t ss r srt - PDF document

▼ ❖ ❉ ❆ ✽ ❲♦r❦s❤♦♣ ♦♥ ▼♦❞❡❧✲❖r✐❡♥t❡❞ ❉❡s✐❣♥ ❛♥❞ ❆♥❛❧②s✐s ❖♣t✐♠❛❧ ❞❡s✐❣♥s ❢♦r ❞✐s❝r✐♠✐♥❛t✐♥❣ ❛♠♦♥❣ s❡✈❡r❛❧ ♥♦♥✲◆♦r♠❛❧ ♠♦❞❡❧s ❈❤✐❛r❛ ❚♦♠♠❛s✐ ❯♥✐✈❡rs✐t② ♦❢ ▼✐❧❛♥♦✱ ■t❛❧② ❏✉♥❡ ✹✕✽✱ ✷✵✵✼

❖❯❚▲■◆❊ • ◆♦t❛t✐♦♥ ❛♥❞ ❇❛❝❦❣r♦✉♥❞✳ • ❑▲✕♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥✿ ❞✐s❝r✐♠✐♥❛t✐♦♥ ❜❡✲ t✇❡❡♥ t✇♦ r✐✈❛❧ ♠♦❞❡❧s✳ • ●❡♥❡r❛❧✐③❡❞ ❑▲✕♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥✿ ❞✐s✲ ❝r✐♠✐♥❛t✐♦♥ ❛♠♦♥❣ ♠♦r❡ t❤❛♥ t✇♦ ♠♦❞✲ ❡❧s✳ • ❖t❤❡r ♣♦ss✐❜❧❡ ❝r✐t❡r✐❛✳ • ❆♥ ❡①❛♠♣❧❡✿ ❞✐❝r✐♠✐♥❛t✐♦♥ ❛♠♦♥❣ t❤r❡❡ ❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳ • ❙♦♠❡ r❡❢❡r❡♥❝❡s✳

◆❖❚❆❚■❖◆ • x ∈ X ✿ ❡①♣❡r✐♠❡♥t❛❧ ❝♦♥❞✐t✐♦♥ ❝❤♦s❡♥ ❜② t❤❡ ❡①♣❡r✐♠❡♥t❡r✳ • ❆♣♣r♦①✐♠❛t❡ ❞❡s✐❣♥ ξ ✿ ❛ ❞✐s❝r❡t❡ ♣r♦❜✲ ❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ X ✇❤✐❝❤ ✐s s✉♣♣♦rt❡❞ ♦♥ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts✿ k � � x 1 x k · · · � ξ = , 0 ≤ ω i ≤ 1 , ω i = 1 . ω 1 ω k · · · i =1 • y = y ( x ) ✿ ♦❜s❡r✈❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧ r❡✲ s♣♦♥s❡ ✳ • f i ( y, x, θ i ) ✿ i − t❤ st❛t✐st✐❝❛❧ ♠♦❞❡❧ ✱ ✇❤❡r❡ R m i ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✉♥✲ θ i ∈ Ω i ⊂ I ❦♥♦✇♥ ♣❛r❛♠❡t❡r ✈❡❝t♦r✳

●❖❆▲ ❆✐♠ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥t ✿ t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦❢ k r✐✈❛❧ st❛t✐st✐❝❛❧ ♠♦❞❡❧s✱ f i ( y, x, θ i ) , i = 1 , . . . , k, ✐s t❤❡ ♠♦r❡ ❛❞❡q✉❛t❡✳

❇❆❈❑●❘❖❯◆❉ • ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ♠♦❞❡❧s ❆t❦✐♥s♦♥ ✭✶✾✼✷✮✿ t❤❡ ♠♦❞❡❧ ✐s ❡♠❜❡❞❡❞ ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♦♥❡ ❛♥❞ t❤❡ ❞❡s✐❣♥ ❧♦♦❦s ❢♦r ❡st✐♠❛t✐♥❣ t❤❡ ❛❞❞✐t✐♦♥❛❧ ♣❛✲ r❛♠❡t❡rs ✐♥ t❤❡ ❜❡st ✇❛②✳ ❆t❦✐♥s♦♥ ❛♥❞ ❈♦① ✭✶✾✼✹✮✿ t❤❡ ♣r❡✈✐✉s ❛♣✲ ♣r♦❛❝❤ ✐s ❣❡♥❡r❛❧✐③❡❞ ❢♦r ❝♦♠♣❛r✐♥❣ s❡✈✲ ❡r❛❧ ❧✐♥❡❛r ♠♦❞❡❧s✳ • ❘❡❣r❡ss✐♦♥ ♠♦❞❡❧s ✭❧✐♥❡❛r ♦r ♥♦t✮ ❆t❦✐♥s♦♥ ❛♥❞ ❋❡❞♦r♦✈ ✭✶✾✼✺❛✱✶✾✼✺❜✮✿ ❚✲ ♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ❢♦r ❝♦♠♣❛r✐♥❣ t✇♦ ♦r ♠♦r❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳ ❯❝✐➠s❦✐ ❛♥❞ ❇♦❣❛❝❦❛ ✭✷✵✵✹✮✿ ●❡♥❡r❛❧✐③❛✲ t✐♦♥ ♦❢ t❤❡ ❚✲❝r✐t❡r✐♦♥ t♦ t❤❡ ❝❛s❡ ♦❢ ❤❡t✲ ❡r♦s❝❡❞❛st✐❝ r❛♥❞♦♠ ❡rr♦rs✳ • ●❡♥❡r❛❧✐③❡❞ ❧✐♥❡❛r ♠♦❞❡❧s ✭❣❧♠✮ P♦♥❝❡ ❞❡ ▲❡♦♥ ❛♥❞ ❆t❦✐♥s♦♥ ✭✶✾✾✷✮✿ ❚❤❡ ❚✲❝r✐t❡r✐♦♥ ✐s ❡①t❡♥❞❡❞ t♦ t❤❡ ❝❛s❡ ♦❢ ❣❧♠✳

❚❍❊ ❑▲✲❖P❚■▼❆▲■❚❨ ❈❘■❚❊❘■❖◆ ❑✉❧❧❜❛❝❦✕▲❡✐❜❧❡r ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ f 1 ( y, x, θ 1 ) ✭t❤❡ ❛ss✉♠❡❞ ✏tr✉❡✑ ♠♦❞❡❧✮ ❛♥❞ f 2 ( y, x, θ 2 ) ✿ � � f 1 ( y, x, θ 1 ) � I [ f 1 ( y, x, θ 1 ) , f 2 ( y, x, θ 2 )] = f 1 ( y, x, θ 1 ) log dy. f 2 ( y, x, θ 2 ) ❑▲✕♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥ ✭ ▲ó♣❡③✕❋✐❞❛❧❣♦✱ ❚♦♠♠❛s✐ ❛♥❞ ❚r❛♥❞❛✜r✱ ✷✵✵✼ ✮✿ � I 2 , 1 ( ξ ) = min X I [ f 1 ( y, x, θ 1 ) , f 2 ( y, x, θ 2 )] ξ ( dx ) θ 2 ∈ Ω 2

P❘❖P❊❘❚■❊❙ ✶✳ ❋♦r r❡❣r❡ss✐♦♥ ♠♦❞❡❧s ✇✐t❤ ◆♦r♠❛❧ ❡rr♦r ❞✐str✐❜✉t✐♦♥✱ t❤❡ ❑▲✲ ♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ❝♦✐♥❝✐❞❡s ✇✐t❤ • t❤❡ ❚✲♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥✱ ✐♥ t❤❡ ❤♦♠♦s❝❤❡❞❛st✐❝ ❝❛s❡❀ • t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❚✲♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ♣r♦✈✐❞❡❞ ❜② ❯❝✐➠s❦✐ ❛♥❞ ❇♦❣❛❝❦❛ ✭✷✵✵✹✮✱ ✐♥ t❤❡ ❤❡t❡r♦s❝❤❡❞❛st✐❝ ❝❛s❡✳ ✷✳ ❋♦r ❣❡♥❡r❛❧✐③❡❞ ❧✐♥❡❛r ♠♦❞❡❧s ✱ t❤❡ ❑▲✲❝r✐t❡r✐♦♥ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❚✲♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ♣r♦✲ ✈✐❞❡❞ ❜② P♦♥❝❡ ❞❡ ▲❡♦♥ ❛♥❞ ❆t❦✐♥s♦♥ ✭✶✾✾✷✮✳

❚❍❊ ●❊◆❊❘❆▲■❩❊❉ ❑▲✲❖P❚■▼❆▲■❚❨ ❈❘■❚❊❘■❖◆ • f k +1 ( y, x, θ k +1 ) ✿ ❡①t❡♥❞❡❞ ♠♦❞❡❧ ✇❤✐❝❤ ✐♥❝❧✉❞❡s t❤❡ k r✐✈❛❧ ♠♦❞❡❧s ❛s s♣❡❝✐❛❧ ❝❛s❡s✳ • I i,k +1 ( ξ ) ✿ ❞❡t❡❝ts ❞❡♣❛rt✉r❡s ❢r♦♠ f i ( y, x, θ i ) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♦t❤❡r ♠♦❞❡❧s✳ • ❊✣❝✐❡♥❝② ♦❢ ξ ❢♦r ❞❡t❡❝t✐♥❣ ❞❡♣❛rt✉r❡s ❢r♦♠ f i ( y, x, θ i ) ✿ Eff i,k +1 ( ξ )= I i,k +1 ( ξ ) ξ ∗ i =argmax I i,k +1 ( ξ ) I i,k +1 ( ξ ∗ i ) ξ • ●❡♥❡r❛❧✐③❡❞ ❑▲✲❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥ ✿ I α ( ξ )= � k i =1 α i · Eff i,k +1 ( ξ ) k ✇❤❡r❡ 0 ≤ α i ≤ 1 , α i = 1 ✳ � i =1

P❘❖P❊❘❚■❊❙ ✶✳ ❚❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥ I α ( ξ ) ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥✳ ✷✳ ■❢ ξ ✐s ❛ r❡❣✉❧❛r ❞❡s✐❣♥ t❤❡♥ k ∂I i,k +1 ( ξ, ¯ ξ ) ∂I α ( ξ, ¯ � ξ ) = α i I i,k +1 ( ξ ∗ i ) i =1 � X ψ α ( x, ξ ) ¯ = ξ ( dx ) , ✇❤❡r❡ ψ α ( x, ξ ) = ∂I α ( ξ, ξ x ) ❛♥❞ ξ x ✐s ❛ ❞❡s✐❣♥ ✇❤✐❝❤ ♣✉ts t❤❡ ✇❤♦❧❡ ♠❛ss ❛t ♣♦✐♥t x ✳ ✸✳ ❊q✉✐✈❛❧❡♥❝❡ ❚❤❡♦r❡♠ ξ ∗ ψ α ( x, ξ ∗ α = argmax I α ( ξ ) α ) ≤ 0 , x ∈ X . ⇔ ξ

❖❚❍❊❘ P❖❙❙■❇▲❊ ❈❘■❚❊❘■❆ • ❚❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥ ♦❢ ❡✣❝✐❡♥❝✐❡s✿ α i � � I GM ( ξ ) = � k Eff i,k +1 ( ξ ) α i =1 I GM log I GM ξ ∗ GM =arg max ( ξ )=arg max ( ξ ) . α α ξ ξ k log I GM � � � ( ξ ) = α i · log I i,k +1 ( ξ ) α i =1 k � � I i,k +1 ( ξ ∗ � α i · log i ) . − i =1 ❚❤✉s✱ ξ ∗ GM ♠❛①✐♠✐③❡s t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥ ♦❢ I i,k +1 ( ξ ) ✱ ✇✐t❤♦✉t t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡✐r ♣♦ss✐❜❧❡ ❞✐✛❡r❡♥t ♠❛❣♥✐t✉❞❡✳ ❋♦r t❤✐s r❡❛s♦♥ t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ k � I α ( ξ )= α i · Eff i,k +1 ( ξ ) i =1 ✐s ♣r❡❢❡rr❡❞ ❤❡r❡✳

• ❚❤❡ ♠✐♥✐♠✉♠ ❡✣❝✐❡♥❝② ❝r✐t❡r✐♦♥ ✿ I m ( ξ ) = i =1 ,...,k Eff i,k +1 ( ξ ) min ❚❤✐s ❝r✐t❡r✐♦♥ ♥❛t✉r❛❧❧② ❣✐✈❡s ❡q✉❛❧ ❡✣✲ ❝✐❡♥❝✐❡s t♦ t❤❡ ♠♦❞❡❧s t❤❛t ❛r❡ t❤❡ ♠♦st ❞✐✣❝✉❧t t♦ ❞✐s❝r✐♠✐♥❛t❡ t❤✉s✱ ✐t s❡❡♠s ✈❡r② ❛♣♣r♦♣r✐❛t❡ ❢♦r ❞✐s❝r✐♠✐♥❛t✐♥❣ ♣✉r♣♦s❡s✳ ❚❤✐s ❝r✐t❡r✐♦♥ ✇✐❧❧ ❜❡ st✉❞✐❡❞ ✐♥ ❞❡♣t❤ ✐♥ ❢✉t✉r❡✳

❊❳❆▼P▲❊ • ❙t❛t✐st✐❝❛❧ ♠♦❞❡❧s ✿ e η i P ( y = 1 , x, θ i ) = F ( η i ) = 1 + e η i , x ∈ [0 , 1] . • ❘✐✈❛❧ ♠♦❞❡❧s ❢♦r t❤❡ ❧♦❣✐t ♦❢ t❤❡ ❡①✲ ♣❡❝t❡❞ r❡s♣♦♥s❡✿ = η 1 β 1 x η 2 = β 0 + β 1 x β 1 x + β 2 x 2 . η 3 = • ❈♦♠❜✐♥❡❞ ♠♦❞❡❧ ❢♦r t❤❡ ❧♦❣✐t ♦❢ t❤❡ ❡①✲ ♣❡❝t❡❞ r❡s♣♦♥s❡✿ β 0 + β 1 x + β 2 x 2 . η 4 =