rt t rts - - PowerPoint PPT Presentation

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r t t r t s r

rt t rts - - PowerPoint PPT Presentation

Mar 21, 2024 152 likes •640 views

rt t rts r trs Ptrs t s

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◆♦♥✲❈❧❛✐r✈♦②❛♥t ❘❡❞✉❝t✐♦♥ ❆❧❣♦r✐t❤♠s ❢♦r ❍❡t❡r♦❣❡♥❡♦✉s P❧❛t❢♦r♠s

❆♥♥❡ ❇❡♥♦✐t✱ ▲♦✉✐s✲❈❧❛✉❞❡ ❈❛♥♦♥ ❛♥❞ ▲♦r✐s ▼❛r❝❤❛❧

❊◆❙ ▲②♦♥✱ ❯♥✐✈❡rs✐té ❞❡ ❋r❛♥❝❤❡✲❈♦♠té ❘♦♠❛ ✭▲■P✮✱ ❉■❙❈ ✭❋❊▼❚❖✲❙❚✮

❆✉❣✉st ✷✻✱ ✷✵✶✸

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶ ✴ ✸✷

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❘❡❞✉❝t✐♦♥ ❊①❛♠♣❧❡

①✐ ①✶ ①✷ ①✸ ①✹ ①✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷ ✴ ✸✷

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❘❡❞✉❝t✐♦♥ ❊①❛♠♣❧❡

①✐ ①✶ ①✷ ①✸ ①✹ ①✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸ ✴ ✸✷

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▼♦t✐✈❛t✐♦♥

❲❤❛t ❛r❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❡①✐st✐♥❣ ❛❧❣♦r✐t❤♠s ✇❤❡♥ tr❛♥s❢❡rs ❛♥❞ ❝♦♠♣✉t❛t✐♦♥s ♠❛② ♦✈❡r❧❛♣ ❛♥❞ ✇❤❡♥ ❝♦sts ❛r❡ ✉♥❦♦✇♥❄

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✹ ✴ ✸✷

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❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ▼♦❞❡❧s ❛♥❞ ❆❧❣♦r✐t❤♠s ❲♦rst✲❝❛s❡ ❆♥❛❧②s✐s ▼❛r❦♦✈ ❈❤❛✐♥ Pr♦❜❛❜✐❧✐st✐❝ ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✺ ✴ ✸✷

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❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ▼♦❞❡❧s ❛♥❞ ❆❧❣♦r✐t❤♠s ❲♦rst✲❝❛s❡ ❆♥❛❧②s✐s ▼❛r❦♦✈ ❈❤❛✐♥ Pr♦❜❛❜✐❧✐st✐❝ ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✻ ✴ ✸✷

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❖❜❥❡❝t✐✈❡

❘❡❞✉❝❡ ❡✣❝✐❡♥t❧② ♥ ❡❧❡♠❡♥ts t❤❛t ❛r❡ ❛✈❛✐❧❛❜❧❡ s✐♠✉❧t❛♥❡♦✉s❧② ✭❡❛❝❤ ♦♥ ❛ ❞✐st✐♥❝t ♣r♦❝❡ss♦r✮✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✼ ✴ ✸✷

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❊①❡❝✉t✐♦♥ ▼♦❞❡❧

◮ ❚❤❡ r❡❞✉❝t✐♦♥ ♦♣❡r❛t✐♦♥ ✐s ❛ss♦❝✐❛t✐✈❡✳ ◮ ❊❛❝❤ ♣r♦❝❡ss♦r ❝❛♥ ♣❡r❢♦r♠ ❛ r❡❞✉❝t✐♦♥ ❛♥❞ ❛♥② ♣❛✐r ♦❢

♣r♦❝❡ss♦rs ❝❛♥ ❝♦♠♠✉♥✐❝❛t❡✳

◮ ❆♥② ♣r♦❝❡ss♦r ✐s ✐♥✈♦❧✈❡❞ ✐♥ ❛t ♠♦st ♦♥❡ tr❛♥s❢❡r ❛t ❛♥② t✐♠❡✳ ◮ ❈♦♠♣✉t❛t✐♦♥ ❛♥❞ tr❛♥s❢❡r ❝❛♥ ♦✈❡r❧❛♣✳ ◮ ❈♦sts ❛r❡ ❤❡t❡r♦❣❡♥❡♦✉s ❛♥❞ t❤❡ s②st❡♠ ✐s ♥♦♥✲❝❧❛✐r✈♦②❛♥t

✭❞✐,❥ ❢♦r t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦st ❜❡t✇❡❡♥ ♣r♦❝❡ss♦r ✐ ❛♥❞ ❥ ❛♥❞ ❝✐ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ❝♦st ♦❢ ♣r♦❝❡ss♦r ✐✮✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✽ ✴ ✸✷

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❙❝❤❡❞✉❧✐♥❣

◮ ❊❛❝❤ ♣r♦❝❡ss♦r r❡❞✉❝❡s ❛❧❧ ✐ts r❡❝❡✐✈❡❞ ❡❧❡♠❡♥ts ✇✐t❤ ✐ts ♦✇♥

♣❛✐r✇✐s❡✳ ❚❤❡♥✱ ✐t s❡♥❞s t❤❡ r❡s✉❧t t♦ ❛♥♦t❤❡r ♣r♦❝❡ss♦r✳

◮ ❙t❛t✐❝ ♦r ❞②♥❛♠✐❝ ♣❧❛♥✳ ◮ ❚❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ str✉❝t✉r❡ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛

s♣❛♥♥✐♥❣ tr❡❡✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✾ ✴ ✸✷

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❙♣❛♥♥✐♥❣ ❚r❡❡ ❊①❛♠♣❧❡

P✶ P✷ P✸ P✹ P✺ P✶ P✷ P✸ P✹ P✺

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✵ ✴ ✸✷

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❘❡❧❛t❡❞ ❲♦r❦

▼P■ ❈♦♥t❡①t

❘❛❜❡♥s❡✐❢♥❡r ❡t ❛❧✳ Pr♦♣♦s❡ t❤❡ ❜✉tt❡r✢② ❛❧❣♦r✐t❤♠ ❢♦r ❛rr❛② r❡❞✉❝t✐♦♥s ✭❈♦♠♣✉t❛t✐♦♥❛❧ ❙❝✐❡♥❝❡✱ ✷✵✵✹✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✷✵✵✹ ❛♥❞ ■❏❍P❈❆✱ ✷✵✵✺✮✳ ❑✐❡❧♠❛♥♥ ❡t ❛❧✳ ❙tr❛t❡❣② ❢♦r ❤✐❡r❛r❝❤✐❝❛❧ ♣❧❛t❢♦r♠s ✭▼P■❉❈✱ ✶✾✾✾✮✳ ❇♦s✐❧❝❛ ❡t ❛❧✳ ❊♠♣✐r✐❝❛❧ ❛♥❞ ❛♥❛❧②t✐❝❛❧ ❝♦♠♣❛r✐s♦♥ ♦❢ s❡✈❡r❛❧ ❡①✐st✐♥❣ ❛❧❣♦r✐t❤♠s ✭❈❧✉st❡r ❈♦♠♣✉t✐♥❣✱ ✷✵✵✼✮✳

▼❛♣❘❡❞✉❝❡ ❈♦♥t❡①t

❆❣❛r✇❛❧ ❡t ❛❧✳ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❆❧❧❘❡❞✉❝❡ ✉s✐♥❣ ❛ s♣❛♥♥✐♥❣ tr❡❡ ✭s✉❜♠✐tt❡❞✮✳ ❙❛♥❞✐❛ ◆❛t✐♦♥❛❧ ▲❛❜♦r❛t♦r✐❡s ▼❛♣❘❡❞✉❝❡✕▼P■ ✐♠♣❧❡♠❡♥ts ▼❛♣❘❡❞✉❝❡ ♦♥ t♦♣ ♦❢ ▼P■✳ ❍♦❡✢❡r✱ ❉♦♥❣❛rr❛ ❡t ❛❧✳ ❍✐❣❤❧✐❣❤t t❤❡ ❜❡♥❡✜ts ♦❢ ▼P■✲✸ ❢❡❛t✉r❡s ❢♦r ▼❛♣❘❡❞✉❝❡ ❛♣♣❧✐❝❛t✐♦♥s ✭❊✉r♦P❱▼✴▼P■ ✷✵✵✾✮✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✶ ✴ ✸✷

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❆❧❣♦r✐t❤♠s ❖✈❡r✈✐❡✇

❇✐♥♦♠✐❛❧✲st❛t ❛♥❞ ❋✐❜♦♥❛❝❝✐✲st❛t str❛t❡❣✐❡s✶

❙t❛t✐❝ ❛♣♣r♦❛❝❤❡s✱ ♦♣t✐♠❛❧ ✇✐t❤ s♣❡❝✐✜❝ ❝♦♠♣✉t❛t✐♦♥ ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦sts✳

❉②♥❛♠✐❝ str❛t❡❣✐❡s

❚r❡❡✲❞②♥ ❆♥② ❝♦♠♠✉♥✐❝❛t✐♦♥ st❛rts ❛s s♦♦♥ ❛s ♣♦ss✐❜❧❡✳ ◆♦♥✲❈♦♠♠✉t✲❚r❡❡✲❞②♥ ❆❞❛♣t❛t✐♦♥ ❢♦r ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♣❡r❛t✐♦♥s✳

✶❈❛♥♦♥✱ ✷✵✶✸ ❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✷ ✴ ✸✷

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❇✐♥♦♠✐❛❧✲st❛t

❉❡✜♥✐t✐♦♥

◮ ❆ ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ✵ ✐s ❛ s✐♥❣❧❡ ♥♦❞❡✳ ◮ ❆ ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ ❝♦♥s✐sts ♦❢ ❛

❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶ ✇❤♦s❡ s✐♥❦ ✐s t❤❡ ♣❛r❡♥t ♦❢ t❤❡ s✐♥❦ ♦❢ ❛♥♦t❤❡r ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶✳ ▼✶ ▼✷ ▼✸ ▼✹

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✸ ✴ ✸✷

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❇✐♥♦♠✐❛❧✲st❛t

❉❡✜♥✐t✐♦♥

◮ ❆ ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ✵ ✐s ❛ s✐♥❣❧❡ ♥♦❞❡✳ ◮ ❆ ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ ❝♦♥s✐sts ♦❢ ❛

❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶ ✇❤♦s❡ s✐♥❦ ✐s t❤❡ ♣❛r❡♥t ♦❢ t❤❡ s✐♥❦ ♦❢ ❛♥♦t❤❡r ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶✳ ▼✶ ▼✷ ▼✸ ▼✹

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✸ ✴ ✸✷

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❇✐♥♦♠✐❛❧✲st❛t

❉❡✜♥✐t✐♦♥

◮ ❆ ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ✵ ✐s ❛ s✐♥❣❧❡ ♥♦❞❡✳ ◮ ❆ ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ ❝♦♥s✐sts ♦❢ ❛

❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶ ✇❤♦s❡ s✐♥❦ ✐s t❤❡ ♣❛r❡♥t ♦❢ t❤❡ s✐♥❦ ♦❢ ❛♥♦t❤❡r ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶✳ ▼✶ ▼✷ ▼✸ ▼✹

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✸ ✴ ✸✷

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❇✐♥♦♠✐❛❧✲st❛t

❉❡✜♥✐t✐♦♥

◮ ❆ ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ✵ ✐s ❛ s✐♥❣❧❡ ♥♦❞❡✳ ◮ ❆ ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ ❝♦♥s✐sts ♦❢ ❛

❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶ ✇❤♦s❡ s✐♥❦ ✐s t❤❡ ♣❛r❡♥t ♦❢ t❤❡ s✐♥❦ ♦❢ ❛♥♦t❤❡r ❜✐♥♦♠✐❛❧ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶✳ ▼✶ ▼✷ ▼✸ ▼✹

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✸ ✴ ✸✷

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SLIDE 26

❋✐❜♦♥❛❝❝✐✲st❛t

❉❡✜♥✐t✐♦♥

◮ ❆ ❦✲❛r② ❋✐❜♦♥❛❝❝✐ tr❡❡ ♦❢ ♦r❞❡r ✵ ♦r ✶ ✐s ❛ s✐♥❣❧❡ ♥♦❞❡✳ ◮ ❆ ❦✲❛r② ❋✐❜♦♥❛❝❝✐ tr❡❡ ♦❢ ♦r❞❡r ❦ ❝♦♥s✐sts ♦❢ ❛ ❦✲❛r② ❋✐❜♦♥❛❝❝✐

tr❡❡ ♦❢ ♦r❞❡r ❦ − ✷ ✇❤♦s❡ s✐♥❦ ✐s t❤❡ ♣❛r❡♥t ♦❢ t❤❡ s✐♥❦ ♦❢ ❛♥♦t❤❡r ❦✲❛r② ❋✐❜♦♥❛❝❝✐ tr❡❡ ♦❢ ♦r❞❡r ❦ − ✶✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✹ ✴ ✸✷

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SLIDE 27

❊①❛♠♣❧❡s ♦❢ ❦✲❛r② ❋✐❜♦♥❛❝❝✐ ❚r❡❡

▼✶ ▼✶ ▼✶ ▼✷ ▼✶ ▼✷ ▼✸ ▼✶ ▼✷ ▼✸ ▼✹ ▼✺ ▼✶ ▼✷ ▼✸ ▼✹ ▼✺ ▼✻ ▼✼ ▼✽

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✺ ✴ ✸✷

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SLIDE 28

❊①❡❝✉t✐♦♥ ❊①❛♠♣❧❡ ✇✐t❤ ❋✐❜♦♥❛❝❝✐✲st❛t

P✶ P✷ P✸ P✹ P✺ P✻ P✼ P✽ P✶ P✷ P✸ P✹ P✺ P✻ P✼ P✽

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✻ ✴ ✸✷

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SLIDE 29

❉②♥❛♠✐❝ ❆♣♣r♦❛❝❤❡s

❚r❡❡✲❞②♥

❆♥② ♣r♦❝❡ss♦r ❜❡❝♦♠✐♥❣ ✐❞❧❡ ✭✐♥✐t✐❛❧❧② ♦r ❛❢t❡r ❛ r❡❞✉❝t✐♦♥✮ s❡♥❞s ✐ts ❡❧❡♠❡♥t t♦ ❛♥② ♦t❤❡r ✐❞❧❡ ♣r♦❝❡ss♦r✳ ■❢ t❤❡r❡ ✐s ♥♦♥❡✱ ✐t ✇❛✐ts✳

◆♦♥✲❈♦♠♠✉t✲❚r❡❡✲❞②♥

■♥ ❛❞❞✐t✐♦♥ t♦ ❜❡✐♥❣ ✐❞❧❡✱ t❤❡ t❛r❣❡t❡❞ ♣r♦❝❡ss♦r ✇✐t❤ ❚r❡❡✲❞②♥ ♠✉st ❜❡ ❛ ✏♥❡✐❣❤❜♦r✑ ✭s✉❝❤ t❤❛t t❤✐s ♣r♦❝❡ss♦r ❝❛♥ t❤❡♥ ♣❡r❢♦r♠ ❛ r❡❞✉❝t✐♦♥ ✇✐t❤ t❤❡ r❡❝❡✐✈❡❞ ❡❧❡♠❡♥t✮✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✼ ✴ ✸✷

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SLIDE 30

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ▼♦❞❡❧s ❛♥❞ ❆❧❣♦r✐t❤♠s ❲♦rst✲❝❛s❡ ❆♥❛❧②s✐s ▼❛r❦♦✈ ❈❤❛✐♥ Pr♦❜❛❜✐❧✐st✐❝ ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✽ ✴ ✸✷

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SLIDE 31

❙✉♠♠❛r②

❝ = ✵ ❛♥② ❝ ❝ = ❞ ❇✐♥♦♠✐❛❧✲st❛t ∆ ∆ + ✶ (∆ + ✶)(✶ + ✶ ❧♦❣✷ ♥) ❧♦❣✷ ϕ ❚r❡❡✲❞②♥ ∆ ∆ + ✶ (∆ + ✶)(✶ + ✶ ❧♦❣✷ ♥) ❧♦❣✷ ϕ ❋✐❜♦♥❛❝❝✐✲st❛t ∆ ❧♦❣✷ ϕ + ∆ ⌈❧♦❣✷ ♥⌉ ∆ ❧♦❣✷ ϕ + ✷∆ ⌈❧♦❣✷ ♥⌉ ∆

✇✐t❤ ∆ = ♠❛①✐,❥ ❞✐,❥

♠✐♥✐,❥ ❞✐,❥ ✱ ❧♦❣✷ ϕ ≈ ✵.✻✾ ❛♥❞ ✶ ❧♦❣✷ ϕ ≈ ✶.✹✹✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✶✾ ✴ ✸✷

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P❡rs♣❡❝t✐✈❡s

◮ ■♠♣r♦✈❡ ♦r s✐♠♣❧✐❢② t❤❡ r❛t✐♦s✳ ◮ ❆♥❛❧②s❡ ◆♦♥✲❈♦♠♠✉t✲❚r❡❡✲❞②♥✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✵ ✴ ✸✷

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SLIDE 33

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ▼♦❞❡❧s ❛♥❞ ❆❧❣♦r✐t❤♠s ❲♦rst✲❝❛s❡ ❆♥❛❧②s✐s ▼❛r❦♦✈ ❈❤❛✐♥ Pr♦❜❛❜✐❧✐st✐❝ ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✶ ✴ ✸✷

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SLIDE 34

❚r❡❡✲❞②♥ ❆♥❛❧②s✐s

▼❡t❤♦❞

◮ ▼♦❞❡❧ t❤❡ ❛❧❣♦r✐t❤♠ ❛s ❛ ♠❡♠♦r②✲❧❡ss ♣r♦❝❡ss✳ ◮ ❆♣♣❧② t❤❡ ✜rst✲st❡♣ ❛♥❛❧②s✐s✳

❘❡s✉❧ts ✇✐t❤ ♥♦ ❝♦♠♣✉t❛t✐♦♥

❆✈❡r❛❣❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡✿

✶ λ❞

✷❍

♥

✷ −✶

+ ✷

♥

✇❤❡r❡ ❍(♥) ✐s t❤❡

♥t❤ ❤❛r♠♦♥✐❝ ♥✉♠❜❡r ❛♥❞ λ❞ ✐s t❤❡ r❛t❡ ♦❢ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦st✳ ❱❛r✐❛♥❝❡ ♦❢ t❤❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡✿

✶ λ✷

❞

✷ ♥

✷ −✶

✐=✶ ✶ ✐✷ + ✹ ♥✷

✳

❉r❛✇❜❛❝❦

❖♥❧② ❚r❡❡✲❞②♥ ❝♦✉❧❞ ❜❡ ❝♦♠♣❧❡t❡❧② ❛♥❛❧②s❡❞✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✷ ✴ ✸✷

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❚r❡❡✲❞②♥ ❆♥❛❧②s✐s

▼❡t❤♦❞

◮ ▼♦❞❡❧ t❤❡ ❛❧❣♦r✐t❤♠ ❛s ❛ ♠❡♠♦r②✲❧❡ss ♣r♦❝❡ss✳ ◮ ❆♣♣❧② t❤❡ ✜rst✲st❡♣ ❛♥❛❧②s✐s✳

❘❡s✉❧ts ✇✐t❤ ♥♦ ❝♦♠♣✉t❛t✐♦♥

❆✈❡r❛❣❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡✿

✶ λ❞

✷❍

♥

✷ −✶

+ ✷

♥

✇❤❡r❡ ❍(♥) ✐s t❤❡

♥t❤ ❤❛r♠♦♥✐❝ ♥✉♠❜❡r ❛♥❞ λ❞ ✐s t❤❡ r❛t❡ ♦❢ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦st✳ ❱❛r✐❛♥❝❡ ♦❢ t❤❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡✿

✶ λ✷

❞

✷ ♥

✷ −✶

✐=✶ ✶ ✐✷ + ✹ ♥✷

✳

❉r❛✇❜❛❝❦

❖♥❧② ❚r❡❡✲❞②♥ ❝♦✉❧❞ ❜❡ ❝♦♠♣❧❡t❡❧② ❛♥❛❧②s❡❞✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✷ ✴ ✸✷

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❚r❡❡✲❞②♥ ❆♥❛❧②s✐s

▼❡t❤♦❞

◮ ▼♦❞❡❧ t❤❡ ❛❧❣♦r✐t❤♠ ❛s ❛ ♠❡♠♦r②✲❧❡ss ♣r♦❝❡ss✳ ◮ ❆♣♣❧② t❤❡ ✜rst✲st❡♣ ❛♥❛❧②s✐s✳

❘❡s✉❧ts ✇✐t❤ ♥♦ ❝♦♠♣✉t❛t✐♦♥

❆✈❡r❛❣❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡✿

✶ λ❞

✷❍

♥

✷ −✶

+ ✷

♥

✇❤❡r❡ ❍(♥) ✐s t❤❡

♥t❤ ❤❛r♠♦♥✐❝ ♥✉♠❜❡r ❛♥❞ λ❞ ✐s t❤❡ r❛t❡ ♦❢ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦st✳ ❱❛r✐❛♥❝❡ ♦❢ t❤❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡✿

✶ λ✷

❞

✷ ♥

✷ −✶

✐=✶ ✶ ✐✷ + ✹ ♥✷

✳

❉r❛✇❜❛❝❦

❖♥❧② ❚r❡❡✲❞②♥ ❝♦✉❧❞ ❜❡ ❝♦♠♣❧❡t❡❧② ❛♥❛❧②s❡❞✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✷ ✴ ✸✷

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❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ▼♦❞❡❧s ❛♥❞ ❆❧❣♦r✐t❤♠s ❲♦rst✲❝❛s❡ ❆♥❛❧②s✐s ▼❛r❦♦✈ ❈❤❛✐♥ Pr♦❜❛❜✐❧✐st✐❝ ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✸ ✴ ✸✷

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❈♦st ▼♦❞❡❧

◮ ❯s❡ ♦❢ ♠✐①t✉r❡ ♦❢ ❣❛♠♠❛ ❞✐str✐❜✉t✐♦♥s ✭✇✐t❤ ✻ ♣❛r❛♠❡t❡rs✮ ❢♦r

♠♦❞❡❧❧✐♥❣ r✉♥t✐♠❡s✷✳

◮ ❚r❛❞❡✲♦✛ ❜❡t✇❡❡♥ t❤❡ ♣r❡❝✐s✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ❛♥❞ t❤❡ ♥✉♠❜❡r

♦❢ ♣❛r❛♠❡t❡rs ✭r✐s❦ ♦❢ ♦✈❡r✜tt✐♥❣ ❛♥❞ ✐♥tr❛❝t❛❜✐❧✐t②✮✳

◮ ●❛♠♠❛ ❞✐str✐❜✉t✐♦♥ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st r❡❧❡✈❛♥t ✇✐t❤ ✷

♣❛r❛♠❡t❡rs ✭♣♦s✐t✐✈❡ ❛♥❞ ❛❞❥✉st❛❜❧❡ ♠❡❛♥ ❛♥❞ ✈❛r✐❛♥❝❡✮✳

✷▲✉❜❧✐♥ ❛♥❞ ❋❡✐t❡❧s♦♥✱ ✷✵✵✸ ❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✹ ✴ ✸✷

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SLIDE 39

▼❈ ❙✐♠✉❧❛t✐♦♥

◮ ❈♦♠♠✉♥✐❝❛t✐♦♥ ✭r❡s♣✳✱ ❝♦♠♣✉t❛t✐♦♥✮ ❝♦sts ❢♦❧❧♦✇ ❛ ❣❛♠♠❛

❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♠❡❛♥ µ❞ ✭r❡s♣✳✱ µ❝✮ ❛♥❞ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ σ❞ ✭r❡s♣✳✱ σ❝✮✳

◮ ❖♥❡ ▼♦♥t❡ ❈❛r❧♦ ✭▼❈✮ s✐♠✉❧❛t✐♦♥ ❝♦♥s✐sts ✐♥ ✐♥st❛♥t✐❛t✐♥❣ ❛❧❧

t❤❡ ❝♦sts ❢♦r ❛ ❣✐✈❡♥ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♥ ❛♥❞ t♦ r✉♥ ❛♥ r❡❞✉❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✇✐t❤ t❤❡s❡ ❝♦sts✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✺ ✴ ✸✷

slide-40

SLIDE 40

❳P ❉❡s✐❣♥

❙✉♠♠❛r②✿

◮ ✺ ♣❛r❛♠❡t❡rs✿ ♥✱ µ❞✱ µ❝✱ σ❞ ❛♥❞ σ❝✳ ◮ ✹ ♠❡t❤♦❞s✿ ❇✐♥♦♠✐❛❧✲st❛t✱ ❋✐❜♦♥❛❝❝✐✲st❛t✱ ❚r❡❡✲❞②♥ ❛♥❞

◆♦♥✲❈♦♠♠✉t✲❚r❡❡✲❞②♥✳ ❚❤r❡❡ ♣❧❛♥s✿ ✶✳ ❚❡st ❛❧❧ ♠❡t❤♦❞s ✇✐t❤ ♥♦ ❝♦♠♣✉t❛t✐♦♥ ❝♦st ❛♥❞ ❞✐✛❡r❡♥t ❈♦❡✣❝✐❡♥ts ♦❢ ❱❛r✐❛t✐♦♥ ✭❈❱✮ ❢♦r t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ✭♥ = ✻✹✱ µ❞ = ✶✱ ✶ ✵✵✵ ✵✵✵ ▼❈ s✐♠✉❧❛t✐♦♥s✮✳ ✷✳ ❚❡st t❤❡ ✐♠♣r♦✈❡♠❡♥t ♦❢ ❚r❡❡✲❞②♥ ♦✈❡r ❋✐❜♦♥❛❝❝✐✲st❛t ✇✐t❤ ✈❛r②✐♥❣ ♦✈❡r❧❛♣♣✐♥❣ ✭✐✳❡✳✱ ❝

❞ ✮ ❛♥❞ ❞✐✛❡r❡♥t ❈❱ ✭ σ❝ µ❝ = σ❞ µ❞ ✮✳

✸✳ ❉❡t❡r♠✐♥❡ t❤❡ ❜❡st ♠❡t❤♦❞ ❛♠♦♥❣ t❤♦s❡ t❤❛t s✉♣♣♦rt ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♣❡r❛t✐♦♥ ✭s❛♠❡ s❡tt✐♥❣s ❛s ♣r❡✈✐♦✉s❧②✮✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✻ ✴ ✸✷

slide-41

SLIDE 41

❳P ❉❡s✐❣♥

❙✉♠♠❛r②✿

◮ ✺ ♣❛r❛♠❡t❡rs✿ ♥✱ µ❞✱ µ❝✱ σ❞ ❛♥❞ σ❝✳ ◮ ✹ ♠❡t❤♦❞s✿ ❇✐♥♦♠✐❛❧✲st❛t✱ ❋✐❜♦♥❛❝❝✐✲st❛t✱ ❚r❡❡✲❞②♥ ❛♥❞

◆♦♥✲❈♦♠♠✉t✲❚r❡❡✲❞②♥✳ ❚❤r❡❡ ♣❧❛♥s✿ ✶✳ ❚❡st ❛❧❧ ♠❡t❤♦❞s ✇✐t❤ ♥♦ ❝♦♠♣✉t❛t✐♦♥ ❝♦st ❛♥❞ ❞✐✛❡r❡♥t ❈♦❡✣❝✐❡♥ts ♦❢ ❱❛r✐❛t✐♦♥ ✭❈❱✮ ❢♦r t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ✭♥ = ✻✹✱ µ❞ = ✶✱ ✶ ✵✵✵ ✵✵✵ ▼❈ s✐♠✉❧❛t✐♦♥s✮✳ ✷✳ ❚❡st t❤❡ ✐♠♣r♦✈❡♠❡♥t ♦❢ ❚r❡❡✲❞②♥ ♦✈❡r ❋✐❜♦♥❛❝❝✐✲st❛t ✇✐t❤ ✈❛r②✐♥❣ ♦✈❡r❧❛♣♣✐♥❣ ✭✐✳❡✳✱ ❝

❞ ✮ ❛♥❞ ❞✐✛❡r❡♥t ❈❱ ✭ σ❝ µ❝ = σ❞ µ❞ ✮✳

✸✳ ❉❡t❡r♠✐♥❡ t❤❡ ❜❡st ♠❡t❤♦❞ ❛♠♦♥❣ t❤♦s❡ t❤❛t s✉♣♣♦rt ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♣❡r❛t✐♦♥ ✭s❛♠❡ s❡tt✐♥❣s ❛s ♣r❡✈✐♦✉s❧②✮✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✻ ✴ ✸✷

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SLIDE 42

❈♦st ❉✐s♣❡rs✐♦♥ ❊✛❡❝t

10 20 50 0.10 1.00 10.00 0.01

Coefficient of variation Schedule length

Method Binomial−stat Fibonacci−stat Tree−dyn Non−Commut−Tree−dyn

Effect of the CV on the performances

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✼ ✴ ✸✷

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SLIDE 43

◆♦♥✲◆❡❣❧✐❣✐❜❧❡ ❈♦♠♣✉t❛t✐♦♥

0.9 1 1.1 1.1 1.2 1.2 1.2 1.2

10% 20% 50% 100% 200% 500% 1,000% 0.01 0.10 1.00 10.00

Coefficient of variation Proportion of computation costs

0.9 1.0 1.1 1.2 Performance ratio

Effect of the CV and computation costs

n the Fibonacci−stat performance

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✽ ✴ ✸✷

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SLIDE 44

◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ❖♣❡r❛t✐♦♥

10% 20% 50% 100% 200% 500% 1,000% 0.01 0.10 1.00 10.00

Coefficient of variation Proportion of computation costs

Method Binomial− stat Fibonacci− stat Non−Commut− Tree−dyn

Expected best method with varying overlapping and variability

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✷✾ ✴ ✸✷

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SLIDE 45

❆♥❛❧②s✐s ❘♦❜✉st♥❡ss

❈♦♥s✐st❡♥t ❝♦♥❝❧✉s✐♦♥s ✇✐t❤ ❛ ❇❡r♥♦✉❧❧✐ ❞✐str✐❜✉t✐♦♥✳

0.0 0.2 0.4 0.6 0.8 1.0 1 2 5 10 20

Ratio of the Fibonacci−stat to the Tree−dyn performance

c/d D/d 0.9 1.0 1.1 1.2 1.3 1.4

. 9 1 1 . 1 1.2 1.3

0.0 0.2 0.4 0.6 0.8 1.0 1 2 5 10 20

Expected best method with varying overlapping and variability

c/d D/d Binomial−stat Fibonacci−stat Non−Commut−Tree−dyn

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸✵ ✴ ✸✷

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SLIDE 46

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ▼♦❞❡❧s ❛♥❞ ❆❧❣♦r✐t❤♠s ❲♦rst✲❝❛s❡ ❆♥❛❧②s✐s ▼❛r❦♦✈ ❈❤❛✐♥ Pr♦❜❛❜✐❧✐st✐❝ ❙✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸✶ ✴ ✸✷

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SLIDE 47

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❉✐r❡❝t✐♦♥s

❈♦♥❝❧✉s✐♦♥

◮ ❘❡❞✉❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ❞✐str✐❜✉t❡❞ ❡❧❡♠❡♥ts ✭▼P■✱ ▼❛♣❘❡❞✉❝❡✮✳ ◮ ❆♥❛❧②s✐s ♦❢ ❢♦✉r str❛t❡❣✐❡s ✭t✇♦ st❛t✐❝ ❛♥❞ t✇♦ ❞②♥❛♠✐❝✮✳ ◮ ❚❤r❡❡ ❞✐✛❡r❡♥t ❛♥❛❧②s✐s✿ ✇♦rst✲❝❛s❡✱ ▼❛r❦♦✈ ❝❤❛✐♥ ❛♥❞

♣r♦❜❛❜✐❧✐st✐❝✳

❋✉t✉r❡ ❉✐r❡❝t✐♦♥s

◮ ▼♦❞❡❧ ❡①t❡♥s✐♦♥s ✭❛rr✐✈❛❧ ❞❛t❡s✱ ♥❡t✇♦r❦ t♦♣♦❧♦❣✐❡s✱ ❛rr❛②

r❡❞✉❝t✐♦♥s✮✳

◮ ■♥tr♦❞✉❝❡ s♦♠❡ ❧✐♠✐t❡❞ ❝❧❛✐r✈♦②❛♥❝❡ ✭s✉❝❤ ❛s ✉♥❝❡rt❛✐♥

♣r❡❞✐❝t✐♦♥s✮✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸✷ ✴ ✸✷

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SLIDE 48

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❉✐r❡❝t✐♦♥s

❈♦♥❝❧✉s✐♦♥

◮ ❘❡❞✉❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ❞✐str✐❜✉t❡❞ ❡❧❡♠❡♥ts ✭▼P■✱ ▼❛♣❘❡❞✉❝❡✮✳ ◮ ❆♥❛❧②s✐s ♦❢ ❢♦✉r str❛t❡❣✐❡s ✭t✇♦ st❛t✐❝ ❛♥❞ t✇♦ ❞②♥❛♠✐❝✮✳ ◮ ❚❤r❡❡ ❞✐✛❡r❡♥t ❛♥❛❧②s✐s✿ ✇♦rst✲❝❛s❡✱ ▼❛r❦♦✈ ❝❤❛✐♥ ❛♥❞

♣r♦❜❛❜✐❧✐st✐❝✳

❋✉t✉r❡ ❉✐r❡❝t✐♦♥s

◮ ▼♦❞❡❧ ❡①t❡♥s✐♦♥s ✭❛rr✐✈❛❧ ❞❛t❡s✱ ♥❡t✇♦r❦ t♦♣♦❧♦❣✐❡s✱ ❛rr❛②

r❡❞✉❝t✐♦♥s✮✳

◮ ■♥tr♦❞✉❝❡ s♦♠❡ ❧✐♠✐t❡❞ ❝❧❛✐r✈♦②❛♥❝❡ ✭s✉❝❤ ❛s ✉♥❝❡rt❛✐♥

♣r❡❞✐❝t✐♦♥s✮✳

❇❡♥♦✐t ✕ ❈❛♥♦♥ ✕ ▼❛r❝❤❛❧ ✸✷ ✴ ✸✷