Evaluation of real time methods for epidemic forecasting 2

❈❤❛♣t❡r ✷ ▼❡t❤♦❞s ❚❤❡ tr❛❝❦✐♥❣ ♦❢ ❛♥ ✐♥✢✉❡♥③❛ ♣❛♥❞❡♠✐❝ ❝❛♥ ❜❡ ❞♦♥❡ ❜② s❡✈❡r❛❧ ♠❡t❤♦❞s✱ ❢♦r ✐♥st❛♥❝❡ s❡r♦❧♦❣✐❝❛❧ ❛♥❛❧②s❡s ♦❢ ❜❧♦♦❞ s❛♠♣❧❡s✱ ✇❤✐❝❤ ✐s ❛♥ ❛❝❝✉r❛t❡ ❜✉t ❝♦st❧② r❡tr♦s♣❡❝t✐✈❡ ♠❡t❤♦❞ t♦ t❡st ✇❤❡t❤❡r ✐♥❢❡❝t✐♦♥ ❤❛s ♦❝❝✉rr❡❞ ✭❈❤❡♥ ❛❧✳✱ ✷✵✶✵✮✳ ❡t ■♥ t❤✐s st✉❞②✱ ✇❡ ❝❤♦♦s❡ t♦ tr❛❝❦ t❤❡ ❞✐s❡❛s❡ ❜② ❝♦❧❧❡❝t✐♥❣ r❡♣♦rts ❜❛s❡❞ ♦♥ s②♥❞r♦♠❡s ❢r♦♠ ❛ ♥❡t✇♦r❦ ♦❢ ❣❡♥❡r❛❧ ♣❤②s✐❝✐❛♥s ♦r ❢❛♠✐❧② ❞♦❝t♦rs ✇❡ s❡t ✉♣ ✭❖♥❣ ❡t ❛❧✳✱ ✷✵✶✵✮✳ ❲❡ ♣r♦♣♦s❡ ❞✐s❡❛s❡ ♠♦❞❡❧✐♥❣ ❛s ❛ ♠❡t❤♦❞ t♦ ♣r❡❞✐❝t t❤❡ s✐③❡ ♦❢ t❤❡ ♣❛♥❞❡♠✐❝ ❛❝r♦ss t✐♠❡✳ ✷✳✶ ❉❛t❛ s♦✉r❝❡ ❚❤❡ ❲❍❖ ■♥✢✉❡♥③❛ ❙✉r✈❡✐❧❧❛♥❝❡ ◆❡t✇♦r❦✱ ❡st❛❜❧✐s❤❡❞ ✐♥ ✶✾✺✷✱ s❡r✈❡s ❛s ❛ ❣❧♦❜❛❧ ♠♦♥✐t♦r✐♥❣ s②st❡♠ ❢♦r t❤❡ ❡♠❡r❣❡♥❝❡ ❛♥❞ ❛❝t✐✈✐t✐❡s ♦❢ ✐♥✢✉❡♥③❛ ✈✐r✉s❡s ✇✐t❤ ♣❛♥❞❡♠✐❝ ♣♦t❡♥t✐❛❧✳ ❚❤❡ ♥❡t✇♦r❦ ❝♦♥s✐sts ♦❢ ✶✸✺ ◆❛t✐♦♥❛❧ ■♥✢✉❡♥③❛ ❈❡♥tr❡s ✇♦r❧❞✇✐❞❡✱ ✇❤✐❝❤ s❛♠♣❧❡ ♣❛t✐❡♥ts ✇✐t❤ ✐♥✢✉❡♥③❛✲❧✐❦❡ ✐❧❧♥❡ss ✭■▲■✮ ❛♥❞ s✉❜♠✐t r❡♣r❡s❡♥t❛t✐✈❡ ✐s♦❧❛t❡s t♦ ❲❍❖ ❈♦❧❧❛❜♦r❛t✐♥❣ ❈❡♥tr❡s ❢♦r ❛♥t✐✲ ✶✺ ✶✻ ❣❡♥✐❝ ❛♥❞ ❣❡♥❡t✐❝ ❛♥❛❧②s❡s ✭❲❍❖✱ ✷✵✵✾✮✳ ❚❤❡ ❲❍❖ ✇✐❧❧ t❤❡♥ ❝♦♠♠✉♥✐❝❛t❡ ♦r ♣r♦✈✐❞❡ ❛❞✈✐❝❡ t♦ ♠❡♠❜❡r st❛t❡s ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ t❤❡ ✜♥❞✐♥❣s✳ ■♥ ❙✐♥❣❛♣♦r❡✱ ❛♥ ❡①✐st✐♥❣ ✐♥✢✉❡♥③❛ s✉r✈❡✐❧❧❛♥❝❡ ♣r♦❣r❛♠♠❡ r❡❧✐❡s ♦♥ s❡♥t✐♥❡❧ ♣❤②s✐❝✐❛♥s ❝♦❧❧❡❝t✐♥❣ t❤r♦❛t ❛♥❞ ♥❛s❛❧ s✇❛❜s ❢r♦♠ ♣❛t✐❡♥ts ♣r❡s❡♥t✐♥❣ ■▲■ ❡t ✇✐t❤ ❢❡✈❡r ≥ 38◦ C ✱ ❛❧✳✱ ✷✵✶✶✮✳ ❚❤✐s ♣r♦❝❡ss ♦❢ s♣❡❝✐♠❡♥ ❝♦❧❧❡❝t✐♦♥ ❞♦❡s ♥♦t✱ ❤♦✇❡✈❡r✱ ♣r♦✈✐❞❡ ❛♥ ❝♦✉❣❤✱ s♦r❡ t❤r♦❛t✱ ❤❡❛❞❛❝❤❡ ❛♥❞ ♠✉s❝❧❡ ❛❝❤❡ ✭❚❡♦ ❡st✐♠❛t❡ ♦❢ t❤❡ ❛❝t✉❛❧ ♥✉♠❜❡r ♦❢ ✐♥❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧s ❛s t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ✉♥❦♥♦✇♥✳ ❖t❤❡r ♣r♦s♣❡❝t✐✈❡ s✉r✈❡✐❧❧❛♥❝❡ ❢♦r ✐♥✢✉❡♥③❛ ❤❛s ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t ✐♥ ❙✐♥❣❛✲ ♣♦r❡ ❜② t❤❡ ❉❡♣❛rt♠❡♥t ♦❢ P❛t❤♦❧♦❣② ✐♥ t❤❡ ▼✐♥✐str② ♦❢ ❍❡❛❧t❤✱ ❙✐♥❣❛♣♦r❡✱ s✐♥❝❡ t❤❡ ❞❡♣❛rt♠❡♥t ✇❛s ❞❡s✐❣♥❛t❡❞ ❛ ◆❛t✐♦♥❛❧ ■♥✢✉❡♥③❛ ❈❡♥tr❡ ❜② t❤❡ ❲❍❖ ✐♥ ❏✉❧② ✶✾✼✷ ✭❉♦r❛✐s✐♥❣❤❛♥ ❡t ❛❧✳✱ ✶✾✽✽✮✳ ❙✉r✈❡✐❧❧❛♥❝❡ ✐s ❞♦♥❡ ❜② t❛❦✲ ✐♥❣ r❛♥❞♦♠ t❤r♦❛t s✇❛❜s ✇❡❡❦❧② ❢r♦♠ ♣❛t✐❡♥ts ❡①❤✐❜✐t✐♥❣ s②♠♣t♦♠s ❢♦r ❛❝✉t❡ r❡s♣✐r❛t♦r② ✐♥❢❡❝t✐♦♥ ✭❆❘■✮ ❛t s❡❧❡❝t❡❞ ❣♦✈❡r♥♠❡♥t ♣♦❧②❝❧✐♥✐❝s ✭▼❖❍✱ ✷✵✵✶✮✳ ❆❘■s ❛r❡ ❞❡✜♥❡❞ ❛s ♣❛t✐❡♥ts ✇❤♦ ♣r❡s❡♥t r❡s♣✐r❛t♦r② s②♠♣t♦♠s ♦❢ ❝♦✉❣❤✱ r❤✐♥♦rr❤❡❛✱ ♥❛s❛❧ ❝♦♥❣❡st✐♦♥ ❛♥❞✴♦r s♦r❡ t❤r♦❛t✱ ✇❤✐❝❤ ♠❛② ❜❡ ❛❝❝♦♠♣❛♥✐❡❞ ❜② ❢❡✈❡r✳ ❍♦✇❡✈❡r✱ t❤✐s ❛♣♣r♦❛❝❤ ✐s ♥♦t s♣❡❝✐✜❝ t♦ ✐♥✢✉❡♥③❛ ❛❧♦♥❡ ❛s t❤❡ s②♠♣t♦♠s ❢♦r ❆❘■ ❝❛♥ ❜❡ ❝❛✉s❡❞ ❜② ♦t❤❡r ❞✐s❡❛s❡s ❛s ✇❡❧❧✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❝♦♥❝❡r♥s✱ ❛s ✇❡❧❧ ❛s ❞❛t❛ ❛❝❝❡ss✐❜✐❧✐t②✱ ✇❡ ❝♦❧❧❡❝t❡❞ t❤❡ ❞❛t❛ ✉s❡❞ ✐♥ t❤✐s st✉❞② ✐♥❞❡♣❡♥❞❡♥t❧②✳ ❲❡ ❝♦♠♠❡♥❝❡❞ ❜② s❡♥❞✐♥❣ r❡❝r✉✐t♠❡♥t ✐♥✈✐t❛t✐♦♥s t♦ ❡✲♠❛✐❧s ♦❢ ❣❡♥❡r❛❧ ♣r❛❝t✐❝❡✴❢❛♠✐❧② ❞♦❝✲ t♦r ❝❧✐♥✐❝s ❝♦♠♣✐❧❡❞ ❢r♦♠ t❤❡ ❈♦❧❧❡❣❡ ♦❢ ❋❛♠✐❧② P❤②s✐❝✐❛♥s ❙✐♥❣❛♣♦r❡ ❛♥❞ t❤❡ ❞✐r❡❝t♦r② ♦❢ P❛♥❞❡♠✐❝ Pr❡♣❛r❡❞♥❡ss ❈❧✐♥✐❝s r❡❣✐st❡r❡❞ ✉♥❞❡r ▼✐♥✐str② ♦❢ ✶✼ ❍❡❛❧t❤ t♦ ♠❛♥❛❣❡ ✐♥✢✉❡♥③❛ ❝❛s❡s✳ ❚❤❡ ✐♥❝❧✉s✐♦♥ ❝r✐t❡r✐❛ ✇❡r❡ t❤❛t ❞♦❝t♦rs ❤❛✈❡ t♦ ❜❡ r❡❣✐st❡r❡❞ ✇✐t❤ t❤❡ ❙✐♥❣❛♣♦r❡ ▼❡❞✐❝❛❧ ❈♦✉♥❝✐❧ ❛♥❞ ✇♦r❦ ❛t ❧❡❛st t❤r❡❡ ❢✉❧❧ ❞❛②s ❛ ✇❡❡❦ ✐♥ ❛ ❣❡♥❡r❛❧ ♣r❛❝t✐❝❡ ♦r ❢❛♠✐❧② ♠❡❞✐❝✐♥❡ ❝❧✐♥✐❝✳ ❉❛t❛ s✉❜♠✐ss✐♦♥ ✇❛s ✈♦❧✉♥t❛r② ❛♥❞ ♣❛rt✐❝✐♣❛t✐♥❣ ❞♦❝t♦rs ✇❡r❡ ❣✐✈❡♥ t❤❡ ♦♣t✐♦♥ t♦ ✇✐t❤❞r❛✇ ❢r♦♠ t❤❡ ♣r♦❥❡❝t ❛t ❛♥② t✐♠❡ ✭❖♥❣ ❡t ❛❧✳✱ ✷✵✶✵✮✳ ❚❤❡ st✉❞② ❞❡s✐❣♥ ✇❛s ❛♣♣r♦✈❡❞ ❜② t❤❡ ✐♥st✐t✉t✐♦♥❛❧ r❡✈✐❡✇ ❜♦❛r❞ ♦❢ t❤❡ ◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❙✐♥❣❛♣♦r❡✳ ❊♥r♦❧❧❡❞ ❞♦❝t♦rs ✇❡r❡ r❡q✉❡st❡❞ t♦ s✉❜♠✐t t❤❡ ♥✉♠❜❡r ♦❢ ❝❧✐♥✐❝❛❧❧② ❞✐❛❣✲ ♥♦s❡❞ ❆❘■s ❜② ❡✲♠❛✐❧ ♦r ❢❛❝s✐♠✐❧❡ ❜② ✷♣♠ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❛②✳ ■♥ ❛❞❞✐t✐♦♥✱ ❜❛s✐❝ ❞❡♠♦❣r❛♣❤✐❝s ❛♥❞ t❤❡ t❡♠♣❡r❛t✉r❡ ❛t ♣r❡s❡♥t❛t✐♦♥ ✇❛s r❡♣♦rt❡❞✳ ❚❤❡♥✱ ✇❡ ❡①tr❛❝t t❤❡ ♥✉♠❜❡r ♦❢ ■▲■ ❝❛s❡s✱ ❞❡✜♥❡❞ ❛s ❆❘■s ✇❤✐❝❤ ❛❧s♦ ❡①❤✐❜✐t ❢❡✈❡r ♦❢ ≥ 37.8◦ C ✱ ❢r♦♠ t❤❡ ❞❛t❛ s✉❜♠✐tt❡❞✳ ■▲■ ✐s ✉s❡❞ ✐♥ ♦✉r r❡♣♦rt ❜❡❝❛✉s❡ ✐t ✐s ♠♦r❡ s♣❡❝✐✜❝ ❢♦r ✐♥✢✉❡♥③❛ ✐♥❢❡❝t✐♦♥s✱ ❛♥❞ ❤❛s ❜❡❡♥ ✉s❡❞ ✇✐❞❡❧② ❛s ❛♥ ✐♥❞✐❝❛✲ t♦r ✐♥ s❡❛s♦♥❛❧ ✐♥✢✉❡♥③❛ s✉r✈❡✐❧❧❛♥❝❡ s②st❡♠s ✐♥ ♠❛♥② ❝♦✉♥tr✐❡s ✭❚❤♦♠♣s♦♥ ❡t ❛❧✳✱ ✷✵✵✻✮✳ ❚❤❡ ❛❞✈❛♥t❛❣❡s ♦❢ ■▲■ s✉r✈❡✐❧❧❛♥❝❡ ♦✈❡r t❤❡ tr❛❞✐t✐♦♥❛❧❧② ✐♠♣❧❡✲ ♠❡♥t❡❞ ♠❡t❤♦❞ ♦❢ ❧❛❜♦r❛t♦r② ❞✐❛❣♥♦s❡s t♦ ♠♦♥✐t♦r ✐♥✢✉❡♥③❛ ❛❝t✐✈✐t② ✐♥❝❧✉❞❡ ❧♦✇❡r ❝♦st✱ s❤♦rt❡r ♣r♦❝❡ss✐♥❣ t✐♠❡✱ ❛♥❞ ♠♦r❡ s❡♥s✐t✐✈✐t② t♦ ❝❤❛♥❣❡s ✐♥ ♥✉♠✲ ❜❡r ♦❢ ❝❛s❡s ❛t t❤❡ ♣❡❛❦ ♦❢ t❤❡ ♣❛♥❞❡♠✐❝ ✭▲❡❡ ❡t ❛❧✳✱ ✷✵✶✶✮✳ ❋✐❣✉r❡ ✷✳✶ s❤♦✇s t❤❡ ✐♥❢❡rr❡❞ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ♣❛t✐❡♥ts ✇✐t❤ ■▲■ ♣❡r ❞❛② r❡♣♦rt❡❞ ❜② t❤❡ ❞♦❝t♦rs ✭❜❧❛❝❦ ❧✐♥❡s✮ ❛♥❞ ✾✺✪ ❝r❡❞✐❜❧❡ ✐♥t❡r✈❛❧ ✭❣r❡②✮✱ ❢r♦♠ ❞❡❝❧❛r❛t✐♦♥s ✐♥ t❤❡ ❣❡♥❡r❛❧ ♣r❛❝t✐❝❡✴❢❛♠✐❧② ❞♦❝t♦r ♥❡t✇♦r❦ ✐♥ t❤✐s st✉❞②✳ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ■▲■s ♣❡r ●P ❛♥❞ t❤❡ ❝r❡❞✐❜❧❡ ✐♥t❡r✈❛❧ ✐s ❣❡♥❡r❛t❡❞ ❛s t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❢r♦♠ ❛ ▼❛r❦♦✈ ❝❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ ❛❧❣♦r✐t❤♠✱ ♦♥ ✶✽ ❋✐❣✉r❡ ✷✳✶✿ ■♥❢❡rr❡❞ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ♣❛t✐❡♥ts ✇✐t❤ ■▲■ ♣❡r ❞❛② ✭❜❧❛❝❦ ❧✐♥❡s✮ ❛♥❞ ✾✺✪ ❝r❡❞✐❜❧❡ ✐♥t❡r✈❛❧ ✭❣r❡② ❧✐♥❡s✮✱ ✇✐t❤ ✇❡❡❦❞❛②s ✐♥❞✐❝❛t❡❞ ✐♥ ❜❧❛❝❦ s♦❧✐❞ Estimated ILIs per GP per day ❝✐r❝❧❡s ❛♥❞ ✇❡❡❦❡♥❞s ♦r ♣✉❜❧✐❝ ❤♦❧✐❞❛②s ✐♥❞✐❝❛t❡❞ ✐♥ ❤♦❧❧♦✇ ❝✐r❝❧❡s✳ 7 ● 6 5 ● ● 4 3 2 1 0 ● ● ● ● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●●●● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ●● ●●● ● ●● ● ● ● ● ● ● ● 25 Jun 5 15 Jul 25 5 15 Aug 25 5 15 Sep t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ♦❜s❡r✈❡❞ ♥✉♠❜❡r ♦❢ ■▲■s ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❞✐str✐❜✉✲ t✐♦♥ ✇✐t❤ ✐ts ♠❡❛♥ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ●P ✇❤♦ r❡♣♦rt❡❞ ♦♥ t❤❛t ❞❛②✳ ❲❡❡❦❡♥❞s ❛♥❞ ♣✉❜❧✐❝ ❤♦❧✐❞❛②s ❛r❡ ✐♥❞✐❝❛t❡❞ ❜② ❤♦❧❧♦✇ ❝✐r❝❧❡s✱ ❛♥❞ ❛r❡ ♦❜s❡r✈❡❞ t♦ ❤❛✈❡ ❧♦✇❡r ❝♦♥s✉❧t❛t✐♦♥ r❛t❡s t❤❛♥ ✇❡❡❦❞❛②s ❢♦❧❧♦✇❡❞ ❜② ❛ ♣♦st✲ ✇❡❡❦❡♥❞ s✉r❣❡ ♦♥ ▼♦♥❞❛②s ❣❡♥❡r❛❧❧②✳ ❲❡ st❛rt❡❞ t❤❡ ❞❛t❛ ❝♦❧❧❡❝t✐♦♥ ♦♥ ✷✻t❤ ❏✉♥❡ s❤♦rt❧② ❛❢t❡r t❤❡ ✜rst ❝❛s❡ ♦❢ ❝♦♠♠✉♥✐t② ✐♥❢❡❝t✐♦♥ ✐♥ ❙✐♥❣❛♣♦r❡✳ ❚❤❡ ■▲■s ♣❡❛❦❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❆✉❣✉st ✇✐t❤ t❤❡ ❤✐❣❤❡st r❡❝♦r❞ ❛t ✻✳✺ ♣❡r ❞♦❝t♦r✱ ♦❝❝✉rr✐♥❣ ♦♥ ✸t❤ ❆✉❣✉st✱ ❛❢t❡r ✇❤✐❝❤ t❤❡② ❢❡❧❧ ❜❛❝❦ t♦ ♥❡❛r ❜❛s❡❧✐♥❡ ❧❡✈❡❧ ❜② t❤❡ ❡♥❞ ♦❢ ❙❡♣t❡♠❜❡r✳ ✶✾ ✷✳✷ ▼♦❞❡❧s ❢♦r ❞✐s❡❛s❡ ❞②♥❛♠✐❝s ▼❛t❤❡♠❛t✐❝❛❧ ❡♣✐❞❡♠✐♦❧♦❣② ❤❛s ❣r♦✇♥ ❡①♣♦♥❡♥t✐❛❧❧② st❛rt✐♥❣ ❢r♦♠ t❤❡ ♠✐❞❞❧❡ ♦❢ t❤❡ ✷✵t❤ ❝❡♥t✉r② ❞✉❡ ✐♥ ♣❛rt t♦ tr❡♠❡♥❞♦✉s ✐♠♣r♦✈❡♠❡♥t ✐♥ ❝♦♠♣✉t✐♥❣ ♣♦✇❡r✱ t❤✉s✱ ❛ ✈❛r✐❡t② ♦❢ ♠♦❞❡❧s ❤❛✈❡ ♥♦✇ ❜❡❡♥ ❢♦r♠✉❧❛t❡❞✱ ♠❛t❤❡♠❛t✐❝❛❧❧② ❛♥❛❧②③❡❞ ❛♥❞ ❛♣♣❧✐❡❞ t♦ ✐♥❢❡❝t✐♦✉s ❞✐s❡❛s❡s✳ ❲❡ ✐♥✈❡st✐❣❛t❡ t✇♦ ❝❧❛ss❡s ♦❢ ♠♦❞❡❧s ✈✐❛ ❛♣♣❧✐❝❛t✐♦♥ t♦ ♦✉r ❡①❛♠♣❧❡ ♦❢ t❤❡ ✐♥✢✉❡♥③❛ ❆✲❍✶◆✶✭✷✵✵✾✮ ♣❛♥❞❡♠✐❝✱ ♥❛♠❡❧② ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s ❛♥❞ st♦❝❤❛st✐❝ ♠♦❞❡❧s✳ ❉❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s ❛r❡ t❤♦s❡ ✇❤✐❝❤ ✉s❡ ❞✐✛❡r❡♥❝❡✱ ❢✉♥❝✲ t✐♦♥❛❧ ♦r ❢✉♥❝t✐♦♥❛❧ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s t♦ ❞❡s❝r✐❜❡ t❤❡ ❝❤❛♥❣❡s ♦❢ ❡❛❝❤ ❡♣✐✲ ❞❡♠✐♦❧♦❣✐❝❛❧ ❝❧❛ss ✐♥ t✐♠❡ ✭❍❡t❤❝♦t❡✱ ✷✵✵✾✮✳ ■♥ ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s✱ ❡✈❡r② s❡t ♦❢ ✈❛r✐❛❜❧❡ st❛t❡s ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ♠♦❞❡❧ ❛♥❞ ❜② s❡ts ♦❢ ♣r❡❝❡❞❡♥t st❛t❡s ♦❢ t❤❡s❡ ✈❛r✐❛❜❧❡s ✭❚❤r✉s❤✱ ✷✵✶✶✮✳ ❉❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s ❛r❡ ✉s❡❞ t♦ ❞❡s❝r✐❜❡ t❤❡ ♦✉t❝♦♠❡s ♦❢ ❞✐s❡❛s❡s ❛t ❛ ♣♦♣✉❧❛t✐♦♥ ❧❡✈❡❧✱ ❛♥❞ t❤❡② ❛❧✇❛②s ❜❡❤❛✈❡ ✐❞❡♥t✐❝❛❧❧② ❢♦r ❛ ♣❛rt✐❝✉❧❛r s❡t ♦❢ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❡①❝❧✉❞✐♥❣ ♥✉♠❡r✐❝❛❧ ♦✈❡r✢♦✇ ✐ss✉❡s ✭▲♦r❡♥③✱ ✶✾✻✸✮✳ ❉❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s ❛r❡ r❡❧❛t✐✈❡❧② ❡❛s② t♦ s❡t ✉♣ ❞✉❡ t♦ ❛✈❛✐❧❛❜✐❧✐t② ♦❢ s♦❢t✇❛r❡✱ ❜✉t ❞♦ ♥♦t r❡✢❡❝t t❤❡ r♦❧❡ ♦❢ ❝❤❛♥❝❡ ✐♥ t❤❡ s♣r❡❛❞ ♦❢ t❤❡ ❞✐s❡❛s❡✳ ❈♦♥✈❡rs❡❧②✱ ❛ st♦❝❤❛st✐❝ ♠♦❞❡❧ ❞♦❡s ♥♦t r❡t✉r♥ ❛ s❡t ♦❢ ✉♥✐q✉❡ s♦❧✉t✐♦♥s ❜✉t ❡♥t❛✐❧s st♦❝❤❛st✐❝✐t② s✉❝❤ t❤❛t t❤❡ ♠♦❞❡❧ ✇✐❧❧ r❡t✉r♥ ❛ r❛♥❣❡ ♦❢ ❞✐✛❡r❡♥t ♦✉t❝♦♠❡s✳ ❚❤❡ r❛t❡s ♦❢ ♠♦✈✐♥❣ ❢r♦♠ ♦♥❡ ❡♣✐❞❡♠✐♦❧♦❣✐❝❛❧ ❝❧❛ss t♦ ❛♥♦t❤❡r ❣✐✈❡ r✐s❡ t♦ ❞②♥❛♠✐❝ r❛♥❞♦♠♥❡ss ❛♥❞ ✈❛r✐❛t✐♦♥ ✐♥ t❤❡ ♠♦❞❡❧✳ ❙t♦❝❤❛st✐❝ ♠♦❞❡❧s ❛r❡ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ t♦ s❡t ✉♣ ❛♥❞ ♠❛② r❡q✉✐r❡ ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣♦✇❡r ✷✵ t♦ ❡①❡❝✉t❡✱ ❛s t❤❡② ♦❢t❡♥ r❡q✉✐r❡ ▼♦♥t❡ ❈❛r❧♦ s❛♠♣❧✐♥❣ t♦ ❞❡r✐✈❡ ❛♥ ♦✉t❝♦♠❡ ❞✐str✐❜✉t✐♦♥ ✭❇❡✐ss✐♥❣❡r ❛♥❞ ❲❡st♣❤❛❧✱ ✶✾✾✽✮✳ ■♥ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧✐♥❣ ♦❢ ✐♥❢❡❝t✐♦✉s ❞✐s❡❛s❡s tr❛♥s♠✐ss✐♦♥✱ t❤❡r❡ ✐s ❛❧✇❛②s ❛ tr❛❞❡✲♦✛ ❜❡t✇❡❡♥ s✐♠♣❧❡ ♠♦❞❡❧s✱ ✇❤✐❝❤ ♦♠✐t ♠♦st ❞❡t❛✐❧s ❛♥❞ ❛r❡ ❞❡s✐❣♥❡❞ ♦♥❧② t♦ ❤✐❣❤❧✐❣❤t ❣❡♥❡r❛❧ q✉❛❧✐t❛t✐✈❡ ♦r q✉❛♥t✐t❛t✐✈❡ ❜❡❤❛✈✐♦✉r✱ ❛♥❞ ❞❡t❛✐❧❡❞ ♠♦❞❡❧s✱ ❞❡s✐❣♥❡❞ ❢♦r ♠♦r❡ s♣❡❝✐✜❝ s✐t✉❛t✐♦♥s ✇❤✐❝❤ ✐♥❝❧✉❞❡ s❤♦rt✲ t❡r♠ q✉❛♥t✐t❛t✐✈❡ ♣r❡❞✐❝t✐♦♥s✱ ❜✉t ❛r❡ ❣❡♥❡r❛❧❧② ❞✐✣❝✉❧t ♦r ✐♠♣♦ss✐❜❧❡ t♦ s♦❧✈❡ ❛♥❛❧②t✐❝❛❧❧② ✭❇r❛✉❡r ❡t ❛❧✳✱ ✷✵✵✽✮✳ ❲❡ ❤❛✈❡ ❢♦❧❧♦✇❡❞ t❤❡ ❖❝❝❛♠✕❇♦① ♣❤✐❧♦s♦♣❤② ♦❢ s❡❧❡❝t✐♥❣ ♠♦❞❡❧ ❝♦♠♣❧❡①✐t② t❤❛t ✐s ❛s ❝♦♠♣❧❡① ❛s ♥❡❡❞ t♦ ❜❡ ✉s❡❢✉❧ ❢♦r ♦✉r ♣✉r♣♦s❡s ✕ ✐✳❡✳ ❢♦r❡❝❛st✐♥❣ ❛❣❣r❡❣❛t❡ ❞✐s❝r❡t❡ ❧❡✈❡❧s ✕ ❜✉t ♥♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞✱ ❛♥❞ ❛s ❛ r❡s✉❧t✱ ✇❡ ❤❛✈❡ ✉s❡❞ q✉✐t❡ s✐♠♣❧❡ ♠♦❞❡❧s ✐♥ t❤✐s r❡♣♦rt✳ ■♥ ♣r✐♥❝✐♣❧❡ ♠♦r❡ ❞❡t❛✐❧❡❞ str✉❝t✉r❡ ❝❛♥ ❜❡ ✐♥❝♦r♣♦r❛t❡❞ ✉s✐♥❣ s✐♠✐❧❛r ❛♣♣r♦❛❝❤❡s ❢♦r r❡s❡❛r❝❤ q✉❡st✐♦♥s t❤❛t ❞❡♠❛♥❞ ✐t✳ ❚❤❡s❡ ❝❛♥ ❜❡ ❡①t❡♥s✐♦♥s t♦ ❝♦♠♣❛rt♠❡♥t❛❧ ♠♦❞❡❧s t♦ ✐♥❝❧✉❞❡ ♠♦r❡ ❝❧❛ss❡s ♦r ✐♥❝❧✉s✐♦♥ ♦❢ ❞❡♠♦❣r❛♣❤✐❝ ❢❛❝t♦rs ✐♥ t❤❡ ♠♦❞❡❧s✳ ❚❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s ✉s❡❞ ✐♥ t❤✐s st✉❞② ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ t❤❡② ❛r❡ t❤❡ ❘✐❝❤❛r❞s ♠♦❞❡❧✱ ❛♥❞ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ❝♦♠♣❛rt✲ ♠❡♥t❛❧ ♠♦❞❡❧✳ ❆❧s♦✱ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ✈❡rs✐♦♥ ♦❢ ❙■❘ ♠♦❞❡❧ ✐s ♠♦❞✐✜❡❞ t♦ ❜❡ st♦❝❤❛st✐❝ t♦ ❛❧❧♦✇ ❢♦r t❤❡ ❡✛❡❝ts ♦❢ st♦❝❤❛st✐❝✐t② ♦❢ t❤❡ ✐♥✢✉❡♥③❛ ♣❛♥❞❡♠✐❝ ❡s♣❡❝✐❛❧❧② ❞✉r✐♥❣ t❤❡ ❡❛r❧② ♣❤❛s❡s ✐♥ ✇❤✐❝❤ t❤❡r❡ ❛r❡ ❢❡✇ ❝❛s❡s✳ ❚❤❡r❡❢♦r❡ ✇❡ ✇✐❧❧ ❛❧s♦ ❞❡s❝r✐❜❡ t❤❡ st♦❝❤❛st✐❝ ❙■❘ ❝♦♠♣❛rt♠❡♥t❛❧ ♠♦❞❡❧✳ ✷✶ ✷✳✷✳✶ ❘✐❝❤❛r❞s ♠♦❞❡❧ ❆♥ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❧♦❣✐st✐❝ ❣r♦✇t❤ ♠♦❞❡❧ ✇❛s ♣r♦♣♦s❡❞ ❜② ❘✐❝❤❛r❞s ✐♥ ✶✾✺✾ t♦ st✉❞② t❤❡ ❣r♦✇t❤ ♣r♦❝❡ss ♦❢ ❜✐♦❧♦❣✐❝❛❧ ♣♦♣✉❧❛t✐♦♥s❀ ❛s ❛♣♣❧✐❡❞ t♦ ♦✉r ✐♥✲ ✢✉❡♥③❛ ❆✲❍✶◆✶✭✷✵✵✾✮ ❞❛t❛✱ ✇❡ t❡r♠ t❤✐s t❤❡ ❘✐❝❤❛r❞s ♠♦❞❡❧✳ ❚❤❡ ♠♦❞❡❧ ✐s ❞❡✈❡❧♦♣❡❞ ❢r♦♠ ❛ ❧♦❣✐st✐❝ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ r❡❧❛t✐✈❡ ❣r♦✇t❤✲r❛t❡ ❞❡❝r❡❛s✲ ✐♥❣ ❧✐♥❡❛r❧② ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ♣♦♣✉❧❛t✐♦♥ s✐③❡ ✭❘✐❝❤❛r❞s✱ ✶✾✺✾✮✳ ❆❧t❤♦✉❣❤ t❤❡ ❘✐❝❤❛r❞s ♠♦❞❡❧ ✐s ♥♦t ❛s ✢❡①✐❜❧❡ ❛s ❛ ❣❡♥❡r❛❧✐③❡❞ ❧♦❣✐st✐❝ ❢✉♥❝t✐♦♥ ✐♥ ♠♦❞❡❧✲ ✐♥❣ ❣r♦✇t❤✱ ✐t ❝♦♥t❛✐♥s ❢❡✇❡r ♣❛r❛♠❡t❡rs ❛♥❞ ✐s t❤❡r❡❢♦r❡ ♠♦r❡ ♣❛rs✐♠♦♥✐♦✉s✳ ❚❤❡ ♠♦❞❡❧ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✱ C (t) = rC(t) 1 − ✇❤❡r❡ C(t) K a , ✭✷✳✶✮ C(t) ✐s t❤❡ ❝✉♠✉❧❛t✐✈❡ ♥✉♠❜❡r ♦❢ ✐♥❢❡❝t❡❞ s✉❜❥❡❝ts ❛t t✐♠❡ t✱ r ✐s t❤❡ ♣❡r ❝❛♣✐t❛ ✐♥tr✐♥s✐❝ ❣r♦✇t❤ r❛t❡ ♦❢ t❤❡ ✐♥❢❡❝t❡❞ ♣♦♣✉❧❛t✐♦♥✱ ♦r ♠❛①✐♠✉♠ ❝❛s❡ ❧♦❛❞ ♦❢ t❤❡ ♦✉t❜r❡❛❦✱ ❛♥❞ ❢r♦♠ t❤❡ st❛♥❞❛r❞ ❧♦❣✐st✐❝ ❝✉r✈❡ ✭❍s✐❡❤ a K ✐s t❤❡ ❧✐♠✐t✐♥❣ ✈❛❧✉❡ ✐s t❤❡ ❡①♣♦♥❡♥t ♦❢ ❞❡✈✐❛t✐♦♥ ❡t ❛❧✳✱ ✷✵✵✽✮✳ ❚❤❡ ♠♦❞❡❧ ✐♥ ❡q✉❛t✐♦♥ ✭✷✳✶✮ ❤❛s ❛♥ ❛♥❛❧②t✐❝ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❢♦r♠ ♦❢ s✐❣✲ ♠♦✐❞ ❢✉♥❝t✐♦♥ ✭❍s✐❡❤ ❡t ❛❧✳✱ ✷✵✵✽✮ ❛s r → ∞✱ C(t) = K/ 1 + e−r(t−tm ) 1/a . ❚❤❡ ✈❡r✐✜❝❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ ✐♥ ✭✷✳✷✮ ✐s ❛s ❢♦❧❧♦✇s✿ ✭✷✳✷✮ ✷✷ C (t) = d dt K · 1 + e−r(t−tm ) −1/a−1 = − Ka 1 + e−r(t−tm ) = −1/a , · e−r(t−tm ) · (−r) , −r(t−tm ) −1 r C(t) 1+e , a 1+e−r(t−tm ) = rC(t) a1 1 − 1 1+e−r(t−tm ) = rC(t) 1 − C a K , 1 . a ■♥ t❤❡ ❘✐❝❤❛r❞s ♠♦❞❡❧✱ t❤❡ ❝✉♠✉❧❛t✐✈❡ ♥✉♠❜❡r ♦❢ ✐♥❢❡❝t❡❞ s✉❜❥❡❝ts ❝❛♥ ❜❡ ❝♦♥✈❡rt❡❞ ✐♥t♦ ❢♦r♠✉❧❛ I(t)✱ C(t) t❤❡ ❞❛✐❧② ♥✉♠❜❡r ♦❢ ✐♥❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧s✱ ❜② t❤❡ I(t) = C(t) − C(t − 1)✱ t❤❡r❡❢♦r❡ r❡♥❞❡r✐♥❣ ✐t ❝♦♠♣❛r❛❜❧❡ t♦ t❤❡ ✐♥❢❡❝t✐♦✉s ❝♦♠♣❛rt♠❡♥t ♦❢ ❙■❘✲t②♣❡ ♠♦❞❡❧✳ ❚❤❡ ❘✐❝❤❛r❞s ♠♦❞❡❧ ❢♦r♠✉❧❛t❡❞ ❛❜♦✈❡ ✐s ❛♣♣❧✐❝❛❜❧❡ t♦ ❛♥ ❡♣✐❞❡♠✐❝ ❝✉r✈❡ ✇✐t❤ ❛ s✐♥❣❧❡ ♣❡❛❦ ♦❢ ❤✐❣❤ ✐♥❝✐❞❡♥❝❡✱ ❛♥❞ ❛ s✐♥❣❧❡ t✉r♥✐♥❣ ♣♦✐♥t ❛t ✇❤✐❝❤ t❤❡ r❛t❡ ♦❢ ❛❝❝✉♠✉❧❛t✐♦♥ ♦❢ ■♥❢❡❝t✐♦✉s ✐♥❞✐✈✐❞✉❛❧s ❝❤❛♥❣❡s ❢r♦♠ ✐♥❝r❡❛s✐♥❣ t♦ ❞❡❝r❡❛s✐♥❣ ✭❍s✐❡❤ ❡t ❛❧✳✱ ✷✵✵✽✮✳ ❚❤✐s t✉r♥✐♥❣ ♣♦✐♥t ✐s ❝r✉❝✐❛❧ ✐♥ t❤❡ ❡♣✐❞❡♠✐♦❧♦❣✐❝❛❧ ❝♦♥t❡①t ♦❢ ❛ ♣❛♥❞❡♠✐❝ ❜❡❝❛✉s❡ ✐t ✐♥❞✐❝❛t❡s t❤❡ ❡♥❞ ♦❢ t❤❡ ❝✉rr❡♥t ✇❛✈❡ ♦❢ ✐♥❢❡❝t✐♦♥✱ ✇❤✐❝❤ s✐❣♥✐✜❡s ❞❡❝r❡❛s✐♥❣ ✐♥❢❡❝t✐♦♥ r❛t❡ ❛♥❞ t❤❛t ❝♦♥tr♦❧ ♠❡❛s✉r❡s✱ ✐❢ ✐♠♣❧❡♠❡♥t❡❞✱ ♠✐❣❤t ❤❛✈❡ st❛rt❡❞ t♦ t❛❦❡ ❡✛❡❝t ♦r t❤❛t ❤❡r❞ ✐♠♠✉♥✐t② ♠✐❣❤t ❤❛✈❡ ❜❡❡♥ ❡st❛❜❧✐s❤❡❞ ✐♥ t❤❡ ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡ s♦❧✉t✐♦♥ ✐♥ ✭✷✳✷✮ ✐s ❛ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ ❧♦❣✐st✐❝ ❢✉♥❝t✐♦♥✱ ❛♥❞ ✐s ❛ss♦❝✐✲ ❛t❡❞ ✇✐t❤ ❛ s✐❣♠♦✐❞ ❣r❛♣❤ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✳✷✳ ❋✐❣✉r❡ ✷✳✷ s❤♦✇s t❤❡ ❞✐✛❡r❡♥t ❡✛❡❝ts ♦❢ t❤❡ ❢♦✉r ♣❛r❛♠❡t❡rs ❢r♦♠ t❤❡ ✜rst ♣❛♥❡❧ t❤❛t K K, r, tm , ❛♥❞ a ♦♥ ❞❡t❡r♠✐♥❡s t❤❡ s✐③❡ ♦❢ C(t) ❛♥❞ C(t)✱ I(t)❀ ✇❡ ❝❛♥ s❡❡ t❤❡ ❞❛✐❧② ♥✉♠❜❡r ♦❢ ✐♥❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧s✱ t❤❡ s❡❝♦♥❞ ♣❛♥❡❧ ♦♥ t❤❡ ❧❡❢t s❤♦✇s t❤❛t ❛s tm ✐♥❝r❡❛s❡✱ ✷✸ t❤❡ t✐♠❡ ✐t t❛❦❡s ❢♦r C(t) t♦ r❡❛❝❤ ✐ts ♣❡❛❦ ❛❧s♦ ✐♥❝r❡❛s❡s✳ ❋r♦♠ t❤❡ ✜rst t✇♦ ♣❛♥❡❧s ♦♥ t❤❡ r✐❣❤t✱ ✇❡ s❡❡ t❤❛t r ❛♥❞ a ❝❤❛♥❣❡ t❤❡ s❧♦♣❡ ♦❢ t❤❡ C(t) ❝✉r✈❡✳ ✷✳✷✳✷ ❉❡t❡r♠✐♥✐st✐❝ ❙■❘ ❝♦♠♣❛rt♠❡♥t❛❧ ♠♦❞❡❧ ❆♣❛rt ❢r♦♠ ❘✐❝❤❛r❞s ♠♦❞❡❧✱ ✇❡ ♣r♦♣♦s❡ ❛♥♦t❤❡r ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧ ✐♥ t❤✐s st✉❞②✱ ❦♥♦✇♥ ❛s t❤❡ ❙■❘ ❝♦♠♣❛rt♠❡♥t❛❧ ♠♦❞❡❧✳ ❚❤❡ ❜❛s✐❝ ❝♦♠♣❛rt♠❡♥t❛❧ ♠♦❞❡❧s ✭❑❡r♠❛❝❦ ❛♥❞ ▼❝❑❡♥❞r✐❝❦✱ ✶✾✷✼✮ ❞✐✲ ✈✐❞❡ ❛t ❛♥② ♣♦✐♥t ♦❢ t✐♠❡ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✉♥❞❡r st✉❞② ✐♥t♦ ❝♦♠♣❛rt♠❡♥ts ❛❧❧♦✇✐♥❣ ❢♦r ❛ss✉♠♣t✐♦♥s ❛❜♦✉t t❤❡ ♥❛t✉r❡ ♦❢ tr❛♥s❢❡r ❢r♦♠ ♦♥❡ ❝♦♠♣❛rt♠❡♥t t♦ ❛♥♦t❤❡r✳ ❈♦♠♣❛rt♠❡♥ts ❛r❡ ♦❢t❡♥ ❧❛❜❡❧❧❡❞ ✇✐t❤ ❧❡tt❡rs ▼✱ ❙✱ ❊✱ ■ ❛♥❞ ❘✱ t♦ ✐♥❞✐❝❛t❡ t❤❡ ❡♣✐❞❡♠✐♦❧♦❣✐❝❛❧ ❝❧❛ss❡s ♦❢ ♣❛ss✐✈❡❧② ✐♠♠✉♥❡✱ s✉s❝❡♣t✐❜❧❡✱ ❡①♣♦s❡❞ ✐♥ ❧❛t❡♥t ♣❡r✐♦❞✱ ✐♥❢❡❝t❡❞✱ r❡♠♦✈❡❞ r❡s♣❡❝t✐✈❡❧②✳ ❊①❛♠♣❧❡s ♦❢ ❡①✐st✲ ✐♥❣ ❝♦♠♣❛rt♠❡♥t❛❧ ♠♦❞❡❧s ✐♥❝❧✉❞❡ t❤❡ ❙■❘✱ ❙■❙✱ ❙■❘❙✱ ❙❊■❙✱ ❙❊■❘✱ ▼❙■❘✱ ▼❙❊■❘ ❛♥❞ ▼❙❊■❘❙ ♠♦❞❡❧s✳ ❚❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ ❝♦♠♣❛rt♠❡♥ts t♦ ❜❡ ✐♥✲ ❝❧✉❞❡❞ ✐♥ ❛ ♠♦❞❡❧ ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ s♣❡❝✐✜❝ ❞✐s❡❛s❡✱ s✉❝❤ ❛s tr❛♥s♠✐ss✐♦♥ ❛♥❞ ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ ❝❧❛ss❡s✳ ❲❡ ✉s❡ ❛ ❜❛s✐❝ ❙■❘ ♠♦❞❡❧ ❢♦r t❤❡ ✐♥✢✉❡♥③❛ ❆✲❍✶◆✶✭✷✵✵✾✮ ✇✐t❤ t❤❡ ♣♦♣✉✲ ❧❛t✐♦♥ ❜❡✐♥❣ ❞✐✈✐❞❡❞ ✐♥t♦ t❤r❡❡ ❝❧❛ss❡s✿ ✐♥❞✐✈✐❞✉❛❧s st❛rt ♦✛ ✐♥ t❤❡ s✉s❝❡♣t✐❜❧❡ ❝❧❛ss ❙✱ ❛♥❞ ♠♦✈❡ s✉❜s❡q✉❡♥t❧② t♦ t❤❡ ✐♥❢❡❝t❡❞ ❝❧❛ss ■ ❛♥❞ t❡r♠✐♥❛t❡ ✐♥ t❤❡ r❡♠♦✈❡❞ ❝❧❛ss ❘✳ ❚❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✉s✐♥❣ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ✷✹ ❋✐❣✉r❡ ✷✳✷✿ ●r❛♣❤s ♦❢ K, r, tm, a = 1 ❛♥❞ I(t) 1.5 1.0 0.0 0 t 5 10 0.0 0.2 0.4 0.6 0.8 1.0 −5 15 0 5 10 15 0 5 10 15 t Effects of a on C(t) a=0.5 a=1 a=2 −5 Effects of K on I(t) 0.0 0.1 0.2 0.3 0.4 0.5 t Effects of r on I(t) r=0.5 r=1 r=2 0.1 0.2 0.3 0.4 K=0.5 K=1 K=2 0.0 I(t) 15 tm=0 tm=5 tm=10 −5 0 5 10 t Effects of tm on I(t) 15 tm=0 tm=5 tm=10 −5 0 5 10 15 0 5 10 15 t Effects of a on I(t) a=0.5 a=1 a=2 0.2 0.1 0.0 0.0 0.1 0.2 0.3 10 t Effects of tm on C(t) −5 I(t) 5 r=0.5 r=1 r=2 0.3 C(t) 0.0 0.2 0.4 0.6 0.8 1.0 0 Effects of r on C(t) 0.0 0.2 0.4 0.6 0.8 1.0 2.0 Effects of K on C(t) K=0.5 K=1 K=2 −5 ❢r♦♠ ❘✐❝❤❛r❞s ♠♦❞❡❧ ✇✐t❤ ♣❛r❛♠❡t❡rs ✉♥❧❡ss ♦t❤❡r✇✐s❡ st❛t❡❞ 0.5 C(t) C(t) −5 0 t 5 10 15 −5 t ✷✺ ❋✐❣✉r❡ ✷✳✸✿ ❚❤❡ ❙■❘ ❝♦♠♣❛rt♠❡♥t❛❧ ♠♦❞❡❧ ✇✐t❤ t❤❡ ❜❧✉❡ ❝✐r❝❧❡s ✐♥❞✐❝❛t✐♥❣ t❤❡ ❝♦♠♣❛rt♠❡♥ts ❙✉s❝❡♣t✐❜❧❡✱ ■♥❢❡❝t❡❞ ❛♥❞ ❘❡❝♦✈❡r❡❞ ❛♥❞ ❛rr♦✇s ✐♥❞✐❝❛t✐♥❣ t❤❡ ♣❡r ❝❛♣✐t❛ r❛t❡ ♦❢ tr❛♥s❢❡r ❢r♦♠ ♦♥❡ ❝♦♠♣❛rt♠❡♥t t♦ t❤❡ s✉❜s❡q✉❡♥t ❝♦♠♣❛rt♠❡♥t✳ ❡q✉❛t✐♦♥s ✭❖❉❊s✮ dS dt = −β SI N dI dt = β SI − γI N dR dt = γI, ✭✷✳✸✮ ❛ss✉♠✐♥❣ t❤❡ s✐③❡s ♦❢ ❡❛❝❤ ❝♦♠♣❛rt♠❡♥t ❝❛♥ ❜❡ ❞✐✛❡r❡♥t✐❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡✱ ❛♥❞ t❤❛t ❝✉rr❡♥t st❛t❡s ❝❛♥ ❜❡ ❢✉❧❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♣r❡✈✐♦✉s st❛t❡s t♦❣❡t❤❡r ✇✐t❤ t❤❡ ♣❛r❛♠❡t❡rs ✐♥✈♦❧✈❡❞✳ ❚❤❡ t❡r♠s dS dI dR ✱ ❛♥❞ r❡♣r❡s❡♥t dt dt dt t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❞✐✈✐❞✉❛❧s ✐♥ t❤❛t ❝❧❛ss ❛t t✐♠❡ t✳ ■♥ t❤❡ ♠♦❞❡❧ ❛❜♦✈❡✱ t❤❡ t♦t❛❧ ♣♦♣✉❧❛t✐♦♥ ❢♦r ❛❧❧ t✐♠❡s t✱ N (t) = S(t)+I(t)+R(t) r❡♠❛✐♥s ❝♦♥st❛♥t s✐♥❝❡ ✇❡ ❛ss✉♠❡ ♥❡❣❧✐❣✐❜❧❡ r❛t❡s ♦❢ ❜✐rt❤ ❛♥❞ ❞❡❛t❤✱ ❛♥❞ t❤❛t t❤❡ ❡♣✐❞❡♠✐❝ ♦❝❝✉rs ✐♥ ❛ r❡❧❛t✐✈❡❧② s❤♦rt t✐♠❡ ❛♥❞ ❝♦♥s❡q✉❡♥t❧② ❞♦ ♥♦t ❣✐✈❡ r✐s❡ t♦ ❡①tr❡♠❡ ❞❡♠♦❣r❛♣❤✐❝ ❝❤❛♥❣❡s ✇❤✐❝❤ ♣❡rt✉r❜ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❝♦♥t❛❝t ❜❡t✇❡❡♥ ✐♥❞✐✈✐❞✉❛❧s ✭■❛♥♥❡❧❧✐✱ ✷✵✵✺✮✱ ❛♥ ❛ss✉♠♣t✐♦♥ t❤❛t ✐s ❛♣♣r♦♣r✐❛t❡ ❢♦r ✐♥✢✉❡♥③❛ ♦✉t❜r❡❛❦s✱ ❜✉t ♥♦t✱ s❛②✱ ❢♦r ❧❡ss ❡①♣❧♦s✐✈❡ ❡♣✐❞❡♠✐❝s s✉❝❤ ❛s ❆■❉❙ ✭❆❧❦❡♠❛ ❡t ❛❧✳✱ ✷✵✵✽✮ ♦r ❈❤❧❛♠②❞✐❛ ✭❆❧t❤❛✉s ♥✉♠❜❡r ♦❢ s✉s❝❡♣t✐❜❧❡ S(0) ❡t ❛❧✳✱ ✷✵✶✵✮✳ ❚❤❡ ✉♥❦♥♦✇♥ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧s I(0) ❛t ✷✻ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❡♣✐❞❡♠✐❝ ❛r❡ tr❡❛t❡❞ ❛s ♣❛r❛♠❡t❡rs✳ ❚❤❡ ♠♦❞❡❧ ✐♠♣❧✐❡s t❤❛t s✉s❝❡♣t✐❜❧❡ ✐♥❞✐✈✐❞✉❛❧s ❢r♦♠ ✐♥❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧s I ❛t ❛ r❛t❡ ♦❢ β SI ✱ N S ✇✐❧❧ ❝♦♥tr❛❝t t❤❡ ❞✐s❡❛s❡ ✇❤❡r❡ t❤❡ ✐♥❢❡❝t✐✈✐t② ♣❛r❛♠❡t❡r β ✐s t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ tr❛♥s♠✐ss✐♦♥s ♣❡r N S−I ♣❛✐r ✐♥ ❛ t✐♠❡ ♣❡r✐♦❞✳ ❚❤❡ ♠♦❞❡❧ ❛ss✉♠❡s ❤♦♠♦❣❡♥❡♦✉s ♠✐①✐♥❣ ✐♥ t❤❡ ♣♦♣✉❧❛t✐♦♥✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❛t ✐♥❞✐✈✐❞✉❛❧s ✐♥t❡r❛❝t ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ❛♥❞ t♦ t❤❡ s❛♠❡ ❡①t❡♥t ❛♥❞ t❤❡r❡ ✐s ♥♦ ❝❧✉st❡r✐♥❣ ♦❢ ✐♥❢❡❝t✐♦♥s ✐♥ s♣❛❝❡ ♦r s♦❝✐❛❧ s♣❛❝❡✳ ❚❤❡r❡❢♦r❡✱ βI/N ✐s t❤❡ r❛t❡ t❤❛t ❛ s✉s❝❡♣t✐❜❧❡ ❣❡ts ✐♥❢❡❝t❡❞ ♣❡r ✉♥✐t t✐♠❡✳ ❚❤❡ r❛t❡ ♦❢ ✐♥❢❡❝t✐♦♥ ✐s ❛ss✉♠❡❞ t♦ ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❢r❛❝t✐♦♥❛❧ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❝✉rr❡♥t❧② ✐♥❢❡❝t❡❞✱ ✐♠♣❧②✐♥❣ t❤❛t ❛s ♣♦♣✉❧❛t✐♦♥ s✐③❡ ✐♥❝r❡❛s❡s✱ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ ❛♥ ✐♥✲ ❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧ ❝♦♠❡ ✐♥t♦ ❝♦♥t❛❝t ✇✐t❤ ♦♥❡ ♣❛rt✐❝✉❧❛r s✉s❝❡♣t✐❜❧❡ ✐♥❞✐✈✐❞✉❛❧ ❞❡❝r❡❛s❡s✳ ■♥❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧s ✇✐❧❧ s❤✐❢t t♦ t❤❡ r❡♠♦✈❡❞ ❝❧❛ss ❛✈❡r❛❣❡ ✐♥❢❡❝t❡❞ ♣❡r✐♦❞ ❜❡✐♥❣ 1/γ ✮✳ R ❛t ❛ r❛t❡ ♦❢ γ ✭t❤❡ ❘❡♠♦✈❛❧ ✐s ❞❡✜♥❡❞ ❜② r❡❝♦✈❡r② ❢r♦♠ t❤❡ ❞✐s❡❛s❡ ❛♥❞ t❤✉s ❣❛✐♥✐♥❣ ✐♠♠✉♥✐t② ❛❣❛✐♥st t❤❡ ✈✐r✉s ❢♦r t❤❡ t✐♠❡ s♣❛♥ ♦❢ t❤❡ ❡♣✐❞❡♠✐❝✱ ♦r ❞❡❛t❤ ❞✐r❡❝t❧② ♦r ✐♥❞✐r❡❝t❧② ❝❛✉s❡❞ ❜② t❤❡ ❞✐s❡❛s❡✳ ❲❡ ✉s❡ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ t♦ ❧♦♦❦ ❢♦r ♠♦❞❡❧ tr❛❥❡❝t♦r✐❡s t❤❛t ✏♠❛t❝❤✑ t❤❡ ❡♣✐❞❡♠✐❝ t✐♠❡ s❡r✐❡s✳ ✷✼ ✷✳✷✳✸ ❙t♦❝❤❛st✐❝ ❙■❘ ❝♦♠♣❛rt♠❡♥t❛❧ ♠♦❞❡❧ ❚❤❡ ❞❡t❡r♠✐♥✐st✐❝ ✈❡rs✐♦♥ ♦❢ ❙■❘ ♠♦❞❡❧ ♦✉t❧✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ✐s ♠♦❞✐✜❡❞ t♦ ❜❡ st♦❝❤❛st✐❝✳ ❲❤✐❧❡ t❤❡ st❛t❡s S✱ I ❛♥❞ R ✐♥ ❛ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧ ❛r❡ ❦♥♦✇♥ ❣✐✈❡♥ t❤❡✐r ♣❛st ✈❛❧✉❡s✱ t❤❡✐r ❝♦✉♥t❡r♣❛rts ✐♥ st♦❝❤❛st✐❝ ♠♦❞❡❧ ❛r❡ ❛❧❧♦✇❡❞ t♦ ✈❛r② r❛♥❞♦♠❧② ❛❝❝♦r❞✐♥❣ t♦ s♣❡❝✐✜❝ ❞✐str✐❜✉t✐♦♥s✳ ❚❤❡ ♣❛r❛♠❡t❡rs β ❛♥❞ γ ✐♥ t❤❡ st♦❝❤❛st✐❝ ❙■❘ ♠♦❞❡❧ ❛r❡ ❛♥❛❧♦❣♦✉s t♦ t❤❛t ✐♥ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧✳ ❚❤❡ ♠♦❞❡❧ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇s✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♥❡✇ ✐♥❢❡❝t✐✈❡s ❜❡t✇❡❡♥ t✐♠❡ t ❛♥❞ t−1 ✐s ❛ss✉♠❡❞ t♦ ❜❡ t−1 ✐s ❛ss✉♠❡❞ t♦ ❜❡ At ∼ ❇✐♥(St−1 , p✐♥❢❡❝t✐♦♥ (t)). ❚❤❡ ♥✉♠❜❡r ♦❢ ♥❡✇ r❡❝♦✈❡r✐❡s ❜❡t✇❡❡♥ t✐♠❡ t ❛♥❞ Bt ∼ ❇✐♥(It−1 , pr❡❝♦✈❡r② (t)). ❚❤❡ ❣❡♥❡r❛❧ ❙■❘ st❛t❡✲s♣❛❝❡ ♠♦❞❡❧ St = St−1 − At , It = It−1 + At − Bt , Rt = Rt−1 + Bt . ■♥ t❤✐s ♠♦❞❡❧✱ ✷✽ ❼ p✐♥❢❡❝t✐♦♥ (t) = 1 − e−βI(t)/N ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ s✉s❝❡♣t✐❜❧❡ ❜❡✐♥❣ ✐♥❢❡❝t❡❞ ♦✈❡r ❛ ♦♥❡✲❞❛② t✐♠❡ ✇✐♥❞♦✇❀ ❼ pr❡❝♦✈❡r② (t) = 1 − e−1/γ ❜❡✐♥❣ r❡♠♦✈❡❞ ❢r♦♠ t❤❡ ❼ At ❛♥❞ Bt I ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ✐♥❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧ ❝❛t❡❣♦r②❀ r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❞✐✈✐❞✉❛❧s ✇❤♦ ❛r❡ ♥❡✇❧② ✐♥❢❡❝t❡❞ ❛♥❞ r❡♠♦✈❡❞✱ r❡s♣❡❝t✐✈❡❧②✱ ❛t t✐♠❡ t✱ ❛♥❞ ✇❡ ❛ss✉♠❡ t❤❡♠ t♦ ❢♦❧❧♦✇ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥s ✇✐t❤ ♣r♦❜❛❜✐❧✐t② p✐♥❢❡❝t✐♦♥ (t) ❛♥❞ pr❡❝♦✈❡r② (t)❀ ❼ t❤❡ ❞❛t❛ ❛r❡ r❡❝♦r❞❡❞ ♦♥ ❛ ❞✐s❝r❡t❡ ❜❛s✐s❀ ❼ t❤❡ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝❛s❡s ♠❛❦❡s ✐t ♣r♦❤✐❜✐t✐✈❡ t♦ s❡❛r❝❤ t❤❡ s♣❛❝❡ ♦❢ ✉♥♦❜s❡r✈❡❞✱ ❝♦♥t✐♥✉♦✉s t✐♠❡ ✐♥❢❡❝t✐♦♥ t✐♠❡s✱ ✇❤✐❝❤ ✇♦✉❧❞ ❜❡ ♥❡❝❡ss❛r② ❢♦r ❛ ❝♦♥t✐♥✉♦✉s t✐♠❡ ♠♦❞❡❧❀ ❼ ✇❡ ❝❤♦♦s❡ t♦ ✇♦r❦ ✇✐t❤ ❛ ❞✐s❝r❡t❡ t✐♠❡ ♠♦❞❡❧ ❜❡❝❛✉s❡ t❤❡ ❞❛✐❧② ❝②❝❧❡ ✐♠♣♦s❡s ♥♦♥✲❝♦♥st❛♥t r❛t❡s ♦❢ ✐♥❢❡❝t✐♦♥ ❛♥②✇❛②✳ ❚❤✐s ❞✐s❝r❡t❡ ♠♦❞❡❧ s❡r✈❡s ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ tr✉❡ ✉♥❞❡r❧②✐♥❣ ❝♦♥✲ t✐♥✉♦✉s t✐♠❡ ♣r♦❝❡ss✳ ✷✳✷✳✹ ❚❤❡ ♦❜s❡r✈❛t✐♦♥ ♠♦❞❡❧ ❚❤❡ ♦❜s❡r✈❛t✐♦♥ ♣r♦❝❡ss ✐s s✉♣♣♦rt❡❞ ❜② ♦❜s❡r✈❛t✐♦♥❛❧ ❞❛t❛ ✇❤✐❝❤ ❛r❡ ♦❜✲ t❛✐♥❡❞ ❛s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✶✳ ❲❡ ✉s❡ t❤❡s❡ ♦❜s❡r✈❛t✐♦♥❛❧ ❞❛t❛ t♦ ♠❛❦❡ ✐♥❢❡r❡♥❝❡s ❛❜♦✉t t❤❡ st❛t❡ ♣r♦❝❡ss ✇❤✐❝❤ ✐♥✈♦❧✈❡s t❤❡ ❙♦♠❡ ♥❡✇ ♣❛r❛♠❡t❡rs ✐♥tr♦❞✉❝❡❞ ❤❡r❡ ❛r❡ δwds ✱ S✱ I ❛♥❞ R ❝❛t❡❣♦r✐❡s✳ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛♥ ✐♥✲ ❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧ ✇✐❧❧ ✈✐s✐t ❛ ❞♦❝t♦r ❢♦r ❝♦♥s✉❧t❛t✐♦♥ ♦♥ ✇❡❡❦❞❛②s ❛♥❞ ❙❛t✲ ✉r❞❛②s✱ δwds × δsph ✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ❛♥ ✐♥❢❡❝t❡❞ ✐♥❞✐✈✐❞✉❛❧ ✇✐❧❧ ✈✐s✐t ❛ ❞♦❝t♦r ❢♦r ❝♦♥s✉❧t❛t✐♦♥ ♦♥ ✇❡❡❦❡♥❞s ❛♥❞ ♣✉❜❧✐❝ ❤♦❧✐❞❛②s✱ ❛♥❞ φ✱ t❤❡ ❝♦♥s✉❧t✐♥❣ r❛t❡ ✷✾ ♦❢ ■▲■s ✇❤✐❝❤ ❛r❡ ♥♦t ❝❛✉s❡❞ ❜② t❤❡ ✐♥✢✉❡♥③❛ ❆✲❍✶◆✶✭✷✵✵✾✮✳ ❚❤❡ ♦❜s❡r✈❡❞ ♥✉♠❜❡r ♦❢ ■▲■s✱ yt ✱ ❢r♦♠ ♦✉r ♥❡t✇♦r❦ ♦❢ ❞♦❝t♦rs ✐s r❡❝♦r❞❡❞ ✐♥ t❤❡ ♦❜s❡r✈❛✲ t✐♦♥❛❧ ❞❛t❛✱ ❤♦✇❡✈❡r✱ It yt ✐s ♥♦t ❡q✉❛❧ t♦ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✐♥❢❡❝t❡❞ s✉❜❥❡❝ts ♣r❡s❡♥t ✐♥ t❤❡ ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ♥❡❡❞ t♦ ✐♥❝♦r♣♦r❛t❡ ♣❛r❛♠❡t❡rs t♦ ❛❞❞r❡ss t❤❛t ❞✐s❝r❡♣❛♥❝② ❛s s❤♦✇♥ ❜❡❧♦✇✱ yt ∼ P♦✐s(Nt ψt ), ✇❤❡r❡ ❞❛② t Nt ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❞♦❝t♦rs ✐♥ ♦✉r ♥❡t✇♦r❦ ✇❤♦ s✉❜♠✐tt❡❞ r❡♣♦rts ♦♥ ❛♥❞ ψt = δwds × (φ + It /2084) δwds × δsph × (φ + It /2084) ✇❤❡♥ ✇❤❡♥ t t ✐s ✇❡❡❦❞❛②s ♦r ❙❛t✉r❞❛②s✱ ✐s ❙✉♥❞❛②s ♦r ♣✉❜❧✐❝ ❤♦❧✐❞❛②s✳ ❚❤❡ ❡st✐♠❛t❡❞ ♥✉♠❜❡r ♦❢ ❞♦❝t♦rs ✐♥ ❙✐♥❣❛♣♦r❡ ✐s 2084✱ ❛♥❞ ✐s ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ t❤❡ ❡st✐♠❛t❡❞ ♥✉♠❜❡r ♦❢ ❣❡♥❡r❛❧ ♣❤②s✐❝✐❛♥s ♦r ❢❛♠✐❧② ❞♦❝t♦rs ✐♥ ❙✐♥✲ ❣❛♣♦r❡ ✐♥ ✷✵✵✾✱ ✇❤✐❝❤ ✇❛s 1730 ✭❖♥❣ ❡t ❛❧✳✱ ✷✵✶✵✮✱ ❛♥❞ t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ ♣❛t✐❡♥ts ✇❤♦ s❡❡❦ ♠❡❞✐❝❛❧ ❛tt❡♥t✐♦♥ ❢r♦♠ t❤❡s❡ ❞♦❝t♦rs ✐♥st❡❛❞ ♦❢ ✈✐s✐t✐♥❣ ❛ ♣♦❧②❝❧✐♥✐❝✱ ✇❛s 83% ✭❊♠♠❛♥✉❡❧ ❡t ❛❧✳✱ ✷✵✵✹✮ ✐♥ ✷✵✵✶✳ ✷✳✸ ❙t❛t✐st✐❝❛❧ ♠❡t❤♦❞♦❧♦❣② ✷✳✸✳✶ ❇❛②❡s✐❛♥ ♣❛r❛❞✐❣♠ ■♥ t❤❡ ❇❛②❡s✐❛♥ ♣❛r❛❞✐❣♠✱ t❤❡ ♣r♦❝❡ss ♦❢ ❧❡❛r♥✐♥❣ ❢r♦♠ t❤❡ ❞❛t❛ ✐s s②st❡♠❛t✐✲ ❝❛❧❧② ✐♠♣❧❡♠❡♥t❡❞ ❜② ♠❛❦✐♥❣ ✉s❡ ♦❢ ❇❛②❡s✬ t❤❡♦r❡♠ t♦ ❝♦♠❜✐♥❡ ❛♥② ❛✈❛✐❧❛❜❧❡ ✸✵ ♣r✐♦r ✐♥❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❞❛t❛ t♦ ♣r♦❞✉❝❡ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✉♥♦❜s❡r✈❡❞ q✉❛♥t✐t✐❡s ❣✐✈❡♥ t❤❡ ♦❜s❡r✈❡❞ ♦♥❡s ✭●❡❧♠❛♥ ❡t ❛❧✳✱ ✷✵✵✸✮✳ ❲❡ ❛r❡ ❛❜❧❡ t♦ ❞❡r✐✈❡ ❛♥❞ ♠❛❦❡ ✐♥❢❡r❡♥❝❡s ♦♥ ✉♥♦❜✲ s❡r✈❡❞ ❡✈❡♥ts ♦r ♣❛r❛♠❡t❡rs✱ θ✱ ❜❛s❡❞ ♦♥ ❛ ❣✐✈❡♥ s❡t ♦❢ ❞❛t❛✱ Y✱ ❛❝❝♦r❞✐♥❣ t♦ ❇❛②❡s✬ t❤❡♦r❡♠ ❛s ❢♦❧❧♦✇s✿ P (θ|Y ) = P (θ)×P (Y |θ) P (Y ) ✭✷✳✹✮ = ✇❤❡r❡ P (θ) ´ P (θ)×P (Y |θ) P (Y |θ)P (θ)dθ ✐s ❦♥♦✇♥ ❛s t❤❡ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✳ ❙✐♥❝❡ ✇❡ ❝♦♥❝❡♣✲ t✉❛❧✐③❡ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡rs t♦ ❜❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✐♥ ❇❛②❡s✐❛♥ st❛t✐s✲ t✐❝s✱ ✇❡ ❤❛✈❡ ❛ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ t❤❛t r❡✢❡❝ts ❛ ❜❡❧✐❡❢ ❛❜♦✉t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❝❛♥ ❜❡ ✐♥❢♦r♠❛t✐✈❡ ♦r ♥♦♥✲✐♥❢♦r♠❛t✐✈❡✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r ✇❡ ✇❛♥t t♦ ❜❛s❡ t❤❡ ♣r✐♦r ♦♥ r❡❧❛t❡❞ ❞❛t❛ ♦r ❡①♣❡r✐❡♥❝❡✱ ♦r ✇❡ ❞♦ ♥♦t ✇❛♥t ❛♥② ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ ♦✉ts✐❞❡ t❤❡ ♣r✐♠❛r② ❞❛t❛s❡t t♦ ❜❡ r❡♣r❡s❡♥t❡❞✳ P (Y |θ) ✐s t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♦❜s❡r✈✐♥❣ t❤❡ ❞❛t❛ ❣✐✈❡♥ t❤❡ s❡t ♦❢ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ❣✐✈❡s r❡❧❛t✐✈❡ ✇❡✐❣❤ts t♦ ❡✈❡r② ♣❛r❛♠❡t❡r ✈❛❧✉❡ ✭❇♦❧st❛❞✱ ✷✵✶✵✮✳ ❆❢t❡r ♠✉❧t✐♣❧②✐♥❣ t❤❡ ♣r✐♦r ❞✐str✐❜✉t✐♦♥ ✇✐t❤ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡♥ ♥♦r♠❛❧✐③✐♥❣ ✇✐t❤ ❛ ❝♦♥st❛♥t ♦❢ ♣r♦♣♦rt✐♦♥❛❧✐t② ❞✐str✐❜✉t✐♦♥ P (θ|Y )✱ ´ P (Y |θ)P (θ)dθ✱ ✇❡ ✇✐❧❧ ♦❜t❛✐♥ t❤❡ ♣♦st❡r✐♦r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❛❢t❡r t❤❡ ❞❛t❛ ❛r❡ ❛❝❝♦✉♥t❡❞ ❢♦r✳ ■♥ ♠❛♥② ❝❛s❡s✱ t❤❡ ❡①❛❝t ❢♦r♠ ♦❢ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ✐s ✐♥tr❛❝t❛❜❧❡ ✸✶ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ ❝♦♥❥✉❣❛t❡ ♣r✐♦rs ❛r❡ ♥♦t ❛✈❛✐❧❛❜❧❡✳ ❚❤❡ ♠❡t❤♦❞s ♦✉t❧✐♥❡❞ ✐♥ t❤❡ s✉❜s❡q✉❡♥t s✉❜s❡❝t✐♦♥s✱ ♥❛♠❡❧② ▼❛r❦♦✈ ❝❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ ✭▼❈▼❈✮✱ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❛♥❞ ♣❛rt✐❝❧❡ ✜❧t❡r✐♥❣✱ ❛❧❧♦✇s ✉s t♦ ❢♦r♠ ❛ s❛♠♣❧❡ ❢r♦♠ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ t❤r♦✉❣❤ s✐♠✉❧❛t✐♦♥s✳ ✷✳✸✳✷ ▼❛r❦♦✈ ❈❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ ▼♦♥t❡ ❈❛r❧♦ s❛♠♣❧✐♥❣ ✐s ♦♥❡ ♦❢ t❤❡ ♠❡t❤♦❞s ✉s❡❞ t♦ ♦❜t❛✐♥ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥s✳ ■t r❡❧✐❡s ♦♥ ❞✐r❡❝t s✐♠✉❧❛t✐♦♥ t♦ ❣❡♥❡r❛t❡ ♣♦st❡r✐♦r ❞✐str✐❜✉✲ t✐♦♥s ♦❢ q✉❛♥t✐t✐❡s ♦❢ ✐♥t❡r❡st✱ ❛♥❞ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ✇❤❡♥ ❛♥❛❧②t✐❝❛❧ ❢♦r♠ ♦❢ ♣♦st❡r✐♦rs ✐s ❦♥♦✇♥✱ ♦r ✇❤❡♥ ♣♦st❡r✐♦rs ❛r❡ st❛♥❞❛r❞ ❞✐str✐❜✉t✐♦♥s ❢♦r t❤❡ ❝♦♥❥✉❣❛t❡ ♣r✐♦rs ✉s❡❞✳ ❚❤❡ q✉❛❧✐t② ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♠♣r♦✈❡s ❛s ✇❡ ✐♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ s✐♠✉❧❛t❡❞ s❛♠♣❧❡s ✭❈❛r❧✐♥ ❛♥❞ ▲♦✉✐s✱ ✷✵✵✵✮✳ ■♥ ❝❛s❡s ✇❤❡r❡ ❛♥❛❧②t✐❝ ♦r ♥✉♠❡r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ✐s ♥♦t ❢❡❛s✐❜❧❡✱ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ ✇✐❧❧ ❜❡ ▼❛r❦♦✈ ❝❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ ✭▼❈▼❈✮ ♠❡t❤♦❞s ✭▼❛♥❧②✱ ✷✵✵✼✮✳ ❚❤❡ ✐❞❡❛ ♦❢ ▼❈▼❈ ❛❧❣♦r✐t❤♠s ✐s t♦ ❝♦♥str✉❝t ❛ ▼❛r❦♦✈ ❝❤❛✐♥ ✇✐t❤ ♦✉r t❛r❣❡t ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❛s ✐ts ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥✱ ✇❛♥t t♦ ✜♥❞ ❛♥ ❛♣♣r♦♣r✐❛t❡ tr❛♥s✐t✐♦♥ ❦❡r♥❡❧✱ P (θ, A), π(θ)✳ ✇❤❡r❡ ❚❤❡r❡❢♦r❡ ✇❡ A ✐s ♠❡❛s✉r❡✲ ❛❜❧❡ s✉❜s❡ts ♦❢ t❤❡ ♣❛r❛♠❡t❡r s♣❛❝❡❀ s✉❝❤ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♦❜s❡r✈❛t✐♦♥s ❣❡♥❡r❛t❡❞ ❢r♦♠ t❤❡ s✐♠✉❧❛t✐♦♥ ❝♦♥✈❡r❣❡s t♦ π(θ) ✭❇♦❧st❛❞✱ ✷✵✶✵✮✳ ▲❡t q(θ, θ ) θ ❣✐✈❡♥ ❛ ❜❡ t❤❡ ✐♠♣r♦♣❡r tr❛♥s✐t✐♦♥ ❦❡r♥❡❧ t❤❛t ❣❡♥❡r❛t❡s ❛ ♣r♦♣♦s❡❞ ✈❛❧✉❡ st❛rt✐♥❣ ✈❛❧✉❡ θ. ■t ✐s s❤♦✇♥ t❤❛t ✐❢ ✸✷ ˆ P (θ, A) = q(θ, θ )dθ + r(θ)δA (θ) A ✇❤❡r❡ δA (θ) r(θ) = 1 − ´ q(θ, θ )dθ ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❡ ❝❤❛✐♥ r❡♠❛✐♥s ❛t ✐s t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ s❡t δA (θ) = ❛♥❞ ✐❢ t❤❡ ❢✉♥❝t✐♦♥ q(θ, θ ) A✱ θ✱ ✇✐t❤    1 ✐❢ θ   0 ✐❢ θ ∈A , ∈ /A s❛t✐s✜❡s t❤❡ r❡✈❡rs✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥ π(θ)q(θ, θ ) = π(θ )q(θ , θ), t❤❡♥ π(θ) ✭✷✳✺✮ ✐s t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ❢♦r t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✭❇♦❧st❛❞✱ ✷✵✶✵✮✳ ❱❡r✐✜❝❛t✐♦♥ t❤❛t t❤❡ tr❛♥s✐t✐♦♥ ❦❡r♥❡❧ ❝♦♥✈❡r❣❡s t♦ ´ ❛♥❞ P (θ, A)π(θ)dθ = = ´ ´ ´ A ´ q(θ, θ )dθ π(θ)dθ + q(θ, θ )π(θ)dθ dθ + ´ ´ ´ ✐s s❤♦✇♥ ❜❡❧♦✇✿ r(θ)δA (θ)π(θ)dθ A r(θ)π(θ)dθ ´ q(θ , θ)π(θ )dθ dθ + A r(θ)π(θ)dθ ´ ´ = A (1 − r(θ ))π(θ )dθ + A r(θ)π(θ)dθ ´ = A π(θ )dθ . = ´ A π(θ) A ❍♦✇❡✈❡r✱ ✇❤❡♥ t❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ✈❡rs✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✐♥ ✭✷✳✺✮ ❛s q(θ, θ ) p(θ, θ ) ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ r❡✲ ❞♦❡s✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠♦✈✐♥❣ ❢r♦♠ ✸✸ θ t♦ θ ✐s ♥♦t t❤❡ s❛♠❡ ❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠♦✈✐♥❣ ❢r♦♠ θ t♦ θ✳ ❚❤❡ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ❛❧❣♦r✐t❤♠ ♣r♦✈✐❞❡s ❛ ♠❡❛♥s t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝❝❡♣t❛♥❝❡ ✐♥ t❤✐s ❝❛s❡✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✇✐❧❧ ❞❡❝✐❞❡ ✇❤❡t❤❡r t♦ ❛❝❝❡♣t ♦r r❡❥❡❝t t❤❡ ♣r♦♣♦s❡❞ ✈❛❧✉❡ θ ❛ ❛❝❝❡♣t❛♥❝❡ ♣r♦❜❛❜✐❧✐t②✱ α(θ, θ )✳ ❢r♦♠ t❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ❚❤❡ ❛❝❝❡♣t❛♥❝❡ ♣r♦❜❛❜✐❧✐t② ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ♣❛r❛♠❡t❡r ✇✐❧❧ ❛ss✉♠❡ t❤❡ ✈❛❧✉❡ ♦❢ ❛❝❝❡♣t❛♥❝❡✱ t❤❡ ♣r♦❝❡ss ✇✐❧❧ ♠♦✈❡ ❢r♦♠ ❛ss✉♠❡ t❤❡ ✈❛❧✉❡ p(θ, θ )✱ θ t♦ θ✳ ❜❛s❡❞ ♦♥ α(θ, θ ) ✐s t❤❡ ■♥ t❤❡ ❝❛s❡ ♦❢ θ ✱ ♦t❤❡r✇✐s❡✱ t❤❡ ♣r♦❝❡ss ✇✐❧❧ r❡✲ θ✱ ❛♥❞ r❡♣❡❛ts ❜② ♣r♦♣♦s✐♥❣ ❛♥♦t❤❡r ✈❛❧✉❡ ❢♦r t❤❡ ♣❛r❛♠❡t❡r✳ ❚❤❡ ✐♠♣r♦♣❡r tr❛♥s✐t✐♦♥ ❦❡r♥❡❧ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s q(θ, θ ) = p(θ, θ ) × α(θ, θ )✳ ❚❤❡ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ❛❧❣♦r✐t❤♠ ✐s ♦✉t❧✐♥❡❞ ❛s ❢♦❧❧♦✇s✿ ✶✳ ❙t❛rt ✇✐t❤ ❛♥ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✷✳ ❘❡♣❡❛t ❢♦r ❛❧❧ s❛♠♣❧❡s ✭❛✮ ❙❛♠♣❧❡ i✿ θ ∼ p(θ, θ )❀ ✭❜✮ ❈❛❧❝✉❧❛t❡ ♣r♦❜❛❜✐❧✐t② ✭❝✮ ❉r❛✇ ✭❞✮ ■❢ θ0 )p(θ ,θ) α(θ, θ ) = ♠✐♥ 1, π(θ π(θ)p(θ,θ ) ❀ u ∼ U [0, 1]❀ u < α(θ, θ ) ❚❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ s❡t α(θ, θ ) θi+1 = θ ✱ ❡❧s❡ s❡t θi+1 = θ✳ ✐♥ t❤❡ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ❛❧❣♦r✐t❤♠ ❞♦❡s ♥♦t r❡q✉✐r❡ t❤❡ ♥♦r♠❛❧✐③✐♥❣ ❝♦♥st❛♥t ♦❢ π(θ) ❛♥❞ π(θ ) ❜❡❝❛✉s❡ t❤❡ ❝♦♥st❛♥t ❝❛♥ ✸✹ ❜❡ ❝❛♥❝❡❧❧❡❞ ♦✛ ❢r♦♠ ❜♦t❤ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r✳ ■❢ t❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ✐s s②♠♠❡tr✐❝✱ r❡❞✉❝❡❞ t♦ p(θ, θ ) = p(θ , θ)✱ ) α(θ, θ ) = ♠✐♥ 1, π(θ π(θ) t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝❝❡♣t❛♥❝❡ ✐s ✳ ❚❤❡ ♠❛❥♦r ❛❞✈❛♥t❛❣❡ ♦❢ ▼♦♥t❡ ❈❛r❧♦ s❛♠♣❧✐♥❣ ✐s t❤❛t ♣♦✐♥ts ❛r❡ s❛♠♣❧❡❞ ♣r❡❢❡r❡♥t✐❛❧❧② ✇❤❡♥ t❤❡② ❛r❡ ❝❧♦s❡ t♦ π(θ)✳ ❇✉t ✐♥ ▼❈▼❈✱ ✇❡ ❤❛✈❡ t❤❡ ✢❡①✐❜✐❧✐t② t♦ ✐♥❝♦r♣♦r❛t❡ ♣r✐♦r ✐♥❢♦r♠❛t✐♦♥ ✇❤❡♥ ❣❡♥❡r❛t✐♥❣ s❛♠♣❧❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ✭❉✉♥s♦♥ ❉✳❇✳✱ ✷✵✵✶✮✳ ❚❤❡ ❞✐s❛❞✈❛♥t❛❣❡ ♦❢ ▼❈▼❈ ✐s t❤❡ ❧❡♥❣t❤ ♦❢ r✉♥ t♦ st❛t✐♦♥❛r✐t② ♠✐❣❤t ❜❡ ❧♦♥❣ ❛♥❞ ❞✐✣❝✉❧t t♦ ✐❞❡♥t✐❢②✳ ✷✳✸✳✸ ■♠♣♦rt❛♥❝❡ ❙❛♠♣❧✐♥❣ ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ✐s ❛♥♦t❤❡r ♠❡t❤♦❞ t♦ s❛♠♣❧❡ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ✇❤❡♥ t❤❡ ❡①❛❝t ❢♦r♠ ✐s ✉♥❦♥♦✇♥✳ ■♥st❡❛❞ ♦❢ s❛♠♣❧✐♥❣ ❞✐r❡❝t❧② ❢r♦♠ t❤❡ ❛❝t✉❛❧ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✱ ✇❡ s❛♠♣❧❡ ❢r♦♠ s♦♠❡ ♦t❤❡r ❞✐str✐❜✉t✐♦♥ t❤❛t ❛♣♣r♦①✐♠❛t❡s t❤❡ ❛❝t✉❛❧ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❛♥❞ ✇❡✐❣❤t t❤❡ s❛♠♣❧❡❞ ✈❛❧✉❡s t♦ r❡✢❡❝t t❤❡✐r ❞❡❣r❡❡ ♦❢ ✐♠♣♦rt❛♥❝❡✱ t❤✉s ❡♥❝♦✉r❛❣✐♥❣ t❤❡ ❞✐str✐❜✉t✐♦♥ t♦ ❝❤❛♥❣❡ ♦✈❡r t✐♠❡ t♦ ❜❡ ♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ t❤❡ tr✉❡ ✉♥❞❡r❧②✐♥❣ ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ ✇❡ ✐t❡r❛t❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ st❡♣s ✭◆❡❛❧✱ ✷✵✵✶✮✳ ■♥ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣✱ t❤❡ s❛♠♣❧❡❞ ♣❛r❛♠❡t❡rs ✇✐t❤ ❤✐❣❤❡r ✇❡✐❣❤ts ❤❛✈❡ ♠♦r❡ ✐♠♣❛❝t ✐♥ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣❛r❛♠❡t❡rs✱ ❛♥❞ ✐❢ t❤❡s❡ ✈❛❧✉❡s ❛r❡ ❡♠♣❤❛s✐③❡❞ ❜② ❜❡✐♥❣ s❛♠♣❧❡❞ ♠♦r❡ ❢r❡q✉❡♥t❧②✱ t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❡st✐♠❛t♦r ❝❛♥ ❜❡ r❡❞✉❝❡❞ ✭❙♠✐t❤ ❡t ❛❧✳✱ ✶✾✾✼✮✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡r❡ ✐s ❛♥ ✐♠♣r♦✈❡❞ ❛❝❝✉r❛❝② ✐♥ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣❛r❛♠❡t❡rs ❛s ✸✺ t❤❡ ✇❡✐❣❤ts ✇✐❧❧ ❝❛✉s❡ ✈❛❧✉❡s ❝❧♦s❡ t♦ t❤❡ ♣❛r❛♠❡t❡r ♦❢ ✐♥t❡r❡st t♦ ♦❝❝✉r ♠♦r❡ ❢r❡q✉❡♥t❧② ✐♥ t❤❡ s❛♠♣❧✐♥❣ ♣r♦❝❡ss t❤❛♥ ✐t ✇♦✉❧❞ ✭●✐✈❡♥s ❛♥❞ ❍♦❡t✐♥❣✱ ✷✵✵✺✮✳ ❆ss✉♠❡ t❤❛t h(θ) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✐♥t❡r❡st ❛♥❞ f (θ) ✐s t❤❡ ♣♦st❡r✐♦r ❞✐str✐✲ ❜✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✱ ✇❤✐❝❤ ✐s ✐♥t❡❣r❛❜❧❡ ❛♥❞ ♥♦♥✲③❡r♦ ♦✈❡r t❤❡ s✉♣♣♦rt ♦❢ θ ✭❆❝❦❡r❜❡r❣✱ ✷✵✵✾✮✳ ❲❡ t❤❡♥ ❡✈❛❧✉❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛♥t✐t② ❜② s✉❜st✐t✉t✐♥❣ t❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ˆ E[h(θ)] = h(θ)f (θ)dθ ˆ = h(θ) ˆ = f (θ) q(θ)dθ q(θ) h(θ)ω(θ)q(θ)dθ ✇❤❡r❡ t❤❡ ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤ts ❛r❡ ω(θ) = f (θ) . q(θ) ✭✷✳✻✮ ✸✻ ❚❤❡r❡❢♦r❡✱ ´ E[h(θ)] = ´ = ´ = h(θ)f (θ)dθ ´ f (θ)dθ (θ) h(θ) fq(θ) q(θ)dθ ´ f (θ) q(θ)dθ q(θ) h(θ)ω(θ)q(θ)dθ ´ . ω(θ)q(θ)dθ ❚❤✐s ✜♥❛❧ ❡①♣r❡ss✐♦♥ ✐♥❞✐❝❛t❡s t❤❛t ✐❢ ✇❡ ❞r❛✇ s❛♠♣❧❡s ❢r♦♠ t❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ✐♥t❡r❡st h(θ) q(θ)✱ ✇❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦❢ ❜② ✉s✐♥❣ E[h(θ)] ≈ = 1 N Σ h(θi )ω(θi ) N i=1 1 N Σ ω(θi ) N i=1 N Σi=1 h(θi )ω (θi ), ✇❤❡r❡ t❤❡ ♥♦r♠❛❧✐③❡❞ ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤ts ω (θi ) = ω ❛r❡ ❣✐✈❡♥ ❜② ω(θi ) . N Σi=1 ω(θi ) ❆ s✐♠♣❧❡ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ❛s ❢♦❧❧♦✇s✿ ✸✼ ✶✳ ❋♦r ❛❧❧ s❛♠♣❧❡s ✭❛✮ ●❡♥❡r❛t❡ ✭❜✮ ❙❡t i ✿ θi ∼ q(θ)❀ ω(θi ) = ✷✳ ❘❡t✉r♥ f (θi ) . q(θi ) N ˆ = Σi=1 h(θi )ω(θi ) h ΣN i=1 ω(θi ) ❛s ❛♥ ❡st✐♠❛t❡ ♦❢ E[h(θ)]✳ ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣✱ ✉♥❧✐❦❡ ▼❈▼❈✱ ❞♦❡s ♥♦t ❤❛✈❡ ❛♥② ✐ss✉❡s ♣❡rt❛✐♥✐♥❣ t♦ t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s ♥❡❡❞❡❞ ❢♦r ❝♦♥✈❡r❣❡♥❝❡ t♦ t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ✭◆✐❡❧s❡♥ ❛♥❞ ❲❛❦❡❧❡②✱ ✷✵✵✶✮✳ ■❢ t❤❡ s❛♠♣❧✐♥❣ ❞✐str✐❜✉t✐♦♥ ✐s s✉✐t❛❜❧② ❝❤♦s❡♥ s✉❝❤ t❤❛t ✐t r❡s❡♠❜❧❡s t❤❡ tr✉❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs✱ t❤❡ s❛♠✲ ♣❧❡ s✐③❡ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ❛❝❤✐❡✈❡ t❤❡ s❛♠❡ ❛❝❝✉r❛❝② ❛s ▼❈▼❈ ✭▼❛❝❡❛❝❤❡r♥ ❡t ❛❧✳✱ ✶✾✾✾✮✳ ❍♦✇❡✈❡r✱ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❝❛♥ ❜❡ ❝❤❛❧❧❡♥❣✲ ✐♥❣ ✇❤❡♥ t❤❡ s❛♠♣❧✐♥❣ ❞✐str✐❜✉t✐♦♥ ❞♦❡s ♥♦t ❤❛✈❡ r❡❛s♦♥❛❜❧❡ ✇❡✐❣❤ts ✐♥ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t② r❡❣✐♦♥s ♦❢ t❤❡ tr✉❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✳ ❚❤✐s ♠✐s♠❛t❝❤ ✐♥ ✇❡✐❣❤ts ✇✐❧❧ r❡s✉❧t ✐♥ ♣♦♦r ❡st✐♠❛t♦r ✇✐t❤ ❛ ❤✐❣❤ ✈❛r✐❛♥❝❡ ❛s ✐t ✐s ❧✐❦❡❧② t❤❛t t❤❡ s❛♠♣❧❡ ✇✐❧❧ ❝♦♥t❛✐♥ ✉♥r❡♣r❡s❡♥t❛t✐✈❡ ♣♦✐♥ts ❜✉t ❛❧s♦ ❜❡ ✐♥t❡r♠✐tt❡♥t❧② ❞♦♠✐♥❛t❡❞ ❜② ❛ ❢❡✇ ♣♦✐♥ts ✇✐t❤ ❤✐❣❤ ✇❡✐❣❤ts ✭❙♦✉t❤❡② ❡t ❛❧✳✱ ✷✵✵✷✮✳ ❚♦ s❡❧❡❝t ❛♥ ❛♣♣r♦♣r✐❛t❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ❢♦r ✐♠♣♦rt❛♥❝❡ s❛♠✲ ♣❧✐♥❣✱ ✇❡ ✜rst ♣❡r❢♦r♠ ▼❈▼❈ t♦ ♦❜t❛✐♥ ❛ s❛♠♣❧❡❞ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✳ ❚❤❡♥✱ ✇❡ ❝♦♥str✉❝t ❛ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ✇❤✐❝❤ ✐s ❝❧♦s❡ t♦ t❤❡ s❛♠♣❧❡❞ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ s♦ t❤❛t t❤✐s ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐❧❧ ✸✽ ❚❛❜❧❡ ✷✳✶✿ ■♥❢♦r♠❛t✐✈❡ ♣r✐♦rs ❢♦r t❤❡ ❘✐❝❤❛r❞s ♠♦❞❡❧ ❛♥❞ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧ ❘✐❝❤❛r❞s ♠♦❞❡❧ K r tm a δsph ∼ ∼ ∼ ∼ ∼ ◆✭200, 100✮ ◆✭0.06, 0.002✮ ❊①♣✭1/8✮ ◆✭5.5, 1✮ ❯❬0, 0.8❪ ❉❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧ β ∼ γ ∼ δwds ∼ δsph ∼ φ ∼ S0 ∼ R0 ∼ ◆✭1.1, 3✮, ◆✭0.9, 3✮, ◆✭0.2, 0.5✮✱ ◆✭0.5, 0.5✮, ◆✭0.8, 0.2✮, ◆✭4790000, 2000✮✱ ◆(10000, 200). ❧❡❛❞ t♦ ❛ s❡t ♦❢ ✈❛❧✉❡s ✇❤✐❝❤ ❤❛s s♠❛❧❧❡r ✈❛r✐❛♥❝❡ ❛♥❞ ✐s ♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ t❤❡ tr✉❡ ♣❛r❛♠❡t❡rs✳ ■♥ t❤✐s ♣r♦❥❡❝t✱ ✇❡ ✉s❡ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ t♦ ♣❡r❢♦r♠ ♣r❡❞✐❝t✐♦♥s ❢♦r ❘✐❝❤❛r❞s ♠♦❞❡❧✱ ❛♥❞ t❤r❡❡ ✐t❡r❛t✐♦♥s ♦❢ ▼❈▼❈ ❢♦❧❧♦✇❡❞ ❜② t❤r❡❡ ✐t❡r❛t✐♦♥s ♦❢ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ t♦ ♣❡r❢♦r♠ ♣r❡❞✐❝t✐♦♥s ❢♦r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧✳ ❚❤✐s ✐s ❛ r❡s✉❧t ♦❢ ✉♥s❛t✐s❢❛❝t♦r② ❡st✐♠❛t❡s ♣r♦❞✉❝❡❞ ❜② ✉s✐♥❣ ♦♥❧② t❤❡ ✐♠✲ ♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ♦♥ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧✳ ❚❤❡ ▼❈▼❈ ✐t❡r❛t✐♦♥s ✇✐❧❧ ♣r♦✈✐❞❡ ✐♥✐t✐❛❧ ❡st✐♠❛t❡s ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣❛r❛♠❡t❡rs s♦ t❤❛t ✇❤❡♥ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ✐s ❡①❡❝✉t❡❞✱ t❤❡ ♣r❡❢❡r❡♥t✐❛❧ s❛♠♣❧✐♥❣ ♦❢ ❤✐❣❤ ♣♦s✲ t❡r✐♦r r❡❣✐♦♥s ✇✐❧❧ ❧❡❛❞ t♦ ❛ ♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❢♦r t❤❡ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ✐♥❢♦r♠❛t✐✈❡ ♣r✐♦rs ❢♦r t❤❡ ♣❛r❛♠❡t❡rs ❛♥❞ ✐♥✐t✐❛❧ st❛t❡s ♦❢ t❤❡ ❘✐❝❤❛r❞s ♠♦❞❡❧ ❛♥❞ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧ ✉s❡❞ ✐♥ t❤❡ ❛❧❣♦r✐t❤♠s ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜❧❡ ✷✳✶✳ ✸✾ ✷✳✸✳✹ P❛rt✐❝❧❡ ✜❧t❡r ❲❤✐❧❡ t❤❡ ▼❈▼❈ ❛♥❞ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠s ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥s ❝❛♥ ♣r♦❞✉❝❡ r❡❛s♦♥❛❜❧❡ ❡st✐♠❛t❡ ❢♦r t❤❡ ♣❛r❛♠❡t❡rs ❣✐✈❡♥ ❛ s❡t ♦❢ ♦❜s❡r✈❛t✐♦♥s✱ ✇❡ ❤❛✈❡ t♦ r❡❝♦♠♣✉t❡ t❤❡ ✇❤♦❧❡ s❡q✉❡♥❝❡ ♦❢ st❛t❡s ❢♦r s✉❜s❡q✉❡♥t ❛✈❛✐❧❛❜❧❡ ♦❜s❡r✈❛t✐♦♥ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛ s❛♠♣❧❡ ♦❢ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ♣❛rt✐❝❧❡ ✜❧t❡r✱ ✇❤✐❝❤ ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s ❛♥ ✐t❡r❛t✐✈❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ✭❉♦✉❝❡t ❡t ❛❧✳✱ ✷✵✵✵✮✱ ✐s ✐♥tr♦❞✉❝❡❞ t♦ ♦✈❡r❝♦♠❡ t❤✐s ♣r♦❜❧❡♠✳ ❯s✐♥❣ t❤❡ ♣❛rt✐❝❧❡ ✜❧t❡r✱ ✇❡ ❝❛♥ ❛❝❤✐❡✈❡ r❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♣♦st❡✲ r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❜❛s❡❞ ♦♥ t❤❡ s❡q✉❡♥t✐❛❧ ✐♠♣♦rt❛♥❝❡ s❛♠✲ ♣❧✐♥❣ ♠❡t❤♦❞✱ s✉❝❤ t❤❛t ✉♣❞❛t✐♥❣ ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r s❡q✉❡♥t✐❛❧❧② ❛✈❛✐❧❛❜❧❡ ♦❜✲ s❡r✈❛t✐♦♥s✳ ❚❤❡ ♣❛rt✐❝❧❡ ✜❧t❡r ✐♥❝r❡❛s❡s t❤❡ ❡✣❝✐❡♥❝② ♦❢ ❡st✐♠❛t✐♦♥ ✐♥ t❡r♠s ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡✱ ❛s ✐t ✇✐❧❧ ♥♦t ❜❡ ♥❡❝❡ss❛r② t♦ r❡♣r♦❝❡ss ❡①✐st✐♥❣ ❞❛t❛ ✐❢ ❛ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❜❡❝♦♠❡s ❛✈❛✐❧❛❜❧❡ ✭❆r✉❧❛♠♣❛❧❛♠ ❡t ❛❧✳✱ ✷✵✵✷✮✱ ❡s♣❡❝✐❛❧❧② ✐♥ ♦✉r ❝❛s❡ ✇❤❡r❡ ✇❡ ❛r❡ ♣❡r❢♦r♠✐♥❣ r❡❛❧✲t✐♠❡ ❡st✐♠❛t✐♦♥ ❢♦r t❤❡ ♣❛♥❞❡♠✐❝✳ ■♥ t❤❡ ♣❛rt✐❝❧❡ ✜❧t❡r✱ ✇❡ ❝❛♥ ❡st✐♠❛t❡ ❜♦t❤ t❤❡ st❛t❡s ❛♥❞ ♣❛r❛♠❡t❡rs ✐♥✲ ✈♦❧✈❡❞ ✐♥ t❤❡ ♣❛♥❞❡♠✐❝✳ ❚❤❡ s❡t ♦❢ ❣❡♥❡r❛t❡❞ ♣❛r❛♠❡t❡rs ❢♦r ❡❛❝❤ s✐♠✉❧❛t✐♦♥ ✈❛r✐❡s ❛❝r♦ss t✐♠❡ ❢♦r t❤❡ ♣❛♥❞❡♠✐❝✱ ✉♥❧✐❦❡ ✐♥ t❤❡ ▼❈▼❈ ❛♥❞ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ r♦✉t✐♥❡s✱ r❡♣r❡s❡♥t✐♥❣ t❤❡ ❝❤❛♥❣✐♥❣ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥✳ ❲❡ r❡✲ s❛♠♣❧❡ t❤❡ s❛♠♣❧❡❞ ♣♦st❡r✐♦r ✐♥ ❛ ❢✉rt❤❡r st❡♣ t♦ ✐♠♣r♦✈❡ t❤❡ ❡st✐♠❛t❡✱ s✉❝❤ t❤❛t ♣❛rt✐❝❧❡s ✇✐t❤ s♠❛❧❧ ✇❡✐❣❤ts ✇✐❧❧ ❜❡ ❡❧✐♠✐♥❛t❡❞ ❛♥❞ ♣❛rt✐❝❧❡s ✇✐t❤ ❧❛r❣❡ ✇❡✐❣❤ts ✇✐❧❧ ❜❡ ♠✉❧t✐♣❧✐❡❞ ✭P✐tt ❛♥❞ ❙❤❡♣❤❛r❞✱ ✶✾✾✾✮✳ ❚❤✐s ♣r♦❝❡❞✉r❡✱ ❦♥♦✇♥ ✹✵ ❛s s❛♠♣❧✐♥❣ ✐♠♣♦rt❛♥❝❡ r❡s❛♠♣❧✐♥❣✱ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐❧ ✐♥ t❤✐s s❡❝t✐♦♥✳ ❙❡q✉❡♥t✐❛❧ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ✭❙■❙✮ q(x1:t , θ)✱ ❚❤❡ s✉♣♣♦rt ♦❢ t❤❡ ♣r♦♣♦s❛❧ ❢✉♥❝t✐♦♥ st❛t❡s ❢r♦♠ t✐♠❡ 1 t♦ i✱ ❛♥❞ θ ✇❤❡r❡ x1:t ❞❡♥♦t❡s t❤❡ ❞❡♥♦t❡s t❤❡ ♣❛r❛♠❡t❡rs❀ ❤❛s t♦ ✐♥❝❧✉❞❡ t❤❡ s✉♣♣♦rt ♦❢ t❤❡ tr✉❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❢♦r ❛ s✉❝❝❡ss❢✉❧ ♣❛rt✐❝❧❡ ✜❧t❡r ✐♠✲ ♣❧❡♠❡♥t❛t✐♦♥✱ ❛s ♣❛rt✐❝❧❡s ❛r❡ ❞r❛✇♥ ❢r♦♠ t❤✐s ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ ♣r♦♣♦s❛❧ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s q(x1:t , θ) = q(x1:t−1 , θ)q(xt |x1:t−1 , θ) t = q(x1 , θ) q(xk |x1:k−1 , θ), k=2 ✇❤✐❝❤ s✐♠♣❧✐✜❡s t❤❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ✉s❡❞ ✐♥ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❛❢t❡r t❛❦✐♥❣ ✹✶ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❞❛t❛ D t♦ ωt = ω(x1:t , θ) = = = = = p(x1:t , θ|D1:t ) q(x1:t , θ) p(Dt , xt |xt−1 , θ)p(x1:t−1 , θ|D1:t−1 ) 1 · p(Dt |D1:t−1 ) q(x1:t , θ) p(Dt , xt |xt−1 , θ) p(x1:t−1 , θ|D1:t−1 ) · q(x1:t−1 , θ) p(Dt |D1:t−1 )q(xt |x1:t−1 , θ) p(Dt , xt |xt−1 , θ) ω(x1:t−1 , θ) · p(Dt |D1:t−1 )q(xt |x1:t−1 , θ) p(Dt , xt |xt−1 , θ) . ωt−1 · p(Dt |D1:t−1 )q(xt |x1:t−1 , θ) ❋r♦♠ t❤✐s ❡①♣r❡ss✐♦♥✱ ✇❡ s❡❡ t❤❛t t❤❡ ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤ts ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ r❡❝✉rs✐✈❡❧② ✐♥ t✐♠❡ ✇✐t❤♦✉t ❤❛✈✐♥❣ t♦ ♠♦❞✐❢② ♣❛st tr❛❥❡❝t♦r✐❡s✳ ❙❛♠♣❧✐♥❣ ✐♠♣♦rt❛♥❝❡ r❡s❛♠♣❧✐♥❣ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤ts ✇✐❧❧ ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ s❦❡✇❡❞ ♦✈❡r t✐♠❡ ❛s ✇❡ r❡✐t❡r❛t❡ t❤❡ ❛❧❣♦r✐t❤♠ ❢♦r ♥❡✇ ❛✈❛✐❧❛❜❧❡ ♦❜s❡r✈❛t✐♦♥ ✭❉♦✉❝❡t ❡t ❛❧✳✱ ✷✵✵✵✮✳ ❚❤❡r❡❢♦r❡✱ ✇❡✐❣❤t ❞❡❣❡♥❡r❛❝② ✇✐❧❧ ♦❝❝✉r ✐♥ t❤❡ ❧♦♥❣ r✉♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❛ s♠❛❧❧ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ♣❛rt✐❝❧❡s ✇✐❧❧ ❝♦♥t❛✐♥ ♥❡❛r❧② ❛❧❧ ♦❢ t❤❡ ✇❡✐❣❤ts ❛♥❞ ♦♥❧② ❢❡✇ ♣❛rt✐❝❧❡s ❤❛✈❡ ♥♦♥✲③❡r♦ ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤t✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ✐s ♥♦t r❡♣r❡s❡♥t❡❞ ❛❞❡q✉❛t❡❧②✳ ✹✷ ■♥st❡❛❞ ♦❢ ❛ss♦❝✐❛t✐♥❣ t❤❡ ♣❛rt✐❝❧❡s ✇✐t❤ t❤❡ ❞❡❣❡♥❡r❛t❡❞ ✇❡✐❣❤ts ❛s t❤❡ s②st❡♠ ❡✈♦❧✈❡s✱ ❛♥ ❛❞❞✐t✐♦♥❛❧ s❡❧❡❝t✐♦♥ r❡s❛♠♣❧✐♥❣ st❡♣ ✐s ✐♥tr♦❞✉❝❡❞ ❛❢t❡r t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤ts ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✳ ■♥ t❤❡ r❡s❛♠♣❧✐♥❣ st❡♣✱ ❛ ♥❡✇ s❡t ♦❢ ♣❛rt✐❝❧❡s ✐s s❛♠♣❧❡❞ ✇✐t❤ r❡♣❧❛❝❡♠❡♥t ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤ts ❞✐str✐❜✉t✐♦♥ ✭▲✐✉ ❛♥❞ ❈❤❡♥✱ ✷✵✵✹✮✳ ❚❤❡♥✱ ❡q✉❛❧ ✇❡✐❣❤ts ❛r❡ ❛ss✐❣♥❡❞ t♦ t❤❡ ♥❡✇ s❡t ♦❢ ♣❛rt✐❝❧❡s✳ ❚❤✐s ❛❞❞✐t✐♦♥❛❧ r❡s❛♠♣❧✐♥❣ st❡♣ ❛❧❧♦✇s ❣♦♦❞ ❡st✐♠❛t❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❛♥❞ st❛t❡s t♦ ❜❡ ❛♠♣❧✐✜❡❞ t♦ ♣r♦✈✐❞❡ ❛ ♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❡✈❡♥t✉❛❧❧②✳ ❑❡r♥❡❧ s♠♦♦t❤✐♥❣ ❆♥♦t❤❡r ♠❡❛s✉r❡ t❛❦❡♥ t♦ ❝♦✉♥t❡r ♣❛rt✐❝❧❡ ❞❡❣❡♥❡r❛❝② ✐s t❤❡ ❦❡r♥❡❧ s♠♦♦t❤✲ ✐♥❣ ♠❡t❤♦❞✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ❦❡r♥❡❧ s♠♦♦t❤✐♥❣ st❡♣ ✐s t♦ ♣❡rt✉r❜ t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ✐♥ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❛t t❤❡ r❡s❛♠♣❧✐♥❣ st❛❣❡✱ s✉❝❤ t❤❛t ♥❡✇ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ✐♥ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡s ❛r❡ ✐♥tr♦❞✉❝❡❞ ✭❚❤♦♠❛s ❡t ❛❧✳✱ ✷✵✵✺✮✳ ❚❤✐s ✐s ❛❝❤✐❡✈❡❞ ❜② ❞r❛✇✐♥❣ ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ t❤❡ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t❡ ♦❢ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥✳ ❑❡r♥❡❧ s♠♦♦t❤✐♥❣ ✐s ❛♣♣❧✐❡❞ t♦ t❤❡ ✈❡❝t♦r ♦❢ s✐♠✉❧❛t❡❞ ♣❛r❛♠❡t❡r ✈❛❧✉❡s t❤❡ ✈❡❝t♦r ♦❢ s♠♦♦t❤❡❞ s✐♠✉❧❛t❡❞ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t✐♠❡ t θt−1 t♦ ❣✐✈❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥ √ θt−1 = θ¯t−1 + h(θt−1 − θ¯t−1 ) + εt 1 − h2 ✇❤❡r❡ ✭✷✳✼✮ ✹✸ i = 1, . . . , k; 0 ≤ h ≤ 1; εt ∼ M V N (0, Σθt−1 ); θ¯t−1 = k j=1 (θj,t−1 , ωj,t ) ✐s t❤❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ ❛❧❧ s✐♠✉❧❛t❡❞ ♣❛r❛♠❡t❡r Σθt−1 ✈❛❧✉❡s❀ ❛♥❞ ✐s t❤❡ ✈❛r✐❛♥❝❡✲❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ❢♦r t❤❡ ♣❛r❛♠❡t❡r s❡t ❡t ❛❧✳✱ ✷✵✵✵✮✳ ✭❚r❡♥❦❡❧ ❚❤❡ ❞❡❣r❡❡ ♦❢ ♣❡rt✉r❜❛t✐♦♥ ✐s ❝♦♥tr♦❧❧❡❞ ❜② t❤❡ s♠♦♦t❤✐♥❣ ♣❛r❛♠❡t❡r h✱ ✇❤❡r❡ h = 1 ❝♦rr❡s♣♦♥❞s t♦ ♥♦ s♠♦♦t❤✐♥❣ ❛♥❞ ♠❛①✐♠✉♠ s♠♦♦t❤✐♥❣✳ ❆s ✐♥ ♦✉r ♣❛♣❡r ✭❖♥❣ s♠♦♦t❤✐♥❣ ♣❛r❛♠❡t❡r✱ h = 0.3 h = 0 ❝♦rr❡s♣♦♥❞s t♦ ❡t ❛❧✳✱ ✷✵✶✵✮✱ ✇❡ ✉s❡ ❛ ♠♦❞❡r❛t❡ ✐♥ t❤✐s ♣r♦❥❡❝t✳ ❆❧❣♦r✐t❤♠ ❚❤❡ ♣❛rt✐❝❧❡ ✜❧t❡r ❛❧❣♦r✐t❤♠ ✐s ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇s ✭❖♥❣ ✶✳ ■♥✐t✐❛❧✐③❛t✐♦♥ st❡♣✿ ❙❡t ❢♦r N t = 0✳ s0 ❛♥❞ ♣❛r❛♠❡t❡rs θ ♣❛rt✐❝❧❡s ❜② ❞r❛✇✐♥❣ ❢r♦♠ ♣r✐♦r ❞✐str✐❜✉t✐♦♥ ❜❛s❡❞ ♦♥ ♣r❡❧✐♠✐♥❛r② ✜♥❞✐♥❣s ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡✳ xnt ❉❡✜♥❡ ✐♥✐t✐❛❧ st❛t❡s ❡t ❛❧✳✱ ✷✵✶✵✮✿ = (st , θ) ❉❡♥♦t❡ ♣❛rt✐❝❧❡ n ❛t t✐♠❡ t ❛s ❛ ✈❡❝t♦r n ✇✐t❤ ❛ss♦❝✐❛t❡❞ ✇❡✐❣❤t wt ✳ ✷✳ ■t❡r❛t✐♦♥ st❡♣✿ ❈❛❧❝✉❧❛t❡ st❛t❡ ✈❛❧✉❡s ❢♦r ♥❡①t t✐♠❡ st❡♣ ❛❝❝♦r❞✐♥❣ t♦ s♣❡❝✐✜❡❞ ❙■❘ ♠♦❞❡❧✳ ❋♦r ❡❛❝❤ ♣❛rt✐❝❧❡ n✱ st+1 ✐s ❞r❛✇♥ ✉s✐♥❣ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥ ❢r♦♠ ✐ts ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ❣✐✈❡♥ ✸✳ ❲❡✐❣❤t✐♥❣ st❡♣✿ ❙❡t xˆnt+1 = (st+1 , θ)✳ Lnt+1 = f (yt+1 |ˆ xnt+1 ) ♦❜s❡r✈❡❞ ❞❛t❛ yt ✳ ❈❛❧❝✉❧❛t❡ ❚❤❡ ❧✐❦❡❧✐❤♦♦❞ ✐s ❝❛❧❝✉❧❛t❡❞ ❜② t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ xnt ✳ ✹✹ ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤ts n wˆt+1 = wtn Lnt+1 ♦❢ ❛❧❧ ♣❛rt✐❝❧❡s✱ ❛♥❞ ♥♦r♠❛❧✐③❡ ✐♠✲ ♣♦rt❛♥❝❡ ✇❡✐❣❤ts s♦ t❤❛t t❤❡② s✉♠ t♦ 1✳ ✹✳ ❘❡s❛♠♣❧✐♥❣ st❡♣✿ ❚❛❦❡ ❛ ✇❡✐❣❤t❡❞ s❛♠♣❧❡ ✇✐t❤ r❡♣❧❛❝❡♠❡♥t ♦♥ ❛❧❧ n ♣❛rt✐❝❧❡s ❛❝❝♦r❞✐♥❣ t♦ ✐♠♣♦rt❛♥❝❡ ✇❡✐❣❤ts ❝❛❧❝✉❧❛t❡❞✱ ❜② ❧❡tt✐♥❣ x ˆnt+1 = xˆm t+1 ✇❤❡r❡ m ✐s ❞r❛✇♥ ❢r♦♠ t❤❡ ✐♥t❡❣❡rs {1, 2, ..., N }✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣r♦♣♦rt✐♦♥❛❧ t♦ m ✳ wˆt+1 ❙❡t n = 1/N wt+1 s♦ t❤❛t ♣❛rt✐❝❧❡s ♥♦✇ ❤❛✈❡ ✉♥✐❢♦r♠ ✇❡✐❣❤ts✳ ✺✳ ❑❡r♥❡❧ s♠♦♦t❤✐♥❣ st❡♣✿ P❡rt✉r❜ t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♣❛rt✐❝❧❡s ❜② ❦❡r♥❡❧ s♠♦♦t❤✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❡q✉❛t✐♦♥ ✭✷✳✼✮✱ t♦ ✐♥✲ ❝r❡❛s❡ t❤❡ ❞✐✈❡rs✐t② ♦❢ ♣❛r❛♠❡t❡r ✈❛❧✉❡s✳ ◆♦✇ ✇❡ ❤❛✈❡ √ h(ˆ xnt+1 − x¯nt+1 ) + 1 − h2 ✳ ✻✳ ❘❡♣❡❛t st❡♣s ✷ t♦ ✺ ✉♣ t♦ t = 120 ❢♦r N = 10000 xnt+1 = x¯nt+1 + ♣❛rt✐❝❧❡s✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ♣❛rt✐❝❧❡ ✜❧t❡r ✐s ✐ts ✢❡①✐❜✐❧✐t② t♦ ❢♦❝✉s ✐♥❝r❡❛s✐♥❣❧② ♦♥ ♣r♦❜❛❜❧❡ r❡❣✐♦♥s ♦❢ st❛t❡✲s♣❛❝❡ ❛t ❡❛❝❤ s✉❜s❡q✉❡♥t ✐t❡r❛t✐♦♥✱ ❛♥❞ ✐t ❝❛♥ ✉♣✲ ❞❛t❡ t❤❡ s❛♠♣❧❡❞ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ r❡❝✉rs✐✈❡❧② ✇✐t❤♦✉t ❤❛✈✐♥❣ t♦ r❡❝♦♠✲ ♣✉t❡ ❢r♦♠ ❛❧❧ t❤❡ ♣❛st ♦❜s❡r✈❛t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✐t ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✐♥t❡♥s✐✈❡ ❛s t❤❡ ♣❛r❛♠❡t❡rs ❛♥❞ st❛t❡s ❛r❡ s❛♠♣❧❡❞ ❛t ❡❛❝❤ t✐♠❡ st❡♣✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡r❡ ❛r❡ ✐ss✉❡s ♦♥ ✇❡✐❣❤t ❞❡❣❡♥❡r❛❝② ❛♥❞ t❤❡ ❧♦ss ♦❢ ❞✐✈❡rs✐t② ♦❢ ♣❛rt✐❝❧❡s✳ ■♥ t❤✐s ♣r♦❥❡❝t✱ ✇❡ ✉s❡ t❤❡ ♣❛rt✐❝❧❡ ✜❧t❡r t♦ ♣❡r❢♦r♠ ♣r❡❞✐❝t✐♦♥s ❢♦r t❤❡ st♦❝❤❛st✐❝ ❙■❘ ♠♦❞❡❧✳ ❚❤❡ ✐♥❢♦r♠❛t✐✈❡ ♣r✐♦rs ❢♦r t❤❡ ♣❛r❛♠❡t❡rs ❛♥❞ ✐♥✐t✐❛❧ st❛t❡s ♦❢ t❤❡ st♦❝❤❛st✐❝ ❙■❘ ♠♦❞❡❧ ✉s❡❞ ✐♥ t❤❡ ❛❧❣♦r✐t❤♠s ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜❧❡ ✷✳✷✳ ✹✺ ❚❛❜❧❡ ✷✳✷✿ ■♥❢♦r♠❛t✐✈❡ ♣r✐♦rs ❢♦r t❤❡ st♦❝❤❛st✐❝ ❙■❘ ♠♦❞❡❧ ❙t♦❝❤❛st✐❝ ❙■❘ ♠♦❞❡❧ β ∼ γ ∼ δwds ∼ δsph ∼ φ ∼ S0 ∼ R0 ∼ ◆✭1, 1✮ ◆✭2, 0.5✮ ◆✭0.4, 0.2✮ ◆✭0.5, 0.1✮ ◆✭0.5, 0.25✮ ◆✭4560000, 2500✮ ◆(24000, 1200) [...]... ♣r✐♦rs ❢♦r t❤❡ ❘✐❝❤❛r❞s ♠♦❞❡❧ ❛♥❞ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧ ❘✐❝❤❛r❞s ♠♦❞❡❧ K r tm a δsph ∼ ∼ ∼ ∼ ∼ ◆ 20 0, 100✮ ◆✭0.06, 0.0 02 ❊①♣✭1/8✮ ◆✭5.5, 1✮ ❯❬0, 0.8❪ ❉❡t❡r♠✐♥✐st✐❝ ❙■❘ ♠♦❞❡❧ β ∼ γ ∼ δwds ∼ δsph ∼ φ ∼ S0 ∼ R0 ∼ ◆✭1.1, 3✮, ◆✭0.9, 3✮, ◆✭0 .2, 0.5✮✱ ◆✭0.5, 0.5✮, ◆✭0.8, 0 .2 , ◆✭4790000, 20 00✮✱ ◆(10000, 20 0) ❧❡❛❞ t♦ ❛ s❡t ♦❢ ✈❛❧✉❡s ✇❤✐❝❤ ❤❛s s♠❛❧❧❡r ✈❛r✐❛♥❝❡ ❛♥❞ ✐s ♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ t❤❡ tr✉❡ ♣❛r❛♠❡t❡rs✳... ❛❧❣♦r✐t❤♠s ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜❧❡ ✷✳✷✳ ✹✺ ❚❛❜❧❡ ✷✳✷✿ ■♥❢♦r♠❛t✐✈❡ ♣r✐♦rs ❢♦r t❤❡ st♦❝❤❛st✐❝ ❙■❘ ♠♦❞❡❧ ❙t♦❝❤❛st✐❝ ❙■❘ ♠♦❞❡❧ β ∼ γ ∼ δwds ∼ δsph ∼ φ ∼ S0 ∼ R0 ∼ ◆✭1, 1✮ ◆ 2, 0.5✮ ◆✭0.4, 0 .2 ◆✭0.5, 0.1✮ ◆✭0.5, 0 .25 ✮ ◆✭4560000, 25 00✮ ◆ (24 000, 120 0) ... ✐♥t❡❣❡rs {1, 2, , N }✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣r♦♣♦rt✐♦♥❛❧ t♦ m ✳ wˆt+1 ❙❡t n = 1/N wt+1 s♦ t❤❛t ♣❛rt✐❝❧❡s ♥♦✇ ❤❛✈❡ ✉♥✐❢♦r♠ ✇❡✐❣❤ts✳ ✺✳ ❑❡r♥❡❧ s♠♦♦t❤✐♥❣ st❡♣✿ P❡rt✉r❜ t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♣❛rt✐❝❧❡s ❜② ❦❡r♥❡❧ s♠♦♦t❤✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❡q✉❛t✐♦♥ ✭✷✳✼✮✱ t♦ ✐♥✲ ❝r❡❛s❡ t❤❡ ❞✐✈❡rs✐t② ♦❢ ♣❛r❛♠❡t❡r ✈❛❧✉❡s✳ ◆♦✇ ✇❡ ❤❛✈❡ √ h(ˆ xnt+1 − x¯nt+1 ) + 1 − h2 ✳ ✻✳ ❘❡♣❡❛t st❡♣s ✷ t♦ ✺ ✉♣ t♦ t = 120 ❢♦r N = 10000... yt ∼ P♦✐s(Nt ψt ), ✇❤❡r❡ ❞❛② t Nt ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❞♦❝t♦rs ✐♥ ♦✉r ♥❡t✇♦r❦ ✇❤♦ s✉❜♠✐tt❡❞ r❡♣♦rts ♦♥ ❛♥❞ ψt = δwds × (φ + It /20 84) δwds × δsph × (φ + It /20 84) ✇❤❡♥ ✇❤❡♥ t t ✐s ✇❡❡❦❞❛②s ♦r ❙❛t✉r❞❛②s✱ ✐s ❙✉♥❞❛②s ♦r ♣✉❜❧✐❝ ❤♦❧✐❞❛②s✳ ❚❤❡ ❡st✐♠❛t❡❞ ♥✉♠❜❡r ♦❢ ❞♦❝t♦rs ✐♥ ❙✐♥❣❛♣♦r❡ ✐s 20 84✱ ❛♥❞ ✐s ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ t❤❡ ❡st✐♠❛t❡❞ ♥✉♠❜❡r ♦❢ ❣❡♥❡r❛❧ ♣❤②s✐❝✐❛♥s ♦r ❢❛♠✐❧② ❞♦❝t♦rs ✐♥ ❙✐♥✲ ❣❛♣♦r❡ ✐♥ ✷✵✵✾✱ ✇❤✐❝❤ ✇❛s... ✜❧t❡r ✐♠✲ ♣❧❡♠❡♥t❛t✐♦♥✱ ❛s ♣❛rt✐❝❧❡s ❛r❡ ❞r❛✇♥ ❢r♦♠ t❤✐s ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ ♣r♦♣♦s❛❧ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s q(x1:t , θ) = q(x1:t−1 , θ)q(xt |x1:t−1 , θ) t = q(x1 , θ) q(xk |x1:k−1 , θ), k =2 ✇❤✐❝❤ s✐♠♣❧✐✜❡s t❤❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ✉s❡❞ ✐♥ ✐♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❛❢t❡r t❛❦✐♥❣ ✹✶ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❞❛t❛ D t♦ ωt = ω(x1:t , θ) = = = = = p(x1:t , θ|D1:t ) q(x1:t , θ) p(Dt , xt |xt−1 , θ)p(x1:t−1 , θ|D1:t−1... s♠♦♦t❤✐♥❣ ✐s ❛♣♣❧✐❡❞ t♦ t❤❡ ✈❡❝t♦r ♦❢ s✐♠✉❧❛t❡❞ ♣❛r❛♠❡t❡r ✈❛❧✉❡s t❤❡ ✈❡❝t♦r ♦❢ s♠♦♦t❤❡❞ s✐♠✉❧❛t❡❞ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t✐♠❡ t θt−1 t♦ ❣✐✈❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥ √ θt−1 = θ¯t−1 + h(θt−1 − θ¯t−1 ) + εt 1 − h2 ✇❤❡r❡ ✭✷✳✼✮ ✹✸ i = 1, , k; 0 ≤ h ≤ 1; εt ∼ M V N (0, Σθt−1 ); θ¯t−1 = k j=1 (θj,t−1 , ωj,t ) ✐s t❤❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ ❛❧❧ s✐♠✉❧❛t❡❞ ♣❛r❛♠❡t❡r Σθt−1 ✈❛❧✉❡s❀ ❛♥❞ ✐s t❤❡ ✈❛r✐❛♥❝❡✲❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ... 0 .2 0.4 0.6 0.8 1.0 −5 15 10 15 10 15 t Effects of a on C(t) a=0.5 a=1 a =2 −5 Effects of K on I(t) 0.0 0.1 0 .2 0.3 0.4 0.5 t Effects of r on I(t) r=0.5 r=1 r =2 0.1 0 .2 0.3 0.4 K=0.5 K=1 K =2 0.0... Effects of tm on I(t) 15 tm=0 tm=5 tm=10 −5 10 15 10 15 t Effects of a on I(t) a=0.5 a=1 a =2 0 .2 0.1 0.0 0.0 0.1 0 .2 0.3 10 t Effects of tm on C(t) −5 I(t) r=0.5 r=1 r =2 0.3 C(t) 0.0 0 .2 0.4 0.6... ❙t♦❝❤❛st✐❝ ❙■❘ ♠♦❞❡❧ β ∼ γ ∼ δwds ∼ δsph ∼ φ ∼ S0 ∼ R0 ∼ ◆✭1, 1✮ ◆ 2, 0.5✮ ◆✭0.4, 0 .2 ◆✭0.5, 0.1✮ ◆✭0.5, 0 .25 ✮ ◆✭4560000, 25 00✮ ◆ (24 000, 120 0)