ANÁLISE FATORIAL COM ROTAÇÃO OBLÍQUA: APLICAÇÃO EM UMA ESCALA PSICOMÉTRICA

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❯❜❡r❧â♥❞✐❛

❋❛❝✉❧❞❛❞❡ ❞❡ ▼❛t❡♠át✐❝❛

❇❛❝❤❛r❡❧❛❞♦ ❡♠ ❊st❛tíst✐❝❛

❆◆➪▲■❙❊ ❋❆❚❖❘■❆▲ ❈❖▼ ❘❖❚❆➬➹❖

❖❇▲❮◗❯❆✿ ❆P▲■❈❆➬➹❖ ❊▼ ❯▼❆

❊❙❈❆▲❆ P❙■❈❖▼➱❚❘■❈❆

▼✐❝❤❛❡❧ ❘♦s❛ ❘❡③❡♥❞❡

▼✐❝❤❛❡❧ ❘♦s❛ ❘❡③❡♥❞❡

❆◆➪▲■❙❊ ❋❆❚❖❘■❆▲ ❈❖▼ ❘❖❚❆➬➹❖

❖❇▲❮◗❯❆✿ ❆P▲■❈❆➬➹❖ ❊▼ ❯▼❆

❊❙❈❆▲❆ P❙■❈❖▼➱❚❘■❈❆

❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ à ❈♦✲ ♦r❞❡♥❛çã♦ ❞♦ ❈✉rs♦ ❞❡ ❇❛❝❤❛r❡❧❛❞♦ ❡♠ ❊st❛tíst✐❝❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❇❛❝❤❛r❡❧ ❡♠ ❊st❛tíst✐❝❛✳

❖r✐❡♥t❛❞♦r✿ P❛trí❝✐❛ ❱✐❛♥❛ ❞❛ ❙✐❧✈❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❯❜❡r❧â♥❞✐❛

❋❛❝✉❧❞❛❞❡ ❞❡ ▼❛t❡♠át✐❝❛

❈♦♦r❞❡♥❛çã♦ ❞♦ ❈✉rs♦ ❞❡ ❇❛❝❤❛r❡❧❛❞♦ ❡♠ ❊st❛tíst✐❝❛

❆ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ❝♦♥❢♦r♠❡ ❛❜❛✐①♦ ❛ss✐♥❛❞♦✱ ❝❡rt✐✜❝❛ ❛ ❛❞❡q✉❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❇❛❝❤❛r❡❧ ❡♠ ❊st❛tíst✐❝❛✳

❯❜❡r❧â♥❞✐❛✱ ❞❡ ❞❡ ✷✵

❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆

P❛trí❝✐❛ ❱✐❛♥❛ ❞❛ ❙✐❧✈❛

❘♦❞r✐❣♦ ▲❛♠❜❡rt

▲❡❛♥❞r♦ ❆❧✈❡s P❡r❡✐r❛

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ s❛ú❞❡ ❡ ❢♦rç❛ ♣❛r❛ s✉♣❡r❛r ❛s ❞✐✜❝✉❧❞❛❞❡s✳ ❆ ❡st❛ ✉♥✐✈❡rs✐❞❛❞❡✱ à ❊st❛tíst✐❝❛ ❡ s❡✉ ❝♦r♣♦ ❞♦❝❡♥t❡✱ ❞✐r❡çã♦ ❡ ❛❞♠✐♥✐str❛çã♦ q✉❡ ♠❡ ♣r♦♣✐✲ ❝✐❛r❛♠ ❛ ❥❛♥❡❧❛ q✉❡ ❤♦❥❡ ✈✐s❧✉♠❜r♦ ✉♠ ❤♦r✐③♦♥t❡ s✉♣❡r✐♦r✱ ❛tr✐❜✉í❞♦ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ♥♦ ♠ér✐t♦ ❡ ét✐❝❛ ❛q✉✐ ♣r❡s❡♥t❡s✳ ❆ ♠✐♥❤❛ ❣r❛♥❞❡ ♦r✐❡♥t❛❞♦r❛ ❡ ♠❡♥t♦r❛ P❛trí❝✐❛ ❱✐❛♥❛ ❞❛ ❙✐❧✈❛ ♥❡ss❡ tr❛❜❛❧❤♦ ár❞✉♦✱ ♣❡❧♦ s✉♣♦rt❡ ♥♦ ♣♦✉❝♦ t❡♠♣♦ q✉❡ ❧❤❡ ❝♦✉❜❡✱ ♣❡❧❛s s✉❛s ❝♦rr❡çõ❡s ❡ ✐♥❝❡♥t✐✈♦s✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♠ã❡ ❙❤❡✐❧❛ ❆♣❛r❡❝✐❞❛ ❘♦s❛✱ ❤❡r♦í♥❛ q✉❡ ♠❡ ❞❡✉ ❛♣♦✐♦✱ ✐♥❝❡♥t✐✈♦ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s✱ ❞❡ ❞❡sâ♥✐♠♦ ❡ ❝❛♥s❛ç♦✳ ❆ ♠✐♥❤❛ ❛✈ó q✉❡ ❛♣❡s❛r ❞❡ t♦❞❛s ❛s ❞✐✜❝✉❧❞❛❞❡s ♠❡ ❢♦rt❛❧❡❝❡✉ ❡ q✉❡ ♣❛r❛ ♠✐♠ ❢♦✐ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✳ ❇❡♠ ❝♦♠♦✱ ♠❡✉ ✐r♠ã♦ ❘♦❞r✐❣♦✱ ♣♦r t♦❞❛s ❛s ♣❛❧❛✈r❛s ❞❡ ✐♥❝❡♥t✐✈♦ ♠❡s♠♦ ❞❡ ❧♦♥❣❡✱ ♠❡ ♠♦str❛♥❞♦ q✉❡ ♥ã♦ ♣♦❞❡r✐❛ ❞❡s✐st✐r ♥✉♥❝❛✳

❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛ ♠✐♥❤❛ ♠❛❞r✐♥❤❛ ●❡✐s❛ ❆♣❛r❡❝✐❞❛ ❘♦s❛ ♣❡❧♦ ❣r❛♥❞❡✱ ❛♣♦✐♦✱ ✉♥✐ã♦ ❡ ❝❛r✐♥❤♦ ❞✉r❛♥t❡ ❡st❛ ❥♦r♥❛❞❛✱ ❡ t❛♠❜é♠ ❛ ✉♠❛ ♠✉❧❤❡r ♥❛ q✉❛❧ s✐♥t♦ ✉♠❛ s❛✉❞❛❞❡ ✐♠❡♥s❛✱ ❏❛♥❛í♥❛ ❆♣❛r❡❝✐❞❛ ❘♦s❛✱ q✉❡ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s ♠❡ ❜r✐♥❞♦✉ ❝♦♠ s✉❛ ❛❧❡❣r✐❛ ❡ ❢♦rç❛ tr❛③❡♥❞♦ ♣r❛ ❞❡♥tr♦ ❞❡ ♠✐♠ ❛ ❝♦♥✜❛♥ç❛ ❡ ❛♣♦✐♦ q✉❡ ❡✉ ♥❡❝❡ss✐t❡✐ ♣❛r❛ ♠❡ ✐♠♣✉❧s✐♦♥❛r ❡ r❡❛❧✐③❛r ❡st❡ s♦♥❤♦✳ ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r t❡r ♠❡ ❞❛❞♦ ♦ s✉♣♦rt❡ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ ❝✉rs♦✳

▼❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s s✐♥❝❡r♦s à ❱í✈✐❛♥ ❘✐❜❡✐r♦ ❇❛rr❡t♦✱ ♦ ♠❡❧❤♦r ♣r❡s❡♥t❡ q✉❡ ❛ ❢❛❝✉❧❞❛❞❡ ♣♦❞❡r✐❛ t❡r ♠❡ ❞❛❞♦✳ P❡❧♦ ❝❛r✐♥❤♦✱ ❝♦♠♣r❡❡♥sã♦✱ ❛♠♦r✱ s♦❧✐❞❛r✐❡❞❛❞❡ ❡ ❝❧❛r♦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❝♦♠ ❛♣♦✐♦ t♦t❛❧✳ ❖❜r✐❣❛❞♦ ♠❡✉ ❛♠♦r ♣♦r t✉❞♦ ♦ q✉❡ ✈♦❝ê tr❛♥s❢♦r♠♦✉ ♥❛ ♠✐♥❤❛ ✈✐❞❛✳ ❖❜r✐❣❛❞♦ ♣❡❧♦ t❡✉ ❝❛r✐♥❤♦✱ t✉❛ ❛❧❡❣r✐❛✱ t✉❛ ❛t❡♥çã♦✱ t✉❛ ✈✐❜r❛çã♦ ❝♦♠ ❛s ♠✐♥❤❛s ❝♦♥q✉✐st❛s ❡ t❡✉ ♦♠❜r♦ ❡♠ ❝❛❞❛ ♠♦♠❡♥t♦ ❞✐❢í❝✐❧ q✉❡ ✈♦❝ê ❛❥✉❞♦✉ ❛ ❛tr❛✈❡ss❛r✳ ❙❡♠ ✈♦❝ê✱ ❡ss❛ ❝♦♥q✉✐st❛ ♥ã♦ t❡r✐❛ ♦ ♠❡s♠♦ ❣♦st♦✳ ❖❜r✐❣❛❞♦ ♠❡✉ ❛♠♦r✳

▼❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛♦s ❛♠✐❣♦s ❈❛✐♦ ❍❡♥r✐q✉❡ ●❛r❝✐❛ ❙✐❧✈❛ ❡ ▲✉✐③ ❈❛r❧♦s ❈♦st❛ ❏✉♥✐♦r✱ ❝♦♠♣❛♥❤❡✐r♦s ❞❡ tr❛❜❛❧❤♦s ❡ ✐r♠ã♦s ♥❛ ❛♠✐③❛❞❡ q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ q✉❡ ✈ã♦ ❝♦♥t✐♥✉❛r ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❝♦♠ ❝❡rt❡③❛✳

❘❡s✉♠♦

❊ss❡ tr❛❜❛❧❤♦ s❡ ♣r♦♣õ❡ ❛ ✉t✐❧✐③❛r ❛s té❝♥✐❝❛s ♠✉❧t✐✈❛r✐❛❞❛s ❞❡ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ♣❛r❛ ✐❞❡♥t✐✜❝❛r r❡❧❛çõ❡s ❡♥tr❡ ❝♦♠♣♦rt❛♠❡♥t♦s ✐♥❞✐❝❛t✐✈♦s ❞❛ ❞✐st♦rçã♦ ❞❛ ✐♠❛❣❡♠ ❝♦r♣♦r❛❧✳ ❖s ❞❛❞♦s ❢♦r❛♠ ♦❜t✐❞♦s ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ◗✉❡st✐♦♥ár✐♦ ❞❡ ■♠❛❣❡♠ ❈♦r♣♦r❛❧ ❡♠ ✶✷✺ ♠✉❧❤❡r❡s ❡♠ ✉♠ ❡st✉❞♦ ❞❡ ✉♠ ❛♠❜✉❧❛tór✐♦ ❞❡ tr❛♥st♦r♥♦s ❛❧✐♠❡♥t❛r❡s✳ ❆ ❆♥á❧✐s❡ ❞❡ ❈♦♠♣♦♥❡♥t❡s Pr✐♥❝✐♣❛✐s ♣♦ss✐❜✐❧✐t♦✉ ❞✐♠✐♥✉✐r ❛ ❞✐♠❡♥sã♦ ❞♦s ❞❛❞♦s ❞❡ ✸✹ ♣❛r❛ ✹ ✈❛r✐á✈❡✐s ❡ ♦ ✉s♦ ❞❛ r♦t❛çã♦ ♦❜❧✐q✉❛ ♦❜❧✐♠✐♥✱ ❛✉①✐❧✐♦✉ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ❝♦♠♣♦♥❡♥t❡s ❝♦♥s✐❞❡r❛❞♦s r❡❧❡✈❛♥t❡s ❡ à ❛❣r✉♣❛r ♦s ❝♦♠♣♦rt❛♠❡♥t♦s ❝♦♠♦ ❛ ✐♥s❛t✐s❢❛çã♦ ❢ís✐❝❛✱ s❡♥t✐♠❡♥t♦ ❞❡ ❡st❛r ❢♦r❛ ❞♦s ♣❛❞rõ❡s ❡stét✐❝♦s✱ ❝♦♠♣❛r❛çã♦ ❛ ♦✉tr❛s ♣❡ss♦❛s ❡ ♦ ✉s♦ ❝♦♥tr♦✈❡rs♦ ❞❡ ♠❡❞✐❝❛♠❡♥t♦s✳

❆❜str❛❝t

❚❤❡ ♠✉❧t✐✈❛r✐❛t❡ t❡❝❤♥✐q✉❡s ♦❢ ❢❛❝t♦r ❛♥❛❧②s✐s ❡♥❛❜❧❡ t♦ ✐❞❡♥t✐❢② r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ❜❡❤❛✲ ✈✐♦rs ♦❢ ❜♦❞② ✐♠❛❣❡ ❞✐st♦rt✐♦♥✳ ❚❤❡ ❞❛t❛ ✇❡r❡ t❤❡ ❛♥s✇❡rs ♦❢ ✶✷✺ ✇♦♠❡♥ ✐♥ ❛ st✉❞② ♦❢ ❛♥ ♦✉t♣❛t✐❡♥t ❡❛t✐♥❣ ❞✐s♦r❞❡r t♦ t❤❡ ❇♦❞② ■♠❛❣❡ ◗✉❡st✐♦♥♥❛✐r❡✳ ❚❤r♦✉❣❤ Pr✐♥❝✐♣❛❧ ❈♦♠♣♦♥❡♥ts ❆♥❛❧②s✐s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❞❛t❛ ✇❛s r❡❞✉❝❡❞ ❢r♦♠ ✸✹ t♦ ✹ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ✉s❡ ♦❢ ♦❜❧✐q✉❡ r♦t❛t✐♦♥ ✭♦❜❧✐♠✐♥✮ ♣r♦✈✐❞❡ r❡❧❡✈❛♥ts ❝❧✉st❡r ♦❢ ❜❡❤❛✈✐♦rs✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❆♥á❧✐s❡ ❋❛t♦r✐❛❧ ✸

✷✳✶ ▼ét♦❞♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ ❢❛t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❊s❝♦❧❤❛ ❞♦ ♥ú♠❡r♦ ❞❡ ❢❛t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❆✈❛❧✐❛çã♦ ❞♦s ❋❛t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✹ ❘♦t❛çã♦ ❞♦s ❋❛t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✸ ❘❡s✉❧t❛❞♦s ✶✶

✹ ❈♦♥❝❧✉sõ❡s ✷✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✷✸

❆♣ê♥❞✐❝❡ ❆ ◗✉❡st✐♦♥ár✐♦ ❞❡ ■♠❛❣❡♠ ❈♦r♣♦r❛❧ ✭❇♦❞② ❙❤❛♣❡ ◗✉❡st✐♦♥♥❛✐r❡✮ ✷✺

■♥tr♦❞✉çã♦ ✶

✶✳ ■♥tr♦❞✉çã♦

❆ ❛♥á❧✐s❡ ♠✉❧t✐✈❛r✐❛❞❛ é ✉♠❛ ár❡❛ ❞❛ ❊st❛tíst✐❝❛ q✉❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛ ❛✈❛❧✐❛çã♦ ❝♦♥❥✉♥t❛ ❞❡ ✈❛r✐á✈❡✐s ♦❜s❡r✈❛❞❛s ♥❛s ♠❡s♠❛s ✉♥✐❞❛❞❡s ❛♠♦str❛✐s✳ ❆❧é♠ ❞✐ss♦✱ ✐♥✈❡st✐❣❛ ❛s ✐♥t❡r✲r❡❧❛çõ❡s ❡♥tr❡ ❡ss❛s ✈❛r✐á✈❡✐s ❡ ❛s s♦❧✉çõ❡s ♣❛r❛ ♦s ♣r♦❜❧❡♠❛s ❛❜♦r❞❛❞♦s sã♦ ♠❛✐s ❝♦♥s✐st❡♥t❡s ❡ út❡✐s ❬✼❪✳ ◆❡ss❡ ❝❛s♦✱ ❛s ✈❛r✐á✈❡✐s ♥ã♦ sã♦ s❡♣❛r❛❞❛s ❡♠ ✈❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠♦ ❡♠ ❛♥á❧✐s❡ ❞❡ r❡❣r❡ssã♦✱ ♠❛s sã♦ ❛❣r✉♣❛❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ s✉❛ ❡str✉t✉r❛ ❞❡ ✐♥t❡r✲r❡❧❛❝✐♦♥❛♠❡♥t♦ ❡♠ ✈❛r✐á✈❡✐s ♥ã♦ ♦❜s❡r✈á✈❡✐s ❞❡♥♦♠✐♥❛❞❛s ❢❛t♦r❡s✳ ❬✽❪✳

❆ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ t❡✈❡ s❡✉s ♣r✐♠❡✐r♦s ❝♦♥❝❡✐t♦s s✉❣❡r✐❞♦s ♣♦r ●❛❧t♦♥ ❬✾❪ ❡ ❡♠ ✶✾✵✹ ❙♣❡❛r✲ ♠❛♥ ❬✷✹❪ ♣r♦♣ôs ✉♠ ♠♦❞❡❧♦✱ ✉s❛❞♦ ❛té ❤♦❥❡✱ ❞❡ ♦r❣❛♥✐③❛çã♦ ❡♠ ❢❛t♦r❡s ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ♠❡♥s✉r❛r ❛ ✐♥t❡❧✐❣ê♥❝✐❛ ❤✉♠❛♥❛✳ ❖ ♠♦❞❡❧♦ ❞❡ ✉♠ ú♥✐❝♦ ❢❛t♦r ❞❡ ❙♣❡❛r♠❛♥ ❢♦✐ ❣❡♥❡r❛❧✐③❛❞♦ ♣♦r ❚❤✉rst♦♥❡ ❬✷✺❪✱ ♣❛r❛ ❝♦♥t❡♠♣❧❛r ♠ú❧t✐♣❧♦s ❢❛t♦r❡s ❬✼❪ ❡ t❡♥t❛r s✐♠♣❧✐✜❝❛r r❡❧❛❝✐♦♥❛♠❡♥t♦s ❝♦♠♣❧❡①♦s✱ ❢♦r♥❡❝❡♥❞♦ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ ❡str✉t✉r❛ s✉❜❥❛❝❡♥t❡ ❞♦s ❞❛❞♦s ❬✻❪✳

❊♠ t❡r♠♦s ❣❡r❛✐s✱ ❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ❛❜♦r❞❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛♥❛❧✐s❛r ❛ ❡str✉t✉r❛ ❞❛s ✐♥t❡r✲ r❡❧❛çõ❡s ❡♥tr❡ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s q✉❡ sã♦ ❛❣r✉♣❛❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ s✉❛ ❡str✉t✉r❛ ❞❡ ✐♥t❡r✲r❡❧❛❝✐♦♥❛♠❡♥t♦ ❡♠ ✈❛r✐á✈❡✐s ♥ã♦ ♦❜s❡r✈á✈❡✐s ❞❡♥♦♠✐♥❛❞❛s ❢❛t♦r❡s✳ ❖s ❞♦✐s ♣r✐♥❝✐♣❛✐s ✉s♦s ❞❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ✲ r❡s✉♠♦ ❡ r❡❞✉çã♦ ❞❡ ❞❛❞♦s ✲ ♣♦❞❡♠ s❡r ❛❧❝❛♥ç❛❞♦s ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❡s❝♦r❡s✱ ✈❛❧♦r❡s ❝❛❧❝✉❧❛❞♦s ♣❛r❛ ♦s ❢❛t♦r❡s q✉❡ ♣♦❞❡♠ s✉❜st✐t✉✐r ❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ❬✶✶❪✳

❯♠ ❞♦s ♠ét♦❞♦s ❞❡ ♦❜t❡♥çã♦ ❞❡ ❢❛t♦r❡s ♠❛✐s ✉t✐❧✐③❛❞♦s é ❛ ❆♥á❧✐s❡ ❞❡ ❈♦♠♣♦♥❡♥t❡s Pr✐♥✲ ❝✐♣❛✐s ✭❆❈P✮✱ ♣♦✐s ♥ã♦ ❡①✐❣❡ s✉♣♦s✐çõ❡s s♦❜r❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s✱ ❝♦♠♦ ❛ ♥♦r♠❛❧✐❞❛❞❡ ♠✉❧t✐✈❛r✐❛❞❛ ❬✽❪✳ ➱ ✉♠❛ té❝♥✐❝❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ✐♥✈❡st✐❣❛r ❡ ❛✉①✐❧✐❛r ❛ ✐♥t❡r♣r❡t❛✲ çã♦ ❞❛ ❡str✉t✉r❛ ❞❡ ✐♥t❡r❞❡♣❡♥❞ê♥❝✐❛ ❞❛s ✈❛r✐á✈❡✐s✳ ❋♦✐ ✐♥tr♦❞✉③✐❞❛ ♣♦r P❡❛rs♦♥ ❡♠ ✶✾✵✶ ❬✷✶❪ ❡ ❞❡s❡♥✈♦❧✈✐❞❛ ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡ ♣♦r ❍♦t❡❧❧✐♥❣ ❡♠ ✶✾✸✸ ❬✶✹❪✳

❊①✐st❡♠ ♦✉tr♦s ♠ét♦❞♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ❢❛t♦r❡s✱ ❞❡♥tr❡ ❡❧❡s✿ ♠ét♦❞♦ ❞❛ ♠á①✐♠❛ ✈❡r♦ss✐✲ ♠✐❧❤❛♥ç❛ q✉❡ ✉t✐❧✐③❛ s✉♣♦s✐çõ❡s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ❡ ❞❡s❡♥✈♦❧✈❡ t❡st❡s ❞❡ ❤✐♣ót❡s❡s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ t❡st❛r ❛ ❛❞❡q✉❛❝✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦❀ ❢❛t♦r❡s ❝♦♠✉♥s tê♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✐❞❡♥t✐✜❝❛r ❛s ❞✐♠❡♥sõ❡s r❡♣r❡s❡♥t❛❞❛s ♥❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s✱ ♣♦ré♠ ❝♦♠ ♣♦✉❝♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ✈❛r✐â♥❝✐❛ ❞❡s❡❥❛♥❞♦ ❡❧✐♠✐♥á✲❧❛❀ ♠ét♦❞♦ ❞♦s ♠í♥✐♠♦s q✉❛❞r❛❞♦s q✉❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♠✐♥✐♠✐③❛r ♦ ❡rr♦❀ ❞❡♥tr❡ ♦✉tr♦s ✭♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✱ ✈❡r ❬✶✹❪✮✳

❆ ❆❈P ♣♦❞❡ s❡r r❡s✉♠✐❞❛ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s✳ ❆s ♥♦✈❛s ✈❛r✐á✈❡✐s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s ♦✉ ❢❛t♦r❡s✳ ❈❛❞❛ ❝♦♠♣♦♥❡♥t❡ ♣r✐♥❝✐♣❛❧ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ❡ ❝❛rr❡❣❛ ✉♠❛ ♣❛r❝❡❧❛ ❞❛ ✈❛r✐❛çã♦ t♦t❛❧ ❞♦s ❞❛❞♦s ♦r✐❣✐♥❛✐s ♥❛ ❢♦r♠❛ ❞❡ s✉❛ ♣ró♣r✐❛ ✈❛r✐â♥❝✐❛✳

❖s ❈P✬s r❡s✉❧t❛♥t❡s sã♦ ❡♠ ♥ú♠❡r♦ ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ♣r❡s❡♥t❡s ♥♦ ❡s✲

✷ ■♥tr♦❞✉çã♦

t✉❞♦ ❡ sã♦ ♦r❣❛♥✐③❛❞♦s ❡♠ ♦r❞❡♠ ❞❡❝r❡s❝❡♥t❡ ❞❡ ✈❛r✐â♥❝✐❛s✱ ♦ ♠❛✐s ✐♥❢♦r♠❛t✐✈♦ ♦✉ ❞❡ ♠❛✐♦r ✈❛r✐â♥❝✐❛ é ♦ ♣r✐♠❡✐r♦ ❡ ♦ ♠❡♥♦s ✐♥❢♦r♠❛t✐✈♦ é ♦ ú❧t✐♠♦ ❬✻❪✳ ❙ã♦ ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞♦s ❛✉t♦✈❡t♦r❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦s ❛✉t♦✈❛❧♦r❡s ❬✶❪ ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ ♦✉ ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦✳ ❖ ♦❜✲ ❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ é q✉❡ ♦s ❢❛t♦r❡s s❡❥❛♠ ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣r❡❡♥sí✈❡✐s ❡ q✉❡ tr❛♥s♠✐t❛♠ ❛ ✐♥❢♦r♠❛çã♦ ❡ss❡♥❝✐❛❧ ❝♦♥t✐❞❛ ♥♦ ❝♦♥❥✉♥t♦ ♦r✐❣✐♥❛❧ ❞❡ ✈❛r✐á✈❡✐s ❬✻❪✳

❖s ❝♦♠♣♦♥❡♥t❡s ✐♥✐❝✐❛❧♠❡♥t❡ ♦❜t✐❞♦s sã♦ ♠✉✐t❛s ✈❡③❡s ❞✐❢í❝❡✐s ❞❡ ✐♥t❡r♣r❡t❛r✱ q✉❛♥❞♦ s❡ t❡♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ♠ét♦❞♦ ❞❡ ❡①tr❛çã♦ ❞❡ ❢❛t♦r❡s ✉t✐❧✐③❛❞♦✳ ❋❡❧✐③♠❡♥t❡✱ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ♥♦✈♦s ❢❛t♦r❡s ❛ ♣❛rt✐r ❞❡ r♦t❛çõ❡s ❛♣❧✐❝❛❞❛s ❛♦s ❝♦♠♣♦♥❡♥t❡s ✐♥✐❝✐❛✐s✳ ❆ ✐❞❡✐❛ ❣❡♦♠étr✐❝❛ é r❡❛❧✐③❛r ✉♠❛ r♦t❛çã♦ ♥♦ s✐st❡♠❛ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ♦s ♥♦✈♦s ❡✐①♦s s❡❥❛♠ ♣♦s✐❝✐♦♥❛❞♦s ♥♦ s❡♥t✐❞♦ ❞❡ ♠❛✐♦r ✈❛r✐❛❜✐❧✐❞❛❞❡ ❬✼❪✳

❊①✐st❡♠ ❞♦✐s t✐♣♦s ❞❡ r♦t❛çã♦✳ ◆♦ ♣r✐♠❡✐r♦✱ ❝❤❛♠❛❞♦ ❞❡ r♦t❛çã♦ ♦rt♦❣♦♥❛❧✱ ♦s ❡✐①♦s sã♦ ♠❛♥t✐❞♦s ♣❡r♣❡♥❞✐❝✉❧❛r❡s✱ ♥♦ s❡❣✉♥❞♦✱ ❝❤❛♠❛❞♦ r♦t❛çã♦ ♦❜❧íq✉❛✱ ♥ã♦ ❡①✐st❡ t❛❧ r❡str✐çã♦ ❡ ♦s ❡✐①♦s ❞♦s ❢❛t♦r❡s ♣♦❞❡♠ s❡r ❣✐r❛❞♦s ❞❡ ❢♦r♠❛ ❧✐✈r❡ ❬✶❪✳

❊st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ❛ té❝♥✐❝❛ ❞❡ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ❡♠ ❞❛❞♦s s♦❜r❡ ❝♦♠♣♦rt❛✲ ♠❡♥t♦s r❡❧❛❝✐♦♥❛❞♦s à ✐♠❛❣❡♠ ❝♦r♣♦r❛❧ ❡♠ ♣❡ss♦❛s q✉❡ ♣♦ss✉❡♠ ❞✐stúr❜✐♦s ❛❧✐♠❡♥t❛r❡s✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❛ ❛♥♦r❡①✐❛ ❡ ❛ ❜✉❧✐♠✐❛✳ ❊❧❛s ❢♦r❛♠ ❞❡s❝r✐t❛s ❤á ♠✉✐t♦s sé❝✉❧♦s ❬✷✵❪ ❡ s❡ t♦r♥❛♠ ❝❛❞❛ ✈❡③ ♠❛✐s ❝♦♠✉♥s ♥❛ s♦❝✐❡❞❛❞❡ ❛t✉❛❧✳

❆ ♣❡ss♦❛ ❝♦♠ ❛♥♦r❡①✐❛ t❡♠ ✉♠❛ ❞✐st♦rçã♦ ❣r❛✈❡ ❞❛ ✐♠❣❡♠ ❞♦ s❡✉ ❝♦r♣♦ ❡ ♣❡r❞❡ ♣r♦♣♦s✐✲ t❛❧♠❡♥t❡ ♦ ♣❡s♦ ♠♦t✐✈❛❞♦ ♣❡❧♦ ❞❡s❡❥♦ ❞❡ ❡♠❛❣r❡❝❡r ❛ss♦❝✐❛❞♦ ❛♦ ♠❡❞♦ ❞❡ ❡♥❣♦r❞❛r✳ ❏á ♣❡ss♦❛s ❝♦♠ ❜✉❧✐♠✐❛ ♥ã♦ ♣♦ss✉❡♠ ✉♠❛ ❞✐st♦rçã♦ tã♦ ❣r❛✈❡ ❞❛ ✐♠❛❣❡♠ ❝♦r♣♦r❛❧✱ ♠❛s ❛♣r❡s❡♥t❛♠ ❝♦♠✲ ♣✉❧s✐✈✐❞❛❞❡ ❡♠ r❡❧❛çã♦ ❛ ❝♦♠✐❞❛ s❡❣✉✐❞❛ ❞❡ ♠❡✐♦s ❝♦♠♣❡♥s❛tór✐♦s ♣❛r❛ ❡✈✐t❛r ♦ ❣❛♥❤♦ ❞❡ ♣❡s♦ ✭✈ô♠✐t♦s ✈♦❧✉♥tár✐♦s✱ ✉s♦ ❞❡ ❧❛①❛♥t❡s ❡ ❞✐✉rét✐❝♦s✱ ❡t❝✳✮ ❬✶✵❪✳

❉✉r❛♥t❡ ✉♠ ❡st✉❞♦ ♣❛r❛ ✐♥✈❡st✐❣❛r ❛ ❝❤❡❝❛❣❡♠ ❞♦ ❝♦r♣♦ ❡♠ tr❛♥st♦r♥♦s ❛❧✐♠❡♥t❛r❡s✱ ♦ ◗✉❡st✐♦♥ár✐♦ ❞❡ ■♠❛❣❡♠ ❈♦r♣♦r❛❧ ✭❆♣ê♥❞✐❝❡❆✮ ❢♦✐ ❛♣❧✐❝❛❞♦ ❛ ✶✷✺ ♠✉❧❤❡r❡s✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♥❛❧✐s❛r ❡ss❡s ❞❛❞♦s ♣❛r❛ ✐❞❡♥t✐✜❝❛r ❝♦♠♣♦rt❛♠❡♥t♦s q✉❡ s❡❥❛♠ r❡❧❛❝✐♦♥❛❞♦s ❡♥tr❡ s✐ ❡ ♠❡❧❤♦r❛r ❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❞✐st♦rçã♦ ❞❛ ✐♠❛❣❡♠ ❝♦r♣♦r❛❧ ❞❛s ♣❛❝✐❡♥t❡s ❞♦ ❡st✉❞♦ ✉t✐❧✐③❛♥❞♦ ❆❈P ❡ r♦t❛çã♦ ♦❜❧íq✉❛✳

◆♦ ❈❛♣✐t✉❧♦ ✷ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ❡ ❞❡ ❝♦♠♣♦♥❡♥t❡s ♣r✐♥✲ ❝✐♣❛✐s✳ ◆❛ s❡çã♦ ✸✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s r❡s✉❧t❛❞♦s ❛ ♣❛rt✐r ❞❛s ❛♥á❧✐s❡s✳ ❊ ♥❛ s❡çã♦ ✹✱ ❛s ❝♦♥❝❧✉sõ❡s ♦❜t✐❞❛s✳

❆♥á❧✐s❡ ❋❛t♦r✐❛❧ ✸

✷✳ ❆♥á❧✐s❡ ❋❛t♦r✐❛❧

❆❧❣✉♥s ❝♦♥❝❡✐t♦s✱ ♥❡❝❡ssár✐♦s à ♥♦t❛çã♦ ❡ à ♠❡t♦❞♦❧♦❣✐❛ s❡rã♦ ✐♥tr♦❞✉③✐❞♦s ❛ s❡❣✉✐r✳

❙❡❥❛ X✱ ❝♦♠ X _{= (X}1, X2, . . . , X_p✮ ✉♠ ✈❡t♦r ❛❧❡❛tór✐♦ ❝♦♥tí♥✉♦ ❝♦♠ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ f(x₎✳ ❉❡✜♥❡✲s❡ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ♦✉ ❡s♣❡r❛♥ç❛ ❞❡st❡ ✈❡t♦r ♣♦r ✉♠ ✐♥t❡❣r❛❧

♠ú❧t✐♣❧❛ ❬✼❪✱

❊(X_{) = µ=}    

E(X1)

✳✳✳

E(X_p)

   =     ✁∞ −∞. . . ✁∞

−∞x1f(x1, . . . , xp)dx1dxp

✳✳✳ ✁∞

−∞. . .

✁∞

−∞xpf(x1, . . . , xp)dx1dxp

   =     µ1 ✳✳✳ µp     ✭✷✳✶✮

❙❡X é ✉♠ ✈❡t♦r ❛❧❡❛tór✐♦ ❞✐s❝r❡t♦ ❛s ✐♥t❡❣r❛✐s sã♦ s♦♠❛s ❞❡✜♥✐❞❛s ♥♦ s✉♣♦rt❡ ❞❛s ✈❛r✐á✈❡✐s✳

❯♠❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛r ❛ ❡str✉t✉r❛ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ✈❛r✐á✈❡✐s é ❛ ❝♦✈❛r✐â♥❝✐❛✳ ❆ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ ❞❡ ✉♠ ✈❡t♦r ❛❧❡❛tór✐♦X é ❞❡✜♥✐❞❛ ♣♦r ❬✼❪✱

Cov(X_{) = Σ =}

     

σ11 σ12 . . . σ1_p

σ21 σ22 . . . σ2_p

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

σp1 σ_p2 . . . σ_pp

      ✭✷✳✷✮

❡♠ q✉❡✱ σij =Cov(Xi, Xj)✱ ♣❛r❛i6=j✳ ❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❝♦✈❛r✐â♥❝✐❛ é ❛ ✈❛r✐â♥❝✐❛ ❞❡ ✉♠❛

✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ V ar(X) = ❊[(X−E(X))2

] =σ2

i =σii✳

❖✉tr❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ✈❛r✐á✈❡✐s é ❛ ❝♦rr❡❧❛çã♦✳ ❆ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ ❞♦ ✈❡t♦r ❛❧❡❛tór✐♦X é ❞❛❞❛ ♣♦r ❬✼❪✱

R=Cor(X_{i) =}

     

1 ρ12 . . . ρ1_p

ρ21 1 . . . ρ2_p

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

ρp1 ρ_p2 . . . 1

     , ✭✷✳✸✮

♥❛ q✉❛❧ ρij =

σij

(σ2

iσ

2

j)

1_/2✱ ♣❛r❛ i 6= j ❡ ρ11 = ρ22 = . . . = ρpp = 1✱ ♣❛r❛ j = 1,2, . . . , p✳ ❆ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ é ♠❛✐s ✉s❛❞❛ q✉❡ ❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛✱ ♣♦✐s s❡✉s ✈❛❧♦r❡s ❡stã♦ ❧✐♠✐t❛❞♦s ✭0≤ρ≤1✮ ♦ q✉❡ ❢❛❝✐❧✐t❛ ❛ ✐♥t❡r♣r❡t❛çã♦✳

❊①✐st❡ ❛✐♥❞❛✱ ♦✉tr♦ ❝♦❡✜❝✐❡♥t❡ q✉❡ ♠❡❞❡ ❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ✈❛r✐á✈❡✐s q✉❛♥❞♦ s❡ ❝♦♥tr♦❧❛ ♦ ❡❢❡✐t♦ ❞❛ ♦✉tr❛ ✈❛r✐á✈❡❧ s♦❜r❡ ❡st❛s✳❙❡❥❛ αX,Y|Z ❛ ❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ❡♥tr❡ ❳ ❡ ❨ ✜①❛♥❞♦ ♦

✹ ❆♥á❧✐s❡ ❋❛t♦r✐❛❧

❡❢❡✐t♦ ❞❡ ❩✱ t❡♠✲s❡ ❬✶✺❪✿

αX,Y|Z =

ρXY −ρXZ ρY Z

p

(1−ρ2

XZ)

p

(1−ρ2

Y Z)

✭✷✳✹✮

❈♦♠ ❜❛s❡ ♥♦s ❝♦♥❝❡✐t♦s ❛♥t❡r✐♦r❡s✱ ❛❧❣✉♠❛s ♠❡❞✐❞❛s ♣♦❞❡♠ s❡r ❛♣r❡s❡♥t❛❞❛s ♣❛r❛ ✈❡r✐✜❝❛r ❛ ✈✐❛❜✐❧✐❞❛❞❡ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ❆♥á❧✐s❡ ❋❛t♦r✐❛❧ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✳ ❯♠❛ ❞❡❧❛s é ❛ ▼❡❞✐❞❛ ❞❡ ❆❞❡q✉❛çã♦ ❆♠♦str❛❧ ✭▼❆❆✮ q✉❡ é ✉t✐❧✐③❛❞❛ ♣❛r❛ q✉❛♥t✐✜❝❛r ♦ ❣r❛✉ ❞❡ ✐♥t❡r✲❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s✳ ❙✉❛ ❞❡✜♥✐çã♦ é ❞❛❞❛ ❛ s❡❣✉✐r✶_✳

❉❡✜♥✐çã♦ ✶ ✿ ❆ ▼❡❞✐❞❛ ❞❡ ❆❞❡q✉❛çã♦ ❆♠♦str❛❧ ♣❛r❛ ❛ ✈❛r✐á✈❡❧ Xi é ❞❛❞❛ ♣♦r ❬✶✼❪✱

M AA(i) = 1−

P j α2 ij P j ρ2 ij

,∀i6=j ✭✷✳✺✮

❡♠ q✉❡✱ ρij é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s Xi ❡Xj ❡♥q✉❛♥t♦ αij é ♦ ❝♦❡✜❝✐❡♥t❡

❞❡ ❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ❡♥tr❡ ❛s ♠❡s♠❛s ✈❛r✐á✈❡✐s ✜①❛♥❞♦ ♦ ❡❢❡✐t♦ ❞❛s ❞❡♠❛✐s ✈❛r✐á✈❡✐s ❞❛ ❛♥á❧✐s❡✳ ❖ ❝r✐tér✐♦ ❞❡ ❑❛✐s❡r✲▼❡②❡r✲❖❧❦✐♥ ✭❑▼❖✮ é ✉♠ ❝r✐tér✐♦ q✉❡ ❝♦♥s✐❞❡r❛ t♦❞❛s ❛s ✈❛r✐á✈❡✐s s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❊❧❡ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ▼❆❆ ❣❧♦❜❛❧ ❬✶✼❪✳

❉❡✜♥✐çã♦ ✷ ✿ ❖ í♥❞✐❝❡ ❞❡ ❑❛✐s❡r✲▼❡②❡r✲❖❧❦✐♥ ✭❑▼❖✮ é ❞❛❞♦ ♣♦r✿

KM O =

P i P j ρ2 ij P i P j ρ2 ij + P i P j α2 ij ✭✷✳✻✮

❡♠ q✉❡✱ ρij é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s Xi ❡Xj ❡♥q✉❛♥t♦ αij é ♦ ❝♦❡✜❝✐❡♥t❡

❞❡ ❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ❡♥tr❡ ❛s ♠❡s♠❛s ✈❛r✐á✈❡✐s ✜①❛♥❞♦ ♦ ❡❢❡✐t♦ ❞❛s ❞❡♠❛✐s ✈❛r✐á✈❡✐s ❞❛ ❛♥á❧✐s❡✳ ❆ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ♥ã♦ é ✈✐á✈❡❧ q✉❛♥❞♦ ♦s ✈❛❧♦r❡s ❞♦ ❑▼❖ ❡ ▼❆❆ ❢♦r❡♠ ❜❛✐①♦s✱ q✉❡ q✉❡r ❞✐③❡r q✉❡ ❛s ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s sã♦ ♠✉✐t♦ ❜❛✐①❛s✳

❈❛s♦ ❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ s❡❥❛ ❝♦♥s✐❞❡r❛❞❛ ✈✐á✈❡❧ ❞❡✜♥❡♠✲s❡ três ❡t❛♣❛s ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧✿ ♦❜t❡♥çã♦ ❞♦s ❢❛t♦r❡s✱ ❞❡t❡r♠✐♥❛çã♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢❛t♦r❡s r❡t✐❞♦s ❡ ❛ r♦t❛çã♦✳

✷✳✶ ▼ét♦❞♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ ❢❛t♦r❡s

◆❛ ❛♥á❧✐s❡ ❞❡ ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s✱ ❞❡s❝r❡✈❡✲s❡ ❛ ✈❛r✐❛çã♦ t♦t❛❧ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ n

♣♦♥t♦s ♥♦ ❡s♣❛ç♦ p✲❞✐♠❡♥s✐♦♥❛❧✱ ✐♥tr♦❞✉③✐♥❞♦ ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ ❞❡p✈❛r✐á✈❡✐s ♦rt♦❣♦♥❛✐s✱ ♥ã♦

❝♦rr❡❧❛❝✐♦♥❛❞❛s✳ ❆s ♥♦✈❛s ✈❛r✐á✈❡✐sY1, Y2, . . . , Y_p✱ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞❛s ❧❛t❡♥t❡s✱ ♦✉ s❡❥❛✱ ♥ã♦ ♦❜s❡r✈á✈❡✐s ♦✉ ♠❡♥s✉rá✈❡✐s ❞✐r❡t❛♠❡♥t❡ ❛ ♣❛rt✐r ❞♦ ❡①♣❡r✐♠❡♥t♦ ♦✉ ❧❡✈❛♥t❛♠❡♥t♦ ❛♠♦str❛❧✳

❆ ♣❛rt✐r ❞♦s ❛✉t♦✈❡t♦r❡s ❡ ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ Σ✭♦✉ ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛✲

çã♦✮ ❞❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ❡♥❝♦♥tr❛♠✲s❡ ♦s ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐sY1, Y2, . . . , Y_p✳ P❛r❛ ❝❛❞❛Y_i

✶_{❊ss❡ í♥❞✐❝❡ ✈❛r✐❛ ❞❡ ✵ ❛ ✶✱ ❛❧❝❛♥ç❛♥❞♦ ✶ q✉❛♥❞♦ ❝❛❞❛ ✈❛r✐á✈❡❧ é ♣❡r❢❡✐t❛♠❡♥t❡ ♣r❡✈✐st❛ s❡♠ ❡rr♦ ♣❡❧❛s}

♦✉tr❛s ✈❛r✐á✈❡✐s✳ ❙❡❣✉♥❞♦ ❍❛✐r ❬✶✶❪ ❛ ♠❡❞✐❞❛ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠ ❛s s❡❣✉✐♥t❡s ♦r✐❡♥t❛çõ❡s✿ ✵✱✽ ♦✉ ❛❝✐♠❛✱

❛❞♠✐rá✈❡❧❀ ✵✱✼ ♦✉ ❛❝✐♠❛✱ ❜♦♠❀ ✵✱✻ ♦✉ ❛❝✐♠❛✱ ♠❡❞✐❛♥♦❀ ✵✱✺ ♦✉ ❛❝✐♠❛✱ r✉✐♠❀ ❡ ❛❜❛✐①♦ ❞❡ ✵✱✺✱ ✐♥❛❝❡✐tá✈❡❧✳ ❙❡❣✉♥❞♦

❋á✈❡r♦ ❬✽❪ q✉❛♥❞♦ ❛ ♠❡❞✐❞❛ ❞❡ ❞❡t❡r♠✐♥❛❞❛ ✈❛r✐á✈❡❧ ❢♦r ❜❛✐①❛✱ ❡st❛ ✈❛r✐á✈❡❧ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞❡✈❡ s❡r

❡❧✐♠✐♥❛❞❛✱ ✉♠❛ ✈❡③ q✉❡ ❡st❛ ✈❛r✐á✈❡❧ ♣♦❞❡ r❡♣r❡s❡♥t❛r ✉♠ ❢❛t♦r ✐s♦❧❛❞❛♠❡♥t❡✳

❆♥á❧✐s❡ ❋❛t♦r✐❛❧ ✺

❜✉s❝❛✲s❡ ✉♠ ✈❡t♦r ❞❡ ❝♦❡✜❝✐❡♥t❡s γ_i′ = (γi1, γ_i2, . . . , γ_ip)✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ✈❛r✐â♥❝✐❛ ❞❡ Y_i =γ_iX s❡❥❛ ♠á①✐♠❛ s♦❜r❡ ❛ ❝❧❛ss❡ ❞❡ t♦❞❛s ❛s ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❞❡X s✉❥❡✐t♦ às r❡str✐çõ❡s_γ′_{γ = 1}

❡ γ_i′γj = 0, i6= j✷❬✼❪✳ ■ss♦ ✐♠♣❡❞❡ ♦ ❛✉♠❡♥t♦ ❛r❜✐trár✐♦ ♥❛ ✈❛r✐â♥❝✐❛ ❞❡γ

′

iX✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡

♦s ❝♦♠♣♦♥❡♥t❡s ❞❡γi s❡❥❛♠ ❣r❛♥❞❡s✸✳

❖ ♣r♦❜❧❡♠❛ ❛❣♦r❛ s❡ t♦r♥❛✿ ♠❛①✐♠✐③❛r γ_i′Σ_γi✱ ❝♦♠ r❡s♣❡✐t♦ ❛_γi✱ s✉❥❡✐t♦ ❛ r❡str✐çã♦_γ′

iγi = 1

❬✶✾❪✳ ❆ss✐♠✱ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡γi ❞❡✈❡♠ s❛t✐s❢❛③❡r ❛s ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❞❡✜♥✐❞❛s ♣♦r ✉♠ s✐st❡♠❛

❝♦♠ ♣ ✈❛r✐á✈❡✐s ❬✶✺❪✱

(Σ_−λiI_{)γi =}0_, ✭✷✳✼✮

❡ ♦ ♠ét♦❞♦ ❞♦ ♠✉❧t✐♣❧✐❝❛❞♦r ▲❛❣r❛♥❣❡ é ✉t✐❧✐③❛❞♦✳ ❙✉♣♦♥❞♦ q✉❡ ❛ s♦❧✉çã♦ é ❞✐❢❡r❡♥t❡ ❞♦ ✈❡t♦r ♥✉❧♦✱ ♦ ✈❛❧♦r ❞❡ λi ❞❡✈❡ s❡r ❡s❝♦❧❤✐❞♦ ♣❛r❛ q✉❡✱

(Σ_−λiI_{) = 0.} ✭✷✳✽✮

❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❛ ❞❡❝♦♠♣♦✲ s✐çã♦ ❡s♣❡❝tr❛❧✳ ❆ss✐♠ s❡ Σé ❞✐❛❣♦♥❛❧✐③á✈❡❧ t❡♠✲s❡

Σ =ΓΛΓ′ _=λ1γ1γ

′

1+λ2γ2γ

′

2+. . .+λpγpγ

′

p, ✭✷✳✾✮

❡♠ q✉❡Λ é ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦λ1, λ2, . . . , λ_p ❡Γ é ✉♠❛ ♠❛tr✐③ ♦rt♦❣♦♥❛❧ ❞❡ ♦r❞❡♠ ♣ ❝✉❥❛s ❝♦❧✉♥❛s sã♦ γ1, γ2, . . . , γ_p ❬✶✺❪✳ ❊♥tã♦✱ ♣♦❞❡✲s❡ ♦❜s❡r✈❛r q✉❡

tr(Σ_{) =tr(}ΓΛΓ′_{) = tr(}ΛΓΓ′_{) =tr(}ΛI_{) = tr(}Λ_{) =}

p

X

i=1

λi. ✭✷✳✶✵✮

◆♦ ❡♥t❛♥t♦✱ tr(Σ₎ é ♦❜✈✐❛♠❡♥t❡ ❞❛❞❛ ♣❡❧❛ s♦♠❛ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❞✐❛❣♦♥❛❧✱ ♦✉ s❡❥❛✱ _tr(Σ_{) =}

p

P

i=1

σ2

i✱ ❞❡ss❛ ❢♦r♠❛

p

X

i=1

σ2_i =

p

X

i=1

λi, ✭✷✳✶✶✮

♦✉ s❡❥❛✱ ❛ ✈❛r✐❛❜✐❧✐❞❛❞❡ t♦t❛❧ ❝♦♥t✐❞❛ ♥❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s é ✐❣✉❛❧ à ✈❛r✐❛❜✐❧✐❞❛❞❡ t♦t❛❧ ❝♦♥t✐❞❛ ♥♦s ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s ❬✼❪✳

P♦r ❞❡✜♥✐çã♦✱ ❛ ✈❛r✐â♥❝✐❛ ❞❡ Yi é ❞❛❞❛ ♣♦r ❬✼❪

V ar(Yi) = γ

′

iΣγi =λiγ

′

iγi =λi, ✭✷✳✶✷✮

✷_{❖s ✈❡t♦r❡s γ}

1, γ2, . . . , γp sã♦ ♦rt♦♥♦r♠❛✐s✱ ♦✉ s❡❥❛✱ ♠✉t✉❛♠❡♥t❡ ♣❡r♣❡♥❞✐❝✉❧❛r❡s (γ

′

iγj = 0, i 6= j) ❡ ❞❡

❝♦♠♣r✐♠❡♥t♦ ✉♥✐tár✐♦(γ_i′γi= 1)✳

✸_{P❛r❛ ✉♠ ❞❛❞♦ ✈❡t♦r γi✱ ♣♦❞❡✲s❡ s❡♠♣r❡ ❡♥❝♦♥tr❛r ♦✉tr♦ ❝♦♠ ✈❛r✐â♥❝✐❛ ♠❛✐♦r ❡s❝♦❧❤❡♥❞♦ ✉♠ ✈❡t♦r ❝♦♠ ❛}

♠❡s♠❛ ❞✐r❡çã♦ ❞❡γi✱ ♠❛s ❝♦♠ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ♠❛✐♦r✳ ■ss♦ ❡q✉✐✈❛❧❡ ❛ ♠✉❧t✐♣❧✐❝❛r γi ♣♦r ✉♠❛ ❝♦♥st❛♥t❡✱ q✉❡

♥ã♦ ❛❧t❡r❛ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❜ás✐❝❛ ❞❡ γ_i′X✳ P♦rt❛♥t♦✱ ❛♣❡♥❛s ❛ ❞✐r❡çã♦ ❞❡ γi ❞❡✈❡ ❞❡t❡r♠✐♥❛r s✉❛ ❛❞❡q✉❛çã♦

❝♦♠♦ s♦❧✉çã♦✱ ❡ ♥ã♦ ♦ s❡✉ ❝♦♠♣r✐♠❡♥t♦✱ q✉❡ ♣♦r ❝♦♥✈❡♥✐ê♥❝✐❛✱ t❛♠❜é♠ ♣♦❞❡ s❡r ✉♠✳

✻ ❆♥á❧✐s❡ ❋❛t♦r✐❛❧

❡ ❛ ❝♦✈❛r✐â♥❝✐❛ ❡♥tr❡Yi ❡Yj✱ ♣♦r

Cov(Yi, Yj) =γ

′

iΣγj =λiγ

′

iγj = 0, i6=j ✭✷✳✶✸✮

✉♠❛ ✈❡③ q✉❡ γi ❡ γj sã♦ ♦rt♦❣♦♥❛✐s✳ ▲♦❣♦✱ ♦s ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s sã♦ ♥ã♦ ❝♦rr❡❧❛❝✐♦♥❛❞♦s✳

❖s ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s✱ Yj, j = 1,2, . . . , p✱ sã♦ ❞❡✜♥✐❞♦s ❝♦♠♦ ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❞❛s

✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s X ❬✻❪✿

Yi =γi1X1 +γ_i2X2+. . .+γ_ipX_p. ✭✷✳✶✹✮

❖s ♣❡s♦s γi1, γ_i2, . . . , γ_ip ❢♦r❛♠ ❞❡t❡r♠✐♥❛❞♦s ♣❛r❛ ♠❛①✐♠✐③❛r ❛ ♣r♦♣♦rçã♦ ❞❛ ✈❛r✐â♥❝✐❛ ❞❡

Yj✱ s✉❥❡✐t❛ à r❡str✐çã♦

_p P

j=1

γ2

1_j = 1

✱ ♣❛r❛ i= 1, i= 2 ❝♦♠ γ1′γ2 = 0✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡ ❬✻❪✳ ❈♦♠ ♦s ❝♦♠♣♦♥❡♥t❡s ❞❡✜♥✐❞♦s é ♣♦ssí✈❡❧ ❝❛❧❝✉❧❛r ♦s ❡s❝♦r❡s✱ ✈❛❧♦r❡s ♥ú♠❡r✐❝♦s ❞♦s ❝♦♠✲ ♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s✱ y1, y2, . . . , y_p✱ ♣❛r❛ ❝❛❞❛ ✐♥❞✐✈í❞✉♦✳ ❊ss❡s ♣♦❞❡♠ s❡r ✉s❛❞♦s ❡♠ ❛♥á❧✐s❡s ♣♦st❡r✐♦r❡s s✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❞❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s✳ ❈❛s♦ s❡❥❛♠ ✉t✐❧✐③❛❞♦s k < p ❝♦♠♣♦✲

♥❡♥t❡s ♣r✐♥❝✐♣❛✐s ✭❞✐♠✐♥✉✐♥❞♦ ❛ ❞✐♠❡♥sã♦ ❞♦s ❞❛❞♦s✮ ❞❡✈❡ ❤❛✈❡r ❣❛r❛♥t✐❛ q✉❡ ❣r❛♥❞❡ ♣r♦♣♦rçã♦ ❞❛ ✈❛r✐❛çã♦ t♦t❛❧ é ❡①♣❧✐❝❛❞❛ ♣❡❧❛ ❡str✉t✉r❛ ❞♦sk ❢❛t♦r❡s r❡t✐❞♦s ❬✶✺❪✳

❖s ♣❡s♦sγi1, γ_i2, . . . , γ_ip✱i= 1,2, . . . , p✱ t❛♠❜é♠ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝❛r❣❛s ❢❛t♦r✐❛✐s ❡ ❝❛❞❛ γ_il ♠❡❞❡ ♦ ❣r❛✉ ❞❡ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛ ❧✲és✐♠❛ ✈❛r✐á✈❡❧ ♦r✐❣✐♥❛❧ ❡ ♦ ✐✲és✐♠♦ ❢❛t♦r✳ ❈♦♠♦ ❛ ♠❛tr✐③ ❞❡ ❝♦♠♣♦♥❡♥t❡s ♣r✐♥❝✐♣❛✐s é Y ₌ ΓX ♣♦❞❡✲s❡ ❡s❝r❡✈❡r ♦ ♠♦❞❡❧♦ ❞❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ❛ ♣❛rt✐r

❞♦s ❝♦♠♣♦♥❡♥t❡s ❬✶✷❪✱

Xi =φbi1Y1+φb_i2Y2+. . .+φb_ikY_k, ✭✷✳✶✺✮

❡♠ q✉❡ φbik sã♦ ♦s ❡❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ Φ=Γ−1✳

◆♦ ❝❛s♦ ❞❡ ♦ ♠♦❞❡❧♦ s♦❢r❡r r♦t❛çã♦✱ ❛s ❝❛r❣❛s ♦❜t✐❞❛s ❛♣ós ❛ r♦t❛çã♦ ❞❡✈❡♠ s❡r ❝♦♥s✐❞❡r❛❞❛s ♥❛ ♦❜t❡♥çã♦ ❞♦s ❡s❝♦r❡s ❡ ♥ã♦ ❛s ❝❛r❣❛s ♦r✐❣✐♥❛✐s ❬✻❪✳

❊①✐st❡ ✉♠ ♠ét♦❞♦ ♦❜t❡♥çã♦ ❞♦s ❢❛t♦r❡s ✉t✐❧✐③❛♥❞♦ r❡❣r❡ssã♦ ❞♦s ❝♦♠♣♦♥❡♥t❡s✱ ♣❡❧♦s ✈❛❧♦r❡s ❞❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s✱ ♣♦ré♠ ❡st❡ ♠ét♦❞♦ ❡①✐❣❡ ♥♦r♠❛❧✐❞❛❞❡ ♠✉❧t✐✈❛r✐❛❞❛ ❬✼❪✳ ◆❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ♣r✐♠❡✐r♦ ❡①tr❛❡♠✲s❡ ♦s ❢❛t♦r❡s ❡ s♦♠❡♥t❡ ✉♠❛ ♣❡q✉❡♥❛ q✉❛♥t✐❞❛❞❡ ❞❡ss❡s é ✉t✐❧✐③❛❞❛ ✭r❡t✐❞❛✮ ♣❛r❛ ❛s ❛♥❛❧✐s❡s ❬✶✶❪✳ ❈♦♠♦ ♦s ❞❛❞♦s ✉t✐❧✐③❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦ sã♦ ❛♣❡♥❛s ♦r❞✐♥❛✐s ❡ ❧✐♠✐t❛❞♦s✱ ♦ ✉s♦ ❞❛s ❞✐str✐❜✉✐çõ❡s ♠✉❧t✐✈❛r✐❛❞❛s é ✐♥✈✐á✈❡❧✳

P❛r❛ ❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ ❛ s♦♠❛ ❞❛ ✈❛r✐❛❜✐❧✐❞❛❞❡ t♦t❛❧ é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s✳ ❉❡ss❛ ❢♦r♠❛✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ♣❛r❛ ❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ sã♦ ✉♠ ♣♦✉❝♦ ❛❧t❡r❛❞❛s✳ ❆ss✐♠✱

tr(R_{) =tr(}ΓΛΓ′_{) =tr(}ΛΓΓ′_{) = tr(}ΛI_{) =tr(}Λ_{) =}

p

X

i=1

λi. ✭✷✳✶✻✮

❊♥tã♦✱

p

X

i=1

λi =

p

X

i=1

1 =p. ✭✷✳✶✼✮

❆♥á❧✐s❡ ❋❛t♦r✐❛❧ ✼

❈♦♠✱

V ar(Yi) = λi, ✭✷✳✶✽✮

♦♥❞❡ λi

p é ❛ ♣r♦♣♦rçã♦ ❞❛ ✈❛r✐â♥❝✐❛ ❡①♣❧✐❝❛❞❛ ♣❡❧♦ ❢❛t♦r✳ ❖s r❡s✉❧t❛❞♦s ♠✉❞❛♠✱ ♠❛s ♠❡❧❤♦r❛

❛ ✐♥t❡r♣r❡t❛çã♦✳

✷✳✷ ❊s❝♦❧❤❛ ❞♦ ♥ú♠❡r♦ ❞❡ ❢❛t♦r❡s

◆❛ ❧✐t❡r❛t✉r❛ ❡①✐st❡♠ ✈ár✐♦s ❝r✐tér✐♦s q✉❡ ❛✉①✐❧✐❛♠ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❢❛t♦r❡s✱ ♣♦❞❡♠♦s ❝✐t❛r ❝♦♠♦ ❡①❡♠♣❧♦s ♦s s❡❣✉✐♥t❡s ❝r✐tér✐♦s✿

• ❈r✐tér✐♦ ❞❡ ❑❛✐s❡r✿ ❝♦♥s✐❞❡r❛♠✲s❡ ❛♣❡♥❛s ❛q✉❡❧❡s ❝♦♠♣♦♥❡♥t❡s q✉❡ ♣♦ss✉❡♠ ❛✉t♦✈❛❧♦r❡s

q✉❡ sã♦ ♠❛✐♦r❡s q✉❡ ♦ ✈❛❧♦r ✶✱ ❞❡st❛ ❢♦r♠❛✱ q✉❛❧q✉❡r ❝♦♠♣♦♥❡♥t❡ ✐♥❞✐✈✐❞✉❛❧ ❞❡✈❡ ❡①♣❧✐❝❛r ❛ ✈❛r✐â♥❝✐❛ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ✈❛r✐á✈❡❧✳

• ❈r✐tér✐♦ ❞❛ ♣♦r❝❡♥t❛❣❡♠ ❞❛ ✈❛r✐â♥❝✐❛ ❡①♣❧✐❝❛❞❛✿ ❞❡t❡r♠✐♥❛ ♦ ♥ú♠❡r♦ ❞❡ ❢❛t♦r❡s q✉❡ ❡①✲

♣❧✐q✉❡ ✉♠❛ ♣♦r❝❡♥t❛❣❡♠ ♠í♥✐♠❛ ♣ré✲❞❡✜♥✐❞❛ ❞❛ ✈❛r✐❛❜✐❧✐❞❛❞❡ ❣❧♦❜❛❧✳ ❊st❛ ♣♦r❝❡♥t❛❣❡♠ ✈❛r✐❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦s ❛✉t♦r❡s ❡ ár❡❛ ❞♦s ♣r♦❜❧❡♠❛s✳

• ❙❝r❡❡ P❧♦t✿ ♠ét♦❞♦ ❣rá✜❝♦ ❜❛s❡❛❞♦ ♥♦s ❛✉t♦✈❛❧♦r❡s✳ ❖ ❣rá✜❝♦ é ❞❡t❡r♠✐♥❛❞♦ ❢❛③❡♥❞♦✲s❡ ♦

❣rá✜❝♦ ❞❛s ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s ❡♠ r❡❧❛çã♦ ❛♦ ♥ú♠❡r♦ ❞❡ ❢❛t♦r❡s ❡♠ s✉❛ ♦r❞❡♠ ❞❡ ❡①tr❛çã♦✱ ❡ ❛ ❢♦r♠❛ ❞❛ ❝✉r✈❛ r❡s✉❧t❛♥t❡ é ✉s❛❞❛ ♣❛r❛ ❛✈❛❧✐❛r ♦ ♣♦♥t♦ ❞❡ ❝♦rt❡✱ ❜✉s❝❛♥❞♦ ✉♠ ✏❝♦t♦✈❡❧♦✑ ♥♦ ❣rá✜❝♦✱ ♦♥❞❡ ❛ ❝✉r✈❛ ❝♦♠❡ç❛ ❛ ✜❝❛r ♥❛ ❤♦r✐③♦♥t❛❧ ❬✶✶❪✳

❊ss❡s ❝r✐tér✐♦s ♣♦❞❡♠ ❝♦♥❞✉③✐r ❛ r❡s✉❧t❛❞♦s ❞✐❢❡r❡♥t❡s ♠❡s♠♦ ❛♣❧✐❝❛❞♦s ♥♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✳ ◆ã♦ ❡①✐st❡ ✉♠ ❝r✐tér✐♦ ❝♦♥s✐❞❡r❛❞♦ ❛❜s♦❧✉t❛♠❡♥t❡ ♠❡❧❤♦r q✉❡ ✉♠ ♦✉tr♦✳ ❆ss✐♠✱ é ✐♥❞✐❝❛❞♦ ✉t✐❧✐③❛r ♠❛✐s ❞❡ ✉♠✱ ❡♥❝♦♥tr❛♥❞♦ ✉♠ ♥ú♠❡r♦ ❝♦♠✉♠ ❡♥tr❡ ♦s ❞✐❢❡r❡♥t❡s t✐♣♦s ❬✶✶❪✳

✷✳✸ ❆✈❛❧✐❛çã♦ ❞♦s ❋❛t♦r❡s

❈♦♠ ♦s ❢❛t♦r❡s ♦❜t✐❞♦s✱ é ♥❡❝❡ssár✐♦ ❛✈❛❧✐❛r ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❡①♣❧✐❝❛çã♦ ❞❛ ❡str✉t✉r❛ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ❢♦r♥❡❝✐❞❛ ♣❡❧❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧✳ ❆ss✐♠✱ ❛❧❣✉♠❛s ♠❡❞✐❞❛s ❞❡✈❡♠ s❡r ❛♥❛❧✐s❛❞❛s✳ ❊ss❛ ♠❡❞✐❞❛s sã♦✿ ❛s ❝❛r❣❛s ❢❛t♦r✐❛✐s✱ ❛s ❝♦♠✉♥❛❧✐❞❛❞❡s ❡ ♦s í♥❞✐❝❡s ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❬✶✶❪✳

❆s ❝❛r❣❛s ❢❛t♦r✐❛✐s r❡♣r❡s❡♥t❛♠ ❛s ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ❡ ♦s ❢❛t♦r❡s✳ ◗✉❛♥t♦ ♠❛✐♦r ❛ ❝❛r❣❛ ❢❛t♦r✐❛❧✱ ♠❛✐♦r s❡rá ❛ ❝♦rr❡❧❛çã♦ ❝♦♠ ❞❡t❡r♠✐♥❛❞♦ ❢❛t♦r✳ ❯♠ ✈❛❧♦r ♥❡❣❛t✐✈♦ ✐♥❞✐❝❛ ✐♠♣❛❝t♦ ✐♥✈❡rs♦ ♥♦ ❢❛t♦r✳ ❊♥tr❡t❛♥t♦✱ ❞✐✈❡rs❛s ♠❡❞✐❞❛s ❡ ✐♥str✉♠❡♥t♦s ✉t✐❧✐③❛✲ ❞♦s ♥❛ ♣s✐❝♦❧♦❣✐❛ ❛♣r❡s❡♥t❛♠ ♣❛❞rõ❡s ❞❡ ❝❛r❣❛s ❢❛t♦r✐❛✐s ❝r✉③❛❞❛s✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ ❛s ✈❛r✐á✈❡✐s s❡ ❝♦rr❡❧❛❝✐♦♥❛♠ ❢♦rt❡♠❡♥t❡ ❝♦♠ ♠❛✐s ❞❡ ✉♠ ❢❛t♦r✱ ❞✐✜❝✉❧t❛♥❞♦ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s r❡s✉❧t❛❞♦s ❬✶✶❪✳ ■ss♦ é ✉♠ ♣♦♥t♦ ♥❡❣❛t✐✈♦ ♥❛ ❛✈❛❧✐❛çã♦ ❞♦s r❡s✉❧t❛❞♦s✳

✽ ❆♥á❧✐s❡ ❋❛t♦r✐❛❧

❆s ❝♦♠✉♥❛❧✐❞❛❞❡s✱ hi✱ i= 1,2, . . . , p✱ ✐♥❞✐❝❛♠ ❛ ♣r♦♣♦rçã♦ ❞❛ ✈❛r✐â♥❝✐❛ ❞❛ ✈❛r✐á✈❡❧ ♦r✐❣✐♥❛❧

Xi q✉❡ ❡stá s❡♥❞♦ ❡①♣❧✐❝❛❞❛ ♣❡❧❛ ❡str✉t✉r❛ ❞♦sk ❢❛t♦r❡s s❡❧❡❝✐♦♥❛❞♦s ❬✶✶❪ ❡ é ❞❛❞❛ ♣♦r

hi =

k

X

j=1

b

γ_ij2. ✭✷✳✶✾✮

◗✉❛♥❞♦ ♦s ✈❛❧♦r❡s ❞❛s ❝♦♠✉♥❛❧✐❞❛❞❡s sã♦ ♠❡♥♦r❡s q✉❡ ✵✱✺ ❛ ❡①♣❧✐❝❛çã♦ ❢♦r♥❡❝✐❞❛ ♣❡❧♦s ❝♦♠♣♦♥❡♥t❡s s❡ t♦r♥❛ ✐♥s❛t✐s❢❛tór✐❛ ❬✶✶❪✳ ❚❛♥t♦ ♣❛r❛ ♦ ❝❛s♦ ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ q✉❛♥t♦ ♥❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛✱ ❛s ❝♦♠✉♥❛❧✐❞❛❞❡s sã♦ ✐♥t❡r♣r❡t❛❞❛s ❝♦♠♦ ♣r♦♣♦rçã♦ ❞❛ ✈❛r✐â♥❝✐❛ ❞❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ❡①♣❧✐❝❛❞❛ ♣❡❧♦s ❢❛t♦r❡s✳

❖ í♥❞✐❝❡ ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ ❍♦❢♠❛♥♥ ❬✶✸❪ r❡❢❡r❡✲s❡ ❛ ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ q✉❡ ✐♥❞✐❝❛✱ ❡♠ ♠é❞✐❛✱ q✉❛♥t♦s ❢❛t♦r❡s sã♦ ✉s❛❞♦s ♣❛r❛ ❡①♣❧✐❝❛r ❝❛❞❛ ✈❛r✐á✈❡❧ ❡♠ ✉♠❛ s♦❧✉çã♦ ❞❡ ❢❛t♦r❡s✹ _{❡ é}

❞❛❞♦ ♣♦r

ci =

k

P

j=1b

γ2

ij

!2

k

P

j=1b

γ4

ij

✭✷✳✷✵✮

◆ã♦ ❤á ❞✐r❡tr✐③❡s ❜❡♠ ❡st❛❜❡❧❡❝✐❞❛s ❝♦♠♦ ❛❞❡q✉❛❞❛s ♣❛r❛ ♦s ✈❛❧♦r❡s ❞♦ í♥❞✐❝❡ ❞❡ ❝♦♠♣❧❡①✐✲ ❞❛❞❡ q✉❛♥❞♦ ❛s ❝❛r❣❛s ❢❛t♦r✐❛✐s ❡stã♦ ❡♠ ❝♦♥❢♦r♠✐❞❛❞❡ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ s✐♠♣❧❡s ♦✉ ❝♦♠♣❧❡①❛ ❬✷✷❪✳

✷✳✹ ❘♦t❛çã♦ ❞♦s ❋❛t♦r❡s

P❛r❛ ♠❡❧❤♦r❛r ❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ❢❛t♦r❡s ❝♦♥s✐❞❡r❛❞♦s r❡❧❡✈❛♥t❡s✱ ❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s ✉t✐❧✐③❛✲s❡ r♦t❛çõ❡s ♥♦s ❝♦♠♣♦♥❡♥t❡s✳ ❊ss❡ ♣r♦❝❡❞✐♠❡♥t♦ ❛❥✉❞❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❡✈✐t❛r ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ❝❛r❣❛s ❢❛t♦r✐❛✐s ❝r✉③❛❞❛s ❡ ❛ ❜❛✐①❛ ❡①♣❧✐❝❛çã♦ ❞❡ ❛❧❣✉♠❛ ✈❛r✐á✈❡❧ ❞❛s ♦r✐❣✐♥❛✐s✳ ❊①✐st❡♠ ❞♦✐s t✐♣♦s ♣r✐♥❝✐♣❛✐s ❞❡ r♦t❛çã♦✿

• ♦rt♦❣♦♥❛❧ q✉❛♥❞♦ ♦s ♥♦✈♦s ❡✐①♦s t❛♠❜é♠ sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ✉♥s ❛♦s ♦✉tr♦s❀

• ♦❜❧íq✉♦s q✉❛♥❞♦ ♦s ♥♦✈♦s ❡✐①♦s ♣♦❞❡♠ t❡r â♥❣✉❧♦s ❞✐✈❡rs♦s ❡♥tr❡ s✐✳

❈♦♠♦ ❛s r♦t❛çõ❡s sã♦ s❡♠♣r❡ r❡❛❧✐③❛❞❛s ❡♠ ✉♠ s✉❜❡s♣❛ç♦ ✭❡s♣❛ç♦ ❞♦s ❢❛t♦r❡s✱ k < p✮✱ ❡❧❛s

❡①♣❧✐❝❛rã♦ ♠❡♥♦s ✈❛r✐â♥❝✐❛ ❞♦ q✉❡ ❛s ✈❛r✐á✈❡✐s ♦r✐❣✐♥❛✐s✳ ❆ ♣❛rt❡ ❞❛ ✈❛r✐â♥❝✐❛ t♦t❛❧ ❡①♣❧✐❝❛❞❛ ♣❡❧♦sk ❢❛t♦r❡s ❛♣ós ❛ r♦t❛çã♦ é ❛ ♠❡s♠❛ q✉❡ ❛♥t❡s ❞❛ r♦t❛çã♦✳ ❆♣❡♥❛s ❛s q✉❛♥t✐❞❛❞❡s ❡①♣❧✐❝❛✲

❞❛s ♣♦r ❝❛❞❛ ❢❛t♦r ♠✉❞❛♠ ❬✶✽❪✳ ❚❤✉rst♦♥❡ ❬✷✺❪ ❡ ❈❛tt❡❧ ❬✹❪ ❞❡❢❡♥❞❡r❛♠ q✉❡ ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ s✐♠♣❧✐✜❝❛ ❛ ❡str✉t✉r❛ ❞♦s ❢❛t♦r❡s ❡✱ ♣♦rt❛♥t♦✱ t♦r♥❛ s✉❛ ✐♥t❡r♣r❡t❛çã♦ ♠❛✐s ❢á❝✐❧ ❡ ❝♦♥✜á✈❡❧ ✭♠❛✐s ❢á❝✐❧ ❞❡ r❡♣❧✐❝❛r ❝♦♠ ❞✐❢❡r❡♥t❡s ❛♠♦str❛s ❞❡ ❞❛❞♦s✮✳

✹_{❯♠❛ ❡str✉t✉r❛ s✐♠♣❧❡s ♣❡r❢❡✐t❛ t❡♠ ❝♦♠♣❧❡①✐❞❛❞❡ ✐❣✉❛❧ ❛ ✶✱ ❝❛❞❛ ✈❛r✐á✈❡❧ ♦r✐❣✐♥❛❧ s❡r✐❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r}

❛♣❡♥❛s ✉♠ ❢❛t♦r✱ ✉♠❛ s♦❧✉çã♦ ❝♦♠ ✐t❡♥s ✉♥✐❢♦r♠❡♠❡♥t❡ ❞✐str✐❜✉í❞♦s t❡♠ ✉♠❛ ❝♦♠♣❧❡①✐❞❛❞❡ ♠❛✐♦r ❞♦ q✉❡ ✶ ❬✷✷❪✳

❆♥á❧✐s❡ ❋❛t♦r✐❛❧ ✾

❊♠ r♦t❛çõ❡s ♦❜❧íq✉❛s ♠❡s♠♦ s❡♠ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r✐❞❛❞❡ ❞♦s ❡✐①♦s ♦ ❣r❛✉ ❞❡ ❝♦rr❡❧❛çã♦ ♣❡r✲ ♠✐t✐❞♦ ❡♥tr❡ ♦s ❢❛t♦r❡s é✱ ❡♠ ❣❡r❛❧✱ ♣❡q✉❡♥♦ ♣♦rq✉❡ ❞♦✐s ❝♦♠♣♦♥❡♥t❡s ❛❧t❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞♦s sã♦ ♠❡❧❤♦r ✐♥t❡r♣r❡t❛❞♦s ❝♦♠♦ ✉♠ ú♥✐❝♦ ❢❛t♦r✳ ❯♠❛ r♦t❛çã♦ ♦rt♦❣♦♥❛❧ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ r♦t❛çõ❡s ♦❜❧íq✉❛s✳

❙❡❥❛ ❛ ❡q✉❛çã♦ ✉♠ ♠♦❞❡❧♦ ❜❛s❡❛❞♦ ❡♠ ❝♦♠♣♦♥❡♥t❡s ♦❜❧íq✉♦s G1, . . . , G_r ❬✺❪✿

X =GΛ′+ǫ ✭✷✳✷✶✮

♦♥❞❡✱ Λ é ✉♠❛ ♠❛tr✐③ ❞❡ ❝❛r❣❛s ❝♦♠ r♦t❛çã♦ ♦❜❧íq✉❛✱ ❡ǫ s✐❣♥✐✜❝❛ ❡rr♦ ❛❧❡❛tór✐♦✳

❆s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s s❡ ♠❛♥tê♠ s✐♠✐❧❛r ❛♦ ❝♦♠♣♦♥❡♥t❡s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛t♦r❡s ✭k < p✮✿

• X′X = ΛΘΛ′ _{♦♥❞❡ Θ =G}′_{G =T}′_{T é ❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ ❞♦s ❝♦♠♣♦♥❡♥t❡s ♦❜❧íq✉♦s✱}

❡ T é ❛ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦ ♦❜❧íq✉❛❀

• X′_{X =GΛ}′_ΛG′_❀

• Λ′ _{= (G}′_G)−1

G′_{X ❡ X = GΛ}′ _{= P}

GX ♦♥❞❡ PG é ❛ ♠❛tr✐③ ❞❡ ✈❡t♦r❡s ❧❛t❡♥t❡s✱ q✉❡ é

✐❞❡♠♣♦t❡♥t❡✺ _{❡ s✐♠étr✐❝❛✳}

■♥✐❝✐❛❧♠❡♥t❡✱ s✉r❣✐r❛♠ ❛s r♦t❛çõ❡s ♦❜❧íq✉❛s ❝❤❛♠❛❞❛s ✏q✉❛rt✐♠✐♥✑ ❡ ✏❝♦✈❛r✐♠✐♥✑ q✉❡ ❧♦❣♦ ❢♦r❛♠ ❣❡♥❡r❛❧✐③❛❞❛s ❬✸❪ ♣❛r❛ ❛ ❢❛♠í❧✐❛ ✏♦❜❧✐♠✐♥✑ q✉❡ ❞❡s❝r❡✈❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠ét♦❞♦s ❡♥✈♦❧✲ ✈❡♥❞♦ r♦t❛çõ❡s ♦❜❧íq✉❛s ❝♦♠ ❜❛s❡ ❡♠ ♠✐♥✐♠✐③❛çã♦ ❞♦s ❝r✐tér✐♦s ❡s♣❡❝í✜❝♦s✳ P❛r❛ ♦ ❛❧❣♦rít♠♦ é ♥❡❝❡ssár✐♦ ❞❡✜♥✐r ✉♠❛ ♠❛tr✐③ ✐♥✐❝✐❛❧ ❞❡ ❝❛r❣❛s ❢❛t♦r✐❛✐s Λp×k ❝♦♠ ❡❧❡♠❡♥t♦s γij✱i = 1,2, . . . , p

❡j = 1,2, . . . , k s❡♥❞♦ ❛ ❝❛r❣❛ ♣❛r❛ ❛ ✐✲és✐♠❛ ✈❛r✐á✈❡❧ ♥♦ ❥✲és✐♠♦ ❢❛t♦r✳ P❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠❛

r♦t❛çã♦ ❞❛ ❢❛♠í❧✐❛ ♦❜❧✐♠✐♥ ❞❡✈❡✲s❡ ♠✐♥✐♠✐③❛r ♦ ❝r✐tér✐♦

Oblimin=X

j6=_l

nX

i

γ_ij2λ2_il−τX

i

λ2_ijX

i

λ2_ijXλ2_il

!

. ✭✷✳✷✷✮

❱❛❧♦r❡s ❞❡ τ ♣❛r❛ ❝❛s♦s ❡s♣❡❝✐❛✐s ❞❛ ❢❛♠í❧✐❛ ♦❜❧✐♠✐♥ sã♦✿

◗✉❛rt✐♠✐♥ τ = 0 é ❛ r♦t❛çã♦ ♠❛✐s ♦❜❧íq✉❛❀

❇✐q✉❛rt✐♠✐♥ τ = 0,5é ❛ r♦t❛çã♦ ♠❡♥♦s ♦❜❧íq✉❛❀ ❡

❈♦✈❛r✐♠✐♥ τ = 1 é ❛ r♦t❛çã♦ ♦❜❧íq✉❛ ♠í♥✐♠❛✳

✺_{❯♠❛ ♠❛tr✐③ é ✐❞❡♠♣♦t❡♥t❡ q✉❛♥❞♦✱ ❛♦ s❡r ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r ♣♦tê♥❝✐❛s ❞❡ s✐ ♠❡s♠❛✱ r❡s✉❧t❛♠ ♥❛ ♣ró♣r✐❛}

♠❛tr✐③✳

✶✵ ❆♥á❧✐s❡ ❋❛t♦r✐❛❧

❘❡s✉❧t❛❞♦s ✶✶

✸✳ ❘❡s✉❧t❛❞♦s

❆s ❛♥á❧✐s❡s ❢♦r❛♠ r❡❛❧✐③❛❞❛s ❝♦♠ ❛✉①í❧✐♦ ❞♦ s♦❢t✇❛r❡ ❘ ✈❡rsã♦ ✸✳✹✳✷ ❬✷✸❪✳

❖s ❞❛❞♦s sã♦ r❡❢❡r❡♥t❡s ❛♦ ◗✉❡st✐♦♥ár✐♦ ❞❡ ■♠❛❣❡♠ ❈♦r♣♦r❛❧ ✭❆♣ê♥❞✐❝❡ ❆✮ r❡❧❛❝✐♦♥❛❞♦ ❛♦ s❡♥t✐♠❡♥t♦ ♥❛s ú❧t✐♠❛s q✉❛tr♦ s❡♠❛♥❛s ❞❡125♠✉❧❤❡r❡s ❞♦ Pr♦❣r❛♠❛ ❞❡ ❚r❛♥st♦r♥♦s ❆❧✐♠❡♥t❛✲

r❡s ✭❆▼❇❯▲■▼✮ ❞♦ ■♥st✐t✉t♦ ❞❡ Ps✐q✉✐❛tr✐❛ ❞♦ ❍♦s♣✐t❛❧ ❞❛s ❈❧í♥✐❝❛s ❞❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ▼❡❞✐❝✐♥❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ✭■Pq ✲ ❍❈✲❋▼❯❙P✮✱ ❞✉r❛♥t❡ ♦ ❡st✉❞♦ ✏❈❤❡❝❛❣❡♠ ❞♦ ❝♦r♣♦ ❡♠ tr❛♥st♦r♥♦s ❛❧✐♠❡♥t❛r❡s✿ r❡❧❛çã♦ ❡♥tr❡ ❝♦♠♣♦rt❛♠❡♥t♦s ❡ ❝♦❣♥✐çõ❡s✑ ❞❡ ❆❞r✐❛♥❛ ❚✳ ❑❛❝❤❛♥✐ ❬✶✻❪✳ ❆ r❡s♣♦st❛ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ♣❡r❣✉♥t❛s é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ♥í✈❡❧ ❞❡ s❛t✐s❢❛çã♦ ❞❛ ♣❛❝✐❡♥t❡ s❡❣✉♥❞♦ ✉♠❛ ❡s❝❛❧❛ ❞❡ ❧✐❦❡rt ✭t✐♣♦ ❞❡ ❡s❝❛❧❛ ♥❛ q✉❛❧ ❛s r❡s♣♦st❛s sã♦ ❝❛t❡❣♦r✐❛s ♦r❞❡✲ ♥❛❞❛s ❞❡ ✐♥t❡♥s✐❞❛❞❡✮✳ ◆❛ ❚❛❜❡❧❛ ✸✳✶✱ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛s ❢r❡q✉ê♥❝✐❛s r❡❧❛t✐✈❛s ❞❛s ❝❛t❡❣♦r✐❛s ♣♦r q✉❡stã♦✳

❚❛❜❡❧❛ ✸✳✶✿ ❋r❡q✉ê♥❝✐❛s r❡❧❛t✐✈❛s ❞❛s r❡s♣♦st❛s✳

◆✉♥❝❛ ❘❛r❛♠❡♥t❡ ➚s ✈❡③❡s ❋r❡q✉❡♥t❡♠❡♥t❡ ▼✉✐t♦ ❙❡♠♣r❡ ❋r❡q✉❡♥t❡♠❡♥t❡

◗✶ 0_,113 0_,129 0_,323 0_,089 0_,065 0_,282

◗✷ 0_,144 0_,120 0_,240 0_,048 0_,136 0_,312

◗✸ 0_,264 0_,168 0_,168 0_,048 0_,040 0_,312

◗✹ 0_,088 0_,040 0_,168 0_,112 0_,072 0_,520

◗✺ 0_,112 0_,080 0_,184 0_,128 0_,096 0_,400

◗✻ 0_,176 0_,136 0_,128 0_,072 0_,096 0_,392

◗✼ 0_,280 0_,072 0_,152 0_,088 0_,056 0_,352

◗✽ 0_,464 0_,128 0_,120 0_,072 0_,064 0_,152

◗✾ 0_,290 0_,081 0_,145 0_,081 0_,065 0_,339

◗✶✵ 0_,331 0_,089 0_,121 0_,073 0_,089 0_,298

◗✶✶ 0_,282 0_,105 0_,161 0_,073 0_,121 0_,258

◗✶✷ 0_,169 0_,121 0_,202 0_,073 0_,129 0_,307

◗✶✸ 0_,379 0_,113 0_,161 0_,057 0_,073 0_,218

◗✶✹ 0_,218 0_,089 0_,234 0_,057 0_,073 0_,331

◗✶✺ 0_,161 0_,121 0_,202 0_,121 0_,161 0_,234

◗✶✻ 0_,210 0_,129 0_,194 0_,097 0_,121 0_,250

◗✶✼ 0_,194 0_,105 0_,137 0_,073 0_,089 0_,403

◗✶✽ 0_,355 0_,089 0_,145 0_,081 0_,105 0_,226

◗✶✾ 0_,282 0_,161 0_,113 0_,113 0_,073 0_,258

◗✷✵ 0_,169 0_,081 0_,194 0_,065 0_,161 0_,331

◗✷✶ 0_,218 0_,186 0_,145 0_,097 0_,089 0_,266

◗✷✷ 0_,266 0_,097 0_,161 0_,097 0_,113 0_,266

◗✷✸ 0_,282 0_,057 0_,145 0_,065 0_,153 0_,298

◗✷✹ 0_,169 0_,081 0_,161 0_,105 0_,105 0_,379

◗✷✺ 0_,455 0_,106 0_,122 0_,008 0_,065 0_,244

◗✷✻ 0_,529 0_,057 0_,041 0_,041 0_,081 0_,252

◗✷✼ 0_,520 0_,064 0_,088 0_,072 0_,080 0_,176

◗✷✽ 0_,096 0_,080 0_,192 0_,104 0_,088 0_,440

◗✷✾ 0_,176 0_,072 0_,208 0_,080 0_,096 0_,368

◗✸✵ 0_,200 0_,152 0_,200 0_,064 0_,112 0_,272

◗✸✶ 0_,208 0_,152 0_,128 0_,064 0_,088 0_,360

◗✸✷ 0_,584 0_,088 0_,144 0_,024 0_,040 0_,120

◗✸✸ 0_,112 0_,096 0_,296 0_,128 0_,104 0_,264

◗✸✹ 0_,128 0_,056 0_,192 0_,144 0_,104 0_,376

◆❛ ❚❛❜❡❧❛ ✸✳✶ ♦❜s❡r✈❛✲s❡✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ ♣❛r❛ ❛ q✉❡stã♦ ✷✺ q✉❡ r❡❢❡r❡✲s❡ ❛ ✏❱♦❝ê ❛❝❤❛

✶✷ ❘❡s✉❧t❛❞♦s

❥✉st♦ q✉❡ ♦✉tr❛s ♣❡ss♦❛s s❡❥❛♠ ♠❛✐s ♠❛❣r❛s q✉❡ ✈♦❝ê❄✑✱ q✉❡ ✹✺✪ ❞❛s ♣❛❝✐❡♥t❡s ♥✉♥❝❛ s❡ s❡♥t❡♠ ✐♥❝♦♠♦❞❛❞❛s ❝♦♠ ❡ss❛ s✐t✉❛çã♦✳ ❏á ✺✷✪ r❡s♣♦♥❞❡r❛♠ ❝♦♠ ♦ ♠❛✐♦r ❣r❛✉ ❞❛ ❡s❝❛❧❛ ❛ ♣❡r❣✉♥t❛ q✉❡ s❡ r❡❢❡r❡ ❛ ✏❱♦❝ê t❡♠ s❡♥t✐❞♦ ♠❡❞♦ ❞❡ ✜❝❛r ❣♦r❞❛ ✲ ♦✉ ♠❛✐s ❣♦r❞❛ ❞♦ q✉❡ ❡stá❄✑✱ q✉❡ ❡stá r❡❧❛❝✐♦♥❛❞♦ à q✉❡stã♦ ✹✳ ❖s ❝♦♠♣♦♥❡♥t❡s ❢♦r❛♠ ❞❡t❡r♠✐♥❛❞♦s ❛tr❛✈és ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦✱ ♣❡❧❛ ♠❛✐♦r ❢❛❝✐❧✐❞❛❞❡ ♥❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦s r❡s✉❧t❛❞♦s✳

❆ ♣❛rt✐r ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ ❞❡ ❙♣❡❛r♠❛♥✱ ♦❜s❡r✈♦✉✲s❡ q✉❡ ❛s ✈❛r✐á✈❡✐s sã♦ t♦❞❛s ♣♦s✐✲ t✐✈❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞❛s✱ ♦ ♠ét♦❞♦ ❣rá✜❝♦ ❛❥✉❞❛ ❛ ♦❜s❡r✈❛r ❛ ♣r❡s❡♥ç❛ ❞❡ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛ ♠❛✐♦r✐❛ ❞❛s ✈❛r✐á✈❡✐s✱ ❛ ✜❣✉r❛ ❡stá ❞✐s♣♦♥í✈❡❧ ♥♦ ❆♣ê♥❞✐❝❡❇✳

▼❡❞✐❞❛s ❞❡ ❛❞❡q✉❛çã♦ ❛♠♦str❛❧ sã♦ ❞❛❞❛s ♣❡❧❛s ❡st❛tíst✐❝❛s ❞❡ ❑▼❖ ❡ ▼❆❆ ❝♦♠ ❜❛s❡ ♥❛s ❝♦rr❡❧❛çõ❡s ♣❛r❝✐❛✐s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s✳ ❆ ❡st❛tíst✐❝❛ ❑▼❖ ❞❡ ✵✱✾✻ ✐♥❞✐❝❛ q✉❡✱ s❡❣✉♥❞♦ ❇❛rr♦s♦ ❬✷❪✱ ♦s ❞❛❞♦s ❛♣r❡s❡♥t❛♠ ❛❞❡q✉❛çã♦ ót✐♠❛ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧✳ P❡❧♦s ❛❧t♦s ✈❛❧♦r❡s ❞❛s ▼❆❆✬s ❞❡ ❝❛❞❛ ✈❛r✐á✈❡❧ ✭❚❛❜❡❧❛ ✸✳✷✮ ❝♦♥❝❧✉✐✲s❡ q✉❡ é ❛❝❡✐tá✈❡❧ ✉t✐❧✐③❛r ❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ❬✶✶❪✳

❚❛❜❡❧❛ ✸✳✷✿ ▼❡❞✐❞❛s ❞❡ ❛❞❡q✉❛çã♦ ❛♠♦str❛❧

❱❛r✐á✈❡✐s ▼❆❆ ❱❛r✐á✈❡✐s ▼❆❆

◗✶ ✵✱✾✹ ◗✶✽ ✵✱✾✼

◗✷ ✵✱✾✹ ◗✶✾ ✵✱✾✻

◗✸ ✵✱✾✺ ◗✷✵ ✵✱✾✽

◗✹ ✵✱✾✽ ◗✷✶ ✵✱✾✺

◗✺ ✵✱✾✹ ◗✷✷ ✵✱✾✻

◗✻ ✵✱✾✼ ◗✷✸ ✵✱✾✻

◗✼ ✵✱✾✺ ◗✷✹ ✵✱✾✻

◗✽ ✵✱✾✹ ◗✷✺ ✵✱✾✷

◗✾ ✵✱✾✼ ◗✷✻ ✵✱✾✻

◗✶✵ ✵✱✾✼ ◗✷✼ ✵✱✾✹ ◗✶✶ ✵✱✾✼ ◗✷✽ ✵✱✾✺ ◗✶✷ ✵✱✾✼ ◗✷✾ ✵✱✾✻ ◗✶✸ ✵✱✾✹ ◗✸✵ ✵✱✾✺ ◗✶✹ ✵✱✾✺ ◗✸✶ ✵✱✾✺ ◗✶✺ ✵✱✾✼ ◗✸✷ ✵✱✾✺ ◗✶✻ ✵✱✾✼ ◗✸✸ ✵✱✾✹ ◗✶✼ ✵✱✾✻ ◗✸✹ ✵✱✾✹

❆ ❚❛❜❡❧❛✸✳✸♠♦str❛ ♦s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦✱ r❡♣r❡s❡♥t❛♥❞♦ ❛s ✈❛r✐â♥❝✐❛s ❞♦s ❝♦♠♣♦♥❡♥t❡s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s ♣r♦♣♦rçõ❡s ❞❛ ✈❛r✐â♥❝✐❛ t♦t❛❧ ❡①♣❧✐❝❛❞❛ ♣♦r ❝❛❞❛ ✉♠ ❞❡❧❡s✳ ➱ ♣♦ssí✈❡❧ ♦❜s❡r✈❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❡❧❛ ✈❛r✐â♥❝✐❛ ❞♦ ❝♦♠♣♦♥❡♥t❡ 1✱ ❞❡ 21,15✱ q✉❡ 21 q✉❡stõ❡s

❞❡♥tr❡ ❛s 34❞♦ q✉❡st✐♦♥ár✐♦ ❡stã♦ s❡♥❞♦ ❡①♣❧✐❝❛❞❛s ♣❡❧♦ ♣r✐♠❡✐r♦ ❝♦♠♣♦♥❡♥t❡✳

❘❡s✉❧t❛❞♦s ✶✸

❚❛❜❡❧❛ ✸✳✸✿ ❆✉t♦✈❛❧♦r❡s ❡ ✈❛r✐â♥❝✐❛ ❡①♣❧✐❝❛❞❛

❈♦♠♣♦♥❡♥t❡s ❆✉t♦✈❛❧♦r Pr♦♣♦rçã♦ ❆❝✉♠✉❧❛❞❛

1 21_,15 0_,622 0_,622

2 1_,14 0_,033 0_,655

3 1_,12 0_,033 0_,688

4 1_,04 0_,031 0_,719

5 0_,93 0_,027 0_,746

6 0_,84 0_,025 0_,771

7 0_,79 0_,023 0_,794

8 0_,71 0_,021 0_,815

9 0_,60 0_,018 0_,833

10 0_,54 0_,016 0_,849

11 0_,53 0_,016 0_,864

12 0_,46 0_,013 0_,878

13 0_,39 0_,012 0_,889

14 0_,39 0_,012 0_,901

15 0_,37 0_,011 0_,912

16 0_,32 0_,009 0_,921

17 0_,30 0_,009 0_,930

18 0_,28 0_,008 0_,938

19 0_,25 0_,007 0_,945

20 0_,20 0_,006 0_,951

21 0_,19 0_,006 0_,957

22 0_,17 0_,005 0_,962

23 0_,17 0_,005 0_,967

24 0_,16 0_,005 0_,971

25 0_,15 0_,005 0_,976

26 0_,13 0_,004 0_,980

27 0_,13 0_,004 0_,983

28 0_,12 0_,004 0_,987

29 0_,10 0_,003 0_,990

30 0_,09 0_,003 0_,992

31 0_,08 0_,002 0_,995

32 0_,07 0_,002 0_,997

33 0_,06 0_,002 0_,999

34 0_,05 0_,001 1_,00

❆ ❡s❝♦❧❤❛ ❞♦ ♥ú♠❡r♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s✱ r❡t✐❞♦s ♣❡❧♦ ❙❝r❡❡ P❧♦t ✭❋✐❣✉r❛ ✸✳✶✮✱ ✈❡r✐✜❝❛ ❛ ❞✐s✲ ♣❡rsã♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ❡♠ r❡❧❛çã♦ ❛♦ ♥ú♠❡r♦ ❞❡ ❢❛t♦r❡s ❛té q✉❡ ❛ ❝✉r✈❛ ❞❛ ✈❛r✐â♥❝✐❛ s❡ t♦r♥❡ ❤♦r✐③♦♥t❛❧ ♦✉ s♦❢r❛ ✉♠❛ q✉❡❞❛ ❛❜r✉♣t❛✳ ■ss♦ ✐♥❞✐❝❛ q✉❡ ♠✉✐t❛ ✈❛r✐â♥❝✐❛ ❢♦✐ ♣❡r❞✐❞❛ ❡✱ ♣♦r ✐ss♦✱ ❞❡✈❡✲s❡ ♣❛r❛r ❞❡ ❡①tr❛✐r ❢❛t♦r❡s✳ P❡❧❛ q✉❛♥t✐❞❛❞❡ ♠í♥✐♠❛ ❞❡ ✈❛r✐â♥❝✐❛ ❡①♣❧✐❝❛❞❛✱ ♠❡s♠♦ ❝♦♠ ❛✉t♦r❡s s✉❣❡r✐♥❞♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❛❝❡✐tá✈❡❧✱ ✜❝❛ ❛ ❝r✐tér✐♦ ❞♦ ♣❡sq✉✐s❛❞♦r ❞❡✜♥✐r ❛ q✉❛♥t✐❞❛❞❡ q✉❡ s❡rá ✉t✐❧✐③❛❞❛✳

0 5 10 15 20 25 30 35

0 5 10 15 20 Screeplot component number Eigen v

alues of components

❋✐❣✉r❛ ✸✳✶✿ ❙❝r❡❡ P❧♦t

◆❡ss❡ tr❛❜❛❧❤♦✱ ❛ ❡s❝♦❧❤❛ ❞♦ ♥ú♠❡r♦ ❞❡ ❢❛t♦r❡s r❡t✐❞♦s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❢❛t♦r✐❛❧ ❢♦✐ r❡❛❧✐③❛❞❛ ❝♦♠ ❜❛s❡ ♥♦ ❈r✐tér✐♦ ❞❡ ❑❛✐s❡r✱ ♦ q✉❛❧ ❞✐③ q✉❡ ❛✉t♦✈❛❧♦r❡s ♠❡♥♦r❡s q✉❡ 1 ✐♥❞✐❝❛♠ q✉❡ ♦