Modelos de regressão sobre dados composicionais

▼♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ s♦❜r❡

❞❛❞♦s ❝♦♠♣♦s✐❝✐♦♥❛✐s

❆♥❞ré P✐❡rr♦ ❞❡ ❈❛♠❛r❣♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛

❛♦

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

♣❛r❛

♦❜t❡♥çã♦ ❞♦ tít✉❧♦

❞❡

▼❡str❡ ❡♠ ❈✐ê♥❝✐❛s

Pr♦❣r❛♠❛✿ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛r❝❡❧♦ ❞❡ ❙♦✉③❛ ▲❛✉r❡tt♦

▼♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ s♦❜r❡ ❞❛❞♦s ❝♦♠♣♦s✐❝✐♦♥❛✐s

❊st❛ ✈❡rsã♦ ❞❡✜♥✐t✐✈❛ ❞❛ ❞✐ss❡rt❛çã♦ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ r❡❛❧✐③❛❞❛ ♣♦r ❆♥❞ré P✐❡rr♦ ❞❡ ❈❛♠❛r❣♦ ❡♠ ✾✴✶✷✴✷✵✶✶✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

• Pr♦❢✳ ❉r✳ ▼❛r❝❡❧♦ ❞❡ ❙♦✉③❛ ▲❛✉r❡tt♦ ✭♦r✐❡♥t❛❞♦r✮ ✲ ❊❆❈❍✲❯❙P • Pr♦❢❛✳ ❉r❛✳ ❉❡❧❤✐ P❛✐✈❛ ❙❛❧✐♥❛s ✲ ❊❆❈❍✲❯❙P

❆❣r❛❞❡❝✐♠❡♥t♦s

• ❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛♦s ♠❡✉s ♣❛✐s ♣♦r t❡r❡♠ ♠❡ ♣r♦♣♦r❝✐♦♥❛❞♦ ✉♠ ❛♠❜✐❡♥t❡ ❢❛♠✐❧✐❛r

❢❛✈♦rá✈❡❧ ❛♦ ♠❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ✐♥t❡❧❡❝t✉❛❧✳

• ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ r❡❧❛①❛♠❡♥t♦

❞❛ ♠❡♥t❡✳

• ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❞q✉✐r✐❞♦ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ❡♠ ❡s♣❡✲

❝✐❛❧ ♠❡✉ ♦r✐❡♥t❛❞♦r ✭▼❛r❝❡❧♦ ▲❛✉r❡tt♦✮✱ ♦ ♣r♦❢❡ss♦r ❏ú❧✐♦ ▼✐❝❤❛❡❧ ❙t❡r♥ ❡ ♠❡✉s ❡①✲♦r✐❡♥t❛❞♦r❡s ❞❡ ✐♥✐❝✐❛çã♦ ❝✐❡♥tí✜❝❛✿ ❊❞✉❛r❞♦ ❞♦ ◆❛s❝✐♠❡♥t♦ ▼❛r❝♦s ❡ P❛✉❧♦ ❆❣♦③③✐♥✐ ▼❛rt✐♥✳

• ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❡s♣♦s❛✱ ❘♦❜❡rt❛✱ ❡ à ♠✐♥❤❛ ♣❡q✉❡♥❛ ✜❧❤❛✱ ▼❛r✐♥❛✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝❛r✐♥❤♦

r❡❝❡❜✐❞♦s ♥❡ss❡ ♣❡rí♦❞♦✳

• ❆❣r❛❞❞❡ç♦ ❛ ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦✳

❘❡s✉♠♦

❉❛❞♦s ❝♦♠♣♦s✐❝✐♦♥❛✐s sã♦ ❝♦♥st✐t✉í❞♦s ♣♦r ✈❡t♦r❡s ❝✉❥❛s ❝♦♠♣♦♥❡♥t❡s r❡♣r❡s❡♥t❛♠ ❛s ♣r♦♣♦rçõ❡s ❞❡ ❛❧❣✉♠ ♠♦♥t❛♥t❡✱ ✐st♦ é✿ ✈❡t♦r❡s ❝♦♠ ❡♥tr❛❞❛s ♣♦s✐t✐✈❛s ❝✉❥❛ s♦♠❛ é ✐❣✉❛❧ ❛ ✶✳ ❊♠ ❞✐✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡st✐♠❛r ❛s ♣❛rt❡s y1, y2, . . . , yD ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦s s❡✲

t♦r❡s SE1, SE2, . . . , SED✱ ❞❡ ✉♠❛ ❝❡rt❛ q✉❛♥t✐❞❛❞❡ Q✱ ❛♣❛r❡❝❡ ❝♦♠ ❢r❡q✉ê♥❝✐❛✳ ❆s ♣♦r❝❡♥t❛❣❡♥s

y1, y2, . . . , yD ❞❡ ✐♥t❡♥çã♦ ❞❡ ✈♦t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦s ❝❛♥❞✐❞❛t♦s Ca1, Ca2, . . . , CaD ❡♠ ❡❧❡✐çõ❡s

❣♦✈❡r♥❛♠❡♥t❛✐s ♦✉ ❛s ♣❛r❝❡❧❛s ❞❡ ♠❡r❝❛❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ✐♥❞✉str✐❛s ❝♦♥❝♦rr❡♥t❡s ❢♦r♠❛♠ ❡①❡♠✲ ♣❧♦s tí♣✐❝♦s✳ ◆❛t✉r❛❧♠❡♥t❡✱ é ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❛♥❛❧✐s❛r ❝♦♠♦ ✈❛r✐❛♠ t❛✐s ♣r♦♣♦rçõ❡s ❡♠ ❢✉♥çã♦ ❞❡ ❝❡rt❛s ♠✉❞❛♥ç❛s ❝♦♥t❡①t✉❛✐s✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❧♦❝❛❧✐③❛çã♦ ❣❡♦❣rá✜❝❛ ♦✉ ♦ t❡♠♣♦✳ ❊♠ q✉❛❧q✉❡r ❛♠✲ ❜✐❡♥t❡ ❝♦♠♣❡t✐t✐✈♦✱ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❡ss❡ ❝♦♠♣♦rt❛♠❡♥t♦ sã♦ ❞❡ ❣r❛♥❞❡ ❛✉①í❧✐♦ ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❛s ❡str❛té❣✐❛s ❞♦s ❝♦♥❝♦rr❡♥t❡s✳

◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ❡ ❞✐s❝✉t✐♠♦s ❛❧❣✉♠❛s ❛❜♦r❞❛❣❡♥s ♣r♦♣♦st❛s ♥❛ ❧✐t❡r❛t✉r❛ ♣❛r❛ r❡❣r❡ssã♦ s♦❜r❡ ❞❛❞♦s ❝♦♠♣♦s✐❝✐♦♥❛✐s✱ ❛ss✐♠ ❝♦♠♦ ❛❧❣✉♥s ♠ét♦❞♦s ❞❡ s❡❧❡çã♦ ❞❡ ♠♦❞❡❧♦s ❜❛s❡❛❞♦s ❡♠ ✐♥❢❡rê♥❝✐❛ ❜❛②❡s✐❛♥❛✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ▼♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦✱ ❉❛❞♦s ❝♦♠♣♦s✐❝✐♦♥❛✐s✱ ❙❡❧❡çã♦ ❞❡ ♠♦❞❡❧♦s✱ ❇■❈✱ ❋❇❙❚✳

❆❜str❛❝t

❈♦♠♣♦s✐t✐♦♥❛❧ ❞❛t❛ ❝♦♥s✐st ♦❢ ✈❡❝t♦rs ✇❤♦s❡ ❝♦♠♣♦♥❡♥ts ❛r❡ t❤❡ ♣r♦♣♦rt✐♦♥s ♦❢ s♦♠❡ ✇❤♦❧❡✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❡st✐♠❛t✐♥❣ t❤❡ ♣♦rt✐♦♥s y1, y2, . . . , yD ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ♣✐❡❝❡s SE1, SE2, . . . , SED

♦❢ s♦♠❡ ✇❤♦❧❡ Q✐s ♦❢t❡♥ r❡q✉✐r❡❞ ✐♥ s❡✈❡r❛❧ ❞♦♠❛✐♥s ♦❢ ❦♥♦✇❧❡❞❣❡✳ ❚❤❡ ♣❡r❝❡♥t❛❣❡sy1, y2, . . . , yD

♦❢ ✈♦t❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝♦♠♣❡t✐t♦rs Ca1, Ca2, . . . , CaD ✐♥ ❣♦✈❡r♥♠❡♥t❛❧ ❡❧❡❝t✐♦♥s ♦r ♠❛r❦❡t s❤❛r❡ ♣r♦❜❧❡♠s ❛r❡ t②♣✐❝❛❧ ❡①❛♠♣❧❡s✳ ❖❢ ❝♦✉rs❡✱ ✐t ✐s ♦❢ ❣r❡❛t ✐♥t❡r❡st t♦ st✉❞② t❤❡ ❜❡❤❛✈✐♦r ♦❢ s✉❝❤ ♣r♦♣♦rt✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ s♦♠❡ ❝♦♥t❡①t✉❛❧ tr❛♥s✐t✐♦♥s✳ ■♥ ❛♥② ❝♦♠♣❡t✐t✐✈❡ ❡♥✈✐r♦♥♠❡t✱ ❛❞❞✐t✐♦♥❛❧ ✐♥❢♦r♠❛t✐♦♥ ♦❢ s✉❝❤ ❜❡❤❛✈✐♦r ❝❛♥ ❜❡ ✈❡r② ❤❡❧♣❢✉❧ ❢♦r t❤❡ str❛t❡❣✐sts t♦ ♠❛❦❡ ♣r♦♣❡r ❞❡❝✐s✐♦♥s✳

■♥ t❤✐s ✇♦r❦ ✇❡ ♣r❡s❡♥t ❛♥❞ ❞✐s❝✉ss s♦♠❡ ❛♣♣r♦❛❝❤❡s ♣r♦♣♦s❡❞ ❜② ❞✐✛❡r❡♥t ❛✉t❤♦rs ❢♦r ❝♦♠♣♦✲ s✐t✐♦♥❛❧ ❞❛t❛ r❡❣r❡ss✐♦♥ ❛s ✇❡❧❧ ❛s s♦♠❡ ♠♦❞❡❧ s❡❧❡❝t✐♦♥ ♠❡t❤♦❞s ❜❛s❡❞ ♦♥ ❜❛②❡s✐❛♥ ✐♥❢❡r❡♥❝❡✳

❑❡②✇♦r❞s✿ ❘❡❣r❡ss✐♦♥ ♠♦❞❡❧s✱ ❈♦♠♣♦s✐t✐♦♥❛❧ ❞❛t❛✱ ▼♦❞❡❧ s❡❧❡❝t✐♦♥✱ ❇■❈✱ ❋❇❙❚✳

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s ✐①

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ①✐

▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐✐✐

▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✈

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✷ ▼♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ❞❡ ❉✐r✐❝❤❧❡t ✺

✷✳✶ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❘❡❣r❡ssã♦ ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ ❊st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸✳✶ ❆❧❣♦r✐t♠♦ ❞❡ ❍✐❥❛③✐✲❏❡r♥✐❣❛♥ ♣❛r❛ s❡❧❡çã♦ ❞❡ ✈❛❧♦r❡s ✐♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸✳✷ ◆♦✈♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ s❡❧❡çã♦ ❞❡ ✈❛❧♦r❡s ✐♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸✳✸ ❘❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✹ ❊❧✐♠✐♥❛çã♦ ❞❛s r❡str✐çõ❡s s♦❜r❡ ♦ ❡s♣❛ç♦ ♣❛r❛♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✸ ❖✉tr♦s ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ✶✸

✸✳✶ ❚r❛♥s❢♦r♠❛çõ❡s SD 7−→RD−1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✸✳✶✳✶ ❆❜r❛♥❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✷ ❈♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✸ ❖✉tr❛s tr❛♥s❢♦r♠❛çõ❡s ♥♦ ❙✐♠♣❧❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸✳✶ ❖❜s❡r✈❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✹ ▼ét♦❞♦ ❞♦s ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♥♦ ❙✐♠♣❧❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✹ ❆♣❧✐❝❛çõ❡s ✷✺

✹✳✶ ▲❛❣♦ ➪rt✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✶✳✶ ▼♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✶✳✷ ▼♦❞❡❧♦ ❞❡ ▼❡❧♦✱ ❱❛s❝♦♥❝❡❧❧♦s ❡ ▲❡♠♦♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✶✳✸ ▼♦❞❡❧♦ ❧✐♥❡❛r ▲♦❣❛r✐t♠♦ ❞❛ ❘❛③ã♦✴ ▲♦❣✲❡s❢ér✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✶✳✹ ▼♦❞❡❧♦ ❧✐♥❡❛r ❚❛♥❣❡♥t❡✲❡s❢ér✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✷ ❉❡s♣❡s❛s ❞♦♠ést✐❝❛s ✭❜✐✈❛r✐❛❞♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✷✳✶ ▼♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✈✐✐✐ ❙❯▼➪❘■❖

✹✳✷✳✷ ▼♦❞❡❧♦ ❞❡ ▼❡❧♦✱ ❱❛s❝♦♥❝❡❧❧♦s ❡ ▲❡♠♦♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✷✳✸ ▼♦❞❡❧♦ ❧✐♥❡❛rT gRatio✭✸✳✷✷✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✹✳✸ ❈❛s❛♠❡♥t♦s ♣♦r ❢❛✐①❛ ❡tár✐❛ ✭❜✐✈❛r✐❛❞♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✸✳✶ ▼♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✸✳✷ ▼♦❞❡❧♦ ❞❡ ▼❡❧♦✱ ❱❛s❝♦♥❝❡❧❧♦s ❡ ▲❡♠♦♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✸✳✸ ▼♦❞❡❧♦ ❧✐♥❛r ▲♦❣✲❡s❢ér✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✹ ❈♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ♦ ❊s♣❛ç♦ P❛r❛♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✹✳✶ ▼♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t✴ ▼í♥✐♠♦s q✉❛❞r❛❞♦s ✭❧✐♥❡❛r✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✺ ❙❡❧❡çã♦ ❞❡ ♠♦❞❡❧♦s ✸✾

✺✳✶ ❇■❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✺✳✶✳✶ ❯♠❛ ♣r♦♣♦st❛ ❞❡ ❢♦r♠❛❧✐③❛çã♦ ❞♦BIC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✺✳✷ ❋❇❙❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✷✳✶ ❉❡✜♥✐çã♦ ❢♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✸ ❚❡st❡ ❞❛ ❘❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭▲❘✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✹ ❋❇❙❚ ✈s ▲❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✺ ❆♣❧✐❝❛çõ❡s ❞♦ ❇■❈ ❛♦s ♠♦❞❡❧♦s ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✺✳✶ ▲❛❣♦ ➪rt✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✺✳✷ ❉❡s♣❡s❛s ❞♦♠ést✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✻ ❈♦♥❝❧✉sõ❡s ✺✶

❆ ❈á❧❝✉❧♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ r❡❣r❡ssã♦ ❧✐♥❡❛r ♣♦r ♠í♥✐♠♦s q✉❛❞r❛❞♦s ✺✸

❇ ❇❛s❡ ❞❡ ❞❛❞♦s ✺✺

❇✳✶ ▲❛❣♦ ➪rt✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ❇✳✷ ❉❡s♣❡s❛s ❉♦♠ést✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ❇✳✸ ❈❛s❛♠❡♥t♦s ♣♦r ❢❛✐①❛✲❡tár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✾

▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s

❇■❈ ❈r✐tér✐♦ ❞❡ ■♥❢♦r♠❛çã♦ ❇❛②❡s✐❛♥♦ ✭❇❛②❡s✐❛♥ ✐♥❢♦r♠❛t✐♦♥ ❝r✐t❡r✐♦♥✮✳

❋❇❙❚ ❚❡st❡ ❞❡ s✐❣♥✐✜❝â♥❝✐❛ ✐♥t❡❣r❛❧♠❡♥t❡ ❜❛②❡s✐❛♥♦ ✭❋✉❧❧ ❇❛②❡s✐❛♥ ❙✐❣♥✐✜❝❛♥t ❚❡st✮✳ ▲❘ ❚❡st❡ ❞❛ ❘❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭▲✐❦❡❧✐❤♦♦❞✲r❛t✐♦ t❡st✮✳

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s

Rq ❊s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ q✲❞✐♠❡♥s✐♦♥❛❧✳

SD ❙✐♠♣❧❡① ✭❡❧❡♠❡♥t♦s ❝♦♠D ❝♦♠♣♦♥❡♥t❡s✮✳

d d=D−1✳

z• ❈♦✈❛r✐á✈❡❧✳

x• ❈♦✈❛r✐á✈❡❧ ❡st❡♥❞✐❞❛✳

X = (xij) ▼❛tr✐③ ❞❛ ❝♦✈❛r✐á✈❡❧ ❡st❡♥❞✐❞❛ ✭❞❡✜♥✐❞❛ ♥❛ ✐♥tr♦❞✉çã♦✮✳ Nd ❉✐str✐❜✉✐çã♦ ♥♦r♠❛❧d✲✈❛r✐❛❞❛✳

Γ ❋✉♥çã♦ ●❛♠❛✳

Dir ❉✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t✳ E ❱❛❧♦r ❡s♣❡r❛❞♦✴ ❊s♣❡r❛♥ç❛✳

β ▼❛tr✐③ ❞❡ ♣❛râ♠❡tr♦s✳

P rev ❋✉♥çã♦ ❞❡ ♣r❡✈✐sã♦ ✭❞❡✜♥✐❞❛ ♥❛ ✐♥tr♦❞✉çã♦✮✳ ψ ❋✉♥çã♦ ❞❡ ♣❛râ♠❡tr♦s ✭❞❡✜♥✐❞❛ ♥❛ ✐♥tr♦❞✉çã♦✮✳ L ❋✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳

δ ◆♦t❛çã♦ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✳

∇ ●r❛❞✐❡♥t❡✳

∆ ❉✐stâ♥❝✐❛ ❞❡ ❆✐t❝❤✐s♦♥✳

|| || ◆♦r♠❛ ❊✉❝❧✐❞✐❛♥❛✳

Md ❚r❛♥s❢♦r♠❛çã♦ ▲♦❣íst✐❝❛ ♠✉❧t✐♣❧✐❝❛t✐✈❛✳ Hd ❚r❛♥s❢♦r♠❛çã♦ ▲♦❣íst✐❝❛ ❤í❜r✐❞❛✳

alr ❚r❛♥s❢♦r♠❛çã♦ ▲♦❣❛r✐t♠♦ ❞❛ r❛③ã♦ ❛ss✐♠étr✐❝❛✳ clr ❚r❛♥s❢♦r♠❛çã♦ ▲♦❣❛r✐t♠♦ ❞❛ r❛③ã♦ ❝❡♥tr❛❞❛✳ Mmod ▼❛tr✐③ ❞❡ ♠♦❞❡❧♦✳

Θ ❊s♣❛ç♦ ♣❛r❛♠étr✐❝♦✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❈♦♠♣❛r❛t✐✈♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❞❡ s❡❧❡çã♦ ❞❡ ✈❛❧♦r❡s ✐♥✐❝✐❛✐s ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✶ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ tr❛♥s❢♦r♠❛çã♦alr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸✳✷ P❛r❛♠❡tr✐③❛çã♦ ❞♦ ✸✲❙✐♠♣❧❡① ♣♦r ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✶ ▼♦❞❡❧♦s ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✷ ▼♦❞❡❧♦s ❞❡ ▼❡❧♦✱ ❱❛s❝♦♥❝❡❧❧♦s ❡ ▲❡♠♦♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✸ ▼♦❞❡❧♦ ❧✐♥❡❛r ▲♦❣✲❡s❢ér✐❝♦ ✈s ▼♦❞❡❧♦ ❧✐♥❡❛r ▲♦❣❛r✐t♠♦ ❞❛ r❛③ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✹ ▼♦❞❡❧♦s ❚❛♥❣❡♥t❡✲❡s❢ér✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✺ ▼♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✻ ▼♦❞❡❧♦ ❞❡ ▼❡❧♦✱ ❱❛s❝♦♥❝❡❧❧♦s ❡ ▲❡♠♦♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✼ ▼♦❞❡❧♦ ❧✐♥❡❛r T gRatio ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✹✳✽ ▼♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✾ ▼♦❞❡❧♦ ❞❡ ▼❡❧♦✱ ❱❛s❝♦♥❝❡❧❧♦s ❡ ▲❡♠♦♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✶✵ ▼♦❞❡❧♦ ▲♦❣✲❡s❢ér✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✶✶ ❙❡♥s✐❜✐❧✐❞❛❞❡ ❞❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞❡ ▼❉◗ s♦❜r❡ ❞✐❧❛t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✶✷ ▼♦❞❡❧♦s q✉❛❞rát✐❝♦s r❡str✐t♦s à ❝❛s❝❛ ❡s❢ér✐❝❛ ✉♥✐tár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✶ ✭❛✮ ❊rr♦ ❚✐♣♦ ■✱ ✭❜✮ ❊rr♦ ❚✐♣♦ ■■✱ ✭❝✮ ❊rr♦ ♠é❞✐♦ ❡ ✭❞✮ ❊rr♦ ❚✐♣♦ ■■ ❡♠♣ír✐❝♦✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✷ ❙✉❜♠♦❞❡❧♦s ❞❡ ▼❉◗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺✳✸ ▼❡❧❤♦r ♠♦❞❡❧♦ s❡❣✉♥❞♦ ❙❝❤✇❛r③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✸✳✶ ❚r❛♥s❢♦r♠❛çõ❡s ❧♦❣íst✐❝❛s ❡❧❡♠❡♥t❛r❡s y ∈ SD 7−→ v ∈ Rd ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✹✳✶ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✷ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✸ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✹ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✺ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✻ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✼ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✽ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✾ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✶✵ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✶✶ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ▼❉◗ ♣❛r❛ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✶✷ P❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✴ ▼í♥✐♠♦s q✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✶ ❆❧❣✉♥s s✉❜♠♦❞❡❧♦s ❞❡ ▼❉◗ ♦r❞❡♥❛❞♦s ❞♦ ♠❡♥♦r ♣❛r❛ ♦ ♠❛✐♦r ❇■❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ❇✳✶ ❈♦♠♣♦s✐çã♦ ❞♦ s♦❧♦ ❞♦ ▲❛❣♦ ➪rt✐❝♦ ❡♠ ❢✉♥çã♦ ❞❛ ♣r♦❢✉♥❞✐❞❛❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ❇✳✷ ❉❡s♣❡s❛s ❞♦♠ést✐❝❛s✿ ❚ = ❚♦t❛❧ ❣❛st♦ ✭❡♠ ❍❑✩✮❀ ❆ = ❆❧✐♠❡♥t❛çã♦❀ ❖ = ❖✉tr♦s

❇❡♥s❀S ❂ ❙❡r✈✐ç♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

❇✳✸ P♦r❝❡♥t❛❣❡♥ ❞❡ ❝❛s❛♠❡♥t♦s ♣♦r ❢❛✐①❛ ❡tár✐❛ ❡ s❡①♦✳ ❋♦♥t❡✿ ■❇●❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❊♠ ❞✐✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡st✐♠❛r ❛s ♣❛rt❡s y1, y2, . . . , yD ❝♦rr❡s♣♦♥✲

❞❡♥t❡s ❛♦s s❡t♦r❡s SE1, SE2, . . . , SED✱ ❞❡ ✉♠❛ ❝❡rt❛ q✉❛♥t✐❞❛❞❡ Q✱ ❛♣❛r❡❝❡ ❝♦♠ ❢r❡q✉ê♥❝✐❛✳ ❆s ♣♦r❝❡♥t❛❣❡♥sy1, y2, . . . , yD ❞❡ ✐♥t❡♥çã♦ ❞❡ ✈♦t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦s ❝❛♥❞✐❞❛t♦sCa1, Ca2, . . . , CaD

❡♠ ❡❧❡✐çõ❡s ❣♦✈❡r♥❛♠❡♥t❛✐s ♦✉ ❛s ♣❛r❝❡❧❛s ❞❡ ♠❡r❝❛❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ✐♥❞✉str✐❛s ❝♦♥❝♦rr❡♥t❡s ❢♦r♠❛♠ ❡①❡♠♣❧♦s tí♣✐❝♦s✳ ◆❛t✉r❛❧♠❡♥t❡✱ é ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❛♥❛❧✐s❛r ❝♦♠♦ ✈❛r✐❛♠ t❛✐s ♣r♦♣♦rçõ❡s ❡♠ ❢✉♥çã♦ ❞❡ ❝❡rt❛s ♠✉❞❛♥ç❛s ❝♦♥t❡①t✉❛✐s✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❧♦❝❛❧✐③❛çã♦ ❣❡♦❣rá✜❝❛ ♦✉ ♦ t❡♠♣♦✳ ❊♠ q✉❛❧q✉❡r ❛♠❜✐❡♥t❡ ❝♦♠♣❡t✐t✐✈♦✱ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❡ss❡ ❝♦♠♣♦rt❛♠❡♥t♦ sã♦ ❞❡ ❣r❛♥❞❡ ❛✉①í❧✐♦ ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❛s ❡str❛té❣✐❛s ❞♦s ❝♦♠♣❡t✐❞♦r❡s✳

❉❡✜♥✐çã♦✿ ❈❤❛♠❛♠♦s ❞❡ ✉♠ ❞❛❞♦ ❝♦♠♣♦s✐❝✐♦♥❛❧ ❛ ❝❛❞❛ D✲✉♣❧❛ (y1, y2, . . . , yD) ❞❡ ♥ú♠❡r♦s

♣♦s✐t✐✈♦s t❛✐s q✉❡ PD

j=1

yj = 1✳ ❖ ❝♦♥❥✉♥t♦

SD ={(y1, y2, . . . , yD)∈RD : D P j=1

yj = 1, yj >0, j = 1,2, . . . , D}

é ❞❡♥♦♠✐♥❛❞♦ ♦ D−1✲❙✐♠♣❧❡①✳

❖❜s❡r✈❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s{(z1•, y1•),(z2•, y2•), . . .(zn•, yn•)}✱ ♦♥❞❡yj•∈SD❡zj•∈Ω⊆R C_✱ j= 1,2, . . . , n✱ ❞❡s❡❥❛✲s❡ ♦❜t❡r ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ♣❛r❛♠étr✐❝♦y• ∼z•✳ ❖ s✉❝❡ss♦ ❞❡ss❛ t❛r❡❢❛

❞❡♣❡♥❞❡✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ❞❡ ❞♦✐s ❢❛t♦r❡s✿ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❢❛♠í❧✐❛ ♣❛r❛♠étr✐❝❛ ❞❡ ♠♦❞❡❧♦s ❡ ❞♦ ♠ét♦❞♦ ❞❡ ✐♥❢❡rê♥❝✐❛ ✉t✐❧✐③❛❞♦✳ ❯♠ ♠ét♦❞♦ ❜❛st❛♥t❡ ✉t✐❧✐③❛❞♦ ❝♦♥s✐st❡ ❡♠✿

✶✳ ❙❡❧❡❝✐♦♥❛r ✉♠❛ ❢❛♠í❧✐❛ ♣❛r❛♠étr✐❝❛ ❞❡ ♠♦❞❡❧♦s ♣r♦❜❛❜✐❧íst✐❝♦s G={g(y•|ψ) :ψ ∈Ψ} ✭❛q✉✐

❞❡s❝r✐t♦s ♣♦r s✉❛s ❢✉♥çõ❡s ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✮✱ ❞❡✜♥✐❞♦s ❡♠ SD ✳

✷✳ ❙✉♣♦r q✉❡✱ ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ❞❡z•✱y•|z• é ❞✐str✐❜✉í❞❛ s❡❣✉♥❞♦G❝♦♠ ♣❛râ♠❡tr♦ψ(z•, β)✱ ❝✉❥❛

❞❡♣❡♥❞❡♥❝✐❛ ❝♦♠z• é ❝❛❧✐❜r❛❞❛ ♣♦r ✉♠ ✈❡t♦r ✭♦✉ ♠❛tr✐③✮β✳

✸✳ ❆❥✉st❛r ♦ ✈❡t♦r✴♠❛tr✐③ ❞❡ ♣❛râ♠❡tr♦s β✳

✹✳ ❊st✐♠❛r ♦ ✈❛❧♦r ❞❡y•❡♠z•✱P rev(z•)✱ ❝♦♠♦ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❛ ❞✐str✐❜✉✐çã♦G❝♦♠ ♣❛râ♠❡tr♦

ψ(z•, β)✳

✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✵

❆ss✉♠✐♥❞♦ q✉❡ ❛s ✈❛r✐á✈❡✐syj•|zj•, j = 1,2, . . . , nsã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♣♦❞❡♠♦s ❢♦r♠❛r ❛ ❢✉♥çã♦

❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✈❡t♦r✴♠❛tr✐③ ❞❡ ♣❛râ♠❡tr♦s β ♣❛r❛ ❡st✐♠❛r s❡✉s ✈❛❧♦r❡s ✈✐❛

♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♦✉ ♦✉tr♦ ♠ét♦❞♦ ❞❡ ✐♥❢❡rê♥❝✐❛✳

◆❡st❡ tr❛❜❛❧❤♦ ✐♥✈❡st✐❣❛♠♦s ❡ ❡st❡♥❞❡♠♦s ❛❧❣✉♠❛s ❛❜♦r❞❛❣❡♥s ♣r♦♣♦st❛s ♥❛ ❧✐t❡r❛t✉r❛ ❜❛s❡❛❞❛s ♥♦ ♠ét♦❞♦ ❞❡s❝r✐t♦ ❛❝✐♠❛✳ ❖ t❡①t♦ ❡stá ❛ss✐♠ ♦r❣❛♥✐③❛❞♦✿

◆♦ s❡❣✉♥❞♦ ❈❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❞♦✐s ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ❞❡s❝r✐t♦s ❡♠ ❈❛♠♣❜❡❧❧ ❡ ▼♦s✐♠❛♥♥ ✭✶✾✽✼✮ ❡ ▼❡❧♦ ❡t ❛❧✳ ✭✷✵✵✾✮✱ ❛♠❜♦s ❜❛s❡❛❞♦s ♥❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ✭❛ ❞✐str✐❜✉✐çã♦ ♠❛✐s ❝♦✲ ♥❤❡❝✐❞❛ ❞❡✜♥✐❞❛ ♥♦ ❙✐♠♣❧❡①✱ P❡r❡✐r❛ ❡ ❙t❡r♥ ✭✷✵✵✽✮✮✳ ❆ ♥♦ss❛ ♣r✐♥❝✐♣❛❧ ❝♦♥tr✐❜✉✐çã♦ r❡❢❡r❡♥t❡ ❛♦ ❝♦♥t❡ú❞♦ ❞❡ss❡ ❈❛♣ít✉❧♦ é ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ♠❛✐s ❡✜❝✐❡♥t❡ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ✈✐á✈❡❧ ♣❛r❛ s❡r ❢♦r♥❡❝✐❞♦ ❛♦s ❛❧❣♦r✐t♠♦s ❞❡ ♠❛①✐♠✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭❙❡çã♦ ✷✳✸✳✷✮✳

◆♦ t❡r❝❡✐r♦ ❈❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ♦s ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ❞❡s❝r✐t♦s ♥♦s tr❛❜❛❧❤♦s ❞❡ ❆✐t❝❤✐♥s♦♥ ✭✶✾✽✻✮ ❡ ❲❛♥❣ ❡t ❛❧✳ ✭✷✵✵✼✮✳ ❖ ♣r✐♠❡✐r♦ ♣r♦♣ô❡ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ à ✉t✐❧✐③❛çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ♥♦ ❙✐♠♣❧❡①✱ tr❛♥s❢♦r♠❛♥❞♦ ♦s ❞❛❞♦s ❝♦♠♣♦s✐❝✐♦♥❛✐s ♣❛r❛ ♦ ❊s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦✱ ♠♦❞❡❧❛♥❞♦ ♦s ❞❛❞♦s tr❛♥s❢♦r♠❛❞♦s ❡ r❡t♦r♥❛♥❞♦ ♣❛r❛ ♦ ❙✐♠♣❧❡①✳ ❖ s❡❣✉♥❞♦ ❞❡s❝r❡✈❡ ♦s ❞❛❞♦s ❝♦♠♣♦s✐❝✐♦♥❛✐s ❡♠ t❡r♠♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ❣❡♥❡r❛❧✐③❛❞❛s ❡ ♠♦❞❡❧❛ ♦s â♥❣✉❧♦s ❞✐r❡t❛♠❡♥t❡ ❡♠ ❢✉♥çã♦ ❞❛ ❝♦✈❛r✐á✈❡❧

X✳ ❆ ♥♦ss❛ ♣r✐♥❝✐♣❛❧ ❝♦♥tr✐❜✉✐çã♦ r❡❢❡r❡♥t❡ ❛♦ ❝♦♥t❡ú❞♦ ❞❡ss❡ ❈❛♣ít✉❧♦ ❝♦♥s✐st❡ ♥❛ ❝♦♠♣❧❡t❛ ❝❛✲

r❛❝t❡r✐③❛çã♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s ✭❙❡çã♦ ✸✳✶✳✶✮ ❡ ♥❛ ❡❧❛❜♦r❛çã♦ ❞❡ ♥♦✈❛s tr❛♥s❢♦r♠❛çõ❡s ❡♥tr❡ ♦ ❙✐♠♣❧❡① ❡ ♦ ❊s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ ✭❙❡çã♦ ✸✳✸✮✳

◆♦ q✉❛rt♦ ❈❛♣ít✉❧♦✱ ❛♣❧✐❝❛♠♦s ♦s ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ❞❡s❝r✐t♦s ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸ ❛ três ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ✭❞✐s♣♦♥í✈❡✐s ♣❛r❛ ❝♦♥s✉❧t❛ ♥♦ ❆♣ê♥❞✐❝❡ ❇✮✿

✶✳ ❖ ♣r✐♠❡✐r♦ é r❡❢❡r❡♥t❡ ❛ ✸✾ ❝♦♠♣♦s✐çõ❡s ❞♦ s♦❧♦ ❞♦ ▲❛❣♦ ➪rt✐❝♦ ✭❛r❡✐❛✱ ❧♦❞♦ ❡ ❛r❣✐❧❛ ❡♠ ❢✉♥çã♦ ❞❛ ♣r♦❢✉♥❞✐❞❛❞❡✮ ❛♣r❡s❡♥t❛❞♦s ♣♦r ❈♦❛❦❧❡② ❡ ❘✉st ❈♦❛❦❧❡② ❡ ❘✉st ✭✶✾✻✽✮ ❡ ❛❞❛♣t❛❞♦s ♣♦r ❆✐t❝❤✐♥s♦♥ ✭✶✾✽✻✮✳ ❆ ❛♥á❧✐s❡ ❞♦ s♦❧♦✱ ❡♠ ❣❡r❛❧✱ é ✉♠ ❢❛t♦r ❞✐s❝r✐♠✐♥❛♥t❡ ♥❛ ár❡❛ ❞❛ ❝♦♥str✉çã♦ ❝✐✈✐❧✳

✷✳ ❖ s❡❣✉♥❞♦ é r❡❢❡r❡♥t❡ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❡♥❝♦♥tr❛❞♦ ❡♠ ❆✐t❝❤✐♥s♦♥ ✭✶✾✽✻✮✱ ❝♦♥st✐t✉í❞♦ ♣❡❧❛s ❞❡s♣❡s❛s ❞♦♠ést✐❝❛s ✭s❡♣❛r❛❞❛s ❡♠ ▼❛♥✉t❡♥çã♦ ❞♦♠✐❝✐❧✐❛r✱ ❆❧✐♠❡♥t❛çã♦✱ ❖✉tr♦s ❜❡♥s ❡ ❙❡r✈✐ç♦s✳✮ ❞❡ ✷✵ ❤♦♠❡♥s ❡ ✷✵ ♠✉❧❤❡r❡s✱ ❡♠ ❢✉♥çã♦ ❞♦ t♦t❛❧ ❣❛st♦✳

✸✳ ❖ t❡r❝❡✐r♦ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❢♦r♥❡❝✐❞♦ ♣❡❧♦ ■❇●❊ ✭■♥st✐t✉t♦ ❇r❛s✐❧❡✐r♦ ❞❡ ●❡♦❣r❛✜❛ ❡ ❊st❛tíst✐❝❛✮ ❡ ❞✐s♣♦♥í✈❡❧ ❡♠ ❤tt♣✿✴✴s❡r✐❡s❡st❛t✐st✐❝❛s✳✐❜❣❡✳❣♦✈✳❜r✴ r❡❧❛t✐✈♦ à ✈❛r✐✲ ❛çã♦ ❞❛s ♣♦r❝❡♥t❛❣❡♥s ❞❡ ❝❛s❛♠❡♥t♦s ♣♦r ❢❛✐①❛ ❡tár✐❛ ❡ s❡①♦ ♥♦ ♣❡rí♦❞♦ ✶✾✽✹✲✷✵✵✷✳

❆ ❛♥á❧✐s❡ ❞❡ss❡s ♦✉tr♦s ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s é ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ ❛s ❝✐ê♥❝✐❛s s♦❝✐❛✐s✳ ❖ s❡❣✉♥❞♦ ❝♦♥st✐t✉✐ ✉♠❛ tí♣✐❝❛ ♣❡sq✉✐s❛ ♠❡r❝❛❞♦❧ó❣✐❝❛ ❡ ♦ ❡st✉❞♦ ❞♦ t❡r❝❡✐r♦ ♣♦❞❡ ❛✉①✐❧✐❛r ♥❛ ❞❡s❝r✐çã♦ ❞❡ ♦✉tr♦s ❢❡♥ô♠❡♥♦s ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞♦s✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❞✐ss❡♠✐♥❛çã♦ ❞❡ ❞♦❡♥ç❛s s❡①✉❛❧♠❡♥t❡ tr❛♥s♠✐ssí✈❡✐s✳

✶✳✶ P❘❊▲■▼■◆❆❘❊❙ ✸

✶✳✶ Pr❡❧✐♠✐♥❛r❡s

❆♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ ❛s ❝♦♠♣♦s✐çõ❡s ♦❜s❡r✈❛❞❛s s❡rã♦ ❞❡♥♦t❛❞❛s ♣♦ryj• = (yj1, yj2, . . . , yjn), j=

1,2, . . . , n ❡ Y = (yij) ❞❡♥♦t❛ ❛ ♠❛tr✐③ ❞❛s ❝♦♠♣♦s✐çõ❡s✳ ❯♠ ❞❛❞♦ ❝♦♠♣♦s✐❝✐♦♥❛❧ ❣❡♥ér✐❝♦ ✭♦✉

✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❛ss✉♠✐♥❞♦ ✈❛❧♦r❡s ♥♦ D−1✲❙✐♠♣❧❡①✮ s❡rá ❞❡♥♦t❛❞♦ ♣♦ry ♦✉ y•✳

❙❡❣✉✐♥❞♦ ❛ ♥♦t❛çã♦ ♣❛❞rã♦ ❡♠ ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦✱ ❡♠ ❧✉❣❛r ❞❛ ❝♦✈❛r✐á✈❡❧ ♦r✐❣✐♥❛❧ z• ∈ Ω✱

tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ❛ ✈❛r✐á✈❡❧ ❡st❡♥❞✐❞❛ x• = (f1(z•), f2(z•), . . . , fk(z•))∈Rk✱ ♦♥❞❡f1, f2, . . . , fk:

Ω−→ Rsã♦ tr❛♥s❢♦r♠❛çõ❡s ❛♣r♦♣r✐❛❞❛s✳ P❛r❛ ❣❛r❛♥t✐r ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ❛❧❣✉♥s ❡st✐♠❛❞♦r❡s ót✐♠♦s

♣❛r❛ ❛ ♠❛tr✐③β✱ ✐r❡♠♦s ❛ss✉♠✐r q✉❡ ❛ ♠❛tr✐③

X= (xij) =    

x1•

x2•

✳✳ ✳

xn•

   =

   

f1(z1•) f2(z1•) . . . fk(z1•)

f1(z2•) f2(z2•) . . . fk(z2•)

✳✳

✳ ✳✳✳ ✳✳✳

f1(zn•) f2(zn•) . . . fk(zn•)

  

 ✭✶✳✶✮

♣♦ss✉✐ ♣♦st♦ k≤n✭✈❡r ❛♣ê♥❞✐❝❡ ❆✮✳

P❛r❛ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦✱ ✈❡t♦r❡s ❞♦ ❊s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ Rq✱ ♣❛r❛ q ❣❡♥ér✐❝♦✱ s❡rã♦ ✐♥t❡r♣r❡t❛✲

❞♦s✱ ❡✈❡♥t✉❛❧♠❡♥t❡✱ ❝♦♠♦ ♠❛tr✐③❡s ❝♦❧✉♥❛s ✭q✉❛♥❞♦ ❝♦✉❜❡r✮✳

❈❛♣ít✉❧♦ ✷

▼♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ❞❡ ❉✐r✐❝❤❧❡t

✷✳✶ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t

❖ ♠♦❞❡❧♦ ♣r♦❜❛❜✐❧íst✐❝♦ ❞❡ ❉✐r✐❝❤❧❡t ✱ Dir(λ1, λ2, . . . , λD)✱ ❞❡✜♥✐❞♦ ♥♦ D−1✲❙✐♠♣❧❡①✱ ♣♦ss✉✐

❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛❞❛ ♣♦r✿

f(y|λ) = Γ(Λ)

D Q i=1

Γ(λi) D Y

i=1

yλi−1

i , ✭✷✳✶✮

♦♥❞❡ Λ =

D

P

i=1

λi✱ λ = (λ1, λ2, . . . , λD)> 0✱ y = (y1, y2, . . . , yD) ∈ SD ❡ Γ r❡♣r❡s❡♥t❛ ❛

❢✉♥çã♦ ●❛♠❛✿ Γ(t) =

∞

R

0

ut−1_e−u_{du✳ ❋❛③❡♥❞♦}

T(y) = (log(y1),log(y2), . . . ,log(yD))✱ A(λ−1) = D

P

i=1

log(Γ(λi))−log(Γ(Λ))✱

♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ✭✷✳✶✮ ❝♦♠♦ f(y|α) = expT(y)×αt−A(α) ✱α= (λ1−1, λ2−1, . . . , λD−1)✱

❞♦♥❞❡ s❡❣✉❡ q✉❡ Dir ♣❡rt❡♥❝❡ à ❋❛♠✐❧✐❛ ❊①♣♦♥❡♥❝✐❛❧✳

❉❛❞❛ ✉♠❛ ❛♠♦str❛ ❛❧❡❛tór✐❛y1•, y2•, . . . , yn• ❞❡ ✉♠❛ ✈❛r✐á✈❡❧y∼Dir(λ1, λ2, . . . , λD)✱ ❛ ❢✉♥çã♦

❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ é ❞❛❞❛ ♣♦r

L(λ|y1•, y2•, . . . , yn•) = exp

( " n P j=1

T(Yj)

#

×αt−n.A(α)

) .

❆ ❞❡♠♦♥str❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞♦ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡rr♦s✐♠✐❧❤❛♥ç❛ ♣❛r❛ ❛ ❉✐s✲ tr✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❘♦♥♥✐♥❣ ✭✶✾✽✾✮✳ ❉❡♠❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ P❡r❡✐r❛ ❡ ❙t❡r♥ ✭✷✵✵✽✮✳

✻ ▼❖❉❊▲❖❙ ❉❊ ❘❊●❘❊❙❙➹❖ ❉❊ ❉■❘■❈❍▲❊❚ ✷✳✷

✷✳✷ ❘❡❣r❡ssã♦ ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r

◆♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r✱ s✉♣õ❡✲s❡ q✉❡ y•|x• ♣♦ss✉✐ ❞✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t

❝♦♠ ♣❛râ♠❡tr♦s ❞❛❞♦s ♣♦r

ψ(x•, β) =

     

λ1(x•)

λ2(x•)

✳✳✳

λD(x•)

     =

     

m11β11 m12β12 . . . m1kβ1k m21β21 m22β22 . . . m2kβ2k

✳✳✳ ✳✳✳ ✳✳✳

mD1βD1 mD2βD2 . . . mDkβDk     

×

x

♦♥❞❡ ❛ ♠❛tr✐③ ❞❡ ♠♦❞❡❧♦Mmod= (mij)✐♥❞✐❝❛ q✉❛✐s ❞♦s ♣❛râ♠❡tr♦sβij ❢❛③❡♠ ♣❛rt❡ ❞♦ ♠♦❞❡❧♦✱

✐st♦ é✿

(

mij = 1, s❡βij ♣❡rt❡♥❝❡ ❛♦ ♠♦❞❡❧♦.

mij = 0, ❝❛s♦ ❝♦♥trár✐♦. ✭✷✳✸✮

❆ ♠❛tr✐③ ❞❡ ♠♦❞❡❧♦ ♣♦ss✉✐ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛s ✈❛r✐á✈❡✐s r❡❧❡✈❛♥t❡s ❞♦ ♠♦❞❡❧♦ ✭✈❡r ❈❛♣ít✉❧♦ ✺✮✳ ◆❡st❡ ❈❛♣ít✉❧♦✱ ♣♦ré♠✱ ✈❛♠♦s ❛ss✉♠✐r ✭❛ ♠❡♥♦s q✉❡ s❡❥❛ ♠❡♥❝✐♦♥❛❞♦✮ q✉❡ ❛ ♠❛tr✐③ ❞❡ ♠♦❞❡❧♦ é ❝♦♠♣❧❡t❛✱ ✐st♦ é✿ mij = 1 ∀ i, j✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ t❛✐s ♠♦❞❡❧♦s ♣♦s✲

s✉❡♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❡s♣❡❝✐❛✐s ✭✈❡r ❛♣ê♥❞✐❝❡ ❆✮✳ ❆ss✐♠✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ✭✷✳✷✮ ♥❛ ❢♦r♠❛ ❝♦♠♣❛❝t❛

ψ(x•, β) =β×x•. ✭✷✳✹✮

❈❛♠♣❜❡❧❧ ❡ ▼♦s✐♠❛♥♥ ✭✶✾✽✼✮ s✉❣❡r✐r❛♠ ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❢❛③❡♥❞♦ ♦s ♣❛râ♠❡tr♦s

λ1(x•), λ2(x•), . . . , λD(x•) ❞❡♣❡♥❞❡r❡♠ ♣♦❧✐♥♦♠✐❛❧♠❡♥t❡ ❞❛ ❝♦✈❛r✐á✈❡❧ z• ✭s✉♣♦♥❞♦ z• ∈ R✮✱ ✐st♦

é✿x•= (1, z•, . . . , z•k−1)✳

❉❡✈✐❞♦ às r❡str✐çõ❡s s♦❜r❡ ♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ✭❞❡✈❡♠ s❡r t♦❞♦s ♣♦s✐t✐✈♦s✮✱ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ❛♣❡♥❛s ♣❛râ♠❡tr♦s βij ❡ ❢✉♥çõ❡s f1, f2, . . . , fk t❛✐s q✉❡ ψ(x•|β)>0 ♣❛r❛ t♦❞♦

✈❛❧♦r ❞❛ ❝♦✈❛r✐á✈❡❧ ❡st❡♥❞✐❞❛ x•✳ ❙❡♥❞♦ x• ✉♠❛ ✈❛r✐á✈❡❧ ❝♦♥tí♥✉❛✱ ❡ss❛ ❝♦♥❞✐çã♦ é ✐♥✈✐á✈❡❧ ❞❡ s❡r

✈❡r✐✜❝❛❞❛ ♥❛ ♣rát✐❝❛ ♣❛r❛ ✉♠❛ ❞❛❞❛ ♠❛tr✐③β✳ ❆ss✐♠✱ ✈❛♠♦s ✐♠♣♦r s♦♠❡♥t❡ q✉❡

ψ(xj•, β) > 0 ∀ j ∈ {1,2, . . . , n}. ✭✷✳✺✮

❊s♣❡r❛✲s❡✶_{✱ ❛ss✐♠✱ q✉❡ψ(x}

j•|β)s❡❥❛ ♣♦s✐t✐✈♦ ♣❛r❛ t♦❞♦ ✈❛❧♦r ❞❛ ❝♦✈❛r✐á✈❡❧ ❡①t❡♥❞✐❞❛x•✭❡♠❜♦r❛

✶_{❆ ❝♦♥❞✐çã♦ ✭✷✳✺✮ ❣❛r❛♥t❡ q✉❡β×x}

•>0s❡♠♣r❡ q✉❡x•❢♦r ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡(x1•),(x2•), . . . ,(xn•)❝♦♠

✷✳✸ ❊❙❚■▼❆➬➹❖ ❉❊ P❆❘➶▼❊❚❘❖❙ ✼

✐ss♦ ♥ã♦ s❡❥❛ ✉♠❛ ❣❛r❛♥t✐❛✮✳

❊st✐♠❛❞♦s ♦s ♣❛râ♠❡tr♦s β✱ ♣♦❞❡♠♦s ♣r❡✈❡r ♦ ✈❛❧♦r ❞♦ ❞❛❞♦ ❝♦♠♣♦s✐❝✐♦♥❛❧ r❡❢❡r❡♥t❡ ❛♦ ✈❛❧♦r x0• ❞❛ ❝♦✈❛r✐á✈❡❧x• t♦♠❛♥❞♦✲s❡ ❝♦♠♦ ❡st✐♠❛t✐✈❛ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ❝♦♠

♣❛râ♠❡tr♦sλ1(x0•), λ2(x0•), . . . , λD(x0•)✱ ✐st♦ é✿

P rev(x0•) =E(y(x0•)), y(x0•)∼Dir(λ1(x0•), λ2(x0•), . . . , λD(x0•)). ✭✷✳✻✮

P❡❧♦ ❚❡♦r❡♠❛ ❞♦s ♠♦♠❡♥t♦s✱ ✈❡r P❡r❡✐r❛ ❡ ❙t❡r♥ ✭✷✵✵✽✮✱ ♦❜t❡♠♦s✿

P rev(x0•) =

λ1(x0•)

Λ(x0•)

,λ2(x0•)

Λ(x0•)

, . . . ,λD(x0•)

Λ(x0•)

, Λ(x0•) =

D X

j=1

λj(x0•). ✭✷✳✼✮

✷✳✸ ❊st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s

❊♠ s❡✉ ❡st✉❞♦ s♦❜r❡ ♦ tr❛❜❛❧❤♦ ❞❡ ❈❛♠♣❜❡❧❧ ❡ ▼♦s✐♠❛♥♥✱ ❍✐❥❛③✐ ❡ ❏❡r♥✐❣❛♥ ✭✷✵✵✾✮ ♣r♦♣õ❡♠ ❡st✐♠❛r ♦s ♣❛râ♠❡tr♦sβij ♣❡❧❛ ♠❛①✐♠✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✿

L(β|Y1, Y2, . . . Yn) = n Q l=1

Γ(Λ(xl•))

D Q i=1

yλi(xl•)−1

li

Γ(λi(xl•))

✳ ❆ss✐♠✱

logL =

n X

l=1

"

log Γ(Λ(xl•)) +

D X

i=1

(λi(xl•)−1)∗logyli−log Γ(λi(xl•))

#

✭✷✳✽✮

∂logL ∂βij = n X l=1  diGamma( D X p=1

λp(xl•) )− diGamma(λi(xl•)) + logyli 

∗xlj ✭✷✳✾✮

∂logL ∂βab∂βij

= n X l=1  triGamma( D X p=1

λp(xl•) )− δai∗triGamma(λi(xl•))



∗xljxlb ✭✷✳✶✵✮

♦♥❞❡ diGamma(u) = ∂log Γ_∂u (u), triGamma(u) = ∂2_∂ulog Γ2 (u) ❡ δai =

1, s❡a=i

0, ❝❛s♦ ❝♦♥trár✐♦ ✳

✽ ▼❖❉❊▲❖❙ ❉❊ ❘❊●❘❊❙❙➹❖ ❉❊ ❉■❘■❈❍▲❊❚ ✷✳✸

✷✳✸✳✶ ❆❧❣♦r✐t♠♦ ❞❡ ❍✐❥❛③✐✲❏❡r♥✐❣❛♥ ♣❛r❛ s❡❧❡çã♦ ❞❡ ✈❛❧♦r❡s ✐♥✐❝✐❛✐s

✶✳ ❊①tr❛✐rr❛♠♦str❛sA1, A2, . . . , Ar✭t♦❞❛s ❞❡ t❛♠❛♥❤♦m, m < n✮ ❞♦ ❝♦♥❥✉♥t♦{Z1, Z2, . . . Zn}✱ Zj = (xj•, yj•), j= 1,2, . . . , n✳

✷✳ P❛r❛ ❝❛❞❛ ❛♠♦str❛ Ai = {Zi1, Zi2, . . . Zim}✱ ❛❥✉st❡ {yi1•, yi2•, . . . , yim•} ♣♦r ✉♠❛ ❉✐r✐❝❤❧❡t

✭♣❡❧♦ ♠ét♦❞♦ ❞❛ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ✈✐❛ ❛❧❣✉♠ ❛❧❣♦r✐t♠♦ ✐t❡r❛t✐✈♦ ❞❡ ♠❛①✐♠✐③❛çã♦✮ ❝♦♠ ♣❛râ♠❡tr♦s λci1,λci2, . . . ,λdiD✱ ✉t✐❧✐③❛♥❞♦ ❛s ❡st✐♠❛t✐✈❛s ♦❜t✐❞❛s ♣❡❧♦ ♠ét♦❞♦ ❞♦s ♠♦♠❡♥✲

t♦s ✭✈❡r P❡r❡✐r❛ ❡ ❙t❡r♥ ✭✷✵✵✽✮✮ ❝♦♠♦ ♣♦♥t♦ ✐♥✐❝✐❛❧✳ ❈❛❧❝✉❧❡ ❛ ♠é❞✐❛ ❛♠♦str❛❧ ❞❛ ❝♦✈❛r✐❛✈❡❧

z• ❡♠ ❝❛❞❛ ❛♠♦str❛✱ ✐st♦ é✿ zi• = _m1

m P j=1

zij• ❡ ❝❛❧❝✉❧❡ ♦s ✈❛❧♦r❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❛ ✈❛r✐á✈❡❧

❡①t❡♥❞✐❞❛xi•✳

❖❜t❡♠✲s❡✱ ❛ss✐♠✱ ❛s ♠❛tr✐③❡s✿

      c

λ11 λc12 . . . λd1D c

λ21 λc22 . . . λd2D

✳✳✳ ✳✳✳ ✳✳✳

c

λr1 λcr2 . . . λdrD       ,      

x1•

x2•

✳✳✳

xr•

     ✳

✸✳ ❖❜t❡r ♦s ♣❛râ♠❡tr♦sβij q✉❡ ♠✐♥✐♠✐③❛♠ ❛ ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛ ❞❛ ❞✐❢❡r❡♥ç❛        d

λ11 dλ21 . . . λdr1

d

λ12 dλ22 . . . λdr2 ✳✳

✳ ✳✳✳ ✳✳✳

d

λ1D λd2D . . . λdrD        −      

m11β11 m12β12 . . . m1kβ1k m21β21 m22β22 . . . m2kβ1k

✳✳

✳ ✳✳✳ ✳✳✳

mD1βD1 mD2βD2 . . . mDkβDk      × h

x1• x2• . . . xr•

i

✳

✹✳ ❯t✐❧✐③❛r ♦s ♣❛râ♠❡tr♦s ♦❜t✐❞♦s ❝♦♠♦ ✈❛❧♦r❡s ✐♥✐❝✐❛✐s✳

❱❛❧❡ r❡ss❛❧t❛r✱ ♣♦ré♠✱ q✉❡ ♦ ♠ét♦❞♦ ✐t❡r❛t✐✈♦ ❞❡ ♠❛①✐♠✐③❛çã♦ ❞❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ r❡str✐t♦ à ❝♦♥❞✐çã♦ ✭✷✳✺✮ r❡q✉❡r q✉❡ ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ t❛♠❜é♠ ❛ s❛t✐s❢❛ç❛ ❡ ♥ã♦ ❤á ❣❛r❛♥t✐❛s ❞❡ q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❍✐❥❛③✐✲❏❡r♥✐❣❛♥ ♣r♦❞✉③❛ s❡♠♣r❡ ♣♦♥t♦s ✐♥❝✐❛✐s ❝♦♠ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡✳ ◆ós ❞❡s❡♥✈♦❧✈❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ♠❛✐s ❡✜❝✐❡♥t❡ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧✳

✷✳✸✳✷ ◆♦✈♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ s❡❧❡çã♦ ❞❡ ✈❛❧♦r❡s ✐♥✐❝✐❛✐s

✶♦_{❝❛s♦✿ ❖ ✐♥t❡r❝❡♣t♦ ✭❢✉♥çã♦ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ ✶✮ ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s{f}

1, f2, . . . , fk}

❞❡✜♥✐❞❛s ♥❛ ❙❡çã♦ ✶✳✶✳

✷✳✸ ❊❙❚■▼❆➬➹❖ ❉❊ P❆❘➶▼❊❚❘❖❙ ✾

✷✳ ■♥tr♦❞✉③❛ ♣❛râ♠❡tr♦s ❛rt✐✜❝✐❛✐s βj11, βj21, . . . , βjl1 ✱ ❞❡ ♠♦❞♦ ❛ ❝♦♠♣❧❡t❛r ❛ ♣r✐♠❡✐r❛

❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ ❞❡ ♠♦❞❡❧♦✱ ❝❛s♦ s❡❥❛ ♥❡❝❡ssár✐♦✳ P❛r❛D= 4, k= 3 ❡



   

λ1(x•) λ2(x•) λ3(x•)

λ4(x_•)     =     

0 β12 β13

β21 0 β23 β31 β32 β33

0 β42 β43

     ×

x

♣♦r ❡①❡♠♣❧♦✱ ❞❡✈❡♠♦s ❝♦♠♣❧❡t❛r ❝♦♠β11 ❡ β4,1✱ ✐st♦ é✿l= 2, j1 = 1 ❡j2 = 4✳

✸✳ ❆❥✉st❡ ♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦sy1•, y2•, . . . yn•♣♦r ✉♠❛ ❉✐r✐❝❤❧❡t ❝♦♠ ♣❛râ♠❡tr♦sβ11, β21, . . . , βD1✳

❖ ♣♦♥t♦ ✐♥❝✐❛❧ β∗

=      

β11 β12= 0 β13= 0 . . . β1k= 0 β21 β22= 0 β23= 0 . . . β1k= 0

✳✳

✳ ✳✳✳ ✳✳✳

βD1 βD2= 0 βD3= 0 . . . βDk= 0     

❝❡rt❛♠❡♥t❡ s❛t✐s❢❛③ ✭✷✳✺✮✳

✹✳ ❖❜t❡♥❤❛ ♦ ♣♦♥t♦ ❞❡ ♠á①✐♠♦✱ ♣♦r ♠❛①✐♠❛③❛çã♦ r❡str✐t❛ à ✭✷✳✺✮✱ ❞❛ ❢✉♥çã♦

e

L = logL(β|y1•, y2•, . . . yn•) − M(η)

l P r=1

(βjr1)2✱ ✉t✐❧✐③❛♥❞♦ ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ β∗✱ ♦♥❞❡ M(η)

é ✉♠❛ q✉❛♥t✐❞❛❞❡ ♣♦s✐t✐✈❛ q✉❡ ❝♦♠❡ç❛ ❝♦♠ ✉♠ ✈❛❧♦r ❜❛✐①♦ ❡ ✈❛✐ ❛✉♠❡♥t❛♥❞♦ ❛♦ ❧♦♥❣♦ ❞♦ ♥ú♠❡r♦ ❞❡ ✐t❡r❛çõ❡sη✳

✺✳ ❉❡s❝❛rt❡ ♦s ✈❛❧♦r❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦s ♣❛râ♠❡tr♦s ❛rt✐✜❝✐❛✐s ❛❞✐❝✐♦♥❛❞♦s ✭♦s ✈❛❧♦r❡s ❞❡s❝❛r✲ t❛❞♦s ❞❡✈❡rã♦ s❡r ♣ró①✐♠♦s ❞❡ ③❡r♦✮✳

✷♦ _{❝❛s♦✿ ❖ ✐♥t❡r❝❡♣t♦ ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s{f}

1, f2, . . . , fk}✳

✶✳ ❘❡♣✐t❛ ♦ ✶♦ _{❝❛s♦ ❛❞✐❝✐♦♥❛♥❞♦ ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ ✶ à ❡ss❡ ❝♦♥❥✉♥t♦ ✭♥♦t❡ q✉❡✱ ♥❡ss❡}

❝❛s♦✱ s❡rã♦ ✐♥tr♦❞✉③✐❞❛s ❉ ✈❛r✐á✈❡✐s ❛rt✐✜❝✐❛✐s✮

❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ Le ♣❡♥❛❧✐③❛ ♦s ♣❛râ♠❡tr♦s ❡✈❡♥t✉❛❧♠❡♥t❡ ❛❞✐❝✐♦♥❛❞♦s ❞❡ t❛❧ ❢♦r♠❛ q✉❡✱

❛♦ ✜♠ ❞♦ ♣r♦❝❡ss♦ ❞❡ ♠❛①✐♠✐③❛çã♦ ✐t❡r❛t✐✈♦✱ ♦s ✈❛❧♦r❡s ❞♦s ♣❛râ♠❡tr♦s ❛❞✐❝✐♦♥❛❞♦s ❞❡✈❡rã♦ s❡r ♣ró①✐♠♦s ❛ ③❡r♦✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❢❛t♦ ❞❡ β∗ s❡♠♣r❡ s❛t✐s❢❛③❡r ✭✷✳✺✮ ❣❛r❛♥t❡ ✉♠ ♠ét♦❞♦ ♠❛✐s ❡stá✈❡❧

❞❡ ♦❜t❡♥çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳

✷✳✸✳✸ ❘❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s

P❛r❛ ❛✈❛❧✐❛r ❛ ❡✜❝✐ê♥❝✐❛ ❞♦s ♠ét♦❞♦s ♣r♦♣♦st♦s ♣❛r❛ s❡❧❡❝✐♦♥❛r ✉♠ ✈❛❧♦r ✐♥✐❝✐❛❧ ♥♦ ♠♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r✱ r❡❛❧✐③❛♠♦s ❛❧❣✉♠❛s s✐♠✉❧❛çõ❡s ❜❛s❡❛❞❛s ❡♠ ♠♦❞❡❧♦s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❣r❛✉ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ✸ ✭k= 4, z• ∈ R, x• = (1, z•, z•2, z•3✮ ❝♦♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s✉❜❛♠♦str❛s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛✲

✶✵ ▼❖❉❊▲❖❙ ❉❊ ❘❊●❘❊❙❙➹❖ ❉❊ ❉■❘■❈❍▲❊❚ ✷✳✹

❡ p = 0.66✳ P❛r❛ ❡✈✐t❛r ♠♦❞❡❧♦s ✐♥❝♦♥s✐st❡♥t❡s✱ t❛♠❜é♠ ❢♦✐ ✐♠♣♦st♦ q✉❡ t❛✐s ♠❛tr✐③❡s ❞❡✈❡r✐❛♠ ❛♣r❡s❡♥t❛r ❛♦ ♠❡♥♦s ✉♠❛ ❡♥tr❛❞❛ ♥ã♦ ♥✉❧❛ ♣♦r ❧✐♥❤❛✳ ❉✉❛s ♠❡❞✐❞❛s ❞❡ ❡✜❝✐ê♥❝✐❛ ❢♦r❛♠ ❛♥❛❧✐s❛❞❛s✿ ✭✶✮ ❆ ♣♦r❝❡♥t❛❣❡♥ ❞♦s ❝❛s♦s ❡♠ q✉❡ ♦s ❛❧❣♦r✐t♠♦s ♥ã♦ ❢♦r❛♠ ❝❛♣❛③❡s ❞❡ ❡♥❝♦♥tr❛r ♣♦♥t♦s ✐♥✐❝✐❛✐s ✈✐á✈❡✐s ✭❧✐♠✐t❛❞♦ ❛ tr✐♥t❛ t❡♥t❛t✐✈❛s✷ _{♣♦r s✉❜❛♠♦str❛✮❀ ✭✷✮ ❖ t❡♠♣♦ ♠é❞✐♦ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡}

❝❛❞❛ ♠ét♦❞♦✳ ❖s r❡s✉❧t❛❞♦s sã♦ ♠♦str❛❞♦s ♥❛ ❋✐❣✉r❛ ✷✳✶✳ ❖ t❛♠❛♥❤♦ ❞❛s s✉❜❛♠♦str❛s ✭✷✵ ♦✉ ✷✼✮ ♣❛r❡❝❡✉ ♥ã♦ ❛❧t❡r❛r ❛ ♣❡r❢♦r♠❛♥❝❡ ❡✱ ❛ss✐♠✱ ♦s r❡s✉❧t❛❞♦s sã♦ ♠♦str❛❞♦s ❥✉♥t♦s✳

0.33 0.5 0.66

Hijazi Nosso Método

Falhas

Completude da Matriz de modelo: Pr(mjk = 1)

%

0

5

10

15

20

✭❛✮

Tempo de processamento

Completude da Matriz de modelo: Pr(mjk = 1)

Segundos (

L

o

g2

)

0.33 0.5 0.66

−2

0

2

4

Hijazi Nosso Método

✭❜✮

❋✐❣✉r❛ ✷✳✶✿ ❈♦♠♣❛r❛t✐✈♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❞❡ s❡❧❡çã♦ ❞❡ ✈❛❧♦r❡s ✐♥✐❝✐❛✐s ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ❉✐r✐❝❤❧❡t ❧✐♥❡❛r

❖ ❣rá✜❝♦ ❞❛ ❡sq✉❡r❞❛ ♠♦str❛ ❝❧❛r❛♠❡♥t❡ ✉♠❛ ♠❛✐♦r ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♥♦ss♦ ❛❧❣♦r✐t♠♦✿ ❡♠ t♦❞♦s ♦s ✻✵✵✵ ❝❛s♦s ❝♦♥s❡❣✉✐♠♦s ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ✈✐á✈❡❧✳ ➱ ❝❧❛r♦ q✉❡ ♦ t❡♠♣♦ ❞❡ ♣r♦❝❡s✲ s❛♠❡♥t♦ ❞❡♣❡♥❞❡ ❞❛ ❢♦r♠❛ ❝♦♠♦ ♦s ❛❧❣♦r✐t♠♦s ❢♦r❛♠ ✐♠♣❧❡♠❡♥t❛❞♦s✳ ❈♦♠ ❛ ♥♦ss❛ ✐♠♣❧❡♠❡♥t❛çã♦✱ ♦ ♥♦ss♦ ❛❧❣♦r✐t♠♦ ♠♦str♦✉ t❛♠❜é♠ ✉♠❛ s✉♣❡r✐♦r✐❞❛❞❡ ❝♦♠ r❡❧❛çã♦ ❛♦ ♦✉tr♦✱ ♠♦str❛❞❛ ♥♦ ❣rá✜❝♦ ❞❛ ❞✐r❡✐t❛✿ ❛ ❞✐str✐❜✉✐çã♦ ❞♦s t❡♠♣♦s ♥♦s três ❝❛s♦s✱ ♣❛r❛ ♦ ♥♦ss♦ ❛❧❣♦r✐t♠♦✱ ❡stá ♠❛✐s ❝♦♥❝❡♥tr❛❞❛ ❡♠ t♦r♥♦ ❞❛s ♠❡❞✐❛♥❛s q✉❡✱ ♣♦r s✉❛ ✈❡③✱ sã♦ ✐♥❢❡r✐♦r❡s às ❝♦rr❡s♣♦♥❞❡♥t❡s ♣r♦❞✉③✐❞❛s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❞❡ ❍✐❥❛③✐✲❏❡r♥✐❣❛♥✳

✷✳✹ ❊❧✐♠✐♥❛çã♦ ❞❛s r❡str✐çõ❡s s♦❜r❡ ♦ ❡s♣❛ç♦ ♣❛r❛♠étr✐❝♦

❯♠❛ ✈❛r✐❛♥t❡ ❞♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r✱ ♣r♦♣♦st♦ ❡♠ ▼❡❧♦ ❡t ❛❧✳ ✭✷✵✵✾✮✱ ❝♦♥s✐st❡ ❡♠ ❡s❝♦❧❤❡r

D ❢✉♥çõ❡s ♣♦s✐t✐✈❛s gj :R−→ R+, j = 1,2, . . . , D ✭✐♥❥❡t♦r❛s ❡ ✸ ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✮ ❡ ❛♣❧✐❝❛r ❛

❢✉♥çã♦gj àj✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡ ❞♦ ♣r♦❞✉t♦ ✭✷✳✹✮ ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s✉♣♦r q✉❡ ❛ ❢✉♥çã♦ ψ(x•|β)✱

é ❞❛ ❢♦r♠❛

✷_{◆♦t❡ q✉❡ ❛♣❡♥❛s ♦ ♠ét♦❞♦ ❞❡ ❍✐❥❛③✐✲❏❡r♥✐❣❛♥ ♣♦ss✉✐ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❡♥✈♦❧✈❡♥❞♦ ❡✈❡♥t♦s ❛❧❡❛tór✐♦s ❡✱ ♣♦rt❛♥t♦✱}

✷✳✹ ❊▲■▼■◆❆➬➹❖ ❉❆❙ ❘❊❙❚❘■➬Õ❊❙ ❙❖❇❘❊ ❖ ❊❙P❆➬❖ P❆❘❆▼➱❚❘■❈❖ ✶✶

ψ(x•, β) =

     

λ1(x•)

λ2(x•)

✳✳✳

λD(x•)

     =

     

g1(m11β11x1+m12β12x2+. . .+m1kβ1kxk ) g2(m21β21x1+m22β22x2+. . .+m2kβ1kxk )

✳✳✳

gD(mD1βD1x1+mD2βD2x1+. . .+mDkβDkxk )

    

, ✭✷✳✶✶✮

x• = (x1, . . . , xk)✳

◆❡st❡ ♠♦❞❡❧♦✱ ❛ ❝♦♥❞✐çã♦ ✭✷✳✺✮ ♣♦❞❡ s❡r ❡❧✐♠✐♥❛❞❛✳ ❙♦❜ ❛s ❤✐♣ót❡s❡s s♦❜r❡ ❛s ❢✉♥çõ❡s gj, j = 1,2, . . . , D❡ s♦❜r❡ ❛ ♠❛tr✐③X❞❡ ✭✶✳✶✮✱ ♦s ❛✉t♦r❡s ❣❛r❛♥t❡♠ ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞♦ ❡st✐♠❛❞♦r ❞❡

♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ❜❛s❡❛❞♦s ❡♠ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❑räts❝❤♠❡r ✭✷✵✵✼✮✳ P♦ré♠✱♦ q✉❡ ❑räts❝❤♠❡r ♠♦str❛ é q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦ssí✈❡✐s ❛♠♦str❛s t❛✐s q✉❡ ♦ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❛ss✉♠❡ ✉♠ ♠á①✐♠♦ ❡♠ ❞♦✐s ♦✉ ♠❛✐s ♣♦♥t♦s ❞✐st✐♥t♦s ♣♦ss✉✐ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛✳ ■ss♦ ❣❛r❛♥t❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✶✱ ♠❛s ♥ã♦ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛✳

◆❛q✉❡❧❡ tr❛❜❛❧❤♦✱ ♦s ❛✉t♦r❡s ✉t✐❧✐③❛♠ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ✭gj(t) = et ∀ j✮✳ ❊♠ ❛❧❣✉♠❛s ❞❡

♥♦ss❛s s✐♠✉❧❛çõ❡s ❝♦♠ ♦s ❞❛❞♦s ❞♦ ▲❛❣♦ ➪rt✐❝♦✱ ♣♦ré♠✱ ♦ ✉s♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❛♣r❡s❡♥t♦✉ ♠✉✐t❛ ✐♥st❛❜✐❧✐❞❛❞❡✳ ❯♠❛ ❛❧t❡r♥❛t✐✈❛ é ✉t✐❧✐③❛r ❛ ❢✉♥çã♦ ❆r❝♦t❛♥❣❡♥t❡ q✉❡✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ ❝♦♥✲ st❛♥t❡ ♣♦s✐t✐✈❛ ♠❛✐♦r ❞♦ q✉❡ π

❈❛♣ít✉❧♦ ✸

❖✉tr♦s ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦

✸✳✶ ❚r❛♥s❢♦r♠❛çõ❡s

S

7−→

R

❙❡❣✉♥❞♦ ❆✐t❝❤✐s♦♥ ✭✶✾✽✷✮✱ ✉♠❛ ❞❛s ❞✐✜❝✉❧❞❛❞❡s ✐♥✐❝✐❛✐s ♣❛r❛ ❛ r❡❣r❡ssã♦ ❞❡ ❞❛❞♦s ❝♦♠♣♦s✐✲ ❝✐♦♥❛✐s é ❛ ❡s❝❛ss❡③ ❞❡ ❢❛♠í❧✐❛s ♣❛r❛♠étr✐❝❛s ❞❡ ♠♦❞❡❧♦s ♣r♦❜❛❜✐❧íst✐❝♦s ❞❡✜♥✐❞♦s ♥♦D−1✲❙✐♠♣❧❡① ✭❛ ♠❛✐s ❝♦♥❤❡❝✐❞❛ é ❛ ❉✐str✐❜✉✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t✮✳ ❆✐t❝❤✐s♦♥ ♣r♦♣õ❡ ❝♦♥tr✉✐r ♥♦✈♦s ♠♦❞❡❧♦s ♣r♦✲ ❜❛❜✐❧íst✐❝♦s ♥♦ ❙✐♠♣❧❡① ✉t✐❧✐③❛♥❞♦ ❞✐str✐❜✉✐çõ❡s ❝♦♥❤❡❝✐❞❛s ❡♠Rd✭d=D−1✮ ♣♦r ♠❡✐♦ ❞❡ tr❛♥s❢♦r✲ ♠❛çõ❡s ❜✐❥❡t♦r❛s ❝♦♥✈❡♥✐❡♥t❡s✳ ❆s ❞✐str✐❜✉✐çõ❡s ◆♦r♠❛✐s✲▲♦❣íst✐❝❛s✱ ❛♣r❡s❡♥t❛❞❛s ❡♠❆✐t❝❤✐s♦♥ ❡ ❙❤❡♥ ✭✶✾✽✵✮✱ ♣♦r ❡①❡♠♣❧♦✱ sã♦ ♣r♦✈❡♥✐❡♥t❡s ❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ❝♦♠ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♠✉❧t✐✲ ✈❛r✐❛❞❛ ✈✐❛ tr❛♥s❢♦r♠❛çã♦ ▲♦❣❛r✐t♠♦ ❞❛ r❛③ã♦ ✭❛❧r✮ ✿

(y1, y2, . . . , yD) ∈ SD 7−→alr (log(y1/yD),log(y2/yD), . . .log(yD−1/yD))∈Rd. ✭✸✳✶✮

❊ss❛ ♥♦✈❛ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡✜♥✐❞❛s ♥♦ ❙✐♠♣❧❡①✱ ♣♦ré♠✱ ♥ã♦ é ❞❡ s❡r✈❡♥t✐❛ ✐♠❡❞✐❛t❛ ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡s❝r✐t♦ ♥♦ ❈❛♣ít✉❧♦ ✐♥tr♦❞✉tór✐♦✱ ♣♦✐s ✧❊♠❜♦r❛ ❡①✐st❛♠ ♦s ♠♦♠❡♥t♦s ❞❡ t♦❞❛s ❛s ♦r❞❡♥s ✳✳✳ s✉❛s ❡①♣r❡ssõ❡s ✐♥t❡❣r❛✐s✶ _{♥ã♦ sã♦ r❡❞✉tí✈❡✐s ❛ ✉♠❛ ❢♦r♠❛ s✐♠♣❧❡s✧❆✐t❝❤✐s♦♥ ❡ ❙❤❡♥}

✭✶✾✽✵✮✳ ❆✐t❝❤✐s♦♥✱ s✉❣❡r❡ tr❛♥s❢♦r♠❛r ♦s ❞❛❞♦s ❞♦ ❙✐♠♣❧❡① ♣❛r❛Rd✱ ❛♣❧✐❝❛r ♦ ♠ét♦❞♦ ♣❛r❛ ♦s ❞❛❞♦s

tr❛♥s❢♦r♠❛❞♦s✱ ❡ r❡t♦r♥❛r ♣❛r❛ ♦ ❙✐♠♣❧❡①✳ ❊♠ ✉♠ ❞♦s s❡✉s ❡st✉❞♦s ❞❡ ❝❛s♦✱ ❡❧❡ s❡ ♣r♦♣õ❡ ❛ ❛♥❛❧✐s❛r ❝♦♠♦ ✈❛r✐❛ ❛ ❝♦♠♣♦s✐çã♦ ❞♦ s♦❧♦ ❞♦ ▲❛❣♦ ➪rt✐❝♦ ❡♠ ❢✉♥çã♦ ❞❛ ♣r♦❢✉♥❞✐❞❛❞❡z ✭❆✐t❝❤✐♥s♦♥ ✭✶✾✽✻✮✮✱

s✉♣♦♥❞♦ q✉❡ ♦ ✈❡t♦r

µ(x) =

"

log(sand_clay)(z) =β11+β12z+β13z2+β14log(z) log(_claysilt)(z) =β21+β22z+β23z2+β24log(z)

#

=β×x, x= (1, z, z2,log(z)) ✭✸✳✷✮

r❡♣r❡s❡♥t❛ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ❜✐✈❛r✐❛❞❛ ◆2_{(µ(z),Σ)✱ ♦♥❞❡Σ✐♥❞❡♣❡♥❞❡} ❞❡z✳ ❆✐t❝❤✐s♦♥ s✉❣❡r❡ ❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦sβij ❡Σ♣♦r ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♠✉❧t✐✈❛r✐❛❞♦ ♦✉✱

s♦❜ ❛ ❤✐♣ót❡s❡ ❞❡ ♥♦r♠❛❧✐❞❛❞❡✱ ♣❡❧❛ ♠❛①✐♠✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭❆✐t❝❤✐♥s♦♥ ✭✶✾✽✻✮✱ ♣á❣✳ ✶✻✵✮✳

✶_{❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ❡①♣r❡ssã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ✈❛❧♦r ❡s♣❡r❛❞♦}

✶✹ ❖❯❚❘❖❙ ▼❖❉❊▲❖❙ ❉❊ ❘❊●❘❊❙❙➹❖ ✸✳✶

◆♦ ❝❛s♦ ❣❡r❛❧✱ s✉♣♦♥❞♦ q✉❡

alr(y•)|x• ∼Nd(µ(x•),Σ), ✭✸✳✸✮

❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ é ❞❛❞❛ ♣♦r✿L(βij,1≤i≤d, 1≤j ≤k,Σ|alr(y1•), . . . alr(yn•)) =

1 (2π)d/2_|Σ|1/2

n n_Y

l=1

exp−(alr(yl•)−µ(xl•))t×Σ−1×(alr(yl•)−µ(xl•)) , ✭✸✳✹✮

♦♥❞❡ µ(x) =ψ(x, β) ♣♦ss✉✐ ❛ ♠❡s♠❛ ❢♦r♠❛ ❞❡ ✭✷✳✷✮✳

❱❛❧❡ r❡ss❛❧t❛r q✉❡✿

• P❛r❛ ♠♦❞❡❧♦s ❝♦♠ ♠❛tr✐③ ❞❡ ♠♦❞❡❧♦ ❝♦♠♣❧❡t❛✱ ♦s ❡st✐♠❛❞♦r❡s ✭♣❛r❛ β✮ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐✲

♠✐❧❤❛♥ç❛ ❡ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s✱ q✉❡ ♠✐♥✐♠✐③❛

Res2(β) =

n X

i=1

|| (alr(yi•)−µ(xi•))||2, ✭✸✳✺✮

sã♦ ✐❞ê♥t✐❝♦s ❡ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ♣♦r ♠❡✐♦ ❞❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ✭✈❡r ❛♣ê♥❞✐❝❡ ❆✮✳

• ◆♦ ❝❛s♦ ❞❡ ❛❥✉st❡ t♦t❛❧ ✭♦♥❞❡ ♦ r❡sí❞✉♦ q✉❛❞rát✐❝♦ ✭✸✳✺✮ é ✐❣✉❛❧ ❛ ③❡r♦✮ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐✲

♠✐❧❤❛♥❛ ✭✸✳✹✮✱ ❝❛❧❝✉❧❛❞❛ ♥♦s ♣❛râ♠❡tr♦sβij ❞❡ ❛❥✉st❡ t♦t❛❧✱ ❞❡♣❡♥❞❡ ❞❡Σ❛♣❡♥❛s ♣❡❧♦ ✐♥✈❡rs♦

❞♦ s❡✉ ❞❡t❡r♠✐♥❛♥t❡✱ ♦ q✉❛❧ ♣♦❞❡ ❛♣r♦①✐♠❛r✲s❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ❞❡ ③❡r♦ ✷_{✳ ■ss♦ ♠♦str❛ q✉❡✱}

♥❡ss❡ ❝❛s♦✱ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♥ã♦ é ❧✐♠✐t❛❞❛ ❡♠ Σ❡✱ ♣♦rt❛♥t♦✱ ❛ ❡st✐♠❛çã♦ ❞❡ Σ ♣♦r ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣♦❞❡✱ ❡✈❡♥t✉❛❧♠❡♥t❡✱ ♥ã♦ ❢❛③❡r s❡♥t✐❞♦✳ ❉❡t❡r♠✐♥❛r q✉❛✐s ❝❛s♦s sã♦ ♣❛t♦❧ó❣✐❝♦s✱ ♣♦ré♠✱ ♣❛r❡❝❡ s❡r ✉♠❛ t❛r❡❢❛ ❝♦♠♣❧❡①❛✳

✸✳✶✳✶ ❆❜r❛♥❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦

❘❡ss❛❧t❛♠♦s q✉❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡s❝r✐t♦ ❛❝✐♠❛ ♣♦❞❡ s❡r ❢❡✐t♦ ♣❛r❛ q✉❛❧q✉❡r tr❛♥s❢♦r♠❛çã♦ ❞❡

SD ♣❛r❛Rd✳ ❆❧é♠ ❞❛ tr❛♥s❢♦r♠❛çã♦alr✱ ❛s s❡❣✉✐♥t❡s tr❛♥s❢♦r♠❛çõ❡s sã♦ ❛♣r❡s❡♥t❛❞❛s ❡♠ ❆✐t❝❤✐s♦♥

✭✶✾✽✷✮✳

✷_{❯♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♠❛tr✐③❡s ♣♦s✐t✐✈❛s✲❞❡✜♥✐❞❛s ❝♦♠ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ é} 1

kI

k∈N✱ ♦♥❞❡I❞❡♥♦t❛