PARTIAL LEAST SQUARE - PATH MODELING

P❆❘❚■❆▲ ▲❊❆❙❚ ❙◗❯❆❘❊ ✲ P❆❚❍ ▼❖❉❊▲■◆●

▼❡t♦❞♦❧♦❣✐❛✱ ❙♦❢t✇❛r❡ ❡ ❆♣❧✐❝❛çã♦

♣♦r

❋r❛♥❝❡s❝♦ ❈♦st✐❣❧✐♦❧❛

❚r❛❜❛❧❤♦ ❞❡ ♣r♦❥❡❝t♦ ❛♣r❡s❡♥t❛❞♦ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥ç❛♦ ❞♦ ❣r❛✉ ❞❡

▼❡str❡ ❡♠ ❊st❛tíst✐❝❛ ❡ ●❡stã♦ ❞❡ ■♥❢♦r♠❛çã♦

♣❡❧♦

■♥st✐t✉t♦ ❙✉♣❡r✐♦r ❞❡ ❊st❛tíst✐❝❛ ❡ ●❡stã♦ ❞❡ ■♥❢♦r♠❛çã♦ ❞❛

P❆❘❚■❆▲ ▲❊❆❙❚ ❙◗❯❆❘❊ ✲ P❆❚❍ ▼❖❉❊▲■◆●

▼❡t♦❞♦❧♦❣✐❛✱ ❙♦❢t✇❛r❡ ❡ ❆♣❧✐❝❛çã♦

♣♦r

❋r❛♥❝❡s❝♦ ❈♦st✐❣❧✐♦❧❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞❡

Pr♦❢❡ss♦r ❉♦✉t♦r P❡❞r♦ ❙✐♠õ❡s ❈♦❡❧❤♦

❚r❛❜❛❧❤♦ ❞❡ ♣r♦❥❡❝t♦ ❛♣r❡s❡♥t❛❞♦ ❝♦♠♦

r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥ç❛♦ ❞♦ ❣r❛✉ ❞❡

▼❡str❡ ❡♠ ❊st❛tíst✐❝❛ ❡ ●❡stã♦ ❞❡ ■♥❢♦r♠❛çã♦

♣❡❧♦

■♥st✐t✉t♦ ❙✉♣❡r✐♦r ❞❡ ❊st❛tíst✐❝❛ ❡ ●❡stã♦ ❞❡ ■♥❢♦r♠❛çã♦

❞❛

❯♥✐✈❡rs✐❞❛❞❡ ◆♦✈❛ ❞❡ ▲✐s❜♦❛

❆✐ ♠✐❡✐ ❣❡♥✐t♦r✐ ❡

❈♦♥t❡ú❞♦

▲✐st❛ ❞❛s ❋✐❣✉r❛s ■■

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❖ ❆❧❣♦r✐t♠♦ P▲❙✲P▼ ✻

✷✳✶ ❊♥q✉❛❞r❛♠❡♥t♦ ●❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✷✳✷ ❖ ▼♦❞❡❧♦ P▲❙✲P▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✷✳✶ ❖ ▼♦❞❡❧♦ ❞❡ ▼❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✷✳✷ ❖ ▼♦❞❡❧♦ ❊str✉t✉r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷✳✸ ❊str✉t✉r❛ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷✳✸✳✶ ❊st✐♠❛çã♦ ❊①t❡r♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷✳✸✳✷ ❊st✐♠❛çã♦ ❞♦s P❡s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷✳✸✳✸ ❊st✐♠❛çã♦ ■♥t❡r♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷✳✸✳✹ Pr♦❝❡ss♦ ■t❡r❛t✐✈♦ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✷✳✹ ▼❡❞✐❞❛s ❞❡ ❋✐❛❜✐❧✐❞❛❞❡ ❡ ❞❡ ❱❛❧✐❞❛❞❡ ❞♦ ▼♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

❈♦♥t❡ú❞♦

✷✳✹✳✶ ❈♦♠✉♥❛❧✐❞❛❞❡ ❡ ❘❡❞✉♥❞â♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✷✳✹✳✷ ❆ ❘❡✲❛♠♦str❛❣❡♠✿ ❏❛❝❦✲❦♥✐❢❡ ❡ ❇♦♦tstr❛♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✸ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❙♦❢t✇❛r❡ ✸✹

✸✳✶ ❊♥q✉❛❞r❛♠❡♥t♦ ●❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✸✳✷ ❊str✉t✉r❛ ❞♦ Pr♦❣r❛♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✸✳✸ ■♥❢♦r♠❛çã♦ ❞❡ ❆rr❛♥q✉❡ ❞♦ ❆❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✸✳✹ ❖✉t♣✉ts ❉❡s❡♥✈♦❧✈✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✹ ❆♣❧✐❝❛çã♦ ❛ ❉❛❞♦s ❘❡❛✐s ✹✾

✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✹✳✷ ❖s ❉❛❞♦s ■♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✹✳✸ ❖s ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✹✳✹ Pr♦❝❡ss♦ ❞❡ ❈♦♠♣❛r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✹✳✹✳✶ ❖ ❙♠❛rtP▲❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✹✳✹✳✷ ❖ ❙♠❛rtP▲❙✿ ♦s ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

✹✳✹✳✸ ❈♦♠♣❛r❛çã♦ ❞♦s ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

✺ ❙í♥t❡s❡ ❡ ❈♦♥❝❧✉sã♦ ✻✵

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ◆♦t❛çã♦ ●rá✜❝❛ ❞♦ ♠♦❞❡❧♦ P▲❙✲P▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✷ ❖ ♠♦❞❡❧♦ P▲❙✲P▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✸ ❖ ▼♦❞♦ ❘❡✢❡❝t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷✳✹ ❖ ▼♦❞♦ ❋♦r♠❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✺ ▼✉❧t✐♣❧❡ ■♥❞✐❝❛t♦rs ❢♦r ▼✉❧t✐♣❧❡ ❈❛✉s❡s ✭▼■▼■❈✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷✳✻ Pr♦❝❡ss♦ ✐t❡r❛t✐✈♦ ❞♦ ❛❧❣♦r✐t♠♦ P▲❙✲P▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✸✳✶ ■♥❢♦r♠❛çã♦ ❞❡ ❆rr❛♥q✉❡ ❞♦ ❆❧❣♦r✐t♠♦✿ ♦ ♠♦❞❡❧♦ ❞❡ ❡①❡♠♣❧♦ ✳ ✹✵

✸✳✷ ■♥❢♦r♠❛çã♦ ❞❡ ❆rr❛♥q✉❡ ❞♦ ❆❧❣♦r✐t♠♦✿ ♦s ❞❛❞♦s ❞❡ ✐♥♣✉t

✭▼♦❞♦ ◆♦r♠❛❧✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✸✳✸ ■♥❢♦r♠❛çã♦ ❞❡ ❆rr❛♥q✉❡ ❞♦ ❆❧❣♦r✐t♠♦✿ ♦s ❞❛❞♦s ❞❡ ✐♥♣✉t

✭▼♦❞♦ ❙✐♠✉❧❛çã♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✸✳✹ ■♥❢♦r♠❛çã♦ ❞❡ ❆rr❛♥q✉❡ ❞♦ ❆❧❣♦r✐t♠♦✿ ♦ ✜❝❤❡✐r♦ ✏s✐♠❴♥✳①❧s✑ ✳ ✹✸

✸✳✺ ■♥❢♦r♠❛çã♦ ❞❡ ❆rr❛♥q✉❡ ❞♦ ❆❧❣♦r✐t♠♦✿ ♦ ✜❝❤❡✐r♦ ✏♠♦❞❡❧✳①❧s✑ ✳ ✹✹

✸✳✻ ■♥❢♦r♠❛çã♦ ❞❡ ❆rr❛♥q✉❡ ❞♦ ❆❧❣♦r✐t♠♦✿ ♦ ✜❝❤❡✐r♦ ✏✐♥♥❡r❴❞❡s✐❣♥✳①❧s✑ ✹✹

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✹✳✶ ▼♦❞❡❧♦ ❞❡ ❙❛t✐s❢❛çã♦ ❝♦♠ ♦s ❖♣❡r❛❞♦r❡s ❞❡ ❘❡❞❡ ▼ó✈❡❧ ✳ ✳ ✳ ✺✵

✹✳✷ ▼♦❞❡❧♦ ❊str✉t✉r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✹✳✸ ▼♦❞❡❧♦ ❞❡ ▼❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✹✳✹ ❖ ❋✐❝❤❡✐r♦ ❞❡ ❖✉t♣✉t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✹✳✺ ▲♦❛❞✐♥❣s ❞♦ ▼♦❞❡❧♦ ❞❡ ▼❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✹✳✻ ❈♦❡✜❝✐❡♥t❡s ❞❛s r❡❧❛çõ❡s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✹✳✼ P❡s♦s ❊①t❡r♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✹✳✽ ❈♦rr❡❧❛çõ❡s ❡♥tr❡ ❛s ❱❛r✐á✈❡✐s ▲❛t❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✹✳✾ ▼❡❞✐❞❛s ❞❡ ✈❛❧✐❞❛❞❡ ❡ ✜❛❜✐❧✐❞❛❞❡ ❞♦s ♠♦❞❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✹✳✶✵ ❙♠❛rtP▲❙✿ ▲♦❛❞✐♥❣s ❞♦ ▼♦❞❡❧♦ ❞❡ ▼❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✹✳✶✶ ❙♠❛rtP▲❙✿ ❈♦❡✜❝✐❡♥t❡s ❞❛s r❡❧❛çõ❡s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s ✺✼

✹✳✶✷ ❙♠❛rtP▲❙✿ P❡s♦s ❊①t❡r♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✹✳✶✸ ❙♠❛rtP▲❙✿ ❈♦rr❡❧❛çõ❡s ❡♥tr❡ ❛s ❱❛r✐á✈❡✐s ▲❛t❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✹✳✶✹ ❙♠❛rtP▲❙✿ ▼❡❞✐❞❛s ❞❡ ✈❛❧✐❞❛❞❡ ❡ ✜❛❜✐❧✐❞❛❞❡ ❞♦s ♠♦❞❡❧♦s ✳ ✳ ✳ ✺✽

❘❡s✉♠♦

❖ tr❛❜❛❧❤♦ ♣r♦♣♦st♦ ❡♥q✉❛❞r❛✲s❡ ♥♦ â♠❜✐t♦ ❞❛ ❡st❛tíst✐❝❛ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ t❡♥❞♦ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ♣r✐♠ár✐♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ s♦❢t✇❛r❡ ❡st❛tís✲ t✐❝♦ ♣❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ P▲❙ ✲ P▼ ✭P❛rt✐❛❧ ▲❡❛st ❙q✉❛r❡ ✲ P❛t❤ ▼♦❞❡❧✐♥❣✮✳ ❖ ❛❧❣♦r✐t♠♦ ❢♦✐ ♣r♦❣r❛♠❛❞♦ ♥❛ s✉❛ t♦t❛❧✐❞❛❞❡ ❡♠ ❙❆❙✲■▼▲ ✭❙❆❙ ✲ ■♥t❡r❛❝t✐✈❡ ▼❛tr✐① ▲❛♥❣✉❛❣❡✮ ❡ ❝♦♠♣❧❡♠❡♥t❛❞♦ ❝♦♠ ❞♦✐s ♠ó❞✉❧♦s ❞❡ s✐♠✉❧❛çã♦ ♣❛r❛ ❛ ❡st✐♠❛çã♦ ❞❛ s✐❣♥✐✜❝â♥❝✐❛ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦✳ ❖ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠❛ ♣❛rt❡ ♠❡t♦❞♦❧ó❣✐❝❛ ❜❛s❡❛❞❛ ♥❛ r❡✈✐sã♦ ❞❛ ❧✐t❡r❛✲ t✉r❛ r❡❧❡✈❛♥t❡ ♥♦ â♠❜✐t♦ ❞♦ P▲❙ ✲ P▼✳ ❉❡ s❡❣✉✐❞❛ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞❛s t♦❞❛s ❛s ❢❛s❡s ❞❡ ❛♥á❧✐s❡ ❡ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ s♦❢t✇❛r❡ ❡♠ q✉❡stã♦✳

❋♦r❛♠ ✐♥❞✐❝❛❞♦s s✉❝❡ss✐✈♦s t❡st❡s r❡❛❧✐③❛❞♦s ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✱ ✉t✐✲ ❧✐③❛♥❞♦ ♦ s♦❢t✇❛r❡ ❞❡s❡♥✈♦❧✈✐❞♦ ❡ ♦✉tr♦ ♣r❡s❡♥t❡ ♥♦ ♠❡r❝❛❞♦✱ ❞❡ ♠♦❞♦ ❛ ❝♦♠♣❛r❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ ❝❛❞❛ ✉♠ ❡ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳ ❆ ❝♦♠♣❛r❛çã♦ ❞♦s ♦✉t♣✉ts ♦r✐❣✐♥♦✉ r❡s✉❧t❛❞♦s ♣♦s✐t✐✈♦s✱ ✐st♦ é✱ ♦s r❡s✉❧t❛❞♦s ♣r♦❞✉③✐❞♦s ❞♦s ❞♦✐s s♦❢t✇❛r❡s sã♦ ✐❣✉❛✐s✳

❆❧é♠ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❜❛s❡ ❛ss♦❝✐❛❞♦s ❛♦ P▲❙✲P▼✱ ♦ s♦❢t✇❛r❡ ❝r✐❛❞♦ ❞✐s✲ ♣♦♥✐❜✐❧✐③❛ ✉♠❛ ♥♦✈❛ té❝♥✐❝❛ ❞❡ s✐♠✉❧❛çã♦ q✉❡ ♣❡r♠✐t❡ ❛❞❡q✉❛r ♦ ❝á❧❝✉❧♦ ❞❛ s✐❣♥✐✜❝â♥❝✐❛ às ❡①✐❣ê♥❝✐❛s ❞♦ ✐♥✈❡st✐❣❛❞♦r✳ ❖✉tr❛ ✈❛♥t❛❣❡♠ q✉❡ ♦ ♣r♦❣r❛♠❛ tr❛③ é ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❡①❡❝✉t❛r ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛✭P▲❙✲P▼✮ q✉❡ ♥ã♦ ❡stá ✐♠♣❧❡♠❡♥t❛❞❛ ♥♦ ❛♠❜✐❡♥t❡ ❙❆❙✳

❆❜str❛❝t

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✇♦r❦ ✇❛s ❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ st❛t✐st✐❝s ❞✐s❝✐♣❧✐♥❡ ❛♥❞ ✐ts ♠❛✐♥ ❣♦❛❧ ✐s t❤❡ ❝r❡❛t✐♦♥ ♦❢ ❛ st❛t✐st✐❝ s♦❢t✇❛r❡ ❢♦r t❤❡ ✐♠✲ ♣❧❡♠❡♥t❛t✐♦♥ ♦❢ P▲❙ ✲ P▼ ✭P❛rt✐❛❧ ▲❡❛st ❙q✉❛r❡ ✲ P❛t❤ ▼♦❞❡❧✐♥❣✮ ❛❧❣♦r✐t❤♠✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❤❛s ❜❡❡♥ ❢✉❧❧② ♣r♦❣r❛♠♠❡❞ ✐♥ ❙❆❙✲■▼▲ ✭❙❆❙ ✲ ■♥t❡r❛❝t✐✈❡ ▼❛tr✐① ▲❛♥❣✉❛❣❡✮ ❛♥❞ ✐t ❝♦♥t❛✐♥s t✇♦ ❞✐✛❡r❡♥t s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s t❤❛t ❛❧❧♦✇ t❤❡ ❡✈❛❧✉❛t✐♦♥ ❛♥❞ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs s✐❣♥✐✜❝❛♥❝❡✳

❚❤✐s t❤❡s✐s ♣r❡s❡♥ts ❛ ♠❡t❤♦❞♦❧♦❣✐❝❛❧ ❝❤❛♣t❡r ❜❛s❡❞ ♦♥ t❤❡ ♠♦st r❡❧❡✈❛♥t ❧✐t❡r❛t✉r❡ ♦♥ P▲❙ ✲ P▼ ❛♥❞ ❛❧❧ t❤❡ ♣❤❛s❡s ♦❢ ❛♥❛❧②s✐s ❛♥❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠❡♥t✐♦♥❡❞ s♦❢t✇❛r❡✳

❙♦♠❡ t❡sts ✇❡r❡ ❛♣♣❧✐❡❞ t♦ ❛ s❡t ♦❢ ❞❛t❛✱ ✉s✐♥❣ t❤❡ ❝✉st♦♠✲❜✉✐❧t s♦❢t✇❛r❡ ❛♥❞ ♦t❤❡r s♦❢t✇❛r❡ ❛✈❛✐❧❛❜❧❡ ♦♥ t❤❡ ♠❛r❦❡t✳ ❚❤♦s❡ t❡sts ✇❡r❡ ♠❛❞❡ t♦ ❝♦♠✲ ♣❛r❡ ❛♥❞ ❛♥❛❧②③❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ r❡s✉❧ts ♦❢ t❤❡ ❞✐✛❡r❡♥t s♦❧✉t✐♦♥s✳ ❚❤❡ ♦✉t♣✉t ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ❞✐✛❡r❡♥t ♣r♦❣r❛♠s ♣r♦❞✉❝❡❞ ♣♦s✐t✐✈❡ r❡s✉❧ts ❜❡❝❛✉s❡ t❤❡✐r ♦✉t♣✉t ✇❛s ❡q✉✐✈❛❧❡♥t✳

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❆ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s é ❤♦❥❡ ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❡st❛tíst✐❝❛ ❝♦♥s♦❧✐❞❛❞❛✱ q✉❡ t❡♠

❝♦♠♦ ♦❜❥❡❝t✐✈♦ ❛ ❛♥á❧✐s❡ ❞❡ ❢❡♥ó♠❡♥♦s r❡❛✐s ❝♦♠♣❧❡①♦s✱ ❛tr❛✈és ❞❛ ♦❜s❡r✲

✈❛çã♦ ❝♦♥❥✉♥t❛ ❞❡ ❞✐✈❡rs❛s ✈❛r✐á✈❡✐s ❛ss♦❝✐❛❞❛s ❛♦ ❢❡♥ó♠❡♥♦ ❡♠ ❡st✉❞♦ ❡

❞❛s r❡❧❛çõ❡s ❡①✐st❡♥t❡s ❡♥tr❡ ❛s ♠❡s♠❛s✳ P❛r❛ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞❛ ❝♦♠♣❧❡✲

①✐❞❛❞❡ ❡①✐st❡♥t❡ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ♦❜s❡r✈❛❞❛s✱ é ❝♦♠ ❢r❡q✉ê♥❝✐❛ ♥❡❝❡ssár✐♦

❡str✉t✉r❛r ✉♠ ♠♦❞❡❧♦✱ ❞❡ ♠♦❞♦ ❛ ❡st✐♠❛r ❛s r❡❧❛çõ❡s q✉❡ ❡①✐st❡♠ ❡♥tr❡ ❛s

♠❡s♠❛s✳ ❖s ♠♦❞❡❧♦s ❡st❛tíst✐❝♦s tê♠ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡s❝r❡✈❡r✱

❡①♣❧✐❝❛r✱ ❡st✐♠❛r✱ r❡♣r♦❞✉③✐r ❡ ✐♥✈❡st✐❣❛r ❢❡♥ó♠❡♥♦s r❡❛✐s ❡ r❡♣r❡s❡♥t❛♠ ✉♠❛

❡①❡♠♣❧✐✜❝❛çã♦ ❞♦ ♠✉♥❞♦ r❡❛❧✳ ❉❡ ❡♥tr❡ ❛s ❞✐❢❡r❡♥t❡s té❝♥✐❝❛s ❡①✐st❡♥t❡s ♣❛r❛

❛ ❛♥á❧✐s❡ ❞❛s r❡❧❛çõ❡s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❞❡ ✉♠ s✐st❡♠❛✱ r❡❛❧ç❛♠✲s❡ ♦s ▼♦❞❡❧♦s

❞❡ ❊q✉❛çõ❡s ❊str✉t✉r❛✐s ✭❙tr✉❝t✉r❛❧ ❊q✉❛t✐♦♥ ▼♦❞❡❧s ✲ ❙❊▼✮ q✉❡ r❡♣r❡s❡♥✲

■♥tr♦❞✉çã♦

❋❛❝t♦r ❆♥❛❧②s✐s ✭❈❋❆✮ ✭❚❤✉rst♦♥❡ ✶✾✸✶✮✳

❖s ♠♦❞❡❧♦s ❞❡ ❊q✉❛çõ❡s ❊str✉t✉r❛✐s ❢♦r❛♠ ✐♥tr♦❞✉③✐❞♦s ♣♦r ❏ör❡s❦♦❣ ❡♠

✶✾✼✵✱ ❢❛③✐❛♠ ♣❛rt❡ ❞♦ ❣r✉♣♦ ❞❡ ♠♦❞❡❧♦s ❝♦♥✜r♠❛tór✐♦s ❡ t✐♥❤❛♠ ♦ ♦❜❥❡❝t✐✈♦

❞❡ ❛✈❛❧✐❛r ❛s r❡❧❛çõ❡s ❞❡ ❝❛✉s❛ ❡❢❡✐t♦ ❡①✐st❡♥t❡s ❡♥tr❡ ❞✉❛s ♦✉ ♠❛✐s ✈❛r✐á✈❡✐s

❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❡ ❡st✐♠❛çã♦ ❞❛ ♠á①✐♠❛ ✈❡r♦s✐♠✐❧❤❛♥ç❛ ✭▼❛①✐♠✉♠ ▲✐✲

❦❡❧✐❤♦♦❞✱ ▼▲✮❀ ❡st❡ ♠ét♦❞♦ é ❞❡s✐❣♥❛❞♦ ♣♦r ❙❊▼✲▼▲ ♦✉ ▲■❙❘❊▲✶_{✳ ❆♦ ❧♦♥❣♦}

❞♦s ❛♥♦s ❡st❡ ❢♦✐ ♦ ú♥✐❝♦ ♠ét♦❞♦ ❞❡ ❡st✐♠❛çã♦ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❞❡ ❊q✉❛✲

çõ❡s ❊str✉t✉r❛✐s✳ ❖ ♣r✐♠❡✐r♦ s♦❢t✇❛r❡ ❛ss♦❝✐❛❞♦ ❛ ❡st❡s ♠♦❞❡❧♦s ❢♦✐ ▲■❙❘❊▲✱

❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❏ör❡s❦♦❣ ❡ ❙ör❜♦♠ ❡♠ ✶✾✾✻✳

❙✉❝❡ss✐✈❛♠❡♥t❡✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ♠✉✐t❛s ♦✉tr❛s té❝♥✐❝❛s ❞❡ ❡st✐♠❛çã♦✱

❝♦♠♦ ●▲❙ ✭●❡♥❡r❛❧✐③❡❞ ▲❡❛st ❙q✉❛r❡s✮ ❡ ❆❉❋ ✭❆s②♠♣t♦t✐❝❛❧❧② ❉✐str✐❜✉t✐♦♥

❋r❡❡✮✳ ❚♦❞❛s ❛s té❝♥✐❝❛s ❡♥✉♥❝✐❛❞❛s ❢❛③❡♠ ♣❛rt❡ ❞♦ ❣r✉♣♦ ❞♦s ♠ét♦❞♦s

❝♦✈❛r✐❛♥❝❡✲❜❛s❡❞✷_✳

❖ P▲❙✲P▼ ♥❛s❝❡✉ ❡♠ ✶✾✼✺ ❝♦♠ ♦ ♦❜❥❡❝t✐✈♦ ❞❡ t♦r♥❛r ♠❛✐s ✢❡①í✈❡❧ ❛ ✉t✐❧✐✲

③❛çã♦ ❞❡st❡s ♠ét♦❞♦s❀ ❞❡ ❢❛❝t♦✱ ❢♦r❛♠ ❛♥✉❧❛❞❛s ❛s ❤✐♣ót❡s❡s ❢♦r♠✉❧❛❞❛s ♥♦s

♠ét♦❞♦s ❝♦✈❛r✐❛♥❝❡✲❜❛s❡❞✳ ◆♦s ❛♥♦s s❡❣✉✐♥t❡s✱ ❢♦r❛♠ ✐♥ú♠❡r❛s ❛s ✈❛r✐❛♥✲

t❡s ❢♦r♠❛❧✐③❛❞❛s ♣❛r❛ ♦s ♠ét♦❞♦s ♠♦❞❡❧✲❜❛s❡❞ ❡ ✐♥ú♠❡r♦s ❢♦r❛♠ t❛♠❜é♠✱

s♦❜r❡t✉❞♦ ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ♦s s♦❢t✇❛r❡s ❛♣r❡s❡♥t❛❞♦s✳

❍♦❥❡✱ ❛ ❛♥á❧✐s❡ P▲❙✲P▼ é r❡❝♦♥❤❡❝✐❞❛ ❡♠ ❞✐✈❡rs❛s ár❡❛s✳ ❖s ❡st✉❞♦s q✉✐✲

✶_{❖ ▲■❙❘❊▲ é ♦ ♣r✐♠❡✐r♦ s♦❢t✇❛r❡ ❞✐s♣♦♥✐❜✐❧✐③❛❞♦ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞♦s ♠♦❞❡❧♦s ❙❊▼✲▼▲✱}

t❡♥❞♦✲s❡ ✈✐♥❞♦ ❛ ❛✜r♠❛r ❝♦♠♦ s✐♥ó♥✐♠♦ ❞❡ss❡s ♠♦❞❡❧♦s ❞❡ ❡q✉❛çõ❡s ❡str✉t✉r❛✐s✳

✷_{❖s ♠ét♦❞♦s ❝♦✈❛r✐❛♥❝❡✲❜❛s❡❞ ❡❢❡❝t✉❛♠ ❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❛tr❛✈és}

❞❛ ♠✐♥✐♠✐③❛çã♦ ❞❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ❛s ♠❛tr✐③❡s ❞❡ ❝♦✈❛r✐â♥❝✐❛ ❡st✐♠❛❞❛ ❡ ❞❛ ❛♠♦str❛✳

■♥tr♦❞✉çã♦

♠✐♦♠étr✐❝♦s ✭❏✐❛♥❣✱ ❇❡rr②✱ ❙✐❡s❧❡r ✫ ❖③❛❦✐ ✷✵✵✷✮✱ ♣♦r ❡①❡♠♣❧♦✱ r❡♣r❡s❡♥t❛♠

♦ ♣r✐♠❡✐r♦ s❡❝t♦r ♦♥❞❡ ♦ P▲❙ ✲ P▼ ❢♦✐ ❛♣❧✐❝❛❞♦✳ ❍♦❥❡ ❡♠ ❞✐❛ ❡st❛ té❝♥✐❝❛

❡♥❝♦♥tr❛ ❛ s✉❛ ♥❛t✉r❛❧ ❝♦❧♦❝❛çã♦ ♥♦ ❝❛♠♣♦ s♦❝✐♦❡❝♦♥ó♠✐❝♦✳ ❉❡ ❢❛❝t♦✱ é

♠✉✐t❛s ✈❡③❡s ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛ ❛✈❛❧✐❛çã♦ ❞❛ s❛t✐s❢❛çã♦ ❝♦♠ ♣r♦❞✉t♦s ❡✴♦✉ s❡r✲

✈✐ç♦s✱ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞❛s r❡❧❛çõ❡s ❝❛✉s❛✐s ❡♥tr❡ ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s ✭❋♦r♥❡❧❧✱

❏♦❤♥s♦♥✱ ❆♥❞❡rs♦♥✱ ❈❤❛ ✫ ❇r②❛♥t ✷✵✵✷✮✳

❆♣❡s❛r ❞❡ t❡r ♥❛s❝✐❞♦ ❡♠ ✶✾✼✺ ❝♦♠ ❲♦❧❞✱ ♦ P▲❙✲P▼ ❡stá✱ ❛✐♥❞❛ ❤♦❥❡✱ ♥♦

❝❡♥tr♦ ❞❡ ✐♥✈❡st✐❣❛çõ❡s q✉❡ tê♠ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ❛ ♠❡❧❤♦r✐❛ ❡ ❞✐✈❡rs✐✜❝❛çã♦

❞❛ té❝♥✐❝❛✳

❊st❡ ♣r♦❥❡❝t♦ ❛♣r❡s❡♥t❛ ✉♠❛ ♣r✐♠❡✐r❛ s❡❝çã♦ ❡str✐t❛♠❡♥t❡ t❡ór✐❝❛✱ ❞❡❞✐❝❛❞❛ à

❞❡✜♥✐çã♦ ❞♦ ♠♦❞❡❧♦ P▲❙✲P▼ ❡ ❞❛s s✉❛s ❞✐✈❡rs❛s ❝♦♠♣♦♥❡♥t❡s❀ ✉♠❛ s❡❣✉♥❞❛

♣❛rt❡ q✉❡ ❡stá t♦t❛❧♠❡♥t❡ ❢♦❝❛❞❛ ♥❛ ❡str✉t✉r❛çã♦ ❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦

❛❧❣♦r✐t♠♦ ❛ss♦❝✐❛❞♦ ❛♦ ♠ét♦❞♦ ❞❡✜♥✐❞♦ ♥❛ ♣r✐♠❡✐r❛ s❡❝çã♦✳

P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ ♣r♦❥❡❝t♦ ❢♦✐ ✉t✐❧✐③❛❞♦ ♦ s♦❢t✇❛r❡ ❡st❛tíst✐❝♦ ❙❆❙✱ ♠❛✐s

❡s♣❡❝✐✜❝❛♠❡♥t❡ ♦ ♠ó❞✉❧♦ ■▼▲ ✭■♥t❡r❛❝t✐✈❡ ▼❛tr✐① ▲❛♥❣✉❛❣❡✮✳ ❉❡ ❡♥tr❡ ♦s

✈ár✐♦s s♦❢t✇❛r❡s✱ ❢♦✐ ❡s❝♦❧❤✐❞♦ ♦ ❙❆❙ ♣♦rq✉❡ ♥ã♦ ♣♦ss✉✐ ♥❡♥❤✉♠ ♣r♦❝❡❞✐♠❡♥t♦

✐♥t❡r♥♦ ♣❛r❛ ❡st❡ t✐♣♦ ❞❡ ❛♥á❧✐s❡ ✭P▲❙✲P▼✮✳ ❉❡♥tr♦ ❞♦ ❙❆❙ ❢♦✐ s❡❧❡❝❝✐♦♥❛❞♦

♦ ♠ó❞✉❧♦ ■▼▲ ♣♦rq✉❡ ❡r❛ ♦ ú♥✐❝♦ q✉❡ ♣❡r♠✐t✐❛ ✉♠❛ ❡①❡❝✉çã♦ ♦♣t✐♠✐③❛❞❛

❞♦ ❛❧❣♦r✐t♠♦✱ s❡♥❞♦ q✉❡✱ ❛tr❛✈és ❞❡st❡ ♠ó❞✉❧♦✱ ❢♦✐ ♣♦ssí✈❡❧ ♣r♦❣r❛♠❛r ♥✉♠❛

❧✐♥❣✉❛❣❡♠ ♠❛tr✐❝✐❛❧ ❜❛s❡❛❞❛ ♥❛ á❧❣❡❜r❛ ❧✐♥❡❛r✳

❊st❡ ♣r♦❥❡❝t♦ ❛♣r❡s❡♥t❛ ❛ ♣❛rt✐❝✉❧❛r✐❞❛❞❡ ❞❡ ♣♦❞❡r s❡r ♣♦t❡♥❝✐❛❞♦✱ t♦r♥❛♥❞♦✲

■♥tr♦❞✉çã♦

s❡ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❞❡ ❛rr❛♥q✉❡ ♣❛r❛ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❢✉t✉r♦s✳ ❊st❛ ❝❛r❛❝✲

t❡ríst✐❝❛ ❞❡✈❡✲s❡ à ❝✐r❝✉♥stâ♥❝✐❛ ❞❡ t❡r s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ❡ ❧✐✈r❡♠❡♥t❡ ❞✐s♣♦✲

♥✐❜✐❧✐③❛❞♦ ♣❛r❛ ❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛✳ ◆❡ss❛ ♣❡rs♣❡❝t✐✈❛✱ r❡♣r❡s❡♥t❛ ✉♠

♣♦♥t♦ ❞❡ ❝❤❡❣❛❞❛ ❡ ✉♠ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛✳

▼❛✐s ❝♦♥❝r❡t❛♠❡♥t❡✱ ❢♦✐ ♣r❡♦❝✉♣❛çã♦ ❝❡♥tr❛❧ ❛ ❞❡ ❝♦♥t❡♠♣❧❛r ♦s s❡❣✉✐♥t❡s

❡❧❡♠❡♥t♦s✿

• ■♥tr♦❞✉çã♦ ❞♦s ❞❛❞♦s ✐♥✐❝✐❛✐s ❡♠ ❊①❝❡❧❀

• ❖♣t✐♠✐③❛çã♦ ❞❡ ❛❧❣✉♠❛s ♣❛rt❡s ❞♦ ❛❧❣♦r✐t♠♦ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ♣❡✲

s❛❞❛s❀

• ❊①♣♦rt❛çã♦ ❞♦s ♦✉t♣✉ts ❛♥❛❧ít✐❝♦s ❡♠ ❊①❝❡❧✳

❘❡❧❛t✐✈❛♠❡♥t❡ à ❡str✉t✉r❛ ❞♦ tr❛❜❛❧❤♦✱ ❡st❛ ❛♣r❡s❡♥t❛✲s❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

• ❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ s❡r✈❡ ❞❡ ✐♥tr♦❞✉çã♦ ❛♦ ❛❧❣♦r✐t♠♦ P▲❙✲P▼✳ ◆❡st❛

s❡❝çã♦ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❡ ❛s

❡st✐♠❛çõ❡s ❞♦ ♠♦❞❡❧♦ ❡①t❡r♥♦ ❡ ❞♦ ♠♦❞❡❧♦ ✐♥t❡r♥♦❀

• ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ s❡rá ❢♦r♠❛❧✐③❛❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝á❧❝✉❧♦ ❞♦ P▲❙✲

P▼✱ t❡♥❞♦ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ s♦❢t✇❛r❡❀

• ◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡s❡♥✈♦❧✲

✈✐❞♦ ❝♦♠ ❞❛❞♦s r❡❛✐s✳ ❖s r❡s✉❧t❛❞♦s s❡rã♦ s✉❝❡ss✈❛♠❡♥t❡ ❝♦♠♣❛r❛❞♦s

❝♦♠ ♦s r❡s✉❧t❛❞♦s ♣r♦❞✉③✐❞♦s ♣❛r❛ ♦✉tr♦ s♦❢t✇❛r❡ ♣r❡s❡♥t❡ ♥♦ ♠❡r❝❛❞♦❀

■♥tr♦❞✉çã♦

• ❆ ú❧t✐♠❛ s❡❝çã♦ ❞❛ t❡s❡ s❡rá ❞❡❞✐❝❛❞❛ à sí♥t❡s❡✱ às ❝♦♥❝❧✉sõ❡s ❡ ❛♦s

♣♦ssí✈❡✐s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❢✉t✉r♦s ❞♦ ♣r♦❥❡❝t♦✳

❈❛♣ít✉❧♦ ✷

❖ ❆❧❣♦r✐t♠♦ P▲❙✲P▼

✷✳✶ ❊♥q✉❛❞r❛♠❡♥t♦ ●❡r❛❧

❉♦s ▼♦❞❡❧♦s ❞❡ ❊q✉❛çõ❡s ❊str✉t✉r❛✐s ❛♦ P▲❙✲P▼

❖s ♠♦❞❡❧♦s ❞❡ ❡q✉❛çõ❡s ❡str✉t✉r❛✐s ✭❙❊▼✮ ✐♥❝❧✉❡♠ ❞✐✈❡rs❛s té❝♥✐❝❛s ❡st❛tís✲

t✐❝❛s q✉❡ ♣❡r♠✐t❡♠ ❛ ❡st✐♠❛çã♦ ❞❡ r❡❧❛çõ❡s ❝❛✉s❛✐s✱ ❞❡✜♥✐❞❛s ❝♦♠ ❜❛s❡ ♥✉♠

♠♦❞❡❧♦ t❡ór✐❝♦✱ ❛tr❛✈és ❞❛s ❧✐❣❛çõ❡s ❡①✐st❡♥t❡s ❡♥tr❡ ❞♦✐s ♦✉ ♠❛✐s ❝♦♥❝❡✐t♦s

❧❛t❡♥t❡s✳ ❊st❡s ❝♦♥❝❡✐t♦s sã♦ ♠❡❞✐❞♦s ❛tr❛✈és ❞❡ ✉♠ ♦✉ ♠❛✐s ✐♥❞✐❝❛❞♦r❡s

♦❜s❡r✈á✈❡✐s✳

❆ ✐❞❡✐❛ q✉❡ s✉❜❥❛③ ❛ ❡st❡s ♠♦❞❡❧♦s é ❛ ❞❡ q✉❡ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❡①✐st❡♥t❡

✷✳✶✳ ❊♥q✉❛❞r❛♠❡♥t♦ ●❡r❛❧

❧❛çõ❡s ❝❛✉s❛✐s ❡♥tr❡ ❝♦♥❝❡✐t♦s ❧❛t❡♥t❡s ❞❡♥♦♠✐♥❛❞♦s ❱❛r✐á✈❡✐s ▲❛t❡♥t❡s ✭❱▲✮✱

❝❛❞❛ ✉♠❛ ♠❡❞✐❞❛ ❛tr❛✈és ❞❡ ✐♥❞✐❝❛❞♦r❡s ❞❡✜♥✐❞♦s ❝♦♠♦ ❱❛r✐á✈❡✐s ▼❛♥✐❢❡st❛s

✭❱▼✮✳

❖s ♠♦❞❡❧♦s ❞❡ ❡q✉❛çõ❡s ❡str✉t✉r❛✐s r❡♣r❡s❡♥t❛♠ ✉♠ ♣♦♥t♦ ❝♦♠✉♠ ❡♥tr❡ ❛

♣❛t❤ ❛♥❛❧②s✐s✭P❆✮ ❡ ❛ ❛♥á❧✐s❡ ❢❛❝t♦r✐❛❧ ❝♦♥✜r♠❛tór✐❛ ✭❈❋❆✮ ✭❚r✐♥❝❤❡r❛ ✷✵✵✼✮✳

❖ ♣♦♥t♦ ❝♦♠✉♠ ❡♥tr❡ ♦s ❙❊▼ ❡ ❛ ❈❋❆ é r❡❧❛t✐✈♦ ❛♦ s✐❣♥✐✜❝❛❞♦ ❞♦s ❜❧♦❝♦s

❞❡ ✈❛r✐á✈❡✐s❀ ❞❡ ❢❛❝t♦✱ ❡♠ ❛♠❜❛s ❛s té❝♥✐❝❛s✱ ❝❛❞❛ ❜❧♦❝♦ ♦✉ s✉❜❝♦♥❥✉♥t♦ ❞❡

✈❛r✐á✈❡✐s é ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ s❡♥❞♦ ❡①♣r❡ssã♦ ❞❡ ✉♠ ú♥✐❝♦ ❝♦♥❝❡✐t♦✳

❆ P❛t❤ ❆♥❛❧②s✐s t❡♠ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ♠♦❞❡❧♦s r❡❧❛❝✐♦♥❛✐s ❝♦♠

❡❢❡✐t♦s ❞✐r❡❝t♦s ❡ ✐♥❞✐r❡❝t♦s ❡♥tr❡ ✈❛r✐á✈❡✐s ♦❜s❡r✈❛❞❛s✳ ◗✉❛♥❞♦ ❛s ✈❛r✐á✈❡✐s

♣r❡s❡♥t❡s ♥♦ ♠♦❞❡❧♦ r❡❧❛❝✐♦♥❛❧ sã♦ ❧❛t❡♥t❡s ❡ sã♦ ♠❡❞✐❞❛s ❛tr❛✈és ❞❡ ✉♠

❝♦♥❥✉♥t♦ ❞❡ ✐♥❞✐❝❛❞♦r❡s ♦❜s❡r✈❛❞♦s✱ ❛ P❛t❤ ❆♥❛❧②s✐s é ❞❡✜♥✐❞❛ ♣♦r ▼♦❞❡❧♦s

❞❡ ❊q✉❛çõ❡s ❊str✉t✉r❛✐s✳

❆s té❝♥✐❝❛s q✉❡ s❡ ❡♥q✉❛❞r❛♠ ❞❡♥tr♦ ❞♦s ♠♦❞❡❧♦s ❞❡ ❡q✉❛çõ❡s ❡str✉t✉r❛✐s✱

♣♦❞❡♠ ❞✐✈✐❞✐r✲s❡ ❡♠ ❞♦✐s ❣r✉♣♦s✿ ❈♦✈❛r✐❛♥❝❡ ❇❛s❡❞ ❡ ❈♦♠♣♦♥❡♥t ❇❛s❡❞✳

◆♦s ♠♦❞❡❧♦s ❈♦✈❛r✐❛♥❝❡ ❇❛s❡❞ ♦s ♠ét♦❞♦s ❞❡ ❡st✐♠❛çã♦ ✉t✐❧✐③❛❞♦s sã♦✱ ❡♥✲

tr❡ ♦✉tr♦s✿ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞✱ ●❡♥❡r❛❧✐③❡❞ ▲❡❛st ❙q✉❛r❡s ❡ ❆ss✐♠♣t♦t✐❝❛❧❧②

❉✐str✐❜✉t✐♦♥ ❋r❡❡✳ ❆s té❝♥✐❝❛s q✉❡ ✉t✐❧✐③❛♠ ❡st❡s ♠ét♦❞♦s ❞❡ ❡st✐♠❛çã♦ tê♠

❝♦♠♦ ♦❜❥❡❝t✐✈♦ ❛ r❡♣r♦❞✉çã♦ ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ ❞❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡s✲

t❛s✳ ❆ ❤✐♣ót❡s❡ ❢✉♥❞❛♠❡♥t❛❧ s✉❜❥❛❝❡♥t❡ ❛ ❡st❛s ❛❜♦r❞❛❣❡♥s é q✉❡ ❛ ♠❛tr✐③

❞❡ ❝♦✈❛r✐â♥❝✐❛ s❡❥❛ ❢✉♥çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦✳ ❊st❛ ❛❜♦r❞❛❣❡♠ é

✷✳✶✳ ❊♥q✉❛❞r❛♠❡♥t♦ ●❡r❛❧

t❛♠❜é♠ ❞❡✜♥✐❞❛ ❝♦♠♦ ❤❛r❞ ♠♦❞❡❧✐♥❣✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ♦✉tr❛s té❝♥✐❝❛s ♠❡♥♦s ❡①✐❣❡♥t❡s ❡♠ t❡r✲

♠♦s ❞❡ ♣r❡ss✉♣♦st♦s ❡st❛tíst✐❝♦s✳ ❊st❛ ❛❜♦r❞❛❣❡♠ é ❞❡s✐❣♥❛❞❛ ❝♦♠♦ ❈♦♠✲

♣♦♥❡♥t ❇❛s❡❞ ✭♦✉ s♦❢t ♠♦❞❡❧✐♥❣✮✳ ❖ P▲❙✲P▼ ❡♥q✉❛❞r❛✲s❡ ♥❡st❡ ❝♦♥❥✉♥t♦

❞❡ té❝♥✐❝❛s✱ ✉t✐❧✐③❛♥❞♦ ♦ P❛rt✐❛❧ ▲❡❛st ❙q✉❛r❡s ❝♦♠♦ ♠ét♦❞♦ ❞❡ ❡st✐♠❛çã♦✳

❊st❛s té❝♥✐❝❛s ✈✐s❛♠ ❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❛tr❛✈és ❞❛ ♠✐♥✐♠✐③❛çã♦ ❞❛

✈❛r✐â♥❝✐❛✳

❊♥q✉❛❞r❛♠❡♥t♦ ❍✐stór✐❝♦

❆ ❡✈♦❧✉çã♦ ❤✐stór✐❝❛ ❞❛ ❛❜♦r❞❛❣❡♠ P▲❙ ✭P❛rt✐❛❧ ▲❡❛st ❙q✉❛r❡s✮ t❡✈❡ ✐♥✐❝✐♦

❡♠ ✶✾✻✻✱ q✉❛♥❞♦ ❍❡r♠❛♥ ❲♦❧❞ ❢♦r♠❛❧✐③♦✉ ❛ ✐❞❡✐❛ ❞❡ P▲❙ ♥✉♠ ❛rt✐❣♦ ❞❡✲

❞✐❝❛❞♦ à ❆♥á❧✐s❡ ❡♠ ❈♦♠♣♦♥❡♥t❡s Pr✐♥❝✐♣❛✐s ✭❲♦❧❞ ✶✾✻✻✮✳ ◆❡st❡ ❛rt✐❣♦ ❢♦✐

✐♥tr♦❞✉③✐❞♦ ♦ ❛❧❣♦r✐t♠♦ ◆■▲❊❙ ✭◆♦♥❧✐♥❡❛r ■t❡r❛t✐✈❡ ▲❊❛st ❙q✉❛r❡s✮✳

❆ ♣❛rt✐r ❞❡ ✶✾✼✸✱ ♦ ♠❡s♠♦ ❛✉t♦r ❡st❡♥❞❡✉ ❡st❛ té❝♥✐❝❛ à ❛♥á❧✐s❡ ❞❛s ❝♦rr❡❧❛✲

çõ❡s ❝❛♥ó♥✐❝❛s ❡ ♦ ❛❧❣♦r✐t♠♦ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ♣♦r ◆■P❆▲❙ ✭◆♦♥❧✐♥❡❛r ■t❡r❛t✐✈❡

P❆rt✐❛❧ ▲❡❛st ❙q✉❛r❡s✮ ✭❲♦❧❞ ✶✾✼✸✱ ❲♦❧❞ ✶✾✼✺✮✳

❆ ♣r✐♠❡✐r❛ ✈❡rsã♦ ✜♥❛❧✐③❛❞❛ ❞♦ ❛❧❣♦r✐t♠♦ P▲❙ ❛♣❧✐❝❛❞♦ às ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s

❢♦✐ ❛♣r❡s❡♥t❛❞❛ ❡♠ ✶✾✼✾✳ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s r❡❧❛t✐✈❛s ❛

❡st❛ ❛❜♦r❞❛❣❡♠ sã♦ ✭❏ör❡s❦♦❣ ✫ ❲♦❧❞ ✶✾✽✷✱ ❲♦❧❞ ✶✾✽✷✮✳

❈♦♠ ❡st❛ ♥♦✈❛ ❛❜♦r❞❛❣❡♠✱ ❲♦❧❞ ❝♦♥s❡❣✉✐✉ ❢❛③❡r ♦♣♦s✐çã♦ ❛♦ ❙❊▼✲▼▲ ✭❏ör❡s❦♦❣

✷✳✶✳ ❊♥q✉❛❞r❛♠❡♥t♦ ●❡r❛❧

✶✾✼✵✮ ❤❛r❞ ♠♦❞❡❧✐♥❣✶ _{❛tr❛✈és ❞♦ P▲❙ s♦❢t ♠♦❞❡❧✐♥❣}✷_{✳ ❊♠ ✶✾✽✷ ❢♦✐ ❛♣r❡✲}

s❡♥t❛❞♦ ✉♠ ❛rt✐❣♦ ❝♦♠ ❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ❛s ❛❜♦r❞❛❣❡♥s ❙❊▼✲▼▲ ❡ P▲❙

✭❏ör❡s❦♦❣ ✫ ❲♦❧❞ ✶✾✽✷✮✳

◆♦s ❛♥♦s s❡❣✉✐♥t❡s✱ ❛ ❛❜♦r❞❛❣❡♠ P▲❙ ❢♦✐ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❢♦r♠❛❧✐③❛r ❛ r❡❣r❡ssã♦

P▲❙ ✭❲♦❧❞✱ ▼❛rt❡♥s ✫ ❲♦❧❞ ✶✾✽✸✮✱ t❡♥❞♦ ❡st❛ té❝♥✐❝❛ t✐❞♦ ✐♠❡♥s♦ s✉❝❡ss♦

♥♦ ❝❛♠♣♦ q✉✐♠✐♦♠étr✐❝♦✳ ❆♠❜❛s ❛s té❝♥✐❝❛s ✭P▲❙ ♣❛r❛ ♦s ♣❛t❤ ♠♦❞❡❧s ❡

r❡❣r❡ssã♦ P▲❙✮ sã♦ ❞❡s✐❣♥❛❞❛s ❝♦♠♦ ❆❜♦r❞❛❣❡♠ P▲❙✳ P❛r❛ ❡✈✐t❛r ❝♦♥❢✉sõ❡s

❡♥tr❡ ❛♠❜❛s✱ ❚❡♥❡♥❤❛✉s✱ ❱✐♥③✐✱ ❈❤❛t❡❧✐♥ ❡ ▲❛✉r♦ ✭❚❡♥❡♥❤❛✉s✱ ❊s♣♦s✐t♦ ❱✐♥③✐✱

❈❤❛t❡❧✐♥ ✫ ▲❛✉r♦ ✷✵✵✺✮ s❡❣✉✐r❛♠ ✉♠❛ s✉❣❡stã♦ ❞❡ ❍✳ ▼❛rt❡♥s ❡ ❞❡❝✐❞✐r❛♠

✉t✐❧✐③❛r ♦ t❡r♠♦ P▲❙ P❛t❤ ▼♦❞❡❧✐♥❣ ♣❛r❛ ❞❡✜♥✐r ♦ P▲❙ ♣❛r❛ ♠♦❞❡❧♦s ❞❡

❡q✉❛çõ❡s ❡str✉t✉r❛✐s✳

❙❡❣✉✐❞❛♠❡♥t❡ à ❢♦r♠❛❧✐③❛çã♦ ❞❛ té❝♥✐❝❛✱ ❢♦r❛♠ ♣♦✉❝❛s ❛s t❡♥t❛t✐✈❛s ❞❡ ♣r♦✲

❣r❛♠❛çã♦ ❞❡ s♦❢t✇❛r❡s ♣❛r❛ ♦s ♠♦❞❡❧♦s ❞❡ ❡q✉❛çõ❡s ❡str✉t✉r❛✐s✳ ❉❡ ❢❛❝t♦✱

❞✉r❛♥t❡ ♠✉✐t♦s ❛♥♦s✱ ♦ ú♥✐❝♦ s♦❢t✇❛r❡ ❞✐s♣♦♥í✈❡❧ ♣❛r❛ ❡st❡s ♠♦❞❡❧♦s ❢♦✐ ♦

▲❱P▲❙ ✶✳✽ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ▲♦❤♠ö❧❧❡r ❡♠ ✶✾✽✼ ✭❞❛t❛ ❞❛ ú❧t✐♠❛ ✈❡rsã♦ ❞✐s✲

♣♦♥✐❜✐❧✐③❛❞❛✮✳ P❛r❛ ❡♥❝♦♥tr❛r ♦✉tr♦ s♦❢t✇❛r❡ ❢♦✐ ♣r❡❝✐s♦ ❡s♣❡r❛r ❛té ✶✾✽✾✱

❛♥♦ ❡♠ q✉❡ ❙❡❧❧✐♥ ✭❙❡❧❧✐♥ ✶✾✽✾✮ ❞❡s❡♥✈♦❧✈❡✉ ♦ P▲❙P❆❚❍✳

❆ s✐t✉❛çã♦ ❛❧t❡r♦✉✲s❡ r❛❞✐❝❛❧♠❡♥t❡ ♥♦s ú❧t✐♠♦s t❡♠♣♦s✳ ❍♦❥❡✱ ♦s ✐♥✈❡st✐❣❛✲

❞♦r❡s ❞❡st❛ ár❡❛ tê♠ ♠✉✐t❛s ♦♣çõ❡s ❞✐❢❡r❡♥t❡s ♥❛ ❡s❝♦❧❤❛ ❞♦ s♦❢t✇❛r❡✱ t❡♥❞♦

✶_{❙ã♦ ❞❡✜♥✐❞❛s ❤❛r❞ ♠♦❞❡❧✐♥❣ ❛s té❝♥✐❝❛s q✉❡ ❛♣r❡s❡♥t❛♠ ❢♦rt❡s ❤✐♣ót❡s❡s ❛❝❡r❝❛ ❞❛}

❞✐str✐❜✉✐çã♦✳

✷_{❈❛r❛❝t❡r✐③❛❞❛ ♣♦r ❛♣r❡s❡♥t❛r ♣♦✉❝❛s ❤✐♣ót❡s❡s ❛❝❡r❝❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛s ✈❛r✐á✈❡✐s ❡♠}

❛♥á❧✐s❡✳

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

❝♦♠ ♦ ♣❛ss❛r ❞♦s ❛♥♦s ♠❡❧❤♦r❛❞♦ ❛ ❢❛❝✐❧✐❞❛❞❡ ❞❡ ✉t✐❧✐③❛çã♦✳ ❖ ❢❛❝t♦ ❞♦s

s♦❢t✇❛r❡s s❡r❡♠ ♠❛✐s ✉s❡r ❢r✐❡♥❞❧② ♣❡r♠✐t✐✉ ❛ ❡✈♦❧✉çã♦ ❞❡st❛ té❝♥✐❝❛ ❡♠ ❞✐✲

✈❡rs❛s ár❡❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♥♦ ♠❛r❦❡t✐♥❣ ❡ ♥❛ ✐♥✈❡st✐❣❛çã♦ ♦r❣❛♥✐③❛❝✐♦♥❛❧

✭❉✐❛♠❛♥t♦♣♦✉❧♦s ✫ ❲✐♥❦❧❤♦❢❡r ✷✵✵✶✮✳

❆❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s s♦❢t✇❛r❡s q✉❡ s❡ ❡♥❝♦♥tr❛♠ ❛❝t✉❛❧♠❡♥t❡ ♥♦ ♠❡r❝❛❞♦

sã♦✿ P▲❙✲●❯■ ✭▲✐ ✷✵✵✺✮✱ ❱✐s✉❛❧P▲❙ ✭❋✉ ✷✵✵✻✮✱ P▲❙✲●r❛♣❤ ✭❈❤✐♥ ✷✵✵✶✮✱

❙P❆❉✲P▲❙ ✭❚❡st ✫ ●♦ ✷✵✵✻✮ ❡ ❙♠❛rtP▲❙ ✭❘✐♥❣❧❡✱ ❲❡♥❞❡ ✫ ❲✐❧❧ ✷✵✵✺✮✳ ◆❛

♠❛✐♦r✐❛ ❞❡st❡s s♦❢t✇❛r❡s✱ ❥á sã♦ ❝♦♥t❡♠♣❧❛❞❛s ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ♠❛✐s

❛✈❛♥ç❛❞❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ tr❛t❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❡♠ ❢❛❧t❛✱ r❡✲❛♠♦str❛❣❡♠✱

✈❛❧✐❞❛çã♦ ❝r✉③❛❞❛✱ ❡t❝✳

❯♠❛ ó♣t✐♠❛ r❡❢❡rê♥❝✐❛ ♣❛r❛ ❛ ❝♦♠♣❛r❛çã♦ ❞♦s ❞✐✈❡rs♦s s♦❢t✇❛r❡s ❞✐s♣♦♥í✈❡✐s

é ✭❚❡♠♠❡✱ ❑r❡✐s ✫ ❍✐❧❞❡❜r❛♥❞t ✷✵✵✺✮✳

✷✳✷ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r ❛ ♠❡t♦❞♦❧♦❣✐❛ r❡❧❛t✐✈❛ ❛♦ P▲❙✲P▼✱ é ♥❡❝❡ssár✐♦ ✐♥tr♦✲

❞✉③✐r ❛❧❣✉♠❛s ♥♦t❛çõ❡s ❣rá✜❝❛s ✭❋✐❣✉r❛ ✷✳✶✮✳ ❖s ♦❜❥❡❝t♦s ❣rá✜❝♦s ♣r❡s❡♥t❡s

♥❛ r❡♣r❡s❡♥t❛çã♦ ❞♦s ♠♦❞❡❧♦s P▲❙✲P▼ sã♦✿

• ❆s ❡❧✐♣s❡s ✭♦✉ ♦s ❝ír❝✉❧♦s✮ sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s❀

• ❖s r❡❝tâ♥❣✉❧♦s ✭♦✉ ♦s q✉❛❞r❛❞♦s✮ sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛s ✈❛r✐á✈❡✐s ♠❛✲

♥✐❢❡st❛s❀

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

• ❆s s❡t❛s r❡♣r❡s❡♥t❛♠ ❛s r❡❧❛çõ❡s ❝❛✉s❛✐s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ✭❧❛t❡♥t❡s

❡✴♦✉ ♠❛♥✐❢❡st❛s✮✱ s❡♥❞♦ q✉❡ ❛ ❞✐r❡❝çã♦ ❞❛s ♠❡s♠❛s ❞❡✜♥❡ ❛ ❞✐r❡❝çã♦

❞❛ r❡❧❛çã♦ ✭♣♦r ❡①❡♠♣❧♦✱ ❞❛❞❛ ✉♠❛ ❡s♣❡❝í✜❝❛ r❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ✈❛r✐á✲

✈❡✐s✱ ❛ ✈❛r✐á✈❡❧ q✉❡ r❡❝❡❜❡ ❛ s❡t❛ é ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ s❡♥❞♦ ❛ ✈❛r✐á✈❡❧

❡♥❞ó❣❡♥❛✮✳

❋✐❣✉r❛ ✷✳✶✿ ◆♦t❛çã♦ ●rá✜❝❛ ❞♦ ♠♦❞❡❧♦ P▲❙✲P▼

◆♦ ♠♦❞❡❧♦ ❞❡st❛❝❛♠✲s❡ ❞♦✐s s✉❜✲♠♦❞❡❧♦s✿ ♦ ♠♦❞❡❧♦ ❞❡ ♠❡❞✐❞❛ ✭♦✉ ♦✉t❡r

♠♦❞❡❧✮ ❡ ♦ ♠♦❞❡❧♦ ❡str✉t✉r❛❧ ✭♦✉ ✐♥♥❡r ♠♦❞❡❧✮✳ ❖ ♣r✐♠❡✐r♦ ❢❛③ r❡❢❡rê♥❝✐❛ às

r❡❧❛çõ❡s q✉❡ ❡①✐st❡♠ ❡♥tr❡ ♦s ✐♥❞✐❝❛❞♦r❡s ❡ ❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s✱ ♦ s❡❣✉♥❞♦ é

r❡❧❛t✐✈♦ às ❧✐❣❛çõ❡s ❡①✐st❡♥t❡s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s ❡♠ ❡st✉❞♦ ✭❋✐❣✉r❛

✷✳✷✮✳

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

❋✐❣✉r❛ ✷✳✷✿ ❖ ♠♦❞❡❧♦ P▲❙✲P▼

✷✳✷✳✶ ❖ ▼♦❞❡❧♦ ❞❡ ▼❡❞✐❞❛

❯♠❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ ξ é ✉♠❛ ✈❛r✐á✈❡❧ ✭♦✉ ❝♦♥❝❡✐t♦✮ ✐♥♦❜s❡r✈á✈❡❧✱ ❞❡s❝r✐t❛

❞❡ ❢♦r♠❛ ✐♥❞✐r❡❝t❛ ♣♦r ✉♠ ❜❧♦❝♦ ❞❡ ✈❛r✐á✈❡✐s ♦❜s❡r✈❛❞❛s xh ❞❡✜♥✐❞❛s ❝♦♠♦

✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s ✭♦✉ ✐♥❞✐❝❛❞♦r❡s✮✳ ❊①✐st❡♠ três ❢♦r♠❛s ❞❡ r❡❧❛❝✐♦♥❛r ❛s

✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s às s✉❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s✿ ▼♦❞♦ ❋♦r♠❛t✐✈♦✱ ▼♦❞♦ ❘❡✲

✢❡❝t✐✈♦ ❡ ▼■▼■❈ ✭▼✉❧t✐♣❧❡ ■♥❞✐❝❛t♦rs ❛♥❞ ▼✉❧t✐♣❧❡ ❈❛✉s❡s✮✳

▼♦❞♦ ❘❡✢❡❝t✐✈♦

◆♦s ♠♦❞❡❧♦s q✉❡ ✉t✐❧✐③❛♠ ♦ ♠♦❞♦ r❡✢❡❝t✐✈♦✱ ❝❛❞❛ ✈❛r✐á✈❡❧ ♠❛♥✐❢❡st❛ ❡stá r❡✲

❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ s✉❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ ❛tr❛✈és ❞❡ ✉♠❛ r❡❣r❡ssã♦ s✐♠♣❧❡s ✭❋✐❣✉r❛

✷✳✸✮✳

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

xh =πh0 +πhξ+ǫh ✭✷✳✶✮

♦♥❞❡ξ t❡♠ ♠é❞✐❛m ❡ ❞❡s✈✐♦ ♣❛❞rã♦ ✐❣✉❛❧ ❛ 1✳ ◆❡st❡ ❡sq✉❡♠❛ ❞❡ r❡❧❛❝✐♦♥❛✲

♠❡♥t♦✱ ❝❛❞❛ ✈❛r✐á✈❡❧ ♠❛♥✐❢❡st❛ xh r❡✢❡❝t❡ ❛ s✉❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ ξ✳

❆ ❡q✉❛çã♦ ✷✳✶ t❡♠ ❞❡ ♦❜❡❞❡❝❡r ❛ ✉♠❛ ❤✐♣ót❡s❡ ❞❡✜♥✐❞❛ ♣r❡❞✐❝t♦r s♣❡❝✐✜❝❛t✐♦♥

♣♦r ❍✳ ❲♦❧❞✿

E(xh|ξ) = πh0+πhξ ✭✷✳✷✮

❊st❛ ❤✐♣ót❡s❡ ✐♠♣❧✐❝❛ q✉❡ ♦ r❡sí❞✉♦ǫh t❡♠ ♠é❞✐❛ ✐❣✉❛❧ ❛ 0❡ ♥ã♦ ❡stá ❝♦rr❡✲

❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ ✈❛r✐á✈❡❧ ξ✳

❋✐❣✉r❛ ✷✳✸✿ ❖ ▼♦❞♦ ❘❡✢❡❝t✐✈♦

◆♦ ♠♦❞♦ r❡✢❡❝t✐✈♦✱ ♦ ❜❧♦❝♦ ❞❡ ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s t❡♠ q✉❡ s❡r ✉♥✐❞✐♠❡♥s✐♦✲

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

♥❛❧ ♥♦ s❡♥t✐❞♦ ❞❛ ❛♥á❧✐s❡ ❢❛❝t♦r✐❛❧✳ ❖ ♦❜❥❡❝t✐✈♦ é q✉❡ ❝❛❞❛ ❜❧♦❝♦ ❞❡ ✈❛r✐á✈❡✐s

♠❛♥✐❢❡st❛s t❡♠ q✉❡ ❝♦♥t❡r ✈❛r✐á✈❡✐s ❢♦rt❡♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞❛s✳

◗✉❛♥❞♦ s❡ tr❛❜❛❧❤❛ ❝♦♠ ❞❛❞♦s r❡❛✐s✱ é ✐♠♣♦rt❛♥tíss✐♠♦ ✈❡r✐✜❝❛r ❛ ✉♥✐❞✐♠❡♥✲

s✐♦♥❛❧✐❞❛❞❡ ❞♦ ❜❧♦❝♦ ❞❡ ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s✳

P❛r❛ ♦ ❝♦♥tr♦❧♦ ❞❛ ✉♥✐❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❡①✐st❡♠ três té❝♥✐❝❛s✿ ❛ ❛♥á❧✐s❡ ❡♠

❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s ❞♦ ❜❧♦❝♦ ❞❡ ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s✱ ♦α ❞❡ ❈r♦♥❜❛❝❤ ❡

♦ρ ❞❡ ❉✐❧❧♦♥✲●♦❧❞st❡✐♥✳

• ❆♥á❧✐s❡ ❡♠ ❈♦♠♣♦♥❡♥t❡s Pr✐♥❝✐♣❛✐s✿ ❡st❛ té❝♥✐❝❛ ♣❡r♠✐t❡ ❝♦♥tr♦❧❛r ❛

✉♥✐❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦s ✈❛❧♦r❡s ♣ró♣r✐♦s✳ ❙❡ ♦ ♣r✐✲

♠❡✐r♦ ✈❛❧♦r ♣ró♣r✐♦ ❢♦r s✉♣❡r✐♦r ❛ 1 ❡ ♦s ♦✉tr♦s ✐♥❢❡r✐♦r❡s ♦✉ ♥♦ ❧✐♠✐t❡

♠✉✐t♦ ❛❢❛st❛❞♦s ❞♦ ♣r✐♠❡✐r♦✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❜❧♦❝♦ ✉♥✐❞✐♠❡♥s✐✲

♦♥❛❧✳ ❆♣ós ❝♦♥tr♦❧♦ ❞♦ ✈❛❧♦r ♣ró♣r✐♦✱ é ♣r❡❝✐s♦ ❝♦♥tr♦❧❛r s❡ t♦❞❛s ❛s

✈❛r✐á✈❡✐s ❞♦ ❜❧♦❝♦ ❡stã♦ ♣♦s✐t✐✈❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ♦ ♣r✐♠❡✐r♦

❢❛❝t♦r✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ✉♠❛ ❞❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s

❡ ♦ ❢❛❝t♦r s❡❥❛ ♥❡❣❛t✐✈❛✱ s✐❣♥✐✜❝❛ q✉❡ ❛ ✈❛r✐á✈❡❧ é ✐♥❛❞❡q✉❛❞❛ ♣❛r❛ ❛

♠❡❞✐çã♦ ❞❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ à q✉❛❧ ❡stá ❛ss♦❝✐❛❞❛✳ ◗✉❛♥❞♦ s❡ ✈❡r✐✜❝❛

❡st❛ s✐t✉❛çã♦ é ❛❝♦♥s❡❧❤á✈❡❧ r❡♠♦✈❡r ❛ ✈❛r✐á✈❡❧ ❞♦ ❜❧♦❝♦ ❞❡ ✐♥❞✐❝❛❞♦r❡s✳

• α ❞❡ ❈r♦♥❜❛❝❤✿ ❡st❛ ❡st❛tíst✐❝❛ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ✈❡r✐✜❝❛r ❛ ✉♥✐✲

❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❞❡ ✉♠ ❜❧♦❝♦ ❞❡ p ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s q✉❛♥❞♦ ❡stã♦

♣♦s✐t✐✈❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞❛s ❡♥tr❡ s✐✳

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

❈r♦♥❜❛❝❤ ♣r♦♣ôs ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ✈❛r✐á✈❡✐s ❡st❛♥❞❛r❞✐③❛✲

❞❛s✳

❆ ✈❛r✐â♥❝✐❛ ❞♦ Pp

h=1xh ❡stá ❞❡s❡♥✈♦❧✈✐❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

V ar p X h=1 xh !

=p+X

h6=_h′

corr(xh, xh′) ✭✷✳✸✮

P♦❞❡♠♦s ❛✜r♠❛r q✉❡ ♦ ❜❧♦❝♦ s❡rá ♠❛✐s ✉♥✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t♦ ♠❛✐♦r

❢♦rP

h6=_h′corr(xh, xh′)✳

❈♦♠ ❜❛s❡ ♥❛ ❡q✉❛çã♦ ✷✳✸ ♣♦❞❡♠♦s ♣❛ss❛r ❛♦ ❝á❧❝✉❧♦ ❞♦ s❡❣✉✐♥t❡ rá❝✐♦✿

α′ =

P

h6=_h′ corr(x_h, x_h′)

p+P

h6=h′ corr(x_h, x_h′) ✭✷✳✹✮

❖ ✈❛❧♦r ♠á①✐♠♦ q✉❡α′ ♣♦❞❡ ❛t✐♥❣✐r é(p−1)pq✉❛♥❞♦ t♦❞❛s ❛scorr(xh, xh′)

sã♦ ✐❣✉❛✐s ❛1✳

❆ss✐♠✱ ♦α❞❡ ❈r♦♥❜❛❝❤ é ♦❜t✐❞♦ ❞✐✈✐❞✐♥❞♦ ♦α′ ♣❡❧♦ s❡✉ ✈❛❧♦r ♠á①✐♠♦✿

α=

P

h6=_h′ corr(x_h, x_h′)

p+P

h6=_h′corr(x_h, x_h′)

× p

p−1 ✭✷✳✺✮

P❛r❛ ❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ♦α ❝❛❧❝✉❧❛✲s❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

α=

P

h6=_h′ corr(x_h, x_h′)

V ar(Pp

h=1xh)

× p

p−1 ✭✷✳✻✮

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

❯♠ ❜❧♦❝♦ é ❝♦♥s✐❞❡r❛❞♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥❞♦ ♦α é ♠❛✐♦r ❞❡0.7✳

• ρ ❞❡ ❉✐❧❧♦♥✲●♦❧❞st❡✐♥✿ P♦r ❝♦♥str✉çã♦✱ ♦ s✐♥❛❧ ❞❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛s

✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛sxh ❡ ❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ξ t❡♠ q✉❡ s❡r ♣♦s✐t✐✈♦✱ ✐st♦

s✐❣♥✐✜❝❛ q✉❡ ♥❛ ❡q✉❛çã♦ ✷✳✶ t♦❞♦s ♦s ❧♦❛❞✐♥❣s πh sã♦ ♣♦s✐t✐✈♦s✳ ❯♠

❜❧♦❝♦ ♣♦❞❡✲s❡ ❞❡✜♥✐r ✉♥✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t♦ ♠❛✐♦r❡s ❢♦r❡♠ ♦s ❧♦❛❞✐♥❣s✳

❖ ♣r✐♠❡✐r♦ ♣❛ss♦ r❡❢❡r❡✲s❡ s❡♠♣r❡ ❛♦ ❝á❧❝✉❧♦ ❞❛ ✈❛r✐â♥❝✐❛ ❞❡ Pp

h=1xh✱

♠❛s ❞❡st❛ ✈❡③ é ❞❡t❡r♠✐♥❛❞❛ ❛ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ ✷✳✶ s✉♣♦♥❞♦ q✉❡ ♦s

t❡r♠♦s r❡s✐❞✉❛✐s ǫh sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✿

V ar p X h=1 xh !

=V ar

p

X

h=1

(πh0+πhξ+ǫh)

! = p X h=1 πh !2

V ar(ξ) +

p

X

h=1

V ar(ǫh)

✭✷✳✼✮

❯♠ ❜❧♦❝♦ r❡s✉❧t❛ ♠❛✐s ✉♥✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t♦ ♠❛✐♦r ❢♦r(Pp

h=1πh) 2

✳

❖ρ ❞❡ ❉✐❧❧♦♥✲●♦❧❞st❡✐♥ é ❞❡t❡r♠✐♥❛❞♦ ❝♦♠♦✿

ρ= (

Pp

h=1πh) 2

V ar(ξ) (Pp

h=1πh) 2

V ar(ξ) +Pp

h=1V ar(ǫh)

✭✷✳✽✮

❉❛❞♦ q✉❡ ♥❡st❛ ❢❛s❡ ♥ã♦ ♣♦ss✉í♠♦s ♦s ✈❛❧♦r❡s ❞❡ ξ✱ s❡rá ♥❡❝❡ssár✐♦

❡st✐♠❛r ♦ s❡✉ ✈❛❧♦r✳

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

❙✉♣♦♥❞♦ q✉❡ t♦❞❛s ❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛sxh ❡ ❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ξ sã♦

❡st❛♥❞❛r❞✐③❛❞❛s✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡

✉t✐❧✐③❛♥❞♦ ♦ ♣r✐♠❡✐r♦ ❢❛❝t♦r ❞❛ ❛♥á❧✐s❡ ❡♠ ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s t1

r❡❧❛t✐✈❛ ❛♦ ❜❧♦❝♦ ❞❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s✳

❖ ❧♦❛❞✐♥❣ πh ♣♦❞❡ s❡r ❡st✐♠❛❞♦ ❛tr❛✈és ❞❛corr(xh, t1) ❡✱ s❡♠♣r❡ ❝♦♠

❜❛s❡ ♥❛ ❡q✉❛çã♦ ✷✳✶✱ ❛ V ar(ǫh)é ❡st✐♠❛❞❛ ❛tr❛✈és ❞❡1−corr2(xh, t1)✳

❖ ρ ❞❡ ❉✐❧❧♦♥✲●♦❧❞st❡✐♥ ❡st✐♠❛❞♦ s❡rá ✐❣✉❛❧ ❛✿

ˆ

ρ= [

Pp

h=1corr(xh, t1)] 2

[Pp

h=1corr(xh, t1)] 2

+Pp

h=1[1−corr 2_(x

h, t1)]

✭✷✳✾✮

❯♠ ❜❧♦❝♦ é ✉♥✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥❞♦ ♦ρˆ❞❡ ❉✐❧❧♦♥✲●♦❧❞st❡✐♥ é ♠❛✐♦r ❞♦

q✉❡ 0.7✳

❙❡❣✉♥❞♦ ❈❤✐♥ ✭❈❤✐♥ ✶✾✾✽✮✱ ❡st❛ ❡st❛tíst✐❝❛ é ❝♦♥s✐❞❡r❛❞❛ ❛ ♠❡❧❤♦r ♣❛r❛

❛ ✈❡r✐✜❝❛çã♦ ❞❛ ✉♥✐❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❞❡ ✉♠ ❜❧♦❝♦ ❞❡ ✈❛r✐á✈❡✐s ♠❛♥✐❢❡s✲

t❛s✳

▼♦❞♦ ❋♦r♠❛t✐✈♦

◆♦ ♠♦❞♦ ❢♦r♠❛t✐✈♦✱ ❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ξé ♦❜t✐❞❛ ❛tr❛✈és ❞❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦

❧✐♥❡❛r ❞❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s ✭❋✐❣✉r❛ ✷✳✹✮ ❛ ❡❧❛ ❛ss♦❝✐❛❞❛s ♠❛✐s ✉♠ t❡r♠♦

r❡s✐❞✉❛❧✿

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

ξ =

p

X

h=1

̟hxh+δ ✭✷✳✶✵✮

◆❡st❡ ❡sq✉❡♠❛✱ ✉♠ ❜❧♦❝♦ ❞❡ ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s ♣♦❞❡ s❡r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧✳

❆ ❤✐♣ót❡s❡ r❡❧❛t✐✈❛ ❛♦ ♣r❡❞✐❝t♦r s♣❡❝✐✜❝❛t✐♦♥ ♣❛r❛ ❛ ❡q✉❛çã♦ ✷✳✶✵ é✿

E(ξ|x1, . . . , x_p) =

p

X

h=1

̟hxh ✭✷✳✶✶✮

❊st❛ ❤✐♣ót❡s❡ ✐♠♣❧✐❝❛ q✉❡ ♦ ✈❡❝t♦r ❞♦s r❡sí❞✉♦s δ t❡♠ ♠é❞✐❛ 0 ❡ ♥ã♦ ❡stá

❝♦rr❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛sxh✳

❋✐❣✉r❛ ✷✳✹✿ ❖ ▼♦❞♦ ❋♦r♠❛t✐✈♦

◆❡st❡ ❡sq✉❡♠❛✱ ❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ é ❣❡r❛❞❛ ❛ ♣❛rt✐r ❞❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛s

✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s ❡ ♦ s✐♥❛❧ ❞❡ ❝❛❞❛ ♣❡s♦̟hé✱ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❝❛❧❝✉❧❛❞♦ ❝♦♠

❜❛s❡ ♥♦ s✐♥❛❧ ❞❛ ❝♦rr❡❧❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ❛ ✈❛r✐á✈❡❧ ♠❛♥✐❢❡st❛xh ❡ ❛ ✈❛r✐á✈❡❧

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

❧❛t❡♥t❡ ξ✳ ◆♦ ❛❧❣♦r✐t♠♦ ❝❧áss✐❝♦ ♥ã♦ ❡①✐st❡♠ ✈í♥❝✉❧♦s ♣❛r❛ ❛ ❞❡✜♥✐çã♦ ❞♦s

❧♦❛❞✐♥❣s ♦✉ ❞♦s ♣❡s♦s✱ ♠❛s ❡①✐st❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❛❞✐❝✐♦♥á✲❧♦s ✭❚❡♥❡♥❤❛✉s

❡t ❛❧✳ ✷✵✵✺✮✳

▼■▼■❈ ✭▼✉❧t✐♣❧❡ ■♥❞✐❝❛t♦rs ❢♦r ▼✉❧t✐♣❧❡ ❈❛✉s❡s✮

❖ ❡sq✉❡♠❛ ▼■▼■❈ é ✉♠❛ ❢✉sã♦ ❡♥tr❡ ♦ ❡sq✉❡♠❛ r❡✢❡❝t✐✈♦ ❡ ♦ ❡sq✉❡♠❛

❢♦r♠❛t✐✈♦ ✭❋✐❣✉r❛ ✷✳✺✮✳

❖ ♠♦❞❡❧♦ ❞❡ ♠❡❞✐❞❛ ♣❛r❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❜❧♦❝♦ é✿

xh =πh0 +πhξ+ǫh ✭✷✳✶✷✮

♦♥❞❡ h∈P1✳

❆ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ξ é ❞❡✜♥✐❞❛ ❝♦♠♦✿

ξ = X

h /∈P1

̟hxh+δ ✭✷✳✶✸✮

♦♥❞❡ P1 é ♦ ❝♦♥❥✉♥t♦ ❞❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s q✉❡ s❡❣✉❡♠ ♦ ❡sq✉❡♠❛ r❡✢❡❝✲

t✐✈♦✳

❆ ❤✐♣ót❡s❡ ❞♦ ♣r❡❞✐❝t♦r s♣❡❝✐✜❝❛t✐♦♥ ♣❛r❛ ❛s ❡q✉❛çõ❡s ✷✳✶✷ ❡ ✷✳✶✸ sã♦ ❛s

♠❡s♠❛s ❞❡✜♥✐❞❛s r❡s♣❡❝t✐✈❛♠❡♥t❡ ♥♦s ❡sq✉❡♠❛s r❡✢❡❝t✐✈♦ ❡ ❢♦r♠❛t✐✈♦✳

✷✳✷✳ ❖ ▼♦❞❡❧♦ P▲❙✲P▼

❋✐❣✉r❛ ✷✳✺✿ ▼✉❧t✐♣❧❡ ■♥❞✐❝❛t♦rs ❢♦r ▼✉❧t✐♣❧❡ ❈❛✉s❡s ✭▼■▼■❈✮

✷✳✷✳✷ ❖ ▼♦❞❡❧♦ ❊str✉t✉r❛❧

❖ ♠♦❞❡❧♦ ❝❛✉s❛❧ ❞❛ ✜❣✉r❛ ✷✳✷ ❛♣r❡s❡♥t❛ ❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s r❡❧❛❝✐♦♥❛❞❛s

❡♥tr❡ s✐✳ ❊st❛s r❡❧❛çõ❡s ❝♦♠♣õ❡♠ ♦ ♠♦❞❡❧♦ ❡str✉t✉r❛❧ ✭♦✉ ✐♥♥❡r ♠♦❞❡❧✮ ❡

♣♦❞❡♠ s❡r ❢♦r♠❛❧✐③❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

ξj =βj0+

X

i

βjiξi+νj ✭✷✳✶✹✮

P❛r❛ ❛ ❡q✉❛çã♦ ✷✳✶✹ t❛♠❜é♠ ❡stá ❞❡✜♥✐❞❛ ❛ ❤✐♣ót❡s❡ ❞♦ ♣r❡❞✐❝t♦r s♣❡❝✐✜❝❛✲

t✐♦♥✳

❆s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s q✉❡ ♥✉♥❝❛ ❛♣❛r❡❝❡♠ ❝♦♠♦ ❞❡♣❡♥❞❡♥t❡s ♥❛ ❡q✉❛çã♦ ✷✳✶✹

sã♦ ❞❡✜♥✐❞❛s ✈❛r✐á✈❡✐s ❡①ó❣❡♥❛s❀ ❛s q✉❡ ❛♣❛r❡❝❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ✈❡③ ❝♦♠♦

❞❡♣❡♥❞❡♥t❡s sã♦ ❞❡✜♥✐❞❛s ✈❛r✐á✈❡✐s ❡♥❞ó❣❡♥❛s✳

✷✳✸✳ ❊str✉t✉r❛ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼

❖ ♠♦❞❡❧♦ ❝❛✉s❛❧ r❡♣r❡s❡♥t❛ ✉♠❛ ❝❛❞❡✐❛ ❝❛✉s❛❧✱ ✐st♦ é✱ ♥ã♦ ❡①✐st❡♠ ❝✐❝❧♦s✳

❊st❡s t✐♣♦s ❞❡ ♠♦❞❡❧♦s sã♦ ❞❡✜♥✐❞♦s ❝♦♠♦ r❡❝✉rs✐✈♦s ❞♦ ▲❛t✐♠ ❘❡❝✉rs✐♦✱ q✉❡

s✐❣♥✐✜❝❛ ❊✉ ♣♦ss♦ ✈♦❧t❛r ✭❚❡♥❡♥❤❛✉s ❡t ❛❧✳ ✷✵✵✺✮✳

❯♠ ♠♦❞❡❧♦ ❡str✉t✉r❛❧ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❛tr❛✈és ❞❡ ✉♠❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r ✐♥✲

❢❡r✐♦r✱ ❞❡ ❞✐♠❡♥sã♦ ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s q✉❡ ♣♦❞❡ ❛♣r❡s❡♥t❛r

③❡r♦s ♦✉ ✉♥s✳ ▲♦❤♠ö❧❧❡r ❞❡✜♥❡ ❡st❛ ♠❛tr✐③ ❝♦♠♦ ✐♥♥❡r ❞❡s✐❣♥ ♠❛tr✐①✳

❆ ❝é❧✉❧❛(i, j)❡st❛rá ♣r❡❡♥❝❤✐❞❛ ❝♦♠ ✉♠1q✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡j ❡①♣❧✐❝❛

❛ ✈❛r✐á✈❡❧ i❀ ❝❛s♦ ❝♦♥trár✐♦ s❡rá ✐❣✉❛❧ ❛ 0✳

✷✳✸ ❊str✉t✉r❛ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼

❉❡ ❡♥tr❡ ♦s ♣r✐♥❝✐♣❛✐s ❛❧❣♦r✐t♠♦s ♣❛r❛ ♦s ♠♦❞❡❧♦s ❞❡ ❡q✉❛çõ❡s ❡str✉t✉r❛✐s✱

❞❡st❛❝❛♠✲s❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❍❡r♠❛♥ ❲♦❧❞ ✭❏ör❡s❦♦❣ ✫ ❲♦❧❞

✶✾✽✷✱ ❲♦❧❞ ✶✾✽✷✮ ❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ▲♦❤♠ö❧❧❡r ✭▲♦❤♠ö❧❧❡r ✶✾✽✼✱ ▲♦❤♠ö❧❧❡r

✶✾✽✾✮✳

P❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ ♣r♦❥❡❝t♦ ❢♦✐ ✉t✐❧✐③❛❞♦ ♦ ❛❧❣♦r✐t♠♦ ♦r✐❣✐♥❛❧ ❞❡

❍✳ ❲♦❧❞✱ q✉❡ s❡rá ❛♣r❡s❡♥t❛❞♦ ❡♠ ❞❡t❛❧❤❡ ♥♦s ♣ró①✐♠♦s ♣❛rá❣r❛❢♦s✳

❆ ❡st✐♠❛çã♦ ❞❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s ξj é ❢❡✐t❛ ❝♦♠ ❜❛s❡ ♥♦s s❡❣✉✐♥t❡s ♣❛ss♦s

q✉❡ s❡rã♦ ✐t❡r❛❞♦s ❛té ❛❧❝❛♥ç❛r ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s ♣❡s♦s✿

✶✳ ❊st✐♠❛çã♦ ❊①t❡r♥❛❀

✷✳ ❊st✐♠❛çã♦ ■♥t❡r♥❛❀

✷✳✸✳ ❊str✉t✉r❛ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼

✸✳ ❊st✐♠❛çã♦ ❞♦s P❡s♦s❀

✷✳✸✳✶ ❊st✐♠❛çã♦ ❊①t❡r♥❛

❆ ♣r✐♠❡✐r❛ ❢❛s❡ t❡♠ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ❛ ❡st✐♠❛çã♦ ❞♦s yj✳ ❆s ✈❛r✐á✈❡✐s ❧❛t❡♥✲

t❡s ❡st❛♥❞❛r❞✐③❛❞❛s ✭♠é❞✐❛ ❂ 0 ❡ ❞❡s✈✐♦ ♣❛❞rã♦ ❂ 1✮ sã♦ ❡st✐♠❛❞❛s ❝♦♠♦

❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛s s✉❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s ❝❡♥tr❛❞❛s✿

yj ∝ ±

hX

wjh+ (xjh−x¯jh)

i

✭✷✳✶✺✮

♦♥❞❡ ♦ sí♠❜♦❧♦ ∝ s✐❣♥✐✜❝❛ q✉❡ ❛ ✈❛r✐á✈❡❧ ❞❛ ❡sq✉❡r❞❛ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦

♦♣❡r❛❞♦r ❞❛ ❞✐r❡✐t❛ ❡ ♦ s✐♥❛❧±r❡♣r❡s❡♥t❛ ❛ ❛♠❜✐❣✉✐❞❛❞❡ ❞♦s s✐♥❛✐s ❛ ✉t✐❧✐③❛r✳

❊st❡ ♣r♦❜❧❡♠❛ é r❡s♦❧✈✐❞♦ s❡❧❡❝❝✐♦♥❛♥❞♦ ♦ s✐♥❛❧ q✉❡ ❢❛③ ❝♦♠ q✉❡ ❛ ✈❛r✐á✈❡❧yj

❡st❡❥❛ ❝♦rr❡❧❛❝✐♦♥❛❞❛ ♣♦s✐t✐✈❛♠❡♥t❡ ❝♦♠ ❛ ♠❛✐♦r✐❛ ❞❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s

xjh✳

❆ j✲és✐♠❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ ❡st✐♠❛❞❛ é ❝❛❧❝✉❧❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

yj =

X

˜

wjh+ (xjh−x¯jh) ✭✷✳✶✻✮

❖s ❝♦❡✜❝✐❡♥t❡s wjh ❡ w˜jh sã♦ ❞❡♥♦♠✐♥❛❞♦s ♦✉t❡r ✇❡✐❣❤ts ✭♣❡s♦s ❡①t❡r♥♦s✮✳

❆ ♠é❞✐❛mj é ❡st✐♠❛❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

ˆ

mj =

X

˜

wjhx¯jh ✭✷✳✶✼✮

✷✳✸✳ ❊str✉t✉r❛ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼

❡ ❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ξj é ❡st✐♠❛❞❛ ❝♦♠♦ s❡❣✉❡

ˆ

ξj =

X

˜

wjhxjh =yj+ ˆmj ✭✷✳✶✽✮

❆ ❡st✐♠❛çã♦ ❞♦s ♣❡s♦sw˜jh ❡stá ❛♣r❡s❡♥t❛❞❛ ♥♦ s❡❣✉✐♥t❡ ♣❛rá❣r❛❢♦✳

✷✳✸✳✷ ❊st✐♠❛çã♦ ❞♦s P❡s♦s

❊①✐st❡♠ ❞✉❛s ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❞❡ ❝❛❧❝✉❧❛r ♦s ♣❡s♦s wjh✿ ♠♦❞♦ ❆ ❡ ♠♦❞♦

❇✳

▼♦❞♦ ❆

◆♦ ♠♦❞♦ ❆✱ ♦s ♣❡s♦s wjh r❡♣r❡s❡♥t❛♠ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ r❡❣r❡ssã♦ ❞❡ zj ♥❛

r❡❣r❡ssã♦ s✐♠♣❧❡s ❞❡xjh s♦❜r❡ zj✿

wjh =cov(xjh, zj) ✭✷✳✶✾✮

♦♥❞❡ ❛ ✈❛r✐á✈❡❧ zj é ❡st❛♥❞❛r❞✐③❛❞❛✳

▼♦❞♦ ❇

◆♦ ♠♦❞♦ ❇✱ ♦ ✈❡❝t♦r ❞♦s ♣❡s♦s wjh r❡♣r❡s❡♥t❛ ♦ ✈❡❝t♦r ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡

r❡❣r❡ssã♦ ♥❛ r❡❣r❡ssã♦ ♠ú❧t✐♣❧❛ ❞❡zj s♦❜r❡ ❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s ❝❡♥tr❛❞❛s

(xjh−x¯jh) r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ξj✿

✷✳✸✳ ❊str✉t✉r❛ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼

wj =

X_j′Xj

−1

X_j′zj ✭✷✳✷✵✮

♦♥❞❡ Xj é ❛ ♠❛tr✐③ q✉❡ t❡♠ ♣♦r ❝♦❧✉♥❛ ❛s ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s ❝❡♥tr❛❞❛s

(xjh−x¯jh) r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛ ♠❡s♠❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡ξj✳

❖ ♠♦❞♦ ❆ é ❛♣r♦♣r✐❛❞♦ ♣❛r❛ ♦s ❜❧♦❝♦s q✉❡ ❛♣r❡s❡♥t❛♠ ✉♠ ♠♦❞❡❧♦ ❞❡ ♠❡❞✐❞❛

r❡✢❡❝t✐✈♦ ❡ ♦ ♠♦❞♦ ❇ ♣❛r❛ ♦s ❢♦r♠❛t✐✈♦s✳ P❛r❛ ♦s ♠♦❞❡❧♦s ▼■▼■❈ sã♦

✉t✐❧✐③❛❞❛s ❛♠❜♦s ♦s ♠♦❞♦s ❞❡ ❡st✐♠❛çã♦ ❞♦s ♣❡s♦s wjh ✭♦ ♠♦❞♦ ❆ ♣❛r❛ ❛

♣❛rt❡ ❢♦r♠❛t✐✈❛ ❡ ♦ ♠♦❞♦ ❇ ♣❛r❛ ❛ r❡✢❡❝t✐✈❛✮✳

❖ ♠♦❞♦ ❇ ♥ã♦ é ✉t✐❧✐③❛❞♦ ❢r❡q✉❡♥t❡♠❡♥t❡ ♣♦r ❝❛✉s❛ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ♠✉❧✲

t✐❝♦❧✐♥❡❛r✐❞❛❞❡ ♥♦s ❜❧♦❝♦s ❞❡ ✈❛r✐á✈❡✐s ♠❛♥✐❢❡st❛s✳ ◆❡st❡s ❝❛s♦s✱ ❡♠ ✈❡③ ❞❛s

r❡❣r❡ssõ❡s ❖▲❙✱ sã♦ ✉t✐❧✐③❛❞❛s ❛s r❡❣r❡ssõ❡s P▲❙✳

◆♦ ❝❛s♦ ❞❡ s❡ t❡r ❛ ♣r✐♦r✐ ❛❧❣✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ r❡❧❛t✐✈♦ à ❡str✉t✉r❛ ❞♦ ♠♦❞❡❧♦✱

♣♦❞❡♠✲s❡ ❛❞✐❝✐♦♥❛r ✈í♥❝✉❧♦s ❞❡ s✐♥❛❧ ❛♦s ♣❡s♦s ❡①✐st❡♥t❡s ♥♦ ♠♦❞❡❧♦✳ ❖ s✐♥❛❧

❞♦s ♣❡s♦s ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

sign(wjh) = sign(cor(xjh, ξj)) ✭✷✳✷✶✮

P❛r❛ ♦ ♠♦❞♦ ❆✱ ❛ ❡q✉❛çã♦ ✷✳✶✾ ♣♦❞❡ s❡r s✉❜st✐t✉í❞❛s ♣♦r

✷✳✸✳ ❊str✉t✉r❛ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼

wjh =cov(xjh, zj) se sign[cov(xjh, zj)] =sign[cor(xjh, ξj)]

= 0 se sign[cov(xjh, zj)]6=sign[cor(xjh, ξj)]

✭✷✳✷✷✮

P❛r❛ ♦ ♠♦❞♦ ❇✱ ❛ r❡❣r❡ssã♦ ❖▲❙ ❞❡zj s♦❜r❡ Xj é s✉❜st✐t✉í❞❛ ♣❡❧❛ r❡❣r❡ssã♦

❞❡zj s♦❜r❡Xj t❡♥❞♦ ❡♠ ❝♦♥t❛ ♦ ✈í♥❝✉❧♦ ❛♣r❡s❡♥t❛❞♦ ♥❛ ❡①♣r❡ssã♦ ✷✳✷✶✳ ❊st❛

r❡❣r❡ssã♦ ♠ú❧t✐♣❧❛ ❧❡✈❛rá ❛ ✉♠❛ r❡❣r❡ssã♦ ♦♥❞❡ ♦s ❝♦❡✜❝✐❡♥t❡s q✉❡ ❛♣r❡s❡♥t❛♠

♦ s✐♥❛❧ ❝♦rr❡❝t♦ s❡rã♦ ❞❡✐①❛❞♦s ✐❣✉❛✐s ❡ ♦s q✉❡ ❛♣r❡s❡♥t❛♠ ♦ s✐♥❛❧ ❝♦♥trár✐♦

❛♦ ❡s♣❡r❛❞♦ s❡rã♦ ♣♦st♦s ✐❣✉❛✐s ❛ ③❡r♦✳

✷✳✸✳✸ ❊st✐♠❛çã♦ ■♥t❡r♥❛

❖ ♦❜❥❡❝t✐✈♦ ❞♦ ♠♦❞❡❧♦ ✐♥t❡r♥♦✱ t❛❧ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠❡✲

❞✐❞❛✱ é ❛ ❡st✐♠❛çã♦ ❞❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s✳ ❆ ❡st✐♠❛çã♦ ✐♥t❡r♥❛zj ❞❛s ✈❛r✐á✲

✈❡✐s ❧❛t❡♥t❡s ❡st❛♥❞❛r❞✐③❛❞❛s (ξj −mj)é ❞❡✜♥✐❞❛ ❝♦♠♦

zj ∝

X

j′ _:ξ

j′ est´a ligada com ξj

e_jj′y_j′ ✭✷✳✷✸✮

♦♥❞❡ ♦s ♣❡s♦s ✐♥t❡r♥♦s e_jj′ ♣♦❞❡♠ s❡r ❡st✐♠❛❞♦s ❛tr❛✈és ❞❛ ✉t✐❧✐③❛çã♦ ❞❡

❞✐✈❡rs❛s té❝♥✐❝❛s✳

❯♠❛ ❞❛s té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ❝♦♠ ♠❛✐♦r ❢r❡q✉ê♥❝✐❛ é ♦ ❡sq✉❡♠❛ ❞♦ ❝❡♥tró✐❞❡

✭❝❡♥tr♦✐❞ s❝❤❡♠❡✮ q✉❡ ❡st✐♠❛ ♦s ♣❡s♦s ✐♥t❡r♥♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

✷✳✸✳ ❊str✉t✉r❛ ❞♦ ❆❧❣♦r✐t♠♦ P▲❙✲P▼

e_jj′ =sign

cor yj, yj′

✭✷✳✷✹✮

◆❡st❡ ❝❛s♦ ♦s ♣❡s♦se_jj′ sã♦ ✐❣✉❛✐s ❛♦s s✐♥❛✐s ❞❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡y_j ❡ ❛ ✈❛r✐á✈❡❧

❧❛t❡♥t❡ y_j′ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ y_j ✭❡q✳ ✷✳✷✹✮✳ ❉✐③❡♠✲s❡ r❡❧❛❝✐♦♥❛❞❛s ❞✉❛s

✈❛r✐á✈❡✐s q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❧✐❣❛çã♦ ❡♥tr❡ ❡❧❛s✱ ✐st♦ é✱ ♥♦ ♠♦❞❡❧♦ ❝❛✉s❛❧✱ ❡①✐st❡

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

❝❛✉s❛❧✐❞❛❞❡ ❡♥tr❡ ❡❧❛s✳

❖ ❡sq✉❡♠❛ ❞♦ ❝❡♥tró✐❞❡ ❛♣r❡s❡♥t❛ ❛❧❣✉♥s ✐♥❝♦♥✈❡♥✐❡♥t❡s q✉❛♥❞♦ s❡ ❡♥❝♦♥✲

tr❛♠ ❝♦rr❡❧❛çõ❡s ♠✉✐t♦ ♣ró①✐♠❛s ❞❡ ③❡r♦✱ s❡♥❞♦ q✉❡ ❡st❛s ❝♦rr❡❧❛çõ❡s ❝♦st✉✲

♠❛♠ ✢✉t✉❛r à ✈♦❧t❛ ❞♦ ③❡r♦✱ ❛♣r❡s❡♥t❛♥❞♦ ✈❛❧♦r❡s ♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s✳ ◆❛s

❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s✱ ❡st❡ ♣r♦❜❧❡♠❛ ♥ã♦ r❡♣r❡s❡♥t❛ ❣r❛♥❞❡s ❝♦♠♣❧✐❝❛çõ❡s ❞♦

♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛s ❡st✐♠❛çõ❡s✳

❖✉tr♦s ❡sq✉❡♠❛s ❞❡ ♣❡s♦s ❞❡✜♥✐❞♦s ♣❛r❛ ❛ ❢❛s❡ ❞❡ ❡st✐♠❛çã♦ ✐♥t❡r♥❛ sã♦ ♦

❡sq✉❡♠❛ ❢❛❝t♦r✐❛❧ ✭❋❛❝t♦r✐❛❧ ❙❝❤❡♠❡✮ ♦♥❞❡ ♦s ♣❡s♦s e_jj′ sã♦ ❝❛❧❝✉❧❛❞♦s ❞❛

s❡❣✉✐♥t❡ ❢♦r♠❛✿

e_jj′ =r_jj′ =cor y_j, y_j′

✭✷✳✷✺✮

❡ ♦ ❡sq✉❡♠❛ ❡str✉t✉r❛❧ ✭❙tr✉❝t✉r❛❧ ❙❝❤❡♠❡✮ ❝❛❧❝✉❧❛❞♦ ❝♦♠♦ s❡❣✉❡✿