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❉❡s✐❣♥✐♥❣ ❙tr❡ss ❙❝❡♥❛r✐♦s

❈❡❝✐❧✐❛ P❛r❧❛t♦r❡ ❛♥❞ ❚❤♦♠❛s P❤✐❧✐♣♣♦♥

◆❨❯ ❙t❡r♥ ❛♥❞ ◆❇❊❘

❙tr❡ss ❚❡st✐♥❣ ❈♦♥❢❡r❡♥❝❡ ❖❝t♦❜❡r ✽✱ ✷✵✷✵

✶ ✴ ✶✺

▼♦t✐✈❛t✐♦♥

◗✉❡st✐♦♥ ❲❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ str❡ss t❡st ❞❡s✐❣♥❄ ❙tr❡ss t❡sts ❛r❡ ✉s❡❞ ✐♥ ❧✐q✉✐❞✐t②✴r✐s❦ ♠❛♥❛❣❡♠❡♥t ❛♥❞ ✜♥❛♥❝✐❛❧ s✉♣❡r✈✐s✐♦♥ ❚❤r❡❡ ❝♦♠♣♦♥❡♥ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ❘✐s❦ ▼♦❞❡❧✐♥❣ ✫ ▲♦ss Pr♦❥❡❝t✐♦♥ ❉✐s❝❧♦s✉r❡ ♦❢ ❘❡s✉❧ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ▲✐t❡r❛t✉r❡ ❢♦❝✉s❡s ♦♥ ❞✐s❝❧♦s✉r❡ ♦❢ r❡s✉❧ts ◆♦ ❣✉✐❞❛♥❝❡ ♦♥ ❤♦✇ t♦ ❞❡s✐❣♥ t❤❡ ❢♦r✇❛r❞✲❧♦♦❦✐♥❣ s❝❡♥❛r✐♦s ❚❤✐s ♣❛♣❡r✿ ❖♣t✐♠❛❧ s❝❡♥❛r✐♦ ❞❡s✐❣♥

✷ ✴ ✶✺

▼♦t✐✈❛t✐♦♥

◗✉❡st✐♦♥ ❲❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ str❡ss t❡st ❞❡s✐❣♥❄ ◮ ❙tr❡ss t❡sts ❛r❡ ✉s❡❞ ✐♥ ❧✐q✉✐❞✐t②✴r✐s❦ ♠❛♥❛❣❡♠❡♥t ❛♥❞ ✜♥❛♥❝✐❛❧ s✉♣❡r✈✐s✐♦♥ ❚❤r❡❡ ❝♦♠♣♦♥❡♥ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ❘✐s❦ ▼♦❞❡❧✐♥❣ ✫ ▲♦ss Pr♦❥❡❝t✐♦♥ ❉✐s❝❧♦s✉r❡ ♦❢ ❘❡s✉❧ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ▲✐t❡r❛t✉r❡ ❢♦❝✉s❡s ♦♥ ❞✐s❝❧♦s✉r❡ ♦❢ r❡s✉❧ts ◆♦ ❣✉✐❞❛♥❝❡ ♦♥ ❤♦✇ t♦ ❞❡s✐❣♥ t❤❡ ❢♦r✇❛r❞✲❧♦♦❦✐♥❣ s❝❡♥❛r✐♦s ❚❤✐s ♣❛♣❡r✿ ❖♣t✐♠❛❧ s❝❡♥❛r✐♦ ❞❡s✐❣♥

✷ ✴ ✶✺

▼♦t✐✈❛t✐♦♥

◗✉❡st✐♦♥ ❲❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ str❡ss t❡st ❞❡s✐❣♥❄ ◮ ❙tr❡ss t❡sts ❛r❡ ✉s❡❞ ✐♥ ❧✐q✉✐❞✐t②✴r✐s❦ ♠❛♥❛❣❡♠❡♥t ❛♥❞ ✜♥❛♥❝✐❛❧ s✉♣❡r✈✐s✐♦♥ ◮ ❚❤r❡❡ ❝♦♠♣♦♥❡♥ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ❘✐s❦ ▼♦❞❡❧✐♥❣ ✫ ▲♦ss Pr♦❥❡❝t✐♦♥ ❉✐s❝❧♦s✉r❡ ♦❢ ❘❡s✉❧ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ▲✐t❡r❛t✉r❡ ❢♦❝✉s❡s ♦♥ ❞✐s❝❧♦s✉r❡ ♦❢ r❡s✉❧ts ◆♦ ❣✉✐❞❛♥❝❡ ♦♥ ❤♦✇ t♦ ❞❡s✐❣♥ t❤❡ ❢♦r✇❛r❞✲❧♦♦❦✐♥❣ s❝❡♥❛r✐♦s ❚❤✐s ♣❛♣❡r✿ ❖♣t✐♠❛❧ s❝❡♥❛r✐♦ ❞❡s✐❣♥

✷ ✴ ✶✺

▼♦t✐✈❛t✐♦♥

◗✉❡st✐♦♥ ❲❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ str❡ss t❡st ❞❡s✐❣♥❄ ◮ ❙tr❡ss t❡sts ❛r❡ ✉s❡❞ ✐♥ ❧✐q✉✐❞✐t②✴r✐s❦ ♠❛♥❛❣❡♠❡♥t ❛♥❞ ✜♥❛♥❝✐❛❧ s✉♣❡r✈✐s✐♦♥ ◮ ❚❤r❡❡ ❝♦♠♣♦♥❡♥ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ❘✐s❦ ▼♦❞❡❧✐♥❣ ✫ ▲♦ss Pr♦❥❡❝t✐♦♥ ❉✐s❝❧♦s✉r❡ ♦❢ ❘❡s✉❧ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ◮ ▲✐t❡r❛t✉r❡ ❢♦❝✉s❡s ♦♥ ❞✐s❝❧♦s✉r❡ ♦❢ r❡s✉❧ts ◮ ◆♦ ❣✉✐❞❛♥❝❡ ♦♥ ❤♦✇ t♦ ❞❡s✐❣♥ t❤❡ ❢♦r✇❛r❞✲❧♦♦❦✐♥❣ s❝❡♥❛r✐♦s ❚❤✐s ♣❛♣❡r✿ ❖♣t✐♠❛❧ s❝❡♥❛r✐♦ ❞❡s✐❣♥

✷ ✴ ✶✺

▼♦t✐✈❛t✐♦♥

◗✉❡st✐♦♥ ❲❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ str❡ss t❡st ❞❡s✐❣♥❄ ◮ ❙tr❡ss t❡sts ❛r❡ ✉s❡❞ ✐♥ ❧✐q✉✐❞✐t②✴r✐s❦ ♠❛♥❛❣❡♠❡♥t ❛♥❞ ✜♥❛♥❝✐❛❧ s✉♣❡r✈✐s✐♦♥ ◮ ❚❤r❡❡ ❝♦♠♣♦♥❡♥ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ❘✐s❦ ▼♦❞❡❧✐♥❣ ✫ ▲♦ss Pr♦❥❡❝t✐♦♥ ❉✐s❝❧♦s✉r❡ ♦❢ ❘❡s✉❧ts ❙❝❡♥❛r✐♦ ❉❡s✐❣♥ ◮ ▲✐t❡r❛t✉r❡ ❢♦❝✉s❡s ♦♥ ❞✐s❝❧♦s✉r❡ ♦❢ r❡s✉❧ts ◮ ◆♦ ❣✉✐❞❛♥❝❡ ♦♥ ❤♦✇ t♦ ❞❡s✐❣♥ t❤❡ ❢♦r✇❛r❞✲❧♦♦❦✐♥❣ s❝❡♥❛r✐♦s ◮ ❚❤✐s ♣❛♣❡r✿ ❖♣t✐♠❛❧ s❝❡♥❛r✐♦ ❞❡s✐❣♥

✷ ✴ ✶✺

❲❤❛t ❛r❡ str❡ss t❡sts ✉s❡❞ ❢♦r❄

❙tr❡ss ❚❡sts ▲❡❛r♥✐♥❣ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t ▲❡❛r♥✐♥❣ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t ❈❛♣✐t❛❧ ❘❡q✉✐r❡♠❡♥ts ❚❤✐s ♣❛♣❡r✿ ♠♦❞❡❧ str❡ss t❡sts ❛s ❛ ❧❡❛r♥✐♥❣ ♠❡❝❤❛♥✐s♠ ▲❡❛r♥ t♦ ♠❛♥❛❣❡ r✐s❦ ❛♥❞ t❛❦❡ ❛ r❡♠❡❞✐❛❧ ❛❝t✐♦♥

✸ ✴ ✶✺

❲❤❛t ❛r❡ str❡ss t❡sts ✉s❡❞ ❢♦r❄

❙tr❡ss ❚❡sts ▲❡❛r♥✐♥❣ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t ▲❡❛r♥✐♥❣ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t ❈❛♣✐t❛❧ ❘❡q✉✐r❡♠❡♥ts ❚❤✐s ♣❛♣❡r✿ ♠♦❞❡❧ str❡ss t❡sts ❛s ❛ ❧❡❛r♥✐♥❣ ♠❡❝❤❛♥✐s♠ ◮ ▲❡❛r♥ t♦ ♠❛♥❛❣❡ r✐s❦ ❛♥❞ t❛❦❡ ❛ r❡♠❡❞✐❛❧ ❛❝t✐♦♥

✸ ✴ ✶✺

▼❛✐♥ r❡s✉❧ts

✶✳ ❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥ ♣r♦❜❧❡♠

■♥❢♦r♠❛t✐♦♥❛❧ ❝♦♥t❡♥t ♦❢ str❡ss t❡st ❞❡♣❡♥❞s ♦♥ str❡ss s❝❡♥❛r✐♦s ❊♥❞♦❣❡♥♦✉s ✐♥❢♦r♠❛t✐♦♥ ♣r♦❝❡ss✐♥❣ ❝♦♥str❛✐♥t

✷✳ ❙♣❡❝✐❛❧✐③❛t✐♦♥ ✈s✳ ❞✐✈❡rs✐✜❝❛t✐♦♥ ✐♥ ❧❡❛r♥✐♥❣

❙tr❡ss ❢❡✇ ❢❛❝t♦rs ✈s✳ ♠❛♥② ❢❛❝t♦rs ❉❡♣❡♥❞s ♦♥ t❤❡ ❝♦st ♦❢ t❤❡ ❡①✲♣♦st r❡♠❡❞✐❛❧ ❛❝t✐♦♥ ❛♥❞ ♣r✐♦rs

✸✳ ❍♦✇ ♠✉❝❤ t♦ str❡ss ❡❛❝❤ ❢❛❝t♦r ❞❡♣❡♥❞s ♦♥

❝♦st ♦❢ r❡♠❡❞✐❛❧ ❛❝t✐♦♥✱ ❜❡❧✐❡❢s ❛❜♦✉t ❡①♣♦s✉r❡s✱ ❤♦✇ s②st❡♠✐❝ t❤❡ ❢❛❝t♦r ✐s

✹ ✴ ✶✺

▼❛✐♥ r❡s✉❧ts

✶✳ ❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥ ♣r♦❜❧❡♠

◮ ■♥❢♦r♠❛t✐♦♥❛❧ ❝♦♥t❡♥t ♦❢ str❡ss t❡st ❞❡♣❡♥❞s ♦♥ str❡ss s❝❡♥❛r✐♦s ◮ ❊♥❞♦❣❡♥♦✉s ✐♥❢♦r♠❛t✐♦♥ ♣r♦❝❡ss✐♥❣ ❝♦♥str❛✐♥t

✷✳ ❙♣❡❝✐❛❧✐③❛t✐♦♥ ✈s✳ ❞✐✈❡rs✐✜❝❛t✐♦♥ ✐♥ ❧❡❛r♥✐♥❣

❙tr❡ss ❢❡✇ ❢❛❝t♦rs ✈s✳ ♠❛♥② ❢❛❝t♦rs ❉❡♣❡♥❞s ♦♥ t❤❡ ❝♦st ♦❢ t❤❡ ❡①✲♣♦st r❡♠❡❞✐❛❧ ❛❝t✐♦♥ ❛♥❞ ♣r✐♦rs

✸✳ ❍♦✇ ♠✉❝❤ t♦ str❡ss ❡❛❝❤ ❢❛❝t♦r ❞❡♣❡♥❞s ♦♥

❝♦st ♦❢ r❡♠❡❞✐❛❧ ❛❝t✐♦♥✱ ❜❡❧✐❡❢s ❛❜♦✉t ❡①♣♦s✉r❡s✱ ❤♦✇ s②st❡♠✐❝ t❤❡ ❢❛❝t♦r ✐s

✹ ✴ ✶✺

▼❛✐♥ r❡s✉❧ts

✶✳ ❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥ ♣r♦❜❧❡♠

◮ ■♥❢♦r♠❛t✐♦♥❛❧ ❝♦♥t❡♥t ♦❢ str❡ss t❡st ❞❡♣❡♥❞s ♦♥ str❡ss s❝❡♥❛r✐♦s ◮ ❊♥❞♦❣❡♥♦✉s ✐♥❢♦r♠❛t✐♦♥ ♣r♦❝❡ss✐♥❣ ❝♦♥str❛✐♥t

✷✳ ❙♣❡❝✐❛❧✐③❛t✐♦♥ ✈s✳ ❞✐✈❡rs✐✜❝❛t✐♦♥ ✐♥ ❧❡❛r♥✐♥❣

◮ ❙tr❡ss ❢❡✇ ❢❛❝t♦rs ✈s✳ ♠❛♥② ❢❛❝t♦rs ◮ ❉❡♣❡♥❞s ♦♥ t❤❡ ❝♦st ♦❢ t❤❡ ❡①✲♣♦st r❡♠❡❞✐❛❧ ❛❝t✐♦♥ ❛♥❞ ♣r✐♦rs

✸✳ ❍♦✇ ♠✉❝❤ t♦ str❡ss ❡❛❝❤ ❢❛❝t♦r ❞❡♣❡♥❞s ♦♥

❝♦st ♦❢ r❡♠❡❞✐❛❧ ❛❝t✐♦♥✱ ❜❡❧✐❡❢s ❛❜♦✉t ❡①♣♦s✉r❡s✱ ❤♦✇ s②st❡♠✐❝ t❤❡ ❢❛❝t♦r ✐s

✹ ✴ ✶✺

▼❛✐♥ r❡s✉❧ts

✶✳ ❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥ ♣r♦❜❧❡♠

◮ ■♥❢♦r♠❛t✐♦♥❛❧ ❝♦♥t❡♥t ♦❢ str❡ss t❡st ❞❡♣❡♥❞s ♦♥ str❡ss s❝❡♥❛r✐♦s ◮ ❊♥❞♦❣❡♥♦✉s ✐♥❢♦r♠❛t✐♦♥ ♣r♦❝❡ss✐♥❣ ❝♦♥str❛✐♥t

✷✳ ❙♣❡❝✐❛❧✐③❛t✐♦♥ ✈s✳ ❞✐✈❡rs✐✜❝❛t✐♦♥ ✐♥ ❧❡❛r♥✐♥❣

◮ ❙tr❡ss ❢❡✇ ❢❛❝t♦rs ✈s✳ ♠❛♥② ❢❛❝t♦rs ◮ ❉❡♣❡♥❞s ♦♥ t❤❡ ❝♦st ♦❢ t❤❡ ❡①✲♣♦st r❡♠❡❞✐❛❧ ❛❝t✐♦♥ ❛♥❞ ♣r✐♦rs

✸✳ ❍♦✇ ♠✉❝❤ t♦ str❡ss ❡❛❝❤ ❢❛❝t♦r ❞❡♣❡♥❞s ♦♥

◮ ❝♦st ♦❢ r❡♠❡❞✐❛❧ ❛❝t✐♦♥✱ ❜❡❧✐❡❢s ❛❜♦✉t ❡①♣♦s✉r❡s✱ ❤♦✇ s②st❡♠✐❝ t❤❡ ❢❛❝t♦r ✐s

✹ ✴ ✶✺

❊♥✈✐r♦♥♠❡♥t

◮ J ♠❛❝r♦❡❝♦♥♦♠✐❝ ❢❛❝t♦rs✱ s = [s1, ..., sJ] ◮ N ❜❛♥❦s✱ i = 1, . . . , N

◮ ▲♦ss❡s ♦❢ ❜❛♥❦ i ❣✐✈❡♥ s yi = xi · s + ηi, ✇❤❡r❡ xi ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡①♣♦s✉r❡s ❛♥❞ ηi ✐s ✐❞✐♦s②♥❝r❛t✐❝ r✐s❦

❖♥❡ r❡❣✉❧❛t♦r ✇✐t❤ ♣r❡❢❡r❡♥❝❡s ♦✈❡r ❛❣❣r❡❣❛t❡ ✇❡❛❧t❤

❘❡♠❡❞✐❛❧ ❛❝t✐♦♥ t♦ r❡❞✉❝❡ ✬s ❡①♣♦s✉r❡ t♦ ❢❛❝t♦r ❛t ❛ ❝♦♥✈❡① ❝♦st

❚❤❡ r❡❣✉❧❛t♦r ❞♦❡s ♥♦t ❦♥♦✇ t❤❡ ❡①♣♦s✉r❡s ❛♥❞ ❝❛♥ ❧❡❛r♥ ❢r♦♠ str❡ss t❡sts

✺ ✴ ✶✺

❊♥✈✐r♦♥♠❡♥t

◮ J ♠❛❝r♦❡❝♦♥♦♠✐❝ ❢❛❝t♦rs✱ s = [s1, ..., sJ] ◮ N ❜❛♥❦s✱ i = 1, . . . , N

◮ ▲♦ss❡s ♦❢ ❜❛♥❦ i ❣✐✈❡♥ s yi = xi · s + ηi, ✇❤❡r❡ xi ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡①♣♦s✉r❡s ❛♥❞ ηi ✐s ✐❞✐♦s②♥❝r❛t✐❝ r✐s❦

◮ ❖♥❡ r❡❣✉❧❛t♦r ✇✐t❤ ♣r❡❢❡r❡♥❝❡s ♦✈❡r ❛❣❣r❡❣❛t❡ ✇❡❛❧t❤ W = ω − ∑

i

yi

◮ ❘❡♠❡❞✐❛❧ ❛❝t✐♦♥ ai,j t♦ r❡❞✉❝❡ i✬s ❡①♣♦s✉r❡ t♦ ❢❛❝t♦r sj ❛t ❛ ❝♦♥✈❡① ❝♦st cj

• ai,j
• ❚❤❡ r❡❣✉❧❛t♦r ❞♦❡s ♥♦t ❦♥♦✇ t❤❡ ❡①♣♦s✉r❡s

❛♥❞ ❝❛♥ ❧❡❛r♥ ❢r♦♠ str❡ss t❡sts

✺ ✴ ✶✺

❊♥✈✐r♦♥♠❡♥t

◮ J ♠❛❝r♦❡❝♦♥♦♠✐❝ ❢❛❝t♦rs✱ s = [s1, ..., sJ] ◮ N ❜❛♥❦s✱ i = 1, . . . , N

◮ ▲♦ss❡s ♦❢ ❜❛♥❦ i ❣✐✈❡♥ s yi = xi · s + ηi, ✇❤❡r❡ xi ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡①♣♦s✉r❡s ❛♥❞ ηi ✐s ✐❞✐♦s②♥❝r❛t✐❝ r✐s❦

◮ ❖♥❡ r❡❣✉❧❛t♦r ✇✐t❤ ♣r❡❢❡r❡♥❝❡s ♦✈❡r ❛❣❣r❡❣❛t❡ ✇❡❛❧t❤ W = ω − ∑

i

yi

◮ ❘❡♠❡❞✐❛❧ ❛❝t✐♦♥ ai,j t♦ r❡❞✉❝❡ i✬s ❡①♣♦s✉r❡ t♦ ❢❛❝t♦r sj ❛t ❛ ❝♦♥✈❡① ❝♦st cj

• ai,j
• ◮ ❚❤❡ r❡❣✉❧❛t♦r ❞♦❡s ♥♦t ❦♥♦✇ t❤❡ ❡①♣♦s✉r❡s {xi}i ❛♥❞ ❝❛♥ ❧❡❛r♥ ❢r♦♠ str❡ss t❡sts

✺ ✴ ✶✺

❙tr❡ss t❡sts

❆ str❡ss t❡st ✐s ✶✳ ❛ s❡t ♦❢ M s❝❡♥❛r✐♦s ˆ S =

• ˆ

s(1)′, ..., ˆ s(M)′′

✐s ❛ r❡❛❧✐③❛t✐♦♥ ♦❢ t❤❡ r✐s❦ ❢❛❝t♦rs ✭❡✳❣✳ ✱ ✱ ✮

✷✳ r❡♣♦rt❡❞ ❧♦ss❡s ˆ y ≡

• ˆ

y(m)

i

• i ❢♦r ❡❛❝❤ s❝❡♥❛r✐♦ m ❢♦r ❡❛❝❤ ❜❛♥❦ i

✇❤❡r❡ ❛♥❞ ❛r❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ❘❡♣♦rt❡❞ ❧♦ss❡s ❛r❡ ♥♦✐s② s✐❣♥❛❧s ❛❜♦✉t ❡①♣♦s✉r❡s t❤❛t ❞❡♣❡♥❞ ♦♥

❲❡✐❣❤t ♦♥ ❡①♣♦s✉r❡s ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❍❛r❞❡r t♦ ♣r❡❞✐❝t ❧♦ss❡s ✉♥❞❡r ♠♦r❡ ❡①tr❡♠❡ s❝❡♥❛r✐♦s✿ ✐♥❝r❡❛s✐♥❣ ✐♥ ❈♦st❧② t♦ ❤❛✈❡ ♠♦r❡ s❝❡♥❛r✐♦s✳ ❚♦❞❛②✿ ❋✐①❡❞

✻ ✴ ✶✺

❙tr❡ss t❡sts

❆ str❡ss t❡st ✐s ✶✳ ❛ s❡t ♦❢ M s❝❡♥❛r✐♦s ˆ S =

• ˆ

s(1)′, ..., ˆ s(M)′′

◮ ˆ s(m) ✐s ❛ r❡❛❧✐③❛t✐♦♥ ♦❢ t❤❡ r✐s❦ ❢❛❝t♦rs s ✭❡✳❣✳ π = 2%✱ u = 10%✱ R = −20%✮

✷✳ r❡♣♦rt❡❞ ❧♦ss❡s ˆ y ≡

• ˆ

y(m)

i

• i ❢♦r ❡❛❝❤ s❝❡♥❛r✐♦ m ❢♦r ❡❛❝❤ ❜❛♥❦ i

✇❤❡r❡ ❛♥❞ ❛r❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ❘❡♣♦rt❡❞ ❧♦ss❡s ❛r❡ ♥♦✐s② s✐❣♥❛❧s ❛❜♦✉t ❡①♣♦s✉r❡s t❤❛t ❞❡♣❡♥❞ ♦♥

❲❡✐❣❤t ♦♥ ❡①♣♦s✉r❡s ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❍❛r❞❡r t♦ ♣r❡❞✐❝t ❧♦ss❡s ✉♥❞❡r ♠♦r❡ ❡①tr❡♠❡ s❝❡♥❛r✐♦s✿ ✐♥❝r❡❛s✐♥❣ ✐♥ ❈♦st❧② t♦ ❤❛✈❡ ♠♦r❡ s❝❡♥❛r✐♦s✳ ❚♦❞❛②✿ ❋✐①❡❞

✻ ✴ ✶✺

❙tr❡ss t❡sts

❆ str❡ss t❡st ✐s ✶✳ ❛ s❡t ♦❢ M s❝❡♥❛r✐♦s ˆ S =

• ˆ

s(1)′, ..., ˆ s(M)′′

◮ ˆ s(m) ✐s ❛ r❡❛❧✐③❛t✐♦♥ ♦❢ t❤❡ r✐s❦ ❢❛❝t♦rs s ✭❡✳❣✳ π = 2%✱ u = 10%✱ R = −20%✮

✷✳ r❡♣♦rt❡❞ ❧♦ss❡s ˆ y ≡

• ˆ

y(m)

i

• i ❢♦r ❡❛❝❤ s❝❡♥❛r✐♦ m ❢♦r ❡❛❝❤ ❜❛♥❦ i

ˆ yi(m) = ˆ s(m) · x′

i + αi (M) ε0 i + σε,i

• ˆ

s(m) · εi, ✇❤❡r❡ ε0

i ❛♥❞ εi ❛r❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞

❘❡♣♦rt❡❞ ❧♦ss❡s ❛r❡ ♥♦✐s② s✐❣♥❛❧s ❛❜♦✉t ❡①♣♦s✉r❡s t❤❛t ❞❡♣❡♥❞ ♦♥

❲❡✐❣❤t ♦♥ ❡①♣♦s✉r❡s ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❍❛r❞❡r t♦ ♣r❡❞✐❝t ❧♦ss❡s ✉♥❞❡r ♠♦r❡ ❡①tr❡♠❡ s❝❡♥❛r✐♦s✿ ✐♥❝r❡❛s✐♥❣ ✐♥ ❈♦st❧② t♦ ❤❛✈❡ ♠♦r❡ s❝❡♥❛r✐♦s✳ ❚♦❞❛②✿ ❋✐①❡❞

✻ ✴ ✶✺

❙tr❡ss t❡sts

❆ str❡ss t❡st ✐s ✶✳ ❛ s❡t ♦❢ M s❝❡♥❛r✐♦s ˆ S =

• ˆ

s(1)′, ..., ˆ s(M)′′

◮ ˆ s(m) ✐s ❛ r❡❛❧✐③❛t✐♦♥ ♦❢ t❤❡ r✐s❦ ❢❛❝t♦rs s ✭❡✳❣✳ π = 2%✱ u = 10%✱ R = −20%✮

✷✳ r❡♣♦rt❡❞ ❧♦ss❡s ˆ y ≡

• ˆ

y(m)

i

• i ❢♦r ❡❛❝❤ s❝❡♥❛r✐♦ m ❢♦r ❡❛❝❤ ❜❛♥❦ i

ˆ yi(m) = ˆ s(m) · x′

i + αi (M) ε0 i + σε,i

• ˆ

s(m) · εi, ✇❤❡r❡ ε0

i ❛♥❞ εi ❛r❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞

◮ ❘❡♣♦rt❡❞ ❧♦ss❡s ❛r❡ ♥♦✐s② s✐❣♥❛❧s ❛❜♦✉t ❡①♣♦s✉r❡s t❤❛t ❞❡♣❡♥❞ ♦♥ ˆ S

◮ ❲❡✐❣❤t ♦♥ ❡①♣♦s✉r❡s ✐s ❞❡t❡r♠✐♥❡❞ ❜② ˆ s ◮ ❍❛r❞❡r t♦ ♣r❡❞✐❝t ❧♦ss❡s ✉♥❞❡r ♠♦r❡ ❡①tr❡♠❡ s❝❡♥❛r✐♦s✿ σε,i ✐♥❝r❡❛s✐♥❣ ✐♥

• ˆ

s(m)

• ◮ ❈♦st❧② t♦ ❤❛✈❡ ♠♦r❡ s❝❡♥❛r✐♦s✳ ❚♦❞❛②✿ ❋✐①❡❞ M = 1

✻ ✴ ✶✺

▲❡❛r♥✐♥❣ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t

❙tr❡ss t❡st ✲ ❙❝❡♥❛r✐♦s ˆ S ✲ ❘❡♣♦rt❡❞ ❧♦ss❡s ˆ y ❘✐s❦ ♠❛♥❛❣❡♠❡♥t ❘❡♠❡❞✐❛❧ ❛❝t✐♦♥ ai,j(ˆ x, ˆ Σ) ✲ ✐♥❝r❡❛s❡s ✇✐t❤ ˆ xij ✲ ✐♥❝r❡❛s❡s ✇✐t❤ ˆ Σjj ▲❡❛r♥✐♥❣ ❢r♦♠ str❡ss t❡sts x|ˆ y, ˆ S ∼ N(ˆ x(ˆ y, ˆ S), ˆ Σx(ˆ S)) ❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❙✐❣♥❛❧ ❞❡s✐❣♥ ▲❡❛r♥✐♥❣ ❆♣♣r♦♣r✐❛t❡ ✐♥t❡r✈❡♥t✐♦♥

✼ ✴ ✶✺

▲❡❛r♥✐♥❣ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t

❙tr❡ss t❡st ✲ ❙❝❡♥❛r✐♦s ˆ S ✲ ❘❡♣♦rt❡❞ ❧♦ss❡s ˆ y ❘✐s❦ ♠❛♥❛❣❡♠❡♥t ❘❡♠❡❞✐❛❧ ❛❝t✐♦♥ ai,j(ˆ x, ˆ Σ) ✲ ✐♥❝r❡❛s❡s ✇✐t❤ ˆ xij ✲ ✐♥❝r❡❛s❡s ✇✐t❤ ˆ Σjj ▲❡❛r♥✐♥❣ ❢r♦♠ str❡ss t❡sts x|ˆ y, ˆ S ∼ N(ˆ x(ˆ y, ˆ S), ˆ Σx(ˆ S)) ❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❙✐❣♥❛❧ ❞❡s✐❣♥ ▲❡❛r♥✐♥❣ ⇓ ❆♣♣r♦♣r✐❛t❡ ✐♥t❡r✈❡♥t✐♦♥

✼ ✴ ✶✺

▲❡❛r♥✐♥❣ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t

❙tr❡ss t❡st ✲ ❙❝❡♥❛r✐♦s ˆ S ✲ ❘❡♣♦rt❡❞ ❧♦ss❡s ˆ y ❘✐s❦ ♠❛♥❛❣❡♠❡♥t ❘❡♠❡❞✐❛❧ ❛❝t✐♦♥ ai,j(ˆ x, ˆ Σ) ✲ ✐♥❝r❡❛s❡s ✇✐t❤ ˆ xij ✲ ✐♥❝r❡❛s❡s ✇✐t❤ ˆ Σjj ▲❡❛r♥✐♥❣ ❢r♦♠ str❡ss t❡sts x|ˆ y, ˆ S ∼ N(ˆ x(ˆ y, ˆ S), ˆ Σx(ˆ S)) ❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ = ❙✐❣♥❛❧ ❞❡s✐❣♥ ▲❡❛r♥✐♥❣ ⇓ ❆♣♣r♦♣r✐❛t❡ ✐♥t❡r✈❡♥t✐♦♥

✼ ✴ ✶✺

▲❡❛r♥✐♥❣ ❢r♦♠ str❡ss t❡sts

◮ ❙tr❡ss t❡st r❡s✉❧ts ✭s✐❣♥❛❧s✮ ˆ y(1)

1

= ˆ s(1) · x′

1 + e(1) 1

✳ ✳ ✳ ˆ y(1)

N = ˆ

s(1) · x′

N + e(1) N

✳ ✳ ✳ ˆ y(M)

1

= ˆ s(M) · x′

1 + e(M) 1

✳ ✳ ✳ ˆ y(M)

N

= ˆ s(M) · x′

N + e(M) N

N × M s✐❣♥❛❧s ✇❤❡r❡ ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡rr♦rs ❆♣♣❧②✐♥❣ t❤❡ ❑❛❧♠❛♥ ✜❧t❡r✱ t❤❡ r❡❣✉❧❛t♦r✬s ♣♦st❡r✐♦r ✐s

✽ ✴ ✶✺

▲❡❛r♥✐♥❣ ❢r♦♠ str❡ss t❡sts

◮ ❙tr❡ss t❡st r❡s✉❧ts ✭s✐❣♥❛❧s✮ ˆ y =

• IN ⊗ ˆ

S

• x + e,

✇❤❡r❡ e ∼ N

• 0, Σe

ˆ S

• ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡rr♦rs

❆♣♣❧②✐♥❣ t❤❡ ❑❛❧♠❛♥ ✜❧t❡r✱ t❤❡ r❡❣✉❧❛t♦r✬s ♣♦st❡r✐♦r ✐s

✽ ✴ ✶✺

▲❡❛r♥✐♥❣ ❢r♦♠ str❡ss t❡sts

◮ ❙tr❡ss t❡st r❡s✉❧ts ✭s✐❣♥❛❧s✮ ˆ y =

• IN ⊗ ˆ

S

• x + e,

✇❤❡r❡ e ∼ N

• 0, Σe

ˆ S

• ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡rr♦rs

◮ ❆♣♣❧②✐♥❣ t❤❡ ❑❛❧♠❛♥ ✜❧t❡r✱ t❤❡ r❡❣✉❧❛t♦r✬s ♣♦st❡r✐♦r ✐s x| ˆ y ∼ N ¯ x + K ˆ S (ˆ y − ¯ x) , ˆ Σx ˆ S

• ✇❤❡r❡

K ˆ S = Σx

• IN ⊗ ˆ

S ′ IN ⊗ ˆ S

• Σx
• IN ⊗ ˆ

S ′ + Σe ˆ S −1 ˆ Σx ˆ S =

• INJ − K

ˆ S IN ⊗ ˆ S

• Σx

✽ ✴ ✶✺

▲❡❛r♥✐♥❣ ❢r♦♠ str❡ss t❡sts

◮ ❙tr❡ss t❡st r❡s✉❧ts ✭s✐❣♥❛❧s✮ ˆ y =

• IN ⊗ ˆ

S

• x + e,

✇❤❡r❡ e ∼ N

• 0, Σe

ˆ S

• ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡rr♦rs

◮ ❆♣♣❧②✐♥❣ t❤❡ ❑❛❧♠❛♥ ✜❧t❡r✱ t❤❡ r❡❣✉❧❛t♦r✬s ♣♦st❡r✐♦r ✐s x| ˆ y ∼ N ¯ x + K ˆ S (ˆ y − ¯ x) , ˆ Σx ˆ S

• ❯♣❞❛t✐♥❣ ✇❡✐❣❤t

⇓ ✐♥❢♦♠❛t✐♦♥ ❛❜♦✉t ✇❤✐❝❤ ❡①♣♦s✉r❡s

✽ ✴ ✶✺

▲❡❛r♥✐♥❣ ❢r♦♠ str❡ss t❡sts

◮ ❙tr❡ss t❡st r❡s✉❧ts ✭s✐❣♥❛❧s✮ ˆ y =

• IN ⊗ ˆ

S

• x + e,

✇❤❡r❡ e ∼ N

• 0, Σe

ˆ S

• ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡rr♦rs

◮ ❆♣♣❧②✐♥❣ t❤❡ ❑❛❧♠❛♥ ✜❧t❡r✱ t❤❡ r❡❣✉❧❛t♦r✬s ♣♦st❡r✐♦r ✐s x| ˆ y ∼ N ¯ x + K ˆ S (ˆ y − ¯ x) , ˆ Σx ˆ S

• ❯♣❞❛t✐♥❣ ✇❡✐❣❤t

⇓ ✐♥❢♦♠❛t✐♦♥ ❛❜♦✉t ✇❤✐❝❤ ❡①♣♦s✉r❡s P♦st❡r✐♦r ✈❛r✐❛♥❝❡ ⇓ ❤♦✇ ♠✉❝❤ ✐♥❢♦r♠❛t✐♦♥

✽ ✴ ✶✺

❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥

◮ ❆ s❝❡♥❛r✐♦ ❝❤♦✐❝❡ ♠❛♣s t♦ ❛ ♣♦st❡r✐♦r ♣r❡❝✐s✐♦♥ ⇒ ❊♥❞♦❣❡♥♦✉s ❢❡❛s✐❜✐❧✐t② s❡t ❢♦r ♣♦st❡r✐♦r ♣r❡❝✐s✐♦♥s ✭❞❡♣❡♥❞s ♦♥❧② ♦♥ ♣r✐♦rs✮ ❊①❛♠♣❧❡✿ ❚✇♦ ❢❛❝t♦rs✱ ♦♥❡ ❜❛♥❦

✾ ✴ ✶✺

❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥

◮ ❆ s❝❡♥❛r✐♦ ❝❤♦✐❝❡ ♠❛♣s t♦ ❛ ♣♦st❡r✐♦r ♣r❡❝✐s✐♦♥ ⇒ ❊♥❞♦❣❡♥♦✉s ❢❡❛s✐❜✐❧✐t② s❡t ❢♦r ♣♦st❡r✐♦r ♣r❡❝✐s✐♦♥s ✭❞❡♣❡♥❞s ♦♥❧② ♦♥ ♣r✐♦rs✮ ◮ ❊①❛♠♣❧❡✿ ❚✇♦ ❢❛❝t♦rs✱ ♦♥❡ ❜❛♥❦

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Pr✐♦r ❝♦rr❡❧❛t✐♦♥ ✐♥ ❡①♣♦s✉r❡s Σx,12 = 0

✾ ✴ ✶✺

❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥

◮ ❆ s❝❡♥❛r✐♦ ❝❤♦✐❝❡ ♠❛♣s t♦ ❛ ♣♦st❡r✐♦r ♣r❡❝✐s✐♦♥ ⇒ ❊♥❞♦❣❡♥♦✉s ❢❡❛s✐❜✐❧✐t② s❡t ❢♦r ♣♦st❡r✐♦r ♣r❡❝✐s✐♦♥s ✭❞❡♣❡♥❞s ♦♥❧② ♦♥ ♣r✐♦rs✮ ◮ ❊①❛♠♣❧❡✿ ❚✇♦ ❢❛❝t♦rs✱ ♦♥❡ ❜❛♥❦

0.2 0.4 0.6 0.8 1

ˆ Σx,11

0.2 0.4 0.6 0.8 1

ˆ Σx,22

Pr✐♦r ❝♦rr❡❧❛t✐♦♥ ✐♥ ❡①♣♦s✉r❡s Σx,12 = 0.5

✾ ✴ ✶✺

❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥

◮ ❆ s❝❡♥❛r✐♦ ❝❤♦✐❝❡ ♠❛♣s t♦ ❛ ♣♦st❡r✐♦r ♣r❡❝✐s✐♦♥ ⇒ ❊♥❞♦❣❡♥♦✉s ❢❡❛s✐❜✐❧✐t② s❡t ❢♦r ♣♦st❡r✐♦r ♣r❡❝✐s✐♦♥s ✭❞❡♣❡♥❞s ♦♥❧② ♦♥ ♣r✐♦rs✮ ◮ ❊①❛♠♣❧❡✿ ❚✇♦ ❢❛❝t♦rs✱ ♦♥❡ ❜❛♥❦

0.2 0.4 0.6 0.8 1

ˆ Σx,11

0.2 0.4 0.6 0.8 1

ˆ Σx,22

Pr✐♦r ❝♦rr❡❧❛t✐♦♥ ✐♥ ❡①♣♦s✉r❡s Σx,12 = 0.8

✾ ✴ ✶✺

❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥

◮ ❘❡❣✉❧❛t♦r✬s ♣r♦❜❧❡♠ max

ˆ Σx∈Σ

E

• E
• U
• W
• a⋆

i,j

ˆ x, ˆ Σx

• i,j
• − ∑

i,j

cj

• a⋆

i,j

ˆ x, ˆ Σx

• ˆ

x, ˆ Σx

• ✇❤❡r❡ Σ ✐s ❡♥❞♦❣❡♥♦✉s✿ ♦✉t❝♦♠❡ ♦❢ ❑❛❧♠❛♥ ✜❧t❡r

❚✇♦ ✇❛②s t♦ r❡❞✉❝❡ r✐s❦✿ ❧❡❛r♥✐♥❣ ✭❡①✲❛♥t❡✮ ✈s✳ ✐♥t❡r✈❡♥✐♥❣ ✭❡①✲♣♦st✮

■♥❝r❡❛s✐♥❣ r❡t✉r♥s t♦ ❧❡❛r♥✐♥❣✿ ▼♦r❡ ❧❡❛r♥✐♥❣ ✐♥t❡r✈❡♥t✐♦♥ r❡s♣♦♥❞s ♠♦r❡ t♦ ❉❡❝r❡❛s✐♥❣ r❡t✉r♥s t♦ ✐♥t❡r✈❡♥✐♥❣✿ ❈♦♥✈❡① ✐♥t❡r✈❡♥t✐♦♥ ❝♦sts

❖♣t✐♠❛❧ ❧❡❛r♥✐♥❣ ♣♦❧✐❝②

❙♣❡❝✐❛❧✐③❛t✐♦♥ ✐❢ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s ❃ ❝♦♥✈❡①✐t② ✐♥ ❝♦sts str❡ss ❢❡✇ ❢❛❝t♦rs ❉✐✈❡rs✐✜❝❛t✐♦♥ ✐❢ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s ❁ ❝♦♥✈❡①✐t② ✐♥ ❝♦sts str❡ss ♠❛♥② ❢❛❝t♦rs

✶✵ ✴ ✶✺

❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥

◮ ❘❡❣✉❧❛t♦r✬s ♣r♦❜❧❡♠ max

ˆ Σx∈Σ

E

• E
• U
• W
• a⋆

i,j

ˆ x, ˆ Σx

• i,j
• − ∑

i,j

cj

• a⋆

i,j

ˆ x, ˆ Σx

• ˆ

x, ˆ Σx

• ✇❤❡r❡ Σ ✐s ❡♥❞♦❣❡♥♦✉s✿ ♦✉t❝♦♠❡ ♦❢ ❑❛❧♠❛♥ ✜❧t❡r

◮ ❚✇♦ ✇❛②s t♦ r❡❞✉❝❡ r✐s❦✿ ❧❡❛r♥✐♥❣ ✭❡①✲❛♥t❡✮ ✈s✳ ✐♥t❡r✈❡♥✐♥❣ ✭❡①✲♣♦st✮

◮ ■♥❝r❡❛s✐♥❣ r❡t✉r♥s t♦ ❧❡❛r♥✐♥❣✿ ▼♦r❡ ❧❡❛r♥✐♥❣ ↔ ✐♥t❡r✈❡♥t✐♦♥ r❡s♣♦♥❞s ♠♦r❡ t♦ ˆ y ◮ ❉❡❝r❡❛s✐♥❣ r❡t✉r♥s t♦ ✐♥t❡r✈❡♥✐♥❣✿ ❈♦♥✈❡① ✐♥t❡r✈❡♥t✐♦♥ ❝♦sts

❖♣t✐♠❛❧ ❧❡❛r♥✐♥❣ ♣♦❧✐❝②

❙♣❡❝✐❛❧✐③❛t✐♦♥ ✐❢ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s ❃ ❝♦♥✈❡①✐t② ✐♥ ❝♦sts str❡ss ❢❡✇ ❢❛❝t♦rs ❉✐✈❡rs✐✜❝❛t✐♦♥ ✐❢ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s ❁ ❝♦♥✈❡①✐t② ✐♥ ❝♦sts str❡ss ♠❛♥② ❢❛❝t♦rs

✶✵ ✴ ✶✺

❙❝❡♥❛r✐♦ ❝❤♦✐❝❡ ❛s s✐❣♥❛❧ ❞❡s✐❣♥

◮ ❘❡❣✉❧❛t♦r✬s ♣r♦❜❧❡♠ max

ˆ Σx∈Σ

E

• E
• U
• W
• a⋆

i,j

ˆ x, ˆ Σx

• i,j
• − ∑

i,j

cj

• a⋆

i,j

ˆ x, ˆ Σx

• ˆ

x, ˆ Σx

• ✇❤❡r❡ Σ ✐s ❡♥❞♦❣❡♥♦✉s✿ ♦✉t❝♦♠❡ ♦❢ ❑❛❧♠❛♥ ✜❧t❡r

◮ ❚✇♦ ✇❛②s t♦ r❡❞✉❝❡ r✐s❦✿ ❧❡❛r♥✐♥❣ ✭❡①✲❛♥t❡✮ ✈s✳ ✐♥t❡r✈❡♥✐♥❣ ✭❡①✲♣♦st✮

◮ ■♥❝r❡❛s✐♥❣ r❡t✉r♥s t♦ ❧❡❛r♥✐♥❣✿ ▼♦r❡ ❧❡❛r♥✐♥❣ ↔ ✐♥t❡r✈❡♥t✐♦♥ r❡s♣♦♥❞s ♠♦r❡ t♦ ˆ y ◮ ❉❡❝r❡❛s✐♥❣ r❡t✉r♥s t♦ ✐♥t❡r✈❡♥✐♥❣✿ ❈♦♥✈❡① ✐♥t❡r✈❡♥t✐♦♥ ❝♦sts

◮ ❖♣t✐♠❛❧ ❧❡❛r♥✐♥❣ ♣♦❧✐❝②

◮ ❙♣❡❝✐❛❧✐③❛t✐♦♥ ✐❢ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s ❃ ❝♦♥✈❡①✐t② ✐♥ ❝♦sts ⇒ str❡ss ❢❡✇ ❢❛❝t♦rs ◮ ❉✐✈❡rs✐✜❝❛t✐♦♥ ✐❢ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s ❁ ❝♦♥✈❡①✐t② ✐♥ ❝♦sts ⇒ str❡ss ♠❛♥② ❢❛❝t♦rs

✶✵ ✴ ✶✺

❖♣t✐♠❛❧ ❙❝❡♥❛r✐♦

◮ ❊①❛♠♣❧❡✿ ▼❡❛♥ ✈❛r✐❛♥❝❡ ♣r❡❢❡r❡♥❝❡s ✰ q✉❛❞r❛t✐❝ ❝♦sts ✰ ♦♥❡ s❝❡♥❛r✐♦ ◮ ❚❤❡ ✇❡✐❣❤t ♦❢ ❛ ❢❛❝t♦r ✐♥ t❤❡ ♦♣t✐♠❛❧ s❝❡♥❛r✐♦

◮ ✐s ♥♦♥✲♠♦♥♦t♦♥❡ ✇✐t❤ r❡s♣❡❝t t♦ ✐ts ❡①✲♣♦st ✐♥t❡r✈❡♥t✐♦♥ ❝♦st ◮ ✐s ♥♦♥✲♠♦♥♦t♦♥❡ ✇✐t❤ r❡s♣❡❝t t♦ ✐ts ❡①♣❡❝t❡❞ ♠❡❛♥ ◮ ✐♥❝r❡❛s❡s ✇✐t❤ ✐ts ♣r✐♦r ✉♥❝❡rt❛✐♥t② ◮ ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ ❝♦rr❡❧❛t✐♦♥ ✇✐t❤ ❡①♣♦s✉r❡s ✇✐t❤✐♥ t❤❡ ❜❛♥❦ ◮ ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ ❝♦rr❡❧❛t✐♦♥ ✇✐t❤ ❡①♣♦s✉r❡s ❛❝r♦ss ❜❛♥❦s ✭s②st❡♠✐❝ ❢❛❝t♦rs✮

✶✶ ✴ ✶✺

■♥t❡r✈❡♥t✐♦♥ ❝♦sts

◮ ❖♥❡ r❡♣r❡s❡♥t❛t✐✈❡ ❜❛♥❦ N = 1✱ t✇♦ r✐s❦ ❢❛❝t♦rs J = 2

5 10 15 20 25 30 35 40 45 50 0.2 0.3 0.4 0.5 0.6 0.7 5 10 15 20 25 30 35 40 45 50 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

✶✷ ✴ ✶✺

❍✐❣❤❡r ❡①♣❡❝t❡❞ ❡①♣♦s✉r❡ t♦ ❛ r✐s❦ ❢❛❝t♦r

◮ ❖♥❡ r❡♣r❡s❡♥t❛t✐✈❡ ❜❛♥❦ N = 1✱ t✇♦ r✐s❦ ❢❛❝t♦rs J = 2

1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 1.5 2 2.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

✶✸ ✴ ✶✺

❙②st❡♠✐❝ ❢❛❝t♦rs

◮ ❚✇♦ ❜❛♥❦s N = 2✱ t✇♦ ❢❛❝t♦rs J = 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

✶✹ ✴ ✶✺

❙✉♠♠❛r②

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

◮ ■♥♣✉ts✿ ❘❡❣✉❧❛t♦r✬s ❜❡❧✐❡❢s ❛♥❞ ♣r❡❢❡r❡♥❝❡s ◮ ❊①t❡♥s✐♦♥s✿ ♥♦♥✲s❡♣❛r❛❜❧❡ ✐♥t❡r✈❡♥t✐♦♥ ❝♦sts✱ ♦t❤❡r ♣r❡❢❡r❡♥❝❡s

◮ ●♦✐♥❣ ❢♦r✇❛r❞✿

◮ ❉②♥❛♠✐❝ str❡ss t❡st✐♥❣✿ ♠✉❧t✐♣❧❡ r♦✉♥❞s ♦❢ ❧❡❛r♥✐♥❣ t❤r♦✉❣❤ str❡ss t❡sts ◮ ❙tr❛t❡❣✐❝ ❡①♣♦s✉r❡s✿ ❊♥❞♦❣❡♥❡✐③❡ ❜❛♥❦ ❡①♣♦s✉r❡s ✭♠♦r❛❧ ❤❛③❛r❞✱ t✐♠❡ ✐♥❝♦♥s✐st❡♥❝②✮

✶✺ ✴ ✶✺