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❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❉❡s✐❣♥ ♦❢ ▲❉P❈ ❈♦❞❡s ❢♦r ❙❧❡♣✐❛♥✲❲♦❧❢ ❝♦❞✐♥❣ ✇✐t❤ ✉♥❝❡rt❛✐♥ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❊❧s❛ ❉✉♣r❛③ ✶ ❆❧✐♥❡ ❘♦✉♠② ✷ ▼✐❝❤❡❧ ❑✐❡✛❡r ✶ , ✸ , ✹ ✶ ▲❙❙ ✲ ❈◆❘❙ ✲ ❙❯P❊▲❊❈ ✲ ❯♥✐✈ P❛r✐s✲❙✉❞ ✷ ■◆❘■❆ ✸ ▲❚❈■ ✲ ❈◆❘❙ ✲ ❚❡❧❡❝♦♠ P❛r✐s❚❡❝❤ ✹ ■♥st✐t✉t ❯♥✐✈❡rs✐t❛✐r❡ ❞❡ ❋r❛♥❝❡ ✶✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❈♦♥t❡①t X Y Sensor Data collection point ✷✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❈♦♥t❡①t ❚❤❡♦r❡t✐❝❛❧ ♣❡r❢♦r♠❛♥❝❡ ❬❙❲✼✸❪ ❈♦❞✐♥❣ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ ❝❤❛♥♥❡❧ ❝♦❞❡s ❬▲❳●✵✷✱ ▲❳●✵✸✱ ▼❯▼✶✵✱ ❳▲❈✵✹❪ ■♥ ❣❡♥❡r❛❧✱ ♣❡r❢❡❝t ❦♥♦✇❧❡❞❣❡ ♦❢ P ( ❳ , ❨ ) P ( ❳ ) ✉♥❦♥♦✇♥ ❬❏❱❲✶✵❪ P ( ❨ | ❳ ) ❣✐✈❡♥ ❛t ❞❡❝♦❞❡r✱ ♣❛rt❧② ✉♥❦♥♦✇♥ ❛t ❡♥❝♦❞❡r ❬❙❣❛✼✼❪ Pr❛❝t✐❝❛❧ s♦❧✉t✐♦♥ ❋❡❡❞❜❛❝❦ ❝❤❛♥♥❡❧ ❬❆❩●✵✷✱ ❊❨✵✺✱ ❱❆●✵✻❪ ✸✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❈♦♥t❡①t ■♥ t❤✐s ✇♦r❦ P ( ❨ | ❳ ) ♣❛rt❧② ✉♥❦♥♦✇♥ t♦ ❜♦t❤ ❡♥❝♦❞❡r ❛♥❞ ❞❡❝♦❞❡r ◆♦ ❢❡❡❞❜❛❝❦ ❖❜❥❡❝t✐✈❡s ❉❡s✐❣♥ ❡✣❝✐❡♥t ❝♦❞✐♥❣✴❞❡❝♦❞✐♥❣ s❝❤❡♠❡s r♦❜✉st t♦ ✉♥❝❡rt❛✐♥t② ♦♥ P ( ❨ | ❳ ) ❙♦❧✉t✐♦♥ ❜❛s❡❞ ♦♥ ♥♦♥✲❜✐♥❛r② ▲❉P❈ ❝♦❞❡s ✹✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❈♦♥t❡①t ✶ ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ✷ ❈♦❞✐♥❣ s❝❤❡♠❡ ✸ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥s ✺ ✺✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❈♦♥t❡①t ✶ ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ✷ ❈♦❞✐♥❣ s❝❤❡♠❡ ✸ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥s ✺ ✻✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ▼♦❞❡❧✐♥❣ t❤❡ ✉♥❝❡rt❛✐♥t② ❋♦✉r s♦✉r❝❡ ♠♦❞❡❧s ❝♦♥s✐❞❡r❡❞ ✐♥ ❬❉❘❑✶✷❪✳ ❍❡r❡✱ ❢♦❝✉s ♦♥ t❤❡ ❙t❛t✐❝ ✇✐t❤♦✉t Pr✐♦r ❙♦✉r❝❡ ✭❙✇P✲❙♦✉r❝❡✮ ❉❡✜♥✐t✐♦♥ ✭❙✇P✲❙♦✉r❝❡✮ ❆ ❙✇P✲❙♦✉r❝❡ ( ❳ , ❨ ) ✱ ♣r♦❞✉❝❡s ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❞✐s❝r❡t❡ s②♠❜♦❧s { ( ❳ ♥ , ❨ ♥ ) } + ∞ ♥ = ✶ ❞r❛✇♥ ❢r♦♠ ❛ ❞✐str✐❜✉t✐♦♥ ❜❡❧♦♥❣✐♥❣ t♦ � � P ( ❳ , ❨ | θ ) = P ( ❳ ) P ( ❨ | ❳ , θ ) θ ∈ P θ θ ✐s ✜①❡❞ ❢♦r { ( ❳ ♥ , ❨ ♥ ) } + ∞ ♥ = ✶ ✳ ✼✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❈♦♥t❡①t ✶ ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ✷ ❈♦❞✐♥❣ s❝❤❡♠❡ ✸ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥s ✺ ✽✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ▲❉P❈✲❜❛s❡❞ ❊♥❝♦❞❡r ✐♥ ●❋✭ q ✮ ❚❤❡♦r❡t✐❝❛❧ P❡r❢♦r♠❛♥❝❡ ❬❈s✐✽✷❪ ❘ = s✉♣ ❍ ( ❳ | ❨ , θ ) ✳ θ ∈ P θ ❊♥❝♦❞✐♥❣ ✐♥ ●❋✭ q ✮ ✭ ♠ < ♥ ✮ s ♠ = ❍ ❚ ① ♥ ❉❡❣r❡❡ ❞✐str✐❜✉t✐♦♥s ( λ ( ① ) , ρ ( ① )) ❢♦r ❍ ❞✐♠❡♥s✐♦♥❡❞ ❢♦r t❤❡ ✇♦rst ❝❛s❡✱ ♥❡❡❞ t♦ ❜❡ ♦♣t✐♠✐③❡❞ ✾✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❉❡❝♦❞❡r ❚✇♦✲st❡♣ ❊▼✲❜❛s❡❞ ❛♣♣r♦❛❝❤✿ ( ① ( ℓ ) , θ ( ℓ ) ) ❛t ✐t❡r❛t✐♦♥ ℓ ▲❉P❈ ❞❡❝♦❞✐♥❣ ✇✐t❤ ❡st✐♠❛t❡ θ ( ℓ ) t♦ ♦❜t❛✐♥ P ( ❳ ♥ = ❦ | ② ♥ , s , θ ( ℓ ) ) ❯♣❞❛t❡ ♦❢ t❤❡ θ ( ℓ ) ❜② ♠❛①✐♠✐③✐♥❣ ◗ ( θ , θ ( ℓ ) ) = ❊ ❳ | ② , s , θ ( ℓ ) [ ❧♦❣ P ( ② | ❳ , s , θ )] q − ✶ ◆ P ( ❳ ♥ = ❦ | ② ♥ , s , θ ( ℓ ) ) ❧♦❣ P ( ② ♥ | ❳ ♥ = ❦ , θ ) = ∑ ∑ ♥ = ✶ ❦ = ✵ ✶✵✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❉❡❝♦❞❡r ❚✇♦✲st❡♣ ❊▼✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ▲❉P❈ ❞❡❝♦❞✐♥❣ ✇✐t❤ ❡st✐♠❛t❡ θ ( ℓ ) t♦ ♦❜t❛✐♥ P ( ❳ ♥ = ❦ | ② ♥ , s , θ ( ℓ ) ) ❯♣❞❛t❡ ♦❢ t❤❡ θ ( ℓ ) ❢♦r ❨ = ❳ ⊕ ❩ ✱ P ( ❩ = ❦ ) = θ ❦ ◆ P ( ❳ ♥ = ② ♥ ⊖ ❦ | ② ♥ , s , θ ( ℓ ) ) ∑ θ ( ℓ + ✶ ) ♥ = ✶ = ❦ q − ✶ ◆ P ( ❳ ♥ = ② ♥ ⊖ ❦ ′ | ② ♥ , s , θ ( ℓ ) ) ∑ ∑ ♥ = ✶ ❦ ′ = ✵ ✶✶✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ■♥✐t✐❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ✭❆❞❞✐t✐✈❡ ▼♦❞❡❧✮ ❆❞❞✐t✐✈❡ ♠♦❞❡❧✿ ❨ = ❳ ⊕ ❩ ✱ P ( ❩ = ❦ ) = θ ❦ ❈♦♠♣✉t❡ ✉ = s ⊖ ❍ ❚ ② = ❍ ❚ ③ ❆ss✉♠♣t✐♦♥✿ t❤❡ ❯ ♠ ❛r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ✐✳✐✳❞✳ ❘✳❱✳s ❩ ( ♠ ) ✳ ❥ ▼❛①✐♠✐③❡ � � ▼ ❞❝ F ( ❲ [ ❤ ( ♠ ) ❧♦❣ F − ✶ ∑ ∏ ▲ ( θ ) = ❧♦❣ P ( ✉ | θ ) = ] θ ) ✉ ♠ ❥ ♠ = ✶ ❥ = ✶ ✶✷✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❈♦♥t❡①t ✶ ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ✷ ❈♦❞✐♥❣ s❝❤❡♠❡ ✸ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥s ✺ ✶✸✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❊①♣❡r✐♠❡♥t❛❧ ❢r❛♠❡✇♦r❦ ❙②♠❜♦❧s ✐♥ ●❋✭✹✮✱ ❨ = ❳ ⊕ ❩ ✱ θ = [ θ ✵ ,..., θ ✸ ] ❛♥❞ Pr ( ❩ = ❦ ) = θ ❦ ✳ P θ s✳t✳ ∀ θ ∈ P θ ✱ θ ✵ ≥ ✵ . ✼✻✳ ❈♦❞❡ t✉♥❡❞ ❢♦r t❤❡ ✇♦rst ❝❛s❡ θ = [ ✵ . ✼✻ , ✵ . ✵✽ , ✵ . ✵✽ , ✵ . ✵✽ ] λ ( ① ) = ✵ . ✹✶✸ ① + ✵ . ✸✼✺ ① ✷ + ✵ . ✵✶✷ ① ✹ ρ ( ① ) = ① ❘ = ✶ . ✻ ❜✐t✴s②♠❜♦❧ ✶✹✴✶✾

❈♦♥t❡①t ❙♦✉r❝❡ ❉❡✜♥✐t✐♦♥ ❈♦❞✐♥❣ s❝❤❡♠❡ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ■♥✐t✐❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ▼❙❊ ♦❢ t❤❡ ❡st✐♠❛t♦rs 10-4 Lemma 4, sums of id. R.V. Lemma 5, sums of id. R.V. Lemma 4, codewords Lemma 5, codewords codewords sums of id. R.V. 10-5 400 600 800 1000 1200 1400 1600 1800 2000 N ✶✺✴✶✾