P❛✐r❡❞ ❈♦♠♣❛r✐s♦♥ ▼♦❞❡❧s ▼♦❞❡❧❧✐♥❣ P❛✐r❡❞ ❈♦♠♣❛r✐s♦♥s ✇✐t❤ ❚❤❡ ♣r❡❢♠♦❞ P❛❝❦❛❣❡ ❘❡❣✐♥❛ ❉✐ttr✐❝❤ ✫ ❘❡✐♥❤♦❧❞ ❍❛t③✐♥❣❡r ■♥st✐t✉t❡ ❢♦r ❙t❛t✐st✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s✱ ❲❯ ❱✐❡♥♥❛ ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
P❛✐r❡❞ ❈♦♠♣❛r✐s♦♥ ▼♦❞❡❧s P❛✐r❡❞ ❈♦♠♣❛r✐s♦♥s • ♠❡t❤♦❞ ♦❢ ❞❛t❛ ❝♦❧❧❡❝t✐♦♥ • ❣✐✈❡♥ ❛ s❡t ♦❢ J ✐t❡♠s • ✐♥❞✐✈✐❞✉❛❧s ❛r❡ ❛s❦❡❞ t♦ ❥✉❞❣❡ ♣❛✐rs ♦❢ ♦❜❥❡❝ts j ♣r❡❢❡rr❡❞ t♦ k k ♣r❡❢❡rr❡❞ t♦ j • ❛✐♠ ✐s t♦ r❛♥❦ ♦❜❥❡❝ts ✐♥t♦ ❛ ♣r❡❢❡r❡♥❝❡ ♦r❞❡r • ♦❜t❛✐♥ ❛♥ ♦✈❡r❛❧❧ r❛♥❦✐♥❣ ♦❢ t❤❡ ♦❜❥❡❝ts ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
P❛✐r❡❞ ❈♦♠♣❛r✐s♦♥ ▼♦❞❡❧s ❖✈❡r✈✐❡✇ • ▲▲❇❚ ♠♦❞❡❧s✿ ❧♦❣❧✐♥❡❛r ❇r❛❞❧②✲❚❡rr② ♠♦❞❡❧s • ❇❛s✐❝ ▲▲❇❚ • ❊①t❡♥❞❡❞ ▲▲❇❚ ✉♥❞❡❝✐❞❡❞ r❡s♣♦♥s❡ s✉❜❥❡❝t ❝♦✈❛r✐❛t❡s ♦❜❥❡❝t s♣❡❝✐✜❝ ❝♦✈❛r✐❛t❡s • P❛tt❡r♥ ▼♦❞❡❧s P❛✐r❡❞ ❝♦♠♣❛r✐s♦♥ ✕ ⊲ ♣❛tt❡r♥ ♠♦❞❡❧s ❘❛♥❦✐♥❣ ✕ ⊲ ♣❛tt❡r♥ ♠♦❞❡❧s ❘❛t✐♥❣ ✕ ⊲ ♣❛tt❡r♥ ♠♦❞❡❧s ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
▲▲❇❚ ❚❤❡ ❇❛s✐❝ ❇r❛❞❧❡②✲❚❡rr② ▼♦❞❡❧ ✭❇❚✮ ❢♦r t❤❡ ❡❛❝❤ ❝♦♠♣❛r✐s♦♥ ( jk ) ♦❢ ♦❜❥❡❝t j t♦ ♦❜❥❡❝t k ✇❡ ♦❜s❡r✈❡✿ • n ( j ≻ k ) ✳ ✳ ✳ t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s j ✐s ♣r❡❢❡rr❡❞ t♦ k • n ( k ≻ j ) ✳ ✳ ✳ t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s k ✐s ♣r❡❢❡rr❡❞ t♦ j t♦t❛❧ ♥✉♠❜❡r ♦❢ r❡s♣♦♥s❡s N ( jk ) = n ( j ≻ k ) + n ( k ≻ j ) t♦ ❝♦♠♣❛r✐s♦♥ ( jk ) t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t j ✐s ♣r❡❢❡rr❡❞ t♦ k ✐♥ ❝♦♠♣❛r✐s♦♥ ( jk ) π ✬s ❛r❡ ❛ ❝❛❧❧❡❞ ✇♦rt❤ ♣❛r❛♠❡t❡rs π j P ( j ≻ k ) = ❛♥❞ ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡ ♥✉♠❜❡rs π j + π k ❞❡s❝r✐❜✐♥❣ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ♦❜❥❡❝ts ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
▲▲❇❚ ❚❤❡ ❇❛s✐❝ ▲♦❣❧✐♥❡❛r ❇❚ ▼♦❞❡❧ ✭▲▲❇❚✮ t❤❡ ♠♦❞❡❧ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ❛ ❧♦❣✲❧✐♥❡❛r ♠♦❞❡❧ ❢♦❧❧♦✇✐♥❣ t❤❡ ✉s✉❛❧ ▼✉❧t✐♥♦♠✐❛❧ ✴ P♦✐ss♦♥ ✲ ❡q✉✐✈❛❧❡♥❝❡✳ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ m ( j ≻ k ) ♦❢ n ( j ≻ k ) ✐s m ( j ≻ k ) = N ( jk ) p ( j ≻ k ) √ π j π j ✇❤❡r❡ c ( jk ) ✐s ❝♦♥st❛♥t P ( j ≻ k ) = = c ( jk ) √ π k π j + π k ❢♦r ❛ ❣✐✈❡♥ ❝♦♠♣❛r✐s♦♥ t❤❡♥ ♦✉r ❜❛s✐❝ ♣❛✐r❡❞ ❝♦♠♣❛r✐s♦♥ ♠♦❞❡❧ ❢♦r ♦♥❡ ❝♦♠♣❛r✐s♦♥ ✐s λ ✬s ❛r❡ t❤❡ ♦❜❥❡❝t ♣❛r❛♠❡t❡rs ln m ( j ≻ k ) = µ ( jk ) + λ j − λ k µ ✬s ❛r❡ ♥✉✐s❛♥❝❡ ♣❛r❛♠❡t❡rs t❤✐s ♠♦❞❡❧ ❢♦r♠✉❧❛t✐♦♥ ✐s ❢❡❛s✐❜❧❡ ❢♦r ❢✉rt❤❡r ❡①t❡♥s✐♦♥s ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
▲▲❇❚ ⊲ ❊①t❡♥s✐♦♥s ⊲ ✉♥❞❡❝✐❞❡❞ ❘❡s♣♦♥s❡ ▲▲❇❚ ✇✐t❤ ❯♥❞❡❝✐❞❡❞ ❘❡s♣♦♥s❡ ❯s✐♥❣ t❤❡ r❡s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s s✉❣❣❡st❡❞ ❜② ❉❛✈✐❞s♦♥ ❛♥❞ ❇❡❛✈❡r ✭✶✾✼✼✮✿ t❤❡ ▲▲❇❚ ♠♦❞❡❧ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❝♦♠♣❛r✐s♦♥ ( jk ) ❛r❡ ♥♦✇✿ ln m ( j ≻ k ) = µ ( jk ) + λ j − λ k ln m ( k ≻ j ) = µ ( jk ) − λ j + λ k ln m ( j = k ) = µ ( jk ) + γ ✇❤❡r❡ γ ✐s t❤❡ ♣❛r❛♠❡t❡r ❢♦r ✉♥❞❡❝✐❞❡❞ r❡s♣♦♥s❡ ✭❝♦✉❧❞ ❛❧s♦ ❜❡ γ ( jk ) ✮ λ ✬s ❛r❡ t❤❡ ♦❜❥❡❝t ♣❛r❛♠❡t❡rs µ ✬s ❛r❡ ♥✉✐s❛♥❝❡ ♣❛r❛♠❡t❡rs ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
▲▲❇❚ t❡r♠s ❛♥❞ r❡❧❛t✐♦♥s • r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ π ❛♥❞ λ ✿ λ j = ln √ π j π j = exp 2 λ j • ✐❞❡♥t✐✜❛❜✐❧✐t② ♦❢ π s ✐s ♦❜t❛✐♥❡❞ ❜② t❤❡ r❡str✐❝t✐♦♥ π J = 1 ✈✐❛ λ J = 0 • t❤❡ ✇♦rt❤ ♣❛r❛♠❡t❡rs ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❜② exp(2 λ j ) π j = j exp(2 λ j ) , j = 1 , 2 , . . . , J � ✇❤❡r❡ � j π j = 1 ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
▲▲❇❚ ⊲ ❊①❛♠♣❧❡ ⊲ ❈❊▼❙ ❊①❛♠♣❧❡✿ ❈❊▼❙ ❡①❝❤❛♥❣❡ ♣r♦❣r❛♠♠❡ st✉❞❡♥ts ♦❢ t❤❡ ❲❯ ❝❛♥ st✉❞② ❛❜r♦❛❞ ✈✐s✐t✐♥❣ ♦♥❡ ♦❢ ❝✉rr❡♥t❧② ✶✼ ❈❊▼❙ ✉♥✐✈❡rs✐t✐❡s ❛✐♠ ♦❢ t❤❡ st✉❞②✿ • ♣r❡❢❡r❡♥❝❡ ♦r❞❡r✐♥❣s ♦❢ st✉❞❡♥ts ❢♦r ❞✐✛❡r❡♥t ❧♦❝❛t✐♦♥s • ✐❞❡♥t✐❢② r❡❛s♦♥s ❢♦r t❤❡s❡ ♣r❡❢❡r❡♥❝❡s ❞❛t❛✿ • P❈✲r❡s♣♦♥s❡s ❛❜♦✉t t❤❡✐r ❝❤♦✐❝❡s ♦❢ ✻ s❡❧❡❝t❡❞ ❈❊▼❙ ✉♥✐✈❡rs✐t✐❡s ❢♦r t❤❡ s❡♠❡st❡r ❛❜r♦❛❞ ✭▲♦♥❞♦♥✱ P❛r✐s✱ ▼✐❧❛♥✱ ❇❛r❝❡❧♦♥❛✱ ❙t✳●❛❧❧✱ ❙t♦❝❦❤♦❧♠✮ • ❛♥s✇❡r✿ ❝❛♥ ♥♦t ❞❡❝✐❞❡ ✇❛s ❛❧❧♦✇❡❞ • s❡✈❡r❛❧ ❝♦✈❛r✐❛t❡s ✭❡✳❣✳✱ ❣❡♥❞❡r✱ ✇♦r❦✐♥❣ st❛t✉s✱ ❧❛♥❣✉❛❣❡ ❛❜✐❧✐t✐❡s✱ ❡t❝✳✮ ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
♣r❡❢♠♦❞ ⊲ ▲▲❇❚ ❈❊▼❙ ❊①❛♠♣❧❡ ❞✐r❡❝t ❡st✐♠❛t✐♦♥✿ ❋✉♥❝t✐♦♥✿ ♣❛ttP❈✳❢✐t✭✮ ⊲ ✜t ❜❛s✐❝ ▲▲❇❚ ♠♦❞❡❧ ❢♦r P❈ ✇✐t❤ ✉♥❞❡❝✐❞❡❞ ❃ ♠✸ ❁✲ ❧❧❜tP❈✳❢✐t✭❝♣❝✱ ♥✐t❡♠s ❂ ✻✱ ✉♥❞❡❝ ❂ ❚❘❯❊✱ ♦❜❥✳♥❛♠❡s ❂ ❝✐t✐❡s✮ ❈❛❧❝✉❧❛t❡ ✇♦rt❤ ❛♥❞ ♣❧♦t ✉s✐♥❣ ❧❧❜t✳✇♦rt❤✭✮ ✱ ♣❧♦t✇♦rt❤✭✮ ❃ ✇♦rt❤✸ ❁✲ ❧❧❜t✳✇♦rt❤✭♠✸✮ ❃ ♣❧♦t✇♦rt❤✭✇♦rt❤✸✮ ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
▲▲❇❚ ⊲ ❊①t❡♥s✐♦♥ ⊲ ❙✉❜❥❡❝t ❈♦✈❛r✐❛t❡s ❙✉❜❥❡❝t ❈♦✈❛r✐❛t❡s ❆r❡ t❤❡ ♣r❡❢❡r❡♥❝❡ ♦r❞❡r✐♥❣s ❞✐✛❡r❡♥t ❢♦r ❞✐✛❡r❡♥t ❣r♦✉♣s ♦❢ s✉❜❥❡❝ts❄ ❋♦r ♦♥❡ s✉❜❥❡❝t ❝♦✈❛r✐❛t❡ ♦♥ s ❧❡✈❡❧s ✇❡ ❤❛✈❡ ♥♦✇ s + ( λ O j + λ O j S ) − ( λ O k + λ O k S ln m ( j ≻ k ) | s = µ ( jk ) s + λ S ) j js k ks ✇❤❡r❡ λ O ♦❜❥❡❝t ♣❛r❛♠❡t❡rs ✭❢♦r s✉❜❥❡❝t ❜❛s❡❧✐♥❡ ❣r♦✉♣✮ λ OS ✐♥t❡r❛❝t✐♦♥ ♣❛r❛♠❡t❡r ❜❡t✇❡❡♥ ♦❜❥❡❝ts ❛♥❞ s✉❜❥❡❝t ❝❛t❡❣♦r② λ S ✜①✐♥❣ t❤❡ ♠❛r❣✐♥ ❢♦r ❝❛t❡❣♦r② s ♦❢ ❝♦✈❛r✐❛t❡ S ✭♥✉✐s❛♥❝❡✮ s µ ✬s ♥✉✐s❛♥❝❡ ♣❛r❛♠❡t❡rs ▲▼❯✱ ✷♥❞ ❲♦r❦s❤♦♣ ♦♥ Ps②❝❤♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣ ✷✵✶✵✲✵✷✲✷✺
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