st t t r t - - PowerPoint PPT Presentation

st t t r t t r s s
SMART_READER_LITE
LIVE PREVIEW

st t t r t - - PowerPoint PPT Presentation

t t t t s tst r t rrss r rsts


slide-1
SLIDE 1

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡

❚❡st✐♥❣ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡

Pr❡s❡♥t❡❞ ❜② ❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

❲♦r❦ ❥♦✐♥t❧② ✇✐t❤ ●✳ ➪❧✈❛r❡③✲Pér❡③✱ ❊✳ ●❛r❝í❛✲P♦rt✉❣✉és✱ ▼✳ ❋❡❜r❡r♦✲❇❛♥❞❡ ❛♥❞ ❲✳ ●♦♥③á❧❡③✲▼❛♥t❡✐❣❛

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶ ✴ ✺✺

slide-2
SLIDE 2

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡

❖✉t❧✐♥❡

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷ ✴ ✺✺

slide-3
SLIDE 3

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❖✉t❧✐♥❡

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸ ✴ ✺✺

slide-4
SLIDE 4

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❚❤❡ st❛t✐st✐❝❛❧ ❛t♦♠s ❛r❡ ♥♦✇ t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ ❢✉♥❝t✐♦♥s✿ ♠❛❦✐♥❣ ✐♥❢❡r❡♥❝❡ ✇✐t❤ t❤❡ ✇❤♦❧❡ ❝✉r✈❡s✳ ❆ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X ✐s ❛ ❢✉♥❝t✐♦♥❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✐❢ ✐t t❛❦❡s ✈❛❧✉❡s ✐♥ ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ ♦r s❡♠✐✲♠❡tr✐❝ s♣❛❝❡ ✭❢✉♥❝t✐♦♥❛❧ s♣❛❝❡✮✳ ❙✐♥❝❡ ❛ ✜♥✐t❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✐s r❡q✉✐r❡❞ ✐♥ ♣r❛❝t✐❝❡✱ ♦♥❡ ♠✐❣❤t t❡♠♣t❡❞ t♦ ❤❛♥❞❧❡ ✐t ✇✐t❤ ▼❱❆ t❡❝❤♥✐q✉❡s✱ ❜✉t ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s ❛r♦s❡✳ ❋❉❆ ✐s ❛❧s♦ s✉❣❣❡st❡❞ ❢♦r ❛♥❛❧②s✐♥❣ t❤❡ ♣❛tt❡r♥ ♦❢ ❝✉r✈❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹ ✴ ✺✺

slide-5
SLIDE 5

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❋❉❆✿ ❛♥t❡❝❡❞❡♥ts

❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ✧❤✐❣❤✕❞✐♠❡♥s✐♦♥❛❧✧ ♠❡❛♥s ✧✐♥✜♥✐t❡✕❞✐♠❡♥s✐♦♥❛❧✧❄

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺ ✴ ✺✺

slide-6
SLIDE 6

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡ ❚❤❡ ♠♦st ❣❡♥❡r❛❧ ❝♦♥t❡①t t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐s ❣✐✈❡♥ ❜② ❛ s❡♠✐✕♠❡tr✐❝ s♣❛❝❡ d (·,·)✳ ❙❡❡ ❬❇❡rr❡♥❞❡r♦ ❡t ❛❧✳✱ ✷✵✶✽❪✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❝❛♥ ❜❡ r❛t❤❡r ❛❜str❛❝t ❛♥❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❝♦✉❧❞ ❜❡ ❛❞♦♣t❡❞ ✐♥ ♣❧❛❝❡✿ ❛ ♥♦r♠❡❞ s♣❛❝❡ (B,·B)✱ ✇❤✐❝❤ ✐s ❝♦♠♣❧❡t❡ r❡s♣❡❝t t♦ ·B✳ ❙❡❡ ❬❘✉✐③✕▼❡❞✐♥❛ ❛♥❞ ➪❧✈❛r❡③✕▲✐é❜❛♥❛✱ ✷✵✶✽❪✳ ◆♦r♠ ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ♣r♦✈✐❞❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t✳ ❇❛♥❛❝❤ s♣❛❝❡s ❛r❡ ✇❡❧❧✕❛❞❛♣t❡❞ ❢♦r ♠❡❛s✉r✐♥❣ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✲ ✐t②✴s✐♥❣✉❧❛r✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ t❤❡ ♠♦st ✉s❡❞ s♣❛❝❡s✿ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s✱ ✇✐t❤ ❛♥ ✐♥♥❡r✲♣r♦❞✉❝t✲❜❛s❡❞ str✉❝t✉r❡✳ ❍✐❧❜❡rt s♣❛❝❡s✱ t❤❛t ✉s✉❛❧❧② ❛r✐s❡ ✐♥ ♣r❛❝t✐❝❡✱ ❛r❡ ♥♦t ✇❡❧❧✕❛❞❛♣t❡❞ t♦ ♠❡❛s✉r❡ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✐t②✴s✐♥❣✉❧❛r✐t②✳ ❯♥❞❡r s❡♣❛r❛❜✐❧✐t② ♦❢ ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ♦r✲ t❤♦❣♦♥❛❧ ❜❛s✐s ✐♥t♦ ✇❤✐❝❤ ✇❡ ❝❛♥ ♣r♦❥❡❝t ♦✉r tr❛❥❡❝t♦r✐❡s✳ ❊①❛♠♣❧❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ ♥♦r♠ s♣❛❝❡s✱ ❡♥❞♦✇❡❞ ✇✐t❤ ✱ ❜❡✐♥❣

✷ s♣❛❝❡s t❤❡ ♦♥❧② ♦♥❡s t❤❛t ❝♦♥st✐t✉t❡ ❛ ❍✐❧❜❡rt s♣❛❝❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✻ ✴ ✺✺

slide-7
SLIDE 7

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡ ❚❤❡ ♠♦st ❣❡♥❡r❛❧ ❝♦♥t❡①t t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐s ❣✐✈❡♥ ❜② ❛ s❡♠✐✕♠❡tr✐❝ s♣❛❝❡ d (·,·)✳ ❙❡❡ ❬❇❡rr❡♥❞❡r♦ ❡t ❛❧✳✱ ✷✵✶✽❪✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❝❛♥ ❜❡ r❛t❤❡r ❛❜str❛❝t ❛♥❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❝♦✉❧❞ ❜❡ ❛❞♦♣t❡❞ ✐♥ ♣❧❛❝❡✿ ❛ ♥♦r♠❡❞ s♣❛❝❡ (B,·B)✱ ✇❤✐❝❤ ✐s ❝♦♠♣❧❡t❡ r❡s♣❡❝t t♦ ·B✳ ❙❡❡ ❬❘✉✐③✕▼❡❞✐♥❛ ❛♥❞ ➪❧✈❛r❡③✕▲✐é❜❛♥❛✱ ✷✵✶✽❪✳ ◆♦r♠ ·B ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ♣r♦✈✐❞❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t✳ ❇❛♥❛❝❤ s♣❛❝❡s ❛r❡ ✇❡❧❧✕❛❞❛♣t❡❞ ❢♦r ♠❡❛s✉r✐♥❣ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✲ ✐t②✴s✐♥❣✉❧❛r✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ t❤❡ ♠♦st ✉s❡❞ s♣❛❝❡s✿ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s✱ ✇✐t❤ ❛♥ ✐♥♥❡r✲♣r♦❞✉❝t✲❜❛s❡❞ str✉❝t✉r❡✳ ❍✐❧❜❡rt s♣❛❝❡s✱ t❤❛t ✉s✉❛❧❧② ❛r✐s❡ ✐♥ ♣r❛❝t✐❝❡✱ ❛r❡ ♥♦t ✇❡❧❧✕❛❞❛♣t❡❞ t♦ ♠❡❛s✉r❡ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✐t②✴s✐♥❣✉❧❛r✐t②✳ ❯♥❞❡r s❡♣❛r❛❜✐❧✐t② ♦❢ ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ♦r✲ t❤♦❣♦♥❛❧ ❜❛s✐s ✐♥t♦ ✇❤✐❝❤ ✇❡ ❝❛♥ ♣r♦❥❡❝t ♦✉r tr❛❥❡❝t♦r✐❡s✳ ❊①❛♠♣❧❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ ♥♦r♠ s♣❛❝❡s✱ ❡♥❞♦✇❡❞ ✇✐t❤ ✱ ❜❡✐♥❣

✷ s♣❛❝❡s t❤❡ ♦♥❧② ♦♥❡s t❤❛t ❝♦♥st✐t✉t❡ ❛ ❍✐❧❜❡rt s♣❛❝❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✻ ✴ ✺✺

slide-8
SLIDE 8

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡ ❚❤❡ ♠♦st ❣❡♥❡r❛❧ ❝♦♥t❡①t t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐s ❣✐✈❡♥ ❜② ❛ s❡♠✐✕♠❡tr✐❝ s♣❛❝❡ d (·,·)✳ ❙❡❡ ❬❇❡rr❡♥❞❡r♦ ❡t ❛❧✳✱ ✷✵✶✽❪✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❝❛♥ ❜❡ r❛t❤❡r ❛❜str❛❝t ❛♥❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❝♦✉❧❞ ❜❡ ❛❞♦♣t❡❞ ✐♥ ♣❧❛❝❡✿ ❛ ♥♦r♠❡❞ s♣❛❝❡ (B,·B)✱ ✇❤✐❝❤ ✐s ❝♦♠♣❧❡t❡ r❡s♣❡❝t t♦ ·B✳ ❙❡❡ ❬❘✉✐③✕▼❡❞✐♥❛ ❛♥❞ ➪❧✈❛r❡③✕▲✐é❜❛♥❛✱ ✷✵✶✽❪✳ ◆♦r♠ ·B ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ♣r♦✈✐❞❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t✳ ❇❛♥❛❝❤ s♣❛❝❡s ❛r❡ ✇❡❧❧✕❛❞❛♣t❡❞ ❢♦r ♠❡❛s✉r✐♥❣ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✲ ✐t②✴s✐♥❣✉❧❛r✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ t❤❡ ♠♦st ✉s❡❞ s♣❛❝❡s✿ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s✱ ✇✐t❤ ❛♥ ✐♥♥❡r✲♣r♦❞✉❝t✲❜❛s❡❞ str✉❝t✉r❡✳ ❍✐❧❜❡rt s♣❛❝❡s✱ t❤❛t ✉s✉❛❧❧② ❛r✐s❡ ✐♥ ♣r❛❝t✐❝❡✱ ❛r❡ ♥♦t ✇❡❧❧✕❛❞❛♣t❡❞ t♦ ♠❡❛s✉r❡ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✐t②✴s✐♥❣✉❧❛r✐t②✳ ❯♥❞❡r s❡♣❛r❛❜✐❧✐t② ♦❢ ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ♦r✲ t❤♦❣♦♥❛❧ ❜❛s✐s ✐♥t♦ ✇❤✐❝❤ ✇❡ ❝❛♥ ♣r♦❥❡❝t ♦✉r tr❛❥❡❝t♦r✐❡s✳ ❊①❛♠♣❧❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ ♥♦r♠ s♣❛❝❡s✱ ❡♥❞♦✇❡❞ ✇✐t❤ ✱ ❜❡✐♥❣

✷ s♣❛❝❡s t❤❡ ♦♥❧② ♦♥❡s t❤❛t ❝♦♥st✐t✉t❡ ❛ ❍✐❧❜❡rt s♣❛❝❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✻ ✴ ✺✺

slide-9
SLIDE 9

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡ ❚❤❡ ♠♦st ❣❡♥❡r❛❧ ❝♦♥t❡①t t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐s ❣✐✈❡♥ ❜② ❛ s❡♠✐✕♠❡tr✐❝ s♣❛❝❡ d (·,·)✳ ❙❡❡ ❬❇❡rr❡♥❞❡r♦ ❡t ❛❧✳✱ ✷✵✶✽❪✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❝❛♥ ❜❡ r❛t❤❡r ❛❜str❛❝t ❛♥❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❝♦✉❧❞ ❜❡ ❛❞♦♣t❡❞ ✐♥ ♣❧❛❝❡✿ ❛ ♥♦r♠❡❞ s♣❛❝❡ (B,·B)✱ ✇❤✐❝❤ ✐s ❝♦♠♣❧❡t❡ r❡s♣❡❝t t♦ ·B✳ ❙❡❡ ❬❘✉✐③✕▼❡❞✐♥❛ ❛♥❞ ➪❧✈❛r❡③✕▲✐é❜❛♥❛✱ ✷✵✶✽❪✳ ◆♦r♠ ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ♣r♦✈✐❞❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t✳ ❇❛♥❛❝❤ s♣❛❝❡s ❛r❡ ✇❡❧❧✕❛❞❛♣t❡❞ ❢♦r ♠❡❛s✉r✐♥❣ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✲ ✐t②✴s✐♥❣✉❧❛r✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❍✐❧❜❡rt s♣❛❝❡s H ❛r❡ t❤❡ ♠♦st ✉s❡❞ s♣❛❝❡s✿ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s✱ ✇✐t❤ ❛♥ ✐♥♥❡r✲♣r♦❞✉❝t✲❜❛s❡❞ str✉❝t✉r❡✳ ❍✐❧❜❡rt s♣❛❝❡s✱ t❤❛t ✉s✉❛❧❧② ❛r✐s❡ ✐♥ ♣r❛❝t✐❝❡✱ ❛r❡ ♥♦t ✇❡❧❧✕❛❞❛♣t❡❞ t♦ ♠❡❛s✉r❡ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✐t②✴s✐♥❣✉❧❛r✐t②✳ ❯♥❞❡r s❡♣❛r❛❜✐❧✐t② ♦❢ ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ♦r✲ t❤♦❣♦♥❛❧ ❜❛s✐s ✐♥t♦ ✇❤✐❝❤ ✇❡ ❝❛♥ ♣r♦❥❡❝t ♦✉r tr❛❥❡❝t♦r✐❡s✳ ❊①❛♠♣❧❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ ♥♦r♠ s♣❛❝❡s✱ ❡♥❞♦✇❡❞ ✇✐t❤ ✱ ❜❡✐♥❣

✷ s♣❛❝❡s t❤❡ ♦♥❧② ♦♥❡s t❤❛t ❝♦♥st✐t✉t❡ ❛ ❍✐❧❜❡rt s♣❛❝❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✻ ✴ ✺✺

slide-10
SLIDE 10

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡ ❚❤❡ ♠♦st ❣❡♥❡r❛❧ ❝♦♥t❡①t t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐s ❣✐✈❡♥ ❜② ❛ s❡♠✐✕♠❡tr✐❝ s♣❛❝❡ d (·,·)✳ ❙❡❡ ❬❇❡rr❡♥❞❡r♦ ❡t ❛❧✳✱ ✷✵✶✽❪✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❝❛♥ ❜❡ r❛t❤❡r ❛❜str❛❝t ❛♥❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❝♦✉❧❞ ❜❡ ❛❞♦♣t❡❞ ✐♥ ♣❧❛❝❡✿ ❛ ♥♦r♠❡❞ s♣❛❝❡ (B,·B)✱ ✇❤✐❝❤ ✐s ❝♦♠♣❧❡t❡ r❡s♣❡❝t t♦ ·B✳ ❙❡❡ ❬❘✉✐③✕▼❡❞✐♥❛ ❛♥❞ ➪❧✈❛r❡③✕▲✐é❜❛♥❛✱ ✷✵✶✽❪✳ ◆♦r♠ ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ♣r♦✈✐❞❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t✳ ❇❛♥❛❝❤ s♣❛❝❡s ❛r❡ ✇❡❧❧✕❛❞❛♣t❡❞ ❢♦r ♠❡❛s✉r✐♥❣ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✲ ✐t②✴s✐♥❣✉❧❛r✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❍✐❧❜❡rt s♣❛❝❡s H ❛r❡ t❤❡ ♠♦st ✉s❡❞ s♣❛❝❡s✿ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s✱ ✇✐t❤ ❛♥ ✐♥♥❡r✲♣r♦❞✉❝t✲❜❛s❡❞ str✉❝t✉r❡✳ ❍✐❧❜❡rt s♣❛❝❡s✱ t❤❛t ✉s✉❛❧❧② ❛r✐s❡ ✐♥ ♣r❛❝t✐❝❡✱ ❛r❡ ♥♦t ✇❡❧❧✕❛❞❛♣t❡❞ t♦ ♠❡❛s✉r❡ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✐t②✴s✐♥❣✉❧❛r✐t②✳ ❯♥❞❡r s❡♣❛r❛❜✐❧✐t② ♦❢ ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ♦r✲ t❤♦❣♦♥❛❧ ❜❛s✐s ✐♥t♦ ✇❤✐❝❤ ✇❡ ❝❛♥ ♣r♦❥❡❝t ♦✉r tr❛❥❡❝t♦r✐❡s✳ ❊①❛♠♣❧❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ ♥♦r♠ s♣❛❝❡s✱ ❡♥❞♦✇❡❞ ✇✐t❤ ✱ ❜❡✐♥❣

✷ s♣❛❝❡s t❤❡ ♦♥❧② ♦♥❡s t❤❛t ❝♦♥st✐t✉t❡ ❛ ❍✐❧❜❡rt s♣❛❝❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✻ ✴ ✺✺

slide-11
SLIDE 11

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡ ❚❤❡ ♠♦st ❣❡♥❡r❛❧ ❝♦♥t❡①t t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐s ❣✐✈❡♥ ❜② ❛ s❡♠✐✕♠❡tr✐❝ s♣❛❝❡ d (·,·)✳ ❙❡❡ ❬❇❡rr❡♥❞❡r♦ ❡t ❛❧✳✱ ✷✵✶✽❪✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❝❛♥ ❜❡ r❛t❤❡r ❛❜str❛❝t ❛♥❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❝♦✉❧❞ ❜❡ ❛❞♦♣t❡❞ ✐♥ ♣❧❛❝❡✿ ❛ ♥♦r♠❡❞ s♣❛❝❡ (B,·B)✱ ✇❤✐❝❤ ✐s ❝♦♠♣❧❡t❡ r❡s♣❡❝t t♦ ·B✳ ❙❡❡ ❬❘✉✐③✕▼❡❞✐♥❛ ❛♥❞ ➪❧✈❛r❡③✕▲✐é❜❛♥❛✱ ✷✵✶✽❪✳ ◆♦r♠ ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ♣r♦✈✐❞❡ ❛♥ ✐♥♥❡r ♣r♦❞✉❝t✳ ❇❛♥❛❝❤ s♣❛❝❡s ❛r❡ ✇❡❧❧✕❛❞❛♣t❡❞ ❢♦r ♠❡❛s✉r✐♥❣ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✲ ✐t②✴s✐♥❣✉❧❛r✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❍✐❧❜❡rt s♣❛❝❡s H ❛r❡ t❤❡ ♠♦st ✉s❡❞ s♣❛❝❡s✿ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s✱ ✇✐t❤ ❛♥ ✐♥♥❡r✲♣r♦❞✉❝t✲❜❛s❡❞ str✉❝t✉r❡✳ ❍✐❧❜❡rt s♣❛❝❡s✱ t❤❛t ✉s✉❛❧❧② ❛r✐s❡ ✐♥ ♣r❛❝t✐❝❡✱ ❛r❡ ♥♦t ✇❡❧❧✕❛❞❛♣t❡❞ t♦ ♠❡❛s✉r❡ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✐t②✴s✐♥❣✉❧❛r✐t②✳ ❯♥❞❡r s❡♣❛r❛❜✐❧✐t② ♦❢ H✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ♦r✲ t❤♦❣♦♥❛❧ ❜❛s✐s ✐♥t♦ ✇❤✐❝❤ ✇❡ ❝❛♥ ♣r♦❥❡❝t ♦✉r tr❛❥❡❝t♦r✐❡s✳ ❊①❛♠♣❧❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡s ❛r❡ ♥♦r♠ s♣❛❝❡s✱ ❡♥❞♦✇❡❞ ✇✐t❤ ✱ ❜❡✐♥❣

✷ s♣❛❝❡s t❤❡ ♦♥❧② ♦♥❡s t❤❛t ❝♦♥st✐t✉t❡ ❛ ❍✐❧❜❡rt s♣❛❝❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✻ ✴ ✺✺

slide-12
SLIDE 12

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

❖✉t❧✐♥❡

✶ ❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

✷ ●♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

✸ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡

❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✼ ✴ ✺✺

slide-13
SLIDE 13

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

❆♥t❡❝❡❞❡♥ts

■♥ t❤❡ ❝❧❛ss✐❝❛❧ r❡❣r❡ss✐♦♥ ❢r❛♠❡✇♦r❦s✱ ✇❡ ❤❛✈❡ ❛t ♦✉r ❞✐s✲

♣♦s❛❧ ❛ ✈❛st r❛♥❣❡ ♦❢ t♦♦❧s ❢♦r ❛ss❡ss✐♥❣ ✐❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ r❡❧❛t❡❞ ❜② ❛ ❧✐♥❡❛r ♠♦❞❡❧ ✭r❡❣r❡ss✐♦♥ ❧✐♥❡✮ ♦r ♥♦t✳ P❧♦tt✐♥❣ t❤❡ r❡❣r❡ss✐♦♥ ❧✐♥❡ ✭s❝❛❧❛r ❢r❛♠❡✇♦r❦✮ ❈♦rr❡❧❛t✐♦♥ ❛♥❛❧②s✐s✳

  • ♦♦❞♥❡ss✲♦❢✲✜t t❡sts✳

❆◆❖❱❆ ♦r ▼❆◆❖❱❆✳ ❘❡s✐❞✉❛❧ ❞✐❛❣♥♦st✐❝s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✽ ✴ ✺✺

slide-14
SLIDE 14

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

❆♥t❡❝❡❞❡♥ts

  • ♦❋ ❧✐t❡r❛t✉r❡ ✐♥ t❤❡ s❝❛❧❛r ❛♥❞ ♠✉❧t✐✈❛r✐❛t❡ r❡❣r❡ss✐♦♥ ❝♦♥t❡①ts

❙♠♦♦t❤✐♥❣✲❜❛s❡❞ t❡sts✿ ❇✐❝❦❡❧ ❛♥❞ ❘♦s❡♥❜❧❛tt ✭✶✾✼✸✮✱ ❍är❞❧❡

❛♥❞ ▼❛♠♠❡♥ ✭✶✾✾✸✮✳ ❆♥ ✐♥❤❡r❡♥t ❞✐♠❡♥s✐♦♥❛❧✐t② ♣r♦❜❧❡♠ s❤♦✉❧❞ ❜❡ s♦❧✈❡❞✳ ❙❡❡ ❍❛rt ✭✶✾✾✼✮ ❢♦r ❛ r❡✈✐❡✇ ♦❢ t❤✐s ❧♦❝❛❧ ❛♣♣r♦❛❝❤✳ ❚❤❡ st❛t✐st✐❝ ✐s r❛t❤❡r s❡♥s✐t✐✈❡ t♦ t❤❡ s♠♦♦t❤✐♥❣ ♣❛r❛♠❡t❡r✳

■♥t❡❣r❛t❡❞✲❜❛s❡❞ t❡sts✿ ❉✉r❜✐♥ ✭✶✾✼✸✮ ❛♥❞ ❙t✉t❡ ✭✶✾✾✼✮✳

P♦♦r ❡♠♣✐r✐❝❛❧ ♣♦✇❡r ✐s ♦❜t❛✐♥❡❞ ✐♥ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥✲ t❡①ts✳ ❆ s✉❜❥❡❝t✐✈❡ ✐♥t❡❣r❛t✐♥❣ ♠❡❛s✉r❡ s❤♦✉❧❞ ❜❡ s❡❧❡❝t❡❞✳ ❊s❝❛♥❝✐❛♥♦ ✭✷✵✵✻✮✿ st❛t✐st✐❝ ✐♥ t❡r♠s ♦❢ ❛ r❡s✐❞✉❛❧ ♠❛r❦❡❞ ❡♠♣✐r✐✲ ❝❛❧ ♣r♦❝❡ss✱ ❜❛s❡❞ ♦♥ t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t❡❞ r❡s✐❞✲ ✉❛❧s ✭❞✐r❡❝t✐♦♥s ♦✈❡r ❊✉❝❧✐❞❡❛♥ s♣❤❡r❡✮✱ ❢♦r ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❞❛t❛✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✾ ✴ ✺✺

slide-15
SLIDE 15

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

❆♥t❡❝❡❞❡♥ts

  • ♦❋ ❧✐t❡r❛t✉r❡ ✐♥ t❤❡ s❝❛❧❛r ❛♥❞ ♠✉❧t✐✈❛r✐❛t❡ r❡❣r❡ss✐♦♥ ❝♦♥t❡①ts

❙♠♦♦t❤✐♥❣✲❜❛s❡❞ t❡sts✿ ❇✐❝❦❡❧ ❛♥❞ ❘♦s❡♥❜❧❛tt ✭✶✾✼✸✮✱ ❍är❞❧❡

❛♥❞ ▼❛♠♠❡♥ ✭✶✾✾✸✮✳ ❆♥ ✐♥❤❡r❡♥t ❞✐♠❡♥s✐♦♥❛❧✐t② ♣r♦❜❧❡♠ s❤♦✉❧❞ ❜❡ s♦❧✈❡❞✳ ❙❡❡ ❍❛rt ✭✶✾✾✼✮ ❢♦r ❛ r❡✈✐❡✇ ♦❢ t❤✐s ❧♦❝❛❧ ❛♣♣r♦❛❝❤✳ ❚❤❡ st❛t✐st✐❝ ✐s r❛t❤❡r s❡♥s✐t✐✈❡ t♦ t❤❡ s♠♦♦t❤✐♥❣ ♣❛r❛♠❡t❡r✳

■♥t❡❣r❛t❡❞✲❜❛s❡❞ t❡sts✿ ❉✉r❜✐♥ ✭✶✾✼✸✮ ❛♥❞ ❙t✉t❡ ✭✶✾✾✼✮✳

P♦♦r ❡♠♣✐r✐❝❛❧ ♣♦✇❡r ✐s ♦❜t❛✐♥❡❞ ✐♥ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥✲ t❡①ts✳ ❆ s✉❜❥❡❝t✐✈❡ ✐♥t❡❣r❛t✐♥❣ ♠❡❛s✉r❡ s❤♦✉❧❞ ❜❡ s❡❧❡❝t❡❞✳ ❊s❝❛♥❝✐❛♥♦ ✭✷✵✵✻✮✿ st❛t✐st✐❝ ✐♥ t❡r♠s ♦❢ ❛ r❡s✐❞✉❛❧ ♠❛r❦❡❞ ❡♠♣✐r✐✲ ❝❛❧ ♣r♦❝❡ss✱ ❜❛s❡❞ ♦♥ t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t❡❞ r❡s✐❞✲ ✉❛❧s ✭❞✐r❡❝t✐♦♥s ♦✈❡r ❊✉❝❧✐❞❡❛♥ s♣❤❡r❡✮✱ ❢♦r ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❞❛t❛✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✾ ✴ ✺✺

slide-16
SLIDE 16

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

❆♥t❡❝❡❞❡♥ts

  • ♦❋ ❧✐t❡r❛t✉r❡ ✐♥ t❤❡ s❝❛❧❛r ❛♥❞ ♠✉❧t✐✈❛r✐❛t❡ r❡❣r❡ss✐♦♥ ❝♦♥t❡①ts

❙♠♦♦t❤✐♥❣✲❜❛s❡❞ t❡sts✿ ❇✐❝❦❡❧ ❛♥❞ ❘♦s❡♥❜❧❛tt ✭✶✾✼✸✮✱ ❍är❞❧❡

❛♥❞ ▼❛♠♠❡♥ ✭✶✾✾✸✮✳ ❆♥ ✐♥❤❡r❡♥t ❞✐♠❡♥s✐♦♥❛❧✐t② ♣r♦❜❧❡♠ s❤♦✉❧❞ ❜❡ s♦❧✈❡❞✳ ❙❡❡ ❍❛rt ✭✶✾✾✼✮ ❢♦r ❛ r❡✈✐❡✇ ♦❢ t❤✐s ❧♦❝❛❧ ❛♣♣r♦❛❝❤✳ ❚❤❡ st❛t✐st✐❝ ✐s r❛t❤❡r s❡♥s✐t✐✈❡ t♦ t❤❡ s♠♦♦t❤✐♥❣ ♣❛r❛♠❡t❡r✳

■♥t❡❣r❛t❡❞✲❜❛s❡❞ t❡sts✿ ❉✉r❜✐♥ ✭✶✾✼✸✮ ❛♥❞ ❙t✉t❡ ✭✶✾✾✼✮✳

P♦♦r ❡♠♣✐r✐❝❛❧ ♣♦✇❡r ✐s ♦❜t❛✐♥❡❞ ✐♥ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥✲ t❡①ts✳ ❆ s✉❜❥❡❝t✐✈❡ ✐♥t❡❣r❛t✐♥❣ ♠❡❛s✉r❡ s❤♦✉❧❞ ❜❡ s❡❧❡❝t❡❞✳

❊s❝❛♥❝✐❛♥♦ ✭✷✵✵✻✮✿ st❛t✐st✐❝ ✐♥ t❡r♠s ♦❢ ❛ r❡s✐❞✉❛❧ ♠❛r❦❡❞ ❡♠♣✐r✐✲

❝❛❧ ♣r♦❝❡ss✱ ❜❛s❡❞ ♦♥ t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t❡❞ r❡s✐❞✲ ✉❛❧s ✭❞✐r❡❝t✐♦♥s ♦✈❡r ❊✉❝❧✐❞❡❛♥ s♣❤❡r❡✮✱ ❢♦r ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❞❛t❛✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✾ ✴ ✺✺

slide-17
SLIDE 17

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

? ❇✉t✳✳✳✇❤❛t ❛❜♦✉t ✇❤❡♥ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥s✐❞❡r❡❞❄ ❲❤❛t

❞♦❡s ✧❧✐♥❡❛r r❡❧❛t✐♦♥✧ ♠❡❛♥s❄ ❙❈❆▲❆❘ ▼❯▲❚■❱❆❘■❆❚❊ ❋❯◆❈❚■❖◆❆▲

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✵ ✴ ✺✺

slide-18
SLIDE 18

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

? ❇✉t✳✳✳✇❤❛t ❛❜♦✉t ✇❤❡♥ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥s✐❞❡r❡❞❄ ❲❤❛t

❞♦❡s ✧❧✐♥❡❛r r❡❧❛t✐♦♥✧ ♠❡❛♥s❄ Y = 〈β,X〉R +ε = βX +ε, ❙❈❆▲❆❘ ▼❯▲❚■❱❆❘■❆❚❊ ❋❯◆❈❚■❖◆❆▲

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✵ ✴ ✺✺

slide-19
SLIDE 19

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

? ❇✉t✳✳✳✇❤❛t ❛❜♦✉t ✇❤❡♥ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥s✐❞❡r❡❞❄ ❲❤❛t

❞♦❡s ✧❧✐♥❡❛r r❡❧❛t✐♦♥✧ ♠❡❛♥s❄ Y = 〈β,X〉R +ε = βX +ε, ❙❈❆▲❆❘ Y = 〈β,X〉Rd +ε = β

TX +ε,

▼❯▲❚■❱❆❘■❆❚❊ ❋❯◆❈❚■❖◆❆▲

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✵ ✴ ✺✺

slide-20
SLIDE 20

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

? ❇✉t✳✳✳✇❤❛t ❛❜♦✉t ✇❤❡♥ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥s✐❞❡r❡❞❄ ❲❤❛t

❞♦❡s ✧❧✐♥❡❛r r❡❧❛t✐♦♥✧ ♠❡❛♥s❄ Y = 〈β,X〉R +ε = βX +ε, ❙❈❆▲❆❘ Y = 〈β,X〉Rd +ε = β

TX +ε,

▼❯▲❚■❱❆❘■❆❚❊ Y = 〈β,X 〉H +ε =

  • β(s)X (s)ds +ε,

❋❯◆❈❚■❖◆❆▲

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✵ ✴ ✺✺

slide-21
SLIDE 21

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

❋✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧✿ ❡st✐♠❛t✐♦♥

❈♦♥t❡①t✿ ❢✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧ ✇✐t❤ s❝❛❧❛r r❡s♣♦♥s❡ ✭❋▲▼❙❘✮ ❨ = ♠β(X )+ε =

  • H

X (s)β(s)ds +ε,

Y ,ε ∈ R, X , β ∈ H.

▼✉❝❤ ♦❢ t❤❡ ❧✐t❡r❛t✉r❡ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧✐♥❣✿

❈❛r❞♦t ❡t ❛❧✳ ✭✶✾✾✾✮ ♣r♦♣♦s❡❞ t❤❡ ❋P❈ ❘❡❣r❡ss✐♦♥ ✭❋P❈❘✮✳ Pr❡❞❛ ❛♥❞ ❙❛♣♦rt❛ ✭✷✵✵✺✮ ♣r♦♣♦s❡❞ t❤❡ ❋✉♥❝t✐♦♥❛❧ P❛rt✐❛❧ ▲❡❛st ❙q✉❛r❡s ❘❡❣r❡ss✐♦♥ ✭❋P▲❙❘✮✳ ❬❇❡rr❡♥❞❡r♦ ❡t ❛❧✳✱ ✷✵✶✽❪ r❡♣❧❛❝❡❞ ❡♥t✐r❡ ♣❛t❤s ❜② r❡❧❡✈❛♥t ✐♥st❛♥ts✳ ❬❋❡❜r❡r♦✲❇❛♥❞❡ ❡t ❛❧✳✱ ✷✵✶✽❪ ❞❡✈❡❧♦♣❡❞ ❛ ❋▲▼❙❘ ❡st✐♠❛t✐♦♥ ♠❡t❤♦❞ ✇❤❡♥ s♦♠❡ ♦❢ t❤❡ r❡s♣♦♥s❡s ❛r❡ ♠✐ss✐♥❣ ❛t r❛♥❞♦♠✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✶ ✴ ✺✺

slide-22
SLIDE 22

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

■♥tr♦❞✉❝t✐♦♥

❋✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡ ✭❋▲▼❋❘✮

Y (t) = ♠β(X )+E(t), Y ,E ∈ H✷, X ∈ H✶, β ❜✐✈❛r✐❛t❡ ❦❡r♥❡❧

❆ ❍✐❧❜❡rt✐❛♥ ❢r❛♠❡✇♦r❦ ✐s ✉s✉❛❧❧② ❛❞♦♣t❡❞✿

♠β(X ) =

  • X (s)β(s,t)❞s,
  • β✷(s,t)dsdt < ∞.

❍♦✇❡✈❡r✱ t❤❡ ❋▲▼❋❘ ❤❛s r❡❝❡✐✈❡❞ ❝♦♥s✐❞❡r❛❜❧② ❧❡ss ❛tt❡♥t✐♦♥✿

❘❛♠s❛② ❛♥❞ ❙✐❧✈❡r♠❛♥ ✭✷✵✵✺✮ ✐♥tr♦❞✉❝❡❞ ❛♥ ❖▲❙ ❡st✐♠❛t♦r✳ ❈r❛♠❜❡s ❛♥❞ ▼❛s ✭✷✵✶✸✮ ♣r♦♣♦s❡❞ ❛ ❑❛r❤✉♥❡♥✕▲♦❡✈❡ ❡st✐♠❛t♦r✳ ❋✉♥❝t✐♦♥❛❧ ❝♦rr❡❧❛t✐♦♥ ❛♥❛❧②s✐s ✇❛s ❛♣♣❧✐❡❞ ✐♥ ❍❡ ❡t ❛❧✳ ✭✷✵✶✵✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✷ ✴ ✺✺

slide-23
SLIDE 23

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥
  • ♦❋ ❧✐t❡r❛t✉r❡ ✐♥ t❤❡ ❋▲▼❙❘ ❝♦♥t❡①t

❊①t❡♥❞✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ❊s❝❛♥❝✐❛♥♦ ✭✷✵✵✻✮✱ ❛ ●♦❋ t❡st ✇❛s ❞❡r✐✈❡❞ ❜②

  • ❛r❝í❛✲P♦rt✉❣✉és ❡t ❛❧✳ ✭✷✵✶✹✮ ❢♦r t❤❡ ❋▲▼❙❘✱ ✐♥ t❡r♠s ♦❢ t❤❡ ♣r♦✲

❥❡❝t❡❞ ❡♠♣✐r✐❝❛❧ ❡st✐♠❛t♦r ♦❢ t❤❡ ✐♥t❡❣r❛t❡❞ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥✳

❈✉❡st❛✲❆❧❜❡rt♦s ❡t ❛❧✳ ✭✷✵✶✾✮ r❡❝❡♥t❧② s✉❣❣❡st ❛ st❛t✐st✐❝ t❡st ❞❡✲

♣❡♥❞✐♥❣ ♦♥ ❛♥ ✉♥✐q✉❡❧② r❛♥❞♦♠❧② ♣r♦❥❡❝t❡❞ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡✱ ♣r♦✈✐❞✐♥❣ ❛ ❧❡ss ♣♦✇❡r❢✉❧ ❜✉t ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡✣❝✐❡♥t t❡st✳

  • ♦❋ t❡sts ❢♦r t❤❡ ❋▲▼❙❘✱ ✇❤❡♥ s♦♠❡ ♦❢ t❤❡ r❡s♣♦♥s❡s ❛r❡ ♠✐ss✲

✐♥❣ ❛t r❛♥❞♦♠✱ ❤❛✈❡ r❡❝❡♥t❧② ♣r♦♣♦s❡❞ ❜② P✳ ●❛❧❡❛♥♦✱ ▼✳ ❋❡❜r❡r♦✲ ❇❛♥❞❡✱ ❲✳ ●♦♥③á❧❡③✲▼❛♥t❡✐❣❛ ❛♥❞ ❊✳ ●❛r❝í❛ P♦rt✉❣✉és ✭t♦ ❛♣♣❡❛r✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✸ ✴ ✺✺

slide-24
SLIDE 24

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥
  • ♦❋ t❡st r❡♠❛✐♥s ♦♣❡♥ ✐♥ t❤❡ ❋▲▼❋❘ s❡t✉♣

❚❤❡ ●♦❋ t❡st ❢r❛♠❡✇♦r❦ r❡♠❛✐♥s ♦♣❡♥ ✐♥ t❤❡ ❋▲▼❋❘ s❡t✉♣✿

❈❤✐♦✉ ❛♥❞ ▼ü❧❧❡r ✭✷✵✵✼✮ ❞❡✈❡❧♦♣❡❞ ♦♥❧② ❛ r❡s✐❞✉❛❧ ❞✐❛❣♥♦st✐❝✳ ❑♦❦♦s③❦❛ ❡t ❛❧✳ ✭✷✵✵✽✮ t❡st❡❞ t❤❡ ❧❛❝❦ ♦❢ ❡✛❡❝t✱ ❜✉t ♥♦♥❧✐♥❡❛r ❛❧t❡r♥❛t✐✈❡s ❝❛♥♥♦t ❜❡ ❞❡t❡❝t❡❞✳ P❛t✐❧❡❛ ❡t ❛❧✳ ✭✷✵✶✻✮ ❢♦r♠✉❧❛t❡❞ ❛ s✐♠♣❧❡ ❤②♣♦t❤❡s✐s t❡st ❝❛♣t✉r✐♥❣ ♥♦♥❧✐♥❡❛r ❛❧t❡r♥❛t✐✈❡s✳ ❲❛♥❣ ❡t ❛❧✳ ✭✷✵✶✽✮ ❞❡✈❡❧♦♣❡❞ ❡♠♣✐r✐❝❛❧ ❧✐❦❡❧✐❤♦♦❞ r❛t✐♦ t❡sts ♦♥❧② ❢♦r ❝♦♥❝✉rr❡♥t ♠♦❞❡❧s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✹ ✴ ✺✺

slide-25
SLIDE 25

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❆◆❉✳✳✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✺ ✴ ✺✺

slide-26
SLIDE 26

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❆◆❉✳✳✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✺ ✴ ✺✺

slide-27
SLIDE 27

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

  • ✐✈❡♥

❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s

X ✳✳✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✻ ✴ ✺✺

slide-28
SLIDE 28

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

  • ✐✈❡♥

❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s

X ✳✳✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✻ ✴ ✺✺

slide-29
SLIDE 29

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

  • ✐✈❡♥ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s

X

❛♥❞

Y ✳✳✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✻ ✴ ✺✺

slide-30
SLIDE 30

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

  • ✐✈❡♥

❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s

X

❛♥❞

Y ✱

✇❡ ✇❛♥t t♦ ❛ss❡s ✐❢ t❤❡r❡ ❡①✐sts ❛ ❧✐♥❡❛r r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠✳✳✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✻ ✴ ✺✺

slide-31
SLIDE 31

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

  • ✐✈❡♥

❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s

X

❛♥❞

Y ✱

✇❡ ✇❛♥t t♦ ❛ss❡s ✐❢ t❤❡r❡ ❡①✐sts ❛ ❧✐♥❡❛r r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠✳✳✳

H✵✿ ❨

✭❛ ❧✐♥❡❛r ♠❛rr✐❛❣❡✮

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✻ ✴ ✺✺

slide-32
SLIDE 32

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

  • ✐✈❡♥

❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s

X

❛♥❞

Y ✱

✇❡ ✇❛♥t t♦ ❛ss❡s ✐❢ t❤❡r❡ ❡①✐sts ❛ ❧✐♥❡❛r r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠✳✳✳

H✵✿ ❨

✭❛ ❧✐♥❡❛r ♠❛rr✐❛❣❡✮

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✻ ✴ ✺✺

slide-33
SLIDE 33

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

  • ✐✈❡♥ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s

X

❛♥❞

Y ✱ ✇❡ ✇❛♥t t♦ ❛ss❡s

✐❢ t❤❡r❡ ❡①✐sts ❛ ❧✐♥❡❛r r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠ ♦r ♥♦t

H✶✿ ❨

✭❛ ♥♦♥❧✐♥❡❛r ❛♥❞ tr♦✉❜❧❡❞ ♠❛rr✐❛❣❡✮

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✻ ✴ ✺✺

slide-34
SLIDE 34

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

▲❡✐t♠♦t✐❢✿ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st

  • ✐✈❡♥ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s

X

❛♥❞

Y ✱ ✇❡ ✇❛♥t t♦ ❛ss❡s

✐❢ t❤❡r❡ ❡①✐sts ❛ ❧✐♥❡❛r r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠ ♦r ♥♦t

H✶✿ ❨

✭❛ ♥♦♥❧✐♥❡❛r ❛♥❞ tr♦✉❜❧❡❞ ♠❛rr✐❛❣❡✮

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✻ ✴ ✺✺

slide-35
SLIDE 35

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

▼❛✐♥ ♦❜❥❡❝t✐✈❡s

❲❡ ❡st❛❜❧✐s❤ ❤❡r❡ ❛ ❈r❛♠ér✕✈♦♥ ▼✐s❡s st❛t✐st✐❝ ❢♦r t❤❡ ♥✉❧❧

❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s ✭❧✐♥❡❛r✐t②✮ H✵ : E[Y |X ] ∈ M =

  • mβ(X )(t) =

b

a

X (s)β(s,t)ds,

  • β✷ < ∞
  • ❆ ♥❡✇ ❤②❜r✐❞ ❛♥❞ ✢❡①✐❜❧❡ ❡st✐♠❛t♦r ♦❢ β ✐s ♣r♦♣♦s❡❞✱ ♣❡r❢♦r♠✲

✐♥❣ ❛ ❧✐♥❡❛r❧② ❝♦♥str❛✐♥❡❞ ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦r ❞r✐✈❡♥ ❜② ▲❆❙❙❖✳

❆♥ ♦♣t✐♠✐③❡❞ ❘ ♣❛❝❦❛❣❡ ✭❣♦❢❢❞❛✮ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞✿ ♦✉r ●♦❋

t❡st ♣r♦♣♦s❛❧ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❛❧❧ ❢✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧s✱ ❡✐t❤❡r s❝❛❧❛r ♦r ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡s✱ ❡✐t❤❡r s❝❛❧❛r ♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss♦rs✳

❚❤❡ ✜♥✐t❡ s❛♠♣❧❡ ❜❡❤❛✈✐♦✉r ✐s ♥✉♠❡r✐❝❛❧❧② ✐❧❧✉str❛t❡❞✱ ✉♥❞❡r

❞✐✛❡r❡♥t ♥✉❧❧ ❤②♣♦t❤❡s❡s✿ ♥♦ ❡✛❡❝ts✱ ❝♦♥❝✉rr❡♥t ♠♦❞❡❧ ❛♥❞ ❋▲▼❋❘✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✼ ✴ ✺✺

slide-36
SLIDE 36

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

▼❛✐♥ ♦❜❥❡❝t✐✈❡s

❲❡ ❡st❛❜❧✐s❤ ❤❡r❡ ❛ ❈r❛♠ér✕✈♦♥ ▼✐s❡s st❛t✐st✐❝ ❢♦r t❤❡ ♥✉❧❧

❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s ✭❧✐♥❡❛r✐t②✮ H✵ : E[Y |X ] ∈ M =

  • mβ(X )(t) =

b

a

X (s)β(s,t)ds,

  • β✷ < ∞
  • ❆ ♥❡✇ ❤②❜r✐❞ ❛♥❞ ✢❡①✐❜❧❡ ❡st✐♠❛t♦r ♦❢ β ✐s ♣r♦♣♦s❡❞✱ ♣❡r❢♦r♠✲

✐♥❣ ❛ ❧✐♥❡❛r❧② ❝♦♥str❛✐♥❡❞ ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦r ❞r✐✈❡♥ ❜② ▲❆❙❙❖✳

❆♥ ♦♣t✐♠✐③❡❞ ❘ ♣❛❝❦❛❣❡ ✭❣♦❢❢❞❛✮ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞✿ ♦✉r ●♦❋

t❡st ♣r♦♣♦s❛❧ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❛❧❧ ❢✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧s✱ ❡✐t❤❡r s❝❛❧❛r ♦r ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡s✱ ❡✐t❤❡r s❝❛❧❛r ♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss♦rs✳

❚❤❡ ✜♥✐t❡ s❛♠♣❧❡ ❜❡❤❛✈✐♦✉r ✐s ♥✉♠❡r✐❝❛❧❧② ✐❧❧✉str❛t❡❞✱ ✉♥❞❡r

❞✐✛❡r❡♥t ♥✉❧❧ ❤②♣♦t❤❡s❡s✿ ♥♦ ❡✛❡❝ts✱ ❝♦♥❝✉rr❡♥t ♠♦❞❡❧ ❛♥❞ ❋▲▼❋❘✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✼ ✴ ✺✺

slide-37
SLIDE 37

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

▼❛✐♥ ♦❜❥❡❝t✐✈❡s

❲❡ ❡st❛❜❧✐s❤ ❤❡r❡ ❛ ❈r❛♠ér✕✈♦♥ ▼✐s❡s st❛t✐st✐❝ ❢♦r t❤❡ ♥✉❧❧

❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s ✭❧✐♥❡❛r✐t②✮ H✵ : E[Y |X ] ∈ M =

  • mβ(X )(t) =

b

a

X (s)β(s,t)ds,

  • β✷ < ∞
  • ❆ ♥❡✇ ❤②❜r✐❞ ❛♥❞ ✢❡①✐❜❧❡ ❡st✐♠❛t♦r ♦❢ β ✐s ♣r♦♣♦s❡❞✱ ♣❡r❢♦r♠✲

✐♥❣ ❛ ❧✐♥❡❛r❧② ❝♦♥str❛✐♥❡❞ ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦r ❞r✐✈❡♥ ❜② ▲❆❙❙❖✳

❆♥ ♦♣t✐♠✐③❡❞ ❘ ♣❛❝❦❛❣❡ ✭❣♦❢❢❞❛✮ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞✿ ♦✉r ●♦❋

t❡st ♣r♦♣♦s❛❧ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❛❧❧ ❢✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧s✱ ❡✐t❤❡r s❝❛❧❛r ♦r ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡s✱ ❡✐t❤❡r s❝❛❧❛r ♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss♦rs✳

❚❤❡ ✜♥✐t❡ s❛♠♣❧❡ ❜❡❤❛✈✐♦✉r ✐s ♥✉♠❡r✐❝❛❧❧② ✐❧❧✉str❛t❡❞✱ ✉♥❞❡r

❞✐✛❡r❡♥t ♥✉❧❧ ❤②♣♦t❤❡s❡s✿ ♥♦ ❡✛❡❝ts✱ ❝♦♥❝✉rr❡♥t ♠♦❞❡❧ ❛♥❞ ❋▲▼❋❘✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✼ ✴ ✺✺

slide-38
SLIDE 38

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

▼❛✐♥ ♦❜❥❡❝t✐✈❡s

❲❡ ❡st❛❜❧✐s❤ ❤❡r❡ ❛ ❈r❛♠ér✕✈♦♥ ▼✐s❡s st❛t✐st✐❝ ❢♦r t❤❡ ♥✉❧❧

❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s ✭❧✐♥❡❛r✐t②✮ H✵ : E[Y |X ] ∈ M =

  • mβ(X )(t) =

b

a

X (s)β(s,t)ds,

  • β✷ < ∞
  • ❆ ♥❡✇ ❤②❜r✐❞ ❛♥❞ ✢❡①✐❜❧❡ ❡st✐♠❛t♦r ♦❢ β ✐s ♣r♦♣♦s❡❞✱ ♣❡r❢♦r♠✲

✐♥❣ ❛ ❧✐♥❡❛r❧② ❝♦♥str❛✐♥❡❞ ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦r ❞r✐✈❡♥ ❜② ▲❆❙❙❖✳

❆♥ ♦♣t✐♠✐③❡❞ ❘ ♣❛❝❦❛❣❡ ✭❣♦❢❢❞❛✮ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞✿ ♦✉r ●♦❋

t❡st ♣r♦♣♦s❛❧ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❛❧❧ ❢✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧s✱ ❡✐t❤❡r s❝❛❧❛r ♦r ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡s✱ ❡✐t❤❡r s❝❛❧❛r ♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss♦rs✳

❚❤❡ ✜♥✐t❡ s❛♠♣❧❡ ❜❡❤❛✈✐♦✉r ✐s ♥✉♠❡r✐❝❛❧❧② ✐❧❧✉str❛t❡❞✱ ✉♥❞❡r

❞✐✛❡r❡♥t ♥✉❧❧ ❤②♣♦t❤❡s❡s✿ ♥♦ ❡✛❡❝ts✱ ❝♦♥❝✉rr❡♥t ♠♦❞❡❧ ❛♥❞ ❋▲▼❋❘✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✼ ✴ ✺✺

slide-39
SLIDE 39

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ✐♥ t❡r♠s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s

▲❡t X ❛♥❞ Y ❜❡ ❢✉♥❝t✐♦♥❛❧ r✳✈✳

✐♥ H✶ = L✷([a,b]) ❛♥❞ H✷ = L✷([c,d])✳ ❙❡❡ ❇❡♥❛t✐❛ ❡t ❛❧✳ ✭✷✵✶✼✮ ❛❜♦✉t ❋▲▼❋❘ ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳

Y = m(X )+E, E[E |X ] = ✵,

m(X ) = E[Y |X ] =

b

a

K (s,t,X (s))ds

❆s ✉s✉❛❧✱ ♠ ✐s ❛ ❍✐❧❜❡rt✕❙❝❤♠✐❞t ♦♣❡r❛t♦r ✭❛♥❞ t❤✉s✱ ❝♦♠♣❛❝t✮✳ ❲❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t❤❡ ♣❛r❛♠❡tr✐❝ ❢r❛♠❡✇♦r❦ ✭♠ ≡ ♠β✮✿

♠β(X ) =

β(s,t)X (s)❞s, d

c

b

a

β✷(s,t)dsdt < ∞

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✽ ✴ ✺✺

slide-40
SLIDE 40

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s ❋✐①❡❞ ❜❛s❡s✿ ✇❛✈❡❧❡ts✱ ❋♦✉r✐❡r ❜❛s❡s✱ ❇✕s♣❧✐♥❡s✱ ❡t❝ ❋✐①❡❞ ❜❛s❡s ❝♦✉❧❞ ♥♦t ❜❡ ✢❡①✐❜❧❡ ❢♦r ❛ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦✳ ▼❛①✐♠✐③✐♥❣ t❤❡ ❡①♣❧❛✐♥❡❞ ✈❛r✐❛♥❝❡✳ ❚❤❡ ♠♦st ❡✛❡❝t✐✈❡ ✇❛② ♦❢ s✉♠♠❛r✐③✐♥❣✿ ❋✉♥❝t✐♦♥❛❧ Pr✐♥❝✐♣❛❧ ❈♦♠♣♦♥❡♥ts ✭❋P❈✮ ❜❛s❡s✳ ▼❛①✐♠✐③✐♥❣ t❤❡ ♣r❡❞✐❝t✐✈❡ ♣❡r❢♦r♠❛♥❝❡✳ ❋✉♥❝t✐♦♥❛❧ P❛rt✐❛❧ ▲❡❛st ❙q✉❛r❡s ✭❋P▲❙✮ ❜❛s❡s✳ ❚❤❡ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❋▲▼❋❘ s❡t✉♣ ✐s ♥♦t tr✐✈✐❛❧ ❛♥❞ ✐t r❡♠❛✐♥s ♦♣❡♥✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✾ ✴ ✺✺

slide-41
SLIDE 41

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s ❋✐①❡❞ ❜❛s❡s✿ ✇❛✈❡❧❡ts✱ ❋♦✉r✐❡r ❜❛s❡s✱ ❇✕s♣❧✐♥❡s✱ ❡t❝ ❋✐①❡❞ ❜❛s❡s ❝♦✉❧❞ ♥♦t ❜❡ ✢❡①✐❜❧❡ ❢♦r ❛ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦✳ ▼❛①✐♠✐③✐♥❣ t❤❡ ❡①♣❧❛✐♥❡❞ ✈❛r✐❛♥❝❡✳ ❚❤❡ ♠♦st ❡✛❡❝t✐✈❡ ✇❛② ♦❢ s✉♠♠❛r✐③✐♥❣✿ ❋✉♥❝t✐♦♥❛❧ Pr✐♥❝✐♣❛❧ ❈♦♠♣♦♥❡♥ts ✭❋P❈✮ ❜❛s❡s✳ ▼❛①✐♠✐③✐♥❣ t❤❡ ♣r❡❞✐❝t✐✈❡ ♣❡r❢♦r♠❛♥❝❡✳ ❋✉♥❝t✐♦♥❛❧ P❛rt✐❛❧ ▲❡❛st ❙q✉❛r❡s ✭❋P▲❙✮ ❜❛s❡s✳ ❚❤❡ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❋▲▼❋❘ s❡t✉♣ ✐s ♥♦t tr✐✈✐❛❧ ❛♥❞ ✐t r❡♠❛✐♥s ♦♣❡♥✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✾ ✴ ✺✺

slide-42
SLIDE 42

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❖♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s ❋✐①❡❞ ❜❛s❡s✿ ✇❛✈❡❧❡ts✱ ❋♦✉r✐❡r ❜❛s❡s✱ ❇✕s♣❧✐♥❡s✱ ❡t❝ ❋✐①❡❞ ❜❛s❡s ❝♦✉❧❞ ♥♦t ❜❡ ✢❡①✐❜❧❡ ❢♦r ❛ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦✳ ▼❛①✐♠✐③✐♥❣ t❤❡ ❡①♣❧❛✐♥❡❞ ✈❛r✐❛♥❝❡✳ ❚❤❡ ♠♦st ❡✛❡❝t✐✈❡ ✇❛② ♦❢ s✉♠♠❛r✐③✐♥❣✿ ❋✉♥❝t✐♦♥❛❧ Pr✐♥❝✐♣❛❧ ❈♦♠♣♦♥❡♥ts ✭❋P❈✮ ❜❛s❡s✳ ▼❛①✐♠✐③✐♥❣ t❤❡ ♣r❡❞✐❝t✐✈❡ ♣❡r❢♦r♠❛♥❝❡✳ ❋✉♥❝t✐♦♥❛❧ P❛rt✐❛❧ ▲❡❛st ❙q✉❛r❡s ✭❋P▲❙✮ ❜❛s❡s✳ ❚❤❡ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❋▲▼❋❘ s❡t✉♣ ✐s ♥♦t tr✐✈✐❛❧ ❛♥❞ ✐t r❡♠❛✐♥s ♦♣❡♥✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✶✾ ✴ ✺✺

slide-43
SLIDE 43

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ✐♥ t❡r♠s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s

❋✉♥❝t✐♦♥❛❧ r❡❣r❡ss♦r X ✐s ❞❡❝♦♠♣♦s❡❞ ✐♥ t❡r♠s ♦❢ ❋P❈

  • Ψ❥

❥=✶

❋✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡ Y ✐s ❞❡❝♦♠♣♦s❡❞ ✐♥ t❡r♠s ♦❢ ❋P❈

  • Φ❦

❦=✶

Xi =

  • j=✶

xi,jΨj, xi,j = 〈Xi,Ψj〉H✶,

Yi =

  • k=✶

yi,kΦk, yi,k = 〈Yi,Φk〉H✷

❆s ✉s✉❛❧✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ tr✉♥❝❛t❡ t❤❡ ❜❛s✐s ❡①♣❛♥s✐♦♥s

X (p)

i

=

p

  • j=✶

xi,jΨj,

Ψ = (〈Ψj, Ψl〉H✶)j,l=✶,...,p , Y (q)

i

=

q

  • k=✶

yi,kΦk

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✵ ✴ ✺✺

slide-44
SLIDE 44

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ✐♥ t❡r♠s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s

X (p)

i

=

p

  • j=✶

xi,jΨj,

Ψ = (〈Ψj, Ψl〉H✶)j,l=✶,...,p , Y (q)

i

=

q

  • k=✶

yi,kΦk

❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ❞❡♥♦t❡ t❤❡ ❧✐♥❡❛r ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ❛s

〈〈·,⋆〉〉: H✶ ×(H✶ ⊗H✷) − → H✷, 〈〈X ,β〉〉 = 〈X ,βt〉H✶, βt := β(·,t)

〈〈X (p),β(p,q)〉〉 = p

  • j=✶

xjΨj,

p

  • l=✶

q

  • k=✶

blkΨl ⊗Φk

  • =

p

  • j=✶

p

  • l=✶

q

  • k=✶

blkxj〈Ψj,Ψl〉H✶Φk

❨q = ❳pΨ❇p,q +❊q, ❨q = (yi,k)k=✶,...,q

i=✶,...,n ,

❳p = (xi,j)j=✶,...,p

i=✶,...,n

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✶ ✴ ✺✺

slide-45
SLIDE 45

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ✐♥ t❡r♠s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s

X (p)

i

=

p

  • j=✶

xi,jΨj,

Ψ = (〈Ψj, Ψl〉H✶)j,l=✶,...,p , Y (q)

i

=

q

  • k=✶

yi,kΦk

❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ❞❡♥♦t❡ t❤❡ ❧✐♥❡❛r ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ❛s

〈〈·,⋆〉〉: H✶ ×(H✶ ⊗H✷) − → H✷, 〈〈X ,β〉〉 = 〈X ,βt〉H✶, βt := β(·,t)

〈〈X (p),β(p,q)〉〉 = p

  • j=✶

xjΨj,

p

  • l=✶

q

  • k=✶

blkΨl ⊗Φk

  • =

p

  • j=✶

p

  • l=✶

q

  • k=✶

blkxj〈Ψj,Ψl〉H✶Φk

❨q = ❳pΨ❇p,q +❊q, ❨q = (yi,k)k=✶,...,q

i=✶,...,n ,

❳p = (xi,j)j=✶,...,p

i=✶,...,n

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✶ ✴ ✺✺

slide-46
SLIDE 46

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ✐♥ t❡r♠s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s

X (p)

i

=

p

  • j=✶

xi,jΨj,

Ψ = (〈Ψj, Ψl〉H✶)j,l=✶,...,p , Y (q)

i

=

q

  • k=✶

yi,kΦk

❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ❞❡♥♦t❡ t❤❡ ❧✐♥❡❛r ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ❛s

〈〈·,⋆〉〉: H✶ ×(H✶ ⊗H✷) − → H✷, 〈〈X ,β〉〉 = 〈X ,βt〉H✶, βt := β(·,t)

〈〈X (p),β(p,q)〉〉 = p

  • j=✶

xjΨj,

p

  • l=✶

q

  • k=✶

blkΨl ⊗Φk

  • =

p

  • j=✶

p

  • l=✶

q

  • k=✶

blkxj〈Ψj,Ψl〉H✶Φk

❨q = ❳pΨ❇p,q +❊q, ❨q = (yi,k)k=✶,...,q

i=✶,...,n ,

❳p = (xi,j)j=✶,...,p

i=✶,...,n

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✶ ✴ ✺✺

slide-47
SLIDE 47

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

  • ✐✈❡♥ {(Xi,Yi)}n

i=✶✱ ❘❛♠s❛② ❛♥❞ ❙✐❧✈❡r♠❛♥ ✭✶✾✾✼✮ ♣r♦♣♦s❡❞ ❛♥ ❡①✲

t❡♥❞❡❞ ❖r❞✐♥❛r② ▲❡❛st ❙q✉❛r❡s ✭❖▲❙ ♦r ❋P❈❘✮ ❡st✐♠❛t♦r

  • ❇p,q

=

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ❨q,

  • ❨q = ❳pΨ

❇p,q = ❍❨q (❨q)i,k

=

yi,k, (❳p)i,j = xi,j, (❇p,q)j,k = bj,k, ❍ ❤❛t ♠❛tr✐①

❍♦✇ ❝❛♥ ✇❡ ❝✐r❝✉♠✈❡♥t t❤❡ ❛ ♣r✐♦r✐ s❡❧❡❝t✐♦♥ ♦❢ ❧❡✈❡❧s p ❛♥❞ q❄

❊①t❡♥❞✐♥❣ t❤❡ P❈❱ ❝r✐t❡r✐♦♥ ✭s❡❡ Pr❡❞❛ ❛♥❞ ❙❛♣♦rt❛ ✭✷✵✵✺✮✮ P❈❱ ✐♥ ❋▲▼❋❘ ❝♦♥t❡①t ②✐❡❧❞s t♦ ❛♥ ✐♥❡✣❝✐❡♥t ♣r♦❜❧❡♠✳ ❊①t❡♥❞✐♥❣ t❤❡ ●❈❱ ♣r♦❝❡❞✉r❡ ✭s❡❡ ❈❛r❞♦t ❡t ❛❧✳ ✭✷✵✵✸✮✮

  • ❈❱ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ ♥♦t tr✐✈✐❛❧ t♦ ❡st✐♠❛t❡

✐♥ t❤❡ ❋▲▼❋❘ s❡t✉♣✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✷ ✴ ✺✺

slide-48
SLIDE 48

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

  • ✐✈❡♥ {(Xi,Yi)}n

i=✶✱ ❘❛♠s❛② ❛♥❞ ❙✐❧✈❡r♠❛♥ ✭✶✾✾✼✮ ♣r♦♣♦s❡❞ ❛♥ ❡①✲

t❡♥❞❡❞ ❖r❞✐♥❛r② ▲❡❛st ❙q✉❛r❡s ✭❖▲❙ ♦r ❋P❈❘✮ ❡st✐♠❛t♦r

  • ❇p,q

=

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ❨q,

  • ❨q = ❳pΨ

❇p,q = ❍❨q (❨q)i,k

=

yi,k, (❳p)i,j = xi,j, (❇p,q)j,k = bj,k, ❍ ❤❛t ♠❛tr✐①

❍♦✇ ❝❛♥ ✇❡ ❝✐r❝✉♠✈❡♥t t❤❡ ❛ ♣r✐♦r✐ s❡❧❡❝t✐♦♥ ♦❢ ❧❡✈❡❧s p ❛♥❞ q❄

❊①t❡♥❞✐♥❣ t❤❡ P❈❱ ❝r✐t❡r✐♦♥ ✭s❡❡ Pr❡❞❛ ❛♥❞ ❙❛♣♦rt❛ ✭✷✵✵✺✮✮ P❈❱ ✐♥ ❋▲▼❋❘ ❝♦♥t❡①t ②✐❡❧❞s t♦ ❛♥ ✐♥❡✣❝✐❡♥t ♣r♦❜❧❡♠✳ ❊①t❡♥❞✐♥❣ t❤❡ ●❈❱ ♣r♦❝❡❞✉r❡ ✭s❡❡ ❈❛r❞♦t ❡t ❛❧✳ ✭✷✵✵✸✮✮

  • ❈❱ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ ♥♦t tr✐✈✐❛❧ t♦ ❡st✐♠❛t❡

✐♥ t❤❡ ❋▲▼❋❘ s❡t✉♣✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✷ ✴ ✺✺

slide-49
SLIDE 49

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

  • ✐✈❡♥ {(Xi,Yi)}n

i=✶✱ ❘❛♠s❛② ❛♥❞ ❙✐❧✈❡r♠❛♥ ✭✶✾✾✼✮ ♣r♦♣♦s❡❞ ❛♥ ❡①✲

t❡♥❞❡❞ ❖r❞✐♥❛r② ▲❡❛st ❙q✉❛r❡s ✭❖▲❙ ♦r ❋P❈❘✮ ❡st✐♠❛t♦r

  • ❇p,q

=

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ❨q,

  • ❨q = ❳pΨ

❇p,q = ❍❨q (❨q)i,k

=

yi,k, (❳p)i,j = xi,j, (❇p,q)j,k = bj,k, ❍ ❤❛t ♠❛tr✐①

❍♦✇ ❝❛♥ ✇❡ ❝✐r❝✉♠✈❡♥t t❤❡ ❛ ♣r✐♦r✐ s❡❧❡❝t✐♦♥ ♦❢ ❧❡✈❡❧s p ❛♥❞ q❄

❊①t❡♥❞✐♥❣ t❤❡ P❈❱ ❝r✐t❡r✐♦♥ ✭s❡❡ Pr❡❞❛ ❛♥❞ ❙❛♣♦rt❛ ✭✷✵✵✺✮✮ P❈❱ ✐♥ ❋▲▼❋❘ ❝♦♥t❡①t ②✐❡❧❞s t♦ ❛♥ ✐♥❡✣❝✐❡♥t ♣r♦❜❧❡♠✳ ❊①t❡♥❞✐♥❣ t❤❡ ●❈❱ ♣r♦❝❡❞✉r❡ ✭s❡❡ ❈❛r❞♦t ❡t ❛❧✳ ✭✷✵✵✸✮✮

  • ❈❱ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ ♥♦t tr✐✈✐❛❧ t♦ ❡st✐♠❛t❡

✐♥ t❤❡ ❋▲▼❋❘ s❡t✉♣✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✷ ✴ ✺✺

slide-50
SLIDE 50

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

  • ❇p,q

=

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ❨q,

  • ❨q = ❳pΨ

❇p,q = ❍❨q ❍

=

❳pΨ

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ,

❍ ❤❛t ♠❛tr✐①

❈✉t✲♦✛ ❧❡✈❡❧s ✇✐❧❧ ❜❡ s❡❧❡❝t❡❞ ❛s t❤❡ ❧♦✇❡st ✈❛❧✉❡s ❝❛♣t✉r✐♥❣ ❛ ♠✐♥✲

✐♠✉♠ ✪ ♦❢ ❡♠♣✐r✐❝❛❧ ✈❛r✐❛♥❝❡ ♦❢ {X }i=✶,...,n ❛♥❞ {Y }i=✶,...,n✳ ❆♥ ❤❛t ♠❛tr✐① ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞✳ ▲❛r❣❡ ❛♥❞ ✭r❡s♣❡❝t t♦ ✮ ❧❡❛❞ t♦ ❛♥ ✐❧❧✲❝♦♥❞✐t✐♦♥❡❞ ♣r♦❜❧❡♠✳ ❙❡❧❡❝t✐♥❣ ❛♥❞ ❝♦♠♣♦♥❡♥ts ❢♦r ❡①♣❧❛✐♥✐♥❣ ❛♥❞ ✱ ❞♦❡s ♥♦t ✐♠♣❧✐❡s t❤❛t t❤❡② ❛r❡ t❤❡ ❜❡st ♦♥❡s ❢♦r ♣r❡❞✐❝t✐♥❣✿ ♥✉❧❧ ❝♦❡✣❝✐❡♥ts ♦❢ ❇ ♠❛② ❜❡ ❜❡✐♥❣ ❡st✐♠❛t❡❞ ✭♦✈❡r✜tt✐♥❣ ♣r♦❜❧❡♠✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✸ ✴ ✺✺

slide-51
SLIDE 51

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

  • ❇p,q

=

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ❨q,

  • ❨q = ❳pΨ

❇p,q = ❍❨q ❍

=

❳pΨ

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ,

❍ ❤❛t ♠❛tr✐①

❈✉t✲♦✛ ❧❡✈❡❧s ✇✐❧❧ ❜❡ s❡❧❡❝t❡❞ ❛s t❤❡ ❧♦✇❡st ✈❛❧✉❡s ❝❛♣t✉r✐♥❣ ❛ ♠✐♥✲

✐♠✉♠ ✪ ♦❢ ❡♠♣✐r✐❝❛❧ ✈❛r✐❛♥❝❡ ♦❢ {X }i=✶,...,n ❛♥❞ {Y }i=✶,...,n✳

❆♥ ❤❛t ♠❛tr✐① ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞✳

▲❛r❣❡ ❛♥❞ ✭r❡s♣❡❝t t♦ ✮ ❧❡❛❞ t♦ ❛♥ ✐❧❧✲❝♦♥❞✐t✐♦♥❡❞ ♣r♦❜❧❡♠✳ ❙❡❧❡❝t✐♥❣ ❛♥❞ ❝♦♠♣♦♥❡♥ts ❢♦r ❡①♣❧❛✐♥✐♥❣ ❛♥❞ ✱ ❞♦❡s ♥♦t ✐♠♣❧✐❡s t❤❛t t❤❡② ❛r❡ t❤❡ ❜❡st ♦♥❡s ❢♦r ♣r❡❞✐❝t✐♥❣✿ ♥✉❧❧ ❝♦❡✣❝✐❡♥ts ♦❢ ❇ ♠❛② ❜❡ ❜❡✐♥❣ ❡st✐♠❛t❡❞ ✭♦✈❡r✜tt✐♥❣ ♣r♦❜❧❡♠✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✸ ✴ ✺✺

slide-52
SLIDE 52

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

  • ❇p,q

=

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ❨q,

  • ❨q = ❳pΨ

❇p,q = ❍❨q ❍

=

❳pΨ

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T ,

❍ ❤❛t ♠❛tr✐①

❈✉t✲♦✛ ❧❡✈❡❧s ✇✐❧❧ ❜❡ s❡❧❡❝t❡❞ ❛s t❤❡ ❧♦✇❡st ✈❛❧✉❡s ❝❛♣t✉r✐♥❣ ❛ ♠✐♥✲

✐♠✉♠ ✪ ♦❢ ❡♠♣✐r✐❝❛❧ ✈❛r✐❛♥❝❡ ♦❢ {X }i=✶,...,n ❛♥❞ {Y }i=✶,...,n✳

❆♥ ❤❛t ♠❛tr✐① ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞✳

− ▲❛r❣❡ p ❛♥❞ q ✭r❡s♣❡❝t t♦ n✮ ❧❡❛❞ t♦ ❛♥ ✐❧❧✲❝♦♥❞✐t✐♦♥❡❞ ♣r♦❜❧❡♠✳ − ❙❡❧❡❝t✐♥❣ p ❛♥❞ q ❝♦♠♣♦♥❡♥ts ❢♦r ❡①♣❧❛✐♥✐♥❣ X ❛♥❞ Y ✱ ❞♦❡s ♥♦t ✐♠♣❧✐❡s t❤❛t t❤❡② ❛r❡ t❤❡ ❜❡st ♦♥❡s ❢♦r ♣r❡❞✐❝t✐♥❣✿ ♥✉❧❧ ❝♦❡✣❝✐❡♥ts ♦❢ ❇p,q ♠❛② ❜❡ ❜❡✐♥❣ ❡st✐♠❛t❡❞ ✭♦✈❡r✜tt✐♥❣ ♣r♦❜❧❡♠✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✸ ✴ ✺✺

slide-53
SLIDE 53

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❆❧t❡r♥❛t✐✈❡s❄ ▲❛ss♦ ✭▲✶✮ ❛♥❞ ❘✐❞❣❡ ✭▲✷✮ t❡❝❤♥✐q✉❡s✿

λ

p,q = argmin ❇p,q

  • ❨q −❳pΨ❇p,q
  • ✷ +λ
  • α
  • ❇p,q
  • ✶ +(✶−α)
  • ❇p,q

  • .

▲✶ ✭ ✶✮ s❤r✐♥❦s t❤❡ ❧❡ss ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s ❝♦❡✣❝✐❡♥ts t♦ ③❡r♦✱ ♣❡r❢♦r♠✐♥❣ ❛ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥✿ ❝♦♠♣♦♥❡♥ts ❛r❡ s❡❧❡❝t❡❞✳ ❆♥ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐① ❝❛♥♥♦t ❜❡ ❝♦♠♣✉t❡❞ ❢♦r ▲✶ ❡st✐♠❛t♦r✳ ❋P❈❘ ❛♥❞ ▲✷ ✭ ✵✮ ❡st✐♠❛t♦rs ♣r♦✈✐❞❡ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐❝❡s✳ ❊①♣❧❛✐♥❡❞ ✈❛r✐❛♥❝❡ t❤r❡s❤♦❧❞s ❛r❡ ❥✉st ✜①❡❞ ❢♦r s❡❧❡❝t✐♥❣ ❛♥❞ ✿ ❛ ❞✐♠❡♥s✐♦♥❛❧✐t② ♣r♦❜❧❡♠ ❛r✐s❡ ❢♦r ❜♦t❤ ❡st✐♠❛t♦rs✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✹ ✴ ✺✺

slide-54
SLIDE 54

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❆❧t❡r♥❛t✐✈❡s❄ ▲❛ss♦ ✭▲✶✮ ❛♥❞ ❘✐❞❣❡ ✭▲✷✮ t❡❝❤♥✐q✉❡s✿

λ

p,q = argmin ❇p,q

  • ❨q −❳pΨ❇p,q
  • ✷ +λ
  • α
  • ❇p,q
  • ✶ +(✶−α)
  • ❇p,q

  • .

▲✶ ✭α = ✶✮ s❤r✐♥❦s t❤❡ ❧❡ss ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s ❝♦❡✣❝✐❡♥ts t♦ ③❡r♦✱ ♣❡r❢♦r♠✐♥❣ ❛ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥✿ p∗ ❝♦♠♣♦♥❡♥ts ❛r❡ s❡❧❡❝t❡❞✳ ❆♥ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐① ❝❛♥♥♦t ❜❡ ❝♦♠♣✉t❡❞ ❢♦r ▲✶ ❡st✐♠❛t♦r✳ ❋P❈❘ ❛♥❞ ▲✷ ✭α = ✵✮ ❡st✐♠❛t♦rs ♣r♦✈✐❞❡ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐❝❡s✳ ❊①♣❧❛✐♥❡❞ ✈❛r✐❛♥❝❡ t❤r❡s❤♦❧❞s ❛r❡ ❥✉st ✜①❡❞ ❢♦r s❡❧❡❝t✐♥❣ p ❛♥❞ q✿ ❛ ❞✐♠❡♥s✐♦♥❛❧✐t② ♣r♦❜❧❡♠ ❛r✐s❡ ❢♦r ❜♦t❤ ❡st✐♠❛t♦rs✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✹ ✴ ✺✺

slide-55
SLIDE 55

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❆❧t❡r♥❛t✐✈❡s❄ ▲❛ss♦ ✭▲✶✮ ❛♥❞ ❘✐❞❣❡ ✭▲✷✮ t❡❝❤♥✐q✉❡s✿

λ

p,q = argmin ❇p,q

  • ❨q −❳pΨ❇p,q
  • ✷ +λ
  • α
  • ❇p,q
  • ✶ +(✶−α)
  • ❇p,q

  • .

▲✶ ✭α = ✶✮ s❤r✐♥❦s t❤❡ ❧❡ss ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s ❝♦❡✣❝✐❡♥ts t♦ ③❡r♦✱ ♣❡r❢♦r♠✐♥❣ ❛ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥✿ p∗ ❝♦♠♣♦♥❡♥ts ❛r❡ s❡❧❡❝t❡❞✳ ❆♥ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐① ❝❛♥♥♦t ❜❡ ❝♦♠♣✉t❡❞ ❢♦r ▲✶ ❡st✐♠❛t♦r✳ ❋P❈❘ ❛♥❞ ▲✷ ✭α = ✵✮ ❡st✐♠❛t♦rs ♣r♦✈✐❞❡ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐❝❡s✳ ❊①♣❧❛✐♥❡❞ ✈❛r✐❛♥❝❡ t❤r❡s❤♦❧❞s ❛r❡ ❥✉st ✜①❡❞ ❢♦r s❡❧❡❝t✐♥❣ p ❛♥❞ q✿ ❛ ❞✐♠❡♥s✐♦♥❛❧✐t② ♣r♦❜❧❡♠ ❛r✐s❡ ❢♦r ❜♦t❤ ❡st✐♠❛t♦rs✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✹ ✴ ✺✺

slide-56
SLIDE 56

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❲❡ ♣r♦♣♦s❡ ❛♥ ❤②❜r✐❞ ❋P❈❘ ❝♦♥str❛✐♥t ✭❋P❈❘✲▲✶✲❈✮ ❡st✐♠❛t♦r✿ ❲❡ ♣❡r❢♦r♠ ❛♥ ▲✶ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥✱ ♣r♦✈✐❞✐♥❣ ❛ ❜✐♥❛r② ♠❛tr✐① ❆

♣r♦✈✐❞✐♥❣ ✉s t❤❡ ✈❛r✐❛❜❧❡s t❤❛t s❤♦✉❧❞ ❜❡ ✐♥❝❧✉❞❡❞✳

❚❤❡ ❋P❈❘ ❡st✐♠❛t✐♦♥ ✐s t❤❡♥ ❛♣♣❧✐❡❞✱ ❛❞♦♣t✐♥❣ ❛s p ✭③❡r♦✐♥❣

♦♥❧② ❤❛♣♣❡♥s ✐♥ r♦✇s ♦❢ ❇p,q✮ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠♣♦♥❡♥ts p∗ ♣r❡✈✐♦✉s❧② s❡❧❡❝t❡❞✿ ♦✈❡r✜tt✐♥❣ ✐s ❛✈♦✐❞❡❞✳

❆♥ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐① ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛♥❞ ✐ts ❝♦♠♣✉t❛t✐♦♥❛❧

❛❞✈❛♥t❛❣❡s ❝❛♥ ❜❡ ❡①♣❧♦✐t❡❞✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✺ ✴ ✺✺

slide-57
SLIDE 57

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❲❡ ♣r♦♣♦s❡ ❛♥ ❤②❜r✐❞ ❋P❈❘ ❝♦♥str❛✐♥t ✭❋P❈❘✲▲✶✲❈✮ ❡st✐♠❛t♦r✿ ❲❡ ♣❡r❢♦r♠ ❛♥ ▲✶ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥✱ ♣r♦✈✐❞✐♥❣ ❛ ❜✐♥❛r② ♠❛tr✐① ❆

♣r♦✈✐❞✐♥❣ ✉s t❤❡ ✈❛r✐❛❜❧❡s t❤❛t s❤♦✉❧❞ ❜❡ ✐♥❝❧✉❞❡❞✳

❚❤❡ ❋P❈❘ ❡st✐♠❛t✐♦♥ ✐s t❤❡♥ ❛♣♣❧✐❡❞✱ ❛❞♦♣t✐♥❣ ❛s p ✭③❡r♦✐♥❣

♦♥❧② ❤❛♣♣❡♥s ✐♥ r♦✇s ♦❢ ❇p,q✮ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠♣♦♥❡♥ts p∗ ♣r❡✈✐♦✉s❧② s❡❧❡❝t❡❞✿ ♦✈❡r✜tt✐♥❣ ✐s ❛✈♦✐❞❡❞✳

❆♥ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐① ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛♥❞ ✐ts ❝♦♠♣✉t❛t✐♦♥❛❧

❛❞✈❛♥t❛❣❡s ❝❛♥ ❜❡ ❡①♣❧♦✐t❡❞✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✺ ✴ ✺✺

slide-58
SLIDE 58

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❲❡ ♣r♦♣♦s❡ ❛♥ ❤②❜r✐❞ ❋P❈❘ ❝♦♥str❛✐♥t ✭❋P❈❘✲▲✶✲❈✮ ❡st✐♠❛t♦r✿ ❲❡ ♣❡r❢♦r♠ ❛♥ ▲✶ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥✱ ♣r♦✈✐❞✐♥❣ ❛ ❜✐♥❛r② ♠❛tr✐① ❆

♣r♦✈✐❞✐♥❣ ✉s t❤❡ ✈❛r✐❛❜❧❡s t❤❛t s❤♦✉❧❞ ❜❡ ✐♥❝❧✉❞❡❞✳

❚❤❡ ❋P❈❘ ❡st✐♠❛t✐♦♥ ✐s t❤❡♥ ❛♣♣❧✐❡❞✱ ❛❞♦♣t✐♥❣ ❛s p ✭③❡r♦✐♥❣

♦♥❧② ❤❛♣♣❡♥s ✐♥ r♦✇s ♦❢ ❇p,q✮ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠♣♦♥❡♥ts p∗ ♣r❡✈✐♦✉s❧② s❡❧❡❝t❡❞✿ ♦✈❡r✜tt✐♥❣ ✐s ❛✈♦✐❞❡❞✳

❆♥ ❡①♣❧✐❝✐t ❤❛t ♠❛tr✐① ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛♥❞ ✐ts ❝♦♠♣✉t❛t✐♦♥❛❧

❛❞✈❛♥t❛❣❡s ❝❛♥ ❜❡ ❡①♣❧♦✐t❡❞✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✺ ✴ ✺✺

slide-59
SLIDE 59

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❲❡ ♣r♦♣♦s❡ ❛♥ ❤②❜r✐❞ ❋P❈❘ ❝♦♥str❛✐♥t ✭❋P❈❘✲▲✶✲❈✮ ❡st✐♠❛t♦r✿

❨q = ❳pΨ❇p,q, s✳t✳ ❆❇p,q = ✵

❝ q = ❳pΨ

❈ p,q = ❍❈❨q

❈ p,q =

  • ■p −
  • (❳pΨ)T ❳pΨ

−✶❆T ❉−✶❆

  • ❇p,q, ❉ = ❆
  • (❳pΨ)T ❳pΨ

−✶❆T

❍❈ = ❳pΨ

  • ■p −
  • (❳pΨ)T ❳pΨ

−✶❆T ❉−✶❆

  • (❳pΨ)T ❳pΨ

−✶(❳pΨ)T

  • ❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛
  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✻ ✴ ✺✺

slide-60
SLIDE 60

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❚♦ ❝♦♠♣❛r❡ t❤❡ ❛❞❡q✉❛❝② t♦ ♦✉r s❡t✉♣ ♦❢ t❤❡ ❡st✐♠❛t♦rs ❞✐s❝✉ss❡❞

s♦ ❢❛r✱ ❛ s✐♠✉❧❛t✐♦♥ st✉❞② ❤❛s ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t✳

❲❡ ❤❛✈❡ ❣❡♥❡r❛t❡❞ ❞✐✛❡r❡♥t s❝❡♥❛r✐♦s✱ ❝♦♥s✐❞❡r✐♥❣ ❞✐✈❡rs❡ ❢✉♥❝✲

t✐♦♥❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✱ ✇✐t❤ s❛♠♣❧❡ s✐③❡ n = ✸✺✵✱ ❜❡✐♥❣ [a,b] = [✵,✶] ❛♥❞ [c,d] = [✷,✸] t❤❡ ✐♥t❡r✈❛❧s ✇❤❡r❡ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛r❡ ✈❛❧✉❡❞✳

❇♦t❤ ✐♥t❡r✈❛❧s ❛r❡ ❞✐s❝r❡t✐③❡❞ ✐♥ lx = ly = ✸✵✶ ❡q✉✐s♣❛❝❡❞ ❣r✐❞ ♣♦✐♥ts✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✼ ✴ ✺✺

slide-61
SLIDE 61

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❈▼✿ ❙❡❡ ❈r❛♠❜❡s ❛♥❞ ▼❛s ✭✷✵✶✸✮✱ ✇❤❡r❡ X = ✺✵

j=✶ λjǫjΨj✱

ǫj ∼ N (✵,✶)✱ λj =

π✷(j− ✶

✷)✷ , Ψj(s) =

  • ✷sin((j −✵.✺)πs)✱ ✇✐t❤ j ≥ ✶✳

❇▼✿ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ❡✐❣❡♥❢✉♥❝t✐♦♥s Ψj(s) =

  • ✷sin((j −✵.✺)πs)✳

■❑✿ ❙❡❡ ■♠❛✐③✉♠✐ ❛♥❞ ❑❛t♦ ✭✷✵✶✽✮✱ ✇❤❡r❡ X = ✺✵

j=✶j−✶.✷/✷UjΨj✱

Uj ∼ U (−

  • ✸,
  • ✸)✱ Ψ✶ ≡ ✶, Ψj(s) =
  • ✷cos(jπs)✳ ❚❤❡ ❛ss♦❝✐❛t❡❞ ❢✉♥❝✲

t✐♦♥❛❧ ❡rr♦r ✇❛s ❣✐✈❡♥ ❜② E(t) = ✺✵

j=✶j−✶.✶/✷ǫjΨj(t)✱ ✇✐t❤ ǫj ∼ N (✵,✶)✳

  • P✿ ●❛✉ss✐❛♥ ♣r♦❝❡ss ✇✐t❤ Σ(s✶,s✷) = (✷.✺)✷exp(− |s✶−s✷|

✵.✷ )✳

❖❯✿ ❖r♥st❡✐♥✲❯❤❧❡♥❜❡❝❦ ♣r♦❝❡ss✱ ✇✐t❤ ❞r✐❢t θ = ✶ ❛♥❞ ❞✐✛✉s✐♦♥

❝♦❡✣❝✐❡♥t γ✱ ✇❤❡r❡ ✐ts ✈❛r✐❛♥❝❡ ✐s ❣✐✈❡♥ ❜② σ✷

E = γ✷

✷θ = ✵.✶✺✵✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✽ ✴ ✺✺

slide-62
SLIDE 62

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❈▼✿ ❙❡❡ ❈r❛♠❜❡s ❛♥❞ ▼❛s ✭✷✵✶✸✮✳ ❇▼✿ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❖❯✿ ❖r♥st❡✐♥✲❯❤❧❡♥❜❡❝❦ ♣r♦❝❡ss✳ ■❑✿ ❙❡❡ ■♠❛✐③✉♠✐ ❛♥❞ ❑❛t♦ ✭✷✵✶✽✮✳ ●P✿ ●❛✉ss✐❛♥ ♣r♦❝❡ss✳

❚❛❜❧❡ ✶✿ ❙✉♠♠❛r② ♦❢ s✐♠✉❧❛t❡❞ s❝❡♥❛r✐♦s✳

❙❝❡♥❛r✐♦ ❑❡r♥❡❧ β(s,t)

X (s) E(t)

❙✶ (s −a)✷ +(t −c)✷ ❈▼ ❇▼ ❙✷

βj,k = ✵, j,k = ✶,...,✽✱ ❛♥❞ βj,k = ✹(−✶)j+kj−✷.✺k−✸

■❑ ■❑ ❙✸ sin(✻πs)+cos(✻πt)

  • P

❖❯

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✷✾ ✴ ✺✺

slide-63
SLIDE 63

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❚❛❜❧❡ ✷✿ ❆✈❡r❛❣❡❞ L✷ ❡rr♦rs ✭✶✵✵ s✐♠✉❧❛t✐♦♥s✮ ❢♦r t❤❡ ❋P❈❘✱ ❋P❈❘✲▲✶✱ ❋P❈❘✲ ▲✷ ❛♥❞ ❋P❈❘✲▲✶✲❈ ❡st✐♠❛t♦rs✳

❙❈❊◆❆❘■❖❙ ❙✶ ❙✷ ❙✸ q − → ✸ ✶✵ ✸ ✶✵ ✸ ✶✵ ✈❛r✳ ❡①♣✳ ✾✾% > ✾✾% ✺✶% ✼✻% ✾✺% > ✾✾% ♣ = ✶✵ p∗ ✹ ✹ ✷ ✸ ✸ ✹ ✈❛r✳ ❡①♣✳ > ✾✾% ✻✵% ✾✵% ❋P❈❘ ✺.✺✻ ✺.✻✽ ✵.✺✹ ✵.✼✶ ✶.✵✸ ✶.✵✺ ❋P❈❘✲▲✶ ✵.✺✶ ✵.✺✵ ✵.✹✻ ✵.✹✽ ✵.✽✶ ✵.✽✷ ❋P❈❘✲▲✷ ✵✳✹✵ ✵✳✹✵ ✵.✸✽ ✵.✹✼ ✵.✼✹ ✵✳✼✹ ❋P❈❘✲▲✶✲❈ ✵.✺✶ ✵.✺✷ ✵✳✸✼ ✵✳✸✽ ✵✳✼✸ ✵✳✼✹ ♣ = ✻✵ p∗ ✹ ✹ ✽ ✽ ✼ ✺ ✈❛r✳ ❡①♣✳ > ✾✾% ✾✼% ✾✾% ❋P❈❘ ✺✺.✾✽ ✺✺.✾✶ ✸.✾✺ ✺.✸✹ ✶✽.✶✽ ✶✽.✻✵ ❋P❈❘✲▲✶ ✵.✺✶ ✵.✺✵ ✵.✹✻ ✵.✹✽ ✵.✽✶ ✵.✽✷ ❋P❈❘✲▲✷ ✵✳✹✵ ✵✳✹✵ ✵.✺✻ ✵.✺✼ ✵.✼✾ ✵.✽✵ ❋P❈❘✲▲✶✲❈ ✵.✺✸ ✵.✺✷ ✵✳✸✼ ✵✳✸✽ ✵✳✼✹ ✵✳✼✹

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✵ ✴ ✺✺

slide-64
SLIDE 64

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

Pr❡❧✐♠✐♥❛r② ❛s♣❡❝ts

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤

❋P❈❘✲▲✶✲❈ ❡st✐♠❛t♦r ♦✉t♣❡r❢♦r♠s t❤❡ ❛❧t❡r♥❛t✐✈❡ ❡st✐♠❛t✐♦♥

❛♣♣r♦❛❝❤❡s ✐♥ t❤❡ s❝❡♥❛r✐♦s ✇✐t❤ ♠♦r❡ ✈❛r✐❛❜✐❧✐t② ✭s❝❡♥❛r✐♦s S✷ ❛♥❞ S✸✮✳ ❋P❈❘✲▲✷ s❡❡♠s t♦ ❜❡ t❤❡ ♠♦r❡ ❛❝❝✉r❛t❡ ❡st✐♠❛t♦r ✐♥ S✶✳

❲❤❡♥ ❛ ❤✉❣❡ ❛♠♦✉♥t ♦❢ ❝♦♠♣♦♥❡♥ts ❛r❡ r❡q✉✐r❡❞ ✭p = ✻✵ ✐s r❡✲

q✉✐r❡❞ ❢♦r ❡①♣❧❛✐♥✐♥❣ ♠♦r❡ t❤❛♥ ✾✾% ✐♥ s❝❡♥❛r✐♦s S✶ ❛♥ S✷✮✱ ❛ ❝r✉❝✐❛❧ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ✐s ❛❝❤✐❡✈❡❞ ✭p∗ = ✽ ❛s ♠❛①✐♠✉♠✮✳

❖✈❡r✜tt✐♥❣ r❡❧❛t❡❞ t♦ ❋P❈❘ ❡st✐♠❛t♦r ❤❛s ❜❡❡♥ ❛✈♦✐❞❡❞✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✶ ✴ ✺✺

slide-65
SLIDE 65

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❖♠♥✐❜✉s t❡st

❖✉r ✐♥t❡r❡st r❡❧✐❡s ♦♥ ✈❡r✐❢②✐♥❣ ✇❤❡t❤❡r t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥

❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss♦rs ❛♥❞ r❡s♣♦♥s❡s ❝❛♥ ❜❡ ❡①♣❧❛✐♥❡❞ ❜② ❛ ❋▲▼❋❘✿ ❍✵ : ♠ = E[Y |X ] ∈ Mβ =

mβ(X ) = 〈〈X ,β〉〉

  • ✭❧✐♥❡❛r✐t②✮

❍✶ :

P m ∈ Mβ

  • > ✵

❚❤❡ s✐♠♣❧❡ ❤②♣♦t❤❡s✐s ❢♦r ❛ ♣❛rt✐❝✉❧❛r β✵ ✭❢♦r ❡①❛♠♣❧❡✱ β✵ = ✵

♥✉❧❧ ♦♣❡r❛t♦r✮ ❝❛♥ ❜❡ ❛❧s♦ t❡st❡❞ ❥✉st r❡♣❧❛❝✐♥❣ t❤❡ ❡st✐♠❛t♦r ♦❢ β ❜② ✐ts ❦♥♦✇♥ ✈❛❧✉❡✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✷ ✴ ✺✺

slide-66
SLIDE 66

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

▼❛✐♥ t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts✿ ✐♥t❡❣r❛t✐♥❣ ♦✈❡r t❤❡ ❍✐❧❜❡rt✐❛♥ s♣❤❡r❡

❚❤❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥❛❧ ♦♣❡r❛t♦r T✱ r❡s♣❡❝t t♦ ❛ ❣✐✈❡♥

❝♦✈❛r✐❛t❡ γ(p) ∈ Sp

H✶✱ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ❛♥ ✐♥t❡❣r❛t✐♦♥ ♦♥ t❤❡ p✲s♣❤❡r❡✿

  • Sp

H✶

T

  • γ(p)

dγ(p) =

  • Sp
  • ❘p
  • −✶T

p

  • j=✶
  • ❘−✶

p gp

  • j Ψj
  • dgp,

gp ❝♦♦r❞✳ ♦❢ γ(p)

❈❤❛r❛❝t❡r✐③✐♥❣ ♥✉❧❧ ❤②♣♦t❤❡s✐s ❜② ♣r♦❥❡❝t✐♦♥s ❚❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✿

✐✳

mβ(X ) = 〈〈X ,β〉〉✱ ∀X ∈ H✶✳

✐✐✳

E

  • Y −〈〈X ,β〉〉|X = X
  • = ✵✱ ❢♦r ❛❧♠♦st ❡✈❡r② ✭❛✳❡✳✮ X ∈ H✶✳

✐✐✐✳

E

  • 〈Y −〈〈X ,β〉〉,γY 〉H✷|〈X ,γX 〉H✶ = u
  • = ✵✱ ❢♦r ❛✳❡✳ u ∈ R✱ ∀γX ∈

SH✶ = x ∈ H✶ : xH✶ = ✶ , γY ∈ SH✷ = x ∈ H✷ : xH✷ = ✶ ✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✸ ✴ ✺✺

slide-67
SLIDE 67

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❍♦✇ ❝❛♥ ✇❡ ❛ss❡s ❤♦✇ ❢❛r ❛r❡ ❞❛t❛ ❢r♦♠ ❍✵❄

❲❡ ✇✐❧❧ ❝❤❛r❛❝t❡r✐③❡ ❍✵ ✭✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✮ ❜② ♠❡❛♥s ♦❢ ✐ts ♣r♦❥❡❝t✐♦♥s ✐♥t♦ s✉✐t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ❞✐r❡❝t✐♦♥s✳ ❚❤✐s ❞❡✈✐❛t✐♦♥ ✐s ♠❡❛s✉r❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❘❡s✐❞✉❛❧ ▼❛r❦❡❞ ❡♠✲ ♣✐r✐❝❛❧ Pr♦❝❡ss ❜❛s❡❞ ♦♥ t✇♦✲s✐❞❡s Pr♦❥❡❝t✐♦♥s ✭❘▼P✷P✮✿

✷ ✶ ✶ ✷

▼❛r❦s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜②

✶ ✭♣r♦❥❡❝t❡❞ r❡s✐❞✉❛❧s✮ ❛♥❞

t❤❡ ❥✉♠♣s ❞r✐✈❡♥ ❜②

✶ ✭♣r♦❥❡❝t❡❞ r❡❣r❡ss♦rs✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✹ ✴ ✺✺

slide-68
SLIDE 68

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❍♦✇ ❝❛♥ ✇❡ ❛ss❡s ❤♦✇ ❢❛r ❛r❡ ❞❛t❛ ❢r♦♠ ❍✵❄ ❲❡ ✇✐❧❧ ❝❤❛r❛❝t❡r✐③❡ ❍✵ ✭✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✮ ❜② ♠❡❛♥s ♦❢ ✐ts

♣r♦❥❡❝t✐♦♥s ✐♥t♦ s✉✐t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ❞✐r❡❝t✐♦♥s✳ ❚❤✐s ❞❡✈✐❛t✐♦♥ ✐s ♠❡❛s✉r❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❘❡s✐❞✉❛❧ ▼❛r❦❡❞ ❡♠✲ ♣✐r✐❝❛❧ Pr♦❝❡ss ❜❛s❡❞ ♦♥ t✇♦✲s✐❞❡s Pr♦❥❡❝t✐♦♥s ✭❘▼P✷P✮✿

✷ ✶ ✶ ✷

▼❛r❦s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜②

✶ ✭♣r♦❥❡❝t❡❞ r❡s✐❞✉❛❧s✮ ❛♥❞

t❤❡ ❥✉♠♣s ❞r✐✈❡♥ ❜②

✶ ✭♣r♦❥❡❝t❡❞ r❡❣r❡ss♦rs✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✹ ✴ ✺✺

slide-69
SLIDE 69

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❍♦✇ ❝❛♥ ✇❡ ❛ss❡s ❤♦✇ ❢❛r ❛r❡ ❞❛t❛ ❢r♦♠ ❍✵❄ ❲❡ ✇✐❧❧ ❝❤❛r❛❝t❡r✐③❡ ❍✵ ✭✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✮ ❜② ♠❡❛♥s ♦❢ ✐ts

♣r♦❥❡❝t✐♦♥s ✐♥t♦ s✉✐t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ❞✐r❡❝t✐♦♥s✳

■♥ t❤❡ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ✭❜✉t ♠✉❧t✐✈❛r✐❛t❡✮ ❢r❛♠❡✇♦r❦✱ ❊s❝❛♥❝✐❛♥♦

✭✷✵✵✻✮ ♣r♦♣♦s❡❞ t♦ ♠❡❛s✉r❡ t❤✐s ❞❡✈✐❛t✐♦♥ ❜② ❛ ♣r♦❥❡❝t❡❞ r❡s✐❞✉❛❧ ♠❛r❦❡❞ ❡♠♣✐r✐❝❛❧ ♣r♦❝❡ss✿ Rn (u,β) = ✶

n

n

  • i=✶
  • εiI{βT X≤u},

u ∈ R, β ∈ Sd ❚❤✐s ❞❡✈✐❛t✐♦♥ ✐s ♠❡❛s✉r❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❘❡s✐❞✉❛❧ ▼❛r❦❡❞ ❡♠✲ ♣✐r✐❝❛❧ Pr♦❝❡ss ❜❛s❡❞ ♦♥ t✇♦✲s✐❞❡s Pr♦❥❡❝t✐♦♥s ✭❘▼P✷P✮✿

✷ ✶ ✶ ✷

▼❛r❦s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜②

✶ ✭♣r♦❥❡❝t❡❞ r❡s✐❞✉❛❧s✮ ❛♥❞

t❤❡ ❥✉♠♣s ❞r✐✈❡♥ ❜②

✶ ✭♣r♦❥❡❝t❡❞ r❡❣r❡ss♦rs✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✹ ✴ ✺✺

slide-70
SLIDE 70

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❍♦✇ ❝❛♥ ✇❡ ❛ss❡s ❤♦✇ ❢❛r ❛r❡ ❞❛t❛ ❢r♦♠ ❍✵❄ ❲❡ ✇✐❧❧ ❝❤❛r❛❝t❡r✐③❡ ❍✵ ✭✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✮ ❜② ♠❡❛♥s ♦❢ ✐ts

♣r♦❥❡❝t✐♦♥s ✐♥t♦ s✉✐t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ ❞✐r❡❝t✐♦♥s✳

❚❤✐s ❞❡✈✐❛t✐♦♥ ✐s ♠❡❛s✉r❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❘❡s✐❞✉❛❧ ▼❛r❦❡❞ ❡♠✲

♣✐r✐❝❛❧ Pr♦❝❡ss ❜❛s❡❞ ♦♥ t✇♦✲s✐❞❡s Pr♦❥❡❝t✐♦♥s ✭❘▼P✷P✮✿

Rn (u,γX ,γY ) = ✶

n

n

  • i=✶

〈 Ei,γY 〉H✷I

〈X ,γX 〉H✶≤u ,

u ∈ R, γX ∈ SH✶, γY ∈ SH✷

▼❛r❦s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② {〈

Ei,γY 〉H✷}n

i=✶ ✭♣r♦❥❡❝t❡❞ r❡s✐❞✉❛❧s✮ ❛♥❞

t❤❡ ❥✉♠♣s ❞r✐✈❡♥ ❜② {〈Xi,γX 〉H✶}n

i=✶ ✭♣r♦❥❡❝t❡❞ r❡❣r❡ss♦rs✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✹ ✴ ✺✺

slide-71
SLIDE 71

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❚❤❡ P❈✈▼n st❛t✐st✐❝ s❤♦✉❧❞ ❜❡ tr✉♥❝❛t❡❞✿

Rn,p,q

  • u,γ(p)

X ,γ(q) Y

  • = ✶

n

n

  • i=✶

〈 E (q)

i

,γ(q)

Y 〉H✷I

〈X (p)

i

,γ(p)

X 〉H✶≤u

, γ(p)

X

∈ Sp

H✶, γ(q)

Y

∈ Sq

H✷

❲❡ ❛ss✉♠❡ ♦rt❤♦♥♦r♠❛❧ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s✿ Ψ = ■p✳ ❍❡♥❝❡❢♦rt❤✱ X (p)

i

❛♥❞ γ(p)

X

r❡♣r❡s❡♥t t❤❡ ♣✲tr✉♥❝❛t❡❞ ❡①♣❛♥s✐♦♥s✱ ✇✐t❤ ❝♦❡✣❝✐❡♥ts ①i,p ❛♥❞ ❣p✿ 〈X (♣)

,γ(♣)

❳ 〉H✶ = ①❚ ✐,♣❣♣✱ i = ✶,...,n✳

❆♥❛❧♦❣♦✉s❧②✱ ✇❡ ✇✐❧❧ ❛❞♦♣t ♥♦t❛t✐♦♥s Y (q)

i

E (q)

i

✱ γ(q)

Y

❛♥❞ ❣q✿

〈 E (q)

,γ(q)

Y 〉H✷ =

❚ ✐,q❣q✱ ❜❡✐♥❣

❊i,q t❤❡ ✈❡❝t♦r ♦❢ ❝♦❡✣❝✐❡♥ts ♦❢

E (q)

i

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✺ ✴ ✺✺

slide-72
SLIDE 72

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❚❤❡ P❈✈▼n st❛t✐st✐❝ s❤♦✉❧❞ ❜❡ tr✉♥❝❛t❡❞✿

Rn,p,q

  • u,γ(p)

X ,γ(q) Y

  • = ✶

n

n

  • i=✶

〈 E (q)

i

,γ(q)

Y 〉H✷I

〈X (p)

i

,γ(p)

X 〉H✶≤u

, γ(p)

X

∈ Sp

H✶, γ(q)

Y

∈ Sq

H✷

❲❡ ❛ss✉♠❡ ♦rt❤♦♥♦r♠❛❧ ❢✉♥❝t✐♦♥❛❧ ❜❛s❡s✿ Ψ = ■p✳ ❍❡♥❝❡❢♦rt❤✱ X (p)

i

❛♥❞ γ(p)

X

r❡♣r❡s❡♥t t❤❡ ♣✲tr✉♥❝❛t❡❞ ❡①♣❛♥s✐♦♥s✱ ✇✐t❤ ❝♦❡✣❝✐❡♥ts ①i,p ❛♥❞ ❣p✿ 〈X (♣)

,γ(♣)

❳ 〉H✶ = ①❚ ✐,♣❣♣✱ i = ✶,...,n✳

❆♥❛❧♦❣♦✉s❧②✱ ✇❡ ✇✐❧❧ ❛❞♦♣t ♥♦t❛t✐♦♥s Y (q)

i

E (q)

i

✱ γ(q)

Y

❛♥❞ ❣q✿

〈 E (q)

,γ(q)

Y 〉H✷ =

❚ ✐,q❣q✱ ❜❡✐♥❣

❊i,q t❤❡ ✈❡❝t♦r ♦❢ ❝♦❡✣❝✐❡♥ts ♦❢

E (q)

i

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✺ ✴ ✺✺

slide-73
SLIDE 73

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❆✐♠❡❞ ❛t ♠❡❛s✉r✐♥❣ ❤♦✇ ❝❧♦s❡ t❤❡ st❛t✐st✐❝ ✐s t♦ ③❡r♦✱ ✇❡ ❛❞♦♣t

t❤❡ Pr♦❥❡❝t❡❞ ❈r❛♠ér✲✈♦♥ ▼✐s❡s ♥♦r♠ ✐♥ Π(p,q) = Sq

H✷ ×Sp H✶ ×R✿

P❈✈▼n,p,q =

  • Sq×Sp×R

n−✶

n

  • i=✶

T i,q❣qI ①T

i,p❣p≤u

Fn,γ(p)

X

(du)ωX (dγ(p)

X )ωY (dγ(q) Y )

✇❤❡r❡ Fn,γ(p)

X (du) ✐s t❤❡ ❡❝❞❢ ♦❢ t❤❡ ♣r♦❥❡❝t❡❞ ❞❛t❛✱ ❜❡✐♥❣ ωX ❛♥❞

ωY ♠❡❛s✉r❡s ✐♥ t❤❡ ✜♥✐t❡ p− ❛♥❞ q− ❞✐♠❡♥s✐♦♥❛❧ ❢✉♥❝t✐♦♥❛❧ s♣❤❡r❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✻ ✴ ✺✺

slide-74
SLIDE 74

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❆ss✉♠✐♥❣ t❤❛t ♠❡❛s✉r❡s ❛r❡ ✉♥✐❢♦r♠ ♦♥ Sp ❛♥❞ Sq✱

P❈✈▼n,p,q = ✶ n✷ ✷πp/✷+q/✷−✶ qΓ

p

  • Γ

q

Tr

T q ❆•

❊q

  • ✇❤❡r❡

❊q =

  • ❊k,i

i=✶,...,n

k=✶,...,q ❛r❡ t❤❡ ❡♠♣✐r✐❝❛❧ r❡s✐❞✉❛❧s ✇❡✐❣❤t❡❞ ❜② ❆•✳

▼❛tr✐① ❆• ✭♦♥❧② O(q ♥✸−♥✷

) ❝♦♠♣✉t❛t✐♦♥s ❛r❡ r❡q✉✐r❡❞ ❛♥❞ ✐t ✐s ✜①❡❞ ❢♦r ❡❛❝❤ s❛♠♣❧❡✮ ✐s ❣✐✈❡♥ ❜② (❆•)✐,❝ =

  • r=✶

❆✐❝r✱ i,c = ✶,...,n✱

Aicr = A(∡)

icr

πp/✷−✶ Γ p

,

A(∡)

icr =

      

✷π, ✐❢ i = c = r,

π,

✐❢ i = c, i = r ♦r c = r,

  • π−❛r❝❝♦s
  • (①i,p−①r,p)T (①c,p−①r,p)

||①i,p−①r,p)||·||①c,p−①r,p||

  • , ♦t❤❡r✇✐s❡

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✼ ✴ ✺✺

slide-75
SLIDE 75

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❈✈▼ st❛t✐st✐❝ t❡st

❆ss✉♠✐♥❣ t❤❛t ♠❡❛s✉r❡s ❛r❡ ✉♥✐❢♦r♠ ♦♥ Sp ❛♥❞ Sq✱

P❈✈▼n,p,q = ✶ n✷ ✷πp/✷+q/✷−✶ qΓ

p

  • Γ

q

Tr

T q ❆•

❊q

  • ✇❤❡r❡

❊q =

  • ❊k,i

i=✶,...,n

k=✶,...,q ❛r❡ t❤❡ ❡♠♣✐r✐❝❛❧ r❡s✐❞✉❛❧s ✇❡✐❣❤t❡❞ ❜② ❆•✳

▼❛tr✐① ❆• ✭♦♥❧② O(q ♥✸−♥✷

) ❝♦♠♣✉t❛t✐♦♥s ❛r❡ r❡q✉✐r❡❞ ❛♥❞ ✐t ✐s ✜①❡❞ ❢♦r ❡❛❝❤ s❛♠♣❧❡✮ ✐s ❣✐✈❡♥ ❜② (❆•)✐,❝ =

  • r=✶

❆✐❝r✱ i,c = ✶,...,n✱

Aicr = A(∡)

icr

πp/✷−✶ Γ p

,

A(∡)

icr =

      

✷π, ✐❢ i = c = r,

π,

✐❢ i = c, i = r ♦r c = r,

  • π−❛r❝❝♦s
  • (①i,p−①r,p)T (①c,p−①r,p)

||①i,p−①r,p)||·||①c,p−①r,p||

  • , ♦t❤❡r✇✐s❡

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✼ ✴ ✺✺

slide-76
SLIDE 76

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❲✐❧❞ ❜♦♦tstr❛♣ r❡s❛♠♣❧✐♥❣ ♣r♦❝❡❞✉r❡

❋♦r ❡❛❝❤ ❜♦♦tstr❛♣ ✐t❡r❛t✐♦♥ b = ✶,...,B✱ ✇❡ ❤❛✈❡ t♦ ❝♦♠♣✉t❡

P❈✈▼∗,b

n,p,q = ✶

n✷ ✷πp/✷+q/✷−✶ qΓ

p

  • Γ

q

Tr

∗,T

q

❆• ❊

q

  • ❆ ✐s ✜①❡❞ ❢♦r ❡❛❝❤ ✐t❡r❛t✐♦♥✳ ❚❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♣✲✈❛❧✉❡ ♦❢

t❤❡ t❡st ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② P❈✈▼ P❈✈▼ ✳

❚❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ❜♦♦tstr❛♣ r❡s✐❞✉❛❧s ❝❛♥ ❜❡ ❡❛s✐❧② ♣❡r❢♦r♠❡❞ ❛s

∗ = ❨∗

q −

q = ❨∗ q −❍❈❨∗ q =

  • ■p −❍❈

❨∗

q

❍❛t ♠❛tr✐① ❍❈ ♦❢ ♦✉r ❤②❜r✐❞ ❡st✐♠❛t♦r r❡♠❛✐♥s t❤❡ s❛♠❡ ❢♦r ❛❧❧ t❤❡ ❜♦♦tstr❛♣ r❡♣❧✐❝❛t❡s✱ ✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ ✐t ❛❣❛✐♥ ❛♥❞ ❛❣❛✐♥✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✽ ✴ ✺✺

slide-77
SLIDE 77

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

❲✐❧❞ ❜♦♦tstr❛♣ r❡s❛♠♣❧✐♥❣ ♣r♦❝❡❞✉r❡

❋♦r ❡❛❝❤ ❜♦♦tstr❛♣ ✐t❡r❛t✐♦♥ b = ✶,...,B✱ ✇❡ ❤❛✈❡ t♦ ❝♦♠♣✉t❡

P❈✈▼∗,b

n,p,q = ✶

n✷ ✷πp/✷+q/✷−✶ qΓ

p

  • Γ

q

Tr

∗,T

q

❆• ❊

q

  • ❆• ✐s ✜①❡❞ ❢♦r ❡❛❝❤ ✐t❡r❛t✐♦♥✳ ❚❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♣✲✈❛❧✉❡ ♦❢

t❤❡ t❡st ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② #{P❈✈▼n,p,q ≤ P❈✈▼∗,b

n,p,q}/B✳

❚❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ❜♦♦tstr❛♣ r❡s✐❞✉❛❧s ❝❛♥ ❜❡ ❡❛s✐❧② ♣❡r❢♦r♠❡❞ ❛s

∗ = ❨∗

q −

q = ❨∗ q −❍❈❨∗ q =

  • ■p −❍❈

❨∗

q

❍❛t ♠❛tr✐① ❍❈ ♦❢ ♦✉r ❤②❜r✐❞ ❡st✐♠❛t♦r r❡♠❛✐♥s t❤❡ s❛♠❡ ❢♦r ❛❧❧

t❤❡ ❜♦♦tstr❛♣ r❡♣❧✐❝❛t❡s✱ ✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ ✐t ❛❣❛✐♥ ❛♥❞ ❛❣❛✐♥✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✽ ✴ ✺✺

slide-78
SLIDE 78

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❖✉t❧✐♥❡

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄ ❆♥t❡❝❡❞❡♥ts ❋❉❆ ❜❛❝❦❣r♦✉♥❞

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

❆♥t❡❝❡❞❡♥ts

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘✿ ♠♦t✐✈❛t✐♦♥

❋▲▼❋❘ ❡st✐♠❛t✐♦♥✿ ❛ ♥♦✈❡❧ ❤②❜r✐❞ ❛♣♣r♦❛❝❤ ❈✈▼ st❛t✐st✐❝ t❡st

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✸✾ ✴ ✺✺

slide-79
SLIDE 79

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘ ♣❛❝❦❛❣❡✿ ❣♦❢❢❞❛ ❆♥ ❘ ♣❛❝❦❛❣❡ ✭❣♦❢❢❞❛ ♣❛❝❦❛❣❡✮ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞✿

  • ❯♥✐✜❡❞

❛♣♣r♦❛❝❤ ❢♦r ❢✉♥❝✲ t✐♦♥❛❧✴s❝❛❧❛r r❡s♣♦♥s❡ ❛♥❞ ❢✉♥❝✲ t✐♦♥❛❧✴s❝❛❧❛r ♣r❡❞✐❝t♦r✳

❋♦❝✉s ♦♥ ♣❡r❢♦r♠❛♥❝❡✿ ❘❝♣♣

❢♦r ❝♦♠♣✉t✐♥❣✲✐♥t❡♥s✐✈❡ ♣❛rts✳

❋P❈❘✱ ❋P❈❘✲▲✶✱ ❋P❈❘✲▲✷

❛♥❞ ❋P❈❘✲▲✶✲❈ ❛r❡ ✐♠♣❧❡♠❡♥t❡❞✳

  • r❛♣❤✐❝❛❧ t♦♦❧s ❢♦r ✈✐s✉❛❧✐s✐♥❣

t❤❡ t❡st ♦✉t♣✉ts✳

❯s✉❛❧ ❞❡✈❡❧♦♣♠❡♥t ♣r❛❝t✐❝❡s✿ ❝♦❞❡ ❝❧❡❛♥♥❡ss✱ ●✐t❍✉❜✱ r♦①②❣❡♥✷✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✵ ✴ ✺✺

slide-80
SLIDE 80

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘ ♣❛❝❦❛❣❡✿ ❣♦❢❢❞❛

❋✐❣✉r❡ ✶✿ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ●♦❋ t❡st ❢♦r ❋▲▼❋❘ ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s✐s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✶ ✴ ✺✺

slide-81
SLIDE 81

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘ ♣❛❝❦❛❣❡✿ ❣♦❢❢❞❛

❋✐❣✉r❡ ✷✿ ●r❛♣❤✐❝❛❧ t♦♦❧s✳ ❖♥ ❧❡❢t✿ ❞❡♥s✐t② ♦❢ ❜♦♦tstr❛♣ st❛t✐st✐❝s✳ ❖♥ r✐❣❤t✿ ❜❧❛❝❦ ❧✐♥❡ r❡♣r❡s❡♥t t❤❡ ♦r✐❣✐♥❛❧ st❛t✐st✐❝s✱ ❣r❛② ❧✐♥❡s t❤❡ ❜♦♦tstr❛♣ st❛t✐st✐❝s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✷ ✴ ✺✺

slide-82
SLIDE 82

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘ ♣❛❝❦❛❣❡✿ ❣♦❢❢❞❛

❋✐❣✉r❡ ✸✿ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ●♦❋ t❡st ❢♦r ❋▲▼❋❘ ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s✐s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✸ ✴ ✺✺

slide-83
SLIDE 83

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘ ♣❛❝❦❛❣❡✿ ❣♦❢❢❞❛

❋✐❣✉r❡ ✹✿ ●r❛♣❤✐❝❛❧ t♦♦❧s✳ ❖♥ ❧❡❢t✿ ❞❡♥s✐t② ♦❢ ❜♦♦tstr❛♣ st❛t✐st✐❝s✳ ❖♥ r✐❣❤t✿ ❜❧❛❝❦ ❧✐♥❡ r❡♣r❡s❡♥t t❤❡ ♦r✐❣✐♥❛❧ st❛t✐st✐❝s✱ ❣r❛② ❧✐♥❡s t❤❡ ❜♦♦tstr❛♣ st❛t✐st✐❝s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✹ ✴ ✺✺

slide-84
SLIDE 84

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❙✐♠✉❧❛t✐♦♥ st✉❞②✿ ♥✉❧❧ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s

❖♥❧② ❝♦♠♣❡t✐♥❣ ♣r♦❝❡❞✉r❡s ♦❢ ❑♦❦♦s③❦❛ ❡t ❛❧✳ ✭✷✵✵✽✮ ❛♥❞ P❛t✐❧❡❛

❡t ❛❧✳ ✭✷✵✶✻✮ ❛r❡ ❝✉rr❡♥t❧② ❛✈❛✐❧❛❜❧❡ ❜✉t ♦♥❧② ❢♦r t❡st✐♥❣ t❤❡ s✐♠♣❧❡ ❤②♣♦t❤❡s✐s ♦❢ ♥♦ ❡✛❡❝ts ✐♥ t❤❡ ❋▲▼❋❘ s❡t✉♣✳

❉✐✛❡r❡♥t ❍✐❧❜❡rt s♣❛❝❡s H✶ = L✷([✵,✶]) ❛♥❞ H✷ = L✷([✷,✸])✱ ❞✐s✲

❝r❡t✐③❡❞ ✐♥ ✼✶ ❡q✉✐s♣❛❝❡❞ ❣r✐❞ ♣♦✐♥ts✱ ❛r❡ ❝♦♥s✐❞❡r❡❞✳

❙❛♠♣❧❡ s✐③❡s n = [✺✵,✷✺✵,✺✵✵]✱ B = ✻✵✵ ❜♦♦tstr❛♣ ✐t❡r❛t✐♦♥s ❛♥❞

MC = ✻✵✵ ▼♦♥t❡ ❈❛r❧♦ r❡♣❧✐❝❛t❡s ❛r❡ ❛❞♦♣t❡❞

❚❤r❡❡ ❞✐✛❡r❡♥t ♥✉❧❧ ❧✐♥❡❛r ❤②♣♦t❤❡s❡s ❛r❡ ❣❡♥❡r❛t❡❞✿ ♥♦ ❡✛❡❝ts✱

❋▲▼❋❘ ❛♥❞ ❝♦♥❝✉rr❡♥t ♠♦❞❡❧✳ ❚✇♦ ❞❡✈✐❛t✐♦♥s ❢r♦♠ ❧✐♥❡❛r✐t② ❛r❡ ❝♦♥s✐❞❡r❡❞✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ♦♥ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ s✐♥✉s♦✐❞❛❧ ❢✉♥❝t✐♦♥s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✺ ✴ ✺✺

slide-85
SLIDE 85

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❙✐♠✉❧❛t✐♦♥ st✉❞②✿ ♥✉❧❧ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s

❚❛❜❧❡ ✸✿ ❙✉♠♠❛r② ♦❢ ♥✉❧❧ ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ❤②♣♦t❤❡s❡s✱ ❢♦r s❝❡♥❛r✐♦s S✶✱ S✷ ❛♥❞ S✸✳

◆❖❚❆❚■❖◆ ❉❊❙❈❘■P❚■❖◆ ▼❖❉❊▲ ❍✵ ◆♦ ❡✛❡❝ts

Y (t) = E(t)✱ ✇✐t❤ β✵ = ✵

❍✵,❋❘ ❋▲▼❋❘

Y (t) = 〈〈X ,β〉〉+E(t)

❍✵,❈ ❈♦♥❝✉rr❡♥t

Y (t) = β(t)X (a+(t −c))+E(t)

❍✶,◆▲.❙ ◆♦♥ ❧✐♥❡❛r✿ s✐♥

Y (t) = 〈〈X ,β〉〉+∆(X )(t)+E(t) ∆(X )(t) = (sin(✷πt)−cos(✷πt))X ✷

H✶

❍✶,◆▲.◗ ◆♦♥ ❧✐♥❡❛r✿ q✉❛❞r❛t✐❝

Y (t) = 〈〈X ,β〉〉+∆(X )(t)+E(t) ∆(X )(t) = X ✷

a+(t −c)∗ b−a

d−c

  • −✶

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✻ ✴ ✺✺

slide-86
SLIDE 86

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❙✐♠✉❧❛t✐♦♥ st✉❞②✿ ♥✉❧❧ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s

❚❛❜❧❡ ✹✿ ●♦❋ ♦♠♥✐❜✉s t❡st ✭❋P❈❘ ❡st✐♠❛t♦r✮✱ ❛t s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ α = ✵.✵✺✱ ❢♦r s❝❡♥❛r✐♦s S✶✱ S✷ ❛♥❞ S✸✳ Pr♦♣♦rt✐♦♥ ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥s ✐♥ ✇❤✐❝❤ ♥✉❧❧ ❤②♣♦t❤❡s✐s ✐s r❡❥❡❝t❡❞✳ ❙❝❡♥❛r✐♦ ❙✶ ❙❝❡♥❛r✐♦ ❙✷ ❙❝❡♥❛r✐♦ ❙✸ ✺✵ ✷✺✵ ✺✵✵ ✺✵ ✷✺✵ ✺✵✵ ✺✵ ✷✺✵ ✺✵✵ ❍✵ ✵.✵✸✸ ✵.✵✹✽ ✵.✵✺✷ ✶.✵✵✵ ✵.✾✻✸ ✵.✶✾✼ ✵.✸✺✺ ✵.✶✺✷ ✵.✵✹✺ ❍✵,FR ✵.✵✺✵ ✵.✵✻✷ ✵.✵✹✽ ✶.✵✵✵ ✵.✾✺✵ ✵.✷✶✺ ✵.✸✷✺ ✵.✶✹✺ ✵.✵✺✶ ❍✵,C ✵.✸✷✽ ✵.✵✻✵ ✵.✵✹✽ ❍✶,NL.S ✵.✶✵✺ ✵.✽✷✽ ✶.✵✵✵ ✵.✾✵✺ ✶.✵✵✵ ✶.✵✵✵ ✵.✾✷✽ ✶.✵✵✵ ✶.✵✵✵ ❍✶,NL.Q ✵.✷✶✺ ✵.✽✻✼ ✶.✵✵✵ ✵.✾✶✵ ✶.✵✵✵ ✶.✵✵✵ ✵.✾✾✽ ✶.✵✵✵ ✶.✵✵✵

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✼ ✴ ✺✺

slide-87
SLIDE 87

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘❡❛❧✲❞❛t❛ ❛♣♣❧✐❝❛t✐♦♥s

❆❊▼❊❚ ❞❛t❛s❡t✳

❉❛✐❧② t❡♠♣❡r❛t✉r❡ ❛✈❡r❛❣❡❞ ❢r♦♠ ✶✾✽✵ t♦ ✷✵✵✾ ✭X ✮ ❛♥❞ t❤❡ ❞❛✐❧② ✇✐♥❞ s♣❡❡❞ ❡✐t❤❡r t❤❡ ❧♦❣ ♣r❡❝✐♣✐t❛t✐♦♥ ✭Y ✮✳ ❙❛♠♣❧❡ s✐③❡ n = ✼✸ ✐s ❝♦♥s✐❞❡r❡❞✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✼✸ ❙♣❛♥✐s❤ ✇❡❛t❤❡r st❛t✐♦♥s✳ ❍✐❧❜❡rt s♣❛❝❡s H✶ = H✷ = L✷([✵,✸✻✺])✱ ✇✐t❤ ✸✻✺ ❣r✐❞ ♣♦✐♥ts✱ ❛r❡ ❛❞♦♣t❡❞✳

100 200 300 5 10 20 30

Temperature: 1980−2009 (mean)

day ºC 100 200 300 2 4 6 8 10

Wind speed: 1980−2009 (mean)

day m/s 100 200 300 −6 −4 −2 2 4

log precipitation: 1980−2009 (mean)

day mm

❋✐❣✉r❡ ✺✿ ❆✈❡r❛❣❡❞ ❞❛✐❧② t❡♠♣❡r❛t✉r❡✱ ❞❛✐❧② ✇✐♥❞ s♣❡❡❞ ❛♥❞ ❞❛✐❧② ♣r❡❝✐♣✐t❛t✐♦♥✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✽ ✴ ✺✺

slide-88
SLIDE 88

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘❡❛❧✲❞❛t❛ ❛♣♣❧✐❝❛t✐♦♥s

❈❛♥❛❞✐❛♥ ✇❡❛t❤❡r ❞❛t❛s❡t✳ ❆ s❡t ♦❢ n = ✸✺ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡s

✭❞❛✐❧② t❡♠♣❡r❛t✉r❡ ❛✈❡r❛❣❡❞ ❢r♦♠ ✶✾✻✵ t♦ ✶✾✾✹✮ ❛♥❞ ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡s ✭❞❛✐❧② ❧♦❣ ♣r❡❝✐♣✐t❛t✐♦♥ ❛✈❡r❛❣❡❞ ❢r♦♠ ✶✾✻✵ t♦ ✶✾✻✹✮✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✸✺ ✇❡❛t❤❡r st❛t✐♦♥s ✐♥ ❈❛♥❛❞❛✳

100 200 300 −30 −10 10

Temperature: 1960−1994 (mean)

days ºC 100 200 300 −1.0 0.0 1.0

Log Precipitation: 1960−1994 (mean)

days ºC

❋✐❣✉r❡ ✻✿ ❉❛✐❧② t❡♠♣❡r❛t✉r❡ ✭◦C✮ ❛✈❡r❛❣❡❞ ❢r♦♠ ✶✾✻✵ t♦ ✶✾✾✹ ❛♥❞ ❞❛✐❧② ♣r❡❝✐♣✐t❛✲

t✐♦♥ ✭✇✐t❤ ❧♦❣❛r✐t❤♠ tr❛♥s❢♦r♠❛t✐♦♥✱ ✐♥ log mm✮ ❛✈❡r❛❣❡❞ ❢r♦♠ ✶✾✻✵ t♦ ✶✾✾✹✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✹✾ ✴ ✺✺

slide-89
SLIDE 89

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘❡❛❧✲❞❛t❛ ❛♣♣❧✐❝❛t✐♦♥s

❖♥t❛r✐♦ ❡❧❡❝tr✐❝✐t② ❝♦♥s✉♠♣t✐♦♥✳ ❋✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡s ✭❛✐r t❡♠✲

♣❡r❛t✉r❡✮ ✈❛❧✉❡❞ ✐♥ H✶ = L✷([−✷✹,✹✽])✱ ✇✐t❤ ✼✸ ❣r✐❞ ♣♦✐♥ts ✭❢r♦♠ ✶ ❞❛② ❜❡❢♦r❡ t♦ ✶ ❧❛t❡r✮✱ ❛♥❞ ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡ ✭❞❛✐❧② ❡❧❡❝tr✐❝✐t② ❝♦♥s✉♠♣t✐♦♥✮✱ ✈❛❧✉❡❞ ✐♥

H✷ = L✷([✵,✷✹])✱ ✇✐t❤ ✷✹ ❣r✐❞ ♣♦✐♥ts✳

−20 −10 10 20 30 40 50 5 10 15 20 25

Temperature: 2010−2014

Time = [Day−1, Current Day, Day+1] ºC 5 10 15 20 12 16 20 24

Electricity conssumption: 2010−2014

hours GW

❋✐❣✉r❡ ✼✿ ❋r♦♠ ❧❡❢t t♦ r✐❣❤t✿ t❤r❡❡✲❞❛②s ❤♦✉r❧② ❛✐r t❡♠♣❡r❛t✉r❡ ✭C✮ ✭♦♥❧② s✉♠♠❡r

♠♦♥t❤s✮❀ ❤♦✉r❧② ❛❣❣r❡❣❛t❡ ❡❧❡❝tr✐❝✐t② ❝♦♥s✉♠♣t✐♦♥ ✭GW ✮ ✭♦♥❧② s✉♠♠❡r ♠♦♥t❤s✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✵ ✴ ✺✺

slide-90
SLIDE 90

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘❡❛❧✲❞❛t❛ ❛♣♣❧✐❝❛t✐♦♥s

❚❛❜❧❡ ✺✿ ❈♦♠♣❛r❛t✐✈❡ st✉❞②✿ ●♦❋ t❡st ❢♦r t❤❡ ♥✉❧❧ s✐♠♣❧❡ ❤②♣♦t❤❡s✐s ❢♦r t❤❡ r❡❛❧ ❞❛t❛s❡ts ❛❜♦✈❡ ❞✐s♣❧❛②❡❞ ✭❋P❈❘ ❡st✐♠❛t♦r✮✳ ❉❛t❛s❡t ❑♦❦♦s③❦❛ P❛t✐❧❡❛ PCvMn,p,q ❆❊▼❊❚ ✭✇✐♥❞ s♣❡❡❞✮ ✵.✵✵✷ ✵.✵✵✾ ✵.✵✵✸ ❆❊▼❊❚ ✭❧♦❣ ♣r❡❝✐♣✐t❛t✐♦♥✮ ✵.✵✵✶ ✵.✵✵✸ ✵.✵✵✶ ❈❛♥❛❞✐❛♥ ✇❡❛t❤❡r ✵.✵✵✺ ✵.✵✵✶ ✵.✵✵✶ ❖♥t❛r✐♦ ❡❧❡❝tr✐❝✐t② ✵.✵✵✷ ✵.✵✵✹ ✵.✵✵✷

❆ ♥♦♥✲tr✐✈✐❛❧ ❢✉♥❝t✐♦♥❛❧ r❡❧❛t✐♦♥ ✐s s✉❣❣❡st❡❞ ✐♥ ❛❧❧ ❝❛s❡s✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✶ ✴ ✺✺

slide-91
SLIDE 91

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❘❡❛❧✲❞❛t❛ ❛♣♣❧✐❝❛t✐♦♥s

❲❡ ❤❛✈❡ ❛❧s♦ ✐♠♣❧❡♠❡♥t❡❞ ♦✉r ●♦❋ t❡st ❢♦r t❤❡ ♥✉❧❧ ❝♦♠♣♦s✐t❡

❤②♣♦t❤❡s✐s✱ t❡st✐♥❣ ✐❢ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❢✉♥❝t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ✐s ❧✐♥❡❛r ♦r ♥♦t✳

❚❛❜❧❡ ✻✿ ❈♦♠♣❛r❛t✐✈❡ st✉❞②✿ ●♦❋ t❡st ❢♦r t❤❡ ♥✉❧❧ s✐♠♣❧❡ ❤②♣♦t❤❡s✐s ❢♦r t❤❡ r❡❛❧ ❞❛t❛s❡ts ❛❜♦✈❡ ❞✐s♣❧❛②❡❞✳ ❉❛t❛s❡t ❋P❈❘ ❋❈P❘✲▲✶ ❋P❈❘✲▲✷ ❋P❈❘✲▲✶✲❈ ❆❊▼❊❚ ✭❧♦❣ r❛✐♥❢❛❧❧✮

< ✷.✷e−✶✻ < ✷.✷e−✶✻ < ✷.✷e−✶✻ < ✷.✷e−✶✻

❈❛♥❛❞✐❛♥ ✇❡❛t❤❡r ✵.✵✵✸✺ ✵.✵✵✶✽ ✵.✵✵✸✺ ✵.✵✵✸✶ ❖♥t❛r✐♦ ❡❧❡❝tr✐❝✐t②

< ✷.✷e−✶✻ < ✷.✷e−✶✻ < ✷.✷e−✶✻ < ✷.✷e−✶✻

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✷ ✴ ✺✺

slide-92
SLIDE 92

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❲❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ ❛ ♥❡✇ ●♦❋ ♦♠♥✐❜✉s t❡st ❢♦r t❤❡ ❋▲▼❋❘✱ ❞r✐✈❡♥ ❜② ❛

❈r❛♠ér✲✈♦♥✲▼✐s❡s t②♣❡ st❛t✐st✐❝✱ ❡❛s✐❧② ✐♠♣❧❡♠❡♥t❡❞ ❛♥❞ ✐♥t❡r♣r❡t❡❞✳ ❚❤❡ st❛t✐st✐❝ ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ r❡s✐❞✉❛❧s ❛♥❞ t❤❡✐r ♣r♦❥❡❝t✐♦♥s✿ ✐t ❝♦✉❧❞ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛❧t❡r♥❛t✐✈❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳

❲❡ ❢♦r♠✉❧❛t❡ ❛ ♥❡✇ ❤②❜r✐❞ ❖▲❙ ❝♦♥str❛✐♥t ❡st✐♠❛t♦r ♦❢ β✳ ❚❤❡ ❝♦♠♣❛r❛t✐✈❡ st✉❞② ✐❧❧✉str❛t❡s ❤♦✇ ♦✉r t❡st ✐s t❤❡ ♠♦r❡ ♣♦✇❡r❢✉❧

♦♣t✐♦♥ ❛✈❛✐❧❛❜❧❡ ❢♦r t❤❡ ♥♦ ❡✛❡❝ts t❡st ✉♥❞❡r ♥♦♥❧✐♥❡❛r ❛❧t❡r♥❛t✐✈❡s✳

◆♦ ❝♦♠♣❡t✐♥❣ ♣r♦❝❡❞✉r❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r t❤❡ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡✲

s✐s✳ ❖✉r t❡st r❡s♣❡❝ts t❤❡ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s❡s✳ ❚❤❡ ❡♠♣✐r✐❝❛❧ ♣♦✇❡r ✉♥❞❡r ❛❧t❡r♥❛t✐✈❡s t❡♥❞s t♦ ✶ ✇❤❡♥ n ✐♥❝r❡❛s❡s✳

❆♥ ❘ ♣❛❝❦❛❣❡ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞ ✭❣♦❢❢❞❛ ♣❛❝❦❛❣❡✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✸ ✴ ✺✺

slide-93
SLIDE 93

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❲❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ ❛ ♥❡✇ ●♦❋ ♦♠♥✐❜✉s t❡st ❢♦r t❤❡ ❋▲▼❋❘✱ ❞r✐✈❡♥ ❜② ❛

❈r❛♠ér✲✈♦♥✲▼✐s❡s t②♣❡ st❛t✐st✐❝✱ ❡❛s✐❧② ✐♠♣❧❡♠❡♥t❡❞ ❛♥❞ ✐♥t❡r♣r❡t❡❞✳ ❚❤❡ st❛t✐st✐❝ ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ r❡s✐❞✉❛❧s ❛♥❞ t❤❡✐r ♣r♦❥❡❝t✐♦♥s✿ ✐t ❝♦✉❧❞ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛❧t❡r♥❛t✐✈❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳

❲❡ ❢♦r♠✉❧❛t❡ ❛ ♥❡✇ ❤②❜r✐❞ ❖▲❙ ❝♦♥str❛✐♥t ❡st✐♠❛t♦r ♦❢ β✳ ❚❤❡ ❝♦♠♣❛r❛t✐✈❡ st✉❞② ✐❧❧✉str❛t❡s ❤♦✇ ♦✉r t❡st ✐s t❤❡ ♠♦r❡ ♣♦✇❡r❢✉❧

♦♣t✐♦♥ ❛✈❛✐❧❛❜❧❡ ❢♦r t❤❡ ♥♦ ❡✛❡❝ts t❡st ✉♥❞❡r ♥♦♥❧✐♥❡❛r ❛❧t❡r♥❛t✐✈❡s✳

◆♦ ❝♦♠♣❡t✐♥❣ ♣r♦❝❡❞✉r❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r t❤❡ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡✲

s✐s✳ ❖✉r t❡st r❡s♣❡❝ts t❤❡ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s❡s✳ ❚❤❡ ❡♠♣✐r✐❝❛❧ ♣♦✇❡r ✉♥❞❡r ❛❧t❡r♥❛t✐✈❡s t❡♥❞s t♦ ✶ ✇❤❡♥ n ✐♥❝r❡❛s❡s✳

❆♥ ❘ ♣❛❝❦❛❣❡ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞ ✭❣♦❢❢❞❛ ♣❛❝❦❛❣❡✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✸ ✴ ✺✺

slide-94
SLIDE 94

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❲❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ ❛ ♥❡✇ ●♦❋ ♦♠♥✐❜✉s t❡st ❢♦r t❤❡ ❋▲▼❋❘✱ ❞r✐✈❡♥ ❜② ❛

❈r❛♠ér✲✈♦♥✲▼✐s❡s t②♣❡ st❛t✐st✐❝✱ ❡❛s✐❧② ✐♠♣❧❡♠❡♥t❡❞ ❛♥❞ ✐♥t❡r♣r❡t❡❞✳ ❚❤❡ st❛t✐st✐❝ ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ r❡s✐❞✉❛❧s ❛♥❞ t❤❡✐r ♣r♦❥❡❝t✐♦♥s✿ ✐t ❝♦✉❧❞ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛❧t❡r♥❛t✐✈❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳

❲❡ ❢♦r♠✉❧❛t❡ ❛ ♥❡✇ ❤②❜r✐❞ ❖▲❙ ❝♦♥str❛✐♥t ❡st✐♠❛t♦r ♦❢ β✳ ❚❤❡ ❝♦♠♣❛r❛t✐✈❡ st✉❞② ✐❧❧✉str❛t❡s ❤♦✇ ♦✉r t❡st ✐s t❤❡ ♠♦r❡ ♣♦✇❡r❢✉❧

♦♣t✐♦♥ ❛✈❛✐❧❛❜❧❡ ❢♦r t❤❡ ♥♦ ❡✛❡❝ts t❡st ✉♥❞❡r ♥♦♥❧✐♥❡❛r ❛❧t❡r♥❛t✐✈❡s✳

◆♦ ❝♦♠♣❡t✐♥❣ ♣r♦❝❡❞✉r❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r t❤❡ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡✲

s✐s✳ ❖✉r t❡st r❡s♣❡❝ts t❤❡ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s❡s✳ ❚❤❡ ❡♠♣✐r✐❝❛❧ ♣♦✇❡r ✉♥❞❡r ❛❧t❡r♥❛t✐✈❡s t❡♥❞s t♦ ✶ ✇❤❡♥ n ✐♥❝r❡❛s❡s✳

❆♥ ❘ ♣❛❝❦❛❣❡ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞ ✭❣♦❢❢❞❛ ♣❛❝❦❛❣❡✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✸ ✴ ✺✺

slide-95
SLIDE 95

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❲❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ ❛ ♥❡✇ ●♦❋ ♦♠♥✐❜✉s t❡st ❢♦r t❤❡ ❋▲▼❋❘✱ ❞r✐✈❡♥ ❜② ❛

❈r❛♠ér✲✈♦♥✲▼✐s❡s t②♣❡ st❛t✐st✐❝✱ ❡❛s✐❧② ✐♠♣❧❡♠❡♥t❡❞ ❛♥❞ ✐♥t❡r♣r❡t❡❞✳ ❚❤❡ st❛t✐st✐❝ ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ r❡s✐❞✉❛❧s ❛♥❞ t❤❡✐r ♣r♦❥❡❝t✐♦♥s✿ ✐t ❝♦✉❧❞ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛❧t❡r♥❛t✐✈❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳

❲❡ ❢♦r♠✉❧❛t❡ ❛ ♥❡✇ ❤②❜r✐❞ ❖▲❙ ❝♦♥str❛✐♥t ❡st✐♠❛t♦r ♦❢ β✳ ❚❤❡ ❝♦♠♣❛r❛t✐✈❡ st✉❞② ✐❧❧✉str❛t❡s ❤♦✇ ♦✉r t❡st ✐s t❤❡ ♠♦r❡ ♣♦✇❡r❢✉❧

♦♣t✐♦♥ ❛✈❛✐❧❛❜❧❡ ❢♦r t❤❡ ♥♦ ❡✛❡❝ts t❡st ✉♥❞❡r ♥♦♥❧✐♥❡❛r ❛❧t❡r♥❛t✐✈❡s✳

◆♦ ❝♦♠♣❡t✐♥❣ ♣r♦❝❡❞✉r❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r t❤❡ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡✲

s✐s✳ ❖✉r t❡st r❡s♣❡❝ts t❤❡ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s❡s✳ ❚❤❡ ❡♠♣✐r✐❝❛❧ ♣♦✇❡r ✉♥❞❡r ❛❧t❡r♥❛t✐✈❡s t❡♥❞s t♦ ✶ ✇❤❡♥ n ✐♥❝r❡❛s❡s✳

❆♥ ❘ ♣❛❝❦❛❣❡ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞ ✭❣♦❢❢❞❛ ♣❛❝❦❛❣❡✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✸ ✴ ✺✺

slide-96
SLIDE 96

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❲❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ ❛ ♥❡✇ ●♦❋ ♦♠♥✐❜✉s t❡st ❢♦r t❤❡ ❋▲▼❋❘✱ ❞r✐✈❡♥ ❜② ❛

❈r❛♠ér✲✈♦♥✲▼✐s❡s t②♣❡ st❛t✐st✐❝✱ ❡❛s✐❧② ✐♠♣❧❡♠❡♥t❡❞ ❛♥❞ ✐♥t❡r♣r❡t❡❞✳ ❚❤❡ st❛t✐st✐❝ ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ r❡s✐❞✉❛❧s ❛♥❞ t❤❡✐r ♣r♦❥❡❝t✐♦♥s✿ ✐t ❝♦✉❧❞ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛❧t❡r♥❛t✐✈❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳

❲❡ ❢♦r♠✉❧❛t❡ ❛ ♥❡✇ ❤②❜r✐❞ ❖▲❙ ❝♦♥str❛✐♥t ❡st✐♠❛t♦r ♦❢ β✳ ❚❤❡ ❝♦♠♣❛r❛t✐✈❡ st✉❞② ✐❧❧✉str❛t❡s ❤♦✇ ♦✉r t❡st ✐s t❤❡ ♠♦r❡ ♣♦✇❡r❢✉❧

♦♣t✐♦♥ ❛✈❛✐❧❛❜❧❡ ❢♦r t❤❡ ♥♦ ❡✛❡❝ts t❡st ✉♥❞❡r ♥♦♥❧✐♥❡❛r ❛❧t❡r♥❛t✐✈❡s✳

◆♦ ❝♦♠♣❡t✐♥❣ ♣r♦❝❡❞✉r❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r t❤❡ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡✲

s✐s✳ ❖✉r t❡st r❡s♣❡❝ts t❤❡ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s❡s✳ ❚❤❡ ❡♠♣✐r✐❝❛❧ ♣♦✇❡r ✉♥❞❡r ❛❧t❡r♥❛t✐✈❡s t❡♥❞s t♦ ✶ ✇❤❡♥ n ✐♥❝r❡❛s❡s✳

❆♥ ❘ ♣❛❝❦❛❣❡ ✐s ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞ ✭❣♦❢❢❞❛ ♣❛❝❦❛❣❡✮✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✸ ✴ ✺✺

slide-97
SLIDE 97

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❘❡❢❡r❡♥❝❡s

❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts

❚❤❡ ✇♦r❦ ♦❢ ❏✳ ➪❧✈❛r❡③✲▲✐é❜❛♥❛ ❤❛s ❜❡❡♥ s✉♣♣♦rt❡❞ ✐♥ ♣❛rt ❜② ♣r♦❥❡❝ts P●❈✷✵✶✽✲✵✾✾✺✹✾✲❇✲■✵✵ ❛♥❞ ▼❚▼✷✵✶✺✲✼✶✽✸✾✲P ♦❢ ▼■◆❊❈❖✱ ❙♣❛✐♥ ✭❝♦✲❢✉♥❞❡❞ ✇✐t❤ ❋❊❉❊❘ ❢✉♥❞s✮✳

❬✶❪ ❈✉❡st❛✲❆❧❜❡rt♦s✱ ❏✳ ❆✳✱ ●❛r❝í❛✲P♦rt✉❣✉és✱ ❊✳✱ ❋❡❜r❡r♦✲❇❛♥❞❡✱ ▼✳ ❛♥❞

  • ♦♥③á❧❡③✲▼❛♥t❡✐❣❛✱ ❲✳ ✭✷✵✶✾✮✳ ●♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❢♦r t❤❡ ❢✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧

❜❛s❡❞ ♦♥ r❛♥❞♦♠❧② ♣r♦❥❡❝t❡❞ ❡♠♣✐r✐❝❛❧ ♣r♦❝❡ss❡s✳ ❆♥♥✳ ❙t❛t✳✱ ✹✼✱ ✹✸✾✕✹✻✼✳ ❬✷❪ ❊s❝❛♥❝✐❛♥♦✱ ❏✳ ❈✳ ✭✷✵✵✻✮✳ ❆ ❝♦♥s✐st❡♥t ❞✐❛❣♥♦st✐❝ t❡st ❢♦r r❡❣r❡ss✐♦♥ ♠♦❞❡❧s ✉s✐♥❣ ♣r♦❥❡❝t✐♦♥s✳ ❊❝♦♥♦♠❡tr✐❝ ❚❤❡♦r②✱ ✷✷✱ ♣♣✳ ✶✵✸✵✕✶✵✺✶✳ ❬✸❪

  • ❛r❝í❛✲P♦rt✉❣✉és✱ ❊✳✱ ●♦♥③á❧❡③✲▼❛♥t❡✐❣❛✱ ❲✳ ❛♥❞ ❋❡❜r❡r♦✲❇❛♥❞❡✱ ▼✳ ✭✷✵✶✹✮✳ ❆

❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ❢♦r t❤❡ ❢✉♥❝t✐♦♥❛❧ ❧✐♥❡❛r ♠♦❞❡❧ ✇✐t❤ s❝❛❧❛r r❡s♣♦♥s❡✳ ❏✳ ❈♦♠♣✳ ●r❛♣❤✳ ❙t❛t✳✱ ✷✸✱ ♣♣✳ ✼✻✶✕✼✼✽✳ ❬✹❪

  • ♦♥③á❧❡③✲▼❛♥t❡✐❣❛✱ ❲✳ ❛♥❞ ❈r✉❥❡✐r❛s✱ ❘✳ ▼✳ ✭✷✵✶✸✮✲ ❆♥ ✉♣❞❛t❡❞ r❡✈✐❡✇ ♦♥
  • ♦♦❞♥❡ss✕♦❢✕❋✐t t❡sts ❢♦r r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳ ❚❡st✱ ✷✷✱ ♣♣✳ ✸✻✶✕✹✹✼✳

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✹ ✴ ✺✺

slide-98
SLIDE 98

❲❤❛t ❛❜♦✉t ❋✉♥❝t✐♦♥❛❧ ❉❛t❛❄

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❢✉♥❝t✐♦♥❛❧ r❡❣r❡ss✐♦♥ ♠♦❞❡❧

◆✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❘ ♣❛❝❦❛❣❡ ❘ ♣❛❝❦❛❣❡ ❙✐♠✉❧❛t✐♦♥ st✉❞② ❘❡❛❧✲❞❛t❛ ❈♦♥❝❧✉s✐♦♥s

❛❧✈❛r❡③❧❥❛✈✐❡r❅✉♥✐♦✈✐✳❡s

  • ❤tt♣s✿✴✴✇✇✇✳r❡s❡❛r❝❤❣❛t❡✳♥❡t✴♣r♦❢✐❧❡✴❏❛✈✐❡r❴❆❧✈❛r❡③❴

▲✐❡❜❛♥❛

❤tt♣s✿✴✴❣✐t❤✉❜✳❝♦♠✴❏❛✈✐❡r❆❧✈❛r❡③▲✐❡❜❛♥❛

❏❛✈✐❡r ➪❧✈❛r❡③ ▲✐é❜❛♥❛

  • ♦❋ ♦♠♥✐❜✉s t❡st ❢♦r ❋▲▼❋❘

✺✺ ✴ ✺✺