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Jan 08, 2023 482 likes •950 views

s s r t tt rt r ss r tt rt r

SLIDE 1

❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ▼❛r❝❤ ✻✱ ✷✵✶✷

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 2

❋❡✇ r❡♠✐♥❞❡rs ❛❜♦✉t ❞✐str✐❜✉t❡❞ ❑✲▼❡❛♥s

❑▼❡❛♥s ❙❡q✉❡♥t✐❛❧ ❚✐♠❡ = (✸❑♥❞ + ❑♥ + ❑❞ + ♥❞) ∗ ■ ∗ ❚ ✢♦♣ ≃ ✸❑♥❞ ∗ ■ ∗ ❚ ✢♦♣

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 3

❙▼P ❈♦st

❑▼❡❛♥s ❙▼P ❉✐str✐❜✉t❡❞ ❈♦st = ❚ ❝♦♠♣

P

= (✸❑♥❞ + ❑♥ + ❑❞ + ♥❞) ∗ ■ ∗ ❚ ✢♦♣ P ≃ ✸❑♥❞ ∗ ■ ∗ ❚ ✢♦♣ P

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 4

❉▼▼ ❈♦st

❑▼❡❛♥s ❉▼▼ ❈♦st = ❚ ❝♦♠♣

P

+ ❚ ❝♦♠♠

P

= (✸❑♥❞ + ❑♥ + ❑❞ + ♥❞) ∗ ■ ∗ ❚ ✢♦♣ P + ❚ ❝♦♠♠

P

≃ ✸❑♥❞ ∗ ■ ∗ ❚ ✢♦♣ P + ❖(❧♦❣(P))

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 5

Worker 1 Worker 2

Centroids local version Centroids local version

Worker 3 Worker 4

Centroids local version Centroids local version

Worker 2

Centroids partial version

Worker 4

Centroids partial version

merge into merge into

merge into

Worker 4

Centroids aggregated version

❋✐❣✉r❡✿ ▼❡r❣✐♥❣ ❝❡♥tr♦ï❞s ❧♦❣✐❝

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 6

❝❧❛ss✐❝❛❧ ✐ss✉❡ ✿ ♣✐♣❡s

❚❤❡ s✐③❡ ♦❢ t❤❡ ❜✐❣❣❡st ♣✐♣❡ ✭♦♥ t♦♣ ♦❢ t❤❡ tr❡❡✮ ❛♥❞ t❤❡ ❛❣❣r❡❣❛t❡❞ ❜❛♥❞✇✐❞t❤✳

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 7

♠♦❞❡❧✐s❛t✐♦♥ ♦❢ ❚ ❝♦♠♠

P

✐♥ ❉▼▼

❚ ❝♦♠♠

P

=

⌈❧♦❣✷(P)⌉

♣=✶

(✷ ■❑❞ ∗ ❙♦❉ ❇❉▼▼,▼P■

P ✷♣

+ ✺■❑❞ ∗ ❚ ✢♦♣)

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 8

♠♦❞❡❧✐s❛t✐♦♥ ♦❢ ❚ ❝♦♠♠

P

✐♥ ❉▼▼ ✭✷✮

❚ ❝♦♠♠

P

= (✷ ■❑❞ ∗ ❙♦❉ ❇P,❛✈❣ ❉▼▼,▼P■ + ✺■❑❞ ∗ ❚ ✢♦♣) ∗ ⌈❧♦❣✷(P)⌉

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 9

❙♣❡❡❞❯♣

❙♣❡❡❞✉♣ = ❑▼❡❛♥s ❙❡q✉❡♥t✐❛❧ ❈♦st ❑▼❡❛♥s❉▼▼❉✐str✐❜✉t❡❞❈♦st = ❑▼❡❛♥s❙❡q✉❡♥t✐❛❧❈♦st ❚ ❝♦♠♣

P

+ ❚ ❝♦♠♠

P

= ✸♥❑❞■❚ ✢♦♣

✸♥❑❞■❚ ✢♦♣ P

+

✷ ■❑❞∗❙♦❉

❇❉▼▼,▼P■

+ ✺■❑❞❚ ✢♦♣

⌈❧♦❣✷(P)⌉

= ✸♥❚ ✢♦♣

✸♥❚ ✢♦♣ P

+

❙♦❉ ❇❉▼▼,▼P■

+ ✺❚ ✢♦♣

⌈❧♦❣✷(P)⌉

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 10

❙♦♠❡ ❝♦♠♠❡♥ts

◮ ❘❆▼ ✐s ♥♦t ♠♦❞❡❧✐③❡ ⇒ s♣❡❡❞ ✉♣ ♠✐❣❤t ❜❡ ❣r❡❛t❡r ◮ s♣❡❡❞✉♣ ✐♥❞❡♣❡♥❞❛♥t ♦❢ ❑✱ ■ ❛♥❞ ❞ ◮ ✇✐♥♥❡r ♣❛♣❡r ♠♦❞❡❧ ✦

❧✐♠

♥→+∞ ❙♣❡❡❞❯♣ = P

✭✶✮ ✭✷✮

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 11

❋✐rst ❝❛s❡ st✉❞② ✿ ❊❉❋

◮ ♥ ❂ ✷✵ ✵✵✵ ✵✵✵ t✐♠❡ s❡r✐❡s ◮ ❞ ❂ ✷✹ ❤♦✉rs ✯ ✸✻✺ ❞❛②s ✯ ✶✵ ②❡❛rs ❂ ✽✼✻✵✵ ✈❛❧✉❡s ♣❡r s❡r✐❡s ◮ ❑ ❂ √♥ ❂ ✹✹✼✷ ❝❧✉st❡rs ◮ ◆♦t❡ ✿ ✈♦❧✉♠❡ ♦❢ t✐♠❡s❡r✐❡s ✐s ✶✷✳✸ ❚❡r❛❇②t❡s✳

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 12

✶ ♠♦♥♦❝♦r❡

✸✼ ②❡❛rs

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 13

❨❛❤♦♦ t❡r❛s♦rt ✇✐♥♥❡r

◮ ✾✶✵ ♥♦❞❡s ◮ ✹ ❞✉❛❧ ❝♦r❡ ❳❡♦♥s ❅ ✷✳✵❣❤③ ♣❡r ❛ ♥♦❞❡ ◮ ✶ ❣✐❣❛❜✐t ❡t❤❡r♥❡t ♦♥ ❡❛❝❤ ♥♦❞❡ ◮ ✹✵ ♥♦❞❡s ♣❡r ❛ r❛❝❦ ◮ ✽ ❣✐❣❛❜✐t ❡t❤❡r♥❡t ✉♣❧✐♥❦s ❢r♦♠ ❡❛❝❤ r❛❝❦ t♦ t❤❡ ❝♦r❡

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 14

❨❛❤♦♦ t❡r❛s♦rt ✇✐♥♥❡r✭✷✮

❚ ❝♦♠♣

P

= (✸♥❑❞ + ♥❑ + ❞❑ + ♥❞) ∗ ■ ∗ ❚ ✢♦♣ P = ✸✷✷✽✾✸s❡❝♦♥❞s ❚ ❝♦♠♠

P

= (✷ ∗ ❇r♦❛❞❝❛st t✐♠❡ ❜❡t✇❡❡♥ ✷ ♣r♦❝❡ss♦rs + ▼❡r❣✐♥❣ t✐♠❡) ∗ ⌈❧♦❣✷ P = (✷ ■❑❞ ∗ ❙♦❉ ❇❉▼▼,▼P■

P,❛✈❣

+ ✺■❑❞ ∗ ❚ ✢♦♣) ∗ ⌈❧♦❣✷(P)⌉ = ✷✷✺✽✻✻s❡❝♦♥❞s

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 15

❨❛❤♦♦ t❡r❛s♦rt ✇✐♥♥❡r✭✷✮ ❜♦✉♥❞❡❞ ❜② ❛❣❣r❡❣❛t❡❞ ❜❛♥❞✇✐❞t❤

❇❉▼▼,▼P■

P,❛✈❣

= ✶✵✾▼❜✐ts ❚ ❝♦♠♠

P

= (✷ ∗ ❇r♦❛❞❝❛st t✐♠❡ ❜❡t✇❡❡♥ ✷ ♣r♦❝❡ss♦rs + ▼❡r❣✐♥❣ t✐♠❡) ∗ ⌈❧♦❣✷ P = (✷ ■❑❞ ∗ ❙♦❉ ❇❉▼▼,▼P■

P,❛✈❣

+ ✺■❑❞ ∗ ❚ ✢♦♣) ∗ ⌈❧♦❣✷(P)⌉ = ✹✸✹✶✺✺s❡❝♦♥❞s

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 16

Pr✐❝✐♥❣ ♦♥ ❆③✉r❡

◮ ❙t♦r❛❣❡ ♣r✐❝❡ ✿ ✶✽✹✺ ❞♦❧❧❛rs✴▼♦♥t❤ ✭✵✱✶✺ ❞♦❧❧❛rs ✴●❇✴▼♦♥t❤✮ ◮ ◆♦ ❝❤❛r❣❡ ❢♦r ✐♥t❡r♥❛❧ tr❛♥s❢❡rs ❄ ◮ ✉♣❞❛t❡ ✿ ✶✵ ●❇②t❡s✴▼♦♥t❤ ✿ ✸ ❞♦❧❧❛rs ✭✵✳✸❞♦❧❧❛rs ✴●❇✮ ◮ ❝♦st ♦❢ r✉♥♥✐♥❣ t❤❡ ❑▼❡❛♥s ✿ ✭✈❡r② r♦✉❣❤✮ ✵✳✶✷❞♦❧❧❛rs✴❤♦✉r ✯

✸✷✹✶✷✵ ❂ ✹✵ ✵✵✵❞♦❧❧❛rs

◮ st♦r❛❣❡ r❡q✉❡sts ✿ ❄ ❢r♦♠ ✵✳✵✶ ❞♦❧❧❛rs t♦ ✶✶ ♠✐❧❧✐♦♥s ❞♦❧❧❛rs

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 17

❋✐❣✉r❡✿ P❛❛s✱ ■❛❛s✱ ❙❛❛❙

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 18

P❛❛s ❛♥❞ ■❛❛s ✿ s♦♠❡ tr❛❞❡♦✛s

◮ ❛❜str❛❝t✐♦♥✴❝♦♥tr♦❧ tr❛❞❡♦✛ ◮ s❝❛❧❛❜✐❧✐t②✴❝♦st t♦ ❞❡✈❡❧♦♣ tr❛❞❡♦✛

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 19

■♥tr♦❞✉❝✐♥❣ ▲♦❦❛❞✳❈❧♦✉❞

◮ ❤✐❣❤❡r ❧❡✈❡❧ ❛❜str❛❝t✐♦♥s

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 20

❆③✉r❡ ❆❜str❛❝t✐♦♥s

◮ ❆③✉r❡ ◗✉❡✉❡s ◮ ❆③✉r❡ ❇❧♦❜❙t♦r❛❣❡ ◮ ❆③✉r❡ ❚❛❜❧❡❙t♦r❛❣❡ ◮ ❆③✉r❡ ❙◗▲❙t♦r❛❣❡

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 21

❆③✉r❡ ◗✉❡✉❡s

◮ ❧♦♦s❡❧② ❝♦✉♣❧❡❞ ❛s②♥❝❤r♦♥♦✉s ❝♦♠♠✉♥✐❝❛t✐♦♥ ♠❡❝❤❛♥✐s♠ ◮ ♥♦t s❝❛❧❛❜❧❡ ②❡t ✭✐♥ s✐③❡ ❛♥❞ ♥✉♠❜❡r ♦❢ ♠❡ss❛❣❡s✮ ◮ ❋■❋❖ ✇❤❡♥ ♥♦t s♦✉❣❤t ◮ ♠❡ss❛❣❡s ❝❛♥ ❜❡ ♣r♦❝❡ss❡❞ s❡✈❡r❛❧ t✐♠❡s ✐❢ ✇♦r❦❡r ❢❛✐❧s ✭s❡❡

✐❞❡♠♣♦t❡♥❝②✮

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 22

❆③✉r❡ ❇❧♦❜❙t♦r❛❣❡

◮ ♦❜❥❡❝ts ✉♣ t♦ ✺✵ ●❇ ◮ ❝♦♥s✐st❡♥❝② ❄ ◮ ❑❡②✴✈❛❧✉❡ s②st❡♠

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 23

❆③✉r❡ ❚❛❜❧❡❙t♦r❛❣❡

◮ ❚❛❜❧❡s✴❊♥t✐t✐❡s✴Pr♦♣❡rt✐❡s ◮ ❦❡②✴✈❛❧✉❡ ♣❛✐r ❛❧s♦

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 24

❆③✉r❡ ◗✉❡✉❡❙❡r✈✐❝❡s

◮ ❙❖❆ ◮ ◗✉❡✉❡ ❝♦♥s✉♠❡rs

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 25

❆❈■❉✴❇❆❙❊ ❝♦♥str❛✐♥ts ♦♥ ❈❧♦✉❞

◮ ❆❈■❉ ◮ ❇❆❙❊ ◮ ❈❆P ❚❤❡♦r❡♠ ◮ ❈▲❆P s②st❡♠

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 26

❆❈■❉✴❇❆❙❊ ❝♦♥str❛✐♥ts ♦♥ ❆③✉r❡

◮ ❆✈❛✐❧❛❜✐❧✐t② ✿ ◆♦ ◮ ■♥t❡r♥❡t ❛♥❞ ❊❧❡❝tr✐❝✐t② ♥❡✐t❤❡r ◮ ❈♦♥s✐st❡♥❝② ✿ ❄

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 27

❈♦♥s✐st❡♥❝② ♦♥ ❆③✉r❡ ✿ r❡♣❧✐❝❛s ❛♥❞ ❝♦♥s✐st❡♥❝②

◮ ❇❧♦❝❦ ❝♦♠♠✐ts ◮ ❊t❛❣ ◮ ✈❡r② ❢❡✇ ♦✈❡r✇r✐t❡s

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 28

❈♦♥s✐st❡♥❝② ♦♥ ❆③✉r❡ ✿ tr❛♥s❛❝t✐♦♥❛❧✐t②

◮ ❚❤❡ ❙◗▲ tr✐❝❦

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 29

❆③✉r❡ tr❛❞❡♦✛s ❝❤♦✐❝❡s

◮ r❡❛❞✴✇r✐t❡ tr❛❞❡♦✛ ◮ ❘❡s♣♦♥s✐✈✐t②✴❡✣❝✐❡♥❝② ◮ ❙②♥❝❤r♦♥♦✉s r❡♣❧✐❝❛t✐♦♥✴❆s②♥❝❤r♦♥♦✉s r❡♣❧✐❝❛t✐♦♥ ❄

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 30

❇❧♦❜❙t♦r❛❣❡ ♦r ❚❛❜❧❡❙t♦r❛❣❡ ❄

◮ P❛rt✐t✐♦♥ ❑❡② ❛♥❞ ❧♦❝❛❧✐t② ◮ s✐③❡ ♦❢ ♦❜❥❡❝ts ◮ ❡✣❝✐❡♥❝② ♦❢ ❇❧♦❜st♦r❛❣❡ ❢♦r ♦❜❥❡❝ts ❧❛r❣❡r t❤❛♥ ✹❑❜②t❡s ◮ Pr✐❝❡

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 31

❆❜str❛❝t✐♦♥s ♣❡r❢♦r♠❛♥❝❡s

◮ ❇❧♦❜❙t♦r❛❣❡ ❘❡❛❞ ✿ ✶✸▼❇②t❡s✴s❡❝✴✇♦r❦❡r✳ ❆❣❣r❡❣❛t❡❞

❜♦✉♥❞❡❞ t♦ ✸✾✸ ▼❇②t❡s✴s❡❝

◮ ❇❧♦❜❙t♦r❛❣❡ ❲r✐t❡ ✿ ✺▼❇②t❡s✴s❡❝✴✇♦r❦❡r✳ ❆❣❣r❡❣❛t❡❞

❜♦✉♥❞❡❞ t♦ ✶✷✹ ▼❇②t❡s✴s❡❝

◮ ❉✐r❡❝t ✇♦r❦❡r ❝♦♠♠✉♥✐❝❛t✐♦♥ ✿ ✺✺▼❇②t❡s✴s❡❝ ♦♥ ✈❡r② ❧❛r❣❡

♠❡ss❛❣❡s

◮ ❱▼ ✿ ❢r♦♠ ✶ t♦ ✽ ❝♦r❡s ◮ ◗✉❡✉❡s t❤r♦✉❣❤♣✉t ✿ ✺✵✵ ♠❡ss❛❣❡s✴s❡❝

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 32

❘❡❞❡s✐❣♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠

◮ ❙❖❆ ◮ ◆♦ ▼❛st❡r ◮ ❆❧❧ ✇♦r❦❡rs ❝❛♥ ❜❡ s✇✐t❝❤❡❞ ◮ ◆♦ ✇♦r❦❡r✴❞❛t❛ ♣❡r ❛✣♥✐t② ❜✉t ✳✳✳ ◮ ❙❝❛❧✐♥❣ ✉♣ ❛s ❛♥ ✐♥✐t✐❛t✐✈❡ ✐♥ P❛❛❙ ◮ ◆♦ ❡①❛❝t ✇♦r❦❡rs ❝♦✉♥t ◮ P✐♥❣✐♥❣ q✉❡✉❡s ❛s ❛ tr❛❞❡♦✛ ❜❡t✇❡❡♥ ❝♦st ❛♥❞ s✐♠♣❧✐❝✐t② ◮ ◗✉❡✉❡ ▼❡ss❛❣❡s ❝♦✉♥t ◮ ✐❞❡♠♣♦t❡♥t ❥♦❜s

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 33

❘❡❞❡s✐❣♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✭✷✮

◮ ■❞❡♠♣♦t❡♥t ❈♦✉♥t❡rs ◮ ❲♦r❦❡rs ❝♦♠♠✉♥✐❝❛t✐♦♥ ❄

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 34

Pr♦♣♦s❡❞ ❆❧❣♦r✐t❤♠

◮ ❙❝❤❡♠❡

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 35

❉✐✣❝✉❧t② t♦ ❛♥t✐❝✐♣❛t❡ s♣❡❡❞❯♣

◮ ❙❡r✐❛❧✐③❛t✐♦♥✴❉❡s❡r✐❛❧✐③❛t✐♦♥✳ ❉❛t❛❈♦♥tr❛❝t ❄ ❇✐♥❛r②❋♦r♠❛t ❄

♦♦❣❧❡Pr♦t♦❇✉❢ ❄

◮ ✈❛r✐❛❜✐❧✐t② ♦❢ t❤❡ ❜❛♥❞✇✐❞t❤ ◮ ❧❛t❡♥❝② ❞✉❡ t♦ ♣✐♥❣✐♥❣ q✉❡✉❡s ◮ ❛ ✇♦r❦❡r ❝❛♥ ❢❛✐❧✱ ❛ ❱▼ ❝❛♥ ❜❡ ❦✐❧❧❡❞ ◮ ❙t♦r❛❣❡ ✉♥❛✈❛✐❧❛❜✐❧✐t② ◮ ❙tr❛❣❣❧❡rs ✐ss✉❡s ❄

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 36

■s t✐♠❡ t♦ ❧♦❛❞ ❞❛t❛ ✐♠♣♦rt❛♥t ❄

❚ ▲♦❛❞

P

= ♥❞ ∗ ❙♦❉ P ∗ ♠❛①(❇❛♥❞✇✐❞t❤r❡❛❞, ▼❛①❘❡❛❞❚❤r♦✉❣❤♣✉t

P

) ❚ ▲♦❛❞

P

<< ❚ ❈♦♠♣

P

■♠♣❧✐❝✐t s✉♣♣♦s✐t✐♦♥ ✿ ❧♦❛❞✐♥❣ ✐s r✉♥ ♦♥❝❡✳ ❙♦❉ ✸■❑ ∗ ❚ ✢♦♣ ∗ ♠❛①(❇❛♥❞✇✐❞t❤r❡❛❞, ▼❛①❘❡❛❞❚❤r♦✉❣❤♣✉t

P

) << ✶

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 37

❈♦♠♠✉♥✐❝❛t✐♦♥ ♠♦❞❡❧✐s❛t✐♦♥

❚ ❝♦♠♠,♣❡r✐t❡r❛t✐♦♥

P

= ❚✐♠❡ t♦ r❡❛❞ ✶ ❜❧♦❜ ”s✐♠✉❧t❛♥❡♦✉s❧②” ❜② P ✇♦r❦❡rs + ❚✐♠❡ t♦ ✇r✐t❡ P ❜❧♦❜s ”s✐♠✉❧t❛♥❡♦✉s❧②” + ❚✐♠❡ t♦ r❡❛❞ P ❜❧♦❜s ❜② ✶ ✇♦r❦❡r + ❚✐♠❡ t♦ ♠❡r❣❡ t❤❡ ❞✐✛❡r❡♥t ❝❡♥tr♦✐❞s ✈❡rs✐♦♥ + ❚✐♠❡ t♦ ✇r✐t❡ ✶ ❜❧♦❜ ♣❡r ✶ ✇♦r❦❡r

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 38

❈♦♠♠✉♥✐❝❛t✐♦♥ ♠♦❞❡❧✐s❛t✐♦♥✭✷✮

❚ ❝♦♠♠,♣❡r✐t❡r❛t✐♦♥

P

= ❑❞ ∗ ❙♦❉ ▼✐♥(❇❛♥❞✇✐❞t❤r❡❛❞, ❇❧♦❜❘❡❛❞❚❤r♦✉❣❤♣✉t

P

) + ❑❞ ∗ ❙♦❉ ▼✐♥(❇❛♥❞✇✐❞t❤✇r✐t❡, ▼❛①❲r✐t❡❚❤r♦✉❣❤♣✉t

P

) + P❑❞ ∗ ❙♦❉ ❇❛♥❞✇✐❞t❤r❡❛❞ + (P − ✶)❑❞ ∗ ✺✢♦❛t❖♣❡r❛t✐♦♥s ∗ ❚ ✢♦♣ + ❑❞ ∗ ❙♦❉ ❇❛♥❞✇✐❞t❤✇r✐t❡

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 39

❈♦♠♠✉♥✐❝❛t✐♦♥ ♠♦❞❡❧✐s❛t✐♦♥✭✸✮

❚ ❝♦♠♠

P

≃ ■P❑❞ ∗ ❙♦❉ ❇❛♥❞✇✐❞t❤r❡❛❞

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 40

❙♣❡❡❞❯♣

❙♣❡❡❞❯♣ = ❚ ❝♦♠♣

✶

❚ ❈♦♠♣

P

+ ❚P❝♦♠♠ ≃ ✸■❑❞♥❚ ✢♦♣

✸■❑❞♥❚ ✢♦♣ P

+

■P❑❞∗❙♦❉ ❇❛♥❞✇✐❞t❤r❡❛❞

≃ ✸♥❚ ✢♦♣

✸♥❚ ✢♦♣ P

+

P∗❙♦❉ ❇❛♥❞✇✐❞t❤r❡❛❞

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 41

❖♣t✐♠❛❧ ◆✉♠❜❡r ♦❢ ✇♦r❦❡rs

P∗ =

✸♥❚ ✢♦♣❇❛♥❞✇✐❞t❤r❡❛❞

❙♦❉

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 42

◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡

◮ ♥ ❂ ✹✸✷ ✵✵✵ ◮ ❑ ❂ ✻✺✼ ◮ ❞ ❂ ✶✻✽ ◮ ■ ❂ ✶✵✵ ◮ P▼❛① ❂ ✻✹ ◮ ❇❛♥❞✇✐❞t❤r❡❛❞ = ✷✵▼❜②t❡s/s❡❝(r❡s♣.✺▼❜②t❡s/s❡❝)

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 43

❘❡s✉❧ts

❙❡q✉❡♥t✐❛❧ ❑▼❡❛♥s t✐♠❡ = ✶✹✸✵✵s❡❝♦♥❞s ❖♣t✐♠❛❧ ♥✉♠❜❡r ♦❢ ✇♦r❦❡rs = ✺✼(r❡s♣✷✽) ❙♣❡❡❞❯♣ = ✷✽, ✺(r❡s♣✶✹)

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 44

■♠♣r♦✈❡♠❡♥ts ✿ ♠♦r❡ r❡❞✉❝❡rs ❄

s❝❤❡♠❡

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞

SLIDE 45

❉✐r❡❝t ✇♦r❦❡r ❝♦♠♠✉♥✐❝❛t✐♦♥ ❄

▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞