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  1. ❙▼❆❈❖❋ ✐♥ ❘ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✐♥ ❘✿ ❙▼❆❈❖❋ P❛tr✐❝❦ ▼❛✐r ■♥st✐t✉t❡ ❢♦r ❙t❛t✐st✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s ❲❯ ❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❊❝♦♥♦♠✐❝s ❛♥❞ ❇✉s✐♥❡ss ❏❛♥ ❞❡ ▲❡❡✉✇ ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧✐❢♦r♥✐❛✱ ▲♦s ❆♥❣❡❧❡s ✭❯❈▲❆✮ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  2. ❙▼❆❈❖❋ ✐♥ ❘ ❲♦r❦s❤♦♣ ❈♦♥t❡♥t ❚❤r♦✉❣❤♦✉t t❤❡ ✇♦r❦s❤♦♣ ✇❡ ✉s❡ t❤❡ ❘ ♣❛❝❦❛❣❡ s♠❛❝♦❢ ❛✈❛✐❧❛❜❧❡ ♦♥ ❈❘❆◆✳ ▼❛t❡r✐❛❧s ❛t ❤tt♣✿✴✴st❛t♠❛t❤✳✇✉✳❛❝✳❛t✴⑦♠❛✐r✴❈❆❘▼❊✴ ✳ • ❙②♠♠❡tr✐❝ ❙▼❆❈❖❋ • ❙♣❤❡r✐❝❛❧ ❙▼❆❈❖❋ • ✸✲❲❛② ❙▼❆❈❖❋ • ❈♦♥str❛✐♥❡❞ ❙▼❆❈❖❋ • ❘❡❝t❛♥❣✉❧❛r ❙▼❆❈❖❋ ❉❡ ▲❡❡✉✇✱ ❏✳ ✫ ▼❛✐r✱ P✳ ✭✷✵✵✾✮✳ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✉s✐♥❣ ▼❛❥♦r✐③❛t✐♦♥✿ ❙▼❆❈❖❋ ✐♥ ❘✳ ❏♦✉r♥❛❧ ♦❢ ❙t❛t✐st✐❝❛❧ ❙♦❢t✇❛r❡✱ ✸✶✭✸✮ ✱ ♣✳ ✶✲✸✵✳ ❯❘▲✿ ❤tt♣✿✴✴✇✇✇✳❥st❛ts♦❢t✳♦r❣ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  3. ❙▼❆❈❖❋ ✐♥ ❘ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✭▼❉❙✮ • ❋❛♠✐❧② ♦❢ ❞❛t❛✲❛♥❛❧②t✐❝ ♠❡t❤♦❞s ✇❤✐❝❤ r❡♣r❡s❡♥t ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ♦❜❥❡❝ts ✐♥ ❛ ❧♦✇✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ • ■♥♣✉t str✉❝t✉r❡✿ ❉✐ss✐♠✐❧❛r✐t② ♠❛tr✐①✳ • ❈♦♠♣✉t❛t✐♦♥✿ ❖♣t✐♠✐③❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❛r❣❡t ❢✉♥❝t✐♦♥✳ • ❖✉t♣✉t✿ ❈♦♥✜❣✉r❛t✐♦♥s ✐♥ ❧♦✇✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ • ❱✐s✉❛❧✐③❛t✐♦♥✿ ❈♦♥✜❣✉r❛t✐♦♥ ♣❧♦t✳ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  4. ❙▼❆❈❖❋ ✐♥ ❘ ▼❉❙ ▼♦❞❡❧s • ❈❧❛ss✐❝❛❧ ✭♠❡tr✐❝✮ s❝❛❧✐♥❣ ✭❚♦r❣❡rs♦♥✱ ✶✾✺✷✮✿ ❞❛t❛ ♦♥ ✐♥t❡r✈❛❧ ❧❡✈❡❧❀ str❛✐♥ ❧♦ss✱ str❡ss ❧♦ss✳ • ◆♦♥♠❡tr✐❝ ▼❉❙ ✭❙❤❡♣❛r❞✱ ✶✾✻✷✮✿ ❞❛t❛ ♦♥ ♦r❞✐♥❛❧ ❧❡✈❡❧❀ ♠♦♥♦t♦♥✐❝✐t② ❝♦♥str❛✐♥t ♦♥ ❞✐ss✐♠✐❧❛r✐t✐❡s✳ • ■♥❞✐✈✐❞✉❛❧ ❞✐✛❡r❡♥❝❡s✿ K ❞✐ss✐♠✐❧❛r✐t② ♠❛tr✐❝❡s✳ • ❯♥❢♦❧❞✐♥❣✿ Pr❡❢❡r❡♥❝❡ ❞❛t❛❀ n 1 r❛t❡rs✱ n 2 ♦❜❥❡❝ts✳ • ❖t❤❡r r❡❧❛t❡❞ ♠♦❞❡❧s✿ Pr♦❝r✉st❡s✱ ❈❆✱ ●✐✜✳ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  5. ❙▼❆❈❖❋ ✐♥ ❘ ❘ ■♠♣❧❡♠❡♥t❛t✐♦♥✿ ❙▼❆❈❖❋ ♣❛❝❦❛❣❡ ❚❤❡ ❘ Pr♦❥❡❝t ❢♦r ❙t❛t✐st✐❝❛❧ ❈♦♠♣✉t✐♥❣ • ❘ ✐s ❛♥ ♦♣❡♥ s♦✉r❝❡ s♦❢t✇❛r❡ ❡♥✈✐r♦♥♠❡♥t ❢♦r st❛t✐st✐❝❛❧ ❝♦♠✲ ♣✉t✐♥❣ ❛♥❞ ❣r❛♣❤✐❝s • ❤tt♣✿✴✴✇✇✇✳❘✲♣r♦❥❡❝t✳♦r❣ • ∼ ✷✽✵✵ ♣❛❝❦❛❣❡s ❛✈❛✐❧❛❜❧❡ ❚❤❡ s♠❛❝♦❢ ♣❛❝❦❛❣❡ ✐s ❛✈❛✐❧❛❜❧❡ ♦♥ • ❈❘❆◆✿ ❤tt♣✿✴✴❈❘❆◆✳❘✲♣r♦❥❡❝t✳♦r❣ • Ps②❝❤♦❘✿ ❤tt♣✿✴✴r✲❢♦r❣❡✳r✲♣r♦❥❡❝t✳♦r❣✴♣r♦❥❡❝ts✴♣s②❝❤♦r ✳ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  6. ❙▼❆❈❖❋ ✐♥ ❘ ❙②♠♠❡tr✐❝ ❙▼❆❈❖❋ • ❉✐st❛♥❝❡ ♠❛tr✐① ∆ ♦❢ ❞✐♠❡♥s✐♦♥ n × n ✇✐t❤ ❡❧❡♠❡♥ts δ ij ✳ • Pr♦❜❧❡♠ t♦ s♦❧✈❡✿ ▲♦❝❛t❡ ♣♦✐♥ts ✭ ❝♦♥✜❣✉r❛t✐♦♥s ✮ ✐♥ ❛ p ✲ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ s✉❝❤ t❤❛t t❤❡ ❞✐st❛♥❝❡s d ij ( X ) ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts ❛♣♣r♦①✐♠❛t❡ δ ij ✳ • ❈♦♥✜❣✉r❛t✐♦♥ ❞✐st❛♥❝❡s✿ � � p � � ( x is − x js ) 2 � d ij ( X ) = s =1 • ▼✐♥✐♠✐③❡ str❡ss ✭ ▼❛❥♦r✐③❛t✐♦♥ ❀ ❞❡ ▲❡❡✉✇✱ ✶✾✼✼✮✿ � w ij ( δ ij − d ij ( X )) 2 → ♠✐♥✦ σ ( X ) = i<j ❊①❛♠♣❧❡ ✶✿ ❋r❡♥❝❤ ▼❛♣ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  7. ❙▼❆❈❖❋ ✐♥ ❘ ◆♦♥♠❡tr✐❝ s②♠♠❡tr✐❝ ❙▼❆❈❖❋ • ❉✐ss✐♠✐❧❛r✐t✐❡s δ ij ♦♥ ♦r❞✐♥❛❧ s❝❛❧❡✳ • ▼♦♥♦t♦♥✐❝ tr❛♥s❢♦r♠❛t✐♦♥ f ✿ δ ij < δ i ′ j ′ ⇒ f ( δ ij ) < f ( δ i ′ j ′ ) ✳ • ❆❝❤✐❡✈❡❞ ❜② ♠♦♥♦t♦♥❡ r❡❣r❡ss✐♦♥ ✜t ✐♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ ✭P❆❱❆✮✳ • ◆♦♥♠❡tr✐❝ str❡ss✿ � d ij − d ij ( X )) 2 σ ( X, � w ij (ˆ D ) = i<j • ❉✐s♣❛r✐t✐❡s ˆ d ij = f ( δ ij ) ❊①❛♠♣❧❡ ✷✿ P♦❧✐t✐❝❛❧ st❛t❡♠❡♥ts ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  8. ❙▼❆❈❖❋ ✐♥ ❘ ✸✲❲❛② ❙▼❆❈❖❋ ❙▼❆❈❖❋ ❢♦r ✐♥❞✐✈✐❞✉❛❧ ❞✐✛❡r❡♥❝❡s✿ • k = 1 , . . . , K s❡♣❛r❛t❡ s②♠♠❡tr✐❝ ❞✐st❛♥❝❡ ♠❛tr✐❝❡s✳ • ❙tr❡ss✿ K � � w ij,k ( δ ij,k − d ij ( X k )) 2 . σ ( X ∗ ) = k =1 i<j • ■◆❉❙❈❆▲ ✭✉♥r❡str✐❝t❡❞❀ ❈❛rr♦❧ ✫ ❈❤❛♥❣✱ ✶✾✼✵✮✳ • ❘❡str✐❝t✐♦♥s ♦♥ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ✇❡✐❣❤t ♠❛tr✐① X k = ZC k ✭❞✐❛❣♦♥❛❧✱ ✐❞❡♥t✐t②✮✳ • ■❉■❖❙❈❆▲ ✭❈❛rr♦❧ ✫ ❲✐s❤✱ ✶✾✼✹✮✳ ❊①❛♠♣❧❡ ✸✿ ❲✐♥❡ t❛st✐♥❣ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  9. ❙▼❆❈❖❋ ✐♥ ❘ ❲✐♥❡ ❚❛st✐♥❣✿ ❉❡s❝r✐♣t✐✈❡s Pr✐❝❡ ❆❧❝♦❤♦❧ ▼❡❛♥ ❘❛t✐♥❣ ❏✉rts❝❤✐ts❝❤ ❈❤❛r❞♦♥♥❛② ✶✹✳✾✾ ✶✸✳✵✵ ✷✳✵✵ ❩✐♥✐❡❧ ❈❤❛r❞♦♥♥❛② ✼✳✵✵ ✶✷✳✵✵ ✷✳✻✵ ▼❛r❦♦✇✐ts❝❤ ❈❤❛r❞♦♥♥❛② ✾✳✾✾ ✶✷✳✺✵ ✷✳✻✵ ❘✐t✐♥✐t✐s ◆♦❜❧❡ ❘❡ts✐♥❛ ✾✳✾✾ ✶✷✳✵✵ ✹✳✸✵ ❘❡ts✐♥❛ ✷✳✾✾ ✶✶✳✺✵ ✹✳✻✵ ❑r❡♠s ❈❤❛r❞♦♥♥❛② ✺✳✾✾ ✶✷✳✺✵ ✷✳✼✵ ❈❛st❡❧ ◆♦✈❛ ❈❤❛r❞♦♥♥❛② ✶✳✾✾ ✶✷✳✵✵ ✷✳✽✵ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  10. ❙▼❆❈❖❋ ✐♥ ❘ ❙♣❤❡r✐❝❛❧ ❙▼❆❈❖❋ ❘❡str✐❝t✐♦♥s ♦♥ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ✭ ✇❡❛❦❧② ❝♦♥str❛✐♥❡❞ ▼❉❙ ✱ ❈♦① ✫ ❈♦①✱ ✶✾✾✶✮✳ x ′ i Λx i + 2 x ′ i β + γ = 0 • R 2 ✿ ❝✐r❝❧❡✱ ❡❧❧✐♣s❡✱ ❤②♣❡r❜♦❧❛✱ ♣❛r❛❜♦❧❛✳ • R 3 ✿ s♣❤❡r❡✱ ❡❧❧✐♣s♦✐❞✱ ❤②♣❡r❜♦❧♦✐❞✱ ♣❛r❛❜♦❧♦✐❞✱ ❝②❧✐♥❞❡r✳ • ❖♣t✐♠✐③❛t✐♦♥✿ Pr✐♠❛❧ ❛♥❞ ❞✉❛❧ ♠❡t❤♦❞s ❛✈❛✐❧❛❜❧❡✳ ❊①❛♠♣❧❡ ✹✿ ❚r❛❞✐♥❣ ✈♦❧✉♠❡ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

  11. ❙▼❆❈❖❋ ✐♥ ❘ ❈♦♥str❛✐♥❡❞ ❙▼❆❈❖❋ ❘❡str✐❝t✐♦♥s ❜② ♠❡❛♥s ♦❢ ❡①t❡r♥❛❧ ❝♦♥str❛✐♥ts ✭ ❝♦♥✜r♠❛t♦r② ▼❉❙ ✱ ❞❡ ▲❡❡✉✇ ✫ ❍❡✐s❡r✱ ✶✾✽✵✮✳ X = ZC ✇❤❡r❡ Z ✐s ❛ ❦♥♦✇♥ ♣r❡❞✐❝t♦r ♠❛tr✐①✳ � � • ❉❡❝♦♠♣♦s✐t✐♦♥✿ X = ✳ X 1 ZC 1 C 2 • X 1 ✐s t❤❡ ✉♥r❡str✐❝t❡❞ ♣❛rt ❛♥❞ ♦❢ ❞✐♠❡♥s✐♦♥ n × q 1 ✳ • ZC 1 ✐s t❤❡ ❧✐♥❡❛r❧② r❡str✐❝t❡❞ ♣❛rt ♦❢ ❞✐♠❡♥s✐♦♥ n × q 2 ✳ • C 2 ✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ♦❢ ♦r❞❡r n ✳ ❈❆❘▼❊ ✷✵✶✶ ✷✵✶✶✲✵✷✲✵✽

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