rt trs P ( X vi = 1 v , i ) = 1 0 1 1 3 1 + - PowerPoint PPT Presentation

●❡♥❡r❛t✐♥❣ ♠❛tr✐❝❡s ❊①❛❝t t❡sts ❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ✭❘❛s❝❤✱ ✶✾✻✵✮ 0 0 1 0 2 1 0 0 1 2 exp ( θ v − β i ) 1 0 0 0 1 ●❡♥❡r❛t✐♥❣ ♠❛tr✐❝❡s P ( X vi = 1 ∣ θ v ,β i ) = 1 0 1 1 3 1 + exp ( θ v − β i ) 0 0 1 0 1 1 1 1 0 3 0 1 0 0 1 ✇✐t❤ ♦r❞✐♥❛❧ r❡s♣♦♥s❡s ❛♥❞ ✜①❡❞ ♠❛r❣✐♥s 1 0 0 0 1 1 1 0 0 2 0 0 0 1 1 ✻ ✸ ✹ ✸ ❑❛t❤r✐♥ ●r✉❜❡r ✫ ❘❡✐♥❤♦❧❞ ❍❛t③✐♥❣❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s✱ ❲❯ ❲✐❡♥ ● ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♠❛r❣✐♥s ● ♠♦❞❡❧ ✜t ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ❜② ♣❛r❛♠❡tr✐❝ ❛♥❞ q✉❛s✐✲❡①❛❝t t❡sts Ps②❝❤♦❝♦ ✷✵✶✷ ✶ Ps②❝❤♦❝♦ ✷✵✶✷ ✷ ❊①❛❝t t❡sts ❊①❛❝t t❡sts ▼♦t✐✈❛t✐♦♥ ❢♦r ❡①❛❝t t❡sts ❙t❛t✐st✐❝❛❧ t❡sts ❛♥❞ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ❛r❡ ❜❛s❡❞ ♦♥ ❡①❛❝t ♣r♦❜❛❜✐❧✐t② st❛t❡♠❡♥ts t❤❛t ❛r❡ ✈❛❧✐❞ ❢♦r ❛♥② s❛♠♣❧❡ s✐③❡✳ 1.0 0.8 ❈♦♥str✉❝t✐♦♥ ♣r✐♥❝✐♣❧❡✿ ● ❘❡❛rr❛♥❣❡ t❤❡ ❧❛❜❡❧s ♦❢ t❤❡ ♦❜s❡r✈❡❞ ❞❛t❛ ♣♦✐♥ts✳ ● ❈❛❧❝✉❧❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ t❤❡ t❡st st❛t✐st✐❝✳ 0.6 quantiles ● ❨✐❡❧❞s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡st st❛t✐st✐❝ ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s✳ 0.4 0.2 0.0 0.30 0.35 0.40 0.45 T(A_0) Ps②❝❤♦❝♦ ✷✵✶✷ ✸ Ps②❝❤♦❝♦ ✷✵✶✷ ✹

❊①❛❝t t❡sts ❊①❛❝t t❡sts ▼♦t✐✈❛t✐♦♥ ❢♦r ❡①❛❝t t❡sts ✭❝♦♥t✳✮ ❇❛❝❦❣r♦✉♥❞ ❱❛r✐♦✉s ❛❧❣♦r✐t❤♠s ❢♦r s❛♠♣❧✐♥❣ 0 − 1 ♠❛tr✐❝❡s ✇✐t❤ ❣✐✈❡♥ ♠❛r❣✐♥❛❧s ✐♥ ❛ ♥♦♥✉♥✐✲ ❆❞✈❛♥t❛❣❡s✿ ❢♦r♠ ✇❛② ❤❛✈❡ ❛❧r❡❛❞② ❜❡❡♥ ♣r♦♣♦s❡❞ ❛♥❞ ❝❛♥ ❣❡♥❡r❛❧❧② ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ ● ◆♦ ♣❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ♥❡❝❡ss❛r②✳ ❝❧❛ss❡s✿ ● ❆r❡ ♥♦t ❜❛s❡❞ ♦♥ ❛s②♠♣t♦t✐❝ ❛♥❞ ❛♣♣r♦①✐♠❛t❡ st❛t✐st✐❝❛❧ ♠❡t❤♦❞s✳ ● ◆♦♥✉♥✐❢♦r♠ s❛♠♣❧✐♥❣ s❝❤❡♠❡s✿ ● ❱❛❧✐❞ ❢♦r s♠❛❧❧ s❛♠♣❧❡ s✐③❡s✳ ✕ ❘❡❝✉rs✐✈❡ s♦❧✈✐♥❣ ♦❢ ❛ ❧✐♥❡❛r ♣r♦❣r❛♠ ✇✐t❤ r❡str✐❝t✐♦♥s t♦ t❤❡ r♦✇ s✉♠s✳ ✕ ❇❛s❡❞ ♦♥ t❤❡ s❡q✉❡♥t✐❛❧ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ✭❙■❙✮ ❛❧❣♦r✐t❤♠ ✭❡✳❣✳ ❙♥✐❥❞❡rs✱ ✶✾✾✶✱ ❈❤❡♥ ❛♥❞ ❙♠❛❧❧✱ ✷✵✵✺✱ ❈❤❡♥✱ ❉✐♥✇♦♦❞ ❛♥❞ ❙✉❧❧✐✈❛♥t✱ ✷✵✵✻✮✳ 1 ✳ ✳ 2 1 ✵ ✶ 2 → 0 ✳ ✳ 1 0 ✶ ✵ 1 ✳ ✳ ✵ ✵ 1 1 1 1 ✷ ✶ ✶ ✷ ✶ ✶ Ps②❝❤♦❝♦ ✷✵✶✷ ✺ Ps②❝❤♦❝♦ ✷✵✶✷ ✻ ❊①❛❝t t❡sts ❊①❛❝t t❡sts ❇❛❝❦❣r♦✉♥❞ ✭❝♦♥t✳✮ ❘❡q✉✐r❡♠❡♥ts ● ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ▼❛r❦♦✈✲❈❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞✿ ✕ ❆❧❧ ♠❛tr✐❝❡s ✐♥ t❤❡ s❛♠♣❧❡ s♣❛❝❡ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s st❛t❡s✳ ✕ ❚❤❡ s❛♠♣❧✐♥❣ s❝❤❡♠❡ ❛♥❞ ❛ s♣❡❝✐❛❧ ♣❡r♠✉t❛t✐♦♥ r✉❧❡ ✐s ❞❡✜♥✐♥❣ t❤❡✐r tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② ✭P♦♥♦❝♥②✱ ✷✵✵✶✱ ❱❡r❤❡❧st✱ ✷✵✵✽✮✳ ● ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡ 0 1 1 0 → ● ■♥❞❡♣❡♥❞❡♥❝❡ 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 ● ❯♥✐❢♦r♠ s❛♠♣❧✐♥❣ Ps②❝❤♦❝♦ ✷✵✶✷ ✼ Ps②❝❤♦❝♦ ✷✵✶✷ ✽

✭✶✮ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡ ✭✶✮ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡ ❚❤❡ ▼❈▼❈ ❛♣♣r♦❛❝❤ ♦❢ ❱❡r❤❡❧st ✐♥ ❣❡♥❡r❛❧ ❚❤❡ ❜✐♥♦♠✐❛❧ r✉❧❡ ❛♥❞ ❜✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s ● ❆❧❧ ❜✐♥❛r② ♠❛tr✐❝❡s ✇✐t❤ ✜①❡❞ r♦✇ ✭ r ✮ ❛♥❞ ❝♦❧✉♠♥ ✭ c ✮ s✉♠s✱ A ∈ Σ rc ❇✐♥♦♠✐❛❧ r✉❧❡✿ ✭✇❤❡r❡❛s Σ rc ❞❡♥♦t❡s t❤❡ s❛♠♣❧❡ s♣❛❝❡ ♦❢ ♣♦ss✐❜❧❡ ♠❛tr✐❝❡s✮ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s st❛t❡s✳ → ❆ss✐❣♥ a ♦♥❡s t♦ m r♦✇s t♦ t❤❡ ✜rst ❝♦❧✉♠♥ ✇✐t❤ r♦✇ t♦t❛❧s ❡q✉❛❧ t♦ ♦♥❡✱ ❛♥❞ ③❡r♦ t♦ t❤❡ m − a r♦✇s✳ ● ❚❤❡ ♦❜s❡r✈❡❞ ❞❛t❛ ♠❛tr✐① ✐s ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ st❛rt✐♥❣ st❛t❡✱ A 0 ✳ → ❨✐❡❧❞s ✜rst ❝♦❧✉♠♥ ♦❢ t❤❡ tr❛♥s❢♦r♠❡❞ ♠❛tr✐①✱ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❥✉st t❤❡ ● A 0 ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥ ♦♥❡ st❡♣ ✐♥t♦ ♦t❤❡r ♠❛tr✐❝❡s A t ∈ Σ rc ✉s✐♥❣ ❛ ❝♦♠♣❧✐♠❡♥t ♦❢ ✐t✳ ✇❡❧❧ ❞❡✜♥❡❞ r✉❧❡ R ✳ ● ❚❤❡ R ✲♥❡✐❣❤❜♦r❤♦♦❞ ✐s t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❝❤❛❜❧❡ ♠❛tr✐❝❡s ✉s✐♥❣ s✉❝❤ ❛ ✶ ✷ ✸ ✹ ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✹ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ A 0 ✱ A R ( A 0 ) ✳ 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 1 ● ❙❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠✿ 1 0 0 0 1 0 1 0 1 0 0 0 → → → 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 A 0 A s +1 A 1 A 2 A s 1 1 1 0 1 1 1 1 1 1 1 0 A R ( A 0 ) A R ( A s ) A R ( A 1 ) 0 1 0 0 0 1 1 0 1 0 0 0 ● ❙❛♠♣❧✐♥❣ s❝❤❡♠❡ ❞❡✜♥❡s t❤❡ tr❛♥s✐t✐♦♥ ♠❛tr✐① P = ( p st ) ✭ ✇✐t❤ lim n → ∞ P n e t = π ✮ 1 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 ♦❢ t❤❡ ▼❛r❦♦✈ ❈❤❛✐♥ 0 0 0 1 0 0 0 0 0 0 0 1 Ps②❝❤♦❝♦ ✷✵✶✷ ✾ Ps②❝❤♦❝♦ ✷✵✶✷ ✶✵ ✭✶✮ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡ ✭✶✮ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡ ❚❤❡ ❜✐♥♦♠✐❛❧ r✉❧❡ ❛♥❞ ❜✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s ✭❝♦♥t✳✮ ❚❤❡ ❜✐♥♦♠✐❛❧ r✉❧❡ ❛♥❞ ❜✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s ✭❝♦♥t✳✮ ❚❤❡ B ij ✲♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ A ∈ Σ rc ✐s ❞❡✜♥❡❞ ❜② A ( i,j ) ( A ) = { A s ∶ A s ✐s ❛ B ij tr❛♥s❢♦r♠ ♦❢ A ❛♥❞ A s ≠ A } ✶ ✷ ✸ ✹ ✶ ✷ ✶ ✸ ✶ ✹ B 1 1 1 0 1 1 1 1 1 0 ❚❤❡ s❡t ♦❢ ❛❧❧ ♠❛tr✐❝❡s t❤❛t ❝❛♥ ❜❡ ❢♦r♠❡❞ ❜② ❛ s✐♥❣❧❡ ❜✐♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥ 1 1 0 1 1 1 1 0 1 1 ♦❢ ❛ s✐♥❣❧❡ ❝♦❧✉♠♥ ♣❛✐r ♦❢ A ✐s 1 0 1 0 1 0 1 1 1 0 A ( i,j ) A B ( A ) = ⋃ ( A ) 0 1 0 0 0 1 0 0 0 0 B 0 1 0 1 0 1 0 0 0 1 ( i,j ) ... 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 ✶ ✷ ✸ ✹ ✶ ✷ ✶ ✸ ✶ ✹ 1 1 1 0 1 1 1 1 1 0 ❚❤❡ ❝♦❧✉♠♥ ♣❛✐r ( i,j ) ✐s ❛ ●✉tt♠❛♥ ♣❛✐r ✐❢ a ij × b ij = 0 ✱ ✐❢ a ij × b ij > 0 t❤❡ ♣❛✐r ✐s 1 1 0 1 1 1 1 0 1 1 ❝❛❧❧❡❞ r❡❣✉❧❛r✳ 1 0 1 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 ❚❤❡ k 2 ✲♠❡❛s✉r❡ ♦❢ A ∈ Σ rc ✐s ❞❡✜♥❡❞ ❛s 0 1 0 1 0 1 0 0 ... 0 1 k 2 ( A ) = { ♯ ( i,j ) ∶ i < j ≤ k, ( i,j ) ✐s ❛ r❡❣✉❧❛r ♣❛✐r } 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 ( 3 1 ) − ( 3 2 ) 1 × 0 1 × 2 1 × 2 Ps②❝❤♦❝♦ ✷✵✶✷ ✶✶ Ps②❝❤♦❝♦ ✷✵✶✷ ✶✷