SLIDE 1 ❖♣t✐♠❛❧ tr❛♥s♣♦rt❛t✐♦♥ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ❲❛ss❡rst❡✐♥ s♣❛❝❡s
❆♥❞r❡✐ ❙♦❜♦❧❡✈s❦✐✶ ❆❧❡❦s❡✐ ❑r♦s❤♥✐♥✶❀✷
✶ ■♥st✐t✉t❡ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠s
♦❢ t❤❡ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ✭❑❤❛r❦❡✈✐❝❤ ■♥st✐t✉t❡✮
✷ ▼♦s❝♦✇ ■♥st✐t✉t❡ ♦❢ P❤②s✐❝s ❛♥❞ ❚❡❝❤♥♦❧♦❣②
◗✉❛s✐❧✐♥❡❛r ❊q✉❛t✐♦♥s✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s
▼■P❚✱ ❙❡♣t❡♠❜❡r ✶✸✱ ✷✵✶✻
SLIDE 3
Pr♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ❛s ♣♦✐♥ts ♦❢ ❣❡♦❞❡s✐❝ s♣❛❝❡
▼ ❞✭①❀ ②✮ ① ② ✖ ✗ ❲✷✭✖❀ ✗✮ ▼ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❣❡♦❞❡s✐❝ s♣❛❝❡ P✭▼✮ s♣❛❝❡ ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ♦♥ ▼
SLIDE 4
Pr♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ❛s ♣♦✐♥ts ♦❢ ❣❡♦❞❡s✐❝ s♣❛❝❡
▼ ❞✭①❀ ②✮ ① ② ✖ ✗ ❲✷✭✖❀ ✗✮ ▼ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❣❡♦❞❡s✐❝ s♣❛❝❡ P✭▼✮ s♣❛❝❡ ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ♦♥ ▼
SLIDE 5
Pr♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ❛s ♣♦✐♥ts ♦❢ ❣❡♦❞❡s✐❝ s♣❛❝❡
▼ ❞✭①❀ ②✮ ① ② ✖ ✗ ❲✷✭✖❀ ✗✮ ▼ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❣❡♦❞❡s✐❝ s♣❛❝❡ P✭▼✮ s♣❛❝❡ ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ♦♥ ▼
SLIDE 6
❲❛ss❡rst❡✐♥ ❞✐st❛♥❝❡ ❲✷✭✖❀ ✗✮
▲❡t ▼ ✚ ❘❞✱ ❞✭①❀ ②✮ ❂ ❥① ②❥ ✭❲✷✭✖❀ ✗✮✮✷ ❂ ✐♥❢ ✌ ✷ P✭▼ ✂ ▼✮ ✖ ✗
❩
❥① ②❥✷ ❞✌✭①❀ ②✮ ■♥☞♠✉♠ ✐s ❛tt❛✐♥❡❞ ❛t tr❛♥s♣♦rt ♣❧❛♥ ✌✄ ❂ ✭✐❞❀ ❚ ✄✮★✖ ✇❤❡r❡ ❚ ✄
★✖ ❂ ✗ ❛♥❞ ❚ ❂ r✟ ✇✐t❤ ✟ ❝♦♥✈❡①
❬❨✳ ❇r❡♥✐❡r ✶✾✾✶❪
SLIDE 7
- ❡♦❞❡s✐❝s ✐♥ P✭▼ ✮✿ ❉✐s♣❧❛❝❡♠❡♥t ✐♥t❡r♣♦❧❛t✐♦♥
✖✵ ❂ ✖ ✖✶ ❂ ✗ ✖t ❂ ✭❚t✮★✖ ❚t ✿ ① ✼✦ ✭✶ t✮① ✰ t ❚ ✄① ❂ ① ✰ t✭❚ ✄① ①✮ ❈✉r❧✲❢r❡❡ ✈❡❧♦❝✐t② ☞❡❧❞✿ ❚ ✄① ① ❂ r✭✟✭①✮ ❥①❥✷ ✷ ✮♥ ❂✿ r✣✭①✮ ❬❘✳ ❏✳ ▼❝❈❛♥♥ ✶✾✾✼❪
SLIDE 8
❋♦r♠❛❧ ❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡ ♦❢ P✭▼ ✮
P✭▼✮ ❚✖P✭▼✮ ✖ r✣ ❚✖P✭▼✮ ✐s ❢♦r♠❡❞ ❜② ❛❧❧ ❝✉r❧✲❢r❡❡ ✈❡❝t♦r ☞❡❧❞s r✣ ♦♥ ▼ ❦r✣❦✷
❚✖P✭▼✮ ❂
❩
▼
❥r✣✭①✮❥✷ ❞✖ ❬❋✳ ❖tt♦ ✷✵✵✶❪ ✖ ▼ ✚ ❘❞ r✣
SLIDE 9
❇❛r②❝❡♥t❡r ♦❢ ❛ ♠❡❛s✉r❡ P
✳ ✳ ✳ ✐♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ❘❞✿ ✖ ① ❂
❩
① ❞P✭①✮ ❂ ❛r❣ ♠✐♥
②
✶ ✷
❩
❥① ②❥✷ ❞P✭①✮ ✳ ✳ ✳ ✐♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ✖ ① ❂ ❛r❣ ♠✐♥
②
P✭②✮❀ ✇❤❡r❡ P✭②✮ ❂ ✶ ✷
❩
❞✭①❀ ②✮✷ ❞P✭①✮ ✖ ① ③❡r♦ ♦❢ rP✭②✮ ❂
❩
❡①♣✶
② ✭①✮ ❞P✭①✮
P✭✁✮ ❧♦❝❛❧❧② ❝♦♥✈❡① ✐❢ s❡❝t✐♦♥❛❧ ❝✉r✈❛t✉r❡ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❛❜♦✈❡ ❬❑✳ ●r♦✈❡✱ ❍✳ ❑❛r❝❤❡r ✶✾✼✸❪
SLIDE 10
❲❛ss❡rst❡✐♥ ❜❛r②❝❡♥t❡rs
❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❞✐s♣❧❛❝❡♠❡♥t ✐♥t❡r♣♦❧❛t✐♦♥ ✖ ✗ ❂ ❛r❣ ♠✐♥
✗
✶ ✷
❳
✐
✕✐ ❲ ✷
✷ ✭✖✐❀ ✗✮
✭❞✐s❝r❡t❡✱ ✕✐ ✕ ✵✱
❳
✐
✕✐ ❂ ✶✮ ✖ ✗ ❂ ❛r❣ ♠✐♥
✗
✶ ✷
❩
❲ ✷
✷ ✭✖❀ ✗✮ ❞P✭✖✮
✭❣❡♥❡r❛❧✮ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ✐♥ t❡r♠s ♦❢
❩
r✣✄
✖ ❞P✭✖✮
❈♦♥✈❡①✐t② ♣r♦♣❡rt✐❡s ❬▼✳ ❆❣✉❡❤✱ ●✳ ❈❛r❧✐❡r ✷✵✶✶❪
SLIDE 11
❲❛ss❡rst❡✐♥ ❜❛r②❝❡♥t❡rs
❈♦♥✈❡♥t✐♦♥❛❧ ✈s ❲❛ss❡rst❡✐♥ ❛✈❡r❛❣✐♥❣ ♦❢ ♠❡❛s✉r❡s ✐♥ ✷❉✿ ❬▼✳ ❈✉t✉r✐✱ ❆✳ ❉♦✉❝❡t ✷✵✶✹❪
SLIDE 12
❋r✓ ❡❝❤❡t ❜❛r②❝❡♥t❡rs ✐♥ ❣❡♥❡r❛❧ ♠❡tr✐❝ s♣❛❝❡s
❬▼✳ ❋r✓ ❡❝❤❡t ✶✾✹✽❪
SLIDE 13
❋r✓ ❡❝❤❡t ❜❛r②❝❡♥t❡r ❢♦r ♠❡❛s✉r❡s
❏✭✖❀ ✗✮ ❂ ✐♥❢ ✌ ✷ P✭❘❞ ✂ ❘❞✮ ✖ ✗
❩
❣✭① ②✮ ❞✌✭①❀ ②✮ ❣✭✁✮ ✕ ✵ str✐❝t❧② ❝♦♥✈❡①✱ ❣✭✵✮ ❂ ✵ ❏ ✕ ✵✱ ❏✭✖❀ ✗✮ ❂ ✵ ✐☛ ✖ ✗ ❜✉t ♥♦ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t② ✖ ✗ ❂ ❛r❣ ♠✐♥
✗
❩
❏✭✖❀ ✗✮ ❞P✭✖✮ ❬❆✳ ❑r♦s❤♥✐♥✱ ❆✳ ❙♦❜♦❧❡✈s❦✐ ✷✵✶✺❪
SLIDE 14
❋r✓ ❡❝❤❡t ❜❛r②❝❡♥t❡rs ❢♦r ♠❡❛s✉r❡s ✐♥ ❘✶
❋ ✶
✖ ✿ ✭✵❀ ✶✮ ✦ ❘✶ q✉❛♥t✐❧❡ ❢✉♥❝t✐♦♥ ♦❢ ♠❡❛s✉r❡ ✖
❚❤❡♦r❡♠
◗✉❛♥t✐❧❡s ♦❢ ❋r✓ ❡❝❤❡t ❜❛r②❝❡♥t❡r ❛r❡ ❣❡♥❡r❛❧✐③❡❞ ♠❡❛♥s ♦❢ q✉❛♥t✐❧❡s✿ ❋ ✶
✖ ✗ ✭☛✮ ❂ ❛r❣ ♠✐♥ ②
❩
❣✭❋ ✶
✖ ✭☛✮ ②✮ ❞P✭✖✮❀
✵ ❁ ☛ ❁ ✶ ❬❆✳ ❑r♦s❤♥✐♥✱ ❆✳ ❙♦❜♦❧❡✈s❦✐ ✷✵✶✺❪
SLIDE 15
▲❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❛♥❞ st❛t✐st✐❝❛❧ ❝♦♥s✐st❡♥❝②
❚❤❡♦r❡♠
✖ ✗♥ ✦ ✖ ✗ ✇❡❛❦❧② ❛s ♥ ✦ ✶ ✖ ✗♥ ❜❛r②❝❡♥t❡r ♦❢ ✐♥❞❡♣❡♥❞❡♥t s❛♠♣❧❡ ♦❢ s✐③❡ ♥ ❢r♦♠ P ✖ ✗ ❜❛r②❝❡♥t❡r ♦❢ P
❚❤❡♦r❡♠
❏✭✖ ✗♥❀ ✖ ✗✮ ✦ ✵ ❛s ♥ ✦ ✶ ✭❭str♦♥❣✧ ❝♦♥✈❡r❣❡♥❝❡✮ ❬❆✳ ❑r♦s❤♥✐♥✱ ❆✳ ❙♦❜♦❧❡✈s❦✐ ✷✵✶✺❪ ❬❚✳ ▲❡ ●♦✉✐❝✱ ❏✳✲▼✳ ▲♦✉❜❡s ✷✵✶✺❪✿ ♠❡❛s✉r❡s ✐♥ ❣❡♥❡r❛❧ ❣❡♦❞❡s✐❝ s♣❛❝❡ ✐♥st❡❛❞ ♦❢ ❘✶❀ ❲❛ss❡rst❡✐♥ ❜❛r②❝❡♥t❡rs ✐♥st❡❛❞ ♦❢ ❣❡♥❡r❛❧ ❋r✓ ❡❝❤❡t ❜❛r②❝❡♥t❡rs
SLIDE 16
❇✐❜❧✐♦❣r❛♣❤② ■
▼✳ ❆❣✉❡❤✱ ●✳ ❈❛r❧✐❡r✳ ❇❛r②❝❡♥t❡rs ✐♥ t❤❡ ❲❛ss❡rst❡✐♥ ❙♣❛❝❡✳ ❙■❆▼ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ✹✸✿✷ ✭✷✵✶✶✮ ✾✵✹④✾✷✹✳ ❨✳ ❇r❡♥✐❡r✳ P♦❧❛r ❢❛❝t♦r✐③❛t✐♦♥ ❛♥❞ ♠♦♥♦t♦♥❡ r❡❛rr❛♥❣❡♠❡♥t ♦❢ ✈❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s✳ ❈♦♠♠✳ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳ ✹✹✿✹ ✭✶✾✾✶✮ ✸✼✺④✹✶✼✳ ▼✳ ❈✉t✉r✐✱ ❆✳ ❉♦✉❝❡t✳ ❋❛st ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❲❛ss❡rst❡✐♥ ❜❛r②❝❡♥t❡rs✳ Pr♦❝✳ ✸✶ ■❈▼▲✱ ❇❡✐❥✐♥❣✱ ✷✵✶✹✳ ▼✳ ❋r✓ ❡❝❤❡t✳ ▲❡s ✓ ❡❧✓ ❡♠❡♥ts ❛❧✓ ❡❛t♦✐r❡s ❞❡ ♥❛t✉r❡ q✉❡❧❝♦♥q✉❡ ❞❛♥s ✉♥ ❡s♣❛❝❡ ❞✐st❛♥❝✐✓ ❡✳ ❆♥♥❛❧❡s ❞❡ ❧✬■❍P ✶✵✿✹ ✭✶✾✹✽✮ ✷✶✺④✸✶✵✳
SLIDE 17
❇✐❜❧✐♦❣r❛♣❤② ■■
❑✳ ●r♦✈❡✱ ❍✳ ❑❛r❝❤❡r✳ ❍♦✇ t♦ ❝♦♥❥✉❣❛t❡ ❈ ✶✲❝❧♦s❡ ❣r♦✉♣ ❛❝t✐♦♥s✳ ▼❛t❤✳ ❩✳ ✶✸✷ ✭✶✾✼✸✮ ✶✶✲✷✵✳ ❆✳ ❑r♦s❤♥✐♥✱ ❆✳ ❙♦❜♦❧❡✈s❦✐✳ ❋r✓ ❡❝❤❡t ❜❛r②❝❡♥t❡rs ❛♥❞ ❛ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ♠❡❛s✉r❡s ♦♥ t❤❡ r❡❛❧ ❧✐♥❡✳ ❛r❳✐✈✿✶✺✶✷✳✵✽✹✷✶✈✷ ❬♠❛t❤✳P❘❪ ✭✷✵✶✺✮ ✶✺ ♣♣✳ ❚✳ ▲❡ ●♦✉✐❝✱ ❏✳✲▼✳ ▲♦✉❜❡s✳ ❊①✐st❡♥❝❡ ❛♥❞ ❝♦♥s✐st❡♥❝② ♦❢ ❲❛ss❡rst❡✐♥ ❜❛r②❝❡♥t❡rs✳ ❛r❳✐✈✿✶✺✵✻✳✵✹✶✺✸ ❬♠❛t❤✳❙❚❪ ✭✷✵✶✺✮ ✶✾ ♣♣✳ ❘✳ ❏✳ ▼❝❈❛♥♥✳ ❆ ❝♦♥✈❡①✐t② ♣r✐♥❝✐♣❧❡ ❢♦r ✐♥t❡r❛❝t✐♥❣ ❣❛s❡s✳ ❆❞✈✳ ▼❛t❤✳ ✶✷✽✿✶ ✭✶✾✾✼✮ ✶✺✸④✶✼✾✳
SLIDE 18
❇✐❜❧✐♦❣r❛♣❤② ■■■
❋✳ ❖tt♦✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ ❞✐ss✐♣❛t✐✈❡ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s✿ t❤❡ ♣♦r♦✉s ♠❡❞✐✉♠ ❡q✉❛t✐♦♥✳ ❈♦♠♠✳ P❉❊ ✷✻✿✶✷ ✭✷✵✵✶✮ ✶✵✶④✶✼✹✳
