❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆ P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❙❚❆❚❮❙❚■❈❆✳
▼❖❉❊▲❖❙ ▼■❙❚❖❙ ❙❊▼■P❆❘❆▼➱❚❘■❈❖❙ P❆❘❈■❆▲▼❊◆❚❊ ◆➹❖ ▲■◆❊❆❘❊❙
❘♦❜s♦♥ ❏♦sé ▼❛r✐❛♥♦ ▼❛❝❤❛❞♦
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆ P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❙❚❆❚❮❙❚■❈❆
▼❖❉❊▲❖❙ ▼■❙❚❖❙ ❙❊▼■P❆❘❆▼➱❚❘■❈❖❙ P❆❘❈■❆▲▼❊◆❚❊ ◆➹❖ ▲■◆❊❆❘❊❙
❘♦❜s♦♥ ❏♦sé ▼❛r✐❛♥♦ ▼❛❝❤❛❞♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ❊st❛tíst✐❝❛ ❝♦♠♦ r❡q✉✐✲ s✐t♦ à ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❊s✲ t❛tíst✐❝❛✳
❖r✐❡♥t❛❞♦r❛✿ ❉r❛✳ ❈✐❜❡❧❡ ▼❛r✐❛ ❘✉ss♦ ❆❣ê♥❝✐❛ ✜♥❛♥❝✐❛❞♦r❛✿ ❈❛♣❡s
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar
M149mm
Machado, Robson José Mariano.
Modelos mistos semiparamétricos parcialmente não lineares / Robson José Mariano Machado. -- São Carlos : UFSCar, 2014.
52 f.
Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2014.
1. Análise de regressão. 2. Modelos semiparamétricos. 3. Modelos mistos não-lineares. 4. Diagnóstico de influência local. I. Título.
❘❡s✉♠♦
❉❛❞♦s ❝♦rr❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❡str✉t✉r❛ ♥ã♦ ❧✐♥❡❛r sã♦ ❝♦♠✉♥s ❡♠ ❜✐♦❡st❛tíst✐❝❛✱ ❡s✲ t✉❞♦s ❢❛r♠❛❝♦❝✐♥ét✐❝♦s ❡ ❧♦♥❣✐t✉❞✐♥❛✐s✳ ▼♦❞❡❧♦s ♠✐st♦s ♥ã♦ ❧✐♥❡❛r❡s sã♦ ❢❡rr❛♠❡♥t❛s út❡✐s ♣❛r❛ s❡ ❛♥❛❧✐s❛r ❡ss❡s t✐♣♦s ❞❡ ♣r♦❜❧❡♠❛s✳ ◆❡st❛ ❞✐ss❡rt❛çã♦✱ ♣r♦♣õ❡✲s❡ ✉♠❛ ❣❡✲ ♥❡r❛❧✐③❛çã♦ ❞❡ss❡s ♠♦❞❡❧♦s✱ ❝❤❛♠❛❞❛ ❞❡ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r ✭▼▼❙P◆▲✮✱ ❝♦♠ ✉♠❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ♣❛r❛ s❡ ♠♦❞❡❧❛r ❛ ♠é❞✐❛ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛✳ ❆ss✉♠❡✲s❡ q✉❡ ❛ ♠é❞✐❛ ❞❛ ✈❛r✐á✈❡❧ ❞❡ ✐♥t❡r❡ss❡ é ❡①♣❧✐❝❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❧✐♥❡❛r✱ q✉❡ ❞❡♣❡♥❞❡ ❞❡ ♣❛râ♠❡tr♦s ❞❡ ❡❢❡✐t♦s ✜①♦s ❡ ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✱ ❡ ♣♦r ✉♠❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛✳ ❚❛❧ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ♣♦ss✉✐ ❣r❛♥❞❡ ✢❡①✐❜✐✲ ❧✐❞❛❞❡✱ ♣❡r♠✐t✐♥❞♦ ✉♠❛ ♠❡❧❤♦r ❛❞❡q✉❛çã♦ à ❢♦r♠❛ ❢✉♥❝✐♦♥❛❧ q✉❡ s✉❜❥❛③ ❛♦s ❞❛❞♦s✳ ❖s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s sã♦ ✐♥❝❧✉í❞♦s ❧✐♥❡❛r♠❡♥t❡ ❛♦ ♠♦❞❡❧♦✱ ♦ q✉❡ s✐♠♣❧✐✜❝❛ ❛ ❡①♣r❡ssã♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❡ ♣❡r♠✐t❡ ❝♦♥s✐❞❡r❛r ❛ ❡str✉t✉r❛ ❞❡ ❝♦rr❡❧❛çã♦ ✐♥tr❛ ❣r✉♣♦✳ ➱ ❛ss✉♠✐❞♦ q✉❡ ♦s ❡rr♦s ❛❧❡❛tór✐♦s ❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ❝♦♥❥✉♥t❛♠❡♥t❡ s❡❣✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♠✉❧t✐✈❛r✐❛❞❛✳ ❊♠ r❡❧❛çã♦ ❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛✱ ✉t✐❧✐③❛✲s❡ ❛ té❝♥✐❝❛ ❞❡ s✉❛✈✐③❛çã♦ ❝♦♠ P✲s♣❧✐♥❡s✳ ❆ ♠❡t♦❞♦❧♦❣✐❛ ♣❛r❛ s❡ ♦❜t❡r❡♠ ❛s ❡st✐♠❛t✐✈❛s ❞♦s ♣❛râ♠❡tr♦s é ♦ ♠ét♦❞♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡♥❛❧✐③❛❞❛✳ ❖s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ✉s❛♥❞♦✲s❡ ♦ ♠ét♦❞♦ ❞❡ ❇❛②❡s ❡♠♣ír✐❝♦✳ ❆ q✉❛❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❡ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ♦❜s❡r✈❛çõ❡s ❛❜❡rr❛♥t❡s é ✈❡r✐✜❝❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ✐♥✢✉ê♥❝✐❛ ❧♦❝❛❧ ❡ ♣♦r ❛♥á❧✐s❡ ❞❡ r❡sí❞✉♦s✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❢❛r♠❛❝♦❝✐♥ét✐❝♦s✱ ❡♠ q✉❡ ♦ ❛♥t✐❛s♠át✐❝♦ t❤❡♦♣❤②❧❧✐♥❡ ❢♦✐ ❛❞♠✐♥✐str❛❞♦ ❛ ✶✷ s✉❥❡✐t♦s ❡ ❝♦♥❝❡♥tr❛çõ❡s sér✐❝❛s ❢♦r❛♠ t♦♠❛❞❛s ❡♠ ✶✶ ✐♥st❛♥t❡s ❞❡ t❡♠♣♦ ❞✉r❛♥t❡ ❛s ✷✺ ❤♦r❛s ✭❛♣ós s❡r ❛❞♠✐♥✐str❛❞♦✮✱ ❢♦✐ r❡❛♥❛❧✐s❛❞♦ ❝♦♠ ♦ ♠♦❞❡❧♦ ♣r♦♣♦st♦✱ ❡①❡♠♣❧✐✜❝❛♥❞♦ s❡✉ ✉s♦ ❡ ♣r♦♣r✐❡❞❛❞❡s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ♠♦❞❡❧♦s ♠✐st♦s ♥ã♦ ❧✐♥❡❛r❡s❀ ♠♦❞❡❧♦s s❡♠✐♣❛r❛♠étr✐❝♦s❀ s✉❛✈✐③❛çã♦❀ ✐♥✢✉ê♥❝✐❛ ❧♦❝❛❧❀P✲s♣❧✐♥❡s✳
❆❜str❛❝t
❈♦rr❡❧❛t❡❞ ❞❛t❛ s❡ts ✇✐t❤ ♥♦♥❧✐♥❡❛r str✉❝t✉r❡ ❛r❡ ❝♦♠♠♦♥ ✐♥ ♠❛♥② ❛r❡❛s s✉❝❤ ❛s ❜✐✲ ♦st❛t✐st✐❝s✱ ♣❤❛r♠❛❝♦❦✐♥❡t✐❝s ❛♥❞ ❧♦♥❣✐t✉❞✐♥❛❧ st✉❞✐❡s✳ ◆♦♥❧✐♥❡❛r ♠✐①❡❞✲❡✛❡❝ts ♠♦❞❡❧s ❛r❡ ✉s❡❢✉❧ t♦♦❧s t♦ ❛♥❛❧②s❡ t❤♦s❡ t②♣❡ ♦❢ ♣r♦❜❧❡♠s✳ ■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ t❤✐s ♠♦❞❡❧s ✐s ♣r♦♣♦s❡❞✱ ♥❛♠❡❧② ❜② s❡♠✐♣❛r❛♠❡tr✐❝ ♣❛rt✐❛❧❧② ♥♦♥❧✐♥❡❛r ♠✐①❡❞✲❡✛❡❝ts ♠♦❞❡❧ ✭▼▼❙P◆▲✮✱ ✇✐t❤ ❛ ♥♦♥♣❛r❛♠❡tr✐❝ ❢✉♥❝t✐♦♥ t♦ ♠♦❞❡❧ t❤❡ ♠❡❛♥ ♦❢ t❤❡ r❡s♣♦♥s❡ ✈❛r✐❛❜❧❡✳ ■t ❛ss✉♠❡s t❤❛t t❤❡ ♠❡❛♥ ♦❢ t❤❡ ✐♥t❡r❡st ✈❛r✐❛❜❧❡ ✐s ❡①♣❧❛✐♥❡❞ ❜② ❛ ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ ✜①❡❞ ❡✛❡❝ts ♣❛r❛♠❡t❡rs ❛♥❞ ❡①♣❧❛♥❛t♦r② ✈❛r✐❛❜❧❡s✱ ❛♥❞ ❜② ❛ ♥♦♥♣❛r❛♠❡tr✐❝ ❢✉♥❝t✐♦♥✳ ❙✉❝❤ ♥♦♥♣❛r❛♠❡t✐❝ ❢✉♥❝t✐♦♥ ✐s q✉✐t❡ ✢❡①✐❜❧❡✱ ❛❧❧♦✇✐♥❣ ❛ ❜❡t✲ t❡r ❛❞❡q✉❛❝② t♦ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❢♦r♠ t❤❛t ✉♥❞❡r❧✐❡s t❤❡ ❞❛t❛✳ ❚❤❡ r❛♥❞♦♠ ❡✛❡❝ts ❛r❡ ✐♥❝❧✉❞❡❞ ❧✐♥❡❛r❧② t♦ t❤❡ ♠♦❞❡❧✱ ✇❤✐❝❤ s✐♠♣❧✐❢② t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ r❡s♣♦♥s❡ ✈❛r✐❛❜❧❡ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ❡♥❛❜❧❡s t❤❡ ♠♦❞❡❧ t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ✇✐t❤✐♥✲❣r♦✉♣ ❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡✳ ■t ✐s ❛ss✉♠❡❞ t❤❛t t❤❡ r❛♥❞♦♠ ❡rr♦rs ❛♥❞ t❤❡ r❛♥❞♦♠ ❡✛❡❝ts ❥♦✐♥t❧② ❢♦❧❧♦✇ ❛ ♠✉❧t✐✈❛r✐❛t❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✳ ❘❡❧❛t❡ t♦ t❤❡ ♥♦♥♣❛r❛♠❡tr✐❝ ❢✉♥❝t✐♦♥✱ ✐t ✐s ❞❡❛❧ ✇✐t❤ t❤❡ P✲s♣❧✐♥❡s s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡✳ ❚❤❡ ♠❡t❤♦❞♦❧♦❣② t♦ ♦❜t❛✐♥ t❤❡ ♣❛r❛♠❡t❡rs ❡st✐♠❛✲ t❡s ✐s ♣❡♥❛❧✐③❡❞ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞✳ ❚❤❡ r❛♥❞♦♠ ❡✛❡❝ts ♠❛② ❜❡ ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ t❤❡ ❊♠♣✐r✐❝❛❧ ❇❛②❡s ♠❡t❤♦❞✳ ❚❤❡ ❣♦♦❞♥❡ss ♦❢ t❤❡ ♠♦❞❡❧ ❛♥❞ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ♣♦t❡♥❝✐❛❧ ✐♥✢✉❡♥t ♦❜s❡r✈❛t✐♦♥ ✐s ✈❡r✐✜❡❞ ✇✐t❤ t❤❡ ❧♦❝❛❧ ✐♥✢✉❡♥❝❡ ♠❡t❤♦❞ ❛♥❞ ❛ r❡s✐❞✉❛❧ ❛♥❛❧②s✐s✳ ❚❤❡ ♣❤❛r♠❛❝♦❦✐♥❡t✐❝ ❞❛t❛ s❡t✱ ✐♥ ✇❤✐❝❤ t❤❡ ❛♥t✐✲❛st❤♠❛t✐❝ ❞r✉❣ t❤❡♦♣❤②❧❧✐♥❡ ✇❛s ❛❞♠✐♥✐st❡r❡❞ t♦ ✶✷ s✉❜❥❡❝ts ❛♥❞ s❡r✉♠ ❝♦♥❝❡♥tr❛t✐♦♥s ✇❡r❡ t❛❦❡♥ ❛t ✶✶ t✐♠❡ ♣♦✐♥ts ♦✈❡r t❤❡ ✷✺ ❤♦✉rs ✭❛❢t❡r ❜❡✐♥❣ ❛❞♠✐♥✐st❡r❡❞✮✱ ✇❛s r❡✲❛♥❛❧②s❡❞ ✇✐t❤ t❤❡ ♣r♦♣♦s❡❞ ♠♦❞❡❧✱ ❡①❡♠♣❧✐❢②✐♥❣ ✐ts ✉s❡s ❛♥❞ ♣r♦♣❡rt✐❡s✳
❑❡② ✇♦r❞s✿ ♥♦♥❧✐♥❡❛r ♠✐①❡❞✲❡✛❡❝ts ♠♦❞❡❧s❀ s❡♠✐♣❛r❛♠❡tr✐❝ ♠♦❞❡❧s❀ s♠♦♦t❤✐♥❣❀ ❧♦✲ ❝❛❧ ✐♥✢✉❡♥❝❡❀ P✲s♣❧✐♥❡s✳
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✷
✷ ▼♦❞❡❧♦s ♥ã♦ ❧✐♥❡❛r❡s ❝♦♠ ❡❢❡✐t♦s ♠✐st♦s ✹
✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❊s♣❡❝✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✸ ❊st✐♠❛çã♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✹ Pr❡❞✐çã♦ ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✸ ▼♦❞❡❧♦ ♠✐st♦s s❡♠✐♣❛r❛♠étr✐❝♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r ✶✵ ✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✷ ❊s♣❡❝✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✸✳✸ ❘❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✹ Pr❡ss✉♣♦st♦s ❞♦ ♠♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✺ ❋✉♥çã♦ ❞❡ ❧♦❣✲✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡♥❛❧✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✻ Pr♦❝❡ss♦ ❞❡ ❡st✐♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✻✳✶ ❖ ♣❛râ♠❡tr♦ ❞❡ s✉❛✈✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✻✳✷ Pr❡❞✐çã♦ ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ❡ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✻✳✸ ❊rr♦s ♣❛❞rã♦ ❞❛s ❡st✐♠❛t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✹ ■♥✢✉ê♥❝✐❛ ❧♦❝❛❧ ✷✵
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✷ ▼ét♦❞♦ ❞❡ ✐♥✢✉ê♥❝✐❛ ❧♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✸ ❉❡r✐✈❛çã♦ ❞❛ ❝✉r✈❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✸✳✶ ▼❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✹ ▼❛tr✐③ ❞❡ ♣❡rt✉r❜❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✹✳✶ P♦♥❞❡r❛çã♦ ❞❡ ❝❛s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✹✳✷ P❡rt✉r❜❛çã♦ ♥❛ ♠❛tr✐③ ❡s❝❛❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✹✳✸ P❡rt✉r❜❛çã♦ ♥❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✺ ❆♣❧✐❝❛çã♦ ✷✾
✺✳✶ ❉❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ t❤❡♦♣❤②❧❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✺✳✷ ▼♦❞❡❧♦ ♣r♦♣♦st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✺✳✸ ❆♥á❧✐s❡ ❞❡ r❡sí❞✉♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✹ ❉✐❛❣♥óst✐❝♦ ❞❡ ■♥✢✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✻ ❉✐s❝✉ssã♦ ❡ ♣r♦♣♦st❛s ❢✉t✉r❛s ✹✶
❆ ❉❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ t❤❡♦♣❤②❧❧✐♥❡ ✹✻
❇ B✲s♣❧✐♥❡s ❡ P✲s♣❧✐♥❡s ✺✵
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✺✳✶ ❉✐❛❣r❛♠❛ ❞❡ ❞✐s♣❡rsã♦ ❞♦s ❞❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ t❤❡♦♣❤②❧❧✐♥❡✳ ✳ ✳ ✳ ✳ ✷✾ ✺✳✷ ❇♦①♣❧♦t ❞♦s ❞❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ t❤❡♦♣❤②❧❧✐♥❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✳✸ ▼♦❞❡❧♦s ♣❛r❛♠étr✐❝♦ ❡ s❡♠✐♣❛r❛♠étr✐❝♦ ❛❥✉st❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✺✳✹ ●rá✜❝♦ ❞❡ í♥❞✐❝❡ ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ♣r❡❞✐t♦s ♣❛r❛ ♦ ♠♦❞❡❧♦ s❡♠✐♣❛r❛✲
♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✺ ❉✐stâ♥❝✐❛ ❞❡ ▼❛❤❛❧❛♥♦❜✐s ♣❛r❛ ♦s ♠♦❞❡❧♦s ♣❛r❛♠étr✐❝♦ ❡ s❡♠✐♣❛r❛♠étr✐❝♦ ✸✹ ✺✳✻ ❘❡sí❞✉♦s ❝♦♥❞✐❝✐♦♥❛✐s ♣❛❞r♦♥✐③❛❞♦s ♣❛r❛ ♦s ♠♦❞❡❧♦s ♣❛r❛♠étr✐❝♦ ❡ s❡♠✐✲
♣❛r❛♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✼ ●rá✜❝♦s ❞❡ q✉❛♥t✐s ❞♦s r❡sí❞✉♦s ❝♦♥❞✐❝✐♦♥❛✐s ♣❛❞r♦♥✐③❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✺✳✽ ❆✉t♦✈❛❧♦r❡s r❡❧❛t✐✈♦s às ❞✐r❡çõ❡s ❞❡ ✐♥✢✉ê♥❝✐❛ s♦❜ ♦ ❡sq✉❡♠❛ ❞❡ ♣♦♥❞❡✲
r❛çã♦ ❞❡ ❝❛s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺✳✾ ●rá✜❝♦s ❞❡ í♥❞✐❝❡s ❞❡ m(0) ♣❛r❛ bθ s♦❜ ♦ ❡sq✉❡♠❛ ❞❡ ♣♦♥❞❡r❛çã♦ ❞❡ ❝❛s♦s✳ ✸✻
✺✳✶✵ ●rá✜❝♦s ❞❡ í♥❞✐❝❡s ❞❡m(2) ♣❛r❛θbs♦❜ ♦ ❡sq✉❡♠❛ ❞❡ ♣♦♥❞❡r❛çã♦ ❞❡ ❝❛s♦s✳ ✸✼
✺✳✶✶ ❆✉t♦✈❛❧♦r❡s r❡❧❛t✐✈♦s às ❞✐r❡çõ❡s ❞❡ ✐♥✢✉ê♥❝✐❛ s♦❜ ♦ ❡sq✉❡♠❛ ❞❡ ♣❡rt✉r✲ ❜❛çã♦ ♥❛ ♠❛tr✐③ ❡s❝❛❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✺✳✶✷ ●rá✜❝♦s ❞❡ í♥❞✐❝❡s ❞❡m(0) ♣❛r❛bθ s♦❜ ♦ ❡sq✉❡♠❛ ♣❡rt✉r❜❛çã♦ ♥❛ ♠❛tr✐③
❡s❝❛❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✶✸ ●rá✜❝♦s ❞❡ í♥❞✐❝❡s ❞❡ m(2) ♣❛r❛ bθ s♦❜ ♦ ❡sq✉❡♠❛ ❞❡ ♣❡rt✉r❜❛çã♦ ♥❛
♠❛tr✐③ ❡s❝❛❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✶✹ ❆✉t♦✈❛❧♦r❡s r❡❧❛t✐✈♦s às ❞✐r❡çõ❡s ❞❡ ✐♥✢✉ê♥❝✐❛ s♦❜ ♦ ❡sq✉❡♠❛ ❞❡ ♣❡rt✉r✲
❜❛çã♦ ♥❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✺✳✶✺ ●rá✜❝♦s ❞❡ í♥❞✐❝❡s ❞❡ m(0) ♣❛r❛ bθ s♦❜ ♦ ❡sq✉❡♠❛ ❞❡ ♣❡rt✉r❜❛çã♦ ♥❛
✈❛r✐á✈❡❧ r❡s♣♦st❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✺✳✶✻ ●rá✜❝♦s ❞❡ í♥❞✐❝❡s ❞❡ m(6) ♣❛r❛ bθ s♦❜ ♦ ❡sq✉❡♠❛ ❞❡ ♣❡rt✉r❜❛çã♦ ♥❛
✈❛r✐á✈❡❧ r❡s♣♦st❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ❇✳✶ ❋✉♥çõ❡s ❜❛s❡ ❞❡ B✲s♣❧✐♥❡s ❞❡ t❡r❝❡✐r♦ ❣r❛✉ ❡ ✸ ♥ós ✐♥t❡r✐♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✺✳✶ ❊st✐♠❛❞♦r❡s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❡rr♦s ♣❛❞rã♦ ♣❛r❛ ♦s ♠♦❞❡❧♦s ♣❛r❛♠étr✐❝♦ ❡ s❡♠✐♣❛r❛♠étr✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ❆✳✶ ❉❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ t❤❡♦♣❤②❧❧✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
▼♦❞❡❧♦s ♥ã♦ ❧✐♥❡❛r❡s sã♦ ❢❡rr❛♠❡♥t❛s út❡✐s ♣❛r❛ s❡ ❛♥❛❧✐s❛r❡♠ ❞❛❞♦s q✉❡ ❛♣r❡s❡♥✲ t❛♠ r❡❧❛çã♦ ♥ã♦ ❧✐♥❡❛r ❡♥tr❡ ❛s ❝♦✈❛r✐á✈❡✐s ❡ ❛s ✈❛r✐á✈❡✐s r❡s♣♦st❛s✱ ❛♦ ❝♦♥s✐❞❡r❛r❡♠ ❡①♣r❡ssõ❡s ♠❛t❡♠át✐❝❛s ♣❛r❛ ❡①♣❧✐❝❛r ♦ ♠❡❝❛♥✐s♠♦ ❣❡r❛❞♦r ❞♦s ❞❛❞♦s✳ ❈♦♠♦ ✈❛♥t❛❣❡♠ ❞❡ss❡s ♠♦❞❡❧♦s t❡♠✲s❡ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❡ ❜♦♥s r❡s✉❧t❛❞♦s ♥❛ ♣r❡❞✐çã♦ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❢♦r❛ ❞♦ ❞♦♠í♥✐♦ ❞♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s ✭P✐♥❤❡✐r♦ ✫ ❇❛t❡s ✷✵✵✵✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❤á q✉❡ s❡ ❝♦♥s✐❞❡r❛r t❛♠❜é♠ q✉❡ ❡ss❡s ♠♦❞❡❧♦s ❛♣r❡s❡♥t❛♠ ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛çã♦✱ ✉♠❛ ✈❡③ q✉❡ ❡♠ ❣❡r❛❧ ❛s ❡st✐♠❛t✐✲ ✈❛s ❞♦s ♣❛râ♠❡tr♦s ♥ã♦ ♣♦ss✉❡♠ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ❡ t♦r♥❛♠✲s❡ ♥❡❝❡ssár✐❛s ❛♣r♦①✐♠❛çõ❡s ♥✉♠ér✐❝❛s✳
❉❛❞♦s ❝♦♠ ♠❡❞✐❞❛s r❡♣❡t✐❞❛s✱ ❝❛r❛❝t❡r✐③❛❞♦s ♣❡❧❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ♦❜s❡r✈❛çõ❡s ✐♥✲ tr❛✉♥✐❞❛❞❡ ❡①♣❡r✐♠❡♥t❛❧✱ t❛♠❜é♠ ♣♦❞❡♠ ❛♣r❡s❡♥t❛r r❡❧❛çã♦ ♥ã♦ ❧✐♥❡❛r ❡♥tr❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❡ ♦s ♣❛râ♠❡tr♦s✳ P❛r❛ ♠♦❞❡❧❛r ❛ ❝♦rr❡❧❛çã♦ ✐♥tr❛✉♥✐❞❛❞❡ ❡①♣❡r✐♠❡♥t❛❧ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ é ❝♦♥s✐❞❡r❛r ♠♦❞❡❧♦s ❝♦♠ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✳ ❉❛✈✐❞✐❛♥ ✫ ●✐❧t✐♥❛♥ ✭✶✾✾✺✮ ❛♣r❡s❡♥t❛♠ ♦ ♠♦❞❡❧♦ ♥ã♦ ❧✐♥❡❛r ❝♦♠ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✱ q✉❡ sã♦ út❡✐s ♣❛r❛ s❡ ❡st✉❞❛r❡♠ t❛✐s ❞❛❞♦s✳ ◆❡ss❡ ❝❛s♦✱ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ é ❡①♣❧✐❝❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❧✐♥❡❛r ♥♦s ♣❛râ♠❡tr♦s ❡ ♥♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✳ ❆ ❞✐✜❝✉❧❞❛❞❡ ❞❡ss❡s ♠♦❞❡❧♦s é ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✐♥t❡❣r❛çõ❡s ♥✉♠ér✐❝❛s ♣❛r❛ s❡ ♦❜t❡r ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ✭♠♦❞❡❧♦ ♠❛r✲ ❣✐♥❛❧✮✳ ❯♠❛ ❛❧t❡r♥❛t✐✈❛ é ❝♦♥s✐❞❡r❛r ♦ ♠♦❞❡❧♦ ♠✐st♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r ✭❱♦♥❡s❤ ✫ ❈❛rt❡r ✶✾✾✷✮ ❡♠ q✉❡ ♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s sã♦ ✐♥❝❧✉í❞♦s ❧✐♥❡❛r♠❡♥t❡✳ ◆❡st❡ tr❛❜❛❧❤♦ ❛ss✉♠❡✲s❡ q✉❡ ♦s ❡rr♦s ❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s s❡❣✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ ❆ ✈❛♥t❛❣❡♠ ❞❡ss❛ ❢♦r♠✉❧❛çã♦ s✉r❣❡ ❛♦ ♣❛ss♦ ❡♠ q✉❡ ♦ ♠♦❞❡❧♦ ♠❛r❣✐♥❛❧ é ♦❜t✐❞♦ ❞❡ ❢♦r♠❛ ❞✐r❡t❛ ♣♦r ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ P♦ré♠✱ ♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ♥ã♦ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ♣❛râ♠❡tr♦s✳ ❘✉ss♦ ✭✷✵✶✵✮ ❡st❡♥❞❡ ❡ss❡ ♠♦❞❡❧♦ ❝♦♥s✐❞❡r❛♥❞♦ ❞✐str✐❜✉✐çõ❡s ❡❧í♣t✐❝❛s✱ q✉❡ ❢♦r♥❡❝❡♠ ❡st✐♠❛t✐✈❛s r♦❜✉st❛s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❡ ♣♦❞❡♠ ❛❝♦♠♦❞❛r ♦❜s❡r✈❛çõ❡s ❛❜❡rr❛♥t❡s✳
❑❡ ✫ ❲❛♥❣ ✭✷✵✵✶✮ ❛♣r❡s❡♥t❛♠ ✉♠ ♠♦❞❡❧♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♠✐st♦ ♥ã♦ ❧✐♥❡❛r ❡♠ q✉❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ é ❡①♣❧✐❝❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ q✉❡ ♣♦❞❡ s❡r ♥ã♦ ❧✐♥❡❛r ❝♦♠ r❡❧❛çã♦ ❛ ♣❛râ♠❡tr♦s✱ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✳ ❆ss✐♠✱ t♦r♥❛✲s❡ ♥❡❝❡ssár✐♦
✉t✐❧✐③❛r ❛♣r♦①✐♠❛çõ❡s ♥✉♠ér✐❝❛s ♣❛r❛ s❡ ♦❜t❡r ♦ ❧♦❣❛r✐t♠♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠❛r❣✐♥❛❧✳ ❆ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ss❡ ♠♦❞❡❧♦ ❞✐✜❝✉❧t❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛çã♦ ❡ ✐♥❢❡rê♥❝✐❛✱ t❛♥t♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ❡❢❡✐t♦s ✜①♦s q✉❛♥t♦ ❞❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛✳ ❊❧♠✐ ❡t ❛❧✳ ✭✷✵✶✶✮ ❞❡s❡♥✈♦❧✈❡r❛♠ ✉♠ ♠ét♦❞♦ ❛❧t❡r♥❛t✐✈♦ ♣❛r❛ ❛❥✉st❛r ♦ ♠♦❞❡❧♦ ❛♣r❡✲ s❡♥t❛❞♦ ♣♦r ❑❡ ✫ ❲❛♥❣ ✭✷✵✵✶✮✱ ♠♦❞❡❧❛♥❞♦ ❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ♣♦r ♠❡✐♦ ❞❡
B✲s♣❧✐♥❡s✳ ❊ss❡ ❢❛t♦ ❢♦r♥❡❝❡ ✈❛♥t❛❣❡♥s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♥❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s✳
■❜❛❝❛❝❤❡✲P✉❧❣❛r ❡t ❛❧✳ ✭✷✵✶✷✮ ❞❡s❡♥✈♦❧✈❡r❛♠ ♦s ♠♦❞❡❧♦s ♠✐st♦s ❧✐♥❡❛r❡s s❡♠✐♣❛r❛♠étr✐✲ ❝♦s ❛❞✐t✐✈♦s ❝♦♥s✐❞❡r❛♥❞♦ ❞✐str✐❜✉✐çõ❡s ❡❧í♣t✐❝❛s ♣❛r❛ ❛s ❝♦♠♣♦♥❡♥t❡s ❛❧❡❛tór✐❛s✳
❉❡st❛ ❢♦r♠❛✱ ♣r♦♣õ❡✲s❡ ♦ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r q✉❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ❣❡♥❡r❛❧✐③❛çã♦ ❞✐r❡t❛ ❞♦ ♠♦❞❡❧♦ ♣r♦♣♦st♦ ♣♦r ❱♦♥❡s❤ ✫ ❈❛r✲ t❡r ✭✶✾✾✷✮✱ ❛♦ ✐♥❝❧✉✐r ✉♠❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ♣❛r❛ ♠♦❞❡❧❛r ❛ ♠é❞✐❛ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛✳ ❖s ♣❛râ♠❡tr♦s ❢♦r♥❡❝❡♠ ✉♠ r❡s✉♠♦ ✐♥t❡r♣r❡tá✈❡❧ ❞♦s ❞❛❞♦s ❡ ❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ❢♦r♥❡❝❡ ✢❡①✐❜✐❧✐❞❛❞❡ ❝♦♠ r❡❧❛çã♦ à ❢♦r♠❛ ❢✉♥❝✐♦♥❛❧ q✉❡ s✉❜❥❛③ ❛♦s ❞❛❞♦s✳
❆❧é♠ ❞✐ss♦✱ ❞❡s❡♥✈♦❧✈❡✲s❡ ♦ ♠ét♦❞♦ ❞❡ ✐♥✢✉ê♥❝✐❛ ❧♦❝❛❧✱ ♣r♦♣♦st♦ ♣♦r ❈♦♦❦ ✭✶✾✽✻✮✱ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡rá ❝♦♥s✐❞❡r❛❞❛ ❛ ❝✉r✈❛t✉r❛ ♥♦r♠❛❧ ❝♦♥❢♦r♠❛❧ ✭P♦♦♥ ✫ P♦♦♥ ✶✾✾✾✮✱ ❛♦ ✐♥✈és ❞❛ ❝✉r✈❛t✉r❛ ♥♦r♠❛❧✱ q✉❡ é ✐♥✈❛r✐❛♥t❡ s♦❜ r❡♣❛r❛♠❡tr✐③❛çõ❡s ❝♦♥❢♦r♠❛✐s ❡ é ✉♠❛ ♠❡❞✐❞❛ ♣❛❞r♦♥✐✲ ③❛❞❛✳ ❊ss❛ té❝♥✐❝❛ ❞❡ ❞✐❛❣♥óst✐❝♦ ♣r♦♣õ❡ ✉♠ ❡♥❢♦q✉❡ ❣❡♦♠étr✐❝♦ ♣❛r❛ s❡ ✈❡r✐✜❝❛r❡♠ ❛s ✈❛r✐❛çõ❡s q✉❡ ♣❡q✉❡♥❛s ♣❡rt✉r❜❛çõ❡s ♥♦s ❞❛❞♦s ♦✉ ♥♦s ♣r❡ss✉♣♦st♦s ❞♦ ♠♦❞❡❧♦ ♣♦❞❡♠ ❝❛✉s❛r ♥❛s ❡st✐♠❛t✐✈❛s ❞♦s ♣❛râ♠❡tr♦s✳ ❆❧❣✉♥s tr❛❜❛❧❤♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❡ss❡ ♠ét♦❞♦ ♥❛ ❧✐t❡r❛t✉r❛ sã♦ ❝✐t❛❞♦s ❛ s❡❣✉✐r✳ ❘✉ss♦ ❡t ❛❧✳ ✭✷✵✵✾✮ r❡❛❧✐③❛r❛♠ ❡st✉❞♦ ❞❡ ✐♥✢✉ê♥❝✐❛ ❧♦❝❛❧ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✐st♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r ❝♦♠ ❞✐str✐❜✉✐çõ❡s ❡❧í♣t✐❝❛s✳ ❖sór✐♦ ❡t ❛❧✳ ✭✷✵✵✼✮ ❞❡s❡♥✈♦❧✈❡r❛♠ ♦ ♠ét♦❞♦ ❞❡ ✐♥✢✉ê♥❝✐❛ ❧♦❝❛❧ ♣❛r❛ ♠♦❞❡❧♦s ♠✐st♦s ❧✐♥❡❛r❡s ❛♣❧✐❝❛❞♦s ❛ ❞❛❞♦s ❝♦♠ ❡str✉t✉r❛ ❧♦♥❣✐t✉❞✐♥❛❧✳ ■❜❛❝❛❝❤❡✲P✉❧❣❛r ❡t ❛❧✳ ✭✷✵✶✷✮ r❡❛❧✐③❛r❛♠ ❡st✉❞♦s ❞❡ ✐♥✢✉ê♥❝✐❛ ❧♦❝❛❧ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ❛❞✐t✐✈♦ ❝♦♥s✐❞❡r❛♥❞♦ ❞✐str✐❜✉✐çõ❡s ❡❧í♣t✐❝❛s ♣❛r❛ ❛s ❝♦♠♣♦♥❡♥t❡s ❛❧❡❛tór✐❛s✳
❆ ♦r❣❛♥✐③❛çã♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ ❡stá ❞❡s❝r✐t❛ ❛ s❡❣✉✐r✳ ❆♣ós ❡st❛ ✐♥tr♦❞✉çã♦✱ ♥♦ ❈❛♣ít✉❧♦ ✷ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ✈✐sã♦ ❣❡r❛❧ ❞❛ t❡♦r✐❛ ❞❡ ♠♦❞❡❧♦s ♥ã♦ ❧✐♥❡❛r❡s ♠✐st♦s ♣❛r❛♠étr✐❝♦s✱ ❞❡st❛❝❛♥❞♦✲s❡ ❛s ✈❛♥t❛❣❡♥s ❡ ❞❡s✈❛♥t❛❣❡♥s✳ ◆♦ ❈❛♣ít✉❧♦ ✸ é ❛♣r❡s❡♥t❛❞♦ ♦ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r ❡ sã♦ r❡❛❧✐③❛❞❛s ❞✐s❝✉ssõ❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ♦s ♠ét♦❞♦s ❞❡ ❡st✐♠❛çã♦ ❡ ✐♥❢❡rê♥❝✐❛s✳ ◆♦ ❈❛♣ít✉❧♦ ✹ é ❞❡s❡♥✈♦❧✈✐❞♦ ♦ ♠ét♦❞♦ ❞❡ ✐♥✢✉ê♥❝✐❛ ❧♦❝❛❧ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♣r♦♣♦st♦✳ ◆♦ ❈❛♣ít✉❧♦ ✺ é ❛♣r❡s❡♥t❛❞❛ ❛ ❛♥á❧✐s❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❢❛r♠❛❝♦❝✐♥ét✐❝♦s t❤❡♦♣❤②❧❧✐♥❡✱ ❡♠ q✉❡ ♦ ❛♥t✐❛s♠át✐❝♦ ❢♦✐ ❛❞♠✐♥✐str❛❞♦ ❛ ✶✷ s✉❥❡✐t♦s ❡ ✶✶ ♠❡❞✐❞❛s ❞❡ss❛ s✉❜stâ♥❝✐❛ ❢♦r❛♠ t♦♠❛❞❛s ♥❛s ✷✺ ❤♦r❛s s❡❣✉✐♥t❡s✳ ◆♦ ❈❛♣ít✉❧♦ ✻ t❡♠✲s❡ ❛ ❞✐s❝✉ssã♦ ❞♦ ❡st✉❞♦ r❡❛❧✐③❛❞♦ ❡ ✐♥❞✐❝❛çã♦ ❞❡ ♣r♦♣♦st❛s ❢✉t✉r❛s✳
❈❛♣ít✉❧♦ ✷
▼♦❞❡❧♦s ♥ã♦ ❧✐♥❡❛r❡s ❝♦♠ ❡❢❡✐t♦s
♠✐st♦s
◆❡st❡ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ✈✐sã♦ ❣❡r❛❧ ❞❛ t❡♦r✐❛ ❞❡ ♠♦❞❡❧♦s ♥ã♦ ❧✐♥❡❛r❡s ❝♦♠ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✳ ◆❛ ❙❡çã♦ ✷✳✶ sã♦ ❞✐s❝✉t✐❞❛s ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s ❡ ❛❧❣✉♥s tr❛❜❛❧❤♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❡ss❛ ❝❧❛ss❡ ❞❡ ♠♦❞❡❧♦s✳ ◆❛ ❙❡çã♦ ✷✳✷ é ❞❡✜♥✐❞♦ ♦ ♠♦❞❡❧♦ ♥ã♦ ❧✐♥❡❛r ❝♦♠ ❡❢❡✐t♦s ♠✐st♦s✱ ❞❡ ❞✉❛s ❢♦r♠❛s ❞✐st✐♥t❛s✱ r❡ss❛❧t❛♥❞♦✲s❡ ❛s ✈❛♥t❛❣❡♥s ❡ ❞❡s✈❛♥t❛❣❡♥s ❞❡ s❡ tr❛❜❛❧❤❛r ❝♦♠ ❝❛❞❛ ♠♦❞❡❧♦✳ ◆❛ ❙❡çã♦ ✷✳✸ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛s ❡q✉❛çõ❡s ♦❜t✐❞❛s ❞❡ ❡st✐♠❛çã♦ ❜❛s❡❛❞❛s ♥♦ ❧♦❣❛r✐t♠♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡ é ❛♣♦♥t❛❞♦ ♦ ❛❧❣♦r✐t♠♦ ❡s❝♦r❡ ❞❡ ❋✐s❤❡r ♣❛r❛ s❡ ♦❜t❡r❡♠ ❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦s ♣❛râ♠❡✲ tr♦s ❛ss♦❝✐❛❞♦s ❛♦ ♠♦❞❡❧♦ ❛♣r❡s❡♥t❛❞♦✳ ◆❛ ❙❡çã♦ ✷✳✹ é ❛♣r❡s❡♥t❛❞♦ ♦ ♠ét♦❞♦ ❞❡ ❇❛②❡s ❡♠♣ír✐❝♦ ♣❛r❛ ♣r❡❞✐çã♦ ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✳
✷✳✶ ■♥tr♦❞✉çã♦
❉❛❞♦s ❝♦♠ ♠❡❞✐❞❛s r❡♣❡t✐❞❛s✱ ❝♦♠✉♥s ❡♠ ❜✐♦❡st❛tíst✐❝❛✱ t❛✐s ❝♦♠♦ ❞❛❞♦s ❢❛r♠❛✲ ❝♦❝✐♥ét✐❝♦s ❡ ❞❛❞♦s ❞❡ ❝r❡s❝✐♠❡♥t♦✱ sã♦✱ ❡♠ ❣❡r❛❧ ❝❛r❛❝t❡r✐③❛❞♦s ♣♦r ✉♠❛ r❡❧❛çã♦ ♥ã♦ ❧✐♥❡❛r ❡♥tr❡ ❛s ❝♦✈❛r✐á✈❡✐s ❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛✳ P♦❞❡♠✲s❡ ❝♦♥s✐❞❡r❛r ♠♦❞❡❧♦s ❞❡ r❡❣r❡s✲ sã♦ ♣♦❧✐♥♦♠✐❛❧ q✉❡ ❛♣r❡s❡♥t❛♠ ❜♦♥s ❛❥✉st❡s ❛♦s ❞❛❞♦s✳ P♦ré♠✱ ♠✉✐t❛s ✈❡③❡s ❡st❡s ♥ã♦ ♣❡r♠✐t❡♠ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❡ ♣♦❞❡♠ ❛♣r❡s❡♥t❛r ♣r♦❜❧❡♠❛s ♣❛r❛ ♣r❡❞✐çã♦ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ♣❛r❛ ✈❛❧♦r❡s ❞♦ ❞♦♠í♥✐♦ ❞♦s ❞❛❞♦s ❛❧é♠ ❞♦ ✐♥t❡r✈❛❧♦ ✭♦✉ r❡❣✐ã♦✮ ❛ q✉❡ ♣❡rt❡♥❝❡♠ ♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s✳ ❯♠❛ ❛❧t❡r♥❛t✐✈❛ sã♦ ♦s ♠♦❞❡❧♦s ♥ã♦ ❧✐♥❡❛r❡s q✉❡ ✈✐s❛♠ ❞❡s❝r❡✈❡r ❛ ♠❡❝â♥✐❝❛ ❣❡r❛❞♦r❛ ❞❛ r❡s♣♦st❛ ❡✱ ♣♦rt❛♥t♦✱ ♣♦ss✉❡♠ ♣❛râ♠❡tr♦s ✐♥t❡r♣r❡tá✈❡✐s✳ ❆❧é♠ ❞✐ss♦✱ ❡ss❡s ♠♦❞❡❧♦s ♣♦ss✉❡♠ ❜♦❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♣r❡❞✐çã♦ ♣❛r❛ ✈❛❧♦r❡s ❞♦ ❞♦♠í♥✐♦ ❞♦s ❞❛❞♦s ❛❧é♠ ❞♦s ♦❜s❡r✈❛❞♦s✱ ❥á q✉❡ ❧❡✈❛♠ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ♠❡❝❛♥✐s♠♦ ❣❡r❛❞♦r ❞♦s ♠❡s♠♦s✳ ❉❛❞♦s ❝♦♠ ♠❡❞✐❞❛s r❡♣❡t✐❞❛s r❡q✉❡r❡♠ ♠♦❞❡❧♦s q✉❡ ❧❡✈❡♠ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ❡str✉t✉r❛ ❞❡ ❝♦rr❡❧❛çã♦ ❡①✐st❡♥t❡ ♥❛s ♦❜s❡r✈❛çõ❡s ✐♥tr❛✉♥✐❞❛❞❡ ❡①♣❡r✐♠❡♥t❛❧✳ ▼♦❞❡❧♦s ❝♦♠ ❡❢❡✐t♦s ♠✐st♦s ❞❡s❡♠♣❡♥❤❛♠ ❜❡♠ ❡ss❛ ❢✉♥çã♦ ❛♦ ❛❞✐❝✐♦♥❛✲
r❡♠ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ❛♦ ♠♦❞❡❧♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♠♦❞❡❧♦s ♥ã♦ ❧✐♥❡❛r❡s ❝♦♠ ❡❢❡✐t♦s ♠✐st♦s sã♦ út❡✐s ♣❛r❛ s❡ ❛♥❛❧✐s❛r❡♠ ❡ss❡s ❞❛❞♦s✱ ✉♠❛ ✈❡③ q✉❡ ♠♦❞❡❧❛♠ t❛♥t♦ s❡✉ ♠❡❝❛♥✐s♠♦ q✉❛♥t♦ ❛ ❡str✉t✉r❛ ❞❡ ❝♦rr❡❧❛çã♦ ✐♥tr❛✉♥✐❞❛❞❡ ❡①♣❡r✐♠❡♥t❛❧✳
❊①✐st❡♠ ❞✐✈❡rs❛s ♠❛♥❡✐r❛s ❞❡ s❡ ❢♦r♠✉❧❛r❡♠ ❡ss❡s ♠♦❞❡❧♦s✳ ▲✐♥❞str♦♠ ✫ ❇❛t❡s ✭✶✾✾✵✮ ❡ P✐♥❤❡✐r♦ ✫ ❇❛t❡s ✭✷✵✵✵✮ ❝♦♥s✐❞❡r❛♠ ♦ ❝❛s♦ ❡♠ q✉❡ ♦s ♣❛râ♠❡tr♦s ❞❡ ❡❢❡✐t♦s ✜①♦s ❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s sã♦ ✐♥❝♦r♣♦r❛❞♦s ❞❡ ❢♦r♠❛ ♥ã♦ ❧✐♥❡❛r✳ ❆ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ✉t✐❧✐③❛r ❡ss❛ ❢♦r♠✉❧❛çã♦ ✈❡♠ ❞♦ ❢❛t♦ ❞❡ q✉❡ ❛s ✐♥❢❡rê♥❝✐❛s sã♦ ❜❛s❡❛❞❛s ♥♦ ♠♦❞❡❧♦ ♠❛r❣✐♥❛❧ q✉❡✱ ❡♠ ❣❡r❛❧✱ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛✳ ❉❡ss❛ ❢♦r♠❛✱ é ♥❡❝❡ssár✐♦ r❡❛❧✐③❛r ❛♣r♦①✐♠❛çõ❡s ♥✉♠ér✐❝❛s ♣❛r❛ ♦ ❧♦❣❛r✐t♠♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ♦ q✉❡ r❡s✉❧t❛ ❡♠ ♣❡r❞❛ ❞❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♣r❡❞✐çã♦ ❞♦ ♠♦❞❡❧♦✳ ❯♠❛ ❛❧t❡r♥❛t✐✈❛ é ❝♦♥s✐❞❡r❛r ♦ ♠♦❞❡❧♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❱♦♥❡s❤ ✫ ❈❛rt❡r ✭✶✾✾✷✮✱ ❡♠ q✉❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ é ❡①♣❧✐❝❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❧✐♥❡❛r ♥♦s ♣❛râ♠❡tr♦s ✜①♦s ❡ ✉♠❛ ❢♦r♠❛ ❧✐♥❡❛r ♥♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✳ ◆❡ss❡ ❝❛s♦✱ ❛ ❞✐str✐❜✉✐çã♦ ♠❛r❣✐♥❛❧ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ é ❝♦♥❤❡❝✐❞❛ ❡ ❛s ✐♥❢❡rê♥❝✐❛s ♣♦❞❡♠ s❡r r❡❛❧✐③❛❞❛s ❞❡ ♠❛♥❡✐r❛ ❞✐r❡t❛✳ ◆❡st❡ ❝❛♣ít✉❧♦ s❡rá ❝♦♥s✐❞❡r❛❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦ ú❧t✐♠♦ ♠♦❞❡❧♦✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦ ✈❡t♦r ❞❡ ❡rr♦s é ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ✈❡t♦r ❞❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ❡ q✉❡ ❛♠❜♦s s❡❣✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♠✉❧t✐✈❛r✐❛❞❛✳
✷✳✷ ❊s♣❡❝✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦
❙❡❥❛yi = (yi1, . . . , yimi)
⊤
♦ ✈❡t♦r ❞❡ ♦❜s❡r✈❛çõ❡s ❞♦ i✲és✐♠♦ ✐♥❞✐✈í❞✉♦ ✭i= 1, . . . , n✮✳
❖ ♠♦❞❡❧♦ ♥ã♦ ❧✐♥❡❛r ✉s✉❛❧ ❝♦♠ ❡❢❡✐t♦s ♠✐st♦s é r❡♣r❡s❡♥t❛❞♦ ♣♦r
yi =ηi(φi,xi) +εi, φi =Aiβ+Ziui,
✭✷✳✶✮
❡♠ q✉❡ ηi(φi,xi) = (ηi(φi1,xi1), . . . , ηi(φimi,ximi))
⊤
é ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❧✐♥❡❛r ❞♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦sφi✱ ❝♦♠φi = (φi1, . . .φimi)⊤
q✉❡ ❞❡♣❡♥❞❡ ❞❛s ♠❛tr✐③❡s ❞❡ ♣❧❛♥❡❥❛♠❡♥t♦
Ai = (Ai1, . . . ,Aimi)⊤
❡ Zi = (Zi1, . . . ,Zimi)⊤
❝♦♠ ❞✐♠❡♥sõ❡s mi ×p ❡ mi ×q✱ r❡s✲
♣❡❝t✐✈❛♠❡♥t❡✱ β = (β1, . . . , βp)
⊤
é ♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s r❡❢❡r❡♥t❡s ❛♦s ❡❢❡✐t♦s ✜①♦s✱
ui é ✉♠ ✈❡t♦r q✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✱xi = (xi1, . . . ,ximi) é ✉♠❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐á✈❡✐s ❡ εi = (εi1, . . . , εimi)
⊤
é ♦ ✈❡t♦r ❞❡ ❡rr♦s ❛❧❡❛tór✐♦s✳
❉❡♥♦t❡ ♣♦r ◆n(µ,Σ) ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♠✉❧t✐✈❛r✐❛❞❛ ❝♦♠ ✈❡t♦r ❞❡ ♠é❞✐❛s µ
n✲❞✐♠❡♥s✐♦♥❛❧ ❡ ♠❛tr✐③ ❞❡ ✈❛r✐â♥❝✐❛s✲❝♦✈❛r✐â♥❝✐❛sΣn×n✳ ➱ ❝♦♠✉♠ s✉♣♦r q✉❡
ui ∼◆q(0,Q) ❡ εi ∼◆mi(0, σ
2I
mi)
sã♦ ♠✉t✉❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ♣❛r❛ i = 1, . . . , n✱ ❡♠ q✉❡ Q é ❛ ♠❛tr✐③ ✈❛r✐â♥❝✐❛s✲
❝♦✈❛r✐â♥❝✐❛s ❞❡ ui ❡Imi r❡♣r❡s❡♥t❛ ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦mi✳
❆s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦ ♠♦❞❡❧♦ ✭✷✳✶✮ sã♦ ❜❛s❡❛❞❛s ♥❛ ❞✐str✐✲
❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡y= (y1, . . . ,yn)⊤
✱ q✉❡ é ❢♦r♥❡❝✐❞❛ ♣♦r
p(y|β, σ2,Q) =
Z
p(y,u|β, σ2,Q)du ✭✷✳✷✮
=
Z
p(y|u,β, σ2)p(u|Q)du,
❡♠ q✉❡ u = (u1, . . . ,uq)⊤✱ p(y|u,β, σ2) é ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡ y ❞❛❞♦s
♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s u ❡ p(u|Q) r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ u✳ ➱ ❝♦♥✈❡♥✐❡♥t❡
❡①♣r❡ss❛r ❛ ♠❛tr✐③ ❞❡ ✈❛r✐â♥❝✐❛s✲❝♦✈❛r✐â♥❝✐❛s ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ♥❛ ❢♦r♠❛ ❞❡ ✉♠ ❢❛t♦r ❞❡ ♣r❡❝✐sã♦ r❡❧❛t✐✈❛✶✱ ∆✱ t❛❧ q✉❡ Q−1
=σ2∆⊤
∆ ✭P✐♥❤❡✐r♦ ✫ ❇❛t❡s ✷✵✵✵✱ ❝❛♣✳ ✼✮✳
❆ss✐♠✱ é ♣♦ssí✈❡❧ ❡①♣r❡ss❛r ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡y ❝♦♠♦ p(y|β, σ2,∆) = |∆|
n
(2πσ2)(m+nq)/2
n Y
i=1
Z
exp
−kyi−ηi(β,ui)k
2+k∆u
ik2
2σ2
dui ✭✷✳✸✮
❡♠ q✉❡ ηi(β,ui) =ηi(φ(β,ui),xi)✳
❈♦♠♦ ηi ♣♦❞❡ s❡r ♥ã♦ ❧✐♥❡❛r ♥♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✱ ❛ ✐♥t❡❣r❛❧ ❡♠ ✭✷✳✸✮ ♣♦❞❡ ♥ã♦ t❡r
s♦❧✉çã♦ ❛♥❛❧ít✐❝❛✳ P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♠❛①✐♠✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ❞✐❢❡r❡♥t❡s ❛♣r♦①✐♠❛çõ❡s ❞❡ss❛ ❡q✉❛çã♦ tê♠ s✐❞♦ ♣r♦♣♦st❛s ✭✈✐❞❡✱ ♣♦r ❡①❡♠♣❧♦✱ P✐♥❤❡✐r♦ ✫ ❇❛t❡s ✷✵✵✵✱ ♣❛❣✳ ✸✶✸✮✳ ❯♠ ❞❡ss❡s ♠ét♦❞♦s✱ ❝♦♥s✐❞❡r❛❞♦ ♣♦r ❱♦♥❡s❤ ✫ ❈❛rt❡r ✭✶✾✾✷✮✱ ❝♦♥s✐st❡ ❡♠ r❡❛❧✐③❛r ✉♠❛ ❡①♣❛♥sã♦ ❞❡ ❚❛②❧♦r ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❞❛ ❢✉♥çã♦ ηi(φi,xi) ❡♠ t♦r♥♦ ❞❡ ui = 0✳ ▲❡✈❛♥❞♦✲s❡ ❡♠ ❝♦♥t❛ q✉❡ ♣❛r❛ ❝❛❞❛ i ∈ {1, . . . , n} ❛ ❢♦r♠❛ ❢✉♥❝✐♦♥❛❧ ❞❡ ηi
é ❛ ♠❡s♠❛✱ ❢❛③ s❡♥t✐❞♦ ❝♦♥s✐❞❡r❛r ηi(φi,xi) = η(φi,xi)✳ ❆ss✐♠✱ ❡ss❛ ❛♣r♦①✐♠❛çã♦ ❞♦
♠♦❞❡❧♦ ✭✷✳✶✮ é ❢♦r♥❡❝✐❞❛ ♣♦r
yi =η(β,xi) +Z
∗
iu
∗
i +εi, ✭✷✳✹✮
❡♠ q✉❡
Z∗i =
∂η(φi,xi)
∂φ⊤i
φi=Aiβ
Ai
❡u∗i =ui✱ ❞❡♥♦♠✐♥❛❞♦ ♠♦❞❡❧♦ ♣s❡✉❞♦ ♥ã♦ ❧✐♥❡❛r ❝♦♠ ❡❢❡✐t♦s ♠✐st♦s✳
❊ss❡ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ♣r♦♣♦st♦ ♣♦r ❱♦♥❡s❤ ✫ ❈❛rt❡r ✭✶✾✾✷✮ ❞❡♥♦♠✐♥❛❞♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ♥ã♦ ❧✐♥❡❛r ❝♦♠ ❡❢❡✐t♦s ♠✐st♦s ❣❡♥❡r❛❧✐③❛❞♦
yi =η(β,xi) +Ziui+εi, i= 1, . . . , n, ✭✷✳✺✮
❝♦♠ η(β,xi) = (η(β,xi1), . . . , η(β,ximi))
⊤
✉♠❛ ❢✉♥çã♦ ♥ã♦ ❧✐♥❡❛r ❞❡ β ❝♦♠ ❞✐♠❡♥sã♦
mi ×1✱ β ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s ✜①♦s✱ xi ✈❡t♦r ❞❡ ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✱ Zi ♠❛tr✐③ ❞❡
♣❧❛♥❡❥❛♠❡♥t♦✷✱ε
i = (εi1, . . . , εimi)
⊤
❡ui = (ui1, . . . , uiq)
⊤
sã♦ ♦s ✈❡t♦r❡s ❞❡ ❡rr♦s ❡ ❡❢❡✐t♦s
✶❙❡Q❢♦r ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✱ ❡♥tã♦ ∆❡①✐st❡✱ ♠❛s ♥ã♦ é ú♥✐❝❛✳
✷❊①❡♠♣❧♦s ❞❡ ❡str✉t✉r❛s ♣❛r❛ ❛ ♠❛tr✐③Zi sã♦ ❛♣♦♥t❛❞♦s ♥❛ ♣á❣✐♥❛ ✶✹✳
❛❧❡❛tór✐♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♠✉t✉❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ◆❡st❡ tr❛❜❛❧❤♦ ♦ ♠♦❞❡❧♦ ✭✷✳✺✮ s❡rá r❡❢❡r✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ♠✐st♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r✳ ❱♦♥❡s❤ ✫ ❈❛rt❡r ✭✶✾✾✷✮ ❝♦♥s✐❞❡r❛♠ ♦ ♠♦❞❡❧♦ ✭✷✳✺✮ s✉♣♦♥❞♦ q✉❡ ♦s ❡rr♦s ❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ s❡❣✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ ❘✉ss♦ ✭✷✵✶✵✮ ❣❡♥❡r❛❧✐③❛ ♦ ú❧t✐♠♦ ♠♦❞❡❧♦ s✉♣♦♥❞♦ ❞✐str✐❜✉✐çõ❡s ❡❧í♣t✐❝❛s ♣❛r❛ ❛s ❝♦♠♣♦♥❡♥t❡s ❛❧❡❛tór✐❛s✱ q✉❡ ♣♦ss✐❜✐❧✐t❛♠ ❛♦ ♠♦❞❡❧♦✱ ♠❛✐♦r ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛❝♦♠♦❞❛r ♦❜s❡r✈❛çõ❡s ❛❜❡rr❛♥t❡s✳ ◆❡st❡ tr❛❜❛❧❤♦ s❡rá ❞❛❞❛ ê♥❢❛s❡ ❛♦ ♠♦❞❡❧♦ ✭✷✳✺✮ ❡♠ q✉❡ ♦s ❡rr♦s ❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s sã♦ ♠✉t✉❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ s❡❣✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✱ ♦ q✉❡ ❡q✉✐✈❛❧❡ ❛ s✉♣♦r q✉❡
yi
ui !
ind. ∼ ◆mi+q
η(β,xi)
0
!
,
"
ZiQZ
⊤
i +σ2Imi ZiQ
QZ⊤i Q
#!
, ✭✷✳✻✮
i = 1, . . . , n✳ ❈♦♠♦ ✈❛♥t❛❣❡♠ ❞❛ ❢♦r♠✉❧❛çã♦ ✭✷✳✺✮ ❡ ✭✷✳✻✮ t❡♠✲s❡ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡
♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡yé ♦❜t✐❞❛ ❞❡ ♠❛♥❡✐r❛ ❞✐r❡t❛ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧
♠✉❧t✐✈❛r✐❛❞❛✱ ♦ q✉❡ ❢❛❝✐❧✐t❛ ❛ ♠❛①✐♠✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ P❡❧❛ ♠❡s♠❛ r❛③ã♦✱ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ui ❞❛❞♦ yi é ❢❛❝✐❧♠❡♥t❡
❞❡t❡r♠✐♥❛❞❛✱ ♦ q✉❡ ♣❡r♠✐t❡ ♣r❡❞✐③❡r ♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ♣❡❧♦ ♠ét♦❞♦ ❇❛②❡s ❡♠♣ír✐❝♦✳ ❆❧é♠ ❞✐ss♦✱ t❛❧ ❢♦r♠✉❧❛çã♦ ♣r❡s❡r✈❛ ❛ ♠é❞✐❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❡♠ s✉❛ ❢♦r♠❛ ♦r✐❣✐♥❛❧ ♥ã♦ ❧✐♥❡❛r ❡ ♣♦ss✐❜✐❧✐t❛ ❛ ✐♥❝❧✉sã♦ ❞❛ ❝♦rr❡❧❛çã♦ ✐♥tr❛❣r✉♣♦✳
✷✳✸ ❊st✐♠❛çã♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛
❆ ❝♦♠♣♦♥❡♥t❡ ❞❛ ❢✉♥çã♦ ❞❡ ❧♦❣✲✈❡r♦ss✐♠✐❧❤❛♥ç❛
n X
i=1
Li(yi;xi,β,Σi) ♣❛r❛ ♦ ♠♦❞❡❧♦
✭✷✳✻✮✱ ❛ss♦❝✐❛❞❛ ❛♦ ❣r✉♣♦ i é ❢♦r♥❡❝✐❞❛ ♣♦r
Li =−
1
2log|Σi|+ 1
2[yi−η(β,xi)]
⊤
Σ−i 1[yi−η(β,xi)] ✭✷✳✼✮ ♣❛r❛ i= 1, . . . , n✱ ❡♠ q✉❡ Σi =ZiQZ⊤
i +σ2Imi✳ ❖ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s ❛ s❡r ❡st✐♠❛❞♦
é ❞❡✜♥✐❞♦ ❝♦♠♦ θ = (β⊤, σ2,τ⊤
)⊤
✱ ❡♠ q✉❡ τ é ♦ ✈❡t♦r ❝♦♠ ❡❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ Q✳
❉❡✜♥❛ γ0 = σ2 ❡ γj = τj, j = 1, . . . , t✳ ❖s ❡❧❡♠❡♥t♦s ❞❛ ❢✉♥çã♦ ❡s❝♦r❡ ✭❘✉ss♦ ❡t ❛❧✳
✷✵✵✾✮ ♣❛r❛θ = (β⊤, γ0, . . . , γt)⊤ = (β
⊤
,γ⊤
)⊤
sã♦ ❞❛❞♦s ♣♦r
Uβ = n X
i=1
J⊤i Σ−i1ri ❡
Uγ = (Uγ0, . . . , Uγt)
⊤
, ❝♦♠ Uγj = −
1 2
n X
i=1
n
trhΣ−i 1Σ˙i(j) i
−r⊤i Σ−i 1Σ˙i−1(j)Σ−i 1ri o
,
❡♠ q✉❡ ri = (yi−η(β,xi))✱ Ji =∂η(β,xi)/∂α⊤
❡ Σ˙i(j) =∂Σi/∂γj, j = 0, . . . , t, i= 1, . . . , n✳
❆ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r ♣❛r❛ θ é ❞❛❞❛ ♣♦r
Kθθ =
Kββ 0 0 Kγγ
!
, ✭✷✳✽✮
❡♠ q✉❡
Kββ =
n X
i=1
J⊤i Σ−i 1Ji
Kγγ = [Kγr,γs], ❝✉❥♦(r, s)✲és✐♠♦ ❡❧❡♠❡♥t♦ é ❞❛❞♦ ♣♦r
Kγr,γs =
1 2
n X
i=1
tr[Σ−i 1Σ˙i(r)Σ
−1
i Σ˙i(s)].
❆s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛θ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❝♦♠ ♦ ❛❧❣♦r✐t♠♦
❡s❝♦r❡ ❞❡ ❋✐s❤❡r✱ q✉❡ é ❞❛❞♦ ♣♦r ˆ
β(k+1) = ˆβ(k)+ [Kββ]−(k1)Uβ(k)
ˆ
γ(k+1) = ˆγ(k)+ [Kγγ]
−1
(k)Uγ(k)
♣❛r❛ k = 0,1,2, . . . ❆s ❡st✐♠❛t✐✈❛s ✐♥✐❝✐❛✐s αˆ(0) ❡ γˆ(0) ♣♦❞❡♠ s❡r ❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s✱ q✉❡ ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ✉t✐❧✐③❛♥❞♦✲s❡ ❛ ❢✉♥çã♦ ♥❧s ❞♦ ❘✳
✷✳✹ Pr❡❞✐çã♦ ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s
❆ ♣r❡❞✐çã♦ ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ♥♦ ♠♦❞❡❧♦ ✭✷✳✺✮ é r❡❛❧✐③❛❞❛ ✉t✐❧✐③❛♥❞♦✲s❡ ♦ ♠ét♦❞♦ ❞❡ ❇❛②❡s ❡♠♣ír✐❝♦✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ♣r❡❞✐③❡r ui ♣❡❧❛ s✉❛ ♠é❞✐❛ ❝♦♥❞✐❝✐♦♥❛❧✱ ❞❛❞❛s ❛s
r❡s♣♦st❛s yi✱ i = 1, . . . , n✳ P❡❧❛ s✉♣♦s✐çã♦ ❢❡✐t❛ ❡♠ ✭✷✳✻✮✱ s❡❣✉❡✱ ♣♦r ♣r♦♣r✐❡❞❛❞❡s ❞❛
❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♠✉❧t✐✈❛r✐❛❞❛ ✭▼✉✐r❤❡❛❞ ✶✾✽✷✱ ❝❛♣✳✶✮✱ q✉❡
ui|yi ∼◆q(QZ
⊤
i Σ
−1
i (yi−η(β,xi)),Q−QZ
⊤
i Σ
−1
i ZiQ).
P❛r❛Σi✜①❛❞❛✱ ❛ ♣r❡❞✐çã♦ ❞♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦sui✉s❛♥❞♦✲s❡ ♦ ♠ét♦❞♦ ❞❡ ❇❛②❡s ❡♠♣ír✐❝♦
é ❞❛❞❛ ♣♦r
b
ui =E[u[i|yi] =QZ
⊤
i Σ
−1
i (yi−η(βb,xi)). ✭✷✳✾✮
❈♦♠♦ Σi é ❞❡s❝♦♥❤❡❝✐❞❛✱ é ✉s✉❛❧ s✉❜st✐t✉í✲❧❛ ♣♦r s✉❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐✲
❧❤❛♥ç❛✳ P♦rt❛♥t♦✱ ❛ ♣r❡❞✐çã♦ ❞❡ yˆi é ❢♦r♥❡❝✐❞❛ ♣♦r
b
yi = η(βb,xi) +Ziubi
= η(βb,xi) +ZiQZb
⊤
i Σb
−1
i (yi−η(βb,xi))
= (Imi −ZiQZb
⊤
i Σb
−1
i )η(βb,xi) +ZiQZb
⊤
i Σb
−1
i yi
= bσ2Σb
−1
i η(βb,xi) + (Imi −bσ
2Σb−1
i )yi,
q✉❡ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠❛ ♠é❞✐❛ ♣♦♥❞❡r❛❞❛ ❞♦ ♣❡r✜❧ ♣♦♣✉❧❛❝✐♦♥❛❧η(βb,xi)❡
❞♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦syi✱ ❝♦♠ ♣❡s♦s σb2Σb
−1
i ❡ (Imi−bσ
2Σb−1
i )✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
◆♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡ é ♣r♦♣♦st❛ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ♠♦❞❡❧♦ ♠✐st♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r ❛♦ s❡ ❛❞✐❝✐♦♥❛r ✉♠❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ♣❛r❛ ♠♦❞❡❧❛r ❛ ♠é❞✐❛ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛✳
❈❛♣ít✉❧♦ ✸
▼♦❞❡❧♦ ♠✐st♦s s❡♠✐♣❛r❛♠étr✐❝♦
♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r
◆❡st❡ ❝❛♣ít✉❧♦ é ♣r♦♣♦st❛ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❱♦♥❡s❤ ✫ ❈❛rt❡r ✭✶✾✾✷✮✱ ❛ q✉❛❧ ♥❡st❡ tr❛❜❛❧❤♦ é ❞❡s✐❣♥❛❞❛ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♣❛r✲ ❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r✳ ◆❛ ❙❡çã♦ ✸✳✶ s❡rã♦ ❞✐s❝✉t✐❞❛s ❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ❡ss❡ ♠♦❞❡❧♦✱ ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s ❡ ❛❧❣✉♥s tr❛❜❛❧❤♦s r❡❧❛❝✐♦♥❛❞♦s✳ ◆❛ ❙❡çã♦ ✸✳✷ ♦ ♠♦❞❡❧♦ é ❞❡✜♥✐❞♦ ❢♦r♠❛❧♠❡♥t❡ ❡ ❛s♣❡❝t♦s r❡❧❡✈❛♥t❡s sã♦ ❞✐s❝✉t✐❞♦s✳ ◆❛ ❙❡çã♦ ✸✳✸ é ❛♣r❡s❡♥t❛❞❛ ❛ r❡♣r❡✲ s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ♣❛r❛ ♦ ♠♦❞❡❧♦✱ ♣❛r❛✱ ♥❛ s❡çã♦ ✸✳✺✱ s❡ ❝♦♥str✉✐r ♦ ❧♦❣❛r✐t♠♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡♥❛❧✐③❛❞❛✳ ◆❛ ❙❡çã♦ ✸✳✻ é ❛♣r❡s❡♥t❛❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s r❡❢❡r❡♥t❡s ❛♦s ❡❢❡✐t♦s ✜①♦s ❡ ❞❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛✳
✸✳✶ ■♥tr♦❞✉çã♦
▼♦❞❡❧♦s ♠✐st♦s ♥ã♦ ❧✐♥❡❛r❡s✱ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✷✱ sã♦ ❢❡rr❛♠❡♥t❛s út❡✐s ♣❛r❛ s❡ ❡st✉❞❛r❡♠ ❞❛❞♦s ❝♦rr❡❧❛❝✐♦♥❛❞♦s q✉❡ ❛♣r❡s❡♥t❛♠ r❡❧❛çã♦ ♥ã♦ ❧✐♥❡❛r ❡♥tr❡ ❛s ❝♦✲ ✈❛r✐á✈✐❡s ❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛✳ P♦ré♠✱ ❡ss❡s ♠♦❞❡❧♦s ♣♦❞❡♠ s❡r ♠✉✐t♦ r❡str✐t✐✈♦s ❡ ❝❡rt♦s ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s sã♦ ♠✉✐t♦ ❝♦♠♣❧❡①♦s ❡ ✐♥❢♦r♠❛t✐✈♦s ♣❛r❛ s❡r❡♠ t♦t❛❧♠❡♥t❡ ♠♦❞❡❧❛✲ ❞♦s ❞❡ ❢♦r♠❛ ♣❛r❛♠étr✐❝❛✳ ◆❡st❡ tr❛❜❛❧❤♦ é ♣r♦♣♦st♦ ♦ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r ✭▼▼❙P◆▲✮ ✭▼❛❝❤❛❞♦ ✫ ❘✉ss♦ ✷✵✶✸✮✱ q✉❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞✐r❡t❛ ❞♦ ♠♦❞❡❧♦ ♠✐st♦ ♥ã♦ ❧✐♥❡❛r ❛♣r❡s❡♥t❛❞♦ ♣♦r ❱♦♥❡s❤ ✫ ❈❛rt❡r ✭✶✾✾✷✮ ❡✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ✈ár✐♦s ♦✉tr♦s ♠♦❞❡❧♦s✳ ❯♠ ▼▼❙P◆▲ ❛ss✉♠❡ q✉❡ ❛ ♠é❞✐❛ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ♣♦❞❡ s❡r ❡①♣❧✐❝❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ♣❛r❛♠étr✐❝❛ ♥ã♦ ❧✐♥❡❛r✱ ❝♦♠ r❡❧❛çã♦ ❛♦s ♣❛râ♠❡tr♦s ❡ às ❝♦✈❛r✐á✈❡✐s✱ ❡ ✉♠❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛✳ ❖s ♣❛râ♠❡tr♦s ♣r♦♣♦r❝✐♦♥❛♠ ✉♠ r❡s✉♠♦ ✐♥t❡r♣r❡tá✈❡❧ ❞♦s ❞❛❞♦s✱ ❡♥q✉❛♥t♦ ❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ❢♦r♥❡❝❡ ✢❡①✐❜✐❧✐❞❛❞❡ ♣❛r❛ ❞❡✐①❛r q✉❡ ♦s ❞❛❞♦s ✏❞❡❝✐❞❛♠✑ s♦❜r❡ ❝♦♠♣♦♥❡♥t❡s ❞❡s❝♦♥❤❡❝✐❞♦s ♦✉ ✐♥❝❡rt♦s✱ ❝♦♠♦ ❛ ❢♦r♠❛ ❢✉♥❝✐♦♥❛❧ ❞❛ ♠é❞✐❛ ❞❛ r❡s♣♦st❛ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳ ❖s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✱ ✉s❛❞♦s ♣❛r❛ ♠♦❞❡❧❛r ❛ ❡str✉t✉r❛ ❞❡ ✈❛r✐â♥❝✐❛s✲❝♦✈❛r✐â♥❝✐❛s
❞♦s ❞❛❞♦s✱ sã♦ ✐♥❝❧✉í❞♦s ❞❡ ❢♦r♠❛ ❧✐♥❡❛r✱ ♦ q✉❡ ♣r♦♣♦r❝✐♦♥❛ ❛ ✈❛♥t❛❣❡♠ ❞❡ s❡ ♦❜t❡r ♦ ♠♦❞❡❧♦ ♠❛r❣✐♥❛❧ ❞❡ ❢♦r♠❛ ❞✐r❡t❛✱ s❡♠ q✉❡ s❡❥❛♠ ♥❡❝❡ssár✐❛s ✐♥t❡❣r❛çõ❡s ♥✉♠ér✐❝❛s✳ ❆ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ é ❛♣r♦①✐♠❛❞❛ ✉s❛♥❞♦✲s❡ s✉❛✈✐③❛çã♦ ♣♦rB✲s♣❧✐♥❡s✳ ❆ ❢✉♥çã♦ ❞❡
✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡♥❛❧✐③❛❞❛ ❜❛s❡❛❞❛ ♥♦ ♠♦❞❡❧♦ ♠❛r❣✐♥❛❧ é ✉t✐❧✐③❛❞❛ ♣❛r❛ s❡ ❡st✐♠❛r❡♠ t♦❞♦s ♦s ♣❛râ♠❡tr♦s ❡ ❛ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛✳ ❆ ♣❡♥❛❧✐❞❛❞❡ é ❜❛s❡❛❞❛ ♥♦ ♦♣❡r❛❞♦r ❞❡ ❞✐❢❡r❡♥ç❛s ❛♣r❡s❡♥t❛❞♦ ♣♦r ❊✐❧❡rs ✫ ▼❛r① ✭✶✾✾✻✮✳
❍á ♣♦✉❝♦s tr❛❜❛❧❤♦s ❡♥✈♦❧✈❡♥❞♦ ♠♦❞❡❧♦s ♠✐st♦s s❡♠✐♣❛r❛♠étr✐❝♦s ♥ã♦ ❧✐♥❡❛r❡s ♥❛ ❧✐t❡r❛t✉r❛✳ ❑❡ ✫ ❲❛♥❣ ✭✷✵✵✶✮ ❛♣r❡s❡♥t❛♠ ✉♠ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♥ã♦ ❧✐✲ ♥❡❛r✱ ❡♠ q✉❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ é ❡①♣❧✐❝❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ q✉❡ ♣♦❞❡ s❡r ♥ã♦ ❧✐♥❡❛r ❝♦♠ r❡❧❛çã♦ ❛ ♣❛râ♠❡tr♦s✱ ❢✉♥çã♦ ♥ã♦ ♣❛r❛♠étr✐❝❛ ❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✳ ❆ss✐♠✱ t♦r♥❛✲s❡ ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ❛♣r♦①✐♠❛çõ❡s ♥✉♠ér✐❝❛s ♣❛r❛ s❡ ♦❜t❡r ❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ♦ ♠♦✲ ❞❡❧♦ ♠❛r❣✐♥❛❧✳ ❆ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ss❡ ♠♦❞❡❧♦ ❞✐✜❝✉❧t❛ ♠✉✐t♦ ♦ ♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛çã♦ ❡ ✐♥❢❡rê♥❝✐❛✳ ▲✐✉ ✫ ❲✉ ✭✷✵✵✽✮ ❛♣r❡s❡♥t❛♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ss❡s ♠♦❞❡❧♦s ♣❛r❛ ❞❛❞♦s ❞❡ ❍■❱ ❝♦♠ ❞❛❞♦s ♣❡r❞✐❞♦s ♥ã♦ ✐❣♥♦rá✈❡✐s ❡ ❝♦♠ ❡rr♦s ❞❡ ♠❡❞✐çã♦✳ ❊❧♠✐ ❡t ❛❧✳ ✭✷✵✶✶✮ ❞❡s❡♥✈♦❧✈❡♠ ✉♠ ♠ét♦❞♦ ❛❧t❡r♥❛t✐✈♦ ♣❛r❛ ❛❥✉st❛r ♦ ♠♦❞❡❧♦ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❑❡ ✫ ❲❛♥❣ ✭✷✵✵✶✮✱ ❜❛s❡❛♥❞♦ ❛ s✉❛✈✐③❛çã♦ ❡♠ t❡r♠♦s ❞❡B✲s♣❧✐♥❡s✳
✸✳✷ ❊s♣❡❝✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦
❈♦♥s✐❞❡r❡ ✉♠❛ ❛♠♦str❛ ❝♦♥s✐st✐♥❞♦ ❞❡ns✉❥❡✐t♦s ❡mi ♦❜s❡r✈❛çõ❡s ❞♦i✲és✐♠♦ s✉❥❡✐t♦
❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳ ❙❡❥❛yij (i= 1, . . . , n;j = 1, . . . , mi) ❛ j✲és✐♠❛ r❡s♣♦st❛ ❞♦ i✲és✐♠♦
s✉❥❡✐t♦✳ ❖ ♠♦❞❡❧♦ ♠✐st♦ s❡♠✐♣❛r❛♠étr✐❝♦ ♣❛r❝✐❛❧♠❡♥t❡ ♥ã♦ ❧✐♥❡❛r ✭▼▼❙P◆▲✮ ❛ss✉♠❡ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛
yij =η(β,xij) +f(tij) +Z
⊤
ijui+εij, ✭✸✳✶✮
❡♠ q✉❡ η é ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❧✐♥❡❛r ❞❡ β ❡ xij✱ xij é ✉♠❛ ❝♦✈❛r✐á✈❡❧ q✉❡ ♣♦❞❡ s❡r ✉♠
❡s❝❛❧❛r ♦✉ ✉♠ ✈❡t♦r✱β= (β1, . . . , βp)
⊤
é ♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s ✜①♦s ❡ ❞❡s❝♦♥❤❡❝✐❞♦s✱Zij
é ✉♠ ✈❡t♦r ❝♦♥❤❡❝✐❞♦ r❡❧❛❝✐♦♥❛❞♦ ❛♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✱ui = (ui1, . . . , uir)
⊤
é ♦ ✈❡t♦r ❞❡ ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡❢❡✐t♦s ❛❧❡❛tór✐♦s ❛ss♦❝✐❛❞♦ ❛♦i✲és✐♠♦ s✉❥❡✐t♦✱ εij sã♦ ♦s ❡rr♦s ❛❧❡❛tór✐♦s✱
f é ✉♠❛ ❢✉♥çã♦ r❡❛❧ ✉♥✐✈❛r✐❛❞❛ ❞♦ t❡♠♣♦ ❡tij sã♦ ♦s ✐♥st❛♥t❡s ❞❡ t❡♠♣♦✳
◆♦ ♠♦❞❡❧♦ ✭✸✳✶✮ ♦s ❡❢❡✐t♦s ✜①♦s✱ ❝♦♥st✐t✉í❞♦s ♣❡❧❛s ❝♦♠♣♦♥❡♥t❡s ♣❛r❛♠étr✐❝❛s ❡ ♥ã♦ ♣❛r❛♠étr✐❝❛s✱ sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♠♦❞❡❧❛r ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ yij✳
❆ ❢✉♥çã♦ f ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ✉♠ ♣❛râ♠❡tr♦ ❞♦ ♠♦❞❡❧♦ ✭✸✳✶✮ ❛ss✐♠ ❝♦♠♦
β✳ ❆ ❞✐❢❡r❡♥ç❛ é q✉❡ ♦ ú❧t✐♠♦ ♣❡rt❡♥❝❡ ❛ ✉♠ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ✭✉s✉❛❧♠❡♥t❡
Rp✮ ❡♥q✉❛♥t♦ f✱ s❡♥❞♦ ✉♠❛ ❢✉♥çã♦✱ ♣❡rt❡♥❝❡ ❛ ✉♠ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✳ ◆♦
♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛çã♦ é ♣r❡❝✐s♦ ❡s♣❡❝✐✜❝❛r ❛ q✉❛❧ ❡s♣❛ç♦ f ♣❡rt❡♥❝❡✳ ❆ ❡s❝♦❧❤❛ ❞❡ss❡
❡s♣❛ç♦ ❞❡♣❡♥❞❡ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❛ ♣r✐♦r✐ s♦❜r❡ ❛ ❢✉♥çã♦ f✱ ❝♦♠♦ ♦ ❞♦♠í♥✐♦✱ ❛ ❢♦r♠❛
❢✉♥❝✐♦♥❛❧✱ r❡str✐çõ❡s ♣❛r❛ ❣❛r❛♥t✐r ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ❡ ♣r♦♣ós✐t♦s ❞❛ ❛♥á❧✐s❡ ✭❑❡ ✫ ❲❛♥❣ ✷✵✵✶✮✳ ❯♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ é ❝♦♥s✐❞❡r❛r q✉❡ f é ✉♠❛ ❢✉♥çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❞❡
♦r❞❡♠2✱ ♦✉ s❡❥❛✱ f ∈W2[a, b]✱ ❡♠ q✉❡
W2[a, b] =
f :f ∈C1[a, b],
Z b
a {
f(2)(x)}2dx <∞
, ✭✸✳✷✮
❡ C1[a, b] r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♥♦ ✐♥t❡r✈❛❧♦ [a, b] q✉❡ ♣♦ss✉❡♠ ❞❡r✐✈❛❞❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♥tí♥✉❛s✳
❙❡❣✉♥❞♦ ❘❡✐♥s❝❤ ✭✶✾✻✼✮ ❡ ❙✐❧✈❡r♠❛♥ ✭✶✾✽✺✮✱ ❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥✲ ç❛ ♣❡♥❛❧✐③❛❞❛ ❞❛ ❢✉♥çã♦f ♣❡rt❡♥❝❡♠ ❛ ✉♠ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡✱
sã♦ ♣♦❧✐♥ô♠✐♦s ❝ú❜✐❝♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ❢❛③ s❡♥t✐❞♦ ❝♦♥s✐❞❡r❛r ❛ ❡①♣❛♥sã♦ ❞❛ ❢✉♥çã♦f ❡♠
✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ♣♦❧✐♥ô♠✐♦s ❝ú❜✐❝♦s✳
❊♠ ♣❛rt✐❝✉❧❛r✱ ✉♠ ♠ét♦❞♦ ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞♦ é ❝♦♥s✐❞❡r❛r B✲s♣❧✐♥❡s ✭✈❡r ❆♣ê♥✲
❞✐❝❡ ❇✮ ♣❛r❛ r❡❛❧✐③❛r t❛❧ ❡①♣❛♥sã♦ ✭❊✐❧❡rs ✫ ▼❛r① ✶✾✾✻✱ ❊❧♠✐ ❡t ❛❧✳ ✷✵✶✶✮✳ ❊❧❛s ❝♦♥s✐st❡♠ ❡♠ ♣❛rt❡s ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ❝♦♥❡❝t❛❞♦s ❞❡ ✉♠❛ ❢♦r♠❛ ❝♦♥✈❡♥✐❡♥t❡ ❡♠ ✈❛❧♦r❡s ❡s♣❡❝✐✜❝❛❞♦s ❞♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ f✱ ♦s ♥ós✳ ❯♠❛ ✈❡③ ❞❛❞♦s ♦s ♥ós✱ ❛sB✲s♣❧✐♥❡s ♣♦❞❡♠ s❡r ❝❛❧❝✉✲
❧❛❞❛s ❢❛❝✐❧♠❡♥t❡ ❞❡ ♠❛♥❡✐r❛ r❡❝✉rs✐✈❛✳ ❊①✐st❡♠ ❢✉♥çõ❡s ❡♠ ❧✐♥❣✉❛❣❡♠ ❘ út❡✐s ♣❛r❛ s❡ ❝♦♥str✉✐r❡♠✱ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡✱ ❛s ♠❛tr✐③❡s ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ss❛s ❢✉♥çõ❡s ❜❛s❡✳ ❆ ❡s❝♦❧❤❛ ❞♦ ♥ú♠❡r♦ ❡ ♣♦s✐çã♦ ❞♦s ♥ós é ✉♠❛ q✉❡stã♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ s❡ ♦❜t❡r ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ❛ ❢✉♥çã♦ f✳ ❯♠❛ q✉❛♥t✐❞❛❞❡ ❣r❛♥❞❡ ❞❡ ♥ós ❧❡✈❛ ❛ ✉♠ s♦❜r❡❛❥✉st❡
❞♦s ❞❛❞♦s✱ ❡♥q✉❛♥t♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ ♣❡q✉❡♥❛ ❞❡❧❡s ❧❡✈❛ ❛ ✉♠ s✉❜❛❥✉st❡ ❞♦s ♠❡s♠♦s✳ P❛r❛ ❝♦♥t♦r♥❛r ❡ss❡ ♣r♦❜❧❡♠❛✱ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ é ❝♦♥s✐❞❡r❛r ✉♠ ♥ú♠❡r♦ r❡❧❛t✐✈❛♠❡♥t❡ ❣r❛♥❞❡ ❞❡ ♥ós✳ P❛r❛ ♣r❡✈❡♥✐r s♦❜r❡❛❥✉st❡✱ ✉♠❛ ♣❡♥❛❧✐❞❛❞❡✱ ❝♦♠ r❡❧❛çã♦ à ❢♦r♠❛ ❢✉♥✲ ❝✐♦♥❛❧ ❞❛ ❝✉r✈❛ ❛❥✉st❛❞❛✱ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ✭✈❡r ❙❡çã♦ ✸✳✺✮✳ ❖ ✉s♦ ❞❡ B✲s♣❧✐♥❡s é
❥✉st✐✜❝❛❞♦ ♣♦r s❡r❡♠ ❡❧❛s ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦ ❞❛ ❢✉♥çã♦ f✱ ❡st✐♠❛❞❛ ❞❡ ❢♦r♠❛ ♥ã♦
♣❛r❛♠étr✐❝❛✱ q✉❡ ❛♣r❡s❡♥t❛♠ ❡✜❝✐ê♥❝✐❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ♥❛s ❡st✐♠❛t✐✈❛s✱ ♥ã♦ ♣♦ss✉❡♠ ❡❢❡✐t♦s ❞❡ ❜♦r❞❛ ❡ ♣❡r♠✐t❡♠ ❡①t❡♥sõ❡s ❞✐r❡t❛s ❞♦s ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ♣❛r❛♠étr✐❝♦s✳
◆❡st❡ tr❛❜❛❧❤♦✱ ❛ ❡①♣❛♥sã♦ ❞❛ ❢✉♥çã♦ f é r❡❛❧✐③❛❞❛ ❝♦♠ B✲s♣❧✐♥❡s ❞❡ ❣r❛✉ ✸ ❝♦♠ l
♥ós ❡q✉✐❞✐st❛♥t❡s✱
f(tij) = α1B1(tij) +. . .+αlBl(tij) ✭✸✳✸✮
❡♠ q✉❡ α = (α1, . . . , αl)⊤ é ✉♠ ✈❡t♦r ❞❡s❝♦♥❤❡❝✐❞♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❞❛s B✲s♣❧✐♥❡s ♣❛r❛
s❡r❡♠ ❡st✐♠❛❞♦s✳
❈♦♠ r❡❧❛çã♦ ❛♦s ❡❢❡✐t♦s ❛❧❡❛tór✐♦s✱ ❡ss❡s sã♦ út❡✐s ♣❛r❛ ♠♦❞❡❧❛r ❛s ❝✉r✈❛s ❡s♣❡❝í✜❝❛s ♣❛r❛ ❝❛❞❛ ✐♥❞✐✈í❞✉♦✱ ❛ss✐♠ ❝♦♠♦ ❛ ❡str✉t✉r❛ ❞❡ ✈❛r✐â♥❝✐❛s✲❝♦✈❛r✐â♥❝✐❛s✳ ❊❧❡s ♣♦❞❡♠ s❡r ♣r❡❞✐t♦s ✉t✐❧✐③❛♥❞♦✲s❡ ♦ ♠ét♦❞♦ ❞❡ ❇❛②❡s ❡♠♣ír✐❝♦✳
❆ t❡r♠✐♥♦❧♦❣✐❛ ▼▼❙P◆▲ ✈❡♠ ❞♦ ❢❛t♦ ❞❡ q✉❡ ♦ ♠♦❞❡❧♦ ✭✸✳✶✮ ❝♦♥té♠ ✉♠❛ ♣❛rt❡ ♣❛r❛✲ ♠étr✐❝❛ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❢✉♥çã♦η q✉❡ ♣♦❞❡ s❡r ♥ã♦ ❧✐♥❡❛r ❝♦♠ r❡❧❛çã♦ ❛♦s ♣❛râ♠❡tr♦s
β ❡ ❛s ❝♦✈❛r✐á✈❡✐s✱ ✉♠❛ ♣❛rt❡ ♥ã♦ ♣❛r❛♠étr✐❝❛ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❢✉♥çã♦ f ❡ ♦s ❡❢❡✐t♦s
❛❧❡❛tór✐♦s ✐♥❝♦r♣♦r❛❞♦s ❞❡ ❢♦r♠❛ ❧✐♥❡❛r ❛♦ ♠♦❞❡❧♦✳ ❖ ♠♦❞❡❧♦ ✭✸✳✶✮ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ✈ár✐♦s ♠♦❞❡❧♦s ❛♠♣❧❛♠❡♥t❡ ❡st✉❞❛❞♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ❝♦♠♦
