Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique

❊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 ✷✵✶✶

mailto:[email protected]




❖✉t❧✐♥❡

✶ ■♥tr♦❞✉❝t✐♦♥

✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s

✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

✺ ❈♦♥❝❧✉s✐♦♥

✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s

✼ ❘❡❢❡r❡♥❝❡s


■♥tr♦❞✉❝t✐♦♥

❖✉t❧✐♥❡

✶ ■♥tr♦❞✉❝t✐♦♥

✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s

✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

✺ ❈♦♥❝❧✉s✐♦♥

✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s

✼ ❘❡❢❡r❡♥❝❡s


■♥tr♦❞✉❝t✐♦♥


■♥tr♦❞✉❝t✐♦♥

❖♥❡ ♠✐❣❤t t❤✐♥❦ ♦❢ ❛♥ ❡♠♣t② ♦✉t❡r s♣❛❝❡✳


■♥tr♦❞✉❝t✐♦♥

❖♥❡ ♠✐❣❤t t❤✐♥❦ ♦❢ ❛♥ ❡♠♣t② ♦✉t❡r s♣❛❝❡✳


■♥tr♦❞✉❝t✐♦♥

❖♥❡ ✇♦✉❧❞ ❜❡ ✇r♦♥❣✦


■♥tr♦❞✉❝t✐♦♥

❖♥❡ ✇♦✉❧❞ ❜❡ ✇r♦♥❣✦


■♥tr♦❞✉❝t✐♦♥

❙♣❛❝❡ ❞❡❜r✐s s✉rr♦✉♥❞ ❊❛rt❤ ❛♥❞ ❛❝t✐✈❡ ♦r❜✐t✐♥❣ ♦❜❥❡❝ts✳


■♥tr♦❞✉❝t✐♦♥

❖♥ ❚✉❡s❞❛② ❋❡❜r✉❛r② t❤❡ ✶✵t❤ ✷✵✵✾✱ s❛t❡❧❧✐t❡s ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s❝♦❧❧✐❞❡❞✳

❲❤❛t ✇❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ✐t ❤❛♣♣❡♥❡❞❄


■♥tr♦❞✉❝t✐♦♥

❖♥ ❚✉❡s❞❛② ❋❡❜r✉❛r② t❤❡ ✶✵t❤ ✷✵✵✾✱ s❛t❡❧❧✐t❡s ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s❝♦❧❧✐❞❡❞✳

❲❤❛t ✇❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ✐t ❤❛♣♣❡♥❡❞❄


■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


■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✐❡♥❝❡✳


■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✐❡♥❝❡✳


■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✐❡♥❝❡✳


■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✐❡♥❝❡✳


■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✳


■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✳


■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✳


■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✳


■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❤❡ ◆❆❙❆ ♠♦❞❡❧✳


■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❤❡ ◆❆❙❆ ♠♦❞❡❧✳


■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❤❡ ◆❆❙❆ ♠♦❞❡❧✳


■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❤❡ ◆❆❙❆ ♠♦❞❡❧✳


■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


■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✳


■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✳


■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✳


■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

❞❝♦❧ ❄


■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

❞❝♦❧ ❄


■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

❞❝♦❧ ❄


■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

❞❝♦❧ ❄


■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

❞❝♦❧ ❄


■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)‖}

❞❝♦❧ ❄


■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[∆ ≤ ❞❝♦❧ ]❄


■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


■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

0.2

0.4

0.6

0.8

1

1.2x 10

−5

Simulation number

Estim

ate

d p

rob

ab

ility

Probability estimation

❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s


■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

0.2

0.4

0.6

0.8

1

1.2x 10

−5

Simulation number

Estim

ate

d p

rob

ab

ility


❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s ✇✐t❤ t❤❡✐r ♠❡❛♥✳


■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

20

30

40

50

60

70

80

Estimated probability (bin)

Nu

mb

er

of

po

ints

in

bin

Probability estimation histogram

❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠


■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

0.2

0.4

0.6

0.8

1

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


Num

ber

of p

oin

ts in b

in


✭❜✮ 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✉❡✦


■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

0.2

0.4

0.6

0.8

1

1.2x 10

−5

Simulation number

Estim

ate

d p

robabili


✭❛✮ 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


Num

ber

of p

oin

ts in b

in


✭❜✮ 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✉❡✦


■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

0.2

0.4

0.6

0.8

1

1.2x 10

−5

Simulation number

Estim

ate

d p

robabili


✭❛✮ 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


Num

ber

of p

oin

ts in b

in


✭❜✮ 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✉❡✦


■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

0.2

0.4

0.6

0.8

1

1.2x 10

−5

Simulation number

Estim

ate

d p

robabili


✭❛✮ 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


Num

ber

of p

oin

ts in b

in


✭❜✮ 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✉❡✦


■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✐❜❧❡✳


■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✐❜❧❡✳


■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✐❜❧❡✳


■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✐❜❧❡✳


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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 ❞♦✳


❆♥ ❛❞❛♣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 ❞♦✳


❆♥ ❛❞❛♣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 ❞♦✳


❆♥ ❛❞❛♣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 ❞♦✳


❆♥ ❛❞❛♣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 ❞♦✳


❆♥ ❛❞❛♣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 ❞♦✳


❆♥ ❛❞❛♣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 ❞♦✳


❆♥ ❛❞❛♣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 ❞♦✳


❆♥ ❛❞❛♣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❡ ❚


❆♥ ❛❞❛♣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❡ ❚


❆♥ ❛❞❛♣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❡ ❚


❆♥ ❛❞❛♣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 = [❚ ,∞[


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

❚❤❡ ❛❧❣♦r✐t❤♠

❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣

Start


law.

k=0



k=k+1




threshold?

Propose transitions with the

Reversible Markov Kernel.

Accept those above the quantile.

Refuse the others

(Repeat ad libitum)

EndCalculate

sought probability

NO

YES


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✦


❆♥ ❛❞❛♣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✦


❆♥ ❛❞❛♣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✦


❆♥ ❛❞❛♣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✦


❆♥ ❛❞❛♣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✦


❆♥ ❛❞❛♣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✦


❆♥ ❛❞❛♣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✦


❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣

❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣

Start


law.

k=0



k=k+1




threshold?




Refuse the others

(Repeat ad libitum)

EndCalculate

sought probability

NO

YES

❆ ❣❧♦❜❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❆❙❚ ✇✐t❤ ♠❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣✦


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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)‖}


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣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✳


❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥

❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s


❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥

❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s

Input points

●❡♥❡r❛t❡ r❛♥❞♦♠ ♥♦✐s❡s(

❳ ✶

❦, · · · ,❳ ♥

❦

)

✳


❆♥ ❛❞❛♣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✈❛❧❧❡ ■ ✳


❆♥ ❛❞❛♣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)


❋✐♥❞ ♠✐♥✐♠❛(

∆✶

❦, · · · , ∆♥

❦

)

✳


❆♥ ❛❞❛♣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)


❋✐♥❞ ♠✐♥✐♠❛(

∆✶

❦, · · · , ∆♥

❦

)

✳


❆♥ ❛❞❛♣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

❋✐♥❞ ♠✐♥✐♠❛(

∆✶

❦, · · · , ∆♥

❦

)

✳


❆♥ ❛❞❛♣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


❙♦rt ♠✐♥✐♠❛ t♦ ❢♦r♠(

∆(✶)❦

, · · · ,∆(♥)❦

)

✳


❆♥ ❛❞❛♣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


❉❡✜♥❡ q✉❛♥t✐❧❡ δ❦+✶✳


❆♥ ❛❞❛♣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


❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳


❆♥ ❛❞❛♣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


❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳


❆♥ ❛❞❛♣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


❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳


❆♥ ❛❞❛♣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


❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳


❆♥ ❛❞❛♣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)


❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳


❆♥ ❛❞❛♣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)


❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳


❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥

❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s

Input points

❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳


❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥

❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s

Input points after discarding

❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳


❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥

❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s

Input points after shaking

❯s❡ t❤❡ ♠❛r❦♦✈ ❦❡r♥❡❧ ❛♥❞ s❡❧❡❝t❡❞ ♣♦✐♥ts t♦ r❡❣❡♥❡r❛t❡ s❛♠♣❧❡✳


❆♥ ❛❞❛♣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


❆♥ ❛❞❛♣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


❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s


❆♥ ❛❞❛♣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


❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s ✇✐t❤ t❤❡✐r ♠❡❛♥✳


❆♥ ❛❞❛♣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


Nu

mb

er

of

po

ints

in

bin


❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠


❆♥ ❛❞❛♣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


✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s

1 2 3 4 5 6 7 8 9 10

x 10−7

0

5

10

15

20

25


Num

ber

of p

oin

ts in b

in


✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠

❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳

▼❡❛♥ ❡st✐♠❛t❡✿ ✹ ✼✽ ✶✵ ✼✳

❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵ ✸✷✸✷✳

◗✉✐t❡ ❘❡❧✐❛❜❧❡✦


❆♥ ❛❞❛♣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


✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s

1 2 3 4 5 6 7 8 9 10

x 10−7

0

5

10

15

20

25


Num

ber

of p

oin

ts in b

in


✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠

❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳

▼❡❛♥ ❡st✐♠❛t❡✿ ✹.✼✽ · ✶✵−✼✳

❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵.✸✷✸✷✳

◗✉✐t❡ ❘❡❧✐❛❜❧❡✦


❆♥ ❛❞❛♣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


✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s

1 2 3 4 5 6 7 8 9 10

x 10−7

0

5

10

15

20

25


Num

ber

of p

oin

ts in b

in


✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠

❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳

▼❡❛♥ ❡st✐♠❛t❡✿ ✹.✼✽ · ✶✵−✼✳

❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵.✸✷✸✷✳

◗✉✐t❡ ❘❡❧✐❛❜❧❡✦


❆♥ ❛❞❛♣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



❆♥ ❛❞❛♣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



❆♥ ❛❞❛♣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


Num

ber

of p

oin

ts in b

in



❆♥ ❛❞❛♣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


❋✐❣✉r❡✿ ❍✐st♦❣r❛♠s

0 0.2 0.4 0.6 0.8 1 1.20

10

20

30

40

50

60

70

80


Numb

er of

point

s in bi

n



❆❙❚✬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


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s

❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣

Start


law.

k=0



k=k+1




threshold?




Refuse the others

(Repeat ad libitum)

EndCalculate

sought probability

NO

YES

❆ ❣❧♦❜❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❆❙❚ ✇✐t❤ ♠❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣✦


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s

❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s

Start


law.

k=0



k=k+1




threshold?




Refuse the others

EndCalculate

sought probability

NO

YES

❍♦✇❡✈❡r✱ ♥♦ r❛t✐♦♥❛❧ q✉❛♥t✐❧❡ ❧❡✈❡❧ ❝❤♦✐❝❡✳✳✳


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s

❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s

Start


law.

k=0



k=k+1




threshold?




Refuse the others

EndCalculate

sought probability

NO

YES

✳✳✳♥♦r ▼❛r❦♦✈✐❛♥ ❝❤♦✐❝❡ ❛♥❞ t✉♥✐♥❣✦


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s

❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s

Start


law.

k=0



k=k+1




threshold?




Refuse the others

EndCalculate

sought probability

NO

YES

❚✇♦ ✐ss✉❡s t♦ t❛❝❦❧❡✦


❆❙❚✬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❤♦❞ ❬✺✱ ✻❪✳


❆❙❚✬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❤♦❞ ❬✺✱ ✻❪✳


❆❙❚✬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❤♦❞ ❬✺✱ ✻❪✳


❆❙❚✬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❤♦❞ ❬✺✱ ✻❪✳


❆❙❚✬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


❆❙❚✬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


❆❙❚✬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 ♥❡❡❞❡❞✳


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧

●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪

■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s

▼(① , ❞②) ∼ N(

①√✶ + ❜✷

,❜✷

✶ + ❜✷

)


●❡♥❡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 ♥❡❡❞❡❞✳


❆❙❚✬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 ♥❡❡❞❡❞✳


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧

●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪

■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s

▼(① , ❞②) ∼ N(

①√✶ + ❜✷

,❜✷

✶ + ❜✷

)


●❡♥❡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 ♥❡❡❞❡❞✳


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧

●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪

■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s

▼(① , ❞②) ∼ N(

①√✶ + ❜✷

,❜✷

✶ + ❜✷

)


●❡♥❡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 ♥❡❡❞❡❞✳


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧

●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪

■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s

▼(① , ❞②) ∼ N(

①√✶ + ❜✷

,❜✷

✶ + ❜✷

)


●❡♥❡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 ♥❡❡❞❡❞✳


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧

●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪

■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s

▼(① , ❞②) ∼ N(

①√✶ + ❜✷

,❜✷

✶ + ❜✷

)


●❡♥❡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 ♥❡❡❞❡❞✳


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧

●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪

■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s

▼(① , ❞②) ∼ N(

①√✶ + ❜✷

,❜✷

✶ + ❜✷

)


●❡♥❡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 ♥❡❡❞❡❞✳


❆❙❚✬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❄


❆❙❚✬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❄


❆❙❚✬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❄


❆❙❚✬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❄


❆❙❚✬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❄


❆❙❚✬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❄


❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

◗✉❛♥t✐❧❡ ❧❡✈❡❧

▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s


❇❡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 ❬✸❪✳



❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

◗✉❛♥t✐❧❡ ❧❡✈❡❧

▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s


❇❡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 ❬✸❪✳



❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

◗✉❛♥t✐❧❡ ❧❡✈❡❧

▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s


❇❡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 ❬✸❪✳



❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

◗✉❛♥t✐❧❡ ❧❡✈❡❧

▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s


❇❡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 ❬✸❪✳



❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

◗✉❛♥t✐❧❡ ❧❡✈❡❧

▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s


❇❡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 ❬✸❪✳



❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

◗✉❛♥t✐❧❡ ❧❡✈❡❧

▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s


❇❡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 ❬✸❪✳



❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

◗✉❛♥t✐❧❡ ❧❡✈❡❧

▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s


❇❡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 ❬✸❪✳



❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

✶ ■♥tr♦❞✉❝t✐♦♥

✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s

✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

✺ ❈♦♥❝❧✉s✐♦♥

✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s

✼ ❘❡❢❡r❡♥❝❡s


❈♦♥❝❧✉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 ♥❡❡❞❡❞✳


❈♦♥❝❧✉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 ♥❡❡❞❡❞✳


❈♦♥❝❧✉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 ♥❡❡❞❡❞✳


❈♦♥❝❧✉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 ♥❡❡❞❡❞✳


❈♦♥❝❧✉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❤❡ ❞♦✉❜❧♦♦♥❄


❈♦♥❝❧✉s✐♦♥

❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②


❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ ❳ ♣ ♣ t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳

❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄


❈♦♥❝❧✉s✐♦♥

❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②


❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ P [❳ ∈ A] ∈ [♣−, ♣+] t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳

❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄


❈♦♥❝❧✉s✐♦♥

❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②


❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ P [❳ ∈ A] ∈ [♣−, ♣+] t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳

❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄


❈♦♥❝❧✉s✐♦♥

P❧❡❛s❡✱ ❧❡t ♠❡ ❛♥s✇❡r ②♦✉r q✉❡st✐♦♥s✳


❈♦♥❝❧✉s✐♦♥

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳


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


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✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳


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❡❧ ✫ ❋✳ ▲❡ ●❧❛♥❞✳


❘❡❢❡r❡♥❝❡s

❖✉t❧✐♥❡

✶ ■♥tr♦❞✉❝t✐♦♥

✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s

✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡

✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s

✺ ❈♦♥❝❧✉s✐♦♥

✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s

✼ ❘❡❢❡r❡♥❝❡s


❘❡❢❡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✳

http://hal.inria.fr/inria-00350762/fr/

http://www.sites.univ-rennes2.fr/laboratoire-statistique/AGUYADER/doc/ghm.pdf

http://www.newton.ac.uk/programmes/SCS/seminars/062114500.pdf

http://sms.cam.ac.uk/media/859487


❘❡❢❡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❧❛❣✱ ✶✾✾✸✳

http://www.math.univ-toulouse.fr/~lagnoux/articlePEIS.pdf

http://www.irisa.fr/aspi/legland/ensta/


❘❡❢❡r❡♥❝❡s

❬✽❪ ▲✉❦❡ ❚■❊❘◆❊❨✳▼❛r❦♦✈ ❝❤❛✐♥s ❢♦r ❡①♣❧♦r✐♥❣ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥s✳❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s✱ ✷✷✿✶✼✵✶✕✶✼✷✽✱ ❉❡❝❡♠❜❡r ✶✾✾✹✳❆✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡✳

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aos/1176325750

Documents

Estimating satellite collision probabilities via the adaptive splitting technique … · 2011-03-27 · Estimating satellite collision rpobabilities via the adaptive splitting technique