estimating satellite collision probabilities via the adaptive splitting technique … ·...
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡
❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❋r♦♠ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❚❤❡♦r②
❘✉❞② P❆❙❚❊▲✱ ❖◆❊❘❆✴❉❈P❙✲P❛❧❛✐s❡❛✉❙✉♣❡r✈✐s♦r ✿ ❏érô♠❡ ▼❖❘■❖✱ ❖◆❊❘❆✴❉❈P❙✲P❛❧❛✐s❡❛✉
❙❝✐❡♥t✐✜❝ ❞✐r❡❝t♦r ✿ ❋r❛♥ç♦✐s ▲❡ ●▲❆◆❉✱ ■◆❘■❆ ❞❡ ❘❡♥♥❡s
❏❛♥✉❛r② ✷✵✶✶
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥❡ ♠✐❣❤t t❤✐♥❦ ♦❢ ❛♥ ❡♠♣t② ♦✉t❡r s♣❛❝❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥❡ ♠✐❣❤t t❤✐♥❦ ♦❢ ❛♥ ❡♠♣t② ♦✉t❡r s♣❛❝❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥❡ ✇♦✉❧❞ ❜❡ ✇r♦♥❣✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥❡ ✇♦✉❧❞ ❜❡ ✇r♦♥❣✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❙♣❛❝❡ ❞❡❜r✐s s✉rr♦✉♥❞ ❊❛rt❤ ❛♥❞ ❛❝t✐✈❡ ♦r❜✐t✐♥❣ ♦❜❥❡❝ts✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥ ❚✉❡s❞❛② ❋❡❜r✉❛r② t❤❡ ✶✵t❤ ✷✵✵✾✱ s❛t❡❧❧✐t❡s ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s❝♦❧❧✐❞❡❞✳
❲❤❛t ✇❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ✐t ❤❛♣♣❡♥❡❞❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥ ❚✉❡s❞❛② ❋❡❜r✉❛r② t❤❡ ✶✵t❤ ✷✵✵✾✱ s❛t❡❧❧✐t❡s ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s❝♦❧❧✐❞❡❞✳
❲❤❛t ✇❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ✐t ❤❛♣♣❡♥❡❞❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s s✐③✐♥❣✿ ❘♦✉❣❤ ❡st✐♠❛t❡s
❘❡❛❧ s✐③❡s ❛r❡ ❝♦♥✜❞❡♥t✐❛❧✳
❋♦r ❝♦♥✈❡♥✐❡♥❝❡ s❛❦❡✱ ✇❡ ❛ss✉♠❡ s♣❤❡r✐❝❛❧ s❛t❡❧❧✐t❡s✳
❘❛❞✐✐ ❞❡❝✐❞❡❞ ❜❛s❡❞ ♦♥ ✜❡❧❞ ❡①♣❡r✐❡♥❝❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s s✐③✐♥❣✿ ❘♦✉❣❤ ❡st✐♠❛t❡s
❘❡❛❧ s✐③❡s ❛r❡ ❝♦♥✜❞❡♥t✐❛❧✳
❋♦r ❝♦♥✈❡♥✐❡♥❝❡ s❛❦❡✱ ✇❡ ❛ss✉♠❡ s♣❤❡r✐❝❛❧ s❛t❡❧❧✐t❡s✳
❘❛❞✐✐ ❞❡❝✐❞❡❞ ❜❛s❡❞ ♦♥ ✜❡❧❞ ❡①♣❡r✐❡♥❝❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s s✐③✐♥❣✿ ❘♦✉❣❤ ❡st✐♠❛t❡s
❘❡❛❧ s✐③❡s ❛r❡ ❝♦♥✜❞❡♥t✐❛❧✳
❋♦r ❝♦♥✈❡♥✐❡♥❝❡ s❛❦❡✱ ✇❡ ❛ss✉♠❡ s♣❤❡r✐❝❛❧ s❛t❡❧❧✐t❡s✳
❘❛❞✐✐ ❞❡❝✐❞❡❞ ❜❛s❡❞ ♦♥ ✜❡❧❞ ❡①♣❡r✐❡♥❝❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s s✐③✐♥❣✿ ❘♦✉❣❤ ❡st✐♠❛t❡s
❘❡❛❧ s✐③❡s ❛r❡ ❝♦♥✜❞❡♥t✐❛❧✳
❋♦r ❝♦♥✈❡♥✐❡♥❝❡ s❛❦❡✱ ✇❡ ❛ss✉♠❡ s♣❤❡r✐❝❛❧ s❛t❡❧❧✐t❡s✳
❘❛❞✐✐ ❞❡❝✐❞❡❞ ❜❛s❡❞ ♦♥ ✜❡❧❞ ❡①♣❡r✐❡♥❝❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ❧♦❝❛❧✐s❛t✐♦♥✿ ❚✇♦ ❧✐♥❡ ❡❧❡♠❡♥t ✭❚▲❊✮
Pr♦✈✐❞❡❞ ❜② t❤❡ ◆❖❘❆❉ ❛♥❞ t❤❡ ◆❆❙❆✳
P♦s✐t✐♦♥ ❛♥❞ ♠♦t✐♦♥ ❛t ♠❡❛s✉r❡♠❡♥t t✐♠❡ ✇✐t❤♦✉t ❛❝❝✉r❛❝②❞❡t❛✐❧s✳
❍♦♠❡ ♠❛❞❡ ♠❡❛♥s ♦❢ t❤❡ ♦r❜✐t ♣❛r❛♠❡t❡rs✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ❧♦❝❛❧✐s❛t✐♦♥✿ ❚✇♦ ❧✐♥❡ ❡❧❡♠❡♥t ✭❚▲❊✮
Pr♦✈✐❞❡❞ ❜② t❤❡ ◆❖❘❆❉ ❛♥❞ t❤❡ ◆❆❙❆✳
P♦s✐t✐♦♥ ❛♥❞ ♠♦t✐♦♥ ❛t ♠❡❛s✉r❡♠❡♥t t✐♠❡ ✇✐t❤♦✉t ❛❝❝✉r❛❝②❞❡t❛✐❧s✳
❍♦♠❡ ♠❛❞❡ ♠❡❛♥s ♦❢ t❤❡ ♦r❜✐t ♣❛r❛♠❡t❡rs✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ❧♦❝❛❧✐s❛t✐♦♥✿ ❚✇♦ ❧✐♥❡ ❡❧❡♠❡♥t ✭❚▲❊✮
Pr♦✈✐❞❡❞ ❜② t❤❡ ◆❖❘❆❉ ❛♥❞ t❤❡ ◆❆❙❆✳
P♦s✐t✐♦♥ ❛♥❞ ♠♦t✐♦♥ ❛t ♠❡❛s✉r❡♠❡♥t t✐♠❡ ✇✐t❤♦✉t ❛❝❝✉r❛❝②❞❡t❛✐❧s✳
❍♦♠❡ ♠❛❞❡ ♠❡❛♥s ♦❢ t❤❡ ♦r❜✐t ♣❛r❛♠❡t❡rs✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ❧♦❝❛❧✐s❛t✐♦♥✿ ❚✇♦ ❧✐♥❡ ❡❧❡♠❡♥t ✭❚▲❊✮
Pr♦✈✐❞❡❞ ❜② t❤❡ ◆❖❘❆❉ ❛♥❞ t❤❡ ◆❆❙❆✳
P♦s✐t✐♦♥ ❛♥❞ ♠♦t✐♦♥ ❛t ♠❡❛s✉r❡♠❡♥t t✐♠❡ ✇✐t❤♦✉t ❛❝❝✉r❛❝②❞❡t❛✐❧s✳
❍♦♠❡ ♠❛❞❡ ♠❡❛♥s ♦❢ t❤❡ ♦r❜✐t ♣❛r❛♠❡t❡rs✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ♣r♦♣❛❣❛t✐♦♥✿ ❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s
❋❛st t♦ ❝♦♠♣✉t❡✳
❊❛s② t♦ ✐♠♣❧❡♠❡♥t✳
▲❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ◆❆❙❆ ♠♦❞❡❧✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ♣r♦♣❛❣❛t✐♦♥✿ ❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s
❋❛st t♦ ❝♦♠♣✉t❡✳
❊❛s② t♦ ✐♠♣❧❡♠❡♥t✳
▲❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ◆❆❙❆ ♠♦❞❡❧✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ♣r♦♣❛❣❛t✐♦♥✿ ❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s
❋❛st t♦ ❝♦♠♣✉t❡✳
❊❛s② t♦ ✐♠♣❧❡♠❡♥t✳
▲❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ◆❆❙❆ ♠♦❞❡❧✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ♣r♦♣❛❣❛t✐♦♥✿ ❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s
❋❛st t♦ ❝♦♠♣✉t❡✳
❊❛s② t♦ ✐♠♣❧❡♠❡♥t✳
▲❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ◆❆❙❆ ♠♦❞❡❧✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
■r✐❞✐✉♠✲❈♦s♠♦s ❞✐st❛♥❝❡ ❛❜♦✉t ❝♦❧❧✐s✐♦♥ t✐♠❡ ❛❝❝♦r❞✐♥❣ t♦ ❚▲❊
16.5 16.55 16.6 16.65 16.7 16.75500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
Hour (h)
Dis
tance (
m)
Iridium−Cosmos Distance on February 10th
2009
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
❘❛♥❞♦♠♥❡ss ❛t ❤❛♥❞✿ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s ✐s ❞❡t❡r♠✐♥✐st✐❝ ♦♥❝❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❛r❡ ❦♥♦✇♥✳
❲❡ ♥♦✐s❡❞ t❤❡ ❚▲❊ t♦ ❝♦♣❡ ✇✐t❤ t❤❡✐r ✉♥❦♥♦✇♥ ✉♥❝❡rt❛✐♥t②✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
❘❛♥❞♦♠♥❡ss ❛t ❤❛♥❞✿ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s ✐s ❞❡t❡r♠✐♥✐st✐❝ ♦♥❝❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❛r❡ ❦♥♦✇♥✳
❲❡ ♥♦✐s❡❞ t❤❡ ❚▲❊ t♦ ❝♦♣❡ ✇✐t❤ t❤❡✐r ✉♥❦♥♦✇♥ ✉♥❝❡rt❛✐♥t②✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
❘❛♥❞♦♠♥❡ss ❛t ❤❛♥❞✿ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s ✐s ❞❡t❡r♠✐♥✐st✐❝ ♦♥❝❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❛r❡ ❦♥♦✇♥✳
❲❡ ♥♦✐s❡❞ t❤❡ ❚▲❊ t♦ ❝♦♣❡ ✇✐t❤ t❤❡✐r ✉♥❦♥♦✇♥ ✉♥❝❡rt❛✐♥t②✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s t♠ ✐ r♠ ✐ ✈♠ ✐ ✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ ♠ ✐ ♠ ✐ ✵✻ t ✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ ✶✻❤✸✵ ✶✻❤✹✺ ✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ ♠ ✐ ♠ ✐ ✵✻ t ✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ ✶✻❤✸✵ ✶✻❤✹✺ ✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ ✶✻❤✸✵ ✶✻❤✹✺ ✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s∀t ∈ ■ , ~❘✐ (t) = ~φ(t, t♠,✐ , (~r♠,✐ , ~✈♠,✐ ) + (~ρ♠,✐ , ~ν♠,✐ ))✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s∀t ∈ ■ , ~❘✐ (t) = ~φ(t, t♠,✐ , (~r♠,✐ , ~✈♠,✐ ) + (~ρ♠,✐ , ~ν♠,✐ ))✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
∆ (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) = ♠✐♥t∈■
{‖~❘✶(t) − ~❘✷(t)‖}
❞❝♦❧ ❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s∀t ∈ ■ , ~❘✐ (t) = ~φ(t, t♠,✐ , (~r♠,✐ , ~✈♠,✐ ) + (~ρ♠,✐ , ~ν♠,✐ ))✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
∆ (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) = ♠✐♥t∈■
{‖~❘✶(t) − ~❘✷(t)‖}
P[∆ ≤ ❞❝♦❧ ]❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ t❤r♦✇s✳
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ t❤r♦✇s✳
0 10 20 30 40 50 60 70 80 90 1000
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1
1.2x 10
−5
Simulation number
Estim
ate
d p
rob
ab
ility
Probability estimation
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ t❤r♦✇s✳
0 10 20 30 40 50 60 70 80 90 1000
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1.2x 10
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Simulation number
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ate
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rob
ab
ility
Probability estimation
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s ✇✐t❤ t❤❡✐r ♠❡❛♥✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ t❤r♦✇s✳
0 0.2 0.4 0.6 0.8 1 1.20
10
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Estimated probability (bin)
Nu
mb
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bin
Probability estimation histogram
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1000
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1.2x 10
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Simulation number
Estim
ate
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robabili
tyProbability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
0 0.2 0.4 0.6 0.8 1 1.20
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Estimated probability (bin)
Num
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of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ ❡①❛❝t❧② s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✶ ✻ ✶✵ ✻✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✶ ✼✷✺✽✳
❯♥r❡❧✐❛❜❧❡✦
◆❡❡❞ ❢♦r ❛ ✇❡❧❧ ❝❤♦s❡♥ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ t❡❝❤♥✐q✉❡✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1000
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1.2x 10
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Simulation number
Estim
ate
d p
robabili
tyProbability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
0 0.2 0.4 0.6 0.8 1 1.20
10
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70
80
Estimated probability (bin)
Num
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of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ ❡①❛❝t❧② s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✶.✻ · ✶✵−✻✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✶.✼✷✺✽✳
❯♥r❡❧✐❛❜❧❡✦
◆❡❡❞ ❢♦r ❛ ✇❡❧❧ ❝❤♦s❡♥ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ t❡❝❤♥✐q✉❡✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1000
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0.6
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1.2x 10
−5
Simulation number
Estim
ate
d p
robabili
tyProbability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
0 0.2 0.4 0.6 0.8 1 1.20
10
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30
40
50
60
70
80
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ ❡①❛❝t❧② s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✶.✻ · ✶✵−✻✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✶.✼✷✺✽✳
❯♥r❡❧✐❛❜❧❡✦
◆❡❡❞ ❢♦r ❛ ✇❡❧❧ ❝❤♦s❡♥ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ t❡❝❤♥✐q✉❡✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1000
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0.4
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0.8
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1.2x 10
−5
Simulation number
Estim
ate
d p
robabili
tyProbability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
0 0.2 0.4 0.6 0.8 1 1.20
10
20
30
40
50
60
70
80
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ ❡①❛❝t❧② s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✶.✻ · ✶✵−✻✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✶.✼✷✺✽✳
❯♥r❡❧✐❛❜❧❡✦
◆❡❡❞ ❢♦r ❛ ✇❡❧❧ ❝❤♦s❡♥ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ t❡❝❤♥✐q✉❡✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❘❛r❡ ❡✈❡♥t t❡❝❤♥✐q✉❡ r❡q✉✐r❡♠❡♥ts
❆♣♣❧✐❝❛❜✐❧✐t②✿ ♠❡t❤♦❞ ♠✉st ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ♦♥ ❛ ❜❧❛❝❦ ❜♦①♠❛♣♣✐♥❣ ❛♥❞ ✇✐t❤♦✉t ♣r✐♦r✳
❆✛♦r❞❛❜✐❧✐t②✿ ♠❡t❤♦❞ s✐♠✉❧❛t✐♦♥ ❝♦st s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s♣♦ss✐❜❧❡✳
❊♥❣✐♥❡❡r✲❢r✐❡♥❞❧✐♥❡ss✿ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❛s s❡❧❢✲t✉♥✐♥❣ ❛s♣♦ss✐❜❧❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❘❛r❡ ❡✈❡♥t t❡❝❤♥✐q✉❡ r❡q✉✐r❡♠❡♥ts
❆♣♣❧✐❝❛❜✐❧✐t②✿ ♠❡t❤♦❞ ♠✉st ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ♦♥ ❛ ❜❧❛❝❦ ❜♦①♠❛♣♣✐♥❣ ❛♥❞ ✇✐t❤♦✉t ♣r✐♦r✳
❆✛♦r❞❛❜✐❧✐t②✿ ♠❡t❤♦❞ s✐♠✉❧❛t✐♦♥ ❝♦st s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s♣♦ss✐❜❧❡✳
❊♥❣✐♥❡❡r✲❢r✐❡♥❞❧✐♥❡ss✿ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❛s s❡❧❢✲t✉♥✐♥❣ ❛s♣♦ss✐❜❧❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❘❛r❡ ❡✈❡♥t t❡❝❤♥✐q✉❡ r❡q✉✐r❡♠❡♥ts
❆♣♣❧✐❝❛❜✐❧✐t②✿ ♠❡t❤♦❞ ♠✉st ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ♦♥ ❛ ❜❧❛❝❦ ❜♦①♠❛♣♣✐♥❣ ❛♥❞ ✇✐t❤♦✉t ♣r✐♦r✳
❆✛♦r❞❛❜✐❧✐t②✿ ♠❡t❤♦❞ s✐♠✉❧❛t✐♦♥ ❝♦st s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s♣♦ss✐❜❧❡✳
❊♥❣✐♥❡❡r✲❢r✐❡♥❞❧✐♥❡ss✿ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❛s s❡❧❢✲t✉♥✐♥❣ ❛s♣♦ss✐❜❧❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❘❛r❡ ❡✈❡♥t t❡❝❤♥✐q✉❡ r❡q✉✐r❡♠❡♥ts
❆♣♣❧✐❝❛❜✐❧✐t②✿ ♠❡t❤♦❞ ♠✉st ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ♦♥ ❛ ❜❧❛❝❦ ❜♦①♠❛♣♣✐♥❣ ❛♥❞ ✇✐t❤♦✉t ♣r✐♦r✳
❆✛♦r❞❛❜✐❧✐t②✿ ♠❡t❤♦❞ s✐♠✉❧❛t✐♦♥ ❝♦st s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s♣♦ss✐❜❧❡✳
❊♥❣✐♥❡❡r✲❢r✐❡♥❞❧✐♥❡ss✿ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❛s s❡❧❢✲t✉♥✐♥❣ ❛s♣♦ss✐❜❧❡✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡❚❤❡ ❛❧❣♦r✐t❤♠❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts❈▼❈ ✈❡rs✉s ❆❙❚
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ ❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ ✳
❣ ❞ ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
s ❳ ❣ ❤ ❳ ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ ✳
❣ ❞ ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
s ❳ ❣ ❤ ❳ ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ ❞ ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
s ❳ ❣ ❤ ❳ ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
s ❳ ❣ ❤ ❳ ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
ξ ≡ s(❳ ) = ❣ ◦ ❤(❳ ) ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
ξ ≡ s(❳ ) = ❣ ◦ ❤(❳ ) ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
A ⊂ R ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
ξ ≡ s(❳ ) = ❣ ◦ ❤(❳ ) ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
A ⊂ R ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
P[ξ ∈ A] = P[❳ ∈ s−✶(A)]❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
ξ ≡ s(❳ ) = ❣ ◦ ❤(❳ ) ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
A ⊂ R ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
P[ξ ∈ A] = P[❳ ∈ s−✶(A)]❄
■❢ P[ξ ∈ A] ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐♥ ❛ ♥✉ts❤❡❧❧
✶ ❘❡♣❤r❛s❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t②
✵
✐ ✶ ✐ ✐ ✶
✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ✐t❡r❛t✐✈❡❧②✳
❋r♦♠ ♥♦✇ ♦♥✱ s✉♣♣♦s❡ ❚
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐♥ ❛ ♥✉ts❤❡❧❧✶ ❘❡♣❤r❛s❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t②
{
R = A✵ ⊃ · · · ⊃ Aκ = AP[ξ ∈ A] =
∏κ✐=✶
P[ξ ∈ A✐ |ξ ∈ A✐−✶]
✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ✐t❡r❛t✐✈❡❧②✳
❋r♦♠ ♥♦✇ ♦♥✱ s✉♣♣♦s❡ ❚
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐♥ ❛ ♥✉ts❤❡❧❧✶ ❘❡♣❤r❛s❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t②
{
R = A✵ ⊃ · · · ⊃ Aκ = AP[ξ ∈ A] =
∏κ✐=✶
P[ξ ∈ A✐ |ξ ∈ A✐−✶]
✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ✐t❡r❛t✐✈❡❧②✳
❋r♦♠ ♥♦✇ ♦♥✱ s✉♣♣♦s❡ ❚
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐♥ ❛ ♥✉ts❤❡❧❧✶ ❘❡♣❤r❛s❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t②
{
R = A✵ ⊃ · · · ⊃ Aκ = AP[ξ ∈ A] =
∏κ✐=✶
P[ξ ∈ A✐ |ξ ∈ A✐−✶]
✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ✐t❡r❛t✐✈❡❧②✳
❋r♦♠ ♥♦✇ ♦♥✱ s✉♣♣♦s❡ A = [❚ ,∞[
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ ❳ ✐
❦ ✶♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦ ✶✳
✷ ❉❡✜♥❡ ❦ ❛s s ❳ ✐
❦ ✶✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡t
❦ ❦ ✳✸ s ❳ ❦ ❦ ✶ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ ❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥
❦ ✳✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ ❦ ❛s s ❳ ✐
❦ ✶✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡t
❦ ❦ ✳✸ s ❳ ❦ ❦ ✶ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ ❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥
❦ ✳✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i−1
Generate ξi−1
points according to fξ|ξ≥ d
i−1
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ s ❳ ❦ ❦ ✶ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ ❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥
❦ ✳✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i−1
Quantile positioning
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳
✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ ❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥
❦ ✳✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i−1
Killing under the threshold
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ A❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥A❦ ✳
✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦
✳✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i−1
Resampling above the threshold uniformly with replacement
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ A❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥A❦ ✳
✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦
✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i
Sampling conditionally to being above the threshold
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ A❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥A❦ ✳
✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦
✳✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i
New quantile positionning
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ A❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥A❦ ✳
✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦
✳✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i
New quantile positionning
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❆❙❚ ❧♦❣✐❣r❛♠
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
"Use" points to generate
conditionally to being
above the quantile.
EndCalculate
sought probability
NO
YES
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
(Repeat ad libitum)
EndCalculate
sought probability
NO
YES
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼ ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
① ▼ ① ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
① ② ❢❳ ① ▼ ① ② ❢❳ ② ▼ ② ①
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
① ② ❢❳ ① ▼ ① ② ❢❳ ② ▼ ② ①
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
∀(① , ②) ∈ X × X, ❢❳ (①)▼(① , ②) = ❢❳ (②)▼(② , ①)
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
∀(① , ②) ∈ X × X, ❢❳ (①)▼(① , ②) = ❢❳ (②)▼(② , ①)
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
∀(① , ②) ∈ X × X, ❢❳ (①)▼(① , ②) = ❢❳ (②)▼(② , ①)
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
∀(① , ②) ∈ X × X, ❢❳ (①)▼(① , ②) = ❢❳ (②)▼(② , ①)
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ❢❳ s ❳ ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼ ❳❦ ✳
✷ ❳❦
❩ ✐❢ s ❩ ❦
❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳❦ ▼❦ ❳❦ ❞② ✇❤❡r❡
▼❦ ❳❦ ❞② ✶❦s ② ▼ ❳❦ ❞② ▼ ❳❦
❝
❦ ❳❦❞②
❦ ❦
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼ ❳❦ ✳
✷ ❳❦
❩ ✐❢ s ❩ ❦
❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳❦ ▼❦ ❳❦ ❞② ✇❤❡r❡
▼❦ ❳❦ ❞② ✶❦s ② ▼ ❳❦ ❞② ▼ ❳❦
❝
❦ ❳❦❞②
❦ ❦
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳❦
❩ ✐❢ s ❩ ❦
❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳❦ ▼❦ ❳❦ ❞② ✇❤❡r❡
▼❦ ❳❦ ❞② ✶❦s ② ▼ ❳❦ ❞② ▼ ❳❦
❝
❦ ❳❦❞②
❦ ❦
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳̃❦ =
{
❩ ✐❢ s(❩ ) ≥ δ❦❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳❦ ▼❦ ❳❦ ❞② ✇❤❡r❡
▼❦ ❳❦ ❞② ✶❦s ② ▼ ❳❦ ❞② ▼ ❳❦
❝
❦ ❳❦❞②
❦ ❦
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳̃❦ =
{
❩ ✐❢ s(❩ ) ≥ δ❦❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳̃❦ ∼ ▼❦(❳❦ , ❞②) ✇❤❡r❡
▼❦(❳❦ , ❞②) = ✶A❦(s(②))▼(❳❦ , ❞②) + ▼(❳❦ ,A❝
❦)δ❳❦
(❞②)A❦ = [δ❦ ,+∞[
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳̃❦ =
{
❩ ✐❢ s(❩ ) ≥ δ❦❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳̃❦ ∼ ▼❦(❳❦ , ❞②) ✇❤❡r❡
▼❦(❳❦ , ❞②) = ✶A❦(s(②))▼(❳❦ , ❞②) + ▼(❳❦ ,A❝
❦)δ❳❦
(❞②)A❦ = [δ❦ ,+∞[
❳̃❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳̃❦ =
{
❩ ✐❢ s(❩ ) ≥ δ❦❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳̃❦ ∼ ▼❦(❳❦ , ❞②) ✇❤❡r❡
▼❦(❳❦ , ❞②) = ✶A❦(s(②))▼(❳❦ , ❞②) + ▼(❳❦ ,A❝
❦)δ❳❦
(❞②)A❦ = [δ❦ ,+∞[
❳̃❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦❛s✇❡❧❧✦
⇒ ❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ |s(❳ )≥δ❦s❡t✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
(Repeat ad libitum)
EndCalculate
sought probability
NO
YES
❆ ❣❧♦❜❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❆❙❚ ✇✐t❤ ♠❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s t♠ ✐ r♠ ✐ ✈♠ ✐ ✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ ♠ ✐ ♠ ✐ ✵✻ t ✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ ✶✻❤✸✵ ✶✻❤✹✺ ✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s∀t ∈ ■ , ~❘✐ (t) = ~φ(t, t♠,✐ , (~r♠,✐ , ~✈♠,✐ ) + (~ρ♠,✐ , ~ν♠,✐ ))✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
∆ (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) = ♠✐♥t∈■
{‖~❘✶(t) − ~❘✷(t)‖}
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡ ■■
❳ ♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ✐s t❤❡ ●❛✉ss✐❛♥ ✐♥♣✉t✳
❳ ✐s t❤❡ r❛♥❞♦♠ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ ■ ✳
✵ ❞❝♦❧ ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡ ■■
❳ = (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) ✐s t❤❡ ●❛✉ss✐❛♥ ✐♥♣✉t✳
❳ ✐s t❤❡ r❛♥❞♦♠ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ ■ ✳
✵ ❞❝♦❧ ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡ ■■
❳ = (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) ✐s t❤❡ ●❛✉ss✐❛♥ ✐♥♣✉t✳
ξ = ∆(❳ ) ✐s t❤❡ r❛♥❞♦♠ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ ■ ✳
✵ ❞❝♦❧ ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡ ■■
❳ = (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) ✐s t❤❡ ●❛✉ss✐❛♥ ✐♥♣✉t✳
ξ = ∆(❳ ) ✐s t❤❡ r❛♥❞♦♠ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ ■ ✳
A = [✵, ❞❝♦❧ ] ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
Input points
●❡♥❡r❛t❡ r❛♥❞♦♠ ♥♦✐s❡s(
❳ ✶
❦, · · · ,❳ ♥
❦
)
✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❈❛❧❝✉❧❛t❡ ■r✐❞✐✉♠✲❑♦s♠♦s ❞✐st❛♥❝❡s ♦✈❡r t✐♠❡ ✐♥t❡r✈❛❧❧❡ ■ ✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❋✐♥❞ ♠✐♥✐♠❛(
∆✶
❦, · · · , ∆♥
❦
)
✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❋✐♥❞ ♠✐♥✐♠❛(
∆✶
❦, · · · , ∆♥
❦
)
✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
10 20 30 40 50 60 70 80 90 100200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❋✐♥❞ ♠✐♥✐♠❛(
∆✶
❦, · · · , ∆♥
❦
)
✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
(10) (20) (30) (40) (50) (60) (70) (80) (90) (100)200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❙♦rt ♠✐♥✐♠❛ t♦ ❢♦r♠(
∆(✶)❦
, · · · ,∆(♥)❦
)
✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
(10) (20) (30) (40) (50) (60) (70) (80) (90) (100)200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉❡✜♥❡ q✉❛♥t✐❧❡ δ❦+✶✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
(10) (20) (30) (40) (50) (60) (70) (80) (90) (100)200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
![Page 100: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/100.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
10 20 30 40 50 60 70 80 90 100200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
![Page 101: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/101.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
10 20 30 40 50 60 70 80 90 100200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
![Page 102: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/102.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
10 20 30 40 50 60 70 80 90 100200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
![Page 103: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/103.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
![Page 104: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/104.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
![Page 105: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/105.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
Input points
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
![Page 106: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/106.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
Input points after discarding
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
![Page 107: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/107.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
Input points after shaking
❯s❡ t❤❡ ♠❛r❦♦✈ ❦❡r♥❡❧ ❛♥❞ s❡❧❡❝t❡❞ ♣♦✐♥ts t♦ r❡❣❡♥❡r❛t❡ s❛♠♣❧❡✳
![Page 108: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/108.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) t❤r♦✇s✳
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
![Page 109: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/109.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) t❤r♦✇s✳
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
rob
ab
ility
Probability estimation
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
![Page 110: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/110.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) t❤r♦✇s✳
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
rob
ab
ility
Probability estimation
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s ✇✐t❤ t❤❡✐r ♠❡❛♥✳
![Page 111: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/111.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) t❤r♦✇s✳
1 2 3 4 5 6 7 8 9 10
x 10−7
0
5
10
15
20
25
Estimated probability (bin)
Nu
mb
er
of
po
ints
in
bin
Probability estimation histogram
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
robabili
ty
Probability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
1 2 3 4 5 6 7 8 9 10
x 10−7
0
5
10
15
20
25
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✹ ✼✽ ✶✵ ✼✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵ ✸✷✸✷✳
◗✉✐t❡ ❘❡❧✐❛❜❧❡✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
robabili
ty
Probability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
1 2 3 4 5 6 7 8 9 10
x 10−7
0
5
10
15
20
25
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✹.✼✽ · ✶✵−✼✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵.✸✷✸✷✳
◗✉✐t❡ ❘❡❧✐❛❜❧❡✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
robabili
ty
Probability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
1 2 3 4 5 6 7 8 9 10
x 10−7
0
5
10
15
20
25
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✹.✼✽ · ✶✵−✼✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵.✸✷✸✷✳
◗✉✐t❡ ❘❡❧✐❛❜❧❡✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈▼❈ ✈❡rs✉s ❆❙❚
❈▼❈ ❱s ❆❙❚
❋✐❣✉r❡✿ ❊st✐♠❛t✐♦♥s
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ate
d p
robability
Probability estimation
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈▼❈ ✈❡rs✉s ❆❙❚
❈▼❈ ❱s ❆❙❚
❋✐❣✉r❡✿ ❊st✐♠❛t✐♦♥s ❛♥❞ ♠❡❛♥s
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ate
d p
robability
Probability estimation
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈▼❈ ✈❡rs✉s ❆❙❚
❈▼❈ ❱s ❆❙❚
❋✐❣✉r❡✿ ❍✐st♦❣r❛♠s
0 0.2 0.4 0.6 0.8 1 1.20
10
20
30
40
50
60
70
80
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
![Page 118: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/118.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈▼❈ ✈❡rs✉s ❆❙❚
❈▼❈ ❱s ❆❙❚
❋✐❣✉r❡✿ ❊st✐♠❛t✐♦♥s
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ated p
robab
ility
Probability estimation
❋✐❣✉r❡✿ ❍✐st♦❣r❛♠s
0 0.2 0.4 0.6 0.8 1 1.20
10
20
30
40
50
60
70
80
Estimated probability (bin)
Numb
er of
point
s in bi
n
Probability estimation histogram
![Page 119: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/119.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s❖✈❡r✈✐❡✇❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧◗✉❛♥t✐❧❡ ❧❡✈❡❧
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
(Repeat ad libitum)
EndCalculate
sought probability
NO
YES
❆ ❣❧♦❜❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❆❙❚ ✇✐t❤ ♠❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
EndCalculate
sought probability
NO
YES
❍♦✇❡✈❡r✱ ♥♦ r❛t✐♦♥❛❧ q✉❛♥t✐❧❡ ❧❡✈❡❧ ❝❤♦✐❝❡✳✳✳
![Page 122: Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique](https://reader030.vdocuments.site/reader030/viewer/2022040908/5e7f3e21d9af05076431c415/html5/thumbnails/122.jpg)
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
EndCalculate
sought probability
NO
YES
✳✳✳♥♦r ▼❛r❦♦✈✐❛♥ ❝❤♦✐❝❡ ❛♥❞ t✉♥✐♥❣✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
EndCalculate
sought probability
NO
YES
❚✇♦ ✐ss✉❡s t♦ t❛❝❦❧❡✦
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐s
❛ ✐♥t❡r❛❝t✐♥❣ ♣❛rt✐❝❧❡ t❡❝❤♥✐q✉❡ ❬✷❪
❛ ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞
✉s❡❞ ❛s ❛ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❬✺✱ ✻❪✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐s
❛ ✐♥t❡r❛❝t✐♥❣ ♣❛rt✐❝❧❡ t❡❝❤♥✐q✉❡ ❬✷❪
❛ ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞
✉s❡❞ ❛s ❛ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❬✺✱ ✻❪✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐s
❛ ✐♥t❡r❛❝t✐♥❣ ♣❛rt✐❝❧❡ t❡❝❤♥✐q✉❡ ❬✷❪
❛ ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞
✉s❡❞ ❛s ❛ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❬✺✱ ✻❪✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐s
❛ ✐♥t❡r❛❝t✐♥❣ ♣❛rt✐❝❧❡ t❡❝❤♥✐q✉❡ ❬✷❪
❛ ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞
✉s❡❞ ❛s ❛ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❬✺✱ ✻❪✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❇❛s✐❝ ■♥❣r❡❞✐❡♥ts ❬✶❪
●✐✈❡♥
❢❳ ■♥♣✉t Pr♦❜❛❜✐❧✐t② ❉❡♥s✐t② ❋✉♥❝t✐♦♥❚ ❚❤r❡s❤♦❧❞ t♦ ❡①❝❡❡❞◆ ❙✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t
❑♥♦✇♥ ♦r ❞❡s✐❣♥❡❞
▼ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❉❡❝✐❞❡❞ ❛♥❞ ❞❡t❡r♠✐♥✐st✐❝
❛❦ ❦t❤ ◗✉❛♥t✐❧❡ ❧❡✈❡❧❜❦ ❦t❤ ❦❡r♥❡❧ ♣❛r❛♠❡t❡r s❡t♥❦ ❦t❤ st❡♣ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t
❖❜s❡r✈❡❞ ❛♥❞ r❛♥❞♦♠
❦ ❦t❤ ❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡❚♦t❛❧ ♥✉♠❜❡r ♦❢ q✉❛♥t✐❧❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❇❛s✐❝ ■♥❣r❡❞✐❡♥ts ❬✶❪
●✐✈❡♥
❢❳ ■♥♣✉t Pr♦❜❛❜✐❧✐t② ❉❡♥s✐t② ❋✉♥❝t✐♦♥❚ ❚❤r❡s❤♦❧❞ t♦ ❡①❝❡❡❞◆ ❙✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t
❑♥♦✇♥ ♦r ❞❡s✐❣♥❡❞
▼(·, ·) ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❉❡❝✐❞❡❞ ❛♥❞ ❞❡t❡r♠✐♥✐st✐❝
❛❦ ❦t❤ ◗✉❛♥t✐❧❡ ❧❡✈❡❧❜❦ ❦t❤ ❦❡r♥❡❧ ♣❛r❛♠❡t❡r s❡t♥❦ ❦t❤ st❡♣ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t
❖❜s❡r✈❡❞ ❛♥❞ r❛♥❞♦♠
δ❦ ❦t❤ ❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡κ ❚♦t❛❧ ♥✉♠❜❡r ♦❢ q✉❛♥t✐❧❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ✵❞ ■❞ ✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼ ① ❞②①
✶ ❜✷
❜✷
✶ ❜✷
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧②
❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳■t❡r❛t✐♥❣ ▼❦ ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳
❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳■t❡r❛t✐♥❣ ▼❦ ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳
■t❡r❛t✐♥❣ ▼❦ ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳■t❡r❛t✐♥❣ ▼❦(·, ·) ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳■t❡r❛t✐♥❣ ▼❦(·, ·) ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ∈ [.✷✵, .✷✺] ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ∈ [.✷✵, .✷✺] ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ∈ [.✷✵, .✷✺] ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❑❡② ✐❞❡❛s t♦ r❡♠❡♠❜❡r
❆❙❚ ✇✐❞❡❧② ♦✉t♣❡r❢♦r♠s ❈▼❈ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❛r❡ ❡✈❡♥ts ✳
❘✉❧❡ ♦❢ t❤❡ t❤✉♠❜ t✉♥✐♥❣ ❝❛♥ ❞♦ t❤❡ tr✐❝❦✳✳✳
✳✳✳❜✉t t❤❡♦r❡t✐❝❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❑❡② ✐❞❡❛s t♦ r❡♠❡♠❜❡r
❆❙❚ ✇✐❞❡❧② ♦✉t♣❡r❢♦r♠s ❈▼❈ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❛r❡ ❡✈❡♥ts ✳
❘✉❧❡ ♦❢ t❤❡ t❤✉♠❜ t✉♥✐♥❣ ❝❛♥ ❞♦ t❤❡ tr✐❝❦✳✳✳
✳✳✳❜✉t t❤❡♦r❡t✐❝❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❑❡② ✐❞❡❛s t♦ r❡♠❡♠❜❡r
❆❙❚ ✇✐❞❡❧② ♦✉t♣❡r❢♦r♠s ❈▼❈ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❛r❡ ❡✈❡♥ts ✳
❘✉❧❡ ♦❢ t❤❡ t❤✉♠❜ t✉♥✐♥❣ ❝❛♥ ❞♦ t❤❡ tr✐❝❦✳✳✳
✳✳✳❜✉t t❤❡♦r❡t✐❝❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❑❡② ✐❞❡❛s t♦ r❡♠❡♠❜❡r
❆❙❚ ✇✐❞❡❧② ♦✉t♣❡r❢♦r♠s ❈▼❈ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❛r❡ ❡✈❡♥ts ✳
❘✉❧❡ ♦❢ t❤❡ t❤✉♠❜ t✉♥✐♥❣ ❝❛♥ ❞♦ t❤❡ tr✐❝❦✳✳✳
✳✳✳❜✉t t❤❡♦r❡t✐❝❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ✐s ♥❡❡❞❡❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②
❆♣♣❧②✐♥❣ ✐t t♦ ❛ ✇✐❞❡r ❝❧❛ss ♦❢ r❛♥❞♦♠ ✐♥♣✉t ✈✐❛▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛♥❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✳
❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ ❳ ♣ ♣ t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳
❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②
❆♣♣❧②✐♥❣ ✐t t♦ ❛ ✇✐❞❡r ❝❧❛ss ♦❢ r❛♥❞♦♠ ✐♥♣✉t ✈✐❛▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛♥❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✳
❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ ❳ ♣ ♣ t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳
❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②
❆♣♣❧②✐♥❣ ✐t t♦ ❛ ✇✐❞❡r ❝❧❛ss ♦❢ r❛♥❞♦♠ ✐♥♣✉t ✈✐❛▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛♥❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✳
❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ P [❳ ∈ A] ∈ [♣−, ♣+] t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳
❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②
❆♣♣❧②✐♥❣ ✐t t♦ ❛ ✇✐❞❡r ❝❧❛ss ♦❢ r❛♥❞♦♠ ✐♥♣✉t ✈✐❛▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛♥❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✳
❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ P [❳ ∈ A] ∈ [♣−, ♣+] t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳
❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
P❧❡❛s❡✱ ❧❡t ♠❡ ❛♥s✇❡r ②♦✉r q✉❡st✐♦♥s✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
❈♦♥❢❡r❡♥❝❡s
✶ ❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ✈❡rs✉s ❞❡❜r✐s ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡✿ ✽t❤ ✐♥t❡r♥❛t✐♦♥❛❧ ✇♦r❦s❤♦♣ ♦♥ r❛r❡
❡✈❡♥t s✐♠✉❧❛t✐♦♥✱ P♦st❡r✱ ✷✵✲✷✸ ❏✉♥❡ ✷✵✶✵ ❈❛♠❜r✐❞❣❡ ✭❯❑✮✱ ❘✳P❛st❡❧✱ ❏✳ ▼♦r✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
✷ ❘❡♣r❡s❡♥t✐♥❣ ❙♣❛t✐❛❧ ❉✐str✐❜✉t✐♦♥ ✈✐❛ t❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣t❡❝❤♥✐q✉❡ ❛♥❞ ■s♦q✉❛♥t✐❧❡ ❈✉r✈❡s ✿ ❊✉r♦♣❡❛♥ ▼❡❡t✐♥❣ ♦❢
❙t❛t✐st✐❝✐❛♥s ✷✵✶✵✱ Prés❡♥t❛t✐♦♥ ❖r❛❧❡ s✉r rés✉♠é✱ ✶✼✲✷✷ ❆✉❣✉st✷✵✶✵ P✐r❛❡✉s ✭●r❡❡❝❡✮✱ ❘✳ P❛st❡❧✱ ❏✳ ▼♦r✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
✸ ❋r♦♠ s❛t❡❧❧✐t❡ ✈❡rs✉s ❞❡❜r✐s ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s t♦ t❤❡ ❛❞❛♣t✐✈❡s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡ ❋r♦♠ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❚❤❡♦r② ✿ ❘❛r❡ ❊✈❡♥t
❙✐♠✉❧❛t✐♦♥ ❲♦r❦s❤♦♣✱ Prés❡♥t❛t✐♦♥ ❖r❛❧❡ s✉r rés✉♠é✱ ✷✽✲✷✾❖❝t♦❜❡r ✷✵✶✵ ❇♦r❞❡❛✉① ✭❋r❛♥❝❡✮✱ ❘✳ P❛st❡❧✱ ❏✳ ▼♦r✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
✹ ❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣t❡❝❤♥✐q✉❡ ✿ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ✐♥ ❈♦♠♣✉t❡r ▼♦❞❡❧✐♥❣ ❛♥❞
❙✐♠✉❧❛t✐♦♥✱ Prés❡♥t❛t✐♦♥ ❖r❛❧❡ s✉r ❛rt✐❝❧❡✱ ✼✲✾ ❏❛♥✉❛r② ✷✵✶✶▼✉♠❜❛✐ ✭■♥❞✐❛✮✱ ❘✳ P❛st❡❧✱ ❏✳ ▼♦r✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
❘❡✈✉❡s
✶ ❙❛♠♣❧✐♥❣ ❚❡❝❤♥✐q✉❡ ❢♦r ❧❛✉♥❝❤❡r ✐♠♣❛❝t s❛❢❡t② ③♦♥❡❡st✐♠❛t✐♦♥ ✿ ❆❝t❛ ❛str♦♥❛✉t✐❝❛✱ ✻✻✭✺✲✻✮✿ ✼✸✻✲✼✹✶✱ ✷✵✶✵✱ ❏✳▼♦r✐♦ ✫ ❘✳ P❛st❡❧✳
✷ ❆♥ ♦✈❡r✈✐❡✇ ♦r ✐♠♣♦rt❛♥❝❡ s♣❧✐tt✐♥❣ ❢♦r r❛r❡ ❡✈❡♥t s✐♠✉❧❛t✐♦♥ ✿❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✱ ✸✶✿✶✷✾✺✲✶✸✵✸✱ ✷✵✶✵✱ ❏✳ ▼♦r✐♦✱ ❘✳P❛st❡❧ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
✸ ❊st✐♠❛t✐♦♥ ❉❡ ♣r♦❜❛❜✐❧✐tés ❡t ❞❡ q✉❛♥t✐❧❡s r❛r❡s ♣♦✉r ❧❛❝❛r❛❝tér✐s❛t✐♦♥ ❞✬✉♥❡ ③♦♥❡ ❞❡ r❡t♦♠❜é❡ ❞✬✉♥ ❡♥❣✐♥ ✿ ❏♦✉r♥❛❧
❞❡ ❧❛ ❙♦❝✐été ❋r❛♥ç❛✐s❡ ❞❡ ❙t❛t✐st✐q✉❡✱ ♣❡♥❞✐♥❣ s✉❜❥❡❝t t♦♠♦❞✐✜❝❛t✐♦♥s✱ ❏✳ ▼♦r✐♦✱ ❘✳ P❛st❡❧ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❘❡❢❡r❡♥❝❡s
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❘❡❢❡r❡♥❝❡s
❬✶❪ ❋ré❞ér✐❝ ❈ér♦✉✱ P✐❡rr❡ ❉❡❧ ▼♦r❛❧✱ ❚❡❞❞② ❋✉r♦♥✱ ❛♥❞ ❆r♥❛✉❞●✉②❛❞❡r✳❘❛r❡ ❡✈❡♥t s✐♠✉❧❛t✐♦♥ ❢♦r ❛ st❛t✐❝ ❞✐str✐❜✉t✐♦♥✳❚❡❝❤♥✐❝❛❧ r❡♣♦rt✱❤tt♣✿✴✴❤❛❧✳✐♥r✐❛✳❢r✴✐♥r✐❛✲✵✵✸✺✵✼✻✷✴❢r✴✱ ✷✵✵✾✳
❬✷❪ P✐❡rr❡ ❉❡❧ ▼❖❘❆▲✳❋❡②♥♠❛♥✲❑❛❝ ❋♦r♠✉❧❛❡✿ ●❡♥❡❛❧♦❣✐❝❛❧ ❛♥❞ ■♥t❡r❛❝t✐♥❣ P❛rt✐❝❧❡
❙②st❡♠s ❲✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✳❙♣r✐♥❣❡r✱ ✷✵✵✹✳
❬✸❪ ❆r♥❛✉❞ ●✉②❛❞❡r✱ ◆✐❝♦❧❛s ❲✳ ❍❡♥❣❛rt♥❡r✱ ❛♥❞ ❊r✐❝▼❛t③♥❡r✲▲ø❜❡r✳■t❡r❛t✐✈❡ ▼♦♥t❡ ❈❛r❧♦ ❢♦r ❡①tr❡♠❡ q✉❛♥t✐❧❡s ❛♥❞ ❡①tr❡♠❡♣r♦❜❛❜✐❧✐t✐❡s✳❚❡❝❤♥✐❝❛❧ r❡♣♦rt✱ ▼❛r❝❤ ✷✵✶✵✳P❛♣❡r ❛♥❞ s❧✐❞❡s❤♦✇ ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❛✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡✳
❬✹❪ ❆❣♥és ▲❛❣♥♦✉①✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❘❡❢❡r❡♥❝❡s
❘❛r❡ ❡✈❡♥t s✐♠✉❧❛t✐♦♥✳P❊■❙✱ ✷✵✭✶✮✿✹✺✕✻✻✱ ❏❛♥✉❛r② ✷✵✵✻✳❆✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡✳
❬✺❪ P✐❡rr❡ ▲✬➱❝✉②❡r✱ ❋r❛♥ç♦✐s ▲❡ ●❧❛♥❞✱ P❛s❝❛❧ ▲❡③❛✉❞✱ ❛♥❞ ❇r✉♥♦❚✉✣♥✳❙♣❧✐tt✐♥❣ ♠❡t❤♦❞s✳■♥ ●❡r❛r❞♦ ❘✉❜✐♥♦ ❛♥❞ ❇r✉♥♦ ❚✉✣♥✱ ❡❞✐t♦rs✱ ▼♦♥t❡ ❈❛r❧♦
▼❡t❤♦❞s ❢♦r ❘❛r❡ ❊✈❡♥t ❆♥❛❧②s✐s✱ ❝❤❛♣t❡r ✸✱ ♣❛❣❡s ✸✾✕✻✶✳ ❏♦❤♥❲✐❧❡② ✫ ❙♦♥s✱ ❈❤✐❝❤❡st❡r✱ ✷✵✵✾✳
❬✻❪ ❋r❛♥ç♦✐s ▲❊●▲❆◆❉✳❋✐❧tr❛❣❡ ❜❛②és✐❡♥ ♦♣t✐♠❛❧ ❡t ❛♣♣r♦①✐♠❛t✐♦♥ ♣❛rt✐❝✉❧❛✐r❡✳▲❡s Pr❡ss❡s ❞❡ ❧✬❊❝♦❧❡ ◆❛t✐♦♥❛❧❡ ❙✉♣ér✐❡✉r❡ ❞❡ ❚❡❝❤♥✐q✉❡s❆✈❛♥❝é❡s✱ ✷✵✵✽✳❆✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡ ✐♥ ❋r❡♥❝❤✳
❬✼❪ ❙✳ P✳ ▼❡②♥ ❛♥❞ ❘✳ ▲✳ ❚✇❡❡❞✐❡✳▼❛r❦♦✇ ❝❤❛✐♥s ❛♥❞ st♦❝❤❛st✐❝ st❛❜✐❧✐t②✳❙♣r✐♥❣❡r✕❱❡r❧❛❣✱ ✶✾✾✸✳
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❘❡❢❡r❡♥❝❡s
❬✽❪ ▲✉❦❡ ❚■❊❘◆❊❨✳▼❛r❦♦✈ ❝❤❛✐♥s ❢♦r ❡①♣❧♦r✐♥❣ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥s✳❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s✱ ✷✷✿✶✼✵✶✕✶✼✷✽✱ ❉❡❝❡♠❜❡r ✶✾✾✹✳❆✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡✳