦s q✉❡ ♣❛r❛ ✉♠ ✈❡t♦rp= (p1, p2, ..., pr)❞❡ ✈❛❧♦r❡s ❞❡s❝♦♥❤❡❝✐❞♦s ❝♦♠Pr_i=1pi =

✶✺

P(p|α) = Γ(α0)

Γ(α1)Γ(α2). . .Γ(αr)

pα11 −1pα22 −1. . . pαrr −1 ✭✶✳✶✶✮

❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ t❡r♠♦ Γ(α0)

Γ(α1)Γ(α2)...Γ(αr) ❝♦♠♦ ♥♦r♠❛❧✐③❛❞♦r✳

❆ss✐♠✱ t❡♠♦s ✶✳✶✷✳

P(p_|α)_∝pα1−1

1 pα

2−1

2 . . . pαrr −1 ✭✶✳✶✷✮

❆ss✉♠✐♥❞♦ ❝♦♠♦ P(p_|α) ♣r✐♦r✐ ❡ P(X1 = x1, . . . , Xr = xr|N, p1, . . . , pr) ❝♦♠♦ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✱ t❡♠♦s q✉❡ ❛ ♣♦st❡r✐♦r✐ P(p|X, α) é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ ✶✳✶✸ ❛

q✉❛❧ t❡♠ ❞✐str✐❜✉✐çã♦ ❉✐r✐❝❤❧❡t ❝♦♠ ♣❛râ♠❡tr♦sα = (α1+x1, ..., αr+xr)❡ E(pi) =

(αi+xi)/(α0+N)✳

P(p|X, α)∝p₁α1+x1−1p₂α2−1. . . pαr_r +xr−1 ✭✶✳✶✸✮

◆♦t❛♠♦s q✉❡✱ ♥❡st❡ ❝❛s♦✱ ❛ ♣♦st❡r✐♦r✐ ♣♦ss✉✐ ♣❡rt❡♥❝❡ à ♠❡s♠❛ ❢❛♠í❧✐❛ ❞❡ ❞✐s✲ tr✐❜✉✐çõ❡s q✉❡ ❛ ♣r✐♦r✐✳ ❆ss✐♠✱ ❞✐③❡♠♦s q✉❡ ❛ ❢❛♠í❧✐❛ ❉✐r✐❝❤❧❡t é ❝♦♥❥✉❣❛❞❛ ♣❛r❛ ❛♠♦str❛s ❝♦♠ ❞✐str✐❜✉✐çã♦ ▼✉❧t✐♥♦♠✐❛❧✳

✶✻

✶✳✸✳✷✳✻ ❉✐str✐❜✉✐çã♦ ◆♦r♠❛❧ ❡ ◆♦r♠❛❧ ▼✉❧t✐✈❛r✐❛❞❛

❆ ❞✐str✐❜✉✐çã♦ ◆♦r♠❛❧ é ✉♠❛ ❞❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❡ ✉t✐❧✐③❛❞❛s ❞✐str✐❜✉✐çõ❡s ❞❡ ♣r♦✲

❜❛❜✐❧✐❞❛❞❡ ✭❈❖❙❚❆ ◆❊❚❖ ❡ ❈❨▼❇❆▲■❙❚❆✱ ✷✵✵✻✮✳ ❈♦♥s✐❞❡r❛♥❞♦X ✉♠❛ ✈❛r✐á✈❡❧

❛❧❡❛tór✐❛ ❝♦♥tí♥✉❛✱ ❞✐③❡♠♦s q✉❡X _∼N(µ, σ2_{)s❡ s✉❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐✲}

❞❛❞❡ é ❡①♣r❡ss❛ ❝♦♠♦ ✶✳✶✹✱ s❡♥❞♦µ ♦ ♣❛râ♠❡tr♦ r❡❧❛t✐✈♦ à ♠é❞✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ σ2

♦ ♣❛râ♠❡tr♦ r❡❧❛t✐✈♦ à ✈❛r✐â♥❝✐❛ ♣♦♣✉❧❛❝✐♦♥❛❧✳

f(x) = √ 1

2πσ2 exp

(

−(x−µ)

2

2σ2

)

, _−∞< x < _∞ ✭✶✳✶✹✮

❊st❛ ❞✐str✐❜✉✐çã♦ t❡♠ s✐❞♦ ✉t✐❧✐③❛❞❛ ❡♠ ❞✐✈❡rs♦s ❝♦♥t❡①t♦s ❡♠ ❘❡❞❡s Pr♦❜❛❜✐✲ ❧íst✐❝❛s ❝♦♥tí♥✉❛s ✭●❊■●❊❘ ❡ ❍❊❈❑❊❘▼❆◆✱ ✶✾✾✹✮✭P➱❘❊❩ ❡t ❛❧✳✱ ✷✵✵✻✮✱ t❛♠❜é♠ sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❘❡❞❡ ●❛✉ss✐❛♥❛ ❈♦♥❞✐❝✐♦♥❛❧ ✭❘●❈✮✳ ❊st❛ ❛❜♦r❞❛❣❡♠ é ✉♠❛ ❛❧t❡r♥❛t✐✈❛ à ❝❛t❡❣♦r✐❛③❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ❝♦♥tí♥✉❛s✳ ❈♦♥t✉❞♦✱ ❛ s✉♣♦s✐çã♦ ❞❡ ♥♦r✲ ♠❛❧✐❞❛❞❡ ♣❛r❛ ✈❛r✐á✈❡✐s ❝♦♥tí♥✉❛s ♣♦❞❡ s❡r ❜❛st❛♥t❡ s❡✈❡r❛✱ ❡st❛ é ❢r❡qü❡♥t❡♠❡♥t❡ ❛❞♦t❛❞❛✱ ♣♦✐s ❣❛r❛♥t❡ ✉♠❛ ❛♣r♦①✐♠❛çã♦ r❛③♦á✈❡❧ ♣❛r❛ ❞✐✈❡rs❛s ❞✐str✐❜✉✐çõ❡s ♥❛t✉r❛✐s ✭❏❖❍◆ ❡ ▲❆◆●▲❊❨✱ ✶✾✾✺✮✳

◆❡st❡ s❡♥t✐❞♦✱ ❝♦♥s✐❞❡r❛♠♦s ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ❡①♣❧✐❝❛t✐✈❛sX = [X1, X2,✳ ✳ ✳, Xk] q✉❡✱ ❡♠ s✉♣♦s✐çã♦✱ ❞❡s❝r❡✈❡♠ ✉♠❛ ♣r♦❜❧❡♠át✐❝❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❡ s❡❣✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ◆♦r♠❛❧ ▼✉❧t✐✈❛r✐❛❞❛ ❞❡ ♦r❞❡♠ k✱ ✐st♦ é✱ X _∼ Nk(µ

∼ ,Σ)✱

s❡♥❞♦µ

∼♦ ✈❡t♦r ❞❡ ♠é❞✐❛s ♣♦♣✉❧❛❝✐♦♥❛✐s ❡ Σ❛ ♠❛tr✐③ ❞❡ ✈❛r✐â♥❝✐❛ ❡ ❝♦✈❛r✐â♥❝✐❛ ♣♦✲

♣✉❧❛❝✐♦♥❛❧✱Σ =

        σ2

1 σ12 · · · σ1k

σ2

2 ✳✳✳

✳✳✳ σ(k−1)k

σ2 k        

❝♦♠σ2

✶✼

Xi ❡σij ✐❣✉❛❧ ❛ ❝♦✈❛r✐â♥❝✐❛ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐sXi ❡Xj s❡♥❞♦1≤i < j ≤k✳ ❆ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ X é ❡①♣r❡ss❛ ♣♦r ✶✳✶✺✱ ♥♦t❡ q✉❡ s❡ k = 1 t❡♠♦s ♦

❝❛s♦ ❡①♣r❡ss♦ ❡♠ ✶✳✶✹✳

f(x) = 1 (2π)k2

|Σ_|12

exp

(

−1₂

x₋µ ∼

t

Σ−1

x₋µ ∼

)

✭✶✳✶✺✮

❆s ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s q✉❡ ❝♦♥s✐❞❡r❛♠ ❡st❡ t✐♣♦ ❞❡ ❡str✉t✉r❛ sã♦ ❛❜♦r❞❛❞❛s ♥♦ ❈❛♣ít✉❧♦ ✹✳

❈♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡✱ ♦s ❝ó❞✐❣♦s ❡♠ ❘ q✉❡ ❝♦♥s✐❞❡r❛♠ ❡st❡ t✐♣♦ ❞❡ ❡str✉t✉r❛ ♣❛r❛ ✉♠❛ r❡❞❡ ♣r♦❜❛❜✐❧íst✐❝❛ sã♦ ❞✐s♣♦♥✐❜✐❧✐③❛❞♦s ♥♦ ❆♣ê♥❞✐❝❡ ❊✳

✶✳✸✳✸ ❆s ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ♣♦❞❡♠ s❡r ❝❤❛♠❛❞❛s ❞❡ ❘❡❞❡s

❇❛②❡s✐❛♥❛s❄

❊①✐st❡ ✉♠❛ ❣r❛♥❞❡ ❞✐s❝✉ssã♦ ♥❛ ❧✐t❡r❛t✉r❛ s♦❜r❡ s❡ ❛s ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s sã♦ r❡❛❧♠❡♥t❡ ❇❛②❡s✐❛♥❛s ♦✉ ♥ã♦✳ ❆❧❡❣❛✲s❡ q✉❡ ❡ss❡ t❡r♠♦ s❡❥❛ ✉♠❛ ♥♦♠❡♥❝❧❛t✉r❛ ✐♥❛✲ ❞❡q✉❛❞❛✳ ❑♦r❜ ❡ ◆✐❝❤♦❧s♦♥ ✭✷✵✵✹✮ ❡✈✐❞❡♥❝✐❛♠ ❛ ♣r♦♥ú♥❝✐❛ ❢♦r♠❛❧ ❞♦ Pr♦❢❡ss♦r ●❡♦✛ ❲❡❜❜✱ ❡s♣❡❝✐❛❧✐st❛ ❡♠ ♠✐♥❡r❛çã♦ ❞❡ ❞❛❞♦s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❆✉str❛❧✐❛♥❛ ❞❡ ▼♦♥❛s❤✱ q✉❡ ❞❡❝❧❛r♦✉ ❞♦✐s ♣♦♥t♦s ❞❡ ✈✐st❛✿

✶✳ ❆ té❝♥✐❝❛ ❞❡ ❘❡❞❡s Pr♦❜❛❜✐❧íst✐❝❛s ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ✉♠ ♠ét♦❞♦ ❞❡ ❉❛t❛ ▼✐♥✐♥❣ q✉❡ ✉t✐❧✐③❛ ♠ét♦❞♦s ♥ã♦✲❇❛②❡s✐❛♥♦s✳