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▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s

◗❯❆▲■▼❆❉❖❙✿ ❆t❡❧✐❡r ◗✉❛❧✐té ❞❡s ♠❛ss❡s ❞❡ ❞♦♥♥é❡s s❝✐❡♥t✐✜q✉❡s

❙✳ ■♦✈❧❡✛

▲❛❜♦r❛t♦✐r❡ P❛✉❧ P❛✐♥❧❡✈é

✷✸ ❏✉✐♥ ✷✵✶✼

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s

❙♦♠♠❛✐r❡

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❲❤❛t ✐s ❈❧✉st❡r✐♥❣ ❄ ❊①❛♠♣❧❡ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s ▼✐①t✉r❡ ▼♦❞❡❧ ❛♥❞ ▼✐①❡❞ ❉❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❲❤❛t ✐s ❈❧✉st❡r✐♥❣ ❄

❈❧✉st❡r✐♥❣ ✐s t❤❡ ❝❧✉st❡r ❜✉✐❧❞✐♥❣ ♣r♦❝❡ss

◮ ❚❤❡ t❡r♠ ❉❛t❛ ❈❧✉st❡r✐♥❣ ✜rst ❛♣♣❡❛r❡❞ ✐♥ ✶✾✺✹ ✭❛❝❝♦r❞✐♥❣ t♦

❏❙❚❖❘✮ ✐♥ ❛♥ ❛rt✐❝❧❡ ❞❡❛❧✐♥❣ ✇✐t❤ ❛♥t❤r♦♣♦❧♦❣✐❝❛❧ ❞❛t❛✱

◮ ▼❛♥②✱ ♠❛♥② ❡①✐st✐♥❣ ♠❡t❤♦❞s

✭❤tt♣s✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴✇✐❦✐✴❈❛t❡❣♦r②✿ ❉❛t❛❴❝❧✉st❡r✐♥❣❴❛❧❣♦r✐t❤♠s✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❲❤❛t ✐s ❈❧✉st❡r✐♥❣ ❄

❈❧✉st❡r✐♥❣ ✐s t❤❡ ❝❧✉st❡r ❜✉✐❧❞✐♥❣ ♣r♦❝❡ss

◮ ❚❤❡ t❡r♠ ❉❛t❛ ❈❧✉st❡r✐♥❣ ✜rst ❛♣♣❡❛r❡❞ ✐♥ ✶✾✺✹ ✭❛❝❝♦r❞✐♥❣ t♦

❏❙❚❖❘✮ ✐♥ ❛♥ ❛rt✐❝❧❡ ❞❡❛❧✐♥❣ ✇✐t❤ ❛♥t❤r♦♣♦❧♦❣✐❝❛❧ ❞❛t❛✱

◮ ▼❛♥②✱ ♠❛♥② ❡①✐st✐♥❣ ♠❡t❤♦❞s

✭❤tt♣s✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴✇✐❦✐✴❈❛t❡❣♦r②✿ ❉❛t❛❴❝❧✉st❡r✐♥❣❴❛❧❣♦r✐t❤♠s✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❲❤❛t ✐s ❈❧✉st❡r✐♥❣ ❄

❈❧✉st❡r✐♥❣ ✐s t❤❡ ❝❧✉st❡r ❜✉✐❧❞✐♥❣ ♣r♦❝❡ss

◮ ❚❤❡ t❡r♠ ❉❛t❛ ❈❧✉st❡r✐♥❣ ✜rst ❛♣♣❡❛r❡❞ ✐♥ ✶✾✺✹ ✭❛❝❝♦r❞✐♥❣ t♦

❏❙❚❖❘✮ ✐♥ ❛♥ ❛rt✐❝❧❡ ❞❡❛❧✐♥❣ ✇✐t❤ ❛♥t❤r♦♣♦❧♦❣✐❝❛❧ ❞❛t❛✱

◮ ▼❛♥②✱ ♠❛♥② ❡①✐st✐♥❣ ♠❡t❤♦❞s

✭❤tt♣s✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴✇✐❦✐✴❈❛t❡❣♦r②✿ ❉❛t❛❴❝❧✉st❡r✐♥❣❴❛❧❣♦r✐t❤♠s✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❲❤❛t ✐s ❈❧✉st❡r✐♥❣ ❄

◆❡✇ ❝❤❛❧❧❡♥❣❡s

◆❡❡❞ t♦ ❛❧❣♦r✐t❤♠s ❢♦r ❇✐❣✲❉❛t❛ ❛♥❞ ❈♦♠♣❧❡① ❉❛t❛✳ ■♥ ♣❛rt✐❝✉❧❛r ♠✐①❡❞ ❢❡❛t✉r❡s ❛♥❞ ♠✐ss✐♥❣ ✈❛❧✉❡s

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✹ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊①❛♠♣❧❡

❆♥ ❡①❛♠♣❧❡

❏♦✐♥t ✇♦r❦s ✇✐t❤ ❈❤r✐st♦♣❤❡ ❇✐❡r♥❛❝❦✐ ✭❤❡❛❞ ♦❢ t❤❡ ■♥r✐❛ ▼♦❞❛❧ t❡❛♠✮✱ ❱✐♥❝❡♥t ❱❛♥❞❡✇❛❧❧❡✱ ❑♦♠✐ ◆❛❣❜❡✱✳✳✳ ❈♦♥tr❛❝t ❢♦r ❛ ❧❛r❣❡ ❧✐♥❣❡r✐❡ st♦r❡✿ ✧❈❧✉st❡r✲ ✐♥❣ ❝❛s❤ r❡❝❡✐♣ts ♦❢ t❤❡ ❈✉st♦♠❡rs ✇✐t❤ ❛ ❧♦②❛❧t② ❝❛r❞✧

◮ ✷✽ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ♣r♦❞✉❝ts✱ ◮ ✻ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ❝♦st✉♠❡rs✱ ◮ ✽ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ st♦r❡s✱ ◮ n = ✷, ✽✾✾, ✵✸✵ r❡❝❡✐♣ts✳

❙♦♠❡ ♠❡❛♥✐♥❣❢✉❧ ✈❛r✐❛❜❧❡s ✇✐t❤ ♠✐ss✐♥❣ ✈❛❧✲ ✉❡s✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✺ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊①❛♠♣❧❡

❆♥ ❡①❛♠♣❧❡

❏♦✐♥t ✇♦r❦s ✇✐t❤ ❈❤r✐st♦♣❤❡ ❇✐❡r♥❛❝❦✐ ✭❤❡❛❞ ♦❢ t❤❡ ■♥r✐❛ ▼♦❞❛❧ t❡❛♠✮✱ ❱✐♥❝❡♥t ❱❛♥❞❡✇❛❧❧❡✱ ❑♦♠✐ ◆❛❣❜❡✱✳✳✳ ❈♦♥tr❛❝t ❢♦r ❛ ❧❛r❣❡ ❧✐♥❣❡r✐❡ st♦r❡✿ ✧❈❧✉st❡r✲ ✐♥❣ ❝❛s❤ r❡❝❡✐♣ts ♦❢ t❤❡ ❈✉st♦♠❡rs ✇✐t❤ ❛ ❧♦②❛❧t② ❝❛r❞✧

◮ ✷✽ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ♣r♦❞✉❝ts✱ ◮ ✻ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ❝♦st✉♠❡rs✱ ◮ ✽ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ st♦r❡s✱ ◮ n = ✷, ✽✾✾, ✵✸✵ r❡❝❡✐♣ts✳

❙♦♠❡ ♠❡❛♥✐♥❣❢✉❧ ✈❛r✐❛❜❧❡s ✇✐t❤ ♠✐ss✐♥❣ ✈❛❧✲ ✉❡s✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✺ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊①❛♠♣❧❡

❆♥ ❡①❛♠♣❧❡

❏♦✐♥t ✇♦r❦s ✇✐t❤ ❈❤r✐st♦♣❤❡ ❇✐❡r♥❛❝❦✐ ✭❤❡❛❞ ♦❢ t❤❡ ■♥r✐❛ ▼♦❞❛❧ t❡❛♠✮✱ ❱✐♥❝❡♥t ❱❛♥❞❡✇❛❧❧❡✱ ❑♦♠✐ ◆❛❣❜❡✱✳✳✳ ❈♦♥tr❛❝t ❢♦r ❛ ❧❛r❣❡ ❧✐♥❣❡r✐❡ st♦r❡✿ ✧❈❧✉st❡r✲ ✐♥❣ ❝❛s❤ r❡❝❡✐♣ts ♦❢ t❤❡ ❈✉st♦♠❡rs ✇✐t❤ ❛ ❧♦②❛❧t② ❝❛r❞✧

◮ ✷✽ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ♣r♦❞✉❝ts✱ ◮ ✻ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ❝♦st✉♠❡rs✱ ◮ ✽ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ st♦r❡s✱ ◮ n = ✷, ✽✾✾, ✵✸✵ r❡❝❡✐♣ts✳

❙♦♠❡ ♠❡❛♥✐♥❣❢✉❧ ✈❛r✐❛❜❧❡s ✇✐t❤ ♠✐ss✐♥❣ ✈❛❧✲ ✉❡s✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✺ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊①❛♠♣❧❡

❆♥ ❡①❛♠♣❧❡

❏♦✐♥t ✇♦r❦s ✇✐t❤ ❈❤r✐st♦♣❤❡ ❇✐❡r♥❛❝❦✐ ✭❤❡❛❞ ♦❢ t❤❡ ■♥r✐❛ ▼♦❞❛❧ t❡❛♠✮✱ ❱✐♥❝❡♥t ❱❛♥❞❡✇❛❧❧❡✱ ❑♦♠✐ ◆❛❣❜❡✱✳✳✳ ❈♦♥tr❛❝t ❢♦r ❛ ❧❛r❣❡ ❧✐♥❣❡r✐❡ st♦r❡✿ ✧❈❧✉st❡r✲ ✐♥❣ ❝❛s❤ r❡❝❡✐♣ts ♦❢ t❤❡ ❈✉st♦♠❡rs ✇✐t❤ ❛ ❧♦②❛❧t② ❝❛r❞✧

◮ ✷✽ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ♣r♦❞✉❝ts✱ ◮ ✻ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ❝♦st✉♠❡rs✱ ◮ ✽ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ st♦r❡s✱ ◮ n = ✷, ✽✾✾, ✵✸✵ r❡❝❡✐♣ts✳

❙♦♠❡ ♠❡❛♥✐♥❣❢✉❧ ✈❛r✐❛❜❧❡s ✇✐t❤ ♠✐ss✐♥❣ ✈❛❧✲ ✉❡s✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✺ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊①❛♠♣❧❡

❆♥ ❡①❛♠♣❧❡ ✭❱❛r✐❛❜❧❡s✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✻ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊①❛♠♣❧❡

❆♥ ❡①❛♠♣❧❡ ✭❱❛r✐❛❜❧❡s✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✼ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊①❛♠♣❧❡

❆♥ ❡①❛♠♣❧❡ ✭❘❡s✉❧ts✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✽ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊①❛♠♣❧❡

❆♥ ❡①❛♠♣❧❡ ✭❘❡s✉❧ts✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✽ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ▼✐①t✉r❡ ▼♦❞❡❧s

▼✐①t✉r❡ ▼♦❞❡❧s

▼❛✐♥ ■❞❡❛

① ✐♥ ❝❧✉st❡r k ⇐ ⇒ ① ❜❡❧♦♥❣s t♦ ❞✐str✐❜✉t✐♦♥ Pk

✞ ✝ ☎ ✆

① = (①✶, . . . , ①n)

− →

❝❧✉st❡r✐♥❣

✞ ✝ ☎ ✆

ˆ ③ = (ˆ ③✶, . . . , ˆ ③n)✱ ˆ K ❝❧✉st❡rs ▼♦❞❡❧ ❇❛s❡❞ ❝❧✉st❡r✐♥❣ ✐s ❛ ♣r♦❜❛❜✐❧✐st✐❝ ❛♣♣r♦❛❝❤✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✾ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ▼✐①t✉r❡ ▼♦❞❡❧s

❘ ♣❛❝❦❛❣❡ ▼✐①❆❧❧ ❛♥❞ ❙❛❛❙ ▼✐①t❈♦♠♣

❚✇♦ s♦❢t✇❛r❡s ❛✈❛✐❧❛❜❧❡

◮ ❘ ♣❛❝❦❛❣❡ ▼✐①❆❧❧

❃ ❧✐❜r❛r②✭▼✐①❆❧❧✮ ❃ ❞❛t❛✭❣❡②s❡r✮ ❃ ★★ ❛❞❞ ✶✵ ♠✐ss✐♥❣ ✈❛❧✉❡s ❛s r❛♥❞♦♠ ❃ ① ❂ ❛s✳♠❛tr✐①✭❣❡②s❡r✮❀ ♥ ❁✲ ♥r♦✇✭①✮❀ ♣ ❁✲ ♥❝♦❧✭①✮❀ ❃ ✐♥❞❡①❡s ❁✲ ♠❛tr✐①✭❝✭r♦✉♥❞✭r✉♥✐❢ ✭✺✱✶✱♥✮✮✱ r♦✉♥❞✭r✉♥✐❢ ✭✺✱✶✱♣✮✮✮✱ ♥❝♦❧ ❂✷✮❀ ❃ ①❬✐♥❞❡①❡s❪ ❁✲ ◆❆❀ ❃ ★★ ❡st✐♠❛t❡ ♠♦❞❡❧ ❃ ♠♦❞❡❧ ❁✲❝❧✉st❡r❉✐❛❣●❛✉ss✐❛♥ ✭ ❞❛t❛❂①✱ ♥❜❈❧✉st❡r ❂✷✿✸ ✱ ♠♦❞❡❧s❂❝✭ ✧❣❛✉ss✐❛♥❴♣❦❴s❥❦✧✮✮ ❃ ♣❧♦t✭♠♦❞❡❧✮ ❃ ♠✐ss✐♥❣❱❛❧✉❡s ✭♠♦❞❡❧✮ r♦✇ ❝♦❧ ✈❛❧✉❡ ✶ ✶✸✸ ✶ ✷✳✵✷✾✻✻✶ ✷ ✹✷ ✷ ✺✹✳✺✻✾✶✹✹ ✸ ✹✾ ✷ ✼✾✳✾✼✵✾✼✸ ✹ ✷✵✾ ✷ ✺✹✳✺✻✾✶✹✹ ✺ ✷✶✸ ✷ ✺✹✳✺✻✾✶✹✹

◮ ❙❛❛❙ s♦❢t✇❛r❡ ▼✐①t❈♦♠♣ ❤tt♣s✿✴✴♠❛ss✐❝❝❝✳❧✐❧❧❡✳✐♥r✐❛✳❢r✴

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✵ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ▼✐①t✉r❡ ▼♦❞❡❧s

❍②♣♦t❤❡s✐s ♦❢ ♠✐①t✉r❡ ♦❢ ♣❛r❛♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥s

◮ ❈❧✉st❡r k ✐s ♠♦❞❡❧❡❞ ❜② ❛ ♣❛r❛♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥

①i|z = k ∼ p(.|αk)

◮ ❈❧✉st❡r k ❤❛s ♣r♦❜❛❜✐❧✐t② πk

zi ∼ M(✶, π✶, . . . , πK).

▼✐①t✉r❡ ♠♦❞❡❧

❚❤❡ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs ❛r❡ θ = (π✶, . . . , πK, α✶, . . . , αK) ❛♥❞ p(①i) =

K

• k=✶

πkp(①i; αk)

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✶ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❊▼ ❆❧❣♦r✐t❤♠

❙t❛rt✐♥❣ ❢r♦♠ ❛♥ ✐♥✐t✐❛❧ ❛r❜✐tr❛r② ♣❛r❛♠❡t❡r θ✵✱ t❤❡ mt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ♦❢ r❡♣❡❛t✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ■✱ ❊ ❛♥❞ ▼ st❡♣s✳

◮ ■ st❡♣✿ ■♠♣✉t❡ ❜② ✉s✐♥❣ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣

①o✱ θr−✶✱ tr−✶

ik

✳

◮ ❊ st❡♣✿ ❈♦♠♣✉t❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s zi = k|①i ✉s✐♥❣ ❝✉rr❡♥t

✈❛❧✉❡ θr−✶ ♦❢ t❤❡ ♣❛r❛♠❡t❡r✿ tr

ik = tr k(①i|θr−✶) =

pr−✶

k

h(①i|αr−✶

k

) K

l=✶ pr−✶ l

h(①i|αr−✶

k

) . ✭✶✮

◮ ▼ st❡♣✿ ❯♣❞❛t❡ ▼▲ ❡st✐♠❛t❡ θr ✉s✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s tr ik ❛s

♠✐①✐♥❣ ✇❡✐❣❤ts L(θ|①✶, . . . , ①n, tr) =

n

• i=✶

K

• k=✶

tr

ik ln [pkh(①i|αk)] , ◮ ■t❡r❛t❡ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✷ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❊▼ ❆❧❣♦r✐t❤♠

❙t❛rt✐♥❣ ❢r♦♠ ❛♥ ✐♥✐t✐❛❧ ❛r❜✐tr❛r② ♣❛r❛♠❡t❡r θ✵✱ t❤❡ mt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ♦❢ r❡♣❡❛t✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ■✱ ❊ ❛♥❞ ▼ st❡♣s✳

◮ ■ st❡♣✿ ■♠♣✉t❡ ❜② ✉s✐♥❣ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣

①o✱ θr−✶✱ tr−✶

ik

✳

◮ ❊ st❡♣✿ ❈♦♠♣✉t❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s zi = k|①i ✉s✐♥❣ ❝✉rr❡♥t

✈❛❧✉❡ θr−✶ ♦❢ t❤❡ ♣❛r❛♠❡t❡r✿ tr

ik = tr k(①i|θr−✶) =

pr−✶

k

h(①i|αr−✶

k

) K

l=✶ pr−✶ l

h(①i|αr−✶

k

) . ✭✶✮

◮ ▼ st❡♣✿ ❯♣❞❛t❡ ▼▲ ❡st✐♠❛t❡ θr ✉s✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s tr ik ❛s

♠✐①✐♥❣ ✇❡✐❣❤ts L(θ|①✶, . . . , ①n, tr) =

n

• i=✶

K

• k=✶

tr

ik ln [pkh(①i|αk)] , ◮ ■t❡r❛t❡ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✷ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❊▼ ❆❧❣♦r✐t❤♠

❙t❛rt✐♥❣ ❢r♦♠ ❛♥ ✐♥✐t✐❛❧ ❛r❜✐tr❛r② ♣❛r❛♠❡t❡r θ✵✱ t❤❡ mt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ♦❢ r❡♣❡❛t✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ■✱ ❊ ❛♥❞ ▼ st❡♣s✳

◮ ■ st❡♣✿ ■♠♣✉t❡ ❜② ✉s✐♥❣ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣

①o✱ θr−✶✱ tr−✶

ik

✳

◮ ❊ st❡♣✿ ❈♦♠♣✉t❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s zi = k|①i ✉s✐♥❣ ❝✉rr❡♥t

✈❛❧✉❡ θr−✶ ♦❢ t❤❡ ♣❛r❛♠❡t❡r✿ tr

ik = tr k(①i|θr−✶) =

pr−✶

k

h(①i|αr−✶

k

) K

l=✶ pr−✶ l

h(①i|αr−✶

k

) . ✭✶✮

◮ ▼ st❡♣✿ ❯♣❞❛t❡ ▼▲ ❡st✐♠❛t❡ θr ✉s✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s tr ik ❛s

♠✐①✐♥❣ ✇❡✐❣❤ts L(θ|①✶, . . . , ①n, tr) =

n

• i=✶

K

• k=✶

tr

ik ln [pkh(①i|αk)] , ◮ ■t❡r❛t❡ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✷ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❊▼ ❆❧❣♦r✐t❤♠

❙t❛rt✐♥❣ ❢r♦♠ ❛♥ ✐♥✐t✐❛❧ ❛r❜✐tr❛r② ♣❛r❛♠❡t❡r θ✵✱ t❤❡ mt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ♦❢ r❡♣❡❛t✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ■✱ ❊ ❛♥❞ ▼ st❡♣s✳

◮ ■ st❡♣✿ ■♠♣✉t❡ ❜② ✉s✐♥❣ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣

①o✱ θr−✶✱ tr−✶

ik

✳

◮ ❊ st❡♣✿ ❈♦♠♣✉t❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s zi = k|①i ✉s✐♥❣ ❝✉rr❡♥t

✈❛❧✉❡ θr−✶ ♦❢ t❤❡ ♣❛r❛♠❡t❡r✿ tr

ik = tr k(①i|θr−✶) =

pr−✶

k

h(①i|αr−✶

k

) K

l=✶ pr−✶ l

h(①i|αr−✶

k

) . ✭✶✮

◮ ▼ st❡♣✿ ❯♣❞❛t❡ ▼▲ ❡st✐♠❛t❡ θr ✉s✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s tr ik ❛s

♠✐①✐♥❣ ✇❡✐❣❤ts L(θ|①✶, . . . , ①n, tr) =

n

• i=✶

K

• k=✶

tr

ik ln [pkh(①i|αk)] , ◮ ■t❡r❛t❡ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✷ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❊▼ ❆❧❣♦r✐t❤♠

❙t❛rt✐♥❣ ❢r♦♠ ❛♥ ✐♥✐t✐❛❧ ❛r❜✐tr❛r② ♣❛r❛♠❡t❡r θ✵✱ t❤❡ mt❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ♦❢ r❡♣❡❛t✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ■✱ ❊ ❛♥❞ ▼ st❡♣s✳

◮ ■ st❡♣✿ ■♠♣✉t❡ ❜② ✉s✐♥❣ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣

①o✱ θr−✶✱ tr−✶

ik

✳

◮ ❊ st❡♣✿ ❈♦♠♣✉t❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s zi = k|①i ✉s✐♥❣ ❝✉rr❡♥t

✈❛❧✉❡ θr−✶ ♦❢ t❤❡ ♣❛r❛♠❡t❡r✿ tr

ik = tr k(①i|θr−✶) =

pr−✶

k

h(①i|αr−✶

k

) K

l=✶ pr−✶ l

h(①i|αr−✶

k

) . ✭✶✮

◮ ▼ st❡♣✿ ❯♣❞❛t❡ ▼▲ ❡st✐♠❛t❡ θr ✉s✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s tr ik ❛s

♠✐①✐♥❣ ✇❡✐❣❤ts L(θ|①✶, . . . , ①n, tr) =

n

• i=✶

K

• k=✶

tr

ik ln [pkh(①i|αk)] , ◮ ■t❡r❛t❡ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✷ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❊▼ ❆❧❣♦r✐t❤♠ ❛♥❞ ✈❛r✐❛t✐♦♥s

❙❊▼✴❙❡♠✐❙❊▼ ❆❧❣♦r✐t❤♠s

❉r❛✇❜❛❝❦s

◮ ❚❤❡ ■ st❡♣ ♠❛② ❜❡ ❞✐✣❝✉❧t ◮ ❊▼ ❛❧❣♦r✐t❤♠ ♠❛② ❝♦♥✈❡r❣❡s s❧♦✇❧② ❛♥❞ ✐s s❧♦✇❡❞ ❞♦✇♥ ❜② t❤❡

✐♠♣✉t❛t✐♦♥ st❡♣

◮ ❇✐❛s❡❞ ❡st✐♠❛t♦rs

❙♦❧✉t✐♦♥✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦

◮ ❘❡♣❧❛❝❡ ■ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ◮ ■❙ st❡♣✿ s✐♠✉❧❛t❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ①m ✉s✐♥❣ ①o✱ θr−✶✱ tr−✶ ik

✳

◮ ❘❡♣❧❛❝❡ ❊ st❡♣ ❜② ❛ s✐♠✉❧❛t✐♦♥ st❡♣ ✭❖♣t✐♦♥❛❧✮ ◮ ❙ st❡♣✿ ❣❡♥❡r❛t❡ ❧❛❜❡❧s ③r = {③r ✶, ..., ③r n} ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛t❡❣♦r✐❝❛❧

❞✐str✐❜✉t✐♦♥ (tr

ik, ✶ ≤ k ≤ K)✳

❙❊▼ ❛♥❞ ❙❡♠✐❙❊▼ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ♣♦✐♥t ✇✐s❡✳ ■t ❣❡♥❡r❛t❡s ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳

◮ ¯

θ = (θr)r=✶,...,R

◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♠♣✉t❡❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ ▼❆P ✈❛❧✉❡ ✭♦r ❡①♣❡❝t❛t✐♦♥✮

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✸ ✴ ✸✵

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ▼✐①t✉r❡ ▼♦❞❡❧ ❛♥❞ ▼✐①❡❞ ❉❛t❛

▼✐①❡❞ ❉❛t❛

▼✐①❡❞ ❞❛t❛ ❛r❡ ❤❛♥❞❧❡❞ ✉s✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✳ ✶✳ ❖❜s❡r✈❛t✐♦♥ s♣❛❝❡ ♦❢ t❤❡ ❢♦r♠ ❳ = ❳✶ × ❳✷ × . . . × ❳L ✷✳ ①i ❛r✐s❡s ❢r♦♠ ❛ ♠✐①t✉r❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ❞❡♥s✐t② f (①i = (①✶i, ①✷i, . . . ①Li)|θ) =

K

• k=✶

πk

L

• l=✶

hl(①li|αlk). ✸✳ ❚❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥s ✭♦r ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s✮ hl(.|αlk) ❝❛♥ ❜❡ ❛♥② ✐♠♣❧❡♠❡♥t❡❞ ♠♦❞❡❧✳ ▼✐①❆❧❧ ✐♠♣❧❡♠❡♥ts ●❛✉ss✐❛♥✱ P♦✐ss♦♥✱ ❈❛t❡❣♦r✐❝❛❧✱ ●❛♠♠❛ ❞✐str✐❜✉t✐♦♥s✳ ▼✐①t❈♦♠♣ ✐♠♣❧❡♠❡♥ts ●❛✉ss✐❛♥✱ P♦✐ss♦♥✱ ❈❛t❡❣♦r✐❝❛❧ ❛♥❞ s♣❡❝✐✜❝ ❞✐str✐❜✉t✐♦♥s ❢♦r r❛♥❦ ❛♥❞ ♦r❞✐♥❛❧ ❞❛t❛✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✹ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s

❙♦♠♠❛✐r❡

❈❧✉st❡r✐♥❣ ✉s✐♥❣ ▼✐①t✉r❡ ▼♦❞❡❧s ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ❈✉❜❡ ♦❢ ❉❛t❛ ▼✐ss✐♥❣ ❉❛t❛✴◆♦✐s② ❉❛t❛✴❙❛♠♣❧✐♥❣ ✭▲♦♥❣ t❡r♠✮ ❖❜❥❡❝t✐✈❡ ▼♦❞❡❧✐♥❣ ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✺ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s

◮ ❉é✜ ▼❛st♦❞♦♥s✿ ❆♣♣❡❧ à Pr♦❥❡t ✷✵✶✻ ✧◗✉❛❧✐té ❞❡s ❞♦♥♥é❡s✧ ◮ ❈r❡❛t✐♦♥ ♦❢ t❤❡ ❈❧♦❍❡ ✭❈▲✉st❡r✐♥❣ ❖❢ ❍❡t❡r♦❣❡♥❡♦✉s ❉❛t❛ ✇✐t❤

❛♣♣❧✐❝❛t✐♦♥s t♦ s❛t❡❧❧✐t❡ ❞❛t❛ r❡❝♦r❞s✮ ♣r♦❥❡❝t

◮ ▼❡♠❜❡rs✿ ▼❛t❤✐❡✉ ❋❛✉✈❡❧ ✭■◆❘❆✮✱ ❙té♣❤❛♥❡ ●✐r❛r❞ ✭■♥r✐❛ ●r❡♥♦❜❧❡✮✱

❱✐♥❝❡♥t ✈❛♥❞❡✇❛❧❧❡ ✭▲✐❧❧❡✷✮✱ ❈r✐s✐t❛♥ Pr❡❞❛ ✭❯♥✐✈❡rs✐té ▲✐❧❧❡ ✶✮ ❤tt♣s✿✴✴♠♦❞❛❧✳❧✐❧❧❡✳✐♥r✐❛✳❢r✴❈❧♦❍❡✴

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✻ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ❈✉❜❡ ♦❢ ❉❛t❛

✞ ✝ ☎ ✆

❋♦r♠♦s❛t✲✷ ✐s ♥♦ ♠♦r❡ ♦♣❡r❛t✐♦♥❛❧

❋✐❣✉r❡✿ ❋♦r♠♦s❛t✲✷ ❢✉r♥✐s❤❡❞ ♠✉❧t✐✲s♣❡❝tr❛❧ ❞❛t❛ ✭❘✱ ●✱ ❇✱ ◆■❘✮ ✇✐t❤ ❛ ✽ ♠❡t❡r r❡s♦❧✉t✐♦♥✳ ✶✼ ❝♦♠♣❧❡t❡ ✐♠❛❣❡s ♦❢ ❋r❛♥❝❡ ❜② ②❡❛r ✞ ✝ ☎ ✆

❙❡♥t✐♥❡❧✲✷❆ st❛rt s❡r✈✐❝❡ ✐♥ ✷✵✶✻✳

❋✐❣✉r❡✿ ❙❡♥t✐♥❡❧✲✷ ❢✉r♥✐s❤ ✶✸ s♣❡❝tr❛❧ ❜❛♥❞✇✐❞t❤s ✇✐t❤ ✹ ❜❛♥❞✇✐❞t❤s ✇✐t❤ ❛ ✶✵ ♠❡t❡rs r❡s♦❧✉t✐♦♥ ❛♥❞ ✻ ❜❛♥❞✇✐❞t❤s ✇✐t❤ ❛ ✷✵ ♠❡t❡rs r❡s♦❧✉t✐♦♥✳ ❆ ❝♦♠♣❧❡t❡ ✐♠❛❣❡ ♦❢ ❋r❛♥❝❡ ❡✈❡r② ✺ ❞❛②s

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✼ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ❈✉❜❡ ♦❢ ❉❛t❛

❉❛t❛ ❈✉❜❡

❋✐❣✉r❡✿ ❙❡♥t✐♥❡❧✲✷ ❢✉r♥✐s❤ ❛♣♣r♦①✐♠❛t❡❧② ✷✵❚❇ ♦❢ ✐♠❛❣❡s✴②❡❛r✱ ❛♥❞ ❝♦✈❡r t❤❡ ❡♥t✐r❡ ❋r❛♥❝❡ ✐♥ ✺ ❞❛②s ✇✐t❤ ✶✳✻ ♠✐❧❧✐❛r❞ ❞❡ ♣✐①❡❧s✳

❉❛t❛ ❈✉❜❡ ❳ = (Xikt), i ∈ I, k ∈ {r✱✈✱❜✱✐r}, ❨ = (Yi), i ∈ J ⊂ I. ✇✐t❤

◮ i = (x, y) ❣❡♦❣r❛♣❤✐❝ ♣♦s✐t✐♦♥✱ ◮ k s♣❡❝tr❛❧ ❜❛♥❞✱ ◮ t ❞❛t❡s✱ ◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✭❝❧♦✉❞s✱ ♣♦rt❡❞

s❤❛❞♦✇s✮ ❛t s♦♠❡ ❞❛t❡s ❛♥❞ s♦♠❡ ♣♦s✐t✐♦♥s✱

◮ ♥♦✐s② ❞❛t❛ ✭✉♥❞❡t❡❝t❡❞

s❤❛❞♦✇s✱ ❝❧♦✉❞ ✈❡✐❧✱ ❡t❝✳✳✳✮✳

◮ ♠✐①❡❧ ✭♠✐①t✉r❡ ♦❢ ♣✐①❡❧✮

♣r❡s❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✽ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ❈✉❜❡ ♦❢ ❉❛t❛

❉❛t❛ ❈✉❜❡

❋✐❣✉r❡✿ ❙❡♥t✐♥❡❧✲✷ ❢✉r♥✐s❤ ❛♣♣r♦①✐♠❛t❡❧② ✷✵❚❇ ♦❢ ✐♠❛❣❡s✴②❡❛r✱ ❛♥❞ ❝♦✈❡r t❤❡ ❡♥t✐r❡ ❋r❛♥❝❡ ✐♥ ✺ ❞❛②s ✇✐t❤ ✶✳✻ ♠✐❧❧✐❛r❞ ❞❡ ♣✐①❡❧s✳

❉❛t❛ ❈✉❜❡ ❳ = (Xikt), i ∈ I, k ∈ {r✱✈✱❜✱✐r}, ❨ = (Yi), i ∈ J ⊂ I. ✇✐t❤

◮ i = (x, y) ❣❡♦❣r❛♣❤✐❝ ♣♦s✐t✐♦♥✱ ◮ k s♣❡❝tr❛❧ ❜❛♥❞✱ ◮ t ❞❛t❡s✱ ◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✭❝❧♦✉❞s✱ ♣♦rt❡❞

s❤❛❞♦✇s✮ ❛t s♦♠❡ ❞❛t❡s ❛♥❞ s♦♠❡ ♣♦s✐t✐♦♥s✱

◮ ♥♦✐s② ❞❛t❛ ✭✉♥❞❡t❡❝t❡❞

s❤❛❞♦✇s✱ ❝❧♦✉❞ ✈❡✐❧✱ ❡t❝✳✳✳✮✳

◮ ♠✐①❡❧ ✭♠✐①t✉r❡ ♦❢ ♣✐①❡❧✮

♣r❡s❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✽ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ❈✉❜❡ ♦❢ ❉❛t❛

❉❛t❛ ❈✉❜❡

❋✐❣✉r❡✿ ❙❡♥t✐♥❡❧✲✷ ❢✉r♥✐s❤ ❛♣♣r♦①✐♠❛t❡❧② ✷✵❚❇ ♦❢ ✐♠❛❣❡s✴②❡❛r✱ ❛♥❞ ❝♦✈❡r t❤❡ ❡♥t✐r❡ ❋r❛♥❝❡ ✐♥ ✺ ❞❛②s ✇✐t❤ ✶✳✻ ♠✐❧❧✐❛r❞ ❞❡ ♣✐①❡❧s✳

❉❛t❛ ❈✉❜❡ ❳ = (Xikt), i ∈ I, k ∈ {r✱✈✱❜✱✐r}, ❨ = (Yi), i ∈ J ⊂ I. ✇✐t❤

◮ i = (x, y) ❣❡♦❣r❛♣❤✐❝ ♣♦s✐t✐♦♥✱ ◮ k s♣❡❝tr❛❧ ❜❛♥❞✱ ◮ t ❞❛t❡s✱ ◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✭❝❧♦✉❞s✱ ♣♦rt❡❞

s❤❛❞♦✇s✮ ❛t s♦♠❡ ❞❛t❡s ❛♥❞ s♦♠❡ ♣♦s✐t✐♦♥s✱

◮ ♥♦✐s② ❞❛t❛ ✭✉♥❞❡t❡❝t❡❞

s❤❛❞♦✇s✱ ❝❧♦✉❞ ✈❡✐❧✱ ❡t❝✳✳✳✮✳

◮ ♠✐①❡❧ ✭♠✐①t✉r❡ ♦❢ ♣✐①❡❧✮

♣r❡s❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✽ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ❈✉❜❡ ♦❢ ❉❛t❛

❉❛t❛ ❈✉❜❡

❋✐❣✉r❡✿ ❙❡♥t✐♥❡❧✲✷ ❢✉r♥✐s❤ ❛♣♣r♦①✐♠❛t❡❧② ✷✵❚❇ ♦❢ ✐♠❛❣❡s✴②❡❛r✱ ❛♥❞ ❝♦✈❡r t❤❡ ❡♥t✐r❡ ❋r❛♥❝❡ ✐♥ ✺ ❞❛②s ✇✐t❤ ✶✳✻ ♠✐❧❧✐❛r❞ ❞❡ ♣✐①❡❧s✳

❉❛t❛ ❈✉❜❡ ❳ = (Xikt), i ∈ I, k ∈ {r✱✈✱❜✱✐r}, ❨ = (Yi), i ∈ J ⊂ I. ✇✐t❤

◮ i = (x, y) ❣❡♦❣r❛♣❤✐❝ ♣♦s✐t✐♦♥✱ ◮ k s♣❡❝tr❛❧ ❜❛♥❞✱ ◮ t ❞❛t❡s✱ ◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✭❝❧♦✉❞s✱ ♣♦rt❡❞

s❤❛❞♦✇s✮ ❛t s♦♠❡ ❞❛t❡s ❛♥❞ s♦♠❡ ♣♦s✐t✐♦♥s✱

◮ ♥♦✐s② ❞❛t❛ ✭✉♥❞❡t❡❝t❡❞

s❤❛❞♦✇s✱ ❝❧♦✉❞ ✈❡✐❧✱ ❡t❝✳✳✳✮✳

◮ ♠✐①❡❧ ✭♠✐①t✉r❡ ♦❢ ♣✐①❡❧✮

♣r❡s❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✽ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ❈✉❜❡ ♦❢ ❉❛t❛

❉❛t❛ ❈✉❜❡

❋✐❣✉r❡✿ ❙❡♥t✐♥❡❧✲✷ ❢✉r♥✐s❤ ❛♣♣r♦①✐♠❛t❡❧② ✷✵❚❇ ♦❢ ✐♠❛❣❡s✴②❡❛r✱ ❛♥❞ ❝♦✈❡r t❤❡ ❡♥t✐r❡ ❋r❛♥❝❡ ✐♥ ✺ ❞❛②s ✇✐t❤ ✶✳✻ ♠✐❧❧✐❛r❞ ❞❡ ♣✐①❡❧s✳

❉❛t❛ ❈✉❜❡ ❳ = (Xikt), i ∈ I, k ∈ {r✱✈✱❜✱✐r}, ❨ = (Yi), i ∈ J ⊂ I. ✇✐t❤

◮ i = (x, y) ❣❡♦❣r❛♣❤✐❝ ♣♦s✐t✐♦♥✱ ◮ k s♣❡❝tr❛❧ ❜❛♥❞✱ ◮ t ❞❛t❡s✱ ◮ ♠✐ss✐♥❣ ✈❛❧✉❡s ✭❝❧♦✉❞s✱ ♣♦rt❡❞

s❤❛❞♦✇s✮ ❛t s♦♠❡ ❞❛t❡s ❛♥❞ s♦♠❡ ♣♦s✐t✐♦♥s✱

◮ ♥♦✐s② ❞❛t❛ ✭✉♥❞❡t❡❝t❡❞

s❤❛❞♦✇s✱ ❝❧♦✉❞ ✈❡✐❧✱ ❡t❝✳✳✳✮✳

◮ ♠✐①❡❧ ✭♠✐①t✉r❡ ♦❢ ♣✐①❡❧✮

♣r❡s❡♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✽ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❉❛t❛✴◆♦✐s② ❉❛t❛✴❙❛♠♣❧✐♥❣

▼✐ss✐♥❣ ❞❛t❛

❋✐❣✉r❡✿ ❱❡r② ❝❧♦✉❞② ❋✐❣✉r❡✿ ❆ ❢❡✇ ♥✉♠❜❡r ♦❢ ❝❧♦✉❞s ❋✐❣✉r❡✿ ✧s❤❡❡♣s✧ ❋✐❣✉r❡✿ ❙♦♠❡ ❝❧♦✉❞s ✇✐t❤ ❛ ✈❡✐❧

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✶✾ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❉❛t❛✴◆♦✐s② ❉❛t❛✴❙❛♠♣❧✐♥❣

◆♦✐s② ❉❛t❛

❋✐❣✉r❡✿ ❝❧♦✉❞s ❛♥❞ t❤❡✐r s❤❛❞♦✇s

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✵ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❉❛t❛✴◆♦✐s② ❉❛t❛✴❙❛♠♣❧✐♥❣

◆♦♥✲❯♥✐❢♦r♠ s❛♠♣❧✐♥❣

❋✐❣✉r❡✿ P❛t❤✲r♦✇ ❣r✐❞ ❢♦r ▲❛♥❞s❛t ❛❝q✉✐s✐t✐♦♥s✳ ❊✈❡r② ♣❛t❤ ✭◆♦rt❤✲❙♦✉t❤ tr❛❝❦✮ ✐s ❛❝q✉✐r❡❞ ♦♥ t❤❡ s❛♠❡ ❞❛t❡ ❡✈❡r② ✶✻ ❞❛②s✳ ❋✐❣✉r❡✿ ▼❛♣ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t ❡✈❡r② ♣✐①❡❧ s❡❡s t❤❡ ❣r♦✉♥❞ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t s❛t❡❧❧✐t❡ r❡✈✐s✐t ❛♥❞ ❝❧♦✉❞ ❝♦✈❡r✳ ❋✐❣✉r❡✿ ❍✐st♦❣r❛♠ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t ❡✈❡r② ♣✐①❡❧ s❡❡s t❤❡ ❣r♦✉♥❞ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t s❛t❡❧❧✐t❡ r❡✈✐s✐t ❛♥❞ ❝❧♦✉❞ ❝♦✈❡r✳

❖♣❡♥ ❆❝❝❡ss✿ ❤tt♣✿✴✴✇✇✇✳♠❞♣✐✳❝♦♠✴✷✵✼✷✲✹✷✾✷✴✾✴✶✴✾✺✴❤t♠

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✶ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✭▲♦♥❣ t❡r♠✮ ❖❜❥❡❝t✐✈❡

❖❜❥❡❝t✐✈❡

❚❤❡ ❛✐♠ ✐s t♦ ❜❡ ❛❜❧❡ t♦ ❝❧✉st❡r t❤❡ ✇❤♦❧❡ ❋r❛♥❝❡ ✉s✐♥❣ ❙❡♥t✐♥❡❧✲✷ ❞❛t❛✳

❋✐❣✉r❡✿ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❋r❛♥❝❡

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✷ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼♦❞❡❧✐♥❣

• ❛✉ss✐❛♥ ♠♦❞❡❧✐♥❣

◮ Yi ∈ {✶, . . . , G}✱ ◮ L(❳i|Yi = g) = N(µg, Σg) ◮ ❚✇♦ ❦✐♥❞s ♦❢ ♣❛rs✐♠♦♥② ❛ss✉♠♣t✐♦♥s ♦♥ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s

◮ ✐♥❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ s♣❡❝tr❛ Σg,k ♦❢ s✐③❡ T × T✱ ✭T = ✶✼✮✱ ◮ ♦r ✐♥❞❡♣❡♥❞❡♥❝❡s ❜❡t✇❡❡♥ t✐♠❡s✱ Σg,t ♦❢ s✐③❡ K × K✱ ✭K = ✹✮✳

◮ ❤❛♥❞❧❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ❢♦r ❜♦t❤ ♠♦❞❡❧s ◮ ■♠♣❧❡♠❡♥t❛t✐♦♥s ❛♥❞ t❡sts ✐♥ ❛ ❘ ♣❛❝❦❛❣❡✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✸ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼♦❞❡❧✐♥❣

• ❛✉ss✐❛♥ ♠♦❞❡❧✐♥❣

◮ Yi ∈ {✶, . . . , G}✱ ◮ L(❳i|Yi = g) = N(µg, Σg) ◮ ❚✇♦ ❦✐♥❞s ♦❢ ♣❛rs✐♠♦♥② ❛ss✉♠♣t✐♦♥s ♦♥ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s

◮ ✐♥❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ s♣❡❝tr❛ Σg,k ♦❢ s✐③❡ T × T✱ ✭T = ✶✼✮✱ ◮ ♦r ✐♥❞❡♣❡♥❞❡♥❝❡s ❜❡t✇❡❡♥ t✐♠❡s✱ Σg,t ♦❢ s✐③❡ K × K✱ ✭K = ✹✮✳

◮ ❤❛♥❞❧❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ❢♦r ❜♦t❤ ♠♦❞❡❧s ◮ ■♠♣❧❡♠❡♥t❛t✐♦♥s ❛♥❞ t❡sts ✐♥ ❛ ❘ ♣❛❝❦❛❣❡✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✸ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼♦❞❡❧✐♥❣

• ❛✉ss✐❛♥ ♠♦❞❡❧✐♥❣

◮ Yi ∈ {✶, . . . , G}✱ ◮ L(❳i|Yi = g) = N(µg, Σg) ◮ ❚✇♦ ❦✐♥❞s ♦❢ ♣❛rs✐♠♦♥② ❛ss✉♠♣t✐♦♥s ♦♥ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s

◮ ✐♥❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ s♣❡❝tr❛ Σg,k ♦❢ s✐③❡ T × T✱ ✭T = ✶✼✮✱ ◮ ♦r ✐♥❞❡♣❡♥❞❡♥❝❡s ❜❡t✇❡❡♥ t✐♠❡s✱ Σg,t ♦❢ s✐③❡ K × K✱ ✭K = ✹✮✳

◮ ❤❛♥❞❧❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ❢♦r ❜♦t❤ ♠♦❞❡❧s ◮ ■♠♣❧❡♠❡♥t❛t✐♦♥s ❛♥❞ t❡sts ✐♥ ❛ ❘ ♣❛❝❦❛❣❡✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✸ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼♦❞❡❧✐♥❣

• ❛✉ss✐❛♥ ♠♦❞❡❧✐♥❣

◮ Yi ∈ {✶, . . . , G}✱ ◮ L(❳i|Yi = g) = N(µg, Σg) ◮ ❚✇♦ ❦✐♥❞s ♦❢ ♣❛rs✐♠♦♥② ❛ss✉♠♣t✐♦♥s ♦♥ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s

◮ ✐♥❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ s♣❡❝tr❛ Σg,k ♦❢ s✐③❡ T × T✱ ✭T = ✶✼✮✱ ◮ ♦r ✐♥❞❡♣❡♥❞❡♥❝❡s ❜❡t✇❡❡♥ t✐♠❡s✱ Σg,t ♦❢ s✐③❡ K × K✱ ✭K = ✹✮✳

◮ ❤❛♥❞❧❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ❢♦r ❜♦t❤ ♠♦❞❡❧s ◮ ■♠♣❧❡♠❡♥t❛t✐♦♥s ❛♥❞ t❡sts ✐♥ ❛ ❘ ♣❛❝❦❛❣❡✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✸ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼♦❞❡❧✐♥❣

▼❡t❤♦❞

▼✐ss✐♥❣ ✈❛❧✉❡s ❢♦r♠❛t✐♦♥ ♣r♦❝❡ss

▼✐ss✐♥❣ ❆t ❘❛♥❞♦♠ ✭▼❆❘✮✿ Pr♦❜❛❜✐❧✐t② ❢♦r ❛ ✈❛❧✉❡ t♦ ❜❡ ♠✐ss✐♥❣ ❞♦❡s ♥♦t ❞❡♣❡♥❞s ❢r♦♠ ✐ts ✈❛❧✉❡ ❝♦♥❞✐t✐♦♥❛❧❧② t♦ t❤❡ ♦t❤❡r ♦❜s❡r✈❛t✐♦♥s✳ ❉❡♥♦t❡ ①+

ik =

①❖

i

①▼+

ik

• ✱ ˜

Σ+

ik =

✵❖

i

✵❖▼

i

✵▼❖

i

˜ Σ▼+

ik

• ✇✐t❤ ✵ ♥✉❧❧ ♠❛tr✐①✱ ❛♥❞

˜ Σ▼+

ik

= Σ▼

ik − Σ▼❖ ik

• Σ❖

ik

−✶ Σ❖▼

ik ✳ t❤❡♥

Σ+

k

= ✶ n+

k n

• i=✶
• (①+

ik − µ+ k )(①+ ik − µ+ k )′ + ˜

Σ+

ik

• ˜

Σ▼+

ik

✐s ❝♦rr❡❝t✐♥❣ t❤❡ ✈❛r✐❛♥❝❡ ❞✉❡ t♦ t❤❡ ✐♠♣✉t❛t✐♦♥ ❜② t❤❡ ♠❡❛♥✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✹ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼♦❞❡❧✐♥❣

• ❛✉ss✐❛♥ ♠♦❞❡❧✐♥❣

Oak Silver fir Black pine Silver fir Silver fir Silver birch Black locust Oak European ash Maritime pine Black pine Black pine Oak Silver birch Oak Douglas fir Silver fir Black pine Silver fir Silver fir Silver fir Douglas fir Oak

Tree Species Classification using Formosat-2 Satellite Image Time Series

Source: Esri, DigitalGlobe, GeoEye, Earthstar Geographics, CNES/Airbus DS, USDA, USGS, AeroGRID, IGN, and the GIS User Community

February 8, 2017

0.6 1.2 0.3 mi 0.7 1.4 0.35 km

1:36,112

❋✐❣✉r❡✿ ❚r❡❡ s♣❡❝✐❡s ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ G = ✶✸

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✺ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

❈♦♥t✐♥✉♦✉s ▼♦❞❡❧

▼❛✐♥ ❛ss✉♠♣t✐♦♥ Y |Z = k ∼ GP(µk, Ck), k = ✶, . . . , K ✭✷✮ ✇❤❡r❡ GP(µk, Ck) ✐s ❛ ●❛✉ss✐❛♥ Pr♦❝❡ss ✇✐t❤ ♠❡❛♥ µk ∈ L✷(I) ❛♥❞ ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♦♣❡r❛t♦r Ck : I × I → R✳

◮ ♠❡❛♥ ❢✉♥❝t✐♦♥s ❜❡❧♦♥❣s t♦ ❛ J−❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡

µk(t) =

J

• j=✶

αkjϕj(t),

◮ ❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥

Ck(s, t)(hk) = θkQ((t − s)/hk),

◮ ❙♣❡❝tr✉♠ ❛r❡ ✐♥❞❡♣❡♥❞❡♥ts✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✻ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

❈♦♥t✐♥✉♦✉s ▼♦❞❡❧

▼❛✐♥ ❛ss✉♠♣t✐♦♥ Y |Z = k ∼ GP(µk, Ck), k = ✶, . . . , K ✭✷✮ ✇❤❡r❡ GP(µk, Ck) ✐s ❛ ●❛✉ss✐❛♥ Pr♦❝❡ss ✇✐t❤ ♠❡❛♥ µk ∈ L✷(I) ❛♥❞ ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♦♣❡r❛t♦r Ck : I × I → R✳

◮ ♠❡❛♥ ❢✉♥❝t✐♦♥s ❜❡❧♦♥❣s t♦ ❛ J−❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡

µk(t) =

J

• j=✶

αkjϕj(t),

◮ ❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥

Ck(s, t)(hk) = θkQ((t − s)/hk),

◮ ❙♣❡❝tr✉♠ ❛r❡ ✐♥❞❡♣❡♥❞❡♥ts✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✻ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

❈♦♥t✐♥✉♦✉s ▼♦❞❡❧

▼❛✐♥ ❛ss✉♠♣t✐♦♥ Y |Z = k ∼ GP(µk, Ck), k = ✶, . . . , K ✭✷✮ ✇❤❡r❡ GP(µk, Ck) ✐s ❛ ●❛✉ss✐❛♥ Pr♦❝❡ss ✇✐t❤ ♠❡❛♥ µk ∈ L✷(I) ❛♥❞ ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♦♣❡r❛t♦r Ck : I × I → R✳

◮ ♠❡❛♥ ❢✉♥❝t✐♦♥s ❜❡❧♦♥❣s t♦ ❛ J−❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡

µk(t) =

J

• j=✶

αkjϕj(t),

◮ ❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥

Ck(s, t)(hk) = θkQ((t − s)/hk),

◮ ❙♣❡❝tr✉♠ ❛r❡ ✐♥❞❡♣❡♥❞❡♥ts✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✻ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

❈♦♥t✐♥✉♦✉s ▼♦❞❡❧

▼❛✐♥ ❛ss✉♠♣t✐♦♥ Y |Z = k ∼ GP(µk, Ck), k = ✶, . . . , K ✭✷✮ ✇❤❡r❡ GP(µk, Ck) ✐s ❛ ●❛✉ss✐❛♥ Pr♦❝❡ss ✇✐t❤ ♠❡❛♥ µk ∈ L✷(I) ❛♥❞ ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♦♣❡r❛t♦r Ck : I × I → R✳

◮ ♠❡❛♥ ❢✉♥❝t✐♦♥s ❜❡❧♦♥❣s t♦ ❛ J−❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡

µk(t) =

J

• j=✶

αkjϕj(t),

◮ ❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥

Ck(s, t)(hk) = θkQ((t − s)/hk),

◮ ❙♣❡❝tr✉♠ ❛r❡ ✐♥❞❡♣❡♥❞❡♥ts✳

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✻ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

❊st✐♠❛t✐♦♥ ♦❢ ❈♦♥t✐♥✉♦✉s ▼♦❞❡❧

❋♦r ❡❛❝❤ i✱ ❧❡t Bi

ℓ,j = ϕj(ti ℓ)✱ mki = Biαk ❛♥❞

Σi

j,j′(hk) = θkQ((ti j − ti j′)/hk) =: θkSi j,j′(hk),

t❤❡♥ yi|Zi = k ∼ NTi(mki, θkSi(hk)), k = ✶, . . . , K, i = ✶, . . . , n ✇❡ ❡♥❞ ✉♣ ✇✐t❤ K ✐♥❞❡♣❡♥❞❡♥t ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✿ (ˆ αk, ˆ hk) = arg max

αk,hk,θk

• Zi=k

log det Si(hk) + Ti log θk + ✶ θk (yi − Biαk)⊤Si(hk)

−✶(yi − Biαk)

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✼ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

❘❡s✉❧ts

❆❜♦✉t ✻✺✪ ✇❡❧❧ ❝❧❛ss✐✜❡❞✳

❋✐❣✉r❡✿ G = ✶✸ s♣❡❝tr✉♠

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✽ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

▼❡❛♥ ✈❛❧✉❡s

❋✐❣✉r❡✿ ✜rst✱ ✹t❤✱ ✼t❤ ❛♥❞ ✶✶t❤ ❝❧❛ss❡s

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✷✾ ✴ ✸✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ▼✐ss✐♥❣ ❱❛❧✉❡s ❄

▲✐♥❦s

◮ ❤tt♣s✿✴✴❝r❛♥✳r✲♣r♦❥❡❝t✳♦r❣✴✇❡❜✴♣❛❝❦❛❣❡s✴▼✐①❆❧❧✴ ◮ ❤tt♣s✿✴✴♠❛ss✐❝❝❝✳❧✐❧❧❡✳✐♥r✐❛✳❢r✴ ◮ ❤tt♣s✿✴✴♠♦❞❛❧✳❧✐❧❧❡✳✐♥r✐❛✳❢r✴❈❧♦❍❡✴ ◮ ❤tt♣✿✴✴✇✇✇✳♠❞♣✐✳❝♦♠✴✷✵✼✷✲✹✷✾✷✴✾✴✶✴✾✺✴❤t♠

❙✳ ■♦✈❧❡✛ ✭▲✐❧❧❡ ✶✮ ▼✐①t✉r❡ ▼♦❞❡❧s ✇✐t❤ ▼✐ss✐♥❣ ❞❛t❛ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙❛t❡❧❧✐t❡ ■♠❛❣❡ ❚✐♠❡ ❙❡r✐❡s ✷✸ ❏✉✐♥ ✷✵✶✼ ✸✵ ✴ ✸✵