❈♦♠♣❛r❛çã♦ ❡ ✈❛❧✐❞❛çã♦ ❞❡ ✈ár✐♦s ♠ét♦❞♦s ❞❡ ♠❛♣❡❛♠❡♥t♦
❝♦rt✐❝❛❧ ♣❛r❛ ♦ ♣❧❛♥❡❛♠❡♥t♦ ♣ré✲❝✐rúr❣✐❝♦ ♣♦r ■♠❛❣❡♠
❋✉♥❝✐♦♥❛❧ ❞❡ ❘❡ss♦♥â♥❝✐❛ ▼❛❣♥ét✐❝❛
❏♦sé ●❛❜r✐❡❧ ▲✐r❛ ●♦♠❡s
❈♦♥t❡ú❞♦
✶ ■♥tr♦❞✉çã♦ ✼
✷ ❖s ❞❛❞♦s ✾
✷✳✶ ❆ ♦r❣❛♥✐③❛çã♦ ❞♦s ❞❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷ ❖ ❞❡s✐❣♥ ❞♦ ❡st✉❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✸ ❖ ❈ér❡❜r♦ ❡ ♦ ♣ré✲♣r♦❝❡ss❛♠❡♥t♦ ❞♦s ❞❛❞♦s ✶✶
✸✳✶ ❖ ❝❛rr❡❣❛♠❡♥t♦ ❞♦s ❞❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✸✳✷ ❆ ♠ás❝❛r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✸✳✷✳✶ ❖ ♠ét♦❞♦ ❞❡ ▼❛❤❛❧❛♥♦❜✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✷✳✷ ❖ ♠ét♦❞♦ ❞❡ ❝❧✉st❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✸ ❖ ❞❡s✐❣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✹ ❖s ♦✉t❧✐❡rs ♦✉ ♣♦♥t♦s ❡s♣úr✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✺ ❆ ✜❧tr❛❣❡♠ ❞♦s ❞❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✻ ❆ ✈✐s✉❛❧✐③❛çã♦ ❞♦s ❞❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✹ ❖ ❈ér❡❜r♦ ❡ ❛ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ✷✶
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✷ ❖ ♠ét♦❞♦ t ❞❡ ❙t✉❞❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✷✳✶ ❚❡♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✷✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✸ ❖ ♠ét♦❞♦ ❞❡ ❝♦rr❡❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✸✳✶ ❚❡♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✸✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✹ ❖ ♠ét♦❞♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✹✳✶ ❚❡♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✹✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✺ ❖ ♠ét♦❞♦ ❞❡ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✺✳✶ ❚❡♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✺✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✻ ❖ ♠ét♦❞♦ ❞❡ ❛♠♣❧✐t✉❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✻✳✶ ❚❡♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✻✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✼ ❖ ♠ét♦❞♦ ❞❡ s♦❜r❡♣♦s✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✼✳✶ ❚❡♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✼✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✽ ❆♥á❧✐s❡ ❝♦♠♣❛r❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✽✳✶ ❊①tr❡♠♦s ❡ ✈♦❧✉♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✽✳✷ ❙❡q✉ê♥❝✐❛s ❡ ❡s♣❡❝tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✽✳✸ P❡r✜s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✽✳✸✳✶ ❋✐❧tr❛❞♦s s❡♠ ❍❘❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✽✳✸✳✷ ◆ã♦ ✜❧tr❛❞♦s✱ s❡♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✹ ❈❖◆❚❊Ú❉❖
✹✳✽✳✸✳✸ ◆ã♦ ✜❧tr❛❞♦s✱ s❡♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✽✳✸✳✹ ◆ã♦ ✜❧tr❛❞♦s✱ s❡♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✽✳✸✳✺ ◆ã♦ ✜❧tr❛❞♦s✱ s❡♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✽✳✸✳✻ ❋✐❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✹✳✽✳✸✳✼ ◆ã♦ ✜❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✹✳✽✳✸✳✽ ◆ã♦ ✜❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✹✳✽✳✸✳✾ ◆ã♦ ✜❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✹✳✽✳✸✳✶✵ ◆ã♦ ✜❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✹✳✽✳✸✳✶✶ ❋✐❧tr❛❞♦s ❡ ❝♦♠ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧ ❝♦♠ ✭✷✼✱✻✱✶✼✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ✹✳✽✳✸✳✶✷ ◆ã♦ ✜❧tr❛❞♦s ❡ ❝♦♠ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧ ❝♦♠ ✭✷✼✱✻✱✶✼✮ ✳ ✳ ✳ ✳ ✳ ✶✶✸ ✹✳✽✳✸✳✶✸ ❈♦♥❝❧✉sõ❡s s♦❜r❡ ♦s ♣❡r✜s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✹✳✽✳✹ ■♠❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✹✳✽✳✹✳✶ ❋✐❧tr❛❞♦s s❡♠ ❍❘❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✹✳✽✳✹✳✷ ◆ã♦ ✜❧tr❛❞♦s✱ s❡♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹ ✹✳✽✳✹✳✸ ◆ã♦ ✜❧tr❛❞♦s✱ s❡♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✾ ✹✳✽✳✹✳✹ ◆ã♦ ✜❧tr❛❞♦s✱ s❡♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✹ ✹✳✽✳✹✳✺ ◆ã♦ ✜❧tr❛❞♦s✱ s❡♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✾ ✹✳✽✳✹✳✻ ❋✐❧tr❛❞♦s ❡ ❝♦♠ ❍❘❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✹ ✹✳✽✳✹✳✼ ◆ã♦ ✜❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✽ ✹✳✽✳✹✳✽ ◆ã♦ ✜❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷ ✹✳✽✳✹✳✾ ◆ã♦ ✜❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻ ✹✳✽✳✹✳✶✵ ◆ã♦ ✜❧tr❛❞♦s✱ ❝♦♠ ❍❘❋ ❡ ❝♦♠ ❛tr❛s♦ ✐❣✉❛❧ ❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✵ ✹✳✽✳✹✳✶✶ ❋✐❧tr❛❞♦s ❡ ❝♦♠ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧ ❝♦♠ ✭✷✼✱✻✱✶✼✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✹ ✹✳✽✳✹✳✶✷ ◆ã♦ ✜❧tr❛❞♦s ❡ ❝♦♠ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧ ❝♦♠ ✭✷✼✱✻✱✶✼✮ ✳ ✳ ✳ ✳ ✳ ✶✻✽ ✹✳✽✳✹✳✶✸ ❈♦♥❝❧✉sõ❡s ❞❛s ✐♠❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽
✺ ❖ ❈ér❡❜r♦ ❡ ❛ s❡❣♠❡♥t❛çã♦ ✶✼✸
✺✳✶ ❚❡♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✸ ✺✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✸
✻ ❈♦♥❥✉❣❛çã♦ ❞❡ r❡s✉❧t❛❞♦s ❞❡ s❡❣♠❡♥t❛çã♦ ❝♦♠ ❢✉♥❝✐♦♥❛✐s ✶✼✺ ✻✳✶ ❚❡♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✺ ✻✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✼
✼ ❯♠ ❝❛s♦ ❝❧í♥✐❝♦ ✶✼✾
✼✳✶ ❖ ❡st✉❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✾ ✼✳✷ ❆♥á❧✐s❡ ❝♦♠♣❛r❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✾ ✼✳✷✳✶ ❊①tr❡♠♦s ❡ ✈♦❧✉♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✾ ✼✳✷✳✷ ❙❡q✉ê♥❝✐❛s ❡ ❡s♣❡❝tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✷ ✼✳✷✳✸ P❡r✜s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✺ ✼✳✷✳✹ ■♠❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✺ ✼✳✷✳✹✳✶ ❉❛❞♦s s❡♠ ❍❘❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✺ ✼✳✷✳✹✳✷ ❉❛❞♦s ❝♦♠ ❍❘❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✽
✽ ❈♦♥❝❧✉sõ❡s ❣❡r❛✐s ❡ tr❛❜❛❧❤♦ ❢✉t✉r♦ ✷✵✶
❇✐❜❧✐♦❣r❛✜❛ ✷✵✸
❆ ❖ ❢♦r♠❛t♦ ▼❛②♦ ❈❧✐♥✐❝ ❆♥❛❧②③❡ ✷✵✺
❇ ❆ ❢✉♥çã♦ ❇❡t❛ ✷✵✼
❈ ❈á❧❝✉❧♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❝✉♠✉❧❛t✐✈❛ ❞❡ ❙t✉❞❡♥t ✷✵✾
❈❖◆❚❊Ú❉❖ ✺
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
◆❛ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ❞❡ ✐♠❛❣❡♥s ❞♦ ❝ér❡❜r♦ ♣♦❞❡♠ ✉t✐❧✐③❛r✲s❡ ❞✐❢❡r❡♥t❡s ♠ét♦❞♦s ♥❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ③♦♥❛s ❞❡ ❛❝t✐✈❛çã♦✳ ❚❡♠ ❤❛✈✐❞♦ ✉♠❛ ❡✈♦❧✉çã♦ ❞❡s❞❡ ♦ ♠ét♦❞♦ ❞❡ ❝♦rr❡❧❛çã♦ ❬✶✾❪✱ ♣❛r❛ ♦✉tr♦s ♠ét♦❞♦s ❬✾❪ ❬✶✹❪ ❛té ♦ ♠ét♦❞♦ ❜❛s❡❛❞♦ ♥♦ ♠♦❞❡❧♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦ q✉❡ é ♠❛✐s ❝♦♠✉♠ s❡r ✉t✐❧✐③❛❞♦ ❤♦❥❡ ❡ q✉❡ ❧❡✈♦✉ ❛♦ ♣❛❝♦t❡ ❞❡ s♦❢t✇❛r❡ ❙P▼ ❬✶✺❪✳ ❉❡✈❡✲s❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ à ✈❡rs❛t✐❧✐❞❛❞❡ q✉❡ ♦ ♠ét♦❞♦ t❡♠ ❡♠ r❡❛❧✐③❛r t❡st❡s ❝♦♠ ❞✐❢❡r❡♥t❡s ♦❜❥❡❝t✐✈♦s✳
❚ê♠ s✐❞♦ ♣✉❜❧✐❝❛❞♦s ❛❧❣✉♥s ❡st✉❞♦s ❝♦♠♣❛r❛t✐✈♦s✳ P♦✉❝♦s tê♠ s✐❞♦ q✉❛♥t✐t❛t✐✈♦s ❬✷✵❪ ❡ q✉❛♥❞♦ ♦ sã♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♠ét♦❞♦s t❡st❛❞♦s é r❡❞✉③✐❞♦❬✷✷❪✳ ❍á ♠✉✐t♦s ❡st✉❞♦s ❝♦♠♣❛r❛t✐✈♦s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❡st❛tíst✐❝❛ ❡♥✈♦❧✈✐❞❛ ✭❞❛ ♠❛t❡♠át✐❝❛✮ ♠❛s q✉❡ tê♠ ❡♠ ❣❡r❛❧ ❛♣❡♥❛s ✜♥s ❛❝❛❞é♠✐❝♦s✳ ❯♠ ♦❜❥❡❝t✐✈♦ ❞❡st❡ ❡st✉❞♦ é ❝♦♠♣❛r❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ❞✐❢❡r❡♥t❡s ♠ét♦❞♦s✳ ➱ ❞❡ ♣❛rt✐✲ ❝✉❧❛r ✐♥t❡r❡ss❡ ❛✈❡r✐❣✉❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ❝❛❞❛ ♠ét♦❞♦ ♥❛ ❢r♦♥t❡✐r❛ ❞♦ ❧♦❝❛❧ ❞❡ ❛❝t✐✈❛çã♦✳ ❆s ❞✐❢❡r❡♥ç❛s s❡rã♦ ❛✈❛❧✐❛❞❛s ♥✉♠❡r✐❝❛♠❡♥t❡ ♣❛r❛ ♦s s❡❣✉✐♥t❡s ♠ét♦❞♦s ❝❧áss✐❝♦s✿ t ❞❡ ❙t✉❞❡♥t✱ ❝♦❡✜✲ ❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❡ ♦ ♠♦❞❡❧♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦✳ ❚rês ♥♦✈♦s ♠ét♦❞♦s sã♦ t❛♠❜é♠ ♣r♦♣♦st♦s ✲ ♦ ♠ét♦❞♦ ❞❡ ♣✐❝♦s ❞❡ ❋♦✉r✐❡r✱ ♦ ♠ét♦❞♦ ❞❡ s♦❜r❡♣♦s✐çã♦ ❡ ♦ ♠ét♦❞♦ ❞❡ ❛♠♣❧✐t✉❞❡✳ ❖ s❡❣✉♥❞♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ♣❛r❛ ♦ ♠❡❧❤♦r❛♠❡♥t♦ ❞♦s ♠ét♦❞♦s ❞❡ t ❞❡ ❙t✉❞❡♥t✱ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❡ ♠♦❞❡❧♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦✳ ❊❧❡ ♣♦❞❡ ♥♦ ❡♥t❛♥t♦✱ t❛♠❜é♠ ♠❛♥t❡r✲s❡ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❞❡ ❛♥á❧✐s❡ ✐♥❞❡✲ ♣❡♥❞❡♥t❡✳ ❆ ✐♥✢✉ê♥❝✐❛ ❡①❡r❝✐❞❛ ❡♠ ❝❛❞❛ ♠ét♦❞♦ ♣❡❧♦s ♣❛râ♠❡tr♦s ♣❡rt✐♥❡♥t❡s é t❛♠❜é♠ ♠❡❞✐❞❛✳ ➱ ❛❞♦♣t❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❝❧í♥✐❝♦s q✉❡ ❡stá ❛♠♣❧❛♠❡♥t❡ ❡st✉❞❛❞♦ ❡ ❞♦❝✉♠❡♥t❛❞♦✳ ❉❡st❛ ❢♦r♠❛ ❡❧✐♠✐♥❛✲s❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s s❡r❡♠ ✐♥t❡r♣r❡t❛❞♦s ❝♦♠♦ s❡♥❞♦ ❡s♣❡❝í✜❝♦s ❞♦ ❝❛s♦ ❡♠ ❡st✉❞♦✳
❍á s✐t✉❛çõ❡s ❡♠ q✉❡ ❛ ✐♥✢✉ê♥❝✐❛ ❞♦ ♠ét♦❞♦ ✉t✐❧✐③❛❞♦ ♥❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛s ár❡❛s ❞❡ ❛❝t✐✈❛çã♦ ❞❡ ✐♠❛❣❡♥s ❢✉♥❝✐♦♥❛✐s ❞♦ ❝ér❡❜r♦ é ❝r✉❝✐❛❧✳ ❚❛❧ ❛❝♦♥t❡❝❡✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ ✉♠ t✉♠♦r ❞❡s❡♥✈♦❧✈❡✲ s❡ ♣❡rt♦ ❞❡ ✉♠❛ ③♦♥❛ ❞❡ ❛❝t✐✈❛çã♦ r❡s♣♦♥sá✈❡❧ ♣♦r ✉♠❛ ❢✉♥çã♦ ✐♠♣♦rt❛♥t❡ ✳ P❛r❛ ♦ ❝✐r✉r❣✐ã♦ t♦r♥❛✲ s❡ ✐♥❞✐s♣❡♥sá✈❡❧ ❛✈❛❧✐❛r s❡ ❡①✐st❡ ❛❧❣✉♠❛ s♦❜r❡♣♦s✐çã♦✳ ❆ ❡s❝♦❧❤❛ ❞❡ ✉♠ ❞♦s ♠ét♦❞♦s ❞✐s♣♦♥í✈❡✐s ♣♦❞❡rá t❡r ✐♥✢✉ê♥❝✐❛ s♦❜r❡ ❛ ❞❡❝✐sã♦ ✜♥❛❧✳ ❙❡ ♦ ♠ét♦❞♦ ❡s❝♦❧❤✐❞♦ ❢♦r ♠❛✐s ❝♦♥s❡r✈❛❞♦r✱ ♣♦❞❡ ✈❡r✐✜❝❛r✲s❡ s♦❜r❡♣♦s✐çã♦ ❡ ❡❧✐♠✐♥❛r✲s❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❝✐r✉r❣✐❛✳ P♦ré♠✱ s❡ ♦ ♠ét♦❞♦ ❢♦r ♠❛✐s r❡str✐t✐✈♦ ❛ ❞❡❝✐sã♦ ✜♥❛❧ ♣♦❞❡ s❡r ❢❛✈♦rá✈❡❧ à ❝✐r✉r❣✐❛✳ ❆rt✐❣♦s r❡❝❡♥t❡s tê♠ s✉♣♦rt❛❞♦ ❛ ✐❞❡✐❛ ❞❡ q✉❡ ❛ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛ ❢✉♥❝✐♦♥❛❧ é ❞❡ ❢❛❝t♦ ♠✉✐t♦ út✐❧ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❞❡❝✐sã♦ ♣ré✲♦♣❡r❛tór✐♦ ❬✶✷❪✳
❖ s❡❣✉♥❞♦ ♦❜❥❡❝t✐✈♦ ❞♦ ❡st✉❞♦ é ❡♥tã♦ ❛✈❛❧✐❛r ❛ s♦❜r❡♣♦s✐çã♦ ❡♥tr❡ ✉♠ ✈♦❧✉♠❡ ❞❡ ❛❝t✐✈❛çã♦ ❡ ♦ ✈♦❧✉♠❡ ❞♦ t✉♠♦r✳
❖s ♣r♦❣r❛♠❛s ✐♥❢♦r♠át✐❝♦s ❞❡ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ❞✐s♣♦♥í✈❡✐s sã♦ ✈❛r✐❛❞♦s ❡♠ ✈ár✐♦s ❛s♣❡❝t♦s✿ ♥❛ ♣❧❛t❛❢♦r♠❛ ❡♠ q✉❡ ❢✉♥❝✐♦♥❛♠ ✭♠❛❝✐♥t♦s❤✱ ❧✐♥✉①✱ ✇✐♥❞♦✇s ♦✉ ♦✉tr❛s✮✱ ♥❛ ❧✐♥❣✉❛❣❡♠ ❡♠ q✉❡ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ✭❡✳❣✳ ❝✰♠♦t✐❢✱ ❝✰♠❛t❧❛❜✱ ♠❛t❧❛❜✱ ❡t❝✳✮ ♥♦ tr❛t❛♠❡♥t♦ ✐♥✐❝✐❛❧ ❞♦s ❞❛❞♦s ✭❛♥t❡s ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ❛♥á❧✐s❡✮✱ ♥♦ ❢♦r♠❛t♦ ❞❛s ✐♠❛❣❡♥s ❡ ♥♦✭s✮ ♠ét♦❞♦✭s✮ ❞❡ ❛♥á❧✐s❡ ❡s❝♦❧❤✐❞♦✭s✮✳ ❊st❡ ❢❛❝t♦ ❞✐✜❝✉❧t❛ q✉❛❧q✉❡r t❡♥t❛t✐✈❛ ❞❡ ❝♦♠♣❛r❛çã♦✳ ➚ ♣❛rt✐❞❛ ❡st❛ ♣♦❞❡rá ❛♣❡♥❛s s❡r q✉❛❧✐t❛t✐✈❛✳ ❯♠❛ ❝♦♠♣❛r❛çã♦ q✉❛♥t✐t❛t✐✈❛ ✐♠♣❧✐❝❛r✐❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♦❝♦rr❡r❡♠ três ❢❛❝t♦s✿ ♦ ✉t✐❧✐③❛❞♦r t❡♠ ❛❝❡ss♦ ❛♦ ❝ó❞✐❣♦ ❞♦ ♣r♦❣r❛♠❛✱ s❛❜❡ ♣r♦❣r❛♠❛r ♥❛s ❞✐❢❡r❡♥t❡s ❧✐♥❣✉❛❣❡♥s ❡ t❡♠ ❧✐❝❡♥ç❛ ❞❡ ✉t✐❧✐③❛çã♦ ❞❡ s♦❢t✇❛r❡ ❝♦♠❡r❝✐❛❧ ✭❡✳❣✳ ♠❛t❧❛❜✮✳
❙❡♥❞♦ ❛ss✐♠ ❢♦✐ ❞❡❝✐❞✐❞♦ ❛❞♦♣t❛r ✉♠❛ ❡str❛té❣✐❛ ✉♥✐✜❝❛❞♦r❛✳ ❖✉ s❡❥❛✱ ❝r✐❛r ✉♠ ♥♦✈♦ ♣r♦❣r❛♠❛
✽ ❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖
❈❛♣ít✉❧♦ ✷
❖s ❞❛❞♦s
✷✳✶ ❆ ♦r❣❛♥✐③❛çã♦ ❞♦s ❞❛❞♦s
❊♠ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛ ❢✉♥❝✐♦♥❛❧ ❞✐s♣♦♠♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ T ✈♦❧✉♠❡s tr✐❞✐♠❡♥s✐♦♥❛✐s V0, V1, ..., VT ❞❡ ✐♠❛❣❡♥s ❞♦ ❝ér❡❜r♦✳ ❈❛❞❛ ✈♦❧✉♠❡ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ✉♥✐❞❛❞❡s ❞❡ ✐♠❛❣❡♠v✳ ❊st❛ ✉♥✐❞❛❞❡ ❝❤❛♠❛✲s❡ ♦ ✈♦①❡❧✳ ❆ ❝❛❞❛ ✈♦①❡❧ ❡stá ❛ss♦❝✐❛❞♦ ✉♠ ✈❛❧♦r ❡ q✉❛tr♦ ❝♦♦r❞❡♥❛❞❛s✳ ❆s três ♣r✐♠❡✐r❛s ✐❞❡♥t✐✜❝❛♠ ❛ ♣♦s✐çã♦ ❞♦ ✈♦①❡❧ ✭✐♥❢♦r♠❛çã♦ ❡s♣❛❝✐❛❧✮ ♥♦ ✈♦❧✉♠❡ ❡ ❛ q✉❛rt❛ ✐❞❡♥t✐✜❝❛ ♦ ✐♥st❛♥t❡ ❛ q✉❡ s❡ r❡❢❡r❡ ♦ ✈❛❧♦r ✭✐♥❢♦r♠❛çã♦ t❡♠♣♦r❛❧✮v=v(x, y, z, t)✳
✷✳✷ ❖ ❞❡s✐❣♥ ❞♦ ❡st✉❞♦
P❛r❛ ❛❧é♠ ❞❛ ✐♥❢♦r♠❛çã♦ ❝♦♥t✐❞❛ ♥♦ ✜❝❤❡✐r♦ ❞❡ ❞❛❞♦s ❢✉♥❝✐♦♥❛✐s é ♥❡❝❡ssár✐♦ t❡r ❡♠ ❝♦♥t❛ ❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❞❡ ❡stí♠✉❧♦s ❛ q✉❡ ♦ ♣❛❝✐❡♥t❡ ❢♦✐ s✉❥❡✐t♦ ❡ ❛ ❢♦r♠❛ ❝♦♠♦ ♦s ❞❛❞♦s ❢♦r❛♠ ❛❞q✉✐r✐❞♦s✳
P❛r❛ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞♦s ❞❛❞♦s é ♥❡❝❡ssár✐♦ s❛❜❡r q✉❛♥t♦s ❡st❛❞♦s ❞❡ ❡stí♠✉❧♦ ❡①✐st❡♠ ♥♦ ❡st✉❞♦✳ P♦r ❡①❡♠♣❧♦✱ ♥✉♠ ❡st✉❞♦ ❡♠ ❜❧♦❝♦s ❞❡ ❞✉❛s t❛r❡❢❛s tí♣✐❝♦ ❡①✐st❡♠ ❞♦✐s ❡st❛❞♦s✿ r❡♣♦✉s♦ ❡ ❛❝t✐✈❛çã♦✳
❊♠ s❡❣✉✐❞❛ é t❛♠❜é♠ ♣r❡❝✐s♦ s❛❜❡r q✉❛♥❞♦ ❝❛❞❛ ❡stí♠✉❧♦ ♦❝♦rr❡✉ ❡ q✉❛❧ ❛ s✉❛ ❞✉r❛çã♦✳ ❊st❛ ✐♥❢♦r♠❛çã♦ ✜❝❛ ❛r♠❛③❡♥❛❞❛ ♥✉♠❛ ♠❛tr✐③ ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❞❡ ❞❡s✐❣♥✳
❈❛❞❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ ❞❡ ❞❡s✐❣♥ r❡❢❡r❡✲s❡ ❛ ✉♠ ❡st❛❞♦✳ ❈❛❞❛ ❧✐♥❤❛ ❝♦♥té♠ ❛ ✐♥❢♦r♠❛çã♦ r❡❢❡r❡♥t❡ ❛ ✉♠ ✐♥st❛♥t❡ ♣❛r❛ t♦❞♦s ♦s ❡st❛❞♦s✳ ❆ss✐♠✱ ❛ ♠❛tr✐③ t❡rá ✉♠ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ✈♦❧✉♠❡s ❞♦ ❡st✉❞♦ ❡ ✉♠ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s ♠❛✐s ✉♠✳ P♦r ❡①❡♠♣❧♦✱ s❡ ♥♦ ❞é❝✐♠♦ ✈♦❧✉♠❡✱ ♦ ♣❛❝✐❡♥t❡ ❡stá s♦❜ ❛ ✐♥✢✉ê♥❝✐❛ ❞♦ ❡st❛❞♦ ❞❡ r❡♣♦✉s♦✱ ❛ ❧✐♥❤❛
10❞❛ ❝♦❧✉♥❛ r❡❢❡r❡♥t❡ ❛♦ ❡st❛❞♦ ❞❡ r❡♣♦✉s♦ t❡rá ♦ ✈❛❧♦r ❞❡1 ❡ ❛ ❧✐♥❤❛10❞❛ ❝♦❧✉♥❛ r❡❢❡r❡♥t❡ ❛♦
❡st❛❞♦ ❞❡ ❛❝t✐✈❛çã♦ t❡rá ♦ ✈❛❧♦r ❞❡0✳
❆ ú❧t✐♠❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ ❞❡ ❞❡s✐❣♥ é ❛♣❡♥❛s ✉♠❛ ❝♦❧✉♥❛ ❞❡1✬s ❡ r❡♣r❡s❡♥t❛ t♦❞♦s ♦s ❡stí♠✉❧t♦s
❝♦♥st❛♥t❡s ❛♦ ❧♦♥❣♦ ❞♦ ❡st✉❞♦✳
❆ ♠❛tr✐③ ❞❡ ❞❡s✐❣♥ é ♣♦rt❛♥t♦ ✉♠❛ ♠❛tr✐③ ❞❡ 0✬s ❡ 1✬s ❛ ♣❛rt✐r ❞❛ q✉❛❧ s❡rá r❡t✐r❛❞❛ t♦❞❛ ❛
✐♥❢♦r♠❛çã♦ ❞❡ ❞❡s✐❣♥ ❞♦ ❡st✉❞♦✳
❋✐♥❛❧♠❡♥t❡✱ ♦ t✐♣♦ ❞❡ ❡st✉❞♦ é ❞❡t❡r♠✐♥❛♥t❡ ♥❛ ❢♦r♠❛ ❝♦♠♦ ♦s ❞❛❞♦s s❡rã♦ tr❛t❛❞♦s✱ ❛ss✐♠ ❝♦♠♦ ♦ t❡♠♣♦ ❞❡ ❛q✉✐s✐çã♦ ❞❡ ❝❛❞❛ ✈♦❧✉♠❡ ✭❚❘✮✳
P❛r❛ ❛ ✜❧tr❛❣❡♠ ❞♦s ❞❛❞♦s ❞❡ ❡st✉❞♦s ❞♦ t✐♣♦ ❜❧♦❝♦ é ♥❡❝❡ssár✐♦ s❛❜❡r t❛♠❜é♠ ♦ ♥ú♠❡r♦ ❞❡ ❛♠♦str❛s ♣♦r ❝✐❝❧♦✳
❈❛♣ít✉❧♦ ✸
❖ ❈ér❡❜r♦ ❡ ♦ ♣ré✲♣r♦❝❡ss❛♠❡♥t♦
❞♦s ❞❛❞♦s
✸✳✶ ❖ ❝❛rr❡❣❛♠❡♥t♦ ❞♦s ❞❛❞♦s
❖ ❈ér❡❜r♦ ❧ê ♦s ✜❝❤❡✐r♦s ❞❡ ❞❛❞♦s ♥♦ ❢♦r♠❛t♦ ▼❛②♦ ❈❧✐♥✐❝ ❆♥❛❧②③❡ ✹❉ ✳ ◆❡st❡ ❢♦r♠❛t♦ ♦s ❞❛❞♦s sã♦ ❞✐str✐❜✉✐❞♦s ♣♦r ❞♦✐s ✜❝❤❡✐r♦s✿ ♦ ✑❤❡❛❞❡r✑ ✭✯✳❤❞r✮ ❡ ❛ ✑✐♠❛❣❡✑ ✭✯✳✐♠❣✮✳ ◆♦ ♣r✐♠❡✐r♦ ❡♥❝♦♥tr❛✲s❡ ♠✉✐t❛ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ❛ ❛q✉✐s✐çã♦ ❞♦s ❞❛❞♦s✳ ❊st❛ ✐♥❢♦r♠❛çã♦ ❡stá ❝♦♥t✐❞❛ ❡♠ ✸✹✽ ❜②t❡s ❡ ❡stá ♦r❣❛♥✐③❛❞❛ s❡❣✉♥❞♦ ♦ ❢♦r♠❛t♦ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❛♥❡①♦ ❆✳ ◆❛ ✑✐♠❛❣❡✑ ❡stá ❛ ✐♥❢♦r♠❛çã♦ ❞❛s ✐♠❛❣❡♥s s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❛rr❛② ❝♦♠ q✉❛tr♦ ❞✐♠❡♥sõ❡s ❡ q✉❡ r❡♣r❡s❡♥t❛ ✈❛❧♦r❡s ❞❛ ❢✉♥çã♦v✳ ❙ó ❞❡♣♦✐s
❞❡ s❛❜❡r ❛s ❞✐♠❡♥sõ❡s ❞♦ ❛rr❛② ❡ ♦ ❢♦r♠❛t♦ ❞♦s ♥ú♠❡r♦s q✉❡ ♦ ❝♦♥st✐t✉❡♠ é q✉❡ é ♣♦ssí✈❡❧ r❡❛❧✐③❛r ❛ ❧❡✐t✉r❛ ❞♦ ♠❡s♠♦✳ ➱ ♣♦rt❛♥t♦ ♥❡❝❡ssár✐♦ q✉❡ ❛ ❧❡✐t✉r❛ ❞♦s ❞❛❞♦s s❡❥❛ ♣r❡❝❡❞✐❞❛ ❞❡ ✉♠❛ ❧❡✐t✉r❛ ❞♦ ✜❝❤❡✐r♦ ✑❤❡❛❞❡r✑✳
❆♣ós ❛ ❧❡✐t✉r❛✱ ♦ ❈ér❡❜r♦ ❛r♠❛③❡♥❛ ❡st❛ ❢✉♥çã♦v s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❛rr❛② ❞❡ ❞✉♣❧❛ ♣r❡❝✐sã♦
q✉❛❞r✐✲❞✐♠❡♥s✐♦♥❛❧✿ ✐♠❛❣❡❉❛t❛❬①❪❬②❪❬③❪❬t❪ ✳
P❛r❛ ❡①tr❛✐r ✐♥❢♦r♠❛çã♦ t❡♠♣♦r❛❧ ✜①❛♠✲s❡ ❛s ✈❛r✐á✈❡✐s x✱y✱ ❡z ♥❛ ♣♦s✐çã♦ ❞❡s❡❥❛❞❛✳ ❋❛③❡♥❞♦
t ✈❛r✐❛r ❞❡ ✵ ❛ ❚✲✶ ✱ ♦ ❛rr❛② ✉♥✐✲❞✐♠❡♥s✐♦♥❛❧ ♦❜t✐❞♦ é ❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❞♦ ✈♦①❡❧ ♥❛ ♣♦s✐çã♦
(x, y, z)✳ T é ♦ ♥ú♠❡r♦ ❞❡ ✈♦❧✉♠❡s ❞♦ ❡st✉❞♦✳
❖s ❞❛❞♦s ✐♠❛❣❡❉❛t❛ ❝♦♥tê♠ ❞♦✐s t✐♣♦s ❞❡ ✐♥❢♦r♠❛çã♦✿ ❡str✉t✉r❛❧ ❡ ❢✉♥❝✐♦♥❛❧✳
❖s ❡str✉t✉r❛✐s sã♦ ❝♦♥st❛♥t❡s ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ❡ ♣❡r♠✐t❡♠✲♥♦s ✐❞❡♥t✐✜❝❛r ❛s ❡str✉t✉r❛s ❛♥❛tó♠✐❝❛s ✭❡✳❣✳ ♠❛tér✐❛ ❝✐♥③❡♥t❛✮✳
❖s ❢✉♥❝✐♦♥❛✐s sã♦ ✈❛r✐á✈❡✐s ♥♦ t❡♠♣♦ ❡ ♠♦str❛♠✲♥♦s ❛ r❡s♣♦st❛ ✜s✐♦❧ó❣✐❝❛ ❞♦ ❝ér❡❜r♦ ❛♦s ❡stí♠✉❧♦s ❡①t❡r♥♦s ❞♦ ❡st✉❞♦✳
◆♦ ♣r♦❣r❛♠❛ ❡st❛ s❡♣❛r❛çã♦ ❞❡ ❞❛❞♦s é ❢❡✐t❛ ❝r✐❛♥❞♦ ❞♦✐s ❛rr❛②s✿ ✉s❡❞■♠❛❣✐♥❣❉❛t❛ ❡ ✉s❡❞❆✲ ♥❛❧②s✐s❉❛t❛✳ ❆♠❜♦s t♦♠❛♠ ✐♥✐❝✐❛❧♠❡♥t❡ ♦s ✈❛❧♦r❡s ❞❡ ✐♠❛❣❡❉❛t❛✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ♠♦❞✐✜❝❛r ♦s ❞❛❞♦s ❢✉♥❝✐♦♥❛✐s s❡♠ q✉❡ ❤❛❥❛ ❛❧t❡r❛çã♦ ❞♦s ❞❛❞♦s ❡str✉t✉r❛✐s ❡ ✈✐❝❡✲✈❡rs❛✳
✸✳✷ ❆ ♠ás❝❛r❛
❆s ✐♠❛❣❡♥s ♦r✐❣✐♥❛✐s ❝♦♥tê♠ ✉♠❛ ❣r❛♥❞❡ ♣❡r❝❡♥t❛❣❡♠ ❞❡ ❞❛❞♦s q✉❡ r❡♣r❡s❡♥t❛♠ ♦ ❢✉♥❞♦✳ ❊st❡ s✉❜✲ ❣r✉♣♦ ❞❡ ❞❛❞♦s é ✐♥út✐❧ ♣❛r❛ ✉♠ ❡st✉❞♦ ❢✉♥❝✐♦♥❛❧✳ ❈♦♠♦ t❛❧ t❡♠♦s t♦❞♦ ♦ ✐♥t❡r❡ss❡ ❡♠ ✐❞❡♥t✐✜❝á✲ ❧♦s✳ ❯♠❛ ✈❡③ ❞✐s♣♦♥í✈❡❧✱ t❛❧ ✐♥❢♦r♠❛çã♦ ♣❡r♠✐t❡✲♥♦s ♣♦✉♣❛r t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❞♦ ♣r♦❣r❛♠❛✳ ❆ ❛♥á❧✐s❡ ♣♦❞❡ s❡r ❢❡✐t❛ ❛♣❡♥❛s ♣❛r❛ ♦s ❞❛❞♦s q✉❡ ♥ã♦ sã♦ ❞❡ ❢✉♥❞♦✳
❆❧é♠ ❞✐ss♦✱ ♦ ♥♦ss♦ ♦❜❥❡❝t♦ ❞❡ ❡st✉❞♦ é ♦ ❝ér❡❜r♦ ❡ ♥ã♦ ♦ ❢✉♥❞♦✳ ◆ã♦ ♣♦❞❡♠♦s t❡r ❡♠ ❝♦♥t❛ ♦s ❞❛❞♦s ❞❡ ❢✉♥❞♦ ♥❛ ❛♥á❧✐s❡ ❡st❛tíst✐❝❛✳
❆ ♠ás❝❛r❛ é ❛r♠❛③❡♥❛❞❛ s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❡①❡♠♣❧❛r ❞❛ ❝❧❛ss❡ ▼❛s❦ ✭❞❛t❛▼❛s❦✮ q✉❡ ❝♦♥té♠ ❛ s✉❛ ✐♥❢♦r♠❛çã♦ s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❛rr❛② ❜♦♦❧❡❛♥♦ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ❊ss❡ ❛rr❛② ❞❡♣❡♥❞❡ ❞❡x✱y ❡ z ❡ t❡♠ ♦ ✈❛❧♦r ❞❡ ✑✈❡r❞❛❞❡✐r♦✑ ♥♦s ✈♦①❡❧s ❞❡ ❞❛❞♦s ❡ ❞❡ ✑❢❛❧s♦✑ ♥♦s ✈♦①❡❧s ❞❡ ❢✉♥❞♦✳
✶✷ ❈❆P❮❚❯▲❖ ✸✳ ❖ ❈➱❘❊❇❘❖ ❊ ❖ P❘➱✲P❘❖❈❊❙❙❆▼❊◆❚❖ ❉❖❙ ❉❆❉❖❙
❖ ♣r♦❣r❛♠❛ ♣❡r♠✐t❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ✈♦①❡❧s ♣♦r ❞♦✐s ♠ét♦❞♦s ❞✐❢❡r❡♥t❡s✿ ♦ ♠ét♦❞♦ ❞❡ ▼❛❤❛✲ ❧❛♥♦❜✐s ❡ ♦ ❞❡ ❝❧✉st❡rs✳
✸✳✷✳✶ ❖ ♠ét♦❞♦ ❞❡ ▼❛❤❛❧❛♥♦❜✐s
❊st❡ ♣♦❞❡ s❡r s❡♣❛r❛❞♦ ❡♠ três ❡t❛♣❛s ♣r✐♥❝✐♣❛✐s✿ ❛♠♦str❛❣❡♠✱ ❛✈❛❧✐❛çã♦ ❡ ❞❡❝✐sã♦✳ ❆ ❛♠♦str❛❣❡♠ é ❛✉t♦♠át✐❝❛ ✭♥ã♦ ❛ss✐st✐❞❛✮ ❡ ❝♦♥s✐st❡ ❡♠ ✐❞❡♥t✐✜❝❛r ❡ r❡❝♦❧❤❡r ♦s ✈❛❧♦r❡s r❡❧❛t✐✈♦s ❛ ✈♦①❡❧s q✉❡ s❛❜❡♠♦s s❡r ❞❡ ❢✉♥❞♦✳ ❆ss✉♠✐♥❞♦ q✉❡ ✉♠ ❝ér❡❜r♦ t❡♠ s✐♠❡tr✐❛ ❡❧✐♣s♦✐❞❛❧✱ ♣♦❞❡♠♦s ❛❝❡✐t❛r ❛ ♣r❡✲ ♠✐ss❛ ❞❡ q✉❡ t♦❞♦s ♦s ✈♦①❡❧s q✉❡ s❡ ❡♥❝♦♥tr❛♠ ♥❛s ❛r❡st❛s ❞♦ ♣❛r❛❧❡❧✐♣í♣❡❞♦ q✉❡ é ♦ ✈♦❧✉♠❡✱ sã♦ ❞❡ ❢✉♥❞♦✳ ❙ã♦ ❡st❡s ♦s ✈♦①❡❧s ❞❡ ❛♠♦str❛❣❡♠ ♥❡st❡ ♠ét♦❞♦✳ ❊♠ ❝❛❞❛ ✈♦❧✉♠❡ é ❢❡✐t❛ ✉♠❛ ❛♠♦s✲ tr❛❣❡♠ ✐♥❞❡♣❡♥❞❡♥t❡✳ ❖❜t❡♠♦s ❛ss✐♠ ✉♠❛ ♠❛tr✐③ ❝♦❧✉♥❛µ❞❡N ♠é❞✐❛s ❡ ✉♠❛ ♠❛tr✐③ ❝♦✈❛r✐â♥❝✐❛ C✱N×N✱ ❡♠ q✉❡N é ♦ ♥ú♠❡r♦ ❞❡ ✈♦❧✉♠❡s✳
❊♠ s❡❣✉✐❞❛ ❝r✐❛✲s❡ ✉♠❛ ♠❛tr✐③ ❝♦❧✉♥❛ ❝♦♠ ♦s ✈❛❧♦r❡s ❞♦ ✈♦①❡❧(0,0,0) ❡ ♠❡❞❡✲s❡ ❛ ❞✐stâ♥❝✐❛
❞❡ ▼❛❤❛❧❛♥♦❜✐s ❞❡st❡ ✈♦①❡❧ ❛té à ♠❛tr✐③ ♠é❞✐❛✳ ❆ ❞✐stâ♥❝✐❛ é ❞❛❞❛ ♣♦r✿
d(µ,µ¯) =
q
(µ−µ¯)TC−1(µ−µ¯) ✭✸✳✶✮
❊st❡ ♣r♦❝❡ss♦ é r❡♣❡t✐❞♦ ♣❛r❛ t♦❞❛s ❛s ♣♦s✐çõ❡s ♣♦ssí✈❡✐s ❞❡ ✈♦①❡❧ ♥♦ ✈♦❧✉♠❡✳
❋✐♥❛❧♠❡♥t❡ ❡st❛❜❡❧❡❝❡♠♦s ✉♠ ❝r✐tér✐♦✳ P♦r ❡①❡♠♣❧♦✱ s❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ ▼❛❤❛❧❛♥♦❜✐s ❢♦r ✐♥❢❡r✐♦r ❛ ✶✵ ♦ ✈♦①❡❧ é ❞❡ ❢✉♥❞♦✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♥ã♦ é✳ ❊st❡ ♠ét♦❞♦ é ❡①❡❝✉t❛❞♦ ❛✉t♦♠❛t✐❝❛♠❡♥t❡ ❧♦❣♦ ❛♣ós ❛ ❧❡✐t✉r❛ ❞♦s ❞❛❞♦s ❞♦ ✜❝❤❡✐r♦ ❆♥❛❧②③❡✳
✸✳✷✳✷ ❖ ♠ét♦❞♦ ❞❡ ❝❧✉st❡rs
❊st❡ ♠ét♦❞♦ ❡①❡❝✉t❛ ✉♠❛ s❡❣♠❡♥t❛çã♦ ❛ss✐st✐❞❛ ❞❛s ✐♠❛❣❡♥s✳ ❱❡r ♦ ❝❛♣ít✉❧♦ ❞❡ s❡❣♠❡♥t❛çã♦ ✭❝❛♣✳✺✮✳
✸✳✸ ❖ ❞❡s✐❣♥
❆ q✉❛♥t✐✜❝❛çã♦ ❞❛ ❢♦r♠❛ ❝♦♠♦ ❢♦✐ ❡①❡❝✉t❛❞❛ ❛ ❛q✉✐s✐çã♦ ❞♦s ❞❛❞♦s é ❡①❡❝✉t❛❞❛ ♣❡❧♦ ❈ér❡❜r♦ ❡♠ três ❡t❛♣❛s✿
• ❞❡✜♥✐çã♦ ❞♦s ❡st❛❞♦s ✲ t♦❞♦s ♦s ❡st❛❞♦s ❞❛ ❛q✉✐s✐çã♦ tê♠ q✉❡ s❡r ✐❞❡♥t✐✜❝❛❞♦s ❞❡ ❞✉❛s ❢♦r♠❛s✿
✉♠❛ ❙✐❣❧❛ ❞❡ três ❧❡tr❛s ❡ ✉♠ ◆♦♠❡ ✭❢r❛s❡ ♦✉ ♣❛❧❛✈r❛ ♠❛✐s ❞❡s❝r✐t✐✈❛ ❛té ✉♠ ♠á①✐♠♦ ❞❡ ✷✺ ❝❛r❛❝t❡r❡s✮✳
• s❡q✉ê♥❝✐❛ ❞❡ ❡st❛❞♦s ✲ ❯♠❛ ❥❛♥❡❧❛ ❝♦♠ ✉♠❛ ❜❛rr❛ ✭✐♥✐❝✐❛❧♠❡♥t❡ t♦❞❛ ❝✐♥③❡♥t❛✮ r❡♣r❡s❡♥t❛
❛ s❡q✉ê♥❝✐❛ ❞♦s ❡st❛❞♦s ❞♦ ❡st✉❞♦✳ ➱ ♣♦ssí✈❡❧ ❛❞✐❝✐♦♥❛r ❡st❛❞♦s ❞❡✜♥✐♥❞♦ ❛ s✐❣❧❛ ♦ ✐♥í❝✐♦ ❡ ❛ ❞✉r❛çã♦✳ ❙❡ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡st❛❞♦s r❡♣❡t✐t✐✈❛ ❜❛st❛ ❞❡s❡♥❤❛r ❡ss❛ s❡q✉ê♥❝✐❛ ❡ ❞❡♣♦✐s ♠❛♥❞❛r r❡♣❡t✐r✳ ❙❡ ❤♦✉✈❡r ❡♥❣❛♥♦ é t❛♠❜é♠ ♣♦ssí✈❡❧ ❛♣❛❣❛r✳ ❯♠❛ ✈❡③ ❝♦♥str✉í❞❛✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❡st❛❞♦s ♣♦❞❡ s❡r ❣✉❛r❞❛❞❛ ♥✉♠ ✜❝❤❡✐r♦ ❝♦♠ ❡①t❡♥sã♦ ✑✳t♠❧✑✳ ❙❡♠♣r❡ q✉❡ ♥❡❝❡ssár✐♦ ♦ ❈ér❡❜r♦ ♣♦❞❡rá ❝❛rr❡❣á✲❧♦✳
• ♣❛râ♠❡tr♦s ✲ ◆❡st❛ ❥❛♥❡❧❛ ❞❡✜♥❡✲s❡ ♦ t❡♠♣♦ ❞❡ ❛q✉✐s✐çã♦ ✭❚❘✮✳ ❖ ♣r♦❣r❛♠❛ r❡❝♦♥❤❡❝❡ ❛✉✲
t♦♠❛t✐❝❛♠❡♥t❡ s❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❡st❛❞♦s é ❞♦ t✐♣♦ ❜❧♦❝♦ ❡✱ ♥❡st❡ ❝❛s♦ ♣❡❞❡ ❛♦ ✉t✐❧✐③❛❞♦r q✉❡ ✐♥s✐r❛ t❛♠❜é♠ ♦ ♥ú♠❡r♦ ❞❡ ❛q✉✐s✐çõ❡s ♣♦r ❝✐❝❧♦✳
✸✳✹✳ ❖❙ ❖❯❚▲■❊❘❙ ❖❯ P❖◆❚❖❙ ❊❙PÚ❘■❖❙ ✶✸
❊st❡ ❛rr❛② ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ♣♦❞❡ t♦♠❛r três ✈❛❧♦r❡s✿ 0✱+1 ❡−1 ❡ t❡♠ ✉♠❛ ❞✐♠❡♥sã♦ ✐❣✉❛❧ ❛♦
♥ú♠❡r♦ ❞❡ ✈♦❧✉♠❡s ❞♦ ❡st✉❞♦✳ ➱+1 q✉❛♥❞♦ ♥✉♠ ✐♥st❛♥t❡ t❡♠♦s ♦ ♣r✐♠❡✐r♦ ❡st❛❞♦✱−1 q✉❛♥❞♦
t❡♠♦s ♦ s❡❣✉♥❞♦ ❡st❛❞♦ ❡0 q✉❛♥❞♦ ♥ã♦ ❡st❛♠♦s ❡♠ q✉❛❧q✉❡r ❞♦s ❞♦✐s ❡st❛❞♦s✳
❖ ✈❡❝t♦r ❞❡ ❞❡s✐❣♥ s❡rá ✉t✐❧✐③❛❞♦ ♣♦r ✈ár✐♦s ♠ét♦❞♦s ❞❡ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ❝♦♠♦ ✉♠❛ ♣r✐♠❡✐r❛ ❛♣r♦①✐♠❛çã♦ ❞♦ s✐♥❛❧ ✜s✐♦❧ó❣✐❝♦ ❞❡ ❛❝t✐✈❛çã♦ q✉❡ ❡s♣❡r❛♠♦s ♠❡❞✐r✳ ➱ ❛♣❡♥❛s ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❞✉❛s r❛③õ❡s✿ ♦ s✐♥❛❧ ✜s✐♦❧ó❣✐❝♦ t❡♠ s❡♠♣r❡ ✉♠ ❛tr❛s♦ ❡♠ r❡❧❛çã♦ ❛♦ ✈❡❝t♦r ❞❡ ❞❡s✐❣♥ ❡ ♦ s✐♥❛❧ r❡s✉❧t❛ ❞❡ ✉♠ ❛✉♠❡♥t♦ ❞❡ ♦①✐❣❡♥❛çã♦ ♣♦r ✈✐❛ ❞♦ ✢✉①♦ s❛♥❣✉í♥❡♦✳ ❊st❡ s✐♥❛❧ é ❝♦♥❤❡❝✐❞♦ ♣♦r ❇❖▲❉ ✭❇❧♦♦❞ ❖①✐❣❡♥❛t✐♦♥ ▲❡✈❡❧ ❉❡♣❡♥❞❡♥t✮✳
➱ ♣♦rt❛♥t♦ ✉♠❛ ♠❛♥❡✐r❛ ✐♥❞✐r❡❝t❛ ❞❡ ♠❡❞✐r ✉♠ ❛✉♠❡♥t♦ ❞❡ ❛❝t✐✈✐❞❛❞❡ ❝❡r❡❜r❛❧✳ ❖ ♣r❡ss✉♣♦st♦ é q✉❡ s❡ ✉♠❛ ③♦♥❛ ❞♦ ❝ér❡❜r♦ ♥❡❝❡ss✐t❛ ❞❡ ♠❛✐s ♦①✐❣❡♥❛çã♦ ❡ ♥✉tr✐❡♥t❡s ❡♥tã♦ ❡❧❛ ❡stá ❡♠ ❛❝t✐✈✐❞❛❞❡✳ P❡r❝❡❜❡✲s❡ ❡♥tã♦ ♣♦rq✉❡ é q✉❡ ❤á ♦ ❛tr❛s♦✳ ❖ ❛✉♠❡♥t♦ ❞❡ ✢✉①♦ s❛♥❣✉í♥❡♦ só ❛✉♠❡♥t❛ q✉❛♥❞♦ ❤á ♥❡❝❡ss✐❞❛❞❡ ✭❞❡♣♦✐s ❞♦s ♥❡✉ró♥✐♦s ❝♦♥s✉♠✐r❡♠ ❛❧❣✉♠❛s r❡s❡r✈❛s✮ ❡ ❞❡♣♦✐s ❞♦s ✈❛s♦s s❛♥❣✉í♥❡♦s t❡r❡♠ t✐❞♦ t❡♠♣♦ ❞❡ ❞✐❧❛t❛r✳
▼❛✐s✱ ❛ r❡s♣♦st❛ ❞♦ ✈❛s♦ s❛♥❣✉í♥❡♦ ❛♣r❡s❡♥t❛ ✉♠ ♣✐❝♦ ✐♥✐❝✐❛❧ ❞❡ ❝♦♠♣❡♥s❛çã♦ q✉❡ ❞❡♣♦✐s r❡❞✉③ ❣r❛❞✉❛❧♠❡♥t❡ ♣❛r❛ ✉♠ ❡st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ❊st❛ ✈❛r✐❛çã♦ ❞♦ ✢✉①♦ s❛♥❣✉í♥❡♦ é ♣r❡✈✐st❛ ♣❡❧♦ ♠♦❞❡❧♦ ❞♦ ❜❛❧ã♦ ❬✹❪✳
❆ ❢✉♥çã♦ q✉❡ ❞❡s❝r❡✈❡ ❡st❛ ✈❛r✐❛çã♦ t❡♠♣♦r❛❧ é ❝♦♥❤❡❝✐❞❛ ♣♦r ❍❘❋ ✭❍❡♠♦❞②♥❛♠✐❝ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥✮ ❡ ♥❡st❡ ♠♦❞❡❧♦ t❡♠ ❛ ❢♦r♠❛ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❣❛♠❛✿
γℓ(t) =
1 Γ (ℓ)t
ℓ−1e−t ✭✸✳✷✮
❖✉tr♦s ♠♦❞❡❧♦s ❬✶✸❪❬✶✻❪ ♣r♦♣õ❡♠ q✉❡ ❛♣ós ♦ ♣✐❝♦ ❞❡ ✢✉①♦ s❛♥❣✉í♥❡♦ ❛ ❢✉♥çã♦ ❍❘❋ ❝❤❡❣❛ ❛ ✐r ❛❜❛✐①♦ ❞♦ ❡st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ❊st❡ ♠♦❞❡❧♦ é ❝♦♥❤❡❝✐❞♦ ♣❡❧♦ ♠♦❞❡❧♦ ❙P▼ ❝❛♥ó♥✐❝♦ ♣♦r s❡r ♦ ♠♦❞❡❧♦ ❛❞♦♣t❛❞♦ ♣❡❧♦ ♣❛❝♦t❡ ❞❡ s♦❢t✇❛r❡ ❙P▼ ❡ é ❞❡s❝r✐t♦ ♣❡❧❛ ❡q✉❛çã♦✿
hC(t) =γ6(t)−1
6γ16(t) ✭✸✳✸✮
❊st✉❞♦s ♠❛✐s r❡❝❡♥t❡s ❬✶✵❪ ❞❡ ❡st✐♠❛t✐✈❛ ❞❛ r❡s♣♦st❛ ❤❡♠♦❞✐♥â♠✐❝❛ s✉❣❡r❡♠ q✉❡ ♦ s✐♥❛❧ ❍❘❋ ✈❛r✐❛ ❞❡ r❡❣✐ã♦ ♣❛r❛ r❡❣✐ã♦✱ ❞❡ t❛r❡❢❛ ♣❛r❛ t❛r❡❢❛ ❡ ❞❡ ♣❛❝✐❡♥t❡ ♣❛r❛ ♣❛❝✐❡♥t❡✳
❙❡ ♦ s✐♥❛❧ ❞❡ ❞❡s✐❣♥ t❡♠ ❛ ♣r❡t❡♥sã♦ ❞❡ r❡♣r♦❞✉③✐r ♦ s✐♥❛❧ ✜s✐♦❧ó❣✐❝♦✱ ❡❧❡ t❡♠ q✉❡ s❡r ♠♦❞✐✜❝❛❞♦✳ ❖ ♣r♦❣r❛♠❛ ❈ér❡❜r♦ ♦❢❡r❡❝❡ ❛♦ ✉t✐❧✐③❛❞♦r ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠♣❡♥s❛r ❡st❛s ❞✐❢❡r❡♥ç❛s✳ ❊❧❡ ♣♦❞❡ ✐♥s❡r✐r ♦ ❛tr❛s♦ q✉❡ q✉✐s❡r ♥♦ s✐♥❛❧✳ P♦❞❡ t❛♠❜é♠ r❡❛❧✐③❛r ❛ ❝♦♥✈♦❧✉çã♦ ❞♦ s✐♥❛❧ ❞❡ ❞❡s✐❣♥ ❝♦♠ ❛ ❢✉♥çã♦ ❍❘❋ ✭❡q✳✸✳✸✮✳ ❯♠❛ ♣♦st❡r✐♦r ✈❡rsã♦ ❞♦ ❈ér❡❜r♦ ✐rá ✐♥❝❧✉✐r ❛ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡ ❞❡ ❡st✐♠❛t✐✈❛ ❞❛ ❢✉♥çã♦ ❍❘❋ ❛ ♣❛rt✐r ❞♦s ❞❛❞♦s ❝♦♠ ♦ ♠ét♦❞♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✶✶❪✳
❆♦ s✐♥❛❧ ❡s❝♦❧❤✐❞♦ ✭s❡q✉ê♥❝✐❛ ❞❡ ❞❡s✐❣♥✱ s❡q✉ê♥❝✐❛ ❞❡ ❞❡s✐❣♥ ❝♦♥✈♦❧✈✐❞❛ ❝♦♠ ♦ ❍❘❋ ♦✉ ♦✉tr♦✮ ❝❤❛♠❛♠♦s s❡q✉ê♥❝✐❛ ❞❡ ♠♦❞❡❧♦ ♦✉ ✈❡❝t♦r ❞❡ ♠♦❞❡❧♦ ✭❢✉♥çã♦u=u(t)✮✳
✸✳✹ ❖s ♦✉t❧✐❡rs ♦✉ ♣♦♥t♦s ❡s♣úr✐♦s
P♦r ✈❡③❡s✱ ♥✉♠❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❛♣❛r❡❝❡ ✉♠ ♣♦♥t♦ ✭♦✉ ✈ár✐♦s✦✮ ❡str❛♥❤♦✳ P♦r ❡①❡♠♣❧♦✱ ❛ s❡q✉ê♥❝✐❛ ❞❛ ✜❣✉r❛ t❡♠ s❡♠♣r❡ ✈❛❧♦r❡s ♥✉❧♦s ❡①❝❡♣t♦ ♥♦ ✐♥st❛♥t❡112✳ ❊st❡ ✈❛❧♦r é ❝❤❛♠❛❞♦ ✉♠
✑♦✉t❧✐❡r✑ ♦✉ ✑♣♦♥t♦ ❡s♣úr✐♦✑✳ ❊st❡s ♣♦♥t♦s ♣♦❞❡♠ t❡r ✉♠❛ ✐♥✢✉ê♥❝✐❛ ♣❡s❛❞❛ s♦❜r❡ ❛s ♠❡❞✐çõ❡s ❡s✲ t❛tíst✐❝❛s ❡❢❡❝t✉❛❞❛s s♦❜r❡ ✉♠❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❡ ❛❧t❡r❛r ❞r❛st✐❝❛♠❡♥t❡ ❛ ✐♥❢❡rê♥❝✐❛ ❡st❛tíst✐❝❛ r❡s✉❧t❛♥t❡✳
❖ ❈ér❡❜r♦ ♣❡r♠✐t❡ ❛♦ ✉t✐❧✐③❛❞♦r r❡t✐r❛r ❡st❡s ♣♦♥t♦s✳ ■st♦ é ❝♦♥s❡❣✉✐❞♦ ❡♠ ✈❛r✐♦s ♣❛ss♦s✿
• ♦r❞❡♥❛ ♣♦r ♦r❞❡♠ ❝r❡s❝❡♥t❡ ❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧✳
• ❝❛❧❝✉❧❛ ❛ ♠❡❞✐❛♥❛ ❡ ♦ ❞❡s✈✐♦ ❛❜s♦❧✉t♦ ❞❛ ♠❡❞✐❛♥❛ ✭❉❆▼✮ ❞♦s ♣r✐♠❡✐r♦s ✾✺✪ ❞♦s ❞❛❞♦s
♦r❞❡♥❛❞♦s✳
• ♠❡❞❡ ❛ q✉❛♥t♦s ❉❆▼s ❞❡ ❞✐stâ♥❝✐❛ ❡stá ❝❛❞❛ ✉♠ ❞♦s r❡st❛♥t❡s ✺✪ ❞❛ ♠❡❞✐❛♥❛✳
• s❡ ❛ ❞✐stâ♥❝✐❛ é ♠✉✐t♦ ❣r❛♥❞❡ ❡♥tã♦ ♠♦❞✐✜❝❛ ❡st❡ ✈❛❧♦r ♣❛r❛ ✉♠ ✈❛❧♦r ✐♥t❡r♣♦❧❛♥❞♦ ❡♥tr❡ ♦s
✶✹ ❈❆P❮❚❯▲❖ ✸✳ ❖ ❈➱❘❊❇❘❖ ❊ ❖ P❘➱✲P❘❖❈❊❙❙❆▼❊◆❚❖ ❉❖❙ ❉❆❉❖❙
❖s ❞❛❞♦s s❡♠ ♦✉t❧✐❡rs sã♦ ❛✉t♦♠❛t✐❝❛♠❡♥t❡ ❡①♣♦rt❛❞♦s ♥♦ ❢♦r♠❛t♦ ❆♥❛❧②③❡ ♣❛r❛ ✉♠ ✜❝❤❡✐r♦ ❝♦♠ ♦ ♥♦♠❡ ❞♦ ✜❝❤❡✐r♦ ♦r✐❣✐♥❛❧ ❝♦♥❝❛t❡♥❛❞♦ ❝♦♠ ❛ ❧❡tr❛ ❝✳ ❙❡ ♦s ❞❛❞♦s ❢♦r❛♠ ♣r❡✈✐❛♠❡♥t❡ ✜❧tr❛❞♦s sã♦ ❡①♣♦rt❛❞♦s ❝♦♠ ♦ t✐♣♦ ❝♦♠♣❧❡①♦✱ ❝❛s♦ ❝♦♥trár✐♦ sã♦ ❡①♣♦rt❛❞♦s ❝♦♠ ♦ t✐♣♦ ❝♦♠♣❧❡①♦✳ ◆❡st❡ ❝❛s♦✱ ♦s ✈❛❧♦r❡s r❡❛✐s sã♦ ♦s ❡str✉t✉r❛✐s ❡ ♦s ✐♠❛❣✐♥ár✐♦s sã♦ ♦s ❢✉♥❝✐♦♥❛✐s✳
✸✳✺ ❆ ✜❧tr❛❣❡♠ ❞♦s ❞❛❞♦s
❖s ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❞✐t♦s ❡stát✐❝♦s ❛♣❧✐❝❛❞♦s ❞✉r❛♥t❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛ ♥❡♠ s❡♠♣r❡ sã♦ ❤♦♠♦❣é♥❡♦s✳ ❖✉ s❡❥❛✱ ♥❡♠ s❡♠♣r❡ sã♦ ❝♦♥st❛♥t❡s ❛♦ ❧♦♥❣♦ ❞♦ ❡s♣❛ç♦ ❡ ❞♦ t❡♠♣♦✳ ❊st❡ ❢❛❝t♦ ❝❛✉s❛ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ r✉í❞♦s ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ♥❛s s❡q✉ê♥❝✐❛s t❡♠♣♦r❛✐s✳ ❖ ❈ér❡❜r♦ ♦❢❡r❡❝❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❛♦ ✉t✐❧✐③❛❞♦r ❞❡ r❡t✐r❛r ❡st❡ r✉í❞♦ ❛♣❧✐❝❛♥❞♦ ✉♠ ✜❧tr♦ ♣❛ss❛✲❛❧t♦✳ ❖ ✜❧tr♦ é ❜❛s❡❛❞♦ ♥❛ ✐❞❡✐❛ ❞❡ r❡❛❧✐③❛r ✉♠❛ ❝♦♥✈♦❧✉çã♦ ❞❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❝♦♠ ✉♠❛ ❢✉♥çã♦g(t)✳
vf(t) =v(t)⊗g(t) ✭✸✳✹✮ ❡♠ q✉❡ g(t) é t❛❧ q✉❡ ❛ s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r é ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞♦ ✜❧tr♦
♣❛ss❛✲❛❧t♦ ✐❞❡❛❧˜g(f)✿
˜
g(f) =F(g(t)) ✭✸✳✺✮
❆ ❢✉♥çã♦ g(t)s❡rá ❡♥tã♦ ✉♠❛ ❢✉♥çã♦ ❞♦ t✐♣♦ s✐♥❝✳ P♦ré♠✱ ♥❡st❡ ❝❛s♦ ♥ã♦ t❡♠♦s ✉♠❛ ❢✉♥çã♦
s✐♥❝ ❝♦♥tí♥✉❛ ❞❡ ✈❛❧♦r❡s ♠❛s s✐♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛❧♦r❡s ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦✳ ❖ ❝á❧❝✉❧♦ ❞❡g(t)
♣❡❧❛ ❚r❛♥s❢♦r♠❛❞❛ ❘á♣✐❞❛ ❞❡ ❋♦✉r✐❡r ✭❚❘❋✮ ✐♠♣❧✐❝❛ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞♦ ❢❡♥ó♠❡♥♦ ❞❡ ✑❛❧✐❛s✐♥❣✑✳ ❖✉ s❡❥❛✱ ❛ s❡q✉ê♥❝✐❛˜g(f)❥á ♥ã♦ é ❡①❛❝t❛♠❡♥t❡ ❛ ❞❡ ✉♠ ✜❧tr♦ ✐❞❡❛❧✳
❊st❡ ♣r♦❜❧❡♠❛ é t♦r♥❡❛❞♦ ❢✉❣✐♥❞♦ ❞❛ ❢✉♥çã♦ s✐♥❝✳ Pr✐♠❡✐r♦ é ❢❡✐t♦ ✉♠ ❝á❧❝✉❧♦ s❡♣❛r❛❞♦ ❞❛ ❚❘❋ ❞❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧✿
˜
v(f) =F(v(t)) ✭✸✳✻✮
❊♠ s❡❣✉♥❞♦ ❧✉❣❛r ❝♦♥str♦✐✲s❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ❞❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞♦ ✜❧tr♦ ♣❛ss❛✲❛❧t♦ ✐❞❡❛❧˜g(f)✳ ❊♠ t❡r❝❡✐r♦ ❧✉❣❛r r❡❛❧✐③❛✲s❡ ♦ ♣r♦❞✉t♦ ❞❛s ❞✉❛s s❡q✉ê♥❝✐❛s ♥♦ ❡s♣❛ç♦
❞❛s ❢r❡q✉ê♥❝✐❛s✳ ❋✐♥❛❧♠❡♥t❡✱ ❝❛❧❝✉❧❛✲s❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❘á♣✐❞❛ ❞❡ ❋♦✉r✐❡r ■♥✈❡rs❛ ✭❚❘❋■✮ ❞♦ r❡s✉❧t❛❞♦ ❞♦ ♣r♦❞✉t♦✿
vf(t) =F−1(˜g(f)·v˜(f)) ✭✸✳✼✮ ◆❛s ✜❣✉r❛s ✸✳✶ ❡ ✸✳✷ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ❛ ✜❧tr❛❣❡♠ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❝♦♠ r✉í❞♦ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✳
❢♦r❛♠ r❡t✐r❛❞❛s ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✳
❖s ❞❛❞♦s ✜❧tr❛❞♦s sã♦ ❛✉t♦♠❛t✐❝❛♠❡♥t❡ ❡①♣♦rt❛❞♦s ♥♦ ❢♦r♠❛t♦ ❆♥❛❧②③❡ ♣❛r❛ ✉♠ ✜❝❤❡✐r♦ ❝♦♠ ♦ ♥♦♠❡ ❞♦ ✜❝❤❡✐r♦ ♦r✐❣✐♥❛❧ ❝♦♥❝❛t❡♥❛❞♦ ❝♦♠ ❛ ❧❡tr❛ ❢✳ ❖s s❡✉s ✈❛❧♦r❡s sã♦ ❛tr✐❜✉í❞♦s ❛♦ ❛rr❛② ✉s❡❞❆♥❛❧②s✐s❉❛t❛✳ ➱ ❞❡ ♥♦t❛r q✉❡ ❛ ♣❛rt✐r ❞❡st❡ ♠♦♠❡♥t♦ ❡st❡s ❞❛❞♦s ❥á ♥ã♦ ❝♦♥tê♠ q✉❛❧q✉❡r ✐♥❢♦r♠❛çã♦ ❡str✉t✉r❛❧✳
❖s ❞❛❞♦s ❡①♣♦rt❛❞♦s sã♦ ❞♦ t✐♣♦ ❝♦♠♣❧❡①♦✳ ❖s ✈❛❧♦r❡s r❡❛✐s sã♦ ♦s ❡str✉t✉r❛✐s ❡ ♦s ✐♠❛❣✐♥ár✐♦s sã♦ ♦s ❢✉♥❝✐♦♥❛✐s✳
✸✳✻ ❆ ✈✐s✉❛❧✐③❛çã♦ ❞♦s ❞❛❞♦s
❖ ❈ér❡❜r♦ ♣♦ss✐❜✐❧✐t❛ ❛ ✈✐③✉❛❧✐③❛çã♦ ❞♦s ❞❛❞♦s s♦❜ q✉❛tr♦ ❢♦r♠❛s ❞✐❢❡r❡♥t❡s✳
✸✳✻✳ ❆ ❱■❙❯❆▲■❩❆➬➹❖ ❉❖❙ ❉❆❉❖❙ ✶✺
❋✐❣✉r❛ ✸✳✶✿ ❙❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ♥ã♦ ✜❧tr❛❞❛✳
✶✻ ❈❆P❮❚❯▲❖ ✸✳ ❖ ❈➱❘❊❇❘❖ ❊ ❖ P❘➱✲P❘❖❈❊❙❙❆▼❊◆❚❖ ❉❖❙ ❉❆❉❖❙
✸✳✻✳ ❆ ❱■❙❯❆▲■❩❆➬➹❖ ❉❖❙ ❉❆❉❖❙ ✶✼
❋✐❣✉r❛ ✸✳✹✿ ❏❛♥❡❧❛ ❞❡ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧✳
❙❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ✲ ✉♠❛ ❥❛♥❡❧❛ ✭✜❣✉r❛ ✸✳✹✮ ♠♦str❛ ✉♠ ❣rá✜❝♦ ❞❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❞❡ ✉♠ ✈♦①❡❧✳ ❚❡♠ ❛s s❡❣✉✐♥t❡s ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s✿ s❡❧❡❝çã♦ ❞♦ ✈♦①❡❧ ♣r❡t❡♥❞✐❞♦❀ ❡①♣♦rt❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ♥♦ ❢♦r♠❛t♦ t①t✳
❊s♣❡❝tr♦ ✲ ✉♠❛ ❥❛♥❡❧❛ ✭✜❣✉r❛ ✸✳✺✮ ♠♦str❛ ♦ ❡s♣❡❝tr♦ ❞❡ ❢r❡q✉ê♥❝✐❛s ❞❡ ❝❛❞❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧✳ ❚❡♠ ❛s s❡❣✉✐♥t❡s ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s✿ ❡s❝♦❧❤❛ ❞♦ ✈♦①❡❧ ♣r❡t❡♥❞✐❞♦❀ ❡①♣♦rt❛çã♦ ❞♦ ❡s♣❡❝tr♦ ♥♦ ❢♦r♠❛t♦ t①t✳ ❖ ❡s♣❡❝tr♦ ♣♦❞❡ s❡r ❛♣r❡s❡♥t❛❞♦ ❝♦♠ ♦✉ s❡♠ ❥❛♥❡❧❛s ✭❍❛♥♥✱ ❍❛♠♠✐♥❣✮✳
❘❡s✉❧t❛❞♦s ✲ ✉♠❛ ❥❛♥❡❧❛ ✭✜❣✉r❛ ✸✳✻✮ ♠♦str❛ ♦s r❡s✉❧t❛❞♦s s♦❜ ❛ ❛❢♦r♠❛ ❞❡ ✐♠❛❣❡♥s ❡ ♣❡r✜s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ s❡❣✉♥❞♦ ❛ três ❞✐r❡❝çõ❡s ♦rt♦❣♦♥❛✐s✳ ❚❡♠ ❛s s❡❣✉✐♥t❡s ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s✿ s❡❧❡❝çã♦ ❞❛ ❢❛t✐❛ ❛♣r❡s❡♥t❛❞❛❀ s❡❧❡❝çã♦ ❞♦ ❧✐♠✐❛r ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❛ ❢❛t✐❛❀ ❡①♣♦rt❛çã♦ ❞❛s ✐♠❛❣❡♥s ♥♦ ❢♦r♠❛t♦ ❥♣❡❣❀ ❡①♣♦rt❛çã♦ ❞♦s ♣❡r✜s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ q✉❛❧q✉❡r ✈♦①❡❧ ♥♦ ❢♦r♠❛t♦ t①t❀ s❡❧❡❝çã♦ ❞♦ ✈♦①❡❧ ❝❧✐❝❛♥❞♦ ♥❛ ✐♠❛❣❡♠❀ ❛♠♣❧✐❛çã♦ ❝♦♠ ❈❚❘▲✲❝❧✐❝❀ r❡❞✉çã♦ ❝♦♠ ❆▲❚✲❝❧✐❝✳
P❡r✜s ✲ ✉♠❛ ❥❛♥❡❧❛ ✭✜❣✉r❛ ✸✳✼✮ ♠♦str❛ ♦s ♣❡r✜s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ s❡♣❛r❛❞❛♠❡♥t❡ ♣❛r❛ ♣❡r♠✐t✐r ❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ♣❡r✜s ❞❡ ✈♦①❡❧s ❞✐❢❡r❡♥t❡s s❡♠ t❡r q✉❡ ❢❛③❡r ♥♦✈♦ ❝á❧❝✉❧♦✳ ❚❡♠ ❛s s❡❣✉✐♥t❡s ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s✿ s❡❧❡❝çã♦ ❞♦ ✈♦①❡❧ ♣♦r ♣♦s✐çã♦❀ ❡①♣♦rt❛çã♦ ❞♦s três ♣❡r✜s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ q✉❛❧q✉❡r ✈♦①❡❧ ♥♦ ❢♦r♠❛t♦ t①t✳
✶✽ ❈❆P❮❚❯▲❖ ✸✳ ❖ ❈➱❘❊❇❘❖ ❊ ❖ P❘➱✲P❘❖❈❊❙❙❆▼❊◆❚❖ ❉❖❙ ❉❆❉❖❙
✸✳✻✳ ❆ ❱■❙❯❆▲■❩❆➬➹❖ ❉❖❙ ❉❆❉❖❙ ✶✾
✷✵ ❈❆P❮❚❯▲❖ ✸✳ ❖ ❈➱❘❊❇❘❖ ❊ ❖ P❘➱✲P❘❖❈❊❙❙❆▼❊◆❚❖ ❉❖❙ ❉❆❉❖❙
❈❛♣ít✉❧♦ ✹
❖ ❈ér❡❜r♦ ❡ ❛ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧
✹✳✶ ■♥tr♦❞✉çã♦
❖ ❈ér❡❜r♦ ♣❡r♠✐t❡ ❛ ❛✈❛❧✐❛çã♦ ❞❡ ❛❝t✐✈❛çã♦ ❢✉♥❝✐♦♥❛❧ ♣♦r s❡✐s ♠ét♦❞♦s ❞✐st✐♥t♦s✿ t ❞❡ ❙t✉❞❡♥t✱ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦✱ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦✱ ♣✐❝♦s ❞❡ ❋♦✉r✐❡r✱ ❛♠♣❧✐t✉❞❡ ❡ s♦❜r❡♣♦s✐çã♦✳
❖s r❡s✉❧t❛❞♦s ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ♠ét♦❞♦ sã♦ ❛♣r❡s❡♥t❛❞♦s s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❛rr❛② tr✐❞✐♠❡♥✲ s✐♦♥❛❧ ✭❞❡♣❡♥❞❡♥t❡ ❞❡x✱y ❡z✮ q✉❡ s❡rá ❞❡s✐❣♥❛❞♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❊st❛ ❢♦r♠❛ ú♥✐❝❛
❞❡ ❛♣r❡s❡♥t❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ♣❡r♠✐t❡ ✉♠❛ ❝♦♠♣❛r❛çã♦ q✉❛♥t✐t❛t✐✈❛ ❞♦s ♠ét♦❞♦s✳
✹✳✷ ❖ ♠ét♦❞♦ t ❞❡ ❙t✉❞❡♥t
✹✳✷✳✶ ❚❡♦r✐❛
❖ ♠ét♦❞♦ t ❞❡ ❙t✉❞❡♥t é ✉♠ ♠ét♦❞♦ ❡st❛tíst✐❝♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❛r❛ ❛✈❛❧✐❛r s❡ ❛s ♠é❞✐❛s ❞❡ ❞✉❛s ♣♦♣✉❧❛çõ❡s sã♦ ❞✐❢❡r❡♥t❡s✳ ❊❧❡ ❜❛s❡✐❛✲s❡ ♥♦ s❡❣✉✐♥t❡ ♣r❡ss✉♣♦st♦✿ ❙❡ ✉♠❛ ✈❛r✐á✈❡❧ ❩ é ♦❜t✐❞❛ ♣❡❧❛ r❛③ã♦ ❡♥tr❡ ❞✉❛s ✈❛r✐❛✈❡✐sX ∼N(0,1) ❡Y ∼χ2❡♥tã♦ ❡❧❛ t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❙t✉❞❡♥t✳
❙❡ ❛ ❡st❡ ♣r❡ss✉♣♦st♦ ❛❧✐❛r♠♦s ✉♠ ❝r✐tér✐♦ ❞❡ ❞❡❝✐sã♦ t❡♠♦s ♦ ♠ét♦❞♦ t ❞❡ ❙t✉❞❡♥t✳
❆ ✈❛r✐á✈❡❧ ❝♦♠ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ❝❡♥tr❛❞❛ ❳ s❡rá ❞❛❞❛ ♣❡❧❛ ❞✐❢❡r❡♥❝❛ ❡♥tr❡ ❛s ♠é❞✐❛s vA ❡
vR✳ ❆ ✈❛r✐á✈❡❧ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞♦ t✐♣♦χ2 s❡rá ❞❛❞❛ ♣❡❧❛ q✉❛♥t✐❞❛❞❡✿
s σ2 A NA + σ 2 R
NR ✭✹✳✶✮
❝♦♠NA é ♦ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s ❞❡ ❛❝t✐✈❛çã♦ ❡NR é ♦ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s ❞❡ r❡♣♦✉s♦✳ ❆ ✈❛r✐á✈❡❧ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ r❛③ã♦ ❡♥tr❡ ❡st❛s ❞✉❛s✿
ϑ= qvA−vR σ2
A
NA+
σ2 R
NR
✭✹✳✷✮
t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ t ❞❡ ❙t✉❞❡♥t✳
φ(ϑ, n) =√ 1
nB 12,n2
1 + ϑ
2 n − n+1 2 ✭✹✳✸✮
❡♠ q✉❡né ♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❡ B é ❛ ❢✉♥çã♦ ❜❡t❛ ✭✈❡r ❛♣ê♥❞✐❝❡ ❇✮✳
◆♦ ❣rá✜❝♦ ❞❛ ✜❣✉r❛ ✹✳✶❛♣r❡s❡♥t❛✲s❡ ❛ ✈❛r✐❛çã♦ ❞❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❝♦♠ t ♣❛r❛ ✈ár✐♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡(1,2,5,100)✿
❖❜s❡r✈❛✲s❡ q✉❡ ❛ ❢✉♥çã♦✿
✲ t❡♠ ❝❛✉❞❛s ♠❛✐s ❡❧❡✈❛❞❛s q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ ✲ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ q✉❛♥❞♦n→ ∞✳
✷✷ ❈❆P❮❚❯▲❖ ✹✳ ❖ ❈➱❘❊❇❘❖ ❊ ❆ ❆◆➪▲■❙❊ ❋❯◆❈■❖◆❆▲
✹✳✷✳ ❖ ▼➱❚❖❉❖ ❚ ❉❊ ❙❚❯❉❊◆❚ ✷✸
❋✐❣✉r❛ ✹✳✷✿ ❉❡♥s✐❞❛❞❡s ❝✉♠✉❧❛t✐✈❛s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ s❡❣✉♥❞♦ t ❞❡ ❙t✉❞❡♥t✳
❆ ❞✐str✐❜✉✐çã♦ ❝✉♠✉❧❛t✐✈❛ t ❞❡ ❙t✉❞❡♥t ✭✈❡r ✜❣✉r❛ ✹✳✷✮ é ❞❛❞❛ ♣♦r✿
Φ (ϑ, n) =
Z ϑ
−∞
φ(x, n)dx=1
2 +sgn(ϑ) 1 2I ϑ2
n+ϑ2 1 2, n 2 ✭✹✳✹✮
❡♠ q✉❡ ■ é ❛ ❢✉♥çã♦ ❜❡t❛ ✐♥❝♦♠♣❧❡t❛ ✭✈❡r ❛♣ê♥❞✐❝❡ ❇✮✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❛ ✐♥t❡❣r❛çã♦ é ❛♣r❡s❡♥t❛❞❛ ♥♦ ❛♣ê♥❞✐❝❡ ❈✳
✹✳✷✳✷ ❆♣❧✐❝❛çã♦
❆❞♠✐t✐♥❞♦ q✉❡ ♦ ❡stí♠✉❧♦ ❞♦ ❝ér❡❜r♦ é ♣❡r✐ó❞✐❝♦ ❡ ❞♦ t✐♣♦ ❞❛ ✜❣✉r❛✹✳✸
♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ❞♦✐s ❡st❛❞♦s✿ ❛❝t✐✈❛çã♦ ❡ r❡♣♦✉s♦✳ P❛r❛ ❝❛❞❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ t♦♠❛♠♦s ❡♥tã♦ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ v ♦❜t✐❞♦s ❞✉r❛♥t❡ ♦ ❡stí♠✉❧♦ ❝♦♠♦ ♣❡rt❡♥❝❡♥t❡s ❛♦ ♣r✐♠❡✐r♦ ❡st❛❞♦ ❡
t♦❞♦s ♦s ♦❜t✐❞♦s ❞✉r❛♥t❡ ♦ r❡♣♦✉s♦ ❝♦♠♦ ♣❡rt❡♥❝❡♥t❡s ❛♦ s❡❣✉♥❞♦ ❡st❛❞♦✳ ❈❛❧❝✉❧❛♠♦s ❛ ♠é❞✐❛
vA ❡ ♦ ❞❡s✈✐♦ ♣❛❞rã♦σA♣❛r❛ ♦s ❞❛❞♦s ❞♦ ♣r✐♠❡✐r♦ ❡st❛❞♦ ❡ r❡♣❡t✐♠♦s ♣❛r❛ ♦s ❞❛❞♦s ❞♦ s❡❣✉♥❞♦ ❡st❛❞♦(vR, σR)✳
❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ♠é❞✐❛s ❞♦s ✈❛❧♦r❡s ♥♦s ❞♦✐s ❡st❛❞♦s t❡rá ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧❄ P♦❞❡✲s❡ ❛r❣✉♠❡♥t❛r q✉❡ ♥ã♦ ♣♦rq✉❡ ♦ s✐♥❛❧ ♥ã♦ é ✉♠❛ ❢✉♥çã♦ ❞❡ ❤❡❛✈②s✐❞❡✳ ➱ ❛♥t❡s s❡ q✉✐s❡r✲ ♠♦s✱ ✉♠❛ ❝♦♥✈♦❧✉çã♦ ❞♦ s✐♥❛❧ ❍❘❋ ❝♦♠ ✉♠❛ ❢✉♥çã♦ ❞❡ ❤❡❛✈②s✐❞❡✳ P♦r ✐ss♦ ❛ ❞✐str✐❜✉✐çã♦ ♣❛r❛ ♦s ❞❛❞♦s ♦r✐❣✐♥❛✐s t❡rá ❞♦✐s ♣✐❝♦s ✭✈❡r ❬✷✶❪✮✳ ❖✉ q✉❡ s❡ ♦ ❛tr❛s♦ ❢♦r ❛♣r❡❝✐á✈❡❧✱ ♦s ❞❛❞♦s ❡s❝♦❧❤✐❞♦s ♣❛r❛ ❝❛❞❛ ❡st❛❞♦ ❛✐♥❞❛ ✐♥❝❧✉❡♠ ♦s ❞♦ ❡st❛❞♦ ❛♥t❡r✐♦r✳
✷✹ ❈❆P❮❚❯▲❖ ✹✳ ❖ ❈➱❘❊❇❘❖ ❊ ❆ ❆◆➪▲■❙❊ ❋❯◆❈■❖◆❆▲
❋✐❣✉r❛ ✹✳✸✿ ❙❡q✉ê♥❝✐❛ ❞❡ ❡stí♠✉❧♦s✳
s❡ t✐✈❡r♠♦s ❡♠ ❝♦♥t❛ q✉❡ ♦s ♥í✈❡✐s ❞❡ r✉í❞♦ sã♦ ♠✉✐t❛s ✈❡③❡s s✉♣❡r✐♦r❡s ❛♦s ❞❛ ♠♦❞❡❧❛çã♦ ✐♥tr♦✲ ❞✉③✐❞❛ ♣❡❧❛ ❢✉♥çã♦ ❍❘❋✳
❙❡♥❞♦ ❛ss✐♠✱ ♦ ❈ér❡❜r♦ ♣❡r♠✐t❡ ❛♦ ✉t✐❧✐③❛❞♦r ❝♦♠♣❡♥s❛r ♣❛r❛ ❛tr❛s♦s ♠❛s ♥ã♦ ❞á ♣r❡s❡♥t❡♠❡♥t❡ ❤✐♣ót❡s❡ ❞❡ r❡t✐r❛r ✭❡✳❣✳ ♣♦r ❞❡❝♦♥✈♦❧✉çã♦ ❞❡ ❍❘❋✮ ❛ ♠♦❞❡❧❛çã♦ ❍❘❋✳
❖ ♣r♦❣r❛♠❛ ❝❛❧❝✉❧❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ❤✐♣ót❡s❡ ♥✉❧❛ s❡r ✈á❧✐❞❛(p)♣❛r❛ ❝❛❞❛ ✈♦①❡❧✳ ❖ r❡s✉❧t❛❞♦
é ❛♣r❡s❡♥t❛❞♦ s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ✈♦❧✉♠❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♠ ♦ ✈❛❧♦r ❞❡1−p♣❛r❛ ❝❛❞❛ ✈♦①❡❧ ❡
❣r❛✈á✈❡❧ ♥✉♠ ✜❝❤❡✐r♦ ❆♥❛❧②③❡ ❛ ♣❛rt✐r ❞❛ ❥❛♥❡❧❛ ❞❡ r❡s✉❧t❛❞♦✳
✹✳✸ ❖ ♠ét♦❞♦ ❞❡ ❝♦rr❡❧❛çã♦
✹✳✸✳✶ ❚❡♦r✐❛
❖ ♠ét♦❞♦ ❞❡ ❝♦rr❡❧❛çã♦ ♣❡r♠✐t❡✲♥♦s q✉❛♥t✐✜❝❛r s❡ ❡①✐st❡ ✉♠❛ r❡❧❛çã♦ ❧✐♥❡❛r ❡♥tr❡ ❞✉❛s ✈❛r✐á✈❡✐s✳ ❆ ♠❡❞✐❞❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ t❛❧ é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❧✐♥❡❛r (r)✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ❝♦♥❥✉♥t♦
❞❡n♣❛r❡s ❞❡ ❞❛❞♦s(ui, vi)✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❧✐♥❡❛r é ❞❛❞♦ ♣♦r✿
r=
Pn−1
i=0 (ui−u¯) (vi−v¯) q
Pn−1
i=0 (ui−u¯)2 q
Pn−1
i=0 (vi−¯v)2
✭✹✳✺✮
❆ss✉♠✐♥❞♦ q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡ v ❡ u é ❞♦ t✐♣♦ v = au+b✱ ♣♦❞❡♠♦s ❡st✐♠❛r ♣❡❧♦ ♠ét♦❞♦
❞♦s ♠í♥✐♠♦s q✉❛❞r❛❞♦s q✉❛✐s sã♦ ♦s ✈❛❧♦r❡s ❞❡ a ❡ b q✉❡ ♠❡❧❤♦r ❛❞❛♣t❛♠ ♦ ♠♦❞❡❧♦ ❛♦s ❞❛❞♦s
✭♠✐♥♦r❛♠ ❛ q✉❛♥t✐❞❛❞❡χ2✮✳
❊♠ s❡❣✉✐❞❛ ❢❛ç❛♠♦s ✉♠❛ ✐♥✈❡rsã♦ ❞♦ ♠♦❞❡❧♦✳ ❖✉ s❡❥❛✱ ✈❛♠♦s ❛ss✉♠✐r ❛❣♦r❛ q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡u❡v é ❞♦ t✐♣♦u=a′v+b′✳ ❙❡ r❡♣❡t✐r♠♦s ♦ ♠ét♦❞♦ ❞♦s ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♦❜t❡♠♦s a′ ❡ b′✳ P♦❞❡♠♦s r❡❢♦r♠✉❧❛r ♦ s❡❣✉♥❞♦ ♠♦❞❡❧♦ ♣❛r❛v= u
a′ −
b′
a′✳
❙❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ❞❛❞♦s ❢♦ss❡ ♣❡r❢❡✐t❛♠❡♥t❡ ❧✐♥❡❛r a❡ a1′ t❡r✐❛♠ ♦ ♠❡s♠♦ ✈❛❧♦r✳ ❖✉ s❡❥❛✱
a×a′ s❡r✐❛ ✐❣✉❛❧ ❛ ✶✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❧✐♥❡❛r é ❞❛❞♦ ♣♦r r=√aa′✳
P♦❞❡♠♦s ❝♦♥❝❧✉ír q✉❡ q✉❛♥❞♦ ❛ r❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ❣r❛♥❞❡③❛s é ♣❡r❢❡✐t❛♠❡♥t❡ ❧✐♥❡❛r ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ é ✉♥✐tár✐♦✳ ◗✉❛♥❞♦x❡y sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ré ♥✉❧♦✳
➱ ♣♦ssí✈❡❧ ❞❡♠♦♥str❛r q✉❡ ❛ q✉❛♥t✐❞❛❞❡✿
r
r
n−2
1−r2 ✭✹✳✻✮
✹✳✹✳ ❖ ▼➱❚❖❉❖ ▲■◆❊❆❘ ●❊◆❊❘❆▲■❩❆❉❖ ✷✺
❋✐❣✉r❛ ✹✳✹✿ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❡♠ ❢✉♥çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ♠♦❞❡❧♦✳
✹✳✸✳✷ ❆♣❧✐❝❛çã♦
◆❛ ❛♣❧✐❝❛çã♦ ❞❡st❡ ♠ét♦❞♦ ♠❡❞❡✲s❡ q✉❛❧ é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❧✐♥❡❛r ❡♥tr❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♠♦❞❡❧♦ ❞❡ ❞♦✐s ❡st❛❞♦s ❡ ❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❞❡ ❝❛❞❛ ✈♦①❡❧ ✭✜❣✉r❛ ✹✳✹✮✳ ❆ s❡q✉ê♥❝✐❛ ❞❡ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ❝♦♠ ❛tr❛s♦ ❡ ♣♦❞❡ t❡r três ❢♦r♠❛s ❞✐st✐♥t❛s✿ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❞❡s✐❣♥✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❞❡s✐❣♥ ❝♦♥✈♦❧✈✐❞❛ ❝♦♠ ❍❘❋ ♦✉ ✉♠❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ ❞❡ ♦✉tr♦ ✈♦①❡❧ à ❡s❝♦❧❤❛✮✳
❉❡t❡r♠✐♥❛✲s❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ❤✐♣ót❡s❡ ♥✉❧❛ s❡r ✈á❧✐❞❛ (p)♣❛r❛ ❝❛❞❛ ✈♦①❡❧✳ ❖ r❡s✉❧t❛❞♦ é
❛♣r❡s❡♥t❛❞♦ s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ✈♦❧✉♠❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♠ ♦ ✈❛❧♦r ❞❡ 1−p♣❛r❛ ❝❛❞❛ ✈♦①❡❧ ❡
❣r❛✈á✈❡❧ ♥✉♠ ✜❝❤❡✐r♦ ❆♥❛❧②③❡ ❛ ♣❛rt✐r ❞❛ ❥❛♥❡❧❛ ❞❡ r❡s✉❧t❛❞♦✳
✹✳✹ ❖ ♠ét♦❞♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦
✹✳✹✳✶ ❚❡♦r✐❛
❖ ♠ét♦❞♦ ❞♦ ♠♦❞❡❧♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦ ✭▼▲●✮ é ✉♠ ♠ét♦❞♦ ❡st❛tíst✐❝♦ q✉❡ s✉r❣❡ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡n♣❛r❡s ❞❡ ❞❛❞♦s(ti, vi)✱ ♥❛ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r q✉❛♥t✐✜❝❛♠✲s❡ ♦s ❝♦❡✜❝✐❡♥t❡sa0 ❡a1 ❞❛ ❡q✉❛çã♦✿
v(t) =a0·1 +a1·t ✭✹✳✼✮
❉✐t♦ ❞❡ ♦✉tr❛ ❢♦r♠❛ ❛ss✉♠✐♠♦s q✉❡ ❛ ✈❛r✐á✈❡❧ v é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛s ❢✉♥çõ❡s1❡t✳
✷✻ ❈❆P❮❚❯▲❖ ✹✳ ❖ ❈➱❘❊❇❘❖ ❊ ❆ ❆◆➪▲■❙❊ ❋❯◆❈■❖◆❆▲
v(t) =
m−1
X j=0
ajuj(t) ✭✹✳✽✮
❆s ❢✉♥çõ❡s uj(t)sã♦ ❞❡s✐❣♥❛❞❛s ❞❡ ❢✉♥çõ❡s ❞❡ ❜❛s❡ ❡ ♣♦❞❡♠ s❡r ♥ã♦ ❧✐♥❡❛r❡s✳
❚❛❧ ❝♦♠♦ ♥❛ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r ♣r♦❝✉r❛♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s q✉❡ ♠✐♥✐♠✐③❛♠ ❛ q✉❛♥t✐❞❛❞❡ χ2✿
χ2=
n−1
X i=0
"
vi−Pmj=0−1ajuj(ti)
σi
#
✭✹✳✾✮
➱ ♣♦ssí✈❡❧ ❞❡♠♦♥str❛r q✉❡ ♦ ♠í♥✐♠♦ ❞❡χ2 ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ♦s ❝♦❡✜❝❡♥t❡s sã♦✿
aj = m−1
X k=0
Cjk "n−1
X i=0
viuk(ti)
σ2
i #
✭✹✳✶✵✮
❡♠ q✉❡Cjk é ❛ ♠❛tr✐③ ❝♦✈❛r✐â♥❝✐❛ ❞♦s ♣❛râ♠❡tr♦s ✭✈❡r ❛♣ê♥❞✐❝❡ ❉✮✳
✹✳✹✳✷ ❆♣❧✐❝❛çã♦
❖ ❈ér❡❜r♦ ❛♣❧✐❝❛ ❡st❡ ♠ét♦❞♦ ❝❛❧❝✉❧❛♥❞♦ ♣❛r❛ ❝❛❞❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧ q✉❛✐s ♦s ❝♦❡✜❝✐❡♥t❡s aj q✉❡ s❛t✐s❢❛③❡♠ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿
vi= T X i=0
ajDij ✭✹✳✶✶✮
❡♠ q✉❡Dij é ♦ ❡❧❡♠❡♥t♦ ❞❛ ❧✐♥❤❛i❡ ❝♦❧✉♥❛j ❞❛ ♠❛tr✐③ ❞❡ ❞❡s✐❣♥✳
❖ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♦❜t✐❞♦ ♥❛ ❡q✳ ✹✳✶✶ é r❡s♦❧✈✐❞♦ ❡♠ ♦r❞❡♠ ❛♦s ❝♦❡✜❝✐❡♥t❡s aj ♥✉♠❡✲ r✐❝❛♠❡♥t❡ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❡❧✐♠✐♥❛çã♦ ❞❡ ●❛✉ss✲❏♦r❞❛♥✳ ❊st❡ ❝á❧❝✉❧♦ é ❢❡✐t♦ ♣❡❧❛ ❝❧❛ss❡ ●❛✉ss✲ ❏♦r❞❛♥❊❧✐♠✐♥❛t✐♦♥✳ ❊st❛ ❝❧❛ss❡ t❡♠ t❛♠❜é♠ ♦ ♠ét♦❞♦ ❣❡t❈♦✈❛r✐❛♥❝❡❆rr❛②✭✮ q✉❡ ❣❡r❛ ♦ ❛rr❛② ❜✐❞✐♠❡♥s✐♦♥❛❧ ❞❛ ♠❛tr✐③ ❝♦✈❛r✐â♥❝✐❛✳
❯♠❛ ✈❡③ ❝❛❧❝✉❧❛❞♦s ♦s ❝♦❡✜❝✐❡♥t❡s ♣❛r❛ ❝❛❞❛ ❡st❛❞♦✱ ♦ ✉t✐❧✐③❛❞♦r ❡s❝♦❧❤❡ ✉♠ ♣❛r❛ ❞❡ ❡st❛❞♦s ✭❡✳❣✳ R ❡A✮✳ ❖s ❝♦❡✜❝✐❡♥t❡s sã♦ ❝♦♠♣❛r❛❞♦s ♣❡❧♦ t❡st❡ t ❞❡ ❙t✉❞❡♥t✿
θ=√ aA−aR
CAA+CRR
✭✹✳✶✷✮
❖❜t❡♠♦s ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ❤✐♣ót❡s❡ ♥✉❧❛ s❡r ✈á❧✐❞❛ ✭p✮ ♣❛r❛ ❝❛❞❛ ✈♦①❡❧✳ ❖ r❡s✉❧t❛❞♦ é
❛♣r❡s❡♥t❛❞♦ s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ✈♦❧✉♠❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♠ ♦ ✈❛❧♦r ❞❡ 1−p♣❛r❛ ❝❛❞❛ ✈♦①❡❧ ❡
❣r❛✈á✈❡❧ ♥✉♠ ✜❝❤❡✐r♦ ❆♥❛❧②③❡ ❛ ♣❛rt✐r ❞❛ ❥❛♥❡❧❛ ❞❡ r❡s✉❧t❛❞♦✳
✹✳✺ ❖ ♠ét♦❞♦ ❞❡ ❋♦✉r✐❡r
✹✳✺✳✶ ❚❡♦r✐❛
❖ ♠ét♦❞♦ ❞❡ ❋♦✉r✐❡r ❜❛s❡✐❛✲s❡ ♥❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ ✉♠ s✐♥❛❧ ♣❡r✐ó❞✐❝♦✳ Pr❡t❡♥❞❡✲s❡ ❝♦♠♣❛r❛r ♦ ♠ó❞✉❧♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ♠♦❞❡❧♦✿
˜
u=|F(u)| ✭✹✳✶✸✮
❝♦♠ ♦ ♠ó❞✉❧♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ ❝❛❞❛ s❡q✉ê♥❝✐❛ t❡♠♣♦r❛❧✿
˜
v=|F(v)| ✭✹✳✶✹✮
