Plausibility of big shocks within a linear state space setting with skewness

Munich Personal RePEc Archive

Plausibility of big shocks within a linear

state space setting with skewness

Koloch, Grzegorz

Warsaw School of Economics

25 January 2016

Online at

https://mpra.ub.uni-muenchen.de/69001/

P❧❛✉s✐❜✐❧✐t② ♦❢ ❜✐❣ s❤♦❝❦s ✇✐t❤✐♥ ❛ ❧✐♥❡❛r st❛t❡ s♣❛❝❡

s❡tt✐♥❣ ✇✐t❤ s❦❡✇♥❡ss

∗

●r③❡❣♦r③ ❑♦❧♦❝❤

❏❛♥✉❛r② ✷✺✱ ✷✵✶✻

❆❜str❛❝t

■♥ t❤✐s ♣❛♣❡r ✇❡ ♣r♦✈✐❞❡ ❢♦r♠✉❧❛❡ ❢♦r ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥✱ ✜❧tr❛t✐♦♥ ❞❡♥s✐t✐❡s ❛♥❞ ♣r❡❞✐❝t✐♦♥ ❞❡♥s✐t✐❡s ♦❢ ❧✐♥❡❛r st❛t❡ s♣❛❝❡ ♠♦❞❡❧ ✐♥ ✇❤✐❝❤ s❤♦❝❦s ❛r❡ ❛❧❧♦✇❡❞ t♦ ❜❡ s❦❡✇❡❞✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ✇♦r❦ ✇✐t❤ t❤❡ ❝❧♦s❡❞ s❦❡✇ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✭❝s♥✮ ✐♥✲ tr♦❞✉❝❡❞ ✐♥ ●♦♥③á❧❡③✲❋❛rí❛s ❡t ❛❧✳ ✭✷✵✵✹✮✱ ✇❤✐❝❤ ♥❡sts ❛ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡✳ ❈❧♦s✉r❡ ♦❢ t❤❡ ❝s♥ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❛❧❧ ♥❡❝❡ss❛r② tr❛♥s❢♦r♠❛✲ t✐♦♥s ✐♥ t❤❡ st❛t❡ s♣❛❝❡ s❡tt✐♥❣ ✐s ❣✉❛r❛♥t❡❡❞ ❜② ❛ s✐♠♣❧❡ st❛t❡ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ✇❤✐❝❤ ❞♦❡s ♥♦t ✐♥✢✉❡♥❝❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥✳ Pr❡s❡♥t❡❞ ❢♦r♠✉❧❛❡ ❛❧❧♦✇ ❢♦r ❡st✐♠❛t✐♦♥✱ ✜❧tr❛t✐♦♥ ❛♥❞ ♣r❡❞✐❝t✐♦♥ ♦❢ ✈❡❝t♦r ❛✉t♦r❡❣r❡ss✐♦♥s ❛♥❞ ✜rst ♦r❞❡r ♣❡rt✉r❜❛t✐♦♥s ♦❢ ❉❙●❊ ♠♦❞❡❧s ✇✐t❤ s❦❡✇❡❞ s❤♦❝❦s✳ ❚❤✐s ❛❧❧♦✇s t♦ ❛ss❡ss ❛s②♠♠❡tr✐❡s ✐♥ s❤♦❝❦s✱ ♦❜s❡r✈❡❞ ❞❛t❛✱ ✐♠♣✉❧s❡ r❡s♣♦♥s❡s ❛♥❞ ❢♦r❡❝❛sts ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s✳ ❙♦♠❡ ♦❢ t❤❡ ❛❞✈❛♥t❛❣❡s ♦❢ ✉s✐♥❣ t❤❡ ♦✉t❧✐♥❡❞ ❛♣♣r♦❛❝❤ ♠❛② ✐♥✈♦❧✈❡ ❝❛♣t✉r✲ ✐♥❣ ❛s②♠♠❡tr✐❝ ✐♥✢❛t✐♦♥ r✐s❦s ✐♥ ❝❡♥tr❛❧ ❜❛♥❦s ❢♦r❡❝❛sts ♦r ♣r♦❞✉❝✐♥❣ ♠♦r❡ ♣❧❛✉s✐❜❧❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❞❡❡♣ ❜✉t r❛r❡ r❡❝❡ss✐♦♥❛r② ❡♣✐s♦❞❡s ✇✐t❤ ❉❙●❊✴❱❆❘ ✜❧tr❛t✐♦♥✳ ❊①✲ ❡♠♣❧❛r② ❡st✐♠❛t✐♦♥ r❡s✉❧ts ❛r❡ ♣r♦✈✐❞❡❞ ✇❤✐❝❤ s❤♦✇ t❤❛t ✇✐t❤✐♥ ❛ ❧✐♥❡❛r s❡tt✐♥❣ ✇✐t❤ s❦❡✇♥❡ss ❢r❡q✉❡♥❝② ♦❢ ❜✐❣ s❤♦❝❦s ❝❛♥ ❜❡ r❛t❤❡r ♣❧❛✉s✐❜❧② ✐❞❡♥t✐✜❡❞✳

❑❡②✇♦r❞s✿ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥✱ ❙t❛t❡ ❙♣❛❝❡ ▼♦❞❡❧s✱ ❈❧♦s❡❞ ❙❦❡✇❡❞ ◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥✱ ❉❙●❊✱ ❱❆❘✳

❏❊▲✿ ❈✺✶✱ ❈✶✸✱ ❊✸✷

∗_{❚❤✐s r❡s❡❛r❝❤ ✇❛s ✜♥❛♥❝❡❞ ❜② t❤❡ ◆❛t✐♦♥❛❧ ❙❝✐❡♥❝❡ ❈❡♥t❡r ❣r❛♥t ◆♦✳ ✷✵✶✷✴✵✼✴❊✴❍❙✹✴✵✶✵✽✵✳} †_{❲❛rs❛✇ ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s✱ ■♥st✐t✉t❡ ♦❢ ❊❝♦♥♦♠❡tr✐❝s✱ ❉❡❝✐s✐♦♥ ❆♥❛❧②s✐s ❛♥❞ ❙✉♣♣♦rt ❯♥✐t✱ ❆❧✳}

◆✐❡♣♦❞❧❡❣❧♦s❝✐ ✶✻✹✱ ✵✵✲✾✺✵ ❲❛rs❛✇✱ P♦❧❛♥❞✳

✶ ■♥tr♦❞✉❝t✐♦♥

❆ ♣r❛❝t✐❝❛❧ ♠♦t✐✈❛t✐♦♥ ❞❡r✐✈❡s ❢r♦♠ ❛♥ ♦❜s❡r✈❛t✐♦♥ ❡♠♣❤❛s✐s❡❞ ❡✳❣✳ ❜② ❈úr❞✐❛✱ ❉❡❧ ◆❡❣r♦ ❛♥❞ ●r❡❡♥✇❛❧❞ ✭✷✵✶✹✮✱ t❤❛t t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ✐❞❡♥t✐✜❡❞ s❤♦❝❦s ✇❤✐❝❤ ❝♦✉❧❞ ❧❡❛❞ t♦ s✐❣♥✐✜❝❛♥t r❡❝❡ss✐♦♥s ✐♥ ✉s✉❛❧ ❧✐♥❡❛r st❛t❡ s♣❛❝❡ s❡tt✐♥❣s ✇✐t❤ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ s❤♦❝❦s✱ ✇✐t❤✐♥ ✇❤✐❝❤ ✜rst ♦r❞❡r ♣❡rt✉r❜❛t✐♦♥s ♦❢ ❉❙●❊ ❡❝♦♥♦♠✐❡s ❛r❡ ♦❢t❡♥ ❡st✐♠❛t❡❞✱ ❛r❡ ✈❡r② ✐♠♣r♦❜❛❜❧❡✳ ❋♦r ✐♥st❛♥❝❡✱ ❛s ✇❡ s❤♦✇ ✐♥ s❡❝t✐♦♥ ✺ ✉s✐♥❣ ❞❛t❛ ✇❤✐❝❤ ❝♦♥t❛✐♥ t❤❡ r❡❝❡♥t ❝r✐s✐s✱ ✐♥ ❛ ❙❝❤♦r❢❤❡✐❞❡ ✭✷✵✵✵✮ ❡❝♦♥♦♠② ✇✐t❤ s❤♦❝❦s ✜tt❡❞ ✇✐t❤ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ ♣r♦❜❛❜✐❧✐t② ♦❢ ❚❋P s❤♦❝❦s ❜❡❧♦✇ t❤❡ ✶st ♣❡r❝❡♥t✐❧❡ ✐s ❛❜♦✉t2_×10−4_{✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t✱ ❢♦r q✉❛rt❡r❧② ❞❛t❛✱ s✉❝❤ ❛ s❤♦❝❦ s❤♦✉❧❞ ❤❛♣♣❡♥} ♦♥❝❡ ❡✈❡r② ✶✷✺✵ ②❡❛rs✳ ❈úr❞✐❛✱ ❉❡❧ ◆❡❣r♦ ❛♥❞ ●r❡❡♥✇❛❧❞ ✭✷✵✶✹✮ ♣r♦♣♦s❡ ❛♥ ❛♣♣r♦❛❝❤ ✐♥ ✇❤✐❝❤ ❧❛r❣❡ s❤♦❝❦s ❝❛♥ ♦❝❝✉r ❛❢t❡r r❡♣❧❛❝✐♥❣ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ❜② ❛ ❙t✉❞❡♥t✬s

t ❞✐str✐❜✉t✐♦♥✳ ❚❤❡② ❛❧s♦ ♣r♦✈✐❞❡ ❛ str♦♥❣ ❡✈✐❞❡♥❝❡ t❤❛t ♥♦r♠❛❧✐t② ♦❢ s❤♦❝❦s ✐s

❝♦✉♥t❡r❢❛❝t✉❛❧✳ ❚❤❡✐r ❛♣♣r♦❛❝❤ ❛❧❧♦✇s ❢♦r ❡①❝❡ss ❦✉rt♦s✐s✱ ❜✉t ♥♦t ❢♦r ❢♦r s❦❡✇♥❡ss✳ ❆s t❤❡② ♣♦✐♥t ♦✉t✱ s❦❡✇♥❡ss ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ s❤♦❝❦s ♠❛② ❛❧s♦ ❜❡ ❛ s❛❧✐❡♥t ❢❡❛t✉r❡ ♦❢ ✐t✱ ❛♥❞ ♥♦t ❛❧❧♦✇✐♥❣ ❢♦r s❦❡✇♥❡ss ♠❛② ❧❡❛❞ t♦ ✉♥❞❡r❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ❢❛t t❛✐❧s ❞✉r✐♥❣ r❡❝❡ss✐♦♥s✳

■♥ t❤✐s ♣❛♣❡r ✇❡ ♣r♦✈✐❞❡ ❛ q✉❛s✐✲♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡ ❢♦r ❛ ❧✐♥❡❛r st❛t❡ s♣❛❝❡ ♠♦❞❡❧ ✇✐t❤ s❦❡✇❡❞ s❤♦❝❦s ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ❡q✉❛t✐♦♥✶_{✳ ❆s ❛ r❡s✉❧t✱}

♣r♦❜❛❜✐❧✐t② ♦❢ ❜✐❣ ♥❡❣❛t✐✈❡ ❚❋P s❤♦❝❦s✱ s✉❝❤ ❛s t❤♦s❡ ✐❞❡♥t✐✜❡❞ ❞✉r✐♥❣ t❤❡ r❡❝❡♥t r❡❝❡ss✐♦♥✱ ❣❡ts r❡❞✉❝❡❞ t♦ ❛❜♦✉t0.5%✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t s✉❝❤ s❤♦❝❦s s❤♦✉❧❞ ❤❛♣♣❡♥

♦♥❝❡ ❡✈❡r②50 ②❡❛rs✱ ✇❤✐❝❤ s❡❡♠s t♦ ❜❡ ❛ ♠♦r❡ ♣❧❛✉s✐❜❧❡ ❢r❡q✉❡♥❝②✳

❙✐♥❝❡ st❛t❡ s♣❛❝❡ ❢♦r♠ ❝♦rr❡s♣♦♥❞s t♦ s♦♠❡ ♣♦♣✉❧❛r ♠❛❝r♦❡❝♦♥♦♠✐❝ t♦♦❧s ✕ ❱❆❘s ❛♥❞ r❡❞✉❝❡❞ ❢♦r♠ ✜rst ♦r❞❡r ♣❡rt✉r❜❛t✐♦♥s ♦❢ ❉❙●❊ ♠♦❞❡❧s✱ ✇❡ ❛❧❧♦✇ ❢♦r r❡♣r❡s❡♥t✲ ✐♥❣ ❞❛t❛ s❦❡✇♥❡ss ✇❤❡♥ ✉s✐♥❣ s✉❝❤ ♠♦❞❡❧s✱ ✇✐t❤✐♥ ❛ ✉s✉❛❧ ❧✐♥❡❛r ❢r❛♠❡✇♦r❦✳

■♥ ♣❛rt✐❝✉❧❛r✱ ❡st✐♠❛t✐♦♥ ♦❢ ❱❆❘s ❛♥❞ ❉❙●❊ ♠♦❞❡❧s ✇✐t❤ s❦❡✇❡❞ s❤♦❝❦s ❜❡❝♦♠❡s ❛♥❛❧♦❣✐❝❛❧ t♦ ❑❛❧♠❛♥ ✜❧t❡r ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ❛♥❞ s❦❡✇♥❡ss ♦❢ ♦❜s❡r✈✲ ❛❜❧❡s✱ st❛t❡s ❛♥❞ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ♦❢ ❢♦r❡❝❛sts ❝❛♥ ❜❡ st❛t✐st✐❝❛❧❧② r❡♣r❡s❡♥t❡❞ ❛♥❞ ✐❞❡♥t✐✜❡❞ ✇✐t❤♦✉t r❡❢❡rr✐♥❣ t♦ ♥♦♥❧✐♥❡❛r✐t✐❡s ✐♥ t❤❡ ♠♦❞❡❧✳ ❙♦♠❡ ♦❢ t❤❡ ❛❞✈❛♥t❛❣❡s ♠❛② ✐♥✈♦❧✈❡ ❝❛♣t✉r✐♥❣ ❛s②♠♠❡tr✐❝ ✐♥✢❛t✐♦♥ r✐s❦s ✐♥ ❝❡♥tr❛❧ ❜❛♥❦s ❢♦r❡❝❛sts ♦r ♣r♦✲ ❞✉❝✐♥❣ ♠♦r❡ ♣❧❛✉s✐❜❧❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❞❡❡♣ ❜✉t r❛r❡ ❝♦♥tr❛❝t✐♦♥s ✇✐t❤ ❧✐♥❡❛r ❱❆❘

✶_{▼❡❛s✉r❡♠❡♥t s❤♦❝❦s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ❜✉t ❡①t❡♥s✐♦♥ t♦ s❦❡✇❡❞ ♠❡❛s✉r❡♠❡♥t}

❡rr♦rs ✐s str❛✐❣❤t❢♦r✇❛r❞✳

♦r ❉❙●❊ s♣❡❝✐✜❝❛t✐♦♥s✳

■♥ t❤✐s ♣❛♣❡r ✇❡ ❢♦❝✉s ♦♥ ❛♥ ❡❝♦♥♦♠❡tr✐❝ ♣❛rt ♦❢ t❤❡ ❛❣❡♥❞❛✱ ✇❤✐❧❡ ♣r♦✈✐❞✐♥❣ ♦♥❧② ❡①❡♠♣❧❛r② r❡s✉❧ts ❝♦♥❝❡r♥✐♥❣ ❡♠♣✐r✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✱ ✇❤✐❝❤ ❛r❡ ❧❡❢t ❢♦r ❢✉rt❤❡r st✉❞②✳

▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧✿

yt = F xt+Hut,

xt = Axt₋1+Bξt,

ut ∼ N(0,Ψu),

ξt ∼ pξ(θξ)

x0 ∼ N(µx0,Ψx0)

❢♦r t_{∈ T} ={1,2, ..., T}✱ ✇❤❡r❡xt∈Rp ❛♥❞ yt ∈Rn ❞❡♥♦t❡ st❛t❡s ❛♥❞ ♦❜s❡r✈❛❜❧❡s

r❡s♣❡❝t✐✈❡❧②✱ ξt _∈ Rnξ ❛♥❞ _{ut ∈ R}nu✱ _n

ξ, nu ≥ 1✱ ❞❡♥♦t❡ s❤♦❝❦s ❛♥❞ ♠❡❛s✉r❡♠❡♥t

❡rr♦rs r❡s♣❡❝t✐✈❡❧②✱ Rp×p _{∋ A 6= 0✱ B ∈ R}p×nξ✱ _F ∈ _Rn×p✱ _H ∈ _Rp×nu✱ ♠♦r❡♦✈❡r

Ψu ∈Rnu×nu✱Ψx0 ∈R

p_×p_{✱ ❛♥❞|Ψ}

u|,|Ψx0| ≥0✳ ❋✐♥❛❧❧②✱pξ(θξ)❞❡♥♦t❡s ❛ ♣r♦❜❛❜✐❧✐t②

❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ♠❛rt✐♥❣❛❧❡ ❞✐✛❡r❡♥❝❡ s❤♦❝❦sξt✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ ❛ ✈❡❝t♦r ♦❢

♣❛r❛♠❡t❡rsθξ✳ ❙❤♦❝❦s ✐♥ t❤❡ ♠❡❛s✉r❡♠❡♥t ❡q✉❛t✐♦♥ ❛r❡ ❛ss✉♠❡❞ t♦ ❢♦❧❧♦✇ ❛ ♥♦r♠❛❧

❞✐str✐❜✉t✐♦♥ ❢♦r s✐♠♣❧✐❝✐t②✱ t❤❡② ❝♦✉❧❞ ❜❡ s❦❡✇❡❞ ❛s ✇❡❧❧✳ ❚❤❡ s❛♠❡ r❡♠❛r❦ ❛♣♣❧✐❡s t♦ ✐♥✐t✐❛❧ st❛t❡sx0✳

❆ ✉s✉❛❧ ❛ss✉♠♣t✐♦♥ ✐s t❤❛tpξ ✐s ❛ ♠✉❧t✐✈❛r✐❛t❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ ✐♥❞❡♣❡♥❞❡♥t

❛❝r♦ss ✐ts ❞✐♠❡♥s✐♦♥s✳ ■♥ s✉❝❤ ❛ ❝❛s❡ ❑❛❧♠❛♥ ✜❧t❡r ❝♦♥st✐t✉t❡s ❛♥ ♦♣t✐♠❛❧ ✜❧tr❛t✐♦♥ ♣r♦❝❡❞✉r❡✷_{✱ s❡❡ ❙✐♠♦♥ ✭✷✵✵✻✮✳ ■❢ ♥♦r♠❛❧✐t② ❛ss✉♠♣t✐♦♥ ✐s r❡❧❛①❡❞✱ ❑❛❧♠❛♥ ✜❧t❡r}

r❡♠❛✐♥s ❛♥ ♦♣t✐♠❛❧ ❧✐♥❡❛r ✜❧t❡r✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ r❡❧❛① ♥♦r♠❛❧✐t② ❛ss✉♠♣t✐♦♥ ❛♥❞ ❛ss✉♠❡ t❤❛t ❡❧❡♠❡♥ts ♦❢ ξt ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ❜✉t✱ ❢♦r s♦♠❡ ✈❛❧✉❡s ♦❢ θξ✱ ♣r♦❜❛❜✐❧✐t②

❞❡♥s✐t② ❢✉♥❝t✐♦♥ pξ ✐s s❦❡✇❡❞ ✭❛s②♠♠❡tr✐❝✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❛ss✉♠❡ t❤❛t s❤♦❝❦s

ξt ❢♦❧❧♦✇ ❛ ❝❧♦s❡❞ s❦❡✇✲ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✭csn ❤❡♥❝❡❢♦rt❤✮✱ ✇❤✐❝❤ ♥❡sts t❤❡ ♥♦r✲

♠❛❧ ❞✐str✐❜✉t✐♦♥ ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡✱ s❡❡ ●♦♥③á❧❡③✲❋❛rí❛s ❡t ❛❧✳ ✭✷✵✵✹✮ ♦r ●❡♥t♦♥ ❡t ❛❧✳ ✭✷✵✵✹✮✳ ❚❤❡ csn ❞✐str✐❜✉t✐♦♥ ✐s ❝❤♦s❡♥✱ ❜❡❝❛✉s❡ ✐t ✐s ❝❧♦s❡❞ ✉♥❞❡r ❛❧♠♦st ❛❧❧

tr❛♥s❢♦r♠❛t✐♦♥s ✐♠♣♦s❡❞ ♦♥ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ st❛t❡ s♣❛❝❡ s❡tt✐♥❣✸_{✳ ■t ✐s ♥♦t ❝❧♦s❡❞✱}

❤♦✇❡✈❡r✱ ✉♥❞❡r r❡❞✉❝❡❞ r❛♥❦ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ❛♥❞ ✇❡ ✇❛♥t t♦ ❛❧❧♦✇ ❢♦r r❛♥❦

✷_{■♥ t❤❡ s❡♥s❡ t❤❛t ✐t ♠✐♥✐♠✐③❡s t❤❡ tr❛❝❡ ♦❢ ♦♥❡✲st❡♣ ❛❤❡❛❞ ✐♥✲s❛♠♣❧❡ ❢♦r❡❝❛st ❡rr♦rs ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✳} ✸_{❉❡t❛✐❧s ❛r❡ ♣r♦✈✐❞❡❞ ✐♥ s❡❝t✐♦♥ ✷✳}

❞❡✜❝✐❡♥❝② ♦❢ tr❛♥s✐t✐♦♥ ♠❛tr✐①A✹✳ ❚❤✐s ❝❛s❡ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ♦❜st❛❝❧❡ ✐♥ ▼▲ ❡st✐✲

♠❛t✐♦♥ s✐♥❝❡ s✐♥❣✉❧❛r✐t② ♦❢A♣r❡❝❧✉❞❡s ♣r♦♣❛❣❛t✐♦♥ ♦❢ t❤❡ csn❞✐str✐❜✉t✐♦♥ t❤r♦✉❣❤

t❤❡ st❛t❡ s♣❛❝❡ s❡tt✐♥❣✳ ❲❡ ✉s❡ ❛ s✐♠♣❧❡ st❛t❡ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ t♦ ❞❡❛❧ ✇✐t❤ t❤✐s ✐ss✉❡✱ s♦ t❤❛t t❤❡csn❞✐str✐❜✉t✐♦♥ ♣r♦♣❛❣❛t❡s t❤r♦✉❣❤ t❤❡ st❛t❡ s♣❛❝❡

s❡tt✐♥❣✳

❚♦ ❛❧❧♦✇ ❢♦r ♣r❡❞✐❝t✐♦♥✱ ✜❧tr❛t✐♦♥ ❛♥❞ ❡st✐♠❛t✐♦♥✱ ✇❡ ♣r♦✈✐❞❡ ❢♦r♠✉❧❛❡ ❢♦rp(yt|Yt−1)✱

p(xt|Yt₋1)✱p(xt|Yt)❛♥❞ ❢♦r t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥p(θ|Yt)✱ ✇❤❡r❡θ= (θF, θH, θu, θA, θF, θξ, θx0)

❛♥❞ Yt ={yt, yt−1, . . . , y1}✳ ❆❧t❤♦✉❣❤ ❡①❛❝t ❢♦r♠✉❧❛❡ ❛r❡ ♣r♦✈✐❞❡❞✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ✐♥✈♦❧✈❡s ♣♦t❡♥t✐❛❧❧② ✈❛r② ❧❛r❣❡ s❝❛❧❡ ♥♦r♠❛❧ ✐♥t❡❣r❛❧s✱ ♣r❛❝✲ t✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ✇❤✐❝❤ ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✐♠♣♦ss✐❜❧❡✳ ❚♦ ♠❛❦❡ t❤❡♠ ♦♣❡r❛t✐♦♥❛❧ ✇❡ ❢❛❝t♦r ♠✉❧t✐✈❛r✐❛t❡ ✐♥t❡❣r❛❧s ✐♥t♦ t❤❡ ♣r♦❞✉❝ts ♦❢ ✉♥✐✈❛r✐❛t❡ ♦♥❡s✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ♦✉t❧✐♥❡❞ ♣r♦❝❡❞✉r❡ ❛ q✉❛s✐✲♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ♦♥❡✳ ❈♦♥s❡q✉❡♥❝❡s ♦❢ t❤✐s ❛♣✲ ♣r♦①✐♠❛t✐♦♥ ❢♦r ❡st✐♠❛t✐♦♥ ❛r❡ ✉♥❦♥♦✇♥ ✐♥ ❛ ❣❡♥❡r❛❧ ❝❛s❡✱ ❤♦✇❡✈❡r ♦✉r ♥✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts s❤♦✇✱ t❤❛t ✐♥ ❝❛s❡ ♦❢ ❉❙●❊ ❡st✐♠❛t✐♦♥✱ t❤✐s ✐s ❛ ✈❡r② r❡❛s♦♥❛❜❧❡ ❛♣✲ ♣r♦①✐♠❛t✐♦♥✱ s✐♥❝❡ t❤❡ ♠✉❧t✐✈❛r✐❛t❡ ♥♦r♠❛❧ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s t❤❛t ❛♣♣❡❛r ❞✉r✐♥❣ ❝❛❧❝✉❧❛t✐♦♥s ❛r❡ ❛❧♠♦st ♣❡r❢❡❝t❧② ✐♥❞❡♣❡♥❞❡♥t ❛❝r♦ss t❤❡ ❞✐♠❡♥s✐♦♥s✳ ❘❡♠❛✐♥❞❡r ♦❢ t❤❡ ♣❛♣❡r ✐s ❛rr❛♥❣❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ s❡❝t✐♦♥ ✷ t❤❡ ❝❧♦s❡❞ s❦❡✇✲ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ✐ts ❜❛s✐❝ ♣r♦♣❡rt✐❡s ❛r❡ ❞✐s❝✉ss❡❞✳ ❙❡❝t✐♦♥ ✸ ♣r♦✈✐❞❡s t❤❡ ✜❧t❡r ❛♥❞ s❡❝t✐♦♥ ✹ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥✳ ■♥ s❡❝t✐♦♥ ✺ ✇❡ s❤♦✇ s♦♠❡ ❡st✐♠❛t✐♦♥ r❡s✉❧ts ✉s✐♥❣ ❛ ❉❙●❊ ❡❝♦♥♦♠② ♦❢ ❙❝❤♦r❢❤❡✐❞❡ ✭✷✵✵✵✮✳

✷ ❚❤❡ s❦❡✇❡❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ♣r♦✈✐❞❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝❧♦s❡❞ s❦❡✇✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ ❞✐s❝✉ss ✐ts ❜❛s✐❝ ♣r♦♣❡rt✐❡s ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❞✐s❝✉ss ✐ts ❝❧♦s✉r❡ ✉♥❞❡r ❛r❜✐tr❛r② ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s✳

✷✳✶ ❉❡✜♥✐t✐♦♥

▲❡t ✉s ❞❡♥♦t❡ ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦❢ ❛p✲❞✐♠❡♥s✐♦♥❛❧ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♠❡❛♥✺

µ ❛♥❞ ❛ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ ❜② φp(z;µ,Σ)✳ ▲❡t ✉s ❛❧s♦ ❞❡♥♦t❡ ❛

✹_{❲❤✐❝❤ ❝❛♥ ❜❡ t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ st❛t❡ s♣❛❝❡ ❢♦r♠ r❡♣r❡s❡♥ts ❛ ✜rst ♦r❞❡r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ❉❙●❊}

♠♦❞❡❧✳

✺_{❆❧❧ ✈❡❝t♦rs ❛r❡ ❝♦❧✉♠♥ ✈❡❝t♦rs ✐♥ t❤✐s ♣❛♣❡r✳}

❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ❛q✲❞✐♠❡♥s✐♦♥❛❧ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♠❡❛♥µ

❛♥❞ ♥♦♥♥❡❣❛t✐✈❡ ❞❡✜♥✐t❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①Σ❜②Φq(z;µ,Σ)✳ ❋♦r q >1❢✉♥❝t✐♦♥ Φq

❞♦❡s ♥♦t ❤❛✈❡ ❛♥ ❛♥❛❧②t✐❝❛❧ ❡①♣r❡ss✐♦♥✳ ▲❡t r(M) ❞❡♥♦t❡ r❛♥❦ ♦❢ ❛ ♠❛tr✐① M✳ ❲❡

✇✐❧❧ ♥♦✇ ❞❡✜♥❡ t❤❡ ❝❧♦s❡❞ s❦❡✇✲♥♦r♠❛❧✱ ♣♦ss✐❜❧② s✐♥❣✉❧❛r✱ ❞✐str✐❜✉t✐♦♥ ❜② ♠❡❛♥s ♦❢ t❤❡ ♠♦♠❡♥t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✭♠❣❢✮ ❛♥❞ t❤❡♥✱ ✉♥❞❡r ♥♦♥s✐♥❣✉❧❛r✐t② ♦❢ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✱ ❜② ♠❡❛♥s ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭♣❞❢✮✳ ❋♦r ❛♥ ❡①t❡♥s✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤✐s ❞✐str✐❜✉t✐♦♥ ✇❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ●❡♥t♦♥ ✭✷✵✵✹✮✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳ ✭❝s♥ ❞✐str✐❜✉t✐♦♥ ✕ ♠❣❢ ✮ ▲❡t µ _∈ Rp _{❛♥❞ ϑ ∈ R}q_{✱ p, q ≥ 1✳ ▲❡t}

Σ _∈ Rp×p _{❛♥❞ ∆ ∈ R}q×q_{✱ |Σ|,|∆| ≥ 0✱ ❛♥❞ ❧❡t D ∈ R}q×p_{✳ ❲❡ s❛② t❤❛t r❛♥❞♦♠}

✈❛r✐❛❜❧❡z❤❛s ❛(p, q)✲❞✐♠❡♥s✐♦♥❛❧ ❝❧♦s❡❞ s❦❡✇ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs

µ✱Σ✱D✱ϑ❛♥❞ ∆✐❢ ♠♦♠❡♥t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ z✱Mz(t)✱ ✐s ❣✐✈❡♥ ❜②✿

Mz(t) =

Φq(DΣt;ϑ,∆ +DΣDT)

Φq(0;ϑ,∆ +DΣDT)

etTµ+1

2t T_Σt

✇❤✐❝❤ ❤❡♥❝❡❢♦rt❤ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜②✿

z_∼csnp,q(µ,Σ, D, ϑ,∆)

■❢ |Σ| > 0✱ ❛ csn r❛♥❞♦♠ ✈❛r✐❛❜❧❡ z ♦❜t❛✐♥s ❛ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ❛❝✲

❝♦r❞✐♥❣ t♦✿

❉❡✜♥✐t✐♦♥ ✷✳✷✳ ✭❝s♥ ❞✐str✐❜✉t✐♦♥ ✕ ♣❞❢ ✮ ■❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ z ❢♦❧❧♦✇s ❛ (p, q)✲

❞✐♠❡♥s✐♦♥❛❧✱p, q_≥1✱ ❝❧♦s❡❞ s❦❡✇❡❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rsµ✱Σ✱D✱ϑ

❛♥❞∆✱ ✇❤❡r❡µ_∈Rp_✱ϑ∈Rq_✱Σ∈Rp×p_{✱|Σ|>0✱∆∈R}q×q_{✱|∆| ≥0❛♥❞D∈R}q×p_✱

t❤❛♥ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦❢z ✐s ❣✐✈❡♥ ❜②✿

p(z) =φp(z;µ,Σ) Φq(D(z−µ);ϑ,∆) Φq(0;ϑ,∆ +DΣDT)

❉❡♥s✐t② ❢✉♥❝t✐♦♥ ✭✶✮ ❞❡✜♥❡s ❛(p, q)✲❞✐♠❡♥s✐♦♥❛❧ ♥♦♥s✐♥❣✉❧❛r ❝❧♦s❡❞ s❦❡✇❡❞ ♥♦r✲

♠❛❧ ❞✐str✐❜✉t✐♦♥ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❤❛s(p, q)✲❞✐♠❡♥s✐♦♥❛❧ ♥♦♥s✐♥✲

❣✉❧❛r ❝❧♦s❡❞ s❦❡✇❡❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs µ✱ Σ✱ D✱ϑ ❛♥❞ ∆✐❢ ❛♥❞

♦♥❧② ✐❢ ✐ts ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ❢♦r ❡✈❡r② z _∈ Rp _{❡q✉❛❧s p(z) ✐♥ ✭✶✮✳ P❛r❛♠❡t❡rs µ✱ Σ}

❛♥❞ D ❤❛✈❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❧♦❝❛t✐♦♥✱ s❝❛❧❡ ❛♥❞ s❦❡✇♥❡ss ♣❛r❛♠❡t❡rs r❡s♣❡❝t✐✈❡❧②✳

P❛r❛♠❡t❡rsϑ❛♥❞∆❛r❡ ❛rt✐✜❝✐❛❧✱ ❜✉t ✐♥❝❧✉s✐♦♥ ♦❢ t❤❡♠ ❛❧❧♦✇s ❢♦r ❝❧♦s✉r❡ ♦❢ t❤❡csn

❞✐str✐❜✉t✐♦♥ ✉♥❞❡r ❝♦♥❞✐t✐♦♥✐♥❣ ❛♥❞ t❛❦✐♥❣ ♠❛r❣✐♥❛❧s r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡q✲❞✐♠❡♥s✐♦♥

✐♥ Φq ✐s ❛❧s♦ ❛rt✐✜❝✐❛❧✱ ❜✉t ✐t ❛❧❧♦✇s ❢♦r ❝❧♦s✉r❡ ♦❢ s✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s

❛♥❞ ❢♦r t❛❦✐♥❣ t❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ✭♥♦t ♥❡❝❡ss❛r✐❧② ✐✐❞✮ ✈❛r✐❛❜❧❡s✳ ❲❤❡♥Σ✱D❛♥❞∆❛r❡ s❝❛❧❛rs✱ t❤❡② ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ r❡s♣❡❝t✐✈❡❧② ❜② σ✱d❛♥❞ δ✳ ❋♦r

D = 0✱ t❤❡ csn ❞✐str✐❜✉t✐♦♥ r❡❞✉❝❡s t♦ ❛ p✲❞✐♠❡♥s✐♦♥❛❧ ♥♦r♠❛❧ ♦♥❡✳ ❉✐♠❡♥s✐♦♥ q

❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ s❦❡✇♥❡ss r❡❧❛t❡❞ ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠ ✐♥ t❤❡ ❞✐str✐❜✉t✐♦♥✳

✷✳✷ Pr♦♣❡rt✐❡s

❚❤✐s s❡❝t✐♦♥s ❞✐s❝✉ss❡s ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡csn❞✐str✐❜✉t✐♦♥✳ ❲❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡

♦♥ t❤r❡❡ ❝r✐t✐❝❛❧ ✐ss✉❡s✱ ✇❤✐❝❤ ❛r❡✿ ❝❧♦s✉r❡ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ✉♥❞❡r st❛t❡✲s♣❛❝❡ tr❛♥s✲ ❢♦r♠❛t✐♦♥s✱ ❝♦♥❥✉❣❛t❡ ✐♥✈❡rs✐♦♥ ❢♦r ❧✐❦❡❧✐❤♦♦❞ ❞❡r✐✈❛t✐♦♥ ❛♥❞ ♦♥ ❧❛r❣❡ s❝❛❧❡ ♥♦r♠❛❧ ✐♥t❡❣r❛t✐♦♥ ✭t❤❡ q✲❞✐♠❡♥s✐♦♥ ❡①♣❛♥s✐♦♥✮✳ ❋✐rst✱ ❤♦✇❡✈❡r✱ ❛❧❧ r❡❧❡✈❛♥t r❡♠❛r❦s ❛♥❞

❝♦r♦❧❧❛r✐❡s ❛r❡ ♦✉t❧✐♥❡❞✳

✷✳✷✳✶ ❈♦r♦❧❧❛r✐❡s ❛♥❞ r❡♠❛r❦s✳

❘❡♠❛r❦ ✷✳✸✳ ❋♦r p =q = 1✱ ϑ = 0❛♥❞ ∆ = 1 t❤❡csn ❞✐str✐❜✉t✐♦♥ r❡❞✉❝❡s t♦ t❤❡

❆③③❛❧✐♥✐ s❦❡✇✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ s❡❡ ❆③③❛❧✐♥✐ ✭✶✾✽✺✮✱ ❆③③❛❧✐♥✐ ✭✶✾✽✻✮✳

❙✉❝❤ ❛ ❝❛s❡ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜②✿

z_∼sn(µ, σ, d)

❈♦r♦❧❧❛r② ✷✳✹✳ ▲❡t z_∼sn(µ, σ, d)✱ t❤❡♥✿

E(z) = µ+

r

2

π dσ

√

1 +d2_σ

D(z) = σ₋ 2 π

d2_σ2

1 +d2_σ

E(z₋E(z))3 = (2₋ π 2)

r

2

π

!3

dσ

(1 +σd2₎1₂

!3

■t ❢♦❧❧♦✇s t❤❛t✿

❘❡♠❛r❦ ✷✳✺✳ ▲❡tz_∼sn(µ, σ, d)✱ t❤❡♥E(z) = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢µ=₋q2

π dσ

√ 1+d2_σ✳

❘❡♠❛r❦ ✷✳✻✳ ❋♦r p _≥ 1✱ q = 1✱ ϑ = 0 ❛♥❞ ∆ = 1 t❤❡ csn ❞✐str✐❜✉t✐♦♥ r❡❞✉❝❡s t♦

t❤❡ ♠✉❧t✐✈❛r✐❛t❡ s❦❡✇✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ s❡❡ ❆③③❛❧✐♥✐ ❛♥❞ ❉❛❧❧❛ ❱❛❧❧❡ ✭✶✾✾✻✮ ♦r ❆③③❛❧✐♥✐ ❛♥❞ ❈❛♣✐t❛♥♦ ✭✶✾✾✾✮✳

❙✉❝❤ ❛ ❝❛s❡ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜②✿

z_∼snp(µ,Σ, d)

❈♦r♦❧❧❛r② ✷✳✼✳ ▲❡t z_∼snp(µ,Σ, d)✱ t❤❡♥✿

E(z) = µ+

r

2

πδ

D(z) = Σ +µTµ+

r

2

π(µδ

T _+δµT₎

✇❤❡r❡ δ= √ Σd

1+dTΣd✳

■t ❢♦❧❧♦✇s t❤❛t✿

❘❡♠❛r❦ ✷✳✽✳ ▲❡tz_∼snp(µ,Σ, d)✱ t❤❡♥E(z) = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢µ=−√ Σd

1+dTΣd✳

❋♦r ❤✐❣❤❡r ♠♦♠❡♥ts ♦❢ t❤❡ ♠✉❧t✐✈❛r✐❛t❡ s❦❡✇✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ s❡❡ ●❡♥t♦♥ ❡t ❛❧✳ ✭✷✵✵✶✮✳

❈♦r♦❧❧❛r② ✷✳✾✳ ▲❡t z_∼csnp,q(µ,Σ, D, ϑ,∆)✱ t❤❡♥✿

E(z) =µ+Φ

⋆

q(Dµ;ϑ,∆ +DΣDT)

Φq(Dµ;ϑ,∆ +DΣDT)

✇❤❡r❡✿

Φ⋆_q(Dµ;ϑ,∆ +DΣDT) =

p

X

i=1

q

X

j=1

(DΣ)ijΦjq(Dµ;ϑ,∆ +DΣDT)ei

❢♦r ei ❜❡✐♥❣ ❛ p✲❞✐♠❡♥s✐♦♥❛❧ ✉♥✐t ✈❡❝t♦r ✇✐t❤ t❤❡i✲t❤ ❡♥tr② ❜❡✐♥❣ ❡q✉❛❧ t♦ 1 ❛♥❞✿

Φj_q(Dµ;ϑ,∆+DΣDT) =φ1((Dµ)j;ϑj,(∆+DΣDT)jj)Φq₋1((Dµ)₋j;ϑ₋j,(∆+DΣDT)|(Dµ)j)

✇❤❡r❡✱ ❢♦r ❛ ❣❡♥❡r✐❝ ✈❡❝t♦r x✱ xj ❞❡♥♦t❡s t❤❡ j✲t❤ ❡❧❡♠❡♥t ♦❢ x ❛♥❞ x₋j ❞❡♥♦t❡s x

✇✐t❤♦✉t t❤❡j✲t❤ ❡❧❡♠❡♥t✳

❋♦r ❤✐❣❤❡r ♠♦♠❡♥ts ♦❢ t❤❡ ❝❧♦s❡❞ s❦❡✇✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ s❡❡ ●♦♥③á❧❡③✲❋❛rí❛s ❡t ❛❧✳ ✭✷✵✵✹✮✳

❈♦r♦❧❧❛r② ✷✳✶✵✳ ▲❡t z _∼ csnp,q(µ,Σ, D, ϑ,∆)✱ p, q _≥1✱ ❢♦r ♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡❢✲

✐♥✐t✐♦♥ ✭✷✳✶✮✳ ❊❧❡♠❡♥ts ♦❢ z ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♠❛tr✐❝❡s Σ ❛♥❞ D ❛r❡

❞✐❛❣♦♥❛❧✳

❈♦r♦❧❧❛r② ✷✳✶✶✳ ▲❡t z _∼csnp,q(µ,Σ, D, ϑ,∆)✱ p, q_≥ 1✱ ❢♦r ♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡✜✲

♥✐t✐♦♥ ✭✷✳✶✮✳ ▲❡t ❛❧s♦ w_∼N(µw,Σw)✱ Σw>0✱ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ z✱ t❤❡♥✿

z+w_∼csnp,q(µ+µw,Σ + Σw, DΣ(Σ + Σw)−1, ϑ,∆ + (D(I−Σ(Σ + Σw)−1))ΣDT)

❈♦r♦❧❧❛r② ✷✳✶✷✳ ▲❡tz_∼csn1,q(µ, σ, d, ϑ, δ)✱ q≥1 ❛♥❞ ❢♦r ♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡✜♥✐✲

t✐♦♥ ✭✷✳✶✮✱ ❧❡t ❛❧s♦ρ₆= 0 ❛♥❞ b_∈R✱ t❤❡♥✿

ρz+b_∼csn1,q(ρµ+b, ρ2σ,

1

ρd, ϑ, δ)

❈♦r♦❧❧❛r② ✷✳✶✸✳ ▲❡t z _∼csnp,q(µ,Σ, D, ϑ,∆)✱ p, q≥ 1✱ ❢♦r ♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡✜✲

♥✐t✐♦♥ ✭✷✳✶✮✱ ❧❡t ❛❧s♦A_∈Rp×p_{✱ A6= 0✱ ❛♥❞b∈R}p_{✱ t❤❡♥✿}

Az+b_∼csnp,q(Aµ+b, AΣAT, DΣA−1, ϑ,∆)

❈♦r♦❧❧❛r② ✷✳✶✹✳ ▲❡t z _∼ csnp,q(µ,Σ, D, ϑ,∆)✱ p, q _≥1✱ ❢♦r ♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡❢✲

✐♥✐t✐♦♥ ✭✷✳✶✮✱ ❧❡t ❛❧s♦ A _∈ Rp×v_{✱ p ≤v ❜❡ ❛ ❢✉❧❧ r♦✇ r❛♥❦ ♠❛tr✐① ✭✐✳❡✳ ♦❢ r❛♥❦ v✮✱}

t❤❡♥✿

Az_∼csnv,q(µA,ΣA, DA, ϑA,∆A)

✇❤❡r❡✿

µA=µA, ΣA=AΣAT, DA=DΣATΣ−_A1,

ϑA=ϑ, ∆A= ∆ +DΣDT −DΣATΣ−_A1AΣDT

❈♦r♦❧❧❛r② ✷✳✶✺✳ ▲❡t z _∼csnp,q(µ,Σ, D, ϑ,∆)✱ p, q≥ 1✱ ❢♦r ♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡✜✲

♥✐t✐♦♥ ✭✷✳✶✮✱ ❧❡t ❛❧s♦A_∈Rv_×p_{✱ v > p❜❡ ❛ ❢✉❧❧ ❝♦❧✉♠♥ r❛♥❦ ♠❛tr✐① ✭✐✳❡✳ ♦❢ r❛♥❦ v✮✱}

t❤❡♥✿

Az_∼csnv,q(µA,ΣA, DA, ϑA,∆A)

✇❤❡r❡✿

µA=µA, ΣA=AΣAT, DA=D(ATA)−1, ϑA=ϑ, ∆A= ∆

◆♦t✐❝❡ t❤❛t ❝♦r♦❧❧❛r② ✭✷✳✶✹✮ r❡❧❛t❡s t♦ ❛♥ ✐s♦♠♦r♣❤✐❝ ❝❛s❡ ♦r ❛ ❝❛s❡ ✐♥ ✇❤✐❝❤ ❞✐♠❡♥s✐♦♥ ♦❢ Z ❣❡ts s❤r✐♥❦❡❞✱ ✇❤❡r❡❛s ❝♦r♦❧❧❛r② ✭✷✳✶✽✮ r❡❧❛t❡s t♦ t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤

❞✐♠❡♥s✐♦♥ ♦❢ z ❣❡ts ❡①♣❛♥❞❡❞✳ ■♥ t❤❡ ❧❛tt❡r ❝❛s❡ ΣA ✐s s✐♥❣✉❧❛r ❛♥❞ t❤❡ r❡s✉❧t✐♥❣

❞✐str✐❜✉t✐♦♥ ♦❢Az ✐s ❝❛❧❧❡❞ s✐♥❣✉❧❛r✳ ❇❡❧♦✇ ✇❡ ❞❡s❝r✐❜❡ ❤♦✇ t♦ t❛❦❡ ❥♦✐♥ts ❛♥❞ s✉♠s

♦❢ ✐♥❞❡♣❡♥❞❡♥t ✭♥♦t ♥❡❝❡ss❛r✐❧② ✐✐❞✮csn✈❛r✐❛❜❧❡s✳

❈♦r♦❧❧❛r② ✷✳✶✻✳ ▲❡t zi ∼ csnpi,qi(µi,Σi, Di, ϑi,∆i)✱ pi, qi ≥ 1✱ i = 1,2, ..., n✱ ❢♦r

♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡✜♥✐t✐♦♥ ✭✷✳✶✮✱ t❤❡♥ t❤❡ ❥♦✐♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡✿ (z1T, zT2, ..., znT)T ∼

csnPn i=1pi,

Pn i=1qi(µ

⋆_,Σ⋆_{, D}⋆_{, ϑ}⋆_,∆⋆_{)✱ ✇❤❡r❡✿}

µ⋆ = (µT1, µT2, ..., µTn)T, Σ⋆ =⊕ni=1Σi, D⋆=⊕ni=1Di,

ϑ⋆ = (ϑT1, ϑT2, ..., ϑTn)T, ∆⋆=⊕ni=1∆i

✇❤❡r❡ ♦♣❡r❛t♦r⊕✱ ❢♦r ❛r❜✐tr❛r② ♠❛tr✐❝❡s A ❛♥❞ B✱ ✐s ❞❡✜♥❡❞ ❛s✿

A_⊕B =

  A 0 0 B  

❈♦r♦❧❧❛r② ✷✳✶✼✳ ▲❡t zi ∼ csnp,qi(µi,Σi, Di, ϑi,∆i)✱ p, qi ≥ 1✱ i = 1,2, ..., n✱ ❢♦r

♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡✜♥✐t✐♦♥ ✭✷✳✶✮✱ t❤❡♥ Pn

i=1zi ∼ csnp,Pn i=1qi(µ

⋆_,Σ⋆_{, D}⋆_{, ϑ}⋆_,∆⋆_)✱

✇❤❡r❡✿

µ⋆ =

n

X

i=1

µi, Σ⋆ = n

X

i=1

Σi, D⋆ = (Σ1DT1, ...,ΣnDnT)T(Σ⋆)−

1

,

ϑ⋆ = (ϑT1, ϑ

T

2, ..., ϑ

T

n)T, ∆⋆= ∆⊕+D⊕Σ⊕D⊕−[⊕ni=1DiΣi](Σ⋆)−1[⊕ni=1DiΣi]−1

❢♦r ∆⊕=⊕n

i=1∆i✱ D⊕=⊕ni=1Di ❛♥❞ Σ⊕=⊕Σi✳

❚❤❡ ❧❛st t❤✐♥❣ ✇❡ ♥❡❡❞ t♦ ♣r♦✈✐❞❡ ✐s t❤❡ ❝♦♥❥✉❣❛t❡ ❇❛②❡s✐❛♥ ✐♥✈❡rs❡✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ✉s❡❞ ❢♦r ✜❧tr❛t✐♦♥ ♦❢ st❛t❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ♠♦❞❡❧ ❣✐✈❡♥ ❜② t❤❡ ♦❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥✳

❈♦r♦❧❧❛r② ✷✳✶✽✳ ▲❡t t❤❡r❡ ❜❡ ❛ ♣r✐♦r ♠♦❞❡❧ z _∼ csnp,q(µ,Σ, D, ϑ,∆)✱ p, q ≥ 1✱

❢♦r ♣❛r❛♠❡t❡rs ❛s ✐♥ ❞❡✜♥✐t✐♦♥ ✭✷✳✶✮✱ ❛♥❞ ❛ ❧✐❦❡❧✐❤♦♦❞ ❡q✉❛t✐♦♥ y = F z +Hu✱ ❢♦r

F _∈Rn×z_{✱ H ∈R}n×nu ❛♥❞ _u∼N(0,Ψ

u)✱ t❤❡♥ ♣♦st❡r✐♦r ♠♦❞❡❧(z|y) ✐s ❛s ❢♦❧❧♦✇s✿

(z_|y)∼csnn,q(µz|y,Σz|y, Dz|y, ϑz|y,∆z|y)

✇❤❡r❡✿

µz_|y =µ+ ΣFT(FΣFT +HΨHT)−

1

(y₋F µ)

Σz_|y = Σ−ΣFT(FΣFT +HΨuHT)−1HΣ

D_z|y =

    DΣ 0  −  

DΣFT

0



(FΣFT +HΨuHT)−1FΣ



Σ−

1

z_|y

ϑ_z|y =

  ϑ 0  −  

DΣFT

0



(FΣFT +HΨuHT)−1FΣ (y−F µ)

∆_z|y =





∆ +DΣDT 0

0 I



− 



DΣFT

0



(FΣFT+HΨ_uHT)−

1





DΣFT

0





T

−D_z|yΣ_z|yDT_z|y

✷✳✷✳✷ ❈❧♦s✉r❡ ❛♥❞ t❤❡ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡

◆♦t❡ t❤❛t ✐♥ ❞❡✜♥✐t✐♦♥2.1 ♠❛tr✐① Σ ✐s ❛❧❧♦✇❡❞ t♦ ❜❡ s✐♥❣✉❧❛r✳ ■❢ Σ ✐s ♥♦t ♣♦s✐t✐✈❡

❞❡✜♥✐t❡✱ ✐✳❡✳ |Σ_|= 0✱ r❡s✉❧t✐♥❣ ❞✐str✐❜✉t✐♦♥ ✐s ❝❛❧❧❡❞ s✐♥❣✉❧❛r✳ ■❢ Σ ✐s ♣♦s✐t✐✈❡ ❞❡✜✲

♥✐t❡✱ ✐✳❡✳ |Σ|>0✱ ❞✐str✐❜✉t✐♦♥ ✐s ❝❛❧❧❡❞ ♥♦♥s✐♥❣✉❧❛r✳ ❚❤❡csn❞✐str✐❜✉t✐♦♥ ✐s ✑❝❧♦s❡❞✑

✐♥ t❤❡ s❡♥s❡✱ t❤❛t ✐t ✐s ❝❧♦s❡❞ ✉♥❞❡r ❢✉❧❧ r❛♥❦ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s✻_{✳ ■s♦♠♦r♣❤✐❝}

✭sq✉❛r❡ ❢✉❧❧ r❛♥❦✮ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s tr❛♥s❢♦r♠ ♥♦♥s✐♥❣✉❧❛r ❝s♥ ✈❛r✐❛❜❧❡s ✐♥t♦ ♥♦♥s✐❣✉❧❛r ♦♥❡s ❛♥❞ s✐♥❣✉❧❛r ✈❛r✐❛❜❧❡s ✐♥t♦ s✐♥❣✉❧❛r ♦♥❡s✳ ❋✉❧❧ r♦✇✱ ❜✉t ❝♦❧✉♠♥ r❛♥❦ ❞❡✜❝✐❡♥t ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ✭❡❣✳ ❞✐♠❡♥s✐♦♥ s❤r✐♥❦❛❣❡✴r❡❞✉❝t✐♦♥✮ tr❛♥s❢♦r♠ ♥♦♥s✐♥❣✉❧❛r ❝s♥ ✈❛r✐❛❜❧❡s ✐♥t♦ ♥♦♥s✐❣✉❧❛r ♦♥❡s✳ ❋✉❧❧ ❝♦❧✉♠♥✱ ❜✉t r♦✇ r❛♥❦ ❞❡✜❝✐❡♥t ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ✭❡❣✳ ❞✐♠❡♥s✐♦♥ ❡①♣❛♥s✐♦♥✮ tr❛♥s❢♦r♠ ♥♦♥s✐♥❣✉❧❛r ❝s♥ ✈❛r✐✲ ❛❜❧❡s ✐♥t♦ s✐♥❣✉❧❛r ♦♥❡s✱ ✇❤❡r❡❛s s✐♥❣✉❧❛r ✈❛r✐❛❜❧❡s r❡♠❛✐♥ s✐♥❣✉❧❛r✳ ❇♦t❤ s✐♥❣✉❧❛r ❛♥❞ ♥♦♥s✐♥❣✉❧❛r ✈❛r✐❛❜❧❡s ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ❛ ♥♦♥✲csn ❞✐str✐❜✉t❡❞ ✈❛r✐❛❜❧❡s

✉♥❞❡r ❛ r❛♥❦ ❞❡✜❝✐❡♥t tr❛♥s❢♦r♠❛t✐♦♥✱ ❤❡♥❝❡✱ t❤❡csn❞✐str✐❜✉t✐♦♥ ✐s ♥♦t ❝❧♦s❡❞ ✉♥❞❡r

s✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥s✱ ✇❤✐❝❤ ♣r❡❝❧✉❞❡s t❤❡csn❞✐str✐❜✉t✐♦♥ ❢r♦♠ ♣r♦♣❛❣❛t✐♦♥ ✐♥ t❤❡

st❛t❡ s♣❛❝❡ s❡tt✐♥❣ ✇❤❡♥ r❛♥❦ ❞❡✜❝✐❡♥t tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ ♣♦ss✐❜❧❡✳ ❚❤✐s ❢❛❝t ✐s ♥❡❣❛t✐✈❡ ❢♦r ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✇❤❡♥ t❤❡ tr❛♥s✐t✐♦♥ ♠❛tr✐①A ✐♥ st❛t❡

s♣❛❝❡ ❡q✉❛t✐♦♥s ✐s s✐♥❣✉❧❛r✱ ✇❤✐❝❤ t②♣✐❝❛❧❧② ✐s t❤❡ ❝❛s❡ ✐♥ ❉❙●❊ ♠♦❞❡❧✐♥❣✳

■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ❞✐s❝✉ss t✇♦ r❡♠❛r❦s ✐♥ t❤✐s r❡s♣❡❝t✳ ❋✐rst✱ ✇❡ ♣r♦✈✐❞❡ ♥❡❝❡s✲ s❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ csn❞✐str✐❜✉t✐♦♥ t♦ ♣r♦♣❛❣❛t❡ ✉♥❞❡r ❛r❜✐tr❛r②

❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s✳ ❙❡❝♦♥❞✱ ✐❢ t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♥♦t s❛t✐s✜❡❞✱ ✇❤✐❝❤ ✐s ❛❧♠♦st ❛❧✇❛②s t❤❡ ❝❛s❡✱ ❛ s✐♠♣❧❡ ❛✉t♦♠❛t❡❞ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ✐s s✉❣❣❡st❡❞✳

❈♦r♦❧❧❛r② ✷✳✶✾✳ ▲❡tη_{∈ R}m _{❜❡ ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❛csn}

m,q ❢♦r s♦♠❡m, q≥1

✇✐t❤ ♣❛r❛♠❡t❡rsµη✱ Ση ≥0✱ Dη✱ ϑη ❛♥❞∆η >0✳ ▲❡t z=Gη✱ G∈Rp×m✳ ❚❤❡♥✱ z

❤❛s ❛csn❞✐str✐❜✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢G❤❛s ❢✉❧❧ r♦✇ r❛♥❦ ♦r ✐❢Im(GT) =Im([GT_|wi])

❢♦r ❛❧❧ i = 1,2, ..., q✱ ✇❤❡r❡ Im(G) ❞❡♥♦t❡s ✐♠❛❣❡ ✭♦r r❛♥❣❡✮ ♦❢ G ❛♥❞ wi ❞❡♥♦t❡s

t❤❡ i✲t❤ r♦✇ Dη✳

❈♦r♦❧❧❛r② ✭✷✳✶✾✮ st❛t❡s✱ t❤❛t ❢♦r ❛ csn ✈❛r✐❛❜❧❡ η✱ ✈❛r✐❛❜❧❡ z = Gη ❤❛s ❛ csn

❞✐str✐❜✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛t ❧❡❛st ♦♥❡ ♦❢ t✇♦ ❝♦♥❞✐t✐♦♥s ❛♣♣❧②✳ ❚❤❡ ✜rst ❝♦♥❞✐t✐♦♥ st❛t❡s t❤❛t r♦✇ r❛♥❦ ♦❢ G ✐s ❢✉❧❧✳ ❚❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ r❡q✉✐r❡s t❤❛t r♦✇s ♦❢ Dη

✻_{❯♥❞❡r ❢✉❧❧ r❛♥❦ tr❛♥s❢♦r♠❛t✐♦♥ ✇❡ ♠❡❛♥ ❢✉❧❧ r♦✇ r❛♥❦ ♦r ❢✉❧❧ ❝♦❧✉♠♥ r❛♥❦ tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ t❤✐s}

❞❡✜♥✐t✐♦♥ ❡♠❜r❛❝❡s t❤❡ ❝❛s❡ ✇❤❡♥ ♠❛tr✐① ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s sq✉❛r❡ ❛♥❞ r❡♣r❡s❡♥ts ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ❲❤❡♥ ❜♦t❤ t❤❡ r♦✇ ❛♥❞ t❤❡ ❝♦❧✉♠♥ r❛♥❦s ❛r❡ ♥♦t ❢✉❧❧✱ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❝❛❧❧❡❞ r❛♥❦ ❞❡✜❝✐❡♥t✳

❜❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ r♦✇s ♦❢G✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❛t r♦✇s ♦❢ Dη ❜❡❧♦♥❣ t♦ t❤❡

✐♠❛❣❡ ♦❢GT✱ ✐✳❡✳ t♦ t❤❡ r♦✇ s♣❛❝❡ ♦❢G✳ ❚❤❡ ✜rst ❝♦♥❞✐t✐♦♥ ❝❛♥ ❜❡ s❛t✐s✜❡❞ ♦♥❧② ❢♦r

p_≤m✳ ❚❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ✐s ❛❧✇❛②s s❛t✐s✜❡❞ ✕ ❢♦r ❛♥②Dη ✕ ✐❢G❤❛s ❛ ❢✉❧❧ ❝♦❧✉♠♥

r❛♥❦✱ ✇❤✐❝❤ ❝❛♥ ♦♥❧② ❜❡ t❤❡ ❝❛s❡ ❢♦r p _≥ m✳ ❋♦r ❛ r❛♥❦ ❞❡✜❝✐❡♥t ♦♣❡r❛t♦r G✱ t❤❡

s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ✐s ❛ ✈❡r② ❞❡♠❛♥❞✐♥❣ ♦♥❡✱ s✐♥❝❡Dη ❝❛♥ ❜❡ ✐♥ ♣r✐♥❝✐♣❧❡ ❛r❜✐tr❛r②✳

❚❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♣r♦♣♦s✐t✐♦♥ ✭✷✳✶✾✮ ❛♥❞ t❤❡ st❛t❡ s♣❛❝❡ ❢♦r♠✉❧❛t✐♦♥ ✭✶✮ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

z = xt

G = [A_|B]

η = (xT_t−1, ξtT)T

❙✐♥❝❡p < m✱ ❛❝❝♦r❞✐♥❣ t♦ ♣r♦♣♦s✐t✐♦♥ ✭✷✳✶✾✮✱ ❢♦r t❤❡csn❞✐str✐❜✉t✐♦♥ t♦ ♣r♦♣❛❣❛t❡✱

✇❡ ♥❡❡❞ G t♦ ❤❛✈❡ ❢✉❧❧ r♦✇ r❛♥❦✳ ❲❤❡♥ ♦♥❡ ✇♦r❦s ✇✐t❤ ♠❡❞✐✉♠✲ ♦r ❧❛r❣❡ s✐③❡

❉❙●❊ ♠♦❞❡❧s✱ t❤❡ r❡❞✉❝❡❞ ❢♦r♠ r❡♣r❡s❡♥t❛t✐♦♥ ♠❛tr✐① A ❝❛♥ ❜❡✱ ❛♥❞ ✉s✉❛❧❧② ✐s✱

r❛♥❦ ❞❡✜❝✐❡♥t✳ ❆❧s♦ ❝♦♠❜✐♥✐♥❣ A ✇✐t❤ B ✉s✉❛❧❧② r❡s✉❧ts ✐♥ G ✇❤✐❝❤ ❞♦❡s♥✬t ❤❛✈❡

❛ ❢✉❧❧ r♦✇ r❛♥❦✳ ❙✐♥❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❣✉♠❡♥t ❛♣♣❧✐❡s t♦ ❢✉❧❧ r♦✇ r❛♥❦ ♠❛tr✐❝❡s G✱

✇❡ ♥❡❡❞ t♦ r❡❢♦r♠✉❧❛t❡ t❤❡ ♠♦❞❡❧ s♦ t❤❛tG❤❛s ❢✉❧❧ r♦✇ r❛♥❦✱ ❜✉t t❤❡ ✈❛❧✉❡ ♦❢ t❤❡

❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ✐s ✉♥❛✛❡❝t❡❞✳ ■❢ G = [A_|B]✐s r❛♥❦ ❞❡✜❝✐❡♥t✱ t❤❡♥ s♦♠❡ ♦❢ t❤❡

st❛t❡sxt ❛r❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ t❤❡ r❡♠❛✐♥✐♥❣ ♦♥❡s✱ ✇❤✐❝❤ ♠❡❛♥s✱ t❤❛t t❤❡②

❝❛♥ ❜❡ s✉❜st✐t✉t❡❞ ♦✉t ❢r♦♠ t❤❡ st❛t❡✲s♣❛❝❡ r❡♣r❡s❡♥t❛t✐♦♥ ✉s✐♥❣ t❤❡ r❡♠❛✐♥✐♥❣ ♦♥❡s ✕ ❜♦t❤ ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ❛♥❞ ✐♥ t❤❡ ♠❡❛s✉r❡♠❡♥t ❡q✉❛t✐♦♥✳ ❚❤✐s ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ❛♥❞✱ ♠♦r❡♦✈❡r✱ t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ❛✉t♦♠❛t✐❝❛❧❧②✳

▲❡t ✉s ❞❡♥♦t❡ ❜②x¯tt❤❡ ✭❛♥②✮ ♠❛①✐♠❛❧ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t s✉❜s❡t ♦❢ st❛t❡s ❢r♦♠

xt✱ ❛♥❞ ❜② xt˜ t❤❡ r❡♠❛✐♥✐♥❣ st❛t❡s✳ ◆✉♠❡r✐❝❛❧❧②✱ ✇❡ ❝❛♥ ✜♥❞ ❛ ♠❛tr✐① K s✉❝❤ t❤❛t

˜

xt = Kx¯t ✭❢♦r ❛❧❧ t∈ T)✳ ❘❡❛rr❛♥❣❡ xt = [¯xTt,x˜Tt]T ❛♥❞ ♣❛rt✐t✐♦♥ ♠♦❞❡❧ ♠❛tr✐❝❡s

❛❝❝♦r❞✐♥❣❧②✱ s♦ t❤❛t✿

yt=

F1 F2

  ¯ xt ˜ xt 

+Hu_t

❛♥❞✿

xt=





A11 A12

A21 A22



 



¯

xt−1

˜

xt₋1

 +   B1 B2 

ξ_t

❚❤❡♥✱ ✜rst t✇♦ st❛t❡ s♣❛❝❡ ❡q✉❛t✐♦♥s ✐♥ ✭✶✮ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿

yt = F¯x¯t+Hut

¯

xt = A¯x¯t−1+ ¯Bξt

✇❤❡r❡_F¯_{= (F1+F2K)✱A}¯_{= (A11+A12K) ❛♥❞B}¯ _{=B1✳ ❆❢t❡r s✉❝❤ ❛ r❡❣✉❧❛r✐③❛t✐♦♥}

♠❛tr✐① G= [ ¯A,B¯]❤❛s ❛ ❢✉❧❧ r♦✇ r❛♥❦✳ ❚❤✐s ❛❧❧♦✇s ❢♦r ❢✉rt❤❡r st❡♣s t♦ ❛♣♣❧②✳ ❙✐♥❝❡

˜

xt=Kx¯t✱ ✇❡ ❛❧s♦ ❤❛✈❡ ❢✉❧❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ st❛t❡s ✇❤✐❝❤ ✇❡r❡ s✉❜st✐t✉t❡❞ ❢♦r✳

✷✳✷✳✸ q✲❞✐♠❡♥s✐♦♥ ❡①♣❛♥s✐♦♥

❇❡❝❛✉s❡Σ ✐s ❛ p_×p ♠❛tr✐① ❛♥❞ D ✐s ❛ q_×p ♠❛tr✐①✱ ❝♦r♦❧❧❛r② ✭✷✳✶✵✮ ✐♠♣❧✐❡s t❤❛t

✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❤❛✈❡ q = 1 ✇❤✐❧❡ ❦❡❡♣✐♥❣ ❡❧❡♠❡♥ts ♦❢ z ✐♥❞❡♣❡♥❞❡♥t ❢♦r p > 1✱

❜❡❝❛✉s❡ ✐t ❤❛s t♦ ❜❡ t❤❡ ❝❛s❡ t❤❛tq =p >1✐♥ ♦r❞❡r ❢♦r D t♦ ❜❡ ❞✐❛❣♦♥❛❧✳ ❚❤✐s ✐s

r❡❧❡✈❛♥t✱ ❜❡❝❛✉s❡ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡xt✱ ✐♥ ❡✈❡r② ♣❡r✐♦❞ ❝♦♥s✐sts ♦❢ t❤❡csn❞✐str✐❜✉t❡❞

st❛t❡ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ♣❡r✐♦❞✱ s❛② xt−1✱ ♣❧✉s t❤❡ csn✲❞✐str✐❜✉t❡❞ ❞✐st✉r❜❛♥❝❡✼ ξt✳

❈♦r♦❧❧❛r② ✷✳✶✼ ✐♠♣❧✐❡s t❤❡♥✱ t❤❛t ✇❤❡♥ ✇❡ ❛❞❞ t✇♦ csn ✈❛r✐❛❜❧❡s ✇❡ ❤❛✈❡ t♦ ❛❞❞

t❤❡✐rq✲❞✐♠❡♥s✐♦♥s✱ s♦ t❤❛t t❤❡q✲❞✐♠❡♥s✐♦♥ ♦❢xt✐s t❤❡ s✉♠ ♦❢ q✲❞✐♠❡♥s✐♦♥s ♦❢xt−1 ❛♥❞ξt✱ t❤❡r❡❢♦r❡✱ ❛❝❝♦r❞✐♥❣ t♦ ❝♦r♦❧❧❛r② ✭✷✳✶✵✮✱ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ξtt♦ t❤❡q✲❞✐♠❡♥s✐♦♥

♦❢ xt ✐♥ ❡✈❡r② ♣❡r✐♦❞ ❝❛♥♥♦t ❜❡ sq✉❡❡③❡❞ t♦ ❡❣✳ 1✱ ❜✉t ♠✉st ❜❡ ❡q✉❛❧ t♦ t❤❡ s✐③❡

♦❢ ξt ✭✐✳❡✳ nξ✮✱ ❤❡♥❝❡ t❤❡ q✲❞✐♠❡♥s✐♦♥ ♦❢ xt q✉✐❝❦❧② ✐♥❝r❡❛s❡s ✇✐t❤ t✱ ✇❤❡♥ t❤❡ csn

❞✐str✐❜✉t✐♦♥ ❣❡ts ♣r♦♣❛❣❛t❡s ❢r♦♠ s❤♦❝❦s t♦ st❛t❡s ❛♥❞ t♦ ♦❜s❡r✈❛t✐♦♥s ✇✐t❤✐♥ t❤❡ st❛t❡✲s♣❛❝❡ s❡tt✐♥❣✳ ▲❡t ✉s t❤❡♥ ♥♦t❡✱ t❤❛t ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭✶✮ ♦❢ t❤❡

csn❞✐str✐❜✉t✐♦♥ ❛♥❞✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ✭s❡❡ s❡❝t✐♦♥ s❡❝t✐♦♥

✹✮✱ t❤❡② ❛❧❧ ✐♥✈♦❧✈❡ ❛ ❝✉♠✉❧❛t✐✈❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ❛q✲❞✐♠❡♥s✐♦♥❛❧

♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✳ ❚♦ ♦✉r ❜❡st ❦♥♦✇❧❡❞❣❡ t❤❡r❡ ✐s ♥♦ ❡✣❝✐❡♥t ✇❛② ♦❢ ❝❛❧❝✉❧❛t✐♥❣ ❧❛r❣❡ s❝❛❧❡ ♥♦r♠❛❧ ✐♥t❡❣r❛❧s ✇✐t❤ ❛♥ ❛r❜✐tr❛r② ❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡✱ t❤❡r❡❢♦r❡ ✇❡ ✇♦r❦ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥✿ Φq(z, ϑ,∆) ≈ Qqj=1Φ1(zj, ϑj,∆jj)✱ ✇❤✐❝❤ ❡❧✐♠✐♥❛t❡s t❤❡ ❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥❛❧✐t②✳ ❆❝❝✉r❛❝② ♦❢ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡ ✐♠♣❧✐❡❞ ❜② t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ∆✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ∆✐s

❞✐❛❣♦♥❛❧✱ t❤❡ r❡s✉❧t ✐s ❡①❛❝t✱ ♥♦t ❛♣♣r♦①✐♠❛t❡❞✳ ■♥ t❤❡ ♥✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ✇✐t❤ ❉❙●❊ ♠♦❞❡❧s ✇❡ ❢♦✉♥❞✱ t❤❛t ♦✛✲❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✱ ✇❤✐❝❤

✼_{❇♦t❤ st❛t❡ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ♣❡r✐♦❞ ❛♥❞ t❤❡ ❞✐st✉r❜❛♥❝❡ ❛r❡ tr❛♥s❢♦r♠❡❞ ❜② t❤❡ ❧✐♥❡❛r tr❛♥s❢♦r✲}

♠❛t✐♦♥A ❛♥❞ B r❡s♣❡❝t✐✈❡❧②✱ ❜✉t ❧❡t ✉s ✐❣♥♦r❡ t❤✐s ❢❛❝t ❢♦r t❤❡ ♣r❡s❡♥t ❛r❣✉♠❡♥t❛t✐♦♥ ✭♦r ❛ss✉♠❡ t❤✐s

tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ ✐❞❡♥t✐t✐❡s✮✳

❛♣♣❡❛rs ✐♥ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ✭❜♦t❤ ✐♥ t❤❡ ♥♦♠✐♥❛t♦r ❛♥❞ ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r✮ ❛r❡ ❝❧♦s❡ t♦ ③❡r♦✱ ✇❤❡r❡❛s ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts ❛r❡ s✐❣♥✐✜❝❛♥t❧② ❜✐❣❣❡r✳ ❋♦r ✐♥st❛♥❝❡✱ ✐♥ t❤❡ ❡st✐♠❛t✐♦♥ ❡①❛♠♣❧❡ ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✺✱ ♠❡❛♥ ❛❜s♦❧✉t❡ ♦✛✲❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥t ♦❢ t❤✐s ♠❛tr✐① ❛♠♦✉♥t❡❞ t♦4.88_×10−4_{✱ ✇❤❡r❡❛s ♠❡❛♥ ♦❢ ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts ✇❛s}

1.09✳

■♥ t❡r♠s ♦❢ ❝♦rr❡❧❛t✐♦♥✱ ♠❡❛♥ ❛❜s♦❧✉t❡ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t ❜❡t✇❡❡♥ ❞✐♠❡♥s✐♦♥s ♦❢ ✈❛r✐❛❜❧❡s ✇✐t❤ s✉❝❤ ❛ ❝♦✈❛r✐❛♥❝❡ str✉❝t✉r❡ ❡q✉❛❧s4.16_×10−4_{✱ ✇❤✐❝❤ ✐s ❝❧♦s❡ t♦ ③❡r♦} ❛s ❢♦r ❛ ❝♦rr❡❧❛t✐♦♥✳

✸ ❚❤❡ ✜❧t❡r

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ♣r♦✈✐❞❡ ❢♦r♠✉❧❛❡ ❢♦r ♣r❡❞✐❝t✐♦♥ ❞❡♥s✐t✐❡s p(yt|Yt₋1)✱ p(xt|Yt₋1) ❛♥❞ ❢♦r ✜❧tr❛t✐♦♥ ❞❡♥s✐t② p(xt|Yt)✳ ❆❧❧ t❤❡ ❞❡♥s✐t✐❡s ❞❡♣❡♥❞ t❤❡ ♣❛r❛♠❡t❡r ✈❡❝t♦r

θ= (θF, θH, θu, θA, θB, θξ, θx0)✳ ❋✐rst✱ ❤♦✇❡✈❡r✱ ✇❡ ❞❡r✐✈❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s

❢♦r st❛t❡s ❛♥❞ ♦❜s❡r✈❛❜❧❡s✳

❚❤❡ st❛t❡ s♣❛❝❡ s❡tt✐♥❣ ✐s ❛ss✉♠❡❞ t♦ ❜❡ ✭✶✮ ❛♥❞ ❞✐str✐❜✉t✐♦♥ pξ ♦❢ s❤♦❝❦s ξt✱

t_{∈ T}✱ ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❛csn❞✐str✐❜✉t✐♦♥✿

ξt ∼ csnnξ,q(µξ,Σξ, Dξ, ϑξ,∆ξ)

❢♦rt= 1,2, ..., T ✇✐t❤Σξ✱Dξ✱ ❛♥❞∆ξ ❜❡✐♥❣ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s✱ ❛♥❞ϑξ = 0q✳ ❋♦rDξ

t♦ ❜❡ ❞✐❛❣♦♥❛❧✱ ✐t ♠✉st ❜❡ t❤❡ ❝❛s❡✱ t❤❛tq =nξ✳ ❘❡♠❛r❦ ✭✷✳✺✮ r❡❞✉❝❡s t❤❡ ❞❡❣r❡❡s ♦❢

❢r❡❡❞♦♠ ✐♥ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ♦❢ s❤♦❝❦s✱ s✐♥❝❡ t♦ ❤❛✈❡E(ξt) = 0✱ ♦♥❡ ♥❡❡❞s

t♦ ✐♠♣♦s❡✿ µξ = −

q

2

π

d_ξiσ_ξi

q

δ_ξi+d2 ξiσξi✱

i = 1,2, ..., nξ✱ ✇❤❡r❡ µξi ✐s t❤❡ i✲t❤ ❡❧❡♠❡♥t ♦❢

µξ✱ σξi ✐s t❤❡ i✲t❤❡ ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥t ♦❢ Σξ ❛♥❞ δξi = 1 ✐s t❤❡ i✲t❤ ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥t

♦❢∆ξ✳ ■♥ ❉❙●❊ ♠♦❞❡❧✐♥❣ ✐t ✉s✉❛❧❧② ✐s t❤❡ ❝❛s❡ t❤❛t nξ< p❛♥❞r(B) =nξ✱ ✇❤✐❝❤ ■

❛ss✉♠❡ t❤❡r❡❛❢t❡r✳

✸✳✶ ❯♥❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s

■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ❝♦♥s✐❞❡r t✇♦ ❝❛s❡s ❢♦r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t❡ ✈❛r✐❛❜❧❡s✳ ❙❡❝t✐♦♥ ✭✸✳✶✳✶✮ ❛ss✉♠❡s t❤❛tA✐s ❢✉❧❧ r❛♥❦ ✭r(A) =p✮✱ s♦ t❤❛t t❤❡csn❞✐str✐❜✉t✐♦♥ ♣r♦♣❛❣❛t❡s

t❤r♦✉❣❤ t❤❡ st❛t❡ s♣❛❝❡ ✇✐t❤♦✉t ♦❜st❛❝❧❡s✳ ❙❡❝t✐♦♥ ✭✸✳✶✳✷✮ ♣r❡s❡♥ts t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✐♥ ✇❤✐❝❤A❝❛♥ ❜❡ r❛♥❦ ❞❡✜❝✐❡♥t✳ ■❢G= [A_|B]✐s ❢✉❧❧ ✭r♦✇✮ r❛♥❦✱ r❡❣❛r❞❧❡ss ✇❤❡t❤❡r

A ✐s ❢✉❧❧ r❛♥❦ ♦r ♥♦t✱ ❢♦r♠✉❧❛❡ ❢r♦♠ s❡❝t✐♦♥ ✭✸✳✶✳✷✮ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ❞✐r❡❝t❧② t♦ t❤❡

s②st❡♠ ✭✶✮ ✇✐t❤♦✉t r❡❣✉❧❛r✐③❛t✐♦♥✳ ■❢G= [A_|B]✐s ✭r♦✇✮ r❛♥❦ ❞❡✜❝✐❡♥t✱ ✐♥ ✇❤✐❝❤ ❝❛s❡

A♠✉st ❜❡ r❛♥❦ ❞❡✜❝✐❡♥t✱ r❡❣✉❧❛r✐③❛t✐♦♥ ❞❡s❝r✐❜❡❞ ✐♥ s❡❝t✐♦♥ ✭✷✳✸✮ ♥❡❡❞s t♦ ❜❡ ❡①❡rt❡❞

♦♥ t❤❡ s②st❡♠ ❜❡❢♦r❡ ♦♥❡ ❡♠♣❧♦②s ❞❡r✐✈❡❞ ❢♦r♠✉❧❛❡✳ ❲❡ ♣r❡s❡♥t s❡❝t✐♦♥ ✭✸✳✶✳✶✮ ♦♥❧② ❜❡❝❛✉s❡ ❢♦r ❢✉❧❧ r❛♥❦ A ✉♥❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s ♦❢ st❛t❡s ❛♥❞ ♦❜s❡r✈❛❜❧❡s ❝❛♥

❜❡ ❞❡r✐✈❡❞ ✇✐t❤♦✉t ✉s✐♥❣ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t❡s ❛♥❞ s❤♦❝❦s✱ ✇❤✐❝❤ ✐s s✐♠♣❧❡r✳ ❘❡❛❞❡rs ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ❝❛♥ s❦✐♣ t♦ s❡❝t✐♦♥ ✭✸✳✶✳✷✮✳ ❙✐♥❝❡ ❢♦r♠✉❧❛❡ ❢♦r ❞✐str✐❜✉t✐♦♥s ♦❢ ♦❜s❡r✈❛❜❧❡s ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❜♦t❤ ❝❛s❡s✱ t❤❡② ❛r❡ ❣✐✈❡♥ s❡♣❛r❛t❡❧② ✐♥ s❡❝t✐♦♥ ✭✸✳✷✮✳

✸✳✶✳✶ ❙t❛t❡ ❞✐str✐❜✉t✐♦♥ ✲ ❢✉❧❧ r❛♥❦ ❝❛s❡

■♥ t❤✐s ♣❛r❛❣r❛♣❤ ✇❡ ❛ss✉♠❡ A ✐s ❢✉❧❧ r❛♥❦✳ ■❢ nξ < p✱ t❤❡♥ Bξt ❞♦❡s ♥♦t ❤❛✈❡ ❛

❞❡♥s✐t② ❢✉♥❝t✐♦♥ s✐♥❝❡ ♠❛tr✐①BΣξBT ✐s s✐♥❣✉❧❛r✳ ■❢r(B) =nξ✱ t❤❡♥Bξt ❢♦❧❧♦✇s✿

Bξt∼csnp,q(µB,ΣB, DB, ϑB,∆B)

✇❤❡r❡✿

µB=Bµξ, ΣB=BΣξBT, DB=Dξ(BTB)−1BT, ϑB =ϑξ, ∆B= ∆ξ

✇❤✐❝❤ ✐s ❛ (p, q)✲❞✐♠❡♥s✐♦♥❛❧ s✐♥❣✉❧❛r csn ❞✐str✐❜✉t✐♦♥ ❢♦r q = nξ✳ ❚❤✐s ✐s tr✉❡ ❢♦r

❡✈❡r②t= 1,2, ..., T✳

❙✐♥❝❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✐s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ csn ❞✐str✐❜✉t✐♦♥✱ ✐t ❝❛♥ ❜❡

✇r✐tt❡♥ t❤❛t✿

x0 ∼csnp,1(µx0,Σx0, Dx0, ϑx0,∆x0)

❢♦r✽_✿

µx0 = ¯x, Σx0 = Ψx, Dx0 = 0, ϑx0 = 0, ∆x0 = 1

●❡♥❡r❛❧❧②✱ ✐✳❡✳ ❢♦r t_{∈ T}✱ ✐❢✿

xt−1∼csnp,qt−1(µxt−1,Σxt−1, Dxt−1, ϑxt−1,∆xt−1)

✇❤✐❝❤ ✐s tr✉❡ ❢♦rt= 1✱ ✐✳❡✳ ❢♦r x0✱ ✇✐t❤q0 = 1✱ t❤❛♥ Axt−1 ❢♦❧❧♦✇s✿

Axt₋1 ∼csnp,qt−1(µA,t−1,ΣA,t−1, DA,t−1, ϑA,t−1,∆A,t−1)

✽_∆

x0 >0 ❝❛♥ ✐♥ ❢❛❝t ❜❡ ❛r❜✐tr❛r②✳

✇❤❡r❡✿

µA,t−1 =Aµx,t−1, ΣA,t−1 =AΣx,t−1AT, DA,t−1 =Dx,t−1Σx,t−1ATΣ−_A,t1₋₁

ϑA,t−1 =ϑx,t−1, ∆A,t−1= ∆x,t−1+Dx,t−1Σx,t−1DT_x,t−1+

−Dx,t−1Σx,t−1ATΣ_A,t−1₋₁AΣx,t−1D_x,tT ₋1

✇❤✐❝❤ ✐s ❛ (p, qt−1)✲❞✐♠❡♥s✐♦♥❛❧ csn ❞✐str✐❜✉t✐♦♥✱ ❛♥❞✱ ✐❢ |Σxt−1| > 0✱ ✐t ✐s ♥♦♥✲

s✐♥❣✉❧❛r✳ ❚❤✐s ✐s tr✉❡ ❢♦rt= 0 ❛♥❞ ❛❧s♦✱ ❜② ✐♥❞✉❝t✐♦♥✱ ❢♦r ❡✈❡r②t_{∈ T}✱ ❜❡❝❛✉s❡✿

xt=Axt₋1+Bξt

✇❤❡r❡Axt−1 ❛♥❞Bξt❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ❢r♦♠ ✇❤✐❝❤ ✐t ❢♦❧❧♦✇s✱ t❤❛t

xt ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦✿

xt_∼csnp,qt(µxt,Σxt, Dxt, ϑxt,∆xt)

❢♦r qt=qt₋1+q✱ ✇❤❡r❡✱ s❡❡ r❡♠❛r❦ ✭✷✳✶✼✮✿

µx,t =µA,t+µB, Σx,t = ΣA,t+ ΣB, Dx,t =

ΣA,tDTA,t,ΣBDTB

T

Σ−_x,t1

ϑA,t=

ϑT_A,t, ϑT_B,tT

, ∆x,t = ∆A,t⊗∆B+ (DA,t⊗DB)(ΣA,t⊗ΣB)(DA,t⊗DB)T+

−(DA,tΣA,t⊕DBΣB)(Σx,t)−1(ΣA,tDA,tT ⊕ΣBDTB)

❙✐♥❝❡ q0 = 1❛♥❞ qt=qt−1+q✱ ✇❡ ❤❛✈❡ qt=tq+q0 =tq+ 1 =tnξ+ 1❛♥❞✿

xt∼csnp,qt(µxt,Σxt, Dxt, ϑxt,∆xt)

✸✳✶✳✷ ❙t❛t❡ ❞✐str✐❜✉t✐♦♥ ✲ r❡❞✉❝❡❞ r❛♥❦ ❝❛s❡

■♥ t❤✐s ♣❛r❛❣r❛♣❤ ■ ❛ss✉♠❡G= [A_|B]✐s ❢✉❧❧ ✭r♦✇✮ r❛♥❦ ✭A❝❛♥ ✐♥ ♣r✐♥❝✐♣❧❡ ❜❡ r❛♥❦

❞❡✜❝✐❡♥t✮✳ ■❢ ✐t ✐s ♥♦t✱ ❜❡❢♦r❡ ❡♠♣❧♦②✐♥❣ ♣r❡s❡♥t❡❞ ❢♦r♠✉❧❛❡✱ r❡❣✉❧❛r✐③❛t✐♦♥ ❞❡s❝r✐❜❡❞ ✐♥ s❡❝t✐♦♥ ✭✷✳✸✮ ♥❡❡❞s t♦ ❜❡ ❛♣♣❧✐❡❞ ✜rst✳

❆ss✉♠❡ t❤❛txt−1∼csnp,qt−1(µxt−1,Σxt−1,∆xt−1, ϑxt−1,∆xt−1)✱ ✇❤✐❝❤ ✐s tr✉❡ ❢♦r

t= 1✱ ❛♥❞q = 1✱ ❧❡t x0 ∼csnp,1(µx0,Σx0, Dx0, ϑx0,∆x0) ❢♦r✾✿

❙✐♥❝❡ xt−1 ❛♥❞ ξt✱t∈ T✱ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✱ ❛❝❝♦r❞✐♥❣ t♦ r❡♠❛r❦ ✭✷✳✶✻✮✱

❥♦✐♥t ❞✐str✐❜✉t✐♦♥gt= (xt₋1, ξt) ✐s✿

gt_∼csnp+nξ,qt+q(µg,t,Σg,t, Dg,t, ϑg,t,∆g,t)

✾_{❙❡❡ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤✳}

✇✐t❤ ♣❛r❛♠❡t❡rs✿

µg,t= (µT_xt

−1, µ

T

ξ)T, Σg,t= Σxt−1 ⊕Σξ, Dg,t=Dxt−1⊕Dξ,

ϑg,t= (ϑT_xt

−1, ϑ

T

ξ)T, ∆g,t= ∆xt−1⊕∆ξ

❯♥❞❡r s✉❝❤ ♥♦t❛t✐♦♥✱xt=Ggt ❢♦r G= [A|B]✱ ❛♥❞ xt ❢♦❧❧♦✇s✿

xt∼csnp,qt(µxt,Σxt, Dxt, ϑxt,∆xt)

❢♦r qt=qt₋1+q✱ ✇❤❡r❡✱ s❡❡ r❡♠❛r❦ ✭✷✳✶✼✮✿

µx,t =Gµg,t, ΣG,t=GΣg,tGT, Dx,t =Dg,tΣg,tGTΣ−G,t1

ϑx,t=ϑg,t, ∆x,t= ∆g,t+Dg,tΣg,tDTg,t+

−Dg,tΣg,tGTΣG,t−1GΣg,tDg,tT

❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤✱ s✐♥❝❡ q0 = 1 ❛♥❞ qt = qt₋1 +q✱ ✇❡ ❤❛✈❡ qt =

tq+q0=tq+ 1 =tnξ+ 1❛♥❞✿

xt_∼csnp,qt(µxt,Σxt, Dxt, ϑxt,∆xt)

✸✳✷ ❖❜s❡r✈❛❜❧❡s

❙♦ ❢❛r ❢♦r♠✉❧❛❡ ❢♦r ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t❡s xt ❢♦r ❛❧❧ t ∈ T ❤❛✈❡ ❜❡❡♥ ❞❡r✐✈❡❞✳

◆♦✇ ❧❡t ✉s ❞♦ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♦❜s❡r✈❛❜❧❡s yt ❢♦r ❛❧❧ t ∈ T✳ ❖♥❝❡ ❛❣❛✐♥✱ s✐♥❝❡

♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✐s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ csn ❞✐str✐❜✉t✐♦♥✱ ✐t ❝❛♥ ❜❡ ✇r✐tt❡♥ t❤❛t✿

ut∼CSNnu,1(µu,Σu, Du, ϑu,∆u)✳

■♥ ❉❙●❊✱ ✐t ✉s✉❛❧❧② ✐s t❤❡ ❝❛s❡ t❤❛t nu =p ❛♥❞H ✐s ❢✉❧❧ r❛♥❦✱ s♦ t❤❛t ♠❡❛s✉r❡✲

♠❡♥t ❡rr♦rs r✉❧❡ ♦✉t st♦❝❤❛st✐❝ s✐♥❣✉❧❛r✐t② ✐♥ t❤❡ ♠❡❛s✉r❡♠❡♥t ❡q✉❛t✐♦♥✳ ■❢ t❤✐s ✐s t❤❡ ❝❛s❡✱ ✇❤✐❝❤ ■ ❛ss✉♠❡✱ ❛ ♠❛tr✐① K = [F, H] ✐s ❢✉❧❧ ✭r♦✇✮ r❛♥❦✱ ♥♦ ♠❛tt❡r ✇❤❛t

r(F) ✐s✱ ❛♥❞ ❤❡r❡ ♥♦ r❡❣✉❧❛r✐③❛t✐♦♥ ♠✉st ❜❡ ✐♥✈♦❧✈❡❞✳

❙✐♥❝❡xt_∼csnp,qt(µxt,Σxt,∆xt, ϑxt,∆xt)✶✵❛♥❞xt❛♥❞ξt✱t∈ T✱ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t

✈❛r✐❛❜❧❡s✱ ❛❝❝♦r❞✐♥❣ t♦ r❡♠❛r❦ ✭✷✳✶✻✮✱ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ kt= (xt, ut) ✐s✿

kt_∼csnp+nu,qt+nu(µk,t,Σk,t, Dk,t, ϑk,t,∆k,t)

✇✐t❤ ♣❛r❛♠❡t❡rs✿

µk,t = (µT_xt, µT_u)T, Σk,t= Σxt ⊕Σu, Dk,t =Dxt ⊕Du,

ϑk,t = (ϑT_xt, ϑT_u)T, ∆k,t = ∆xt⊕∆u

✶✵_{❙❡❡ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤✳}

❯♥❞❡r s✉❝❤ ♥♦t❛t✐♦♥✱yt=Kkt ❢♦r K= [F|H]✱ ❛♥❞ yt❢♦❧❧♦✇s✿

yt_∼csnp,qt+nu(µyt,Σyt, Dyt, ϑyt,∆yt)

✇❤❡r❡✱ s❡❡ r❡♠❛r❦ ✭✷✳✶✼✮✿

µy,t=Kµk,t, ΣK,t=KΣk,tGT, Dy,t=Dk,tΣk,tKTΣ−K,t1

ϑy,t=ϑk,t, ∆y,t= ∆k,t+Dk,tΣk,tDk,tT +

−Dk,tΣk,tKTΣK,t−1KΣk,tDk,tT

◆♦t✐❝❡ t❤❛t q✲❞✐♠❡♥s✐♦♥ ♦❢yt ✐s ❡✈❡♥ ❜✐❣❣❡r t❤❛t ♦❢ xt ✲ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ♠❡❛✲

s✉r❡♠❡♥t ❡rr♦rsnu =n✳

✸✳✸ ❈♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s

❋♦r t _{∈ T}✱ ❧❡t ✉s ❞❡✜♥❡ ❛♥ ✐♥❢♦r♠❛t✐♦♥ s❡t Yt = {y1, y2, ..., yt} ✇❤✐❝❤ ❝♦♥s✐sts ♦❢

♦❜s❡r✈❛❜❧❡s ✉♣ t♦ t✐♠❡t✳ ■ ✇✐❧❧ ❞❡r✐✈❡ t❤❡ ❛ ♣♦st❡r✐♦r✐ ❞✐str✐❜✉t✐♦♥(xt|Yt) ✐♥ ❛ ✉s✉❛❧

✇❛②✱ ✐✳❡✳ ❜② ❝♦♥str✉❝t✐♥❣ t❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥(xt, yt|Yt−1)✇✐t❤ t❤❡ ✑r❡s✐❞✉❛❧ tr✐❝❦✑ ❛♥❞ t❤❛♥ ❝♦♥❞✐t✐♦♥✐♥❣ ✉♣♦♥ yt✳

❆ss✉♠❡✱ t❤❛t t❤❡ ❛ ♣♦st❡r✐♦r✐ ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t❡s xt−1✱ ✐✳❡✳ ❝♦♥❞✐t✐♦♥❛❧ ❞✐s✲ tr✐❜✉t✐♦♥ ♦❢ st❛t❡s xt₋1 ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥❢♦r♠❛t✐♦♥ s❡t Yt₋1✱ t❤❡r❡❢♦r❡ ❛❢t❡r ♦❜s❡r✈✐♥❣ yt−1✱ ✐s ❣✐✈❡♥ ❜②✿

(xt−1|Yt−1)∼csnp,qt−1(µt−1,Σt−1, Dt−1, ϑt−1,∆t−1)

❢♦r s♦♠❡ ♣❛r❛♠❡t❡rs µt₋1✱ Σt₋1✱ Dt₋1✱ ϑt₋1 ❛♥❞ ∆t₋1✳ ■❢ s♦✱ t❤❡ ❛ ♣r✐♦r✐ r❛♥❞♦♠ ✈❛r✐❛❜❧❡(xt|Yt−1) ✐s ❣✐✈❡♥ ❜②✿

(xt|Yt−1) = (Axt−1+Bξt|Yt−1) = (Axt−1|Yt−1) +Bξt∼csn(µ,Σ, D, ϑ,∆)

❢♦r✿

µ=µA+µB, Σ = ΣA+ ΣB, D=

ΣADAT,ΣBDTB

T

Σ−1, ϑ=

ϑT_A,0TT

,

∆ = ∆A⊕∆B+ (DA⊕DH)(ΣA⊕ΣD)(DA⊕DH)T+

−(DAΣA⊕DHΣH)(Σ)−1(DAΣA⊕DHΣH)T

✇❤❡r❡✿

µA=Aµt−1, ΣA=AΣt−1AT, DA=Dt−1Σt−1ATΣ−_A1,

ϑA=ϑt−1, ∆ = ∆t−1+Dt−1Σt−1DT_t−1−Dt−1Σt−1ATΣ_A−1AΣt−1DT_t−1

❖❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥ ✐♥ ✭✶✮ ❞❡✜♥❡s ❛ ❧✐❦❡❧✐❤♦♦❞ ♠♦❞❡❧ ❢♦r yt✱ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ xt

❛♥❞Yt₋1✱ ✇❤✐❝❤ ✐s ✐s ❣✐✈❡♥ ❜②✿

yt=F xt+Hut

❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ ❝♦r♦❧❧❛r② ✭✷✳✶✽✮✱ t❤❡ ❛ ♣♦st❡r✐♦r✐ ❞✐str✐❜✉t✐♦♥ ♦❢(xt_|Yt₋1✱ ❝♦♥❞✐✲ t✐♦♥❛❧ ♦♥yt✱ ✐✳❡✳ ❞✐str✐❜✉t✐♦♥ ♦❢(xt|Yt) ✐s ❛s ❢♦❧❧♦✇s✿

(xt|Yt)∼csnn,q(µx|y,Σx|y, Dx|y, ϑx|y,∆x|y)

✇❤❡r❡✿

µx_|y =µ+ ΣFT(FΣFT +HΨHT)−

1

(y₋F µ)

Σx_|y = Σ−ΣFT(FΣFT +HΨuHT)−1HΣ

D_x|y =

    DΣ 0  −  

DΣFT

0



(FΣFT +HΨuHT)−1FΣ



Σ−

1

x_|y

ϑ_x|y =

  ϑ 0  −  

DΣFT

0



(FΣFT +HΨuHT)−1FΣ (yt−F µ)

∆_x|y =





∆ +DΣDT 0

0 I



− 



DΣFT

0



(FΣFT+HΨ_uHT)−

1





DΣFT

0





T

−D_x|yΣ_x|yD_xT_|y

◆♦✇ ✇❡ ❝❛♥ ♠♦✈❡ ♦♥ t♦ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤✱ ❣✐✈❡♥ ❢♦r♠✉❧❛❡ ❢♦r ❞✐str✐✲ ❜✉t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s(xt|Yt)✱t= 1,2, ..., T✱ ✐s st❛♥❞❛r❞✳

✹ ▲✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥

❚❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ st❛t❡ s♣❛❝❡ ♠♦❞❡❧ ✭✶✮ ✐s ❣✐✈❡♥ ❜②✿

L=p(y0)

T

Y

t=2

p(yt_|Yt₋1)

❇❡❝❛✉s❡Hut ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢Yt₋1✱ ✉s✐♥❣ t❤❡ ♠❡❛s✉r❡♠❡♥t ❡q✉❛t✐♦♥ ✇❡ ❣❡t✿

(yt_|Yt₋1) = (F xt+Hut|Yt₋1) = (F xt|Yt₋1+Hut) =F(xt|Yt₋1) +H(ut)

❙✐♥❝❡✱ ✐♥ t❤❡ ♥♦t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤✱ ❞✐str✐❜✉t✐♦♥ ♦❢ (xt_|Yt₋1) ✐s✿

(xt_|Yt₋1)∼csnp,qt(µ,Σ, D, ϑ,∆)

t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❛ ♣r✐♦r✐ ❞✐str✐❜✉t✐♦♥ ♦❢(yt|Yt₋1) =F(xt|Yt₋1) +H(ut) ✐s✿

(yt|Yt−1)∼csnqt+p(µy,Σy, Dy, ϑy,∆y)

✇✐t❤ ♣❛r❛♠❡t❡rs✿

µy =µF +µH, Σy = ΣF + ΣH, Dy =

ΣFDTF,ΣHDTH

T

Σ−_y1,

ϑy =

ϑT_F, ϑT_HT , ∆y = ∆F ⊗∆H + (DF ⊗DH)(ΣF ⊗ΣH)(DF ⊗DH)T+

−(DFΣF ⊕DHΣH) (Σy)−1 ΣFDFT ⊕ΣHDHT

✇❤❡r❡✿

µF =F µx, ΣF =FΣxFT, DF =DxΣxFTΣ−_F1

ϑF =ϑx, ∆F = ∆x+DxΣxDxT −DxΣxFTΣ_F−1FΣxDxT

t❤❡r❡❢♦r❡✿

p(yt|Yt₋1) =φp(yt;µy,Σy)

Φqt+p(Dy(yt−µy);ϑy,∆y) Φqt+p(0;ϑy,∆y+DyΣyDTy)

✭✶✮

❱❛❧✉❡ ♦❢ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ Lθ = p(Y|θ) ❝❛♥ ♥♦✇ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❢♦r ❣✐✈❡♥ θ✳

❱❛❧✉❡ ♦❢ Lθ ❝❛♥ ❜❡ ❢❡❡❞❡❞ ✐♥t♦ ❛♥② ♥✉♠❡r✐❝❛❧ ♦♣t✐♠✐③❛t✐♦♥ r♦✉t✐♥❡✳ ❚❤❡ ♠♦❞❡❧ ❝❛♥

❜❡ ❡st✐♠❛t❡❞✳

✺ ❊st✐♠❛t✐♦♥ ❡①❛♠♣❧❡

❆s ❛♥ ❡①❡♠♣❧❛r② ❡♠♣✐r✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡ ✇❡ ✉s❡❞ ❛ ♠♦❞❡❧ ❞❡s❝r✐❜❡❞ ✐♥ ❙❝❤♦r❢❤❡✐❞❡ ✭✷✵✵✵✮✳ ■t ✐s ❛ ♠❡❞✐✉♠ s❝❛❧❡ ❝❛s❤ ✐♥ ❛❞✈❛♥❝❡ ❉❙●❊ ❡❝♦♥♦♠② ✇✐t❤ t✇♦ ❞r✐✈❡rs✿ t❤❡ t❡❝❤♥♦❧♦❣② s❤♦❝❦ ❛♥❞ t❤❡ ♠♦♥❡t❛r② ♣♦❧✐❝② s❤♦❝❦ ✭♠♦♥❡② st♦❝❦ ❣r♦✇t❤ r❛t❡ s❤♦❝❦✮✳ ❚❤❡ t❡❝❤♥♦❧♦❣② s❤♦❝❦ǫA,t ❡♥t❡rs t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ Yt =

Kα

t(AtNt)1−α ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿

logAt=γ+ logAt−1+ǫA,t

❛♥❞ t❤❡ ♠♦♥❡t❛r② ♣♦❧✐❝② s❤♦❝❦ǫM,t❞✐st✉r❜s t❤❡ ♣❛t❤ ♦❢ ❣r♦✇t❤ ♦❢ ♠♦♥❡②mt= M_Mt+1_t

❛s ❢♦❧❧♦✇s✿

lnmt= (1−ρ) lnm⋆+ρlnmt−1+ǫM,t

✇❤❡r❡m⋆ ❞❡♥♦t❡s ❡q✉✐❧✐❜r✐✉♠ ❣r♦✇t❤ r❛t❡ ♦❢ ♠♦♥❡② s✉♣♣❧② ✐♥ t❤❡ ❡❝♦♥♦♠②✳ ❲❤❡♥

♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✐s ✉s❡❞✱ ✇❡ ❛ss✉♠❡ t❤❛t✿





ǫA,t

ǫM,t



∼ N     0 0  ,   σ2 A 0

0 σ2_M



 

