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■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs

❉❛♥✐❡❧ ❙❝❤❛❞❡♥✱ ❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❲♦r❦s❤♦♣ ✏❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ■♥✈❡rs✐♦♥ ✉♥❞❡r ❯♥❝❡rt❛✐♥t②✑ ❘■❈❆▼ ▲✐♥③

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❖✉t❧✐♥❡

■ ▲✐♥❡❛r ♠♦❞❡❧s✱ ❜❡st ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦rs ✭❇▲❯❊s✮ ■ ❱❛r✐❛♥❝❡ ❜♦✉♥❞✱ ♠♦♥♦♠✐❛❧ ❡①❛♠♣❧❡ ■ ❙❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥✱ P❉❊ ❡①❛♠♣❧❡ ■ ❆s②♠♣t♦t✐❝ ❛♥❛❧②s✐s ♦❢ ♠✉❧t✐❧❡✈❡❧ ❇▲❯❊s

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▼♦t✐✈❛t✐♦♥ ✭✶✴✸✮

■ ●♦❛❧✿ ❊st✐♠❛t❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❊[❩] ♦❢ s❝❛❧❛r ✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❩ ■ ❈❛♥ ❣❡♥❡r❛t❡ s❛♠♣❧❡s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ❩❵ ✙ ❩✱ ❵ = ✶❀ ✿ ✿ ✿ ❀ ▲ ■ ❊①♣❡❝t❛t✐♦♥s ✖ = (✖✶❀ ✿ ✿ ✿ ❀ ✖▲)❃ ✷ ❘▲ ■ ❈♦✈❛r✐❛♥❝❡ ▼❛tr✐① ❈ = (Cov(❩❦❀ ❩❵))▲

❦❀❵=✶ ✷ ❘▲✂▲

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✸

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▼♦t✐✈❛t✐♦♥ ✭✷✴✸✮

■ ▼❛♥② ❧✐♥❡❛r✱ ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦rs ❢♦r ✖▲ = ❊[❩▲] ✙ ❊[❩]

■ ▼♦♥t❡ ❈❛r❧♦

• ✖▼❈

▲

= ✶ ♠▲

♠▲

• ✐=✶

❩▲(✦✐) ■ ▼✉❧t✐❧❡✈❡❧ ▼♦♥t❡ ❈❛r❧♦ ✭●✐❧❡s ✷✵✵✽✮

• ✖▼▲▼❈

▲

=

▲

• ❵=✶

✶ ♠❵

♠❵

• ✐=✶

(❩❵(✦❵

✐ ) ❩❵✶(✦❵ ✐ ))

■ ▼✉❧t✐✜❞❡❧✐t② ▼♦♥t❡ ❈❛r❧♦ ✭P❡❤❡rst♦r❢❡r ❡t ❛❧✳ ✷✵✶✻✮ ■ ❆♣♣r♦①✐♠❛t❡ ❈♦♥tr♦❧ ❱❛r✐❛t❡s ✭●♦r♦❞❡ts❦② ❡t ❛❧✳ ✷✵✶✽✮

■ ❆r❡ t❤❡s❡ ❡st✐♠❛t♦rs ♦♣t✐♠❛❧❄ ✦ ♠✐♥✐♠❛❧ ✈❛r✐❛♥❝❡

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✹

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▼♦t✐✈❛t✐♦♥ ✭✸✴✸✮

■ ▲✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t♦rs✿

• ✖▲ =

❑

• ❦=✶
• ❵✷❙❦

☞❦

❵

✶ ♠❦

♠❦

• ✐=✶

❩❵(✦❦

✐ )

■ ❯♥❜✐❛s❡❞♥❡ss ✭r♦✇ s✉♠ ❝♦♥str❛✐♥t✮✿ ❑

❦=✶ ☞❦ ❵ = ✍❵❀▲

1.00 1.00 -1.00 1.00 -1.00 1.00 -1.00 1.00 -1.00 1.00 -1.00 1.00 2.00

• 1.00
• 2.00
• 3.00

1.00 0.25 3.00

• 1.00 1.00
• 1.00
• 0.50

2.00 0.25

• 1.00

1.00 ❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✺

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❊①❛♠♣❧❡ ✭✶✴✸✮

■ ▼✉❧t✐❧❡✈❡❧ ▼♦♥t❡ ❈❛r❧♦ ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t ✐♥♣✉ts ✦✷

✶❀ ✦✷ ✷❀ ✦✶ ✶

• ✖▼▲▼❈

✷

= ✶ ✷

✷

• ✐=✶

(❩✷(✦✷

✐ ) ❩✶(✦✷ ✐ )) + ❩✶(✦✶ ✶)

✙ (✖✷ ✖✶) + ✖✶ ■ ❚✇♦ s❛♠♣❧❡s t♦ ❡st✐♠❛t❡ ✖✷ ✖✶ ❛♥❞ ♦♥❡ s❛♠♣❧❡ t♦ ❡st✐♠❛t❡ ✖✶ ■ ❘❡❛rr❛♥❣❡✿

• ✖▼▲▼❈

✷

= ✶ ✷

• (❩✷(✦✷

✶) ❩✶(✦✷ ✶)) + (❩✷(✦✷ ✷) ❩✶(✦✷ ✷)

• + ❩✶(✦✶

✶)

= ✶ ✷(❩✷(✦✷

✶) + ❩✷(✦✷ ✷)) + ❩✶(✦✶ ✶) ✶

✷(❩✶(✦✷

✶) + ❩✶(✦✷ ✷))

■ ❚❤r❡❡ s❛♠♣❧❡s ♦❢ ❩✶ ❛♥❞ t✇♦ s❛♠♣❧❡s ♦❢ ❩✷

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✻

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❊①❛♠♣❧❡ ✭✷✴✸✮

■ ❑❡② ✐❞❡❛✿ tr❡❛t ♠♦❞❡❧ ❡✈❛❧✉❛t✐♦♥s ❛s ♦❜s❡r✈❛t✐♦♥s ♦❢ ✉♥❦♥♦✇♥ ✧tr✉t❤✧ ❊[❩] ■ ❊①❛♠♣❧❡✿ ✜✈❡ ✭❝♦rr❡❧❛t❡❞✮ ♦❜s❡r✈❛t✐♦♥s ♦❢ ❩✶ ❛♥❞ ❩✷ ■ ❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❧✐♥❡❛r ♠♦❞❡❧✿       ❩✶(✦✷

✶)

❩✷(✦✷

✶)

❩✶(✦✷

✷)

❩✷(✦✷

✷)

❩✶(✦✶

✶)

      =       ✖✶ ✖✷ ✖✶ ✖✷ ✖✶       +       ❩✶(✦✷

✶) ✖✶

❩✷(✦✷

✶) ✖✷

❩✶(✦✷

✷) ✖✶

❩✷(✦✷

✷) ✖✷

❩✶(✦✶

✶) ✖✶

      =       ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵       ✖✶ ✖✷

• +

  ✑✷(✦✷

✶)

✑✷(✦✷

✷)

✑✶(✦✶

✶)

  ■ ❖❜s❡r✈❛t✐♦♥ ✈❡❝t♦r ❨ ✱ ❉❡s✐❣♥ ♠❛tr✐① ❍✱ ♣❛r❛♠❡t❡r ✖✱ ♠❡❛♥ ③❡r♦ ♥♦✐s❡ ✧ ❨ = ❍✖ + ✧

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✼

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❊①❛♠♣❧❡ ✭✸✴✸✮

■ ▼♦❞❡❧ ❣r♦✉♣s ❙✶ = ❢✶❣ ❛♥❞ ❙✷ = ❢✶❀ ✷❣ ✇✐t❤ ❙✶❀ ❙✷ ✒ ❢✶❀ ✷❣✱ ▲ = ✷ ■ ♠✶ = ✶✱ ♠✷ = ✷ ❩ ✶ =

• ❩✶
• =
• ✖✶
• +
• ❩✶ ✖✶
• =
• ✶

✵

• ✖ + ✑✶❀

❘✶ =

• ✶

✵

• ❩ ✷ =

❩✶ ❩✷

• =

✖✶ ✖✷

• +

❩✶ ✖✶ ❩✷ ✖✷

• =

✶ ✵ ✵ ✶

• ✖ + ✑✷❀

❘✷ = ✶ ✵ ✵ ✶

• 

     ❩✶(✦✷

✶)

❩✷(✦✷

✶)

❩✶(✦✷

✷)

❩✷(✦✷

✷)

❩✶(✦✶

✶)

      =       ✖✶ ✖✷ ✖✶ ✖✷ ✖✶       +       ❩✶(✦✷

✶) ✖✶

❩✷(✦✷

✶) ✖✷

❩✶(✦✷

✷) ✖✶

❩✷(✦✷

✷) ✖✷

❩✶(✦✶

✶) ✖✶

      =       ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵       ✖✶ ✖✷

• +

  ✑✷(✦✷

✶)

✑✷(✦✷

✷)

✑✶(✦✶

✶)

 

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✽

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▲✐♥❡❛r ▼♦❞❡❧ ✭✶✴✸✮

■ ▼♦❞❡❧ ❣r♦✉♣s ❙❦ ✷ ✷❢✶❀✿✿✿❀▲❣ ♥ ❢❀❣✱ ❦ = ✶❀ ✿ ✿ ✿ ❀ ✷▲ ✶ =: ❑

▲✐♥❡❛r ♠♦❞❡❧ ✇✐t❤✐♥ ❣r♦✉♣

❩ ❦ = ❘❦✖ + ✑❦❀ ✭❙❛♠♣❧❡ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ♣r♦❜❧❡♠✮ ❩ ❦ := (❩❵)❵✷❙❦ ❀ ✭❖❜s❡r✈❛t✐♦♥s✮ ✑❦ := (❩❵ ✖❵)❵✷❙❦ ❀ ✭▼❡❛♥ ③❡r♦ ♥♦✐s❡✱ ❈♦✈❛r✐❛♥❝❡ ❈❙❦ ❀❙❦ ✮ ❘❦✈ := (✈❵)❵✷❙❦ ❀ ❢♦r ❛❧❧ ✈ ✷ ❘▲ ✭❘❡str✐❝t✐♦♥ ♦♣❡r❛t♦r✱ ❉❡s✐❣♥ ♠❛tr✐①✮ ■ ■❞❡❛✿ ❈♦❧❧❡❝t ❣r♦✉♣ ❧✐♥❡❛r ♠♦❞❡❧s ❛♥❞ s❛♠♣❧❡s ✐♥ ❛ ✭❤✉❣❡✮ ❜❧♦❝❦ ❧✐♥❡❛r ♠♦❞❡❧

❇❧♦❝❦ ❧✐♥❡❛r ♠♦❞❡❧

♠❦ ✷ ◆✵ s❛♠♣❧❡s ♦❢ ♠♦❞❡❧s ✐♥ ❙❦ ✷ ✷❢✶❀✿✿✿❀▲❣ ♥ ❢❀❣ ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts ✦❦

✐

❨ = ❍✖ + ✧❀ ✭▲✐♥❡❛r ♠♦❞❡❧✮ ❨ := ((❩ ❦(✦❦

✐ ))♠❦ ✐=✶)❑ ❦=✶❀

✭❖❜s❡r✈❛t✐♦♥s✮ ❍ := ((❘❦)♠❦

✐=✶)❑ ❦=✶

✭❉❡s✐❣♥ ♠❛tr✐①✮ ✧ := ((✑❦(✦❦

✐ ))♠❦ ✐=✶)❑ ❦=✶

✭▼❡❛♥ ③❡r♦ ♥♦✐s❡✮

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✾

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▲✐♥❡❛r ▼♦❞❡❧ ✭✷✴✸✮

■ ❨ = ❍✖ + ✧ ■ ② := ❍❃ ❈ ✶❨ = ❑

❦=✶ P❦(❈ ❦)✶ ♠❦ ✐=✶ ❩ ❦(✦❦ ✐ )

■ ✠ := ❍❃ ❈ ✶❍ = ❑

❦=✶ ♠❦P❦(❈ ❦)✶❘❦

■ ❇▲❯❊

• ✖❇

▲ := (❍❃

❈ ✶❍)✶❍❃ ❈ ✶❨ = ❡❚

▲ ✠✶②

■ ❈ ❦ := ❈❙❦❀❙❦ ■ P❦ := (❘❦)❚ ❚❤❡♦r❡♠ ✭●❛✉ss✲▼❛r❦♦✈✲❆✐t❦❡♥✮ ▲❡t ❈ ❜❡ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❛♥❞ ❛ss✉♠❡ ❡❛❝❤ ♠♦❞❡❧ ❩❵ ✐s ❡✈❛❧✉❛t❡❞ ❛t ❧❡❛st ♦♥❝❡ ❢❵ ✷ ❢✶❀ ✿ ✿ ✿ ❀ ▲❣ ❥ ✾♠❦ ❃ ✵ ✇✐t❤ ❵ ✷ ❙❦❣ = ❢✶❀ ✿ ✿ ✿ ❀ ▲❣✿ ❚❤❡♥ ✖❇

▲ ✐s ✇❡❧❧ ❞❡✜♥❡❞ ❛♥❞ ✐s t❤❡ ✭✉♥✐q✉❡✮ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦r

✭❇▲❯❊✮ ✇✐t❤ ✈❛r✐❛♥❝❡ Var( ✖❇

▲ ) = ❡❚ ▲ ✠✶❡▲✳

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✵

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▲✐♥❡❛r ▼♦❞❡❧ ✭✸✴✸✮

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❇▲❯❊✿

• ✖❇

▲ = ❡❚ ▲ ✠✶②

■ ◆♦r♠❛❧ ❡q✉❛t✐♦♥s ✭s♠❛❧❧✦✮✿ ✠ ✖❇ = ②✱ ✠ ✷ ❘▲✂▲✱ ② ✷ ❘▲ ■ ✖❇

▲ ✐s t❤❡ ❧❛st ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✈❡❝t♦r ♦❢ t❤❡ ♥♦r♠❛❧ ❡q✉❛t✐♦♥s

■ Var( ✖❇

▲ ) ✔ Var(

✖▲) ❢♦r ❛♥② ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦r ✖▲ ✇✐t❤ t❤❡ s❛♠❡ s❛♠♣❧❡ s❡q✉❡♥❝❡ (♠✶❀ ✿ ✿ ✿ ❀ ♠❑) ■ ▲✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t♦rs✿

• ✖❇

▲ = ❑

• ❦=✶
• ❵✷❙❦

☞❦

❵

✶ ♠❦

♠❦

• ✐=✶

❩❵(✦❦

✐ )

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✶

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❖✉t❧✐♥❡

■ ▲✐♥❡❛r ♠♦❞❡❧s✱ ❜❡st ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦rs ✭❇▲❯❊s✮ ■ ❱❛r✐❛♥❝❡ ❜♦✉♥❞✱ ♠♦♥♦♠✐❛❧ ❡①❛♠♣❧❡ ■ ❙❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥✱ P❉❊ ❡①❛♠♣❧❡ ■ ❆s②♠♣t♦t✐❝ ❛♥❛❧②s✐s ♦❢ ♠✉❧t✐❧❡✈❡❧ ❇▲❯❊s

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✷

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▲♦✇❡r ❜♦✉♥❞ ❢♦r ❡st✐♠❛t♦r ✈❛r✐❛♥❝❡

■ ❋♦❧❧♦✇ s❡t✉♣ ✐♥ ❬●♦r♦❞❡ts❦② ❡t ❛❧✳✱ ✷✵✶✽❪ ■ ❙♣❧✐t ✐♥❞✐❝❡s✿ ◗❀ ◗✶ ✒ ❢✶❀ ✿ ✿ ✿ ❀ ▲❣✱ ◗ ❬ ◗✶ = ❢✶❀ ✿ ✿ ✿ ❀ ▲❣✱ ◗ ✭ ◗✶ ■ ▼♦❞❡❧ ❣r♦✉♣s ✐♥ ◗ ❛r❡ ❡✈❛❧✉❛t❡❞ ▼✲t✐♠❡s ■ ▼♦❞❡❧ ❣r♦✉♣s ✐♥ ◗✶ ❛r❡ ❡✈❛❧✉❛t❡❞ ◆✲t✐♠❡s✱ ❝♦♥s✐❞❡r ❧✐♠✐t ◆ ✦ +✶ ■ ▲♦✇❡r ❜♦✉♥❞ ❢♦r ✈❛r✐❛♥❝❡ ♦❢ ❇▲❯❊ ❢♦r ◗ = ❢✶❀ ✿ ✿ ✿ ❀ ▲❣ ❛♥❞ ◗✶ = ❢✶❀ ✿ ✿ ✿ ❀ ▲ ✶❣✿ Var( ✖❇

▲ ) ✕ (❈▲❀▲ ❈▲❀✶:▲✶❈ ✶ ✶:▲✶❀✶:▲✶❈✶:▲✶❀▲)❂▼

✭s❛♠❡ ❛s ❢♦r ❆♣♣r♦①✐♠❛t❡ ❈♦♥tr♦❧ ❱❛r✐❛t❡s ✐♥ ❬●♦r♦❞❡ts❦② ❡t ❛❧✳✱ ✷✵✶✽❪✮

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✸

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▼♦♥♦♠✐❛❧ ❊①❛♠♣❧❡ ✭✶✴✸✮

❬●♦r♦❞❡ts❦② ❡t ❛❧✳✱ ✷✵✶✽❪ ■ ❩❵(✦) = ✦❵ ❢♦r ❵ = ✶❀ ✿ ✿ ✿ ❀ ✺ ✇✐t❤ ✦ ✘ ❯(✵❀ ✶)✳ ■ ❋✐① t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♠♦❞❡❧ ❡✈❛❧✉❛t✐♦♥s ♥❵ = ✷◆✷▲❵❀ ❢♦r ❩✶❀ ✿ ✿ ✿ ❀ ❩✹❀ ♥▲ = ✶❀ ❢♦r ❩✺✿

5 10 15 10 -3 10 -2 5 10 15 20 25 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✹

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▼♦♥♦♠✐❛❧ ❊①❛♠♣❧❡ ✭✷✴✸✮

■ ❙❛♠❡ ❡①❛♠♣❧❡ ❜✉t ✇✐t❤ ♥♦✐s❡ ■ ❩❵(✦❀ ✘) = ✦❵✶ + ✘ ❢♦r ❵ = ✶❀ ✿ ✿ ✿ ❀ ✺ ✇✐t❤ ✦ ✘ ❯(✵❀ ✶)✱ ✘ ✘ ◆(✵❀ ✷)✳ ■ ❩✻(✦❀ ✘) = ✦✺ ■ ❙♠❛❧❧ ✭❡♠♣✐r✐❝❛❧✮ ❝♦rr❡❧❛t✐♦♥ ♦❢ ❩✶❀ ✿ ✿ ✿ ❀ ❩✺ ✇✐t❤ ❩✻ ❞✉❡ t♦ ♥♦✐s❡ ▼♦❞❡❧ ❩✶ ❩✷ ❩✸ ❩✹ ❩✺ ❩✻ ❩✶ ✶✳✵✵✵✵ ✵✳✾✽✾✽ ✵✳✾✽✾✶ ✵✳✾✾✵✷ ✵✳✾✾✶✸ ✵✳✵✵✶✷ ❩✷ s②♠ ✶✳✵✵✵✵ ✵✳✾✾✾✸ ✵✳✾✾✽✸ ✵✳✾✾✼✹ ✵✳✶✶✽✷ ❩✸ s②♠ s②♠ ✶✳✵✵✵✵ ✵✳✾✾✾✼ ✵✳✾✾✾✶ ✵✳✶✸✼✹ ❩✹ s②♠ s②♠ s②♠ ✶✳✵✵✵✵ ✵✳✾✾✾✽ ✵✳✶✸✼✹ ❩✺ s②♠ s②♠ s②♠ s②♠ ✶✳✵✵✵✵ ✵✳✶✸✶✾ ❩✻ s②♠ s②♠ s②♠ s②♠ s②♠ ✶✳✵✵✵✵

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✺

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

▼♦♥♦♠✐❛❧ ❊①❛♠♣❧❡ ✭✸✴✸✮

5 10 15 20 25 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 BLUE MLMC MFMC ACV-MF ACV-KL MC 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✻

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❖✉t❧✐♥❡

■ ▲✐♥❡❛r ♠♦❞❡❧s✱ ❜❡st ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦rs ✭❇▲❯❊s✮ ■ ❱❛r✐❛♥❝❡ ❜♦✉♥❞✱ ♠♦♥♦♠✐❛❧ ❡①❛♠♣❧❡ ■ ❙❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥✱ P❉❊ ❡①❛♠♣❧❡ ■ ❆s②♠♣t♦t✐❝ ❛♥❛❧②s✐s ♦❢ ♠✉❧t✐❧❡✈❡❧ ❇▲❯❊s

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✼

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❙❛♠♣❧❡ ❆❧❧♦❝❛t✐♦♥

■ ▼♦t✐✈❛t✐♦♥✿ ❋♦r ✜①❡❞ ♠❦ t❤❡ ❡st✐♠❛t♦r ✖❇

▲ ✐s ✈❛r✐❛♥❝❡ ♠✐♥✐♠❛❧✳ ♠❦ =❄

■ ❉❡✜♥❡ ❝♦st ✇❵ := Cost(❩❵)✱ ❲ ❦ := Cost(❩ ❦) =

❵✷❙❦ ✇❵

■ ❋✐①❡❞ ❜✉❞❣❡t ♣ ❃ ✵✱ ❡①❝❧✉❞❡ ❝♦st ♦❢ ❡st✐♠❛t✐♥❣ ❈ ■ ❈♦♠♣✉t❡ ❛ ♠✐♥✐♠✐③❡r ♠✄ ♦❢ t❤❡ ❙❛♠♣❧❡ ❆❧❧♦❝❛t✐♦♥ Pr♦❜❧❡♠ min

♠✷◆❑

✵

Var( ✖❇

▲ ) = ❡❚ ▲ ✠(♠)✶❡▲

s✳ t✳

❑

• ❦=✶

♠❦❲ ❦ ✔ ♣✿ ■ ❉❡✜♥❡ ❙❛♠♣❧❡ ❆❧❧♦❝❛t✐♦♥ ❖♣t✐♠❛❧ ❇▲❯❊

• ✖❙❆❖❇

▲

:= ❡❚

▲ ✠(♠✄)✶②(♠✄)

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✽

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

Pr♦♣❡rt✐❡s ♦❢ ❙❆❖❇

■ ❚❤❡ ❙❆❖❇ ❤❛s t❤❡ s♠❛❧❧❡st ✈❛r✐❛♥❝❡ ❛♠♦♥❣ ❛❧❧ ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦rs t❤❛t ♦♥❧② ✉s❡ s❛♠♣❧❡s ♦❢ ❩✶❀ ✿ ✿ ✿ ❀ ❩▲ ❛♥❞ ❤❛✈❡ ❝♦sts ❜♦✉♥❞❡❞ ❜② ♣✳ ■ ❚❤❡ ❙❆❖❇ ❡st✐♠❛t♦r ❤❛s t❤❡ s♠❛❧❧❡st ❝♦st ❛♠♦♥❣ ❛❧❧ ❧✐♥❡❛r ✭✉♥❜✐❛s❡❞✮ ❡st✐♠❛t♦rs t❤❛t ♦♥❧② ✉s❡ s❛♠♣❧❡s ♦❢ ❩✶❀ ✿ ✿ ✿ ❀ ❩▲✳ ■ ❚❤❡r❡ ❡①✐sts ❛ s♦❧✉t✐♦♥ ♠✄ ♦❢ t❤❡ s❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ ✉s✐♥❣ ❛t ♠♦st ▲ ♠♦❞❡❧ ❣r♦✉♣s✳

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✶✾

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❙❆❖❇ ✭✶✴✷✮

❚❤❡♦r❡♠ ▲❡t ❈ ❜❡ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ❧❡t ✖▲ ❜❡ ❛ ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦r ✉s✐♥❣ ♦♥❧② s❛♠♣❧❡s ♦❢ ❩✶❀ ✿ ✿ ✿ ❀ ❩▲ ✇✐t❤ ❝♦st ❜♦✉♥❞❡❞ ❜② Cost( ✖▲) ✔ ♣✳ ❚❤❡♥ Var( ✖▲) ✕ Var( ✖❙❆❖❇

▲

)✿ Pr♦♦❢✳ ■ ❯s❡ ●❛✉ss✲▼❛r❦♦✈✲❆✐t❦❡♥ t♦ s❤♦✇ Var( ✖▲) ✕ Var( ✖❇

▲ (♠)) ✇✐t❤ t❤❡ s❛♠♣❧❡

❛❧❧♦❝❛t✐♦♥ ♦❢ ♠ = ♠( ✖▲) ■ ❙✐♥❝❡ Cost( ✖▲) = ❑

❦=✶ ♠❦❲ ❦ ✔ ♣✱ ♠(

✖▲) ✐s ❛ ❢❡❛s✐❜❧❡ s❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥✳ ♠✄ ✐s t❤❡ ♠✐♥✐♠✐③❡r✱ t❤✉s Var( ✖❇

▲ (♠)) ✕ Var(

✖❇

▲ (♠✄)) = Var(

✖❙❆❖❇

▲

)✳

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✵

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❙❆❖❇ ✭✷✴✷✮

❈♦r♦❧❧❛r② ▲❡t ❈ ❜❡ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ❧❡t ✖▲ ❜❡ ❛ ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦r ✉s✐♥❣ ♦♥❧② s❛♠♣❧❡s ♦❢ ❩✶❀ ✿ ✿ ✿ ❀ ❩▲ s✉❝❤ t❤❛t ❊[❥ ✖▲ ❊[❩]❥✷] ✔ ✧✷ ✇✐t❤ Cost( ✖▲) ✔ ✣(✧✷)✿ ❚❤❡♥ t❤❡ ❙❆❖❇ ❡st✐♠❛t♦r s❛t✐s✐❢❡s ❊[❥ ✖❙❆❖❇

▲

❊[❩]❥✷] ✔ ✧✷ ✇✐t❤ Cost( ✖❙❆❖❇

▲

) ✔ ✣(✧✷)✿ Pr♦♦❢✳ ■ ❇✐❛s✲✈❛r✐❛♥❝❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❊[❥ ✖▲ ❊[❩]❥✷] = ❥❊[✖▲] ❊[❩]❥✷ + Var( ✖▲) ■ ❙❆❖❇ ❤❛s s❛♠❡ ❜✐❛s ❊[ ✖❙❆❖❇

▲

] = ❊[ ✖▲] = ✖▲ ❜✉t s♠❛❧❧❡r ♦r ❡q✉❛❧ ✈❛r✐❛♥❝❡ s✐♥❝❡ ❜✉❞❣❡t ♣ = Cost( ✖▲)✳

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✶

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

P❉❊ ❊①❛♠♣❧❡ ✭✶✴✸✮

■ ❩(✦) =

✶ ❥❉♦❜s❥

• ❉♦❜s ②(①❀ ✦)❞①

■ ❉♦❜s := ✸

✹❀ ✼ ✽

• ✂

✼

✽❀ ✶

• ✒ ❉ := (✵❀ ✶)✷

■ ② s♦❧✈❡s ❡❧❧✐♣t✐❝ P❉❊ div(❛(①❀ ✦)r②(①❀ ✦)) = ✶❀ ① ✷ ❉❀ ②(①❀ ✦) = ✵❀ ① ✷ ❅❉✿ ■ ▲♦❣♥♦r♠❛❧ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ❛(①❀ ✦) = exp(✔(①❀ ✦)) ■ ✔ ♠❡❛♥ ③❡r♦ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✜❡❧❞ ✇✐t❤ ❲❤✐tt❧❡✕▼❛tér♥ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥✱ s♠♦♦t❤♥❡ss ✗ = ✸❂✷✱ ✈❛r✐❛♥❝❡ ✛✷ = ✶✱ ❝♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤ ✚ = ✵✿✶✳

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✷

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

P❉❊ ❊①❛♠♣❧❡ ✭✷✴✸✮

■ ❉✐s❝r❡t✐s❛t✐♦♥s ❩✶❀ ✿ ✿ ✿ ❀ ❩✻ ✇✐t❤ ❧✐♥❡❛r ❋❊s✱ ✉♥✐❢♦r♠ ♠❡s❤ r❡✜♥❡♠❡♥t ■ ❈♦♠♣✉t❡❞ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ✙ ❈ ✉s✐♥❣ ✶✵✺ ♣✐❧♦t s❛♠♣❧❡s ■ ●♦❛❧✿ ▼✐♥✐♠✐③❡ r♦♦t ♠❡❛♥ sq✉❛r❡ ❡rr♦r ✧ = (❊[( ✖❵ ❊[❩])✷])✶❂✷

10 -4 10 -3 10 -2 10 -2 10 0 10 2 10 4 10 6 10 -4 10 -3 10 -2 1 1.5 2 2.5 3 3.5

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✸

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

P❉❊ ❊①❛♠♣❧❡ ✭✸✴✸✮

■ ✖❙❆❖❇

▲

= ❑

❦=✶

• ❵✷❙❦ ☞❦

❵ ✶ ♠❦

♠❦

✐=✶ ❩❵(✦❦ ✐ )

■ ❈♦❡✣❝✐❡♥ts ❢♦r ✧ = ✷✿✺ ✂ ✶✵✺

1.00 1.00 -1.00 1.00 -1.00 1.00 -1.00 1.00 -1.00 1.00 -1.00 1.00 1.18

• 0.05

0.03 -1.93 1.64 0.95

• 2.66

1.81 0.04

• 0.26

1.24

• 2.79

1.78 0.00 0.02

• 0.22

0.98

• 1.78

1.00 ❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✹

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❙✉♠♠❛r② ✭✶✴✷✮

■ ❘❡❢♦r♠✉❧❛t❡❞ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ❊[❩] ❛s ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ♣r♦❜❧❡♠ ■ ❈♦♥str✉❝t❡❞ ♥♦✈❡❧ ♠✉❧t✐❧❡✈❡❧ ❇▲❯❊ ✦ ♠✐♥✐♠❛❧ ✈❛r✐❛♥❝❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♥✉♠❜❡r ♦❢ s❛♠♣❧❡s ■ ◆♦✐s② ♠♦♥♦♠✐❛❧ ❡①❛♠♣❧❡ s❤♦✇s ❤✐❣❤ ❝♦rr❡❧❛t✐♦♥ ✇✐t❤ ❤✐❣❤ ✜❞❡❧✐t② ♠♦❞❡❧ ✐s ♥♦t ♥❡❝❡ss❛r② ❢♦r ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ■ ❆❞❞✐t✐♦♥❛❧ ❜✉❞❣❡t ❝♦♥str❛✐♥t ❧❡❛❞s t♦ ❙❆❖❇ ■ ✺✵% ❝♦♠♣✉t✳ ❝♦st ❝♦♠♣❛r❡❞ t♦ ▼▲▼❈ ❢♦r ❛ s❝❛❧❛r ✈❛❧✉❡❞ ◗✉❛♥t✐t② ♦❢ ■♥t❡r❡st ❞❡r✐✈❡❞ ❢r♦♠ ❛♥ ❡❧❧✐♣t✐❝ P❉❊ ✇✐t❤ r❛♥❞♦♠ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✺

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❖✉t❧✐♥❡

■ ▲✐♥❡❛r ♠♦❞❡❧s✱ ❜❡st ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦rs ✭❇▲❯❊s✮ ■ ❱❛r✐❛♥❝❡ ❜♦✉♥❞✱ ♠♦♥♦♠✐❛❧ ❡①❛♠♣❧❡ ■ ❙❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥✱ P❉❊ ❡①❛♠♣❧❡ ■ ❆s②♠♣t♦t✐❝ ❛♥❛❧②s✐s ♦❢ ♠✉❧t✐❧❡✈❡❧ ❇▲❯❊s ✦ ◆❡✇ ♣r❡♣r✐♥t✦

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✻

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

P❛r❛♠❡tr✐❝ ♠♦❞❡❧ ❢❛♠✐❧②

■ ❉✐s❝r❡t✐③❛t✐♦♥ ♣❛r❛♠❡t❡r ❤❵ = s❵+✶❤✱ ❤ ❃ ✵✱ s ❃ ✶ ✜①❡❞✱ ❵ = ✶❀ ✿ ✿ ✿ ❀ ▲ ■ ❩❵(✦) ✥ ✦ ❩✦(❤❵) ■ ❆ss✉♠♣t✐♦♥✿ ❆s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ ❩✦(❤) = ❩✦(✵) +

• ❝✷(✦)r ✶(❤)❀

q = ✶❀ q

❦=✷ ❝❦(✦)❤☞❦ + ❝q+✶r q(❤)❀

q ✕ ✷✿ ■ ❙✐♥❝❡ t❤❡ ❇▲❯❊ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♠♦❞❡❧ ❡✈❛❧✉❛t✐♦♥s ❩✦(❤❵)✱ t❤✐s ❛ss✉♠♣t✐♦♥ ❛❧❧♦✇s ✉s t♦ st✉❞② t❤❡ ✈❛r✐❛♥❝❡ ✭❛♥❞ ❝♦♠♣❧❡①✐t②✮ ♦❢ t❤❡ ❇▲❯❊✦

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✼

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

P❉❊ ❊①❛♠♣❧❡ ✕ r❡✈✐s✐t❡❞

10 -4 10 -3 10 -2 10 -1 10 0 10 -2 10 0 10 2 10 4 10 6 10 8 10 10 10 -4 10 -3 10 -2 10 -1 10 0 10 -5 10 0 10 5 10 10 10 15

❋✐❣✉r❡✿ P❉❊ ❡①❛♠♣❧❡ ✭✚ = ✵✿✺✱ ✛✷ = ✸✮ ✇✐t❤ tr✉❡ ❝♦st r❛t❡ ☞❈♦st = ✷ ✭❧❡❢t ✐♠❛❣❡✮ ❛♥❞ ♠❛♥✉❢❛❝t✉r❡❞ ❝♦st r❛t❡ ☞❈♦st = ✻ ✭r✐❣❤t ✐♠❛❣❡✮✳ ❲❡ ❝❛♥ ♣r♦✈❡ t❤❡ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞ ❢♦r t❤❡ ❙❆❖❇ ❢♦r t❤❡ tr✉❡ ❝♦st r❛t❡ ☞❈♦st = ✷✱ ❜✉t ♥♦t ②❡t t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❙❆❖❇ ❢♦r t❤❡ ❤✐❣❤❡r ✭♠❛♥✉❢❛❝t✉r❡❞✮ ❝♦st r❛t❡✳

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✽

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❙✉♠♠❛r② ✭✷✴✷✮

■ ❈♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ♦❢ ❙❆❖❇ ❢♦r P❉❊✲❜❛s❡❞ ♠♦❞❡❧s ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❛♥ ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ❢❛♠✐❧② t♦❣❡t❤❡r ✇✐t❤ ❘✐❝❤❛r❞s♦♥ ❡①tr❛♣♦❧❛t✐♦♥ ■ ❈♦♠♣❧❡①✐t② ♦❢ ❙❆❖❇ ♣r♦✈❛❜❧② ♥♦t ✇♦rs❡ t❤❛♥ ❝♦♠♣❧❡①✐t② ♦❢ ▼▲▼❈ ■ ▼❛♥✉❢❛❝t✉r❡❞ ❡①❛♠♣❧❡ ✇✐t❤ ♦♣t✐♠❛❧ ✭❛♥❞ ❜❡tt❡r t❤❛♥ ▼▲▼❈✮ ❝♦♠♣❧❡①✐t② ❖(✧✷) ✦ Pr♦♦❢ ✐♥ ♣r♦❣r❡ss

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✷✾

■♥t❡r♥❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❚r❛✐♥✐♥❣ ●r♦✉♣ ■●❉❑ ▼✉♥✐❝❤ ✕ ●r❛③

❘❡❢❡r❡♥❝❡s

■ ❆✳❆✳ ●♦r♦❞❡ts❦②✱ ●✳ ●❡r❛❝✐✱ ▼✳ ❊❧❞r❡❞✱ ❏✳❉✳ ❏❛❦❡♠❛♥✿ ❆ ●❡♥❡r❛❧✐③❡❞ ❋r❛♠❡✇♦r❦ ❢♦r ❆♣♣r♦①✐♠❛t❡ ❈♦♥tr♦❧ ❱❛r✐❛t❡s✱ ❛r❳✐✈✿✶✽✶✶✳✵✹✾✽✽✳ ■ ❉✳ ❙❝❤❛❞❡♥✱ ❊✳ ❯❧❧♠❛♥♥✿ ❖♥ ♠✉❧t✐❧❡✈❡❧ ❜❡st ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦rs✱ ✉♥❞❡r r❡✈✐❡✇ ❛t ❏❯◗✱ Pr❡♣r✐♥t ◆♦✳ ■●❉❑✲✷✵✶✾✲✶✶✱ ❤tt♣s✿✴✴✇✇✇✳✐❣❞❦✳ ❡✉✴❢♦s✇✐❦✐✴♣✉❜✴■●❉❑✶✼✺✹✴Pr❡♣r✐♥ts✴✶✾✵✺❴❙❝❤❛❞❡♥❯❧❧♠❛♥♥✳♣❞❢ ■ ❉✳ ❙❝❤❛❞❡♥✱ ❊✳ ❯❧❧♠❛♥♥✿ ❆s②♠♣t♦t✐❝ ❛♥❛❧②s✐s ♦❢ ♠✉❧t✐❧❡✈❡❧ ❜❡st ❧✐♥❡❛r ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦rs✱ Pr❡♣r✐♥t ◆♦✳ ■●❉❑✲✷✵✶✾✲✷✷✱ ❤tt♣s✿✴✴✇✇✇✳✐❣❞❦✳❡✉✴ ❢♦s✇✐❦✐✴♣✉❜✴■●❉❑✶✼✺✹✴Pr❡♣r✐♥ts✴✶✾✶✶❴s❛♦❜❴❛s②♠♣t♦t✐❝❴s✐❛♠✳♣❞❢✱ s✉❜♠✐tt❡❞✳

❊❧✐s❛❜❡t❤ ❯❧❧♠❛♥♥ ❖♥ ▼✉❧t✐❧❡✈❡❧ ❇❡st ▲✐♥❡❛r ❯♥❜✐❛s❡❞ ❊st✐♠❛t♦rs ✶✶ ◆♦✈❡♠❜❡r ✷✵✶✾ ✸✵