trt t t srs ss - - PowerPoint PPT Presentation

■♥tr♦❞✉❝t✐♦♥ t♦ t✐♠❡ s❡r✐❡s ❛♥❛❧②s✐s

❙t❛t✐st✐❝❛❧ ♠♦❞❡❧❧✐♥❣✿ ❚❤❡♦r② ❛♥❞ ♣r❛❝t✐❝❡

• ✐❧❧❡s ●✉✐❧❧♦t

❣✐❣✉❅✐♠♠✳❞t✉✳❞❦

◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶ ✴ ✸✺

✶

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❛♥❞ ♦✉t❧✐♥❡

✷

Pr♦❜❛❜✐❧t② ❝♦♥❝❡♣ts ❢♦r t✐♠❡ s❡r✐❡s

✸

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

✹

❆✉t♦✲r❡❣r❡ss✐✈❡ ♠♦❞❡❧s

✺

❘❡❢❡r❡♥❝❡s

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷ ✴ ✸✺

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❛♥❞ ♦✉t❧✐♥❡

❋✐♥❛♥❝❡ ❞❛t❛

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✸ ✴ ✸✺

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❛♥❞ ♦✉t❧✐♥❡

• ❡♦♣❤②s✐❝❛❧ ❞❛t❛
• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✹ ✴ ✸✺

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❛♥❞ ♦✉t❧✐♥❡

❇✐♦♠❡❞✐❝❛❧ ❞❛t❛

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✺ ✴ ✸✺

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❛♥❞ ♦✉t❧✐♥❡

❈✐✈✐❧ ❡♥❣✐♥❡❡r✐♥❣ ❞❛t❛

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✻ ✴ ✸✺

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❛♥❞ ♦✉t❧✐♥❡

❉✐✛❡r❡♥t s❝✐❡♥t✐✜❝✴t❡❝❤♥✐❝❛❧ ✜❡❧❞s ❜✉t ❝♦♠♠♦♥ ❛s♣❡❝ts ✐♥ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡s ■

❈♦♠♠♦♥ q✉❡st✐♦♥s✿ ❆ ❝❡♥tr❛❧ q✉❡st✐♦♥✿ ♣r❡❞✐❝t✐♦♥ ♦❢ ❢✉t✉r❡ ✈❛❧✉❡s ❖t❤❡r q✉❡st✐♦♥s✿

✐s t❤❡ ♣❤❡♥♦♠❡♥♦♥ ♣❡r✐♦❞✐❝❄ ✇❤❛t ✐s t❤❡ ♣❡r✐♦❞❄ ✐s t❤❡r❡ ❛♥② tr❡♥❞ ❄ ❝❛♥ ✐t ❜❡ ❡st✐♠❛t❡❞ ✭✜❧t❡r✐♥❣ ♣r♦❜❧❡♠✮ ✐s t❤❡r❡ ❛ t✐♠❡ str✉❝t✉r❡❄

❈♦♠♠♦♥ ❞❛t❛ ❢❡❛t✉r❡s✿ ❖❜s❡r✈❛t✐♦♥s ❝♦♠❡ ✐♥ ❛ s♣❡❝✐✜❝ ♦r❞❡r ✭t✐♠❡✦ ❤❡r❡ ♦❜s✳ ✐♥❞❡① ♥♦t ❥✉st ❛ ❧❛❜❡❧✦✮ ❚❤❡r❡ ✐s ❛ ♣❛tt❡r♥ ✐♥ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ❛❝r♦ss t✐♠❡ ❚❤❡r❡ ✐s ❡rr❛t✐❝ ✈❛r✐❛t✐♦♥ ❛✇❛② ❢r♦♠ t❤❡ ♠❛✐♥ ✏♣❛tt❡r♥✑

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✼ ✴ ✸✺

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❛♥❞ ♦✉t❧✐♥❡

❉✐✛❡r❡♥t s❝✐❡♥t✐✜❝✴t❡❝❤♥✐❝❛❧ ✜❡❧❞s ❜✉t ❝♦♠♠♦♥ ❛s♣❡❝ts ✐♥ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡s ■■

• ♦❛❧ ♦❢ t❤❡ ❧❡❝t✉r❡✿ ❣✐✈❡ ❛ ✢❛✈♦✉r ♦❢ st❛t✐st✐❝❛❧ t♦♦❧s t♦ ♠♦❞❡❧ ✈❛r✐❛t✐♦♥

❛❝r♦ss t✐♠❡✳ ■♥ ❧❡❝t✉r❡s ♦♥ ❧✐♥❡❛r ♠♦❞❡❧✱ ✇❡ ❤❛❞ y = βx + ε✳ ❚❤❡ ❢♦❝✉s ✇❛s ♦♥ t❤❡ ✏♣❛tt❡r♥✑✱ t❤❡ r❡st ✇❛s ✐✳✐✳❞ ♥♦✐s❡✳ ■♥ t❤✐s ❧❡❝t✉r❡✱ t❤❡ ✏r❡s✐❞✉❛❧ ✈❛r✐❛t✐♦♥✑ ✐s ❛s ✐♠♣♦rt❛♥t ❛s t❤❡ ♠❛✐♥ ✏tr❡♥❞✑✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✽ ✴ ✸✺

Pr♦❜❛❜✐❧t② ❝♦♥❝❡♣ts ❢♦r t✐♠❡ s❡r✐❡s

❙♦♠❡ ❞❡✜♥✐t✐♦♥s ■

❚✐♠❡ s❡r✐❡s✿ ❛ s❡t ♦❢ ♦❜s❡r✈❛t✐♦♥s ♦❢ ❛ ✈❛r✐❛❜❧❡ ❛t ❞✐✛❡r❡♥t t✐♠❡ ♣♦✐♥ts (yt)t=1,...,T = (y1, ..., yT ) ❙t♦❝❤❛st✐❝ ♣r♦❝❡ss✿ ❛ s❡t ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦❜s❡r✈❡❞ ❛t ❞✐✛❡r❡♥t t✐♠❡ ♣♦✐♥ts (Yt)t=1,...,T = (Y1, ..., YT )✳ ■❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛r❡ ♠✉t✉❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✱ t❤❡ ♣r♦❝❡ss ✐s ❥✉st ❛ ♥♦✐s❡ s❡❡♥ ❜❡❢♦r❡✳ ❘❡❛❧✐③❛t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss✿ ❛ s✐♥❣❧❡ s❡t ♦❢ ♦❜s❡r✈❛t✐♦♥s (yt)t=1,...,T = (y1, ..., yT )✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✾ ✴ ✸✺

Pr♦❜❛❜✐❧t② ❝♦♥❝❡♣ts ❢♦r t✐♠❡ s❡r✐❡s

❙♦♠❡ ❞❡✜♥✐t✐♦♥s ■■

❈♦♥t✐♥✉♦✉s✲t✐♠❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss✿ ♠♦❞❡❧ ❢♦r ❛ t✐♠❡ s❡r✐❡s ✇❤✐❝❤ ✐s ✭♦r ❝♦✉❧❞ ❜❡ ✐♥ ♣r✐♥❝✐♣❧❡✮ ♦❜s❡r✈❡❞ ❝♦♥t✐♥✉♦✉s❧② ✐♥ t✐♠❡✳ ❊①❛♠♣❧❡✿t❡♠♣❡r❛t✉r❡ ✐♥ ▲②♥❣❜② ❉✐s❝r❡t❡✲t✐♠❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss✿ ♠♦❞❡❧ ❢♦r ❛ t✐♠❡ s❡r✐❡s ✇❤✐❝❤ ✐s ❜② ♥❛t✉r❡ ❞❡✜♥❡❞ ♦♥❧② ❛t s♦♠❡ ❞✐s❝r❡t❡ t✐♠❡ ♣♦✐♥ts✳ ❊①❛♠♣❧❡✿ ❡①❝❤❛♥❣❡ r❛t❡ ❡✉r♦✴❞♦❧❧❛r✱ ●❉P ♦❢ ❛ ❝♦✉♥tr②✱ ♠❛①✳ ❞❛✐❧② t❡♠♣❡r❛t✉r❡ ❙❛♠♣❧✐♥❣ ❢r❡q✉❡♥❝②✿ ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❡❞ ❞❛t❛ ♣♦✐♥t ♣❡r t✐♠❡ ✉♥✐t ❙✉♣♣♦rt✿ t❤❡ t✐♠❡ ♣❡r✐♦❞ ♦✈❡r ✇❤✐❝❤ t❤❡ ✈❛r✐❛❜❧❡ ✐s ❛✈❡r❛❣❡❞✳ ❉♦ ♥♦t ♠✐① ✉♣ ❛✈❡r❛❣❡ ❞❛✐❧② t❡♠♣❡❛rt✉r❡s ✇✐t❤ ✏✐♥st❛♥t✑ t❡♠♣❡r❛t✉r❡ ♠❡❛s✉r❡❞ ❛t ♥♦♦♥ ❡✈❡r② ❞❛②✳ ❇♦t❤ ❛r❡ ♦❜s❡r✈❡❞ ❛t t❤❡ s❛♠❡ ❢r❡q✉❡♥❝② ❜✉t ♥♦t ♦✈❡r t❤❡ s❛♠❡ t✐♠❡ s✉♣♣♦rt✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✵ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❋✐rst ❛♥❞ s❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ■

❉❡✜♥✐t✐♦♥✿ ▼❡❛♥ ❢✉♥❝t✐♦♥ ❋♦r (Yt) ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss✱ t❤❡ ♠❡❛♥ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s µ(t) = E[Yt]

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✶ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❋✐rst ❛♥❞ s❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ■■

◆❇✿ ■❢ n r❡❛❧✐③❛t✐♦♥s y(k)

t

♦❢ (Yt) ❤❛✈❡ ❜❡❡♥ ♦❜s❡r✈❡❞ ✭❡✳❣✳ ✈❛r✐❛t✐♦♥ ♦❢ ❣❧✉❝♦s❡ ❛❝r♦ss t✐♠❡ ❢♦r n ♣❛t✐❡♥ts✮✱ t❤❡♥ µ(t) ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❛s 1/n n

k=1 y(k) t

✳ ❚❤❡r❡ ✐s ♦❢t❡♥ ♦♥❧② ♦♥❡ r❡❛❧✐③❛t✐♦♥ ❛✈❛✐❧❛❜❧❡ ❛♥❞ t❤❡ ❡st✐♠❛t♦r ❛❜♦✈❡ ❞♦❡s ♥♦t ♠❛❦❡ s❡♥s❡✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ♠❡❛♥ ❢✉♥❝t✐♦♥ ✐s s♦♠❡❤♦✇ ❛r❜✐tr❛r②✳ ❚❤✐♥❦ ♦❢ t❤❡ ♠❡❛♥ ❡①❝❤❛♥❣❡ r❛t❡ ❉♦❧❧❛r✴❊✉r♦ ✐♥ ◆♦✈❡♠❜❡r ✷✵ ✷✵✶✷❄

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✷ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❉❡✜♥✐t✐♦♥✿ ❱❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❋♦r (Yt) ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss✱ t❤❡ ✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s σ2(t) = V ar[Yt] ❲✐t❤ s❡✈❡r❛❧ r❡❛❧✐③❛t✐♦♥s ❛t ❤❛♥❞✱ ♦♥❡ ❝♦✉❧❞ ❡st✐♠❛t❡ σ2(t) ❜②

• σ2(t) = 1/(n − 1) n

k=1(y(k) t

− ¯ yt)2

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✸ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ■

❉❡✜♥✐t✐♦♥✿ ❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss (Yt)✱ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s C(t, t′) = Cov[Yt, Y ′

t ]

◆❇✿ C(t, t) = Cov[Yt, Yt] = σ2(t)

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✹ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ■■

❉❡✜♥✐t✐♦♥✿ ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ❋♦r ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss (Yt)✱ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s ρ(t, t′) = Cor[Yt, Y ′

t ]

◆❇✿ ρ(t, t) = Cor[Yt, Yt] = 1

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✺ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❙t❛t✐♦♥❛r✐t② ■

❉❡✜♥✐t✐♦♥✿ ❋✐rst✲♦r❞❡r st❛t✐♦♥❛r✐t② ❆ st♦❝❤❛st✐❝ ♣r♦❝❡ss (Yt) ✐s ✜rst✲♦r❞❡r st❛t✐♦♥❛r② ✐❢ ✐ts ♠❡❛♥ µ(t) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t✐♠❡✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✻ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❙t❛t✐♦♥❛r✐t② ■■

❉❡✜♥✐t✐♦♥✿ ❙❡❝♦♥❞✲♦r❞❡r st❛t✐♦♥❛r✐t② ❆ st♦❝❤❛st✐❝ ♣r♦❝❡ss (Yt) ✐s s❡❝♦♥❞✲♦r❞❡r st❛t✐♦♥❛r② ✐❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♦❜s❡r✈❛t✐♦♥s ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ t✐♠❡ ❧❛❣ ❜❡t✇❡❡♥ t❤❡♠✳ ❋♦r♠❛❧❧②✱ t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥ K s✉❝❤ t❤❛t C(t, t′) = K(t − t′)✳ ■♥ t❤✐s❡ ❝❛s❡✱ ✇✐t❤ ❛ s❧✐❣❤t ❛❜✉s❡✱ ✇❡ ♦❢t❡♥ ✇r✐t❡s✿ C(t, t′) = C(t − t′) = C(k)✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✼ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

• ❛✉ss✐❛♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ■

❉❡✜♥✐t✐♦♥✿ ●❛✉ss✐❛♥ ♣r♦❝❡ss ❆ st♦❝❤❛st✐❝ ♣r♦❝❡ss (Yt) ✐s ●❛✉ss✐❛♥ ✐❢ ❢♦r ❛♥② s✉s❜❡t ♦❢ t✐♠❡ ♣♦✐♥ts t1, ..., tn ❛♥❞ ❛♥② ❢❛♠✐❧② ♦❢ ✇❡✐❣❤ts λ1, ..., λn t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡

i λiYti

✐s ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✽ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

• ❛✉ss✐❛♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ■■

❚❤❡♦r❡♠✿ ❉❡♥s✐t② ♦❢ ❛ ♥♦♥✲❞❡❣❡♥❡r❛t❡❞ ●❛✉ss✐❛♥ ♣r♦❝❡ss ■❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ ♦❢ ❛ ●❛✉ss✐❛♥ ♣r♦❝❡ss (Yt) ♦❜s❡r✈❡❞ ❛t t✐♠❡ ♣♦✐♥ts t1, ..., tn ✐s ♥♦♥✲s✐♥❣✉❧❛r✱ (Yt1, ..., Ytn) ❛❞♠✐ts ❛ ❞❡♥s✐t② ♦❢ t❤❡ ❢♦r♠✿ f(y) = 1 |Σ|1/2(2π)n/2 exp

• −1

2(y − µ)tΣ−1(y − µ)

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✶✾ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❊①❛♠♣❧❡ ♦❢ ●❛✉ss✐❛♥ ♣r♦❝❡ss❡s

❲❡ ❝♦♥s✐❞❡r ❜❡❧♦✇ ❛ s❡❝♦♥❞✲♦r❞❡r st❛t✐♦♥❛r② ●❛✉ss✐❛♥ ♣r♦❝❡ss ✇✐t❤ ♠❡❛♥ ✵ ❛♥❞ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ C(t, t′) = C(t − t′) = C(k) = exp(−|k|/α)✳ ❲✐t❤ t❤r❡❡ ❞✐✛❡r❡♥t ✈❛❧✉❡s ❢♦r t❤❡ s❝❛❧❡ ♣❛r❛♠❡t❡r α = 3, 100, 1000

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✵ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 h C(h) 3 100 1000 Time Y 200 400 600 800 1000 −3 −1 1 2 3 4

❘❡❛❧✐③❛t✐♦♥s ♦❢ t❤r❡❡ ●❛✉ss✐❛♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ ❝♦✈❛r✐❛♥❝❡s✳ ❇❧❛❝❦ ❝✉r✈❡✿ ✏s❤♦rt r❛♥❣❡✑ ❞❡♣❡♥❞❡♥❝❡ ✭α = 3✮✳ ■t ✐s ✈❡r② ❡rr❛t✐❝ ❛♥❞ t❤❡ ❝✉r✈❡ ✈✐s✐ts ❛ ❜r♦❛❞ r❛♥❣❡ ♦❢ ✈❛❧✉❡s ✇✐t❤✐♥ ❛ s❤♦rt ♣❡r✐♦❞ ♦❢ t✐♠❡✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✶ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❊st✐♠❛t✐♥❣ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥

❆ss✉♠✐♥❣ t❤❛t Yt ✐s s❡❝♦♥❞✲♦r❞❡r st❛t✐♦♥❛r② ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ C(k) ✇❡ ❝❛♥ ❡st✐♠❛t❡ t❤❡ ✭❝♦♥st❛♥t✮ ♠❡❛♥ ❜② ˆ µ = ¯ Y =

T

• t=1

Yt/T ❛♥❞ C(k) ❜② ˆ C(k) = 1 Nt,t′

• |t−t′|=k

(Yt − ¯ Y )(Yt′ − ¯ Y ) ✇❤❡r❡ Nt,t′ = #{|t − t′| = k}

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✷ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❊st✐♠❛t✐♥❣ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ✇✐t❤ ❘

❚❤❡ ❘ ❢✉♥❝t✐♦♥ ❛❝❢✿ ❛❝❢✭②✱t②♣❡✱❧❛❣✳♠❛①✱♣❧♦t❂❚❘❯❊✮ ✇✐t❤ t②♣❡❂✬❝♦✈❛r✐❛♥❝❡✬ ♦r t②♣❡❂✬❝♦rr❡❧❛t✐♦♥✬ P❧♦t t❤❡ ❡♠♣✐r✐❝❛❧ ❝♦✈❛r✐❛♥❝❡ ♦r ❝♦rr❡❧❛t✐♦♥✳ ❚❤❡ ❘ ♦❜❥❡❝t r❡t✉r♥❡❞ ✐s ♦❢ ❝❧❛ss ❛❝❢ ❛♥❞ ❝♦♥t❛✐♥s ❝♦✈❛r✐❛♥❝❡ ♦r ❝♦rr❡❧❛t✐♦♥ ✈❛❧✉❡s ❛t ❞✐✛❡r❡♥t t✐♠❡ ❧❛❣s✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✸ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❘ ❝♦❞❡ ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥

♣❛r✭♠❢r♦✇❂❝✭✸✱✶✮✱♠❛r❂❝✭✹✱✹✱✶✱✷✮✮ ❛❝❢✭②✶✱❧❛❣✳♠❛①❂✶✵✵✱t②♣❡❂✬❝♦✈✬✮ ❧✐♥❡s✭t✐♠❡s✱❡①♣✭✲t✐♠❡s✴❛❧♣❤❛✶✮✮ ❛❝❢✭②✷✱❧❛❣✳♠❛①❂✶✵✵✱t②♣❡❂✬❝♦✈✬✮ ❧✐♥❡s✭t✐♠❡s✱❡①♣✭✲t✐♠❡s✴❛❧♣❤❛✷✮✱❝♦❧❂✷✮ ❛❝❢✭②✸✱❧❛❣✳♠❛①❂✶✵✵✱t②♣❡❂✬❝♦✈✬✮ ❧✐♥❡s✭t✐♠❡s✱❡①♣✭✲t✐♠❡s✴❛❧♣❤❛✸✮✱❝♦❧❂✸✮

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✹ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❊st✐♠❛t❡❞ ❝♦✈❛r✐❛♥❝❡s

20 40 60 80 100 −0.2 0.2 0.4 0.6 0.8 1.0 Lag ACF (cov) 20 40 60 80 100 0.3 0.4 0.5 0.6 0.7 0.8 Lag ACF (cov) 20 40 60 80 100 0.3 0.4 0.5 0.6 0.7 Lag ACF (cov)

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✺ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❊st✐♠❛t❡❞ ❝♦rr❡❧❛t✐♦♥s

20 40 60 80 100 −0.2 0.2 0.4 0.6 0.8 1.0 Lag ACF 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✻ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❊st✐♠❛t✐♥❣ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥✱ ✇❤❛t ❢♦r❄

P✉r❡❧② ❞❡s❝r✐♣t✐✈❡ ♣✉r♣♦s❡✿ ❤♦✇ ❢❛r ✐s t❤❡ t✐♠❡ s❡r✐❡s ❢r♦♠ ❛ ♣✉r❡ ♥♦✐s❡❄ ❆t ✇❤❛t ❧❛❣ ❞♦❡s ❞❡❝♦rr❡❧❛t✐♦♥ ♦❝❝✉r❄ ■❢ Z(t) = X(t) + Y (t) ✇❤❡r❡ Y ❛♥❞ Z ❛r❡ ✉♥♦❜s❡r✈❡❞ ♣r♦❝❡ss❡s ✇✐t❤ ❦♥♦✇♥ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥✱ ♦♥❡ ❝❛♥ ♣r❡❞✐❝t Xt ❢r♦♠ ♦❜s❡r✈❛t✐♦♥s ♦❢ Zt ❛t t✐♠❡ 1, ..., T ✭✜❧t❡r✐♥❣✮✳ ■❢ Y ✐s ♦❜s❡r✈❡❞ ❛t t✐♠❡s 1, ..., T ♦♥❡ ❝❛♥ ♣r❡❞✐❝t YT+k ❢♦r ❛♥② k✳ ❚❤❡ ❧❛tt❡r ♣r♦❜❧❡♠s r❡q✉✐r❡ s♦♠❡ ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ ❝♦✈❛r✐❛♥❝❡ str✉❝t✉r❡ ✭❧✐♥❡❛r ♣r❡❞✐❝t✐♦♥ ♠❡t❤♦❞s✮✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✼ ✴ ✸✺

❙❡❝♦♥❞ ♦r❞❡r ❝❤❛r❛❝t❡r✐❝s ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ♦♥ s❡❝♦♥❞✲♦r❞❡r ❝❤❛r❛❝t❡r✐st✐❝s

❚❤❡ ❝♦✈❛r✐❛♥❝❡ ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ♣r♦✈✐❞❡ ❛ ❣❧♦❜❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss✳ ❚❤❡② ❛r❡ ♣♦✇❡r❢✉❧❧ ❜✉t ❝❛♥ ❜❡❝♦♠❡ ♥✉♠❡r✐❝❛❧❧② ❞❡♠❛♥❞✐♥❣ ❢♦r ❧❛r❣❡ ❞❛t❛s❡ts ❚❤❡ ♥❡①t s❧✐❞❡s ♣r♦✈✐❞❡ ❛♥ ❛❧t❡r♥❛t✐✈❡ ♠♦❞❡❧❧✐♥❣ ♣♦✐♥t ♦❢ ✈✐❡✇ ❜❛s❡❞ ♦♥ ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❧♦❝❛❧ t✐♠❡ str✉❝t✉r❡✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✽ ✴ ✸✺

❆✉t♦✲r❡❣r❡ss✐✈❡ ♠♦❞❡❧s

❋✐rst ♦r❞❡r ❛✉t♦✲r❡❣r❡ss✐✈❡ ♠♦❞❡❧s ■

■♥t✉✐t✐✈❡❧②✱ t♦♠♦rr♦✇✬s ❊✉r♦✴❉♦❧❧❛r ❡①❝❤❛♥❣❡ r❛t❡ ✐s t♦❞❛②✬s r❛t❡ ✰ ❛ s♠❛❧❧ ♣❡rt✉❜❛t✐♦♥✳ t♦♠♦rr♦✇✬s t❡♠♣❡r❛t✉r❡ ✐s t♦❞❛②✬s t❡♠♣❡r❛t✉r❡ ✰ ❛ s♠❛❧❧ ♣❡rt✉❜❛t✐♦♥ ❉❡✜♥✐t✐♦♥✿ ❋✐rst ♦r❞❡r ❛✉t♦✲r❡❣r❡ss✐✈❡ ♣r♦❝❡ss ❆ ✜rst ♦r❞❡r ❛✉t♦✲r❡❣r❡ss✐✈❡ ♣r♦❝❡ss ✭❆❘✭✶✮ ❢♦r s❤♦rt✮ ✐s ❛ ❞✐s❝r❡t❡✲t✐♠❡ ♣r♦❝❡ss s✉❝❤ t❤❛t Yt = c + φYt−1 + εt ✇❤❡r❡ c ❛♥❞ φ ❛r❡ ❝♦♥st❛♥t ♣❛r❛♠❡t❡rs ❛♥❞ (εt) ✐s ❛♥ ③❡r♦✲♠❡❛♥ ✐✳✐✳❞ r❛♥❞♦♠ ♥♦✐s❡✳ ❆ss✉♠✐♥❣ t❤❛t εt ∼ N(0, σ2

ε) ♠❛❦❡s t❤❡ ♣r♦❝❡ss ❛❜♦✈❡ ❛ ●❛✉ss✐❛♥ ❆❘✶✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✷✾ ✴ ✸✺

❆✉t♦✲r❡❣r❡ss✐✈❡ ♠♦❞❡❧s

❙❡❝♦♥❞ ♦r❞❡r ♣r♦♣❡rt✐❡s ♦❢ ❆❘✭✶✮ ♣r♦❝❡ss❡s ■

❯♥❞❡r ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ✐s ❛♥ ❆❘✭✶✮ ♣r♦❝❡ss st❛t✐♦♥❛r②❄ ▼❡❛♥✿ Yt = c + φYt−1 + εt = ⇒ E(Yt) = c + φE(Yt−1) + 0 ❋✐rst✲♦r❞❡r st❛r✐♦♥❛r✐t② r❡q✉✐r❡s t❤❛t µ = c + φµ ❤❡♥❝❡ µ = c/(1 − φ)

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✸✵ ✴ ✸✺

❆✉t♦✲r❡❣r❡ss✐✈❡ ♠♦❞❡❧s

❙❡❝♦♥❞ ♦r❞❡r ♣r♦♣❡rt✐❡s ♦❢ ❆❘✭✶✮ ♣r♦❝❡ss❡s ■■

❱❛r✐❛♥❝❡✿ Yt = c + φYt−1 + εt = ⇒ V (Yt) = φ2V (Yt−1) + σ2

ε

❙❡❝♦♥❞✲♦r❞❡r st❛t✐♦♥❛r✐t② r❡q✉✐r❡s t❤❛t σ2

Y = φ2σ2 Y + σ2 ε✳

■❢ |φ| < 1 t❤❡ ❛❜♦✈❡ ✐s ❛❝❤✐❡✈❡❞ ✐❢ σ2

Y = σ2 ε/(1 − φ2)

❈♦♥✈❡rs❡❧②✱ ❢♦r φ s✉❝❤ t❤❛t |φ| < 1✱ ✐❢ E(Y0) = c/(1 − φ) ❛♥❞ V (Y0) = σ2

ε/(1 − φ2)✱

t❤❡♥ (Yt) ❤❛s st❛t✐♦♥❛r② ♠❡❛♥ ❛♥❞ ✈❛r✐❛♥❝❡s✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✸✶ ✴ ✸✺

❆✉t♦✲r❡❣r❡ss✐✈❡ ♠♦❞❡❧s

❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ♦❢ ❛♥ ❆❘✭✶✮✿ ❋♦r k > 0✱ Cov(Yt, Yt+k) = Cov(Yt, c + φYt+k−1 + εt+k) = φCov(Yt, Yt+k−1) ❋♦r k = 1✱ ❛ss✉♠✐♥❣ st❛t✐♦♥❛r✐t② ♦❢ t❤❡ ✈❛r✐❛♥❝❡✱ Cov(Yt, Yt+1) = φσ2

Y

❇② ✐♥❞✉❝t✐♦♥✱ Cov(Yt, Yt+k) = φkσ2

Y ∝ ek ln φ

❚❤❡ ❝♦✈❛r✐❛♥❝❡ ❞❡❝❛②s ❛t ❛♥ ❡①♣♦♥❡♥t✐❛❧ r❛t❡✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✸✷ ✴ ✸✺

❆✉t♦✲r❡❣r❡ss✐✈❡ ♠♦❞❡❧s

❈♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ❛ ●❛✉ss✐❛♥ ❆❘✭✶✮ ♣r♦❝❡ss

❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t εt ∼ N(0, σ2

ε)✱

t❤❡ ❆❘✭✶✮ str✉❝t✉r❡ Yt = c + φYt−1 + εt ✐♠♣❧✐❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥✿ Yt|Yt−1 ∼ N(c + φYt−1, σ2

ε)

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✸✸ ✴ ✸✺

❆✉t♦✲r❡❣r❡ss✐✈❡ ♠♦❞❡❧s

▼❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ✐♥❢❡r❡♥❝❡ ❢♦r ❛ ●❛✉ss✐❛♥ ❆❘✭✶✮ ♣r♦❝❡ss

❉❛t❛✿ y = (y1, ..., yT )✳ P❛r❛♠❡t❡r✿ θ = (c, φ, σε)✳ L(y; θ) = L(y1, ...., yT ; c, φ, σε) = f(y1, ...., yT ) = f(y1)f(y2, ...., yT |y1) = f(y1)

T

• t=2

f(yt|yt−1) = f(y1)

T

• t=2

exp

• − 1

2σ2

ε

(yt − c − φyt−1)2

• ❚❤❡ t❡r♠ f(y1) ❝❛♥ ❜❡ ❞❡❛❧t ✇✐t❤ ❡✐t❤❡r ❜② ❞✐sr❡❣❛r❞✐♥❣ ✐t ♦r ❜② ♠❛❦✐♥❣

❛ss✉♠♣t✐♦♥s ♦♥ t❤✐s ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞ ❝❛♥ ❜❡ ♠❛①✐♠✐③❡❞ ❡✳❣✳ ✇✐t❤ t❤❡ ❘ ❢✉♥❝t✐♦♥ ♦♣t✐♠✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✸✹ ✴ ✸✺

❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s

❍✳ ▼❛❞s❡♥✱ ❚✐♠❡ ❙❡r✐❡s ❆♥❛❧②s✐s✱ ❈❤❛♣♠❛♥ ✫ ❍❛❧❧✴❈❘❈✱ ✷✽✵ ♣♣✳✱ ✷✵✵✽✳

• ✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅✐♠♠✳❞t✉✳❞❦✮

■♥tr♦ t✐♠❡ s❡r✐❡s ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✹ ✸✺ ✴ ✸✺