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

• ❡♥❡r❛t✐♥❣ ♠❛tr✐❝❡s
• ❡♥❡r❛t✐♥❣ ♠❛tr✐❝❡s

✇✐t❤ ♦r❞✐♥❛❧ r❡s♣♦♥s❡s ❛♥❞ ✜①❡❞ ♠❛r❣✐♥s

❑❛t❤r✐♥ ●r✉❜❡r ✫ ❘❡✐♥❤♦❧❞ ❍❛t③✐♥❣❡r

❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s✱ ❲❯ ❲✐❡♥ Ps②❝❤♦❝♦ ✷✵✶✷ ✶ ❊①❛❝t t❡sts

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ✭❘❛s❝❤✱ ✶✾✻✵✮

P(Xvi = 1∣θv,βi) = exp(θv − βi) 1 + exp(θv − βi) 1 2 1 1 2 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1 1 2 1 1 ✻ ✸ ✹ ✸

• ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♠❛r❣✐♥s
• ♠♦❞❡❧ ✜t ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ❜② ♣❛r❛♠❡tr✐❝ ❛♥❞ q✉❛s✐✲❡①❛❝t t❡sts

Ps②❝❤♦❝♦ ✷✵✶✷ ✷ ❊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❡✳ ❈♦♥str✉❝t✐♦♥ ♣r✐♥❝✐♣❧❡✿

• ❘❡❛rr❛♥❣❡ t❤❡ ❧❛❜❡❧s ♦❢ t❤❡ ♦❜s❡r✈❡❞ ❞❛t❛ ♣♦✐♥ts✳
• ❈❛❧❝✉❧❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ t❤❡ t❡st st❛t✐st✐❝✳
• ❨✐❡❧❞s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡st st❛t✐st✐❝ ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s✳

Ps②❝❤♦❝♦ ✷✵✶✷ ✸ ❊①❛❝t t❡sts

0.30 0.35 0.40 0.45 0.0 0.2 0.4 0.6 0.8 1.0 T(A_0) quantiles

Ps②❝❤♦❝♦ ✷✵✶✷ ✹

❊①❛❝t t❡sts

▼♦t✐✈❛t✐♦♥ ❢♦r ❡①❛❝t t❡sts ✭❝♦♥t✳✮

❆❞✈❛♥t❛❣❡s✿

• ◆♦ ♣❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ♥❡❝❡ss❛r②✳
• ❆r❡ ♥♦t ❜❛s❡❞ ♦♥ ❛s②♠♣t♦t✐❝ ❛♥❞ ❛♣♣r♦①✐♠❛t❡ st❛t✐st✐❝❛❧ ♠❡t❤♦❞s✳
• ❱❛❧✐❞ ❢♦r s♠❛❧❧ s❛♠♣❧❡ s✐③❡s✳

Ps②❝❤♦❝♦ ✷✵✶✷ ✺ ❊①❛❝t t❡sts

❇❛❝❦❣r♦✉♥❞

❱❛r✐♦✉s ❛❧❣♦r✐t❤♠s ❢♦r s❛♠♣❧✐♥❣ 0−1 ♠❛tr✐❝❡s ✇✐t❤ ❣✐✈❡♥ ♠❛r❣✐♥❛❧s ✐♥ ❛ ♥♦♥✉♥✐✲ ❢♦r♠ ✇❛② ❤❛✈❡ ❛❧r❡❛❞② ❜❡❡♥ ♣r♦♣♦s❡❞ ❛♥❞ ❝❛♥ ❣❡♥❡r❛❧❧② ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ ❝❧❛ss❡s✿

• ◆♦♥✉♥✐❢♦r♠ s❛♠♣❧✐♥❣ s❝❤❡♠❡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 1 ✳ ✳ 1 ✷ ✶ ✶

→

1 ✵ ✶ 2 ✶ ✵ 1 1 ✵ ✵ 1 ✷ ✶ ✶ Ps②❝❤♦❝♦ ✷✵✶✷ ✻ ❊①❛❝t t❡sts

❇❛❝❦❣r♦✉♥❞ ✭❝♦♥t✳✮

• ❆♣♣❧✐❝❛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✱ ✷✵✵✽✮✳ 1 1 1 1 1

→

1 1 1 1 1 Ps②❝❤♦❝♦ ✷✵✶✷ ✼ ❊①❛❝t t❡sts

❘❡q✉✐r❡♠❡♥ts

• ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡
• ■♥❞❡♣❡♥❞❡♥❝❡
• ❯♥✐❢♦r♠ s❛♠♣❧✐♥❣

Ps②❝❤♦❝♦ ✷✵✶✷ ✽

✭✶✮ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡

❚❤❡ ▼❈▼❈ ❛♣♣r♦❛❝❤ ♦❢ ❱❡r❤❡❧st ✐♥ ❣❡♥❡r❛❧

• ❆❧❧ ❜✐♥❛r② ♠❛tr✐❝❡s ✇✐t❤ ✜①❡❞ r♦✇ ✭r✮ ❛♥❞ ❝♦❧✉♠♥ ✭c✮ s✉♠s✱ A ∈ Σrc

✭✇❤❡r❡❛s Σrc ❞❡♥♦t❡s t❤❡ s❛♠♣❧❡ s♣❛❝❡ ♦❢ ♣♦ss✐❜❧❡ ♠❛tr✐❝❡s✮ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s st❛t❡s✳

• ❚❤❡ ♦❜s❡r✈❡❞ ❞❛t❛ ♠❛tr✐① ✐s ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ st❛rt✐♥❣ st❛t❡✱ A0✳
• A0 ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥ ♦♥❡ st❡♣ ✐♥t♦ ♦t❤❡r ♠❛tr✐❝❡s At ∈ Σrc ✉s✐♥❣ ❛

✇❡❧❧ ❞❡✜♥❡❞ r✉❧❡ R✳

• ❚❤❡ R✲♥❡✐❣❤❜♦r❤♦♦❞ ✐s t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❝❤❛❜❧❡ ♠❛tr✐❝❡s ✉s✐♥❣ s✉❝❤ ❛

tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ A0✱ AR(A0)✳

• ❙❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠✿

A0 A1 A2 As+1 AR(A0) AR(A1) AR(As) As

• ❙❛♠♣❧✐♥❣ s❝❤❡♠❡ ❞❡✜♥❡s t❤❡ tr❛♥s✐t✐♦♥ ♠❛tr✐① P = (pst) ✭ ✇✐t❤ lim

n→∞P net = π✮

♦❢ t❤❡ ▼❛r❦♦✈ ❈❤❛✐♥ Ps②❝❤♦❝♦ ✷✵✶✷ ✾ ✭✶✮ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡

❚❤❡ ❜✐♥♦♠✐❛❧ r✉❧❡ ❛♥❞ ❜✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s

❇✐♥♦♠✐❛❧ r✉❧❡✿ → ❆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 ✜rst ❝♦❧✉♠♥ ♦❢ t❤❡ tr❛♥s❢♦r♠❡❞ ♠❛tr✐①✱ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❥✉st t❤❡ ❝♦♠♣❧✐♠❡♥t ♦❢ ✐t✳ ✶ ✷ ✸ ✹ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

→

✶ ✷ 1 1 1 1 1 1 1 1 1

→

✶ ✷ 1 1 1 1 1 1 1 1 1

→

✶ ✷ ✸ ✹ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Ps②❝❤♦❝♦ ✷✵✶✷ ✶✵ ✭✶✮ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡

❚❤❡ ❜✐♥♦♠✐❛❧ r✉❧❡ ❛♥❞ ❜✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s ✭❝♦♥t✳✮

❚❤❡ Bij✲♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ A ∈ Σrc ✐s ❞❡✜♥❡❞ ❜② A(i,j)

B

(A) = {As ∶ As ✐s ❛ Bij tr❛♥s❢♦r♠ ♦❢ A ❛♥❞ As ≠ A} ❚❤❡ s❡t ♦❢ ❛❧❧ ♠❛tr✐❝❡s t❤❛t ❝❛♥ ❜❡ ❢♦r♠❡❞ ❜② ❛ s✐♥❣❧❡ ❜✐♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ s✐♥❣❧❡ ❝♦❧✉♠♥ ♣❛✐r ♦❢ A ✐s AB(A) = ⋃

(i,j)

A(i,j)

B

(A) ✶ ✷ ✸ ✹ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ✶ ✷ 1 1 1 1 1 1 1 1 1 (3 1) 1 × 2 ✶ ✸ 1 1 1 1 1 1 1 − 1 × 0 ... ✶ ✹ 1 1 1 1 1 1 1 (3 2) 1 × 2 Ps②❝❤♦❝♦ ✷✵✶✷ ✶✶ ✭✶✮ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡

❚❤❡ ❜✐♥♦♠✐❛❧ r✉❧❡ ❛♥❞ ❜✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s ✭❝♦♥t✳✮

✶ ✷ ✸ ✹ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ✶ ✷ 1 1 1 1 1 1 1 1 1 ✶ ✸ 1 1 1 1 1 1 1 ... ✶ ✹ 1 1 1 1 1 1 1 ❚❤❡ ❝♦❧✉♠♥ ♣❛✐r (i,j) ✐s ❛ ●✉tt♠❛♥ ♣❛✐r ✐❢ aij × bij = 0✱ ✐❢ aij × bij > 0 t❤❡ ♣❛✐r ✐s ❝❛❧❧❡❞ r❡❣✉❧❛r✳ ❚❤❡ k2✲♠❡❛s✉r❡ ♦❢ A ∈ Σrc ✐s ❞❡✜♥❡❞ ❛s k2(A) = {♯(i,j) ∶ i < j ≤ k,(i,j) ✐s ❛ r❡❣✉❧❛r ♣❛✐r} Ps②❝❤♦❝♦ ✷✵✶✷ ✶✷

✭✷✮ ■♥❞❡♣❡♥❞❡♥❝❡

❚❤❡ ▼❈▼❈✲♠❡t❤♦❞ ✐♥ ❣❡♥❡r❛❧

A0 As As+1 AS Stationary distribution Burn-in Step size

Ps②❝❤♦❝♦ ✷✵✶✷ ✶✸ ✭✸✮ ❯♥✐❢♦r♠ s❛♠♣❧✐♥❣

❚❤❡ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ❛❧❣♦r✐t❤♠

• ❙t❛rt t❤❡ ❝❤❛✐♥ ✐♥ t❤❡ ♦❜s❡r✈❡❞ ❞❛t❛ ♠❛tr✐①✱ A0✳
• ❙❡❧❡❝t r❛♥❞♦♠❧② ❛ ♣❛✐r ♦❢ ❝♦❧✉♠♥s (i,j) ❢r♦♠ t❤❡ k2(As) r❡❣✉❧❛r ❝♦❧✉♠♥

♣❛✐rs ♦❢ As✳

• ❆♣♣❧② ❛ r❛♥❞♦♠ ❜✐♥♦♠✐❛❧ ♦♣❡r❛t✐♦♥ t♦ t❤❡ s❡❧❡❝t❡❞ ♣❛✐r✱ ②✐❡❧❞✐♥❣ As+1✳

✕ ■❢ As+1 = As r❡♣❡❛t st❡♣ ✷✳ ✕ ❖t❤❡r✇✐s❡✿ As As+1 k2(As+1) ≤ k2(As) k2(As+1) > k2(As) As Ps②❝❤♦❝♦ ✷✵✶✷ ✶✹ P♦❧②t♦♠♦✉s ❞❛t❛

❚❤❡ P❛rt✐❛❧ ❈r❡❞✐t ▼♦❞❡❧ ✭▼❛st❡rs✱ ✶✾✽✷✮

❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❢♦r ❛ r❡s♣♦♥s❡ ♦❢ ♣❡rs♦♥ v ♦♥ ❝❛t❡❣♦r② h ✭h = 0,...,mi✮ ♦❢ ✐t❡♠ i✱ ✇❤❡r❡❛s mi ✐s t❤❡ ♥✉♠❜❡r ♦❢ r❡s♣♦♥s❡ ❝❛t❡❣♦r✐❡s✳ P(Xvih = 1) = exp(hθv + βih) ∑mi

l=0 exp(lθv + βil)

Xvih ... ♣❡rs♦♥ v s❝♦r❡s ✐♥ ❝❛t❡❣♦r② h ♦❢ ✐t❡♠ i θv ... ❧♦❝❛t✐♦♥ ♦❢ ♣❡rs♦♥ v ♦♥ ❧❛t❡♥t tr❛✐t h βih ... ✐t❡♠ ❝❛t❡❣♦r② ❝♦♠❜✐♥❛t✐♦♥ ✭❛❧❧♦✇s ❞✐✛❡r❡♥t ♥✉♠❜❡rs ♦❢ r❡s♣♦♥s❡ ❝❛t✲ ❡❣♦r✐❡s✮ Ps②❝❤♦❝♦ ✷✵✶✷ ✶✺

• ❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ▼❈▼❈ ❛♣♣r♦❛❝❤

❘❡q✉✐r❡♠❡♥ts

✶✳ ❈♦✈❡r❛❣❡ ♦❢ t❤❡ ✇❤♦❧❡ s❛♠♣❧❡ s♣❛❝❡ ✷✳ ■♥❞❡♣❡♥❞❡♥❝❡ ✸✳ ❯♥✐❢♦r♠ s❛♠♣❧✐♥❣ ✹✳ ❋r❡q✉❡♥❝② ❞✐str✐❜✉t✐♦♥ ❢♦r ❝❛t❡❣♦r✐❡s ♠✉st ❜❡ ♠❛✐♥t❛✐♥❡❞ ❢♦r ❡❛❝❤ ✐t❡♠✳ Ps②❝❤♦❝♦ ✷✵✶✷ ✶✻

• ❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ▼❈▼❈ ❛♣♣r♦❛❝❤

❇✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s ✐♥ ♦r❞✐♥❛❧ ❞❛t❛

❆♣♣❧② ❛ ❜✐♥♦♠✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ♦♥ ❛ s✐♥❣❧❡ ❝❛t❡❣♦r② t✉♣❡❧ (g,h) ♦❢ ❛ ❞❡✜♥❡❞ s❡t ♦❢ ✐♥t❡❣❡r t✉♣❡❧s C(i,j) ✭❂ ❛❧❧ ♣♦ss✐❜❧❡ ❝❛t❡❣♦r② ❝♦♠❜✐♥❛t✐♦♥s✮✳ ✶ ✷ ✸ ✹ 2 1 1 2 1 1 2 2 1 1 2 1 2 2 2 2 1 1 2 1

→

✶ ✷ 2 1 2 1 2 2 2 2 2 1

→

✶ ✷ 2 1 2 1 2 2 2 2 2 1

→

✶ ✷ ✸ ✹ 2 1 1 2 1 1 2 2 1 1 2 1 2 2 2 2 1 1 2 1 Ps②❝❤♦❝♦ ✷✵✶✷ ✶✼

• ❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ▼❈▼❈ ❛♣♣r♦❛❝❤

❇✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s ✐♥ ♦r❞✐♥❛❧ ❞❛t❛ ✭❝♦♥t✳✮

❚❤❡ ❣❡♥❡r❛❧✐③❡❞ Bij✲♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛ ♠❛tr✐① B ∈ Σrc ✐s ❞❡✜♥❡❞ ❛s AB(B) = ⋃

(i,j)

(AB(i,j)(B) × C(i,j)) B ✐s ❛ ♠❛tr✐① ✇✐t❤ ♠❛①✐♠✉♠ x ❝❛t❡❣♦r✐❡s ♦❢ ✐♥t❡❣❡rs ✐♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r✳ C(i,j) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❝❛t❡❣♦r② t✉♣❡❧s (g,h),g < h ♦❢ ❛ ❝♦❧✉♠♥ ♣❛✐r (i,j) ∈ B✳ ✭✶✱✷✮ ✭✶✱✸✮ ✭✶✱✹✮ ✭✷✱✸✮ ✭✷✱✹✮ ✭✸✱✹✮ ✭✐✱❥✮ ✭✵✱✶✮ ① ① ✭✵✱✷✮ ① ① ✭✶✱✷✮ ① ① ✭❣✱❤✮ Ps②❝❤♦❝♦ ✷✵✶✷ ✶✽

• ❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ▼❈▼❈ ❛♣♣r♦❛❝❤

❊①t❡♥s✐♦♥ t♦ ♠✉❧t✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s

❆♣♣❧② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r ❡❛❝❤ ❝❛t❡❣♦r② t✉♣❡❧ (g,h),g < h✱ s✐♠✉❧t❛♥❡♦✉s❧②✳ ✶ ✷ ✸ ✹ 2 1 1 2 1 1 2 2 1 1 2 1 2 2 2 2 1 1 2 1

→

✶ ✷ 2 1 2 1 2 2 2 2 2 1

→

✶ ✷ 2 2 1 1 2 2 2 2 2 1

→

✶ ✷ ✸ ✹ 2 1 2 1 1 1 2 2 1 1 2 1 2 2 2 2 1 1 2 1 Ps②❝❤♦❝♦ ✷✵✶✷ ✶✾

• ❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ▼❈▼❈ ❛♣♣r♦❛❝❤

❊①t❡♥s✐♦♥ t♦ ♠✉❧t✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞s ✭❝♦♥t✳✮

❚❤❡ Mij✲♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ♠❛tr✐① B ∈ Σrc ✐s ❞❡✜♥❡❞ ❛s BM(B) = P (AB

(i,j) (g,h)(B))

❙❡t ♦❢ s✐♠✉❧t❛♥❡♦✉s ❜✐♥♦♠✐❛❧ tr❛♥s❢♦r♠s ♦♥ ❛ s✐♥❣❧❡ ❝♦❧✉♠♥ ♣❛✐r (i,j) ❛♣♣❧✐❡❞ t♦ ❡❛❝❤ ♦❢ ✐ts ❝❛t❡❣♦r② t✉♣❡❧s (g,h)✳ ❚❤❡ ♠✉❧t✐♥♦♠✐❛❧ ♥❡✐❣❤❜♦r❤♦♦❞ ✐s t❤❡ ♣♦✇❡r s❡t ♦❢ t❤❡ ❝♦❧✉♠♥ ♣❛✐r s✉❜s❡t D ❛♥❞ t❤❡ ❝❛t❡❣♦r② t✉♣❡❧ s✉❜s❡t C(i,j)✳ AB

(i,j) (g,h)(B) = D × C(i,j)

✭✶✱✷✮ ✭✶✱✸✮ ✭✶✱✹✮ ✭✷✱✸✮ ✭✷✱✹✮ ✭✸✱✹✮ ✭✐✱❥✮ ✭✵✱✶✮ ① ① ✭✵✱✷✮ ① ① ✭✶✱✷✮ ① ① ✭❣✱❤✮ Ps②❝❤♦❝♦ ✷✵✶✷ ✷✵

• ❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ▼❈▼❈ ❛♣♣r♦❛❝❤

❖✉t❧♦♦❦

• ■♥✈❡st✐❣❛t❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✳

✕ ❋✐rst r✉❧❡ ❢♦r s♠❛❧❧ ❞❛t❛ s✐③❡s✳ ✕ ❙❡❝♦♥❞ r✉❧❡ ❢♦r ❜✐❣ ❞❛t❛ s✐③❡s✳

• ❈❤❡❝❦ r❡q✉✐r❡❞✳✳✳

✕ ❇✉r♥✲✐♥ ♣❡r✐♦❞ ✭st❛t✐♦♥❛r✐t②✮ ✕ ❘❡❥❡❝t✐♦♥ r❛t❡s ♦❢ t❤❡ ▼❍ ✕ ❙t❡♣ s✐③❡

• ❉❡✈❡❧♦♣ q✉❛s✐✲❡①❛❝t t❡sts ❢♦r t❤❡ ❢❛♠✐❧② ♦❢ ♣❛rt✐❛❧ ❝r❡❞✐t ♠♦❞❡❧s✳

Ps②❝❤♦❝♦ ✷✵✶✷ ✷✶ ▲✐t❡r❛t✉r❡ ❨✳ ❈❤❡♥✱ P✳ ❉✐❛❝♦♥✐s✱ ❙✳P✳ ❍♦❧♠❡s✱ ❛♥❞ ❏✳❙✳ ▲✐✉✳ ❙❡q✉❡♥t✐❛❧ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s ❢♦r ❙t❛t✐st✐❝❛❧ ❆♥❛❧②s✐s ♦❢ ❚❛❜❧❡s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❙t❛t✐st✐❝❛❧ ❆ss♦❝✐❛t✐♦♥✱ ✶✵✵✭✹✻✾✮✿✶✵✾✕✶✷✵✱ ✷✵✵✺✳ ❨✳ ❈❤❡♥✱ ■✳❍✳ ❉✐♥✇♦♦❞✐❡✱ ❛♥❞ ❙✳ ❙✉❧❧✐✈❛♥t✳ ❙❡q✉❡♥t✐❛❧ ■♠♣♦rt❛♥❝❡ ❙❛♠♣❧✐♥❣ ❢♦r ▼✉❧t✐✇❛② ❚❛❜❧❡s✳ ❚❤❡ ❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s✱ ✸✹✭✶✮✿✺✷✸✕✺✹✺✱ ✷✵✵✻✳ ❨✳ ❈❤❡♥ ❛♥❞ ❉✳ ❙♠❛❧❧✳ ❊①❛❝t ❚❡sts ❢♦r t❤❡ ❘❛s❝❤ ▼♦❞❡❧ ✈✐❛ ❙❡q✉❡♥t✐❛❧ ■♠♣♦rt❛♥❝❡ ❙❛♠♣❧✐♥❣✳ Ps②❝❤♦♠❡tr✐❦❛✱ ✼✵✭✶✮✿✶✶✕✸✵✱ ✷✵✵✺✳

• ✳◆✳ ▼❛st❡rs✳

❆ ❘❛s❝❤ ▼♦❞❡❧ ❢♦r P❛rt✐❛❧ ❈r❡❞✐t ❙❝♦r✐♥❣✳ Ps②❝❤♦♠❡tr✐❦❛✱ ✹✼✭✷✮✿✶✹✾✕✶✼✹✱ ✶✾✽✷✳ ■✳ P♦♥♦❝♥②✳ ◆♦♥♣❛r❛♠❡tr✐❝ ❣♦♦❞♥❡ss✲♦❢ ✜t t❡sts ❢♦r t❤❡ ❘❛s❝❤ ♠♦❞❡❧✳ Ps②✲ ❝❤♦♠❡tr✐❦❛✱ ✻✻✿✹✸✼✕✹✻✵✱ ✷✵✵✶✳ ❚✳ ❙♥✐❥❞❡rs✳ ❊♥✉♠❡r❛t✐♦♥ ❛♥❞ ❙✐♠✉❧❛t✐♦♥ ▼❡t❤♦❞s ❢♦r ✵✲✶ ▼❛tr✐❝❡s ✇✐t❤

• ✐✈❡♥ ▼❛r❣✐♥❛❧s✳ Ps②❝❤♦♠❡tr✐❦❛✱ ✺✻✭✸✮✿✸✾✼✕✹✶✼✱ ✶✾✾✶✳

◆✳❉✳ ❱❡r❤❡❧st✳ ❆♥ ❡✣❝✐❡♥t ▼❈▼❈ ❛❧❣♦r✐t❤♠ t♦ s❛♠♣❧❡ ❜✐♥❛r② ♠❛tr✐❝❡s ✇✐t❤ ✜①❡❞ ♠❛r❣✐♥❛❧s✳ Ps②❝❤♦♠❡tr✐❦❛✱ ✼✸✿✼✵✺✕✼✷✽✱ ✷✵✵✽✳ Ps②❝❤♦❝♦ ✷✵✶✷ ✷✷