Estudio comparativo de técnicas basadas en 'Machine Learning' para la discriminación de partículas en los detectores de materia oscura PICO

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Índice

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𝑥(𝑡, 𝑆) 𝑆

𝛺 𝑆 ∈ 𝛺 → 𝑥(𝑡, 𝑆) ∈ ℝ 𝑡

𝑇 𝑡 ∈ 𝑇 ⊂ ℝ

𝑥(𝑡, 𝑆0) ≡ 𝑥(𝑡)

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𝑅𝑀𝑆{𝑥[𝑛]} = √1

𝑁∑ 𝑥[𝑛]

𝑁−1

𝑖=0

𝑆𝑃𝐿{𝑥[𝑛]} = 20 log₁₀ 𝑅𝑀𝑆 20𝜇𝑃𝑎

𝐴𝑆{𝑥[𝑛]} =𝐸[𝑥 − 𝜇{𝑥}]³ 𝜎{𝑥}³

𝜇 𝜎 𝐸[𝑦]

𝑦

𝜇

𝑦𝑖

𝑝(𝑦𝑖) 𝐸[𝑦] = ∑ 𝑦𝑖𝑝(𝑦𝑖)

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𝑓

𝐹{𝑥(𝑡)} ≡ 𝑋_𝐹(𝑓) = ∫ 𝑥(𝑡) · 𝑒^{−2𝜋𝑓𝑡}𝑑𝑡

+∞

−∞

𝑆𝑥𝑥(𝑓) = | 𝑋_𝐹(𝑓)|²

𝑆_𝑥𝑥 𝑓

𝑥[𝑛] 𝑁

𝑥[𝑛]

𝐹{𝑥(𝑡)} ≡ 𝑋𝐹[𝑘/𝑁] =1

𝑁∑ 𝑥[𝑛] · 𝑒^{−2𝜋𝑛𝑘/𝑁}

𝑁−1

𝑛=0

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𝑆𝑀0{𝑥[𝑛]} = ∑ 𝑃𝑆𝐷_𝑖

𝑁−1

𝑖=0

𝑆𝑀𝑗{𝑥[𝑛]} = ∑ 𝑃𝑖𝑓_𝑖^𝑗

𝑁−1

𝑖=0

𝑉𝐹𝐶{𝑥[𝑛]} =𝑆𝑀2

𝑆𝑀0− (𝑆𝑀1 𝑆𝑀0)

2

∑ 𝑃𝑆𝐷_𝑖

𝑀𝐷𝐹

𝑖=0

= ∑ 𝑃𝑆𝐷_𝑖

𝑁−1

𝑖=𝑀𝐷𝐹

=1

2∑ 𝑃𝑆𝐷_𝑖

𝑁−1

𝑖=0

𝑀𝑁𝐹{𝑥[𝑛]} =∑^𝑁−1_𝑖=0 𝑃𝑆𝐷_𝑖𝑓_𝑖

∑^𝑁−1_𝑖=0 𝑃𝑆𝐷_𝑖

𝑃𝑆𝐷_𝑖

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𝑥(𝑡)

𝑃𝑣𝑎𝑟[𝑛] = ∑ 𝑉𝐴𝑅{𝑥[𝑚]}

𝑛

0

𝑉𝐴𝑅{𝑥[𝑛]} = 𝑅𝑀𝑆²− 𝑀𝑉²=¹

𝑁∑^𝑁−1𝑖=0(𝑋[𝑛] − 𝑀𝑉)²

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𝑥(𝑡) −1/𝜋𝑡

𝑖 · 𝑠𝑔𝑛(𝑓) +𝜋/2

−𝜋/2

𝐻𝑇{𝑥[𝑛]} ≡ 𝑋𝐻[𝑛] = ∑^𝑥[𝑙] 𝑛 − 𝑙

𝑁−1

𝑙=0

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▪

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𝑃𝑟𝑒𝑐𝑖𝑠𝑖ó𝑛 = 𝑇𝑃

𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁

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𝑾

[𝑑 𝑥 𝑘] 𝒙

𝑘 𝑑

𝑥 = [𝑥

₁

, 𝑥

₂

, … , 𝑥

_𝑑

], 𝑥 ∈ ℝ

^𝑑

↓ 𝒙𝑾, 𝑾 𝑥 ∈ ℝ

^𝑑

𝑧 = [𝑧

₁

, 𝑧

₂

, … , 𝑧

_𝑘

], 𝑧 ∈ ℝ

^𝑘

≪ á

á ó

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𝑾

𝒛 = 𝒙𝑾 𝑧 ∈ ℝ

^𝑘

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≤

≤ ≥

≤

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𝐼𝐺(𝐷_𝑝, 𝑓) = 𝐼(𝐷_𝑝) − ∑𝑁_𝑗 𝑁_𝑝 𝐼(𝐷_𝑗)

𝑚

𝑗=1

𝑓 𝐷_𝑝 𝐷_𝑗

𝑗 𝐼 𝑁_𝑝

𝑁_𝑗 𝑗

(𝐼_𝐻)

(𝐼_𝐺) (𝐼_𝐸)

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𝐼𝐻(𝑡) = − ∑ 𝑝(𝑖|𝑡) log₂𝑝(𝑖|𝑡)

𝑐

𝑖=1

▪ 𝐼_𝐺(𝑡) = 1 − ∑ 𝑝(𝑖|𝑡)²

𝑐

𝑖=1

▪ 𝐼𝐸(𝑡) = 1 − max {𝑝(𝑖|𝑡)}

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𝑆 = {(𝑥₁, 𝑦₁), … , (𝑥_𝑛, 𝑦_𝑛)} 𝑥_𝑖∈ ℝ^𝑑 𝑦_𝑖∈ {+1, −1}

𝐷(𝑥) = (𝑤₁𝑥₁+ ⋯ + 𝑤_𝑑𝑥_𝑑) + 𝑏 = < 𝑤, 𝑥 > +𝑏

< 𝑤, 𝑥𝑖> +𝑏 ≥ 0 𝑠𝑖 𝑦𝑖= +1

𝑝𝑎𝑟𝑎 𝑖 = 1, … , 𝑛

< 𝑤, 𝑥_𝑖> +𝑏 ≤ 0 𝑠𝑖 𝑦_𝑖= −1

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𝑦 = 𝑓 (∑ 𝑤

_𝑗

𝑥

_𝑗

𝑁

𝑗=1

)

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𝑥_𝑗

𝑔𝑖(𝑥𝑗) = exp (−∥ 𝑥𝑗− 𝜇𝑖∥² 2𝜎_𝑖² )

𝜇_𝑖 𝜎_𝑖²

𝐶_𝑖)

𝑤_𝑖𝑘

𝑦_𝑘(𝑥_𝑗) = ∑ 𝑤_𝑖𝑘𝑔_𝑖(𝑥_𝑗)

𝑖=0

𝑛_ℎ

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▪ - -

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𝑋

𝑁[𝜇, 𝜎]

𝑥′

𝑃(𝜎 − 𝑥^′≤ 𝑋 ≤ 𝜎 + 𝑥′) = ∫ 𝑁[𝜇, 𝜎]𝑑𝑋

𝜎+𝑥^′ 𝜎−𝑥^′

𝑥^′⁄𝜎

1𝜎

𝑁(𝜇, 𝜎) = ¹

𝜎√2𝜋𝑒⁻^{(𝑥−𝜇)2}^2𝜎2

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𝜆

𝛽

𝜆

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▪

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𝑠𝑡𝑑 = √∑^𝑛_𝑖=1(𝑋_𝑖− 𝑋̅)² (𝑛 − 1)

𝑠𝑡𝑑²= ∑^𝑛_𝑖=1(𝑋_𝑖− 𝑋̅)² (𝑛 − 1)

𝑐𝑜𝑣(𝑋, 𝑌) =∑^𝑛_𝑖=1(𝑋_𝑖− 𝑋̅)(𝑌𝑖− 𝑌̅) 𝑛 − 1

𝜆 ∈ ℂ 𝑣 ∈ ℂ^𝑚, 𝑣 ≠ 0 𝐴𝑣 = 𝜆𝑣

𝑣 𝜆. 𝑣 𝐴

𝜆 𝑣 𝐴

𝜆 𝑣₁, 𝑣₂ 𝜆

𝐴 𝜆

(𝝈) [𝑑 𝑥 𝑑]

𝑐𝑜𝑣(𝒙𝒋 𝒙𝒌)

𝜎𝑗𝑘(𝑥𝑖, 𝑥𝑗) =1

𝑛∑(𝑥_𝑗^(𝑖)− 𝜇𝑗)(𝑥_𝑘^(𝑖)− 𝜇𝑘)

𝑛

𝑖=1

, 𝑑𝑜𝑛𝑑𝑒 𝜇 =1 𝑛∑ 𝑥𝑖

𝑖=1

Σ = [ 𝜎

₁²

𝜎

₁₂

𝜎

₂₁

𝜎

₂²

]

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(𝜆)

Σν = λν

𝜆_𝑗

λ

_j

∑

^𝑑_𝑗=1

𝜆

_𝑗

𝑾 𝒛,

𝒛 = 𝒙𝑾 𝑧 ∈ ℝ

^𝑘

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▪

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𝐸(𝑊, 𝑏) =1

2‖𝑦 − 𝑥‖² 𝑚

𝐸(𝑊, 𝑏) = [1 𝑚∑ (1

2‖𝑦 − 𝑥‖²)

𝑚

𝑖=1

] +𝜆

2 ∑ ∑ ∑ (𝑊_𝑖𝑗^(𝑙))²

𝑆_𝑙+1

𝑗=1 𝑠_𝑙

𝑖=1 𝑛_𝑙−1

𝑙=1

𝜆

𝑊, 𝑏 𝑊_𝑖𝑗^(𝑙)= 𝑊_𝑖𝑗^(𝑙)− 𝛼 𝜕

𝜕𝑊_𝑖𝑗^(𝑙)𝐸(𝑊, 𝑏) 𝑏_(𝑙)= 𝑏_(𝑙)− 𝛼 𝜕

𝜕𝑏(𝑙)

𝐸(𝑊, 𝑏) 𝛼

𝜕

𝜕𝑊_𝑖𝑗^(𝑙)𝐸(𝑊, 𝑏) = [1

𝑚∑ 𝜕

𝜕𝑊_𝑖𝑗^(𝑙)𝐸(𝑊, 𝑏, 𝑥, 𝑦)

𝑚

𝑖=1

] + 𝜆𝑊_𝑖𝑗^(𝑙)

𝜕

𝜕𝑏_(𝑙)𝐸(𝑊, 𝑏) = 1

𝑚∑ 𝜕

𝜕𝑏_(𝑙)𝐸(𝑊, 𝑏, 𝑥, 𝑦)

𝑚

𝑖=1

𝛿^𝑜𝑢𝑡 = −(𝑦 − 𝑥)

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𝛿^𝑙 = ((𝑊^(𝑙))^𝑇𝛿^(𝑙+1)) · 𝑓′(𝑧^(𝑙))

𝜕

𝜕𝑊_𝑖𝑗^(𝑙)𝐸(𝑊, 𝑏, 𝑥, 𝑦) = 𝛿^(𝑙+1)(𝑎^(𝑙))^𝑇

𝜕

𝜕𝑏_(𝑙)𝐸(𝑊, 𝑏, 𝑥, 𝑦) = 𝛿^(𝑙+1)

𝛿𝑗

𝑊^(𝑙)= 𝑊^(𝑙)− 𝛼 [(1

𝑚Δ𝑊^(𝑙)) + 𝜆𝑊^(𝑙)]

𝑏(𝑙)= 𝑏(𝑙)− 𝛼 [(1

𝑚Δ𝑏(𝑙) )]

Δ𝑊^(𝑙) 𝑦 Δ𝑏(𝑙)

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{𝑥₁, … , 𝑥𝑁} 𝑁 𝑥

𝜇^(𝑗), 𝑗 ∈ {1, … , 𝑘}

𝑑(𝑥, 𝑦)²= ∑(𝑥_𝑗− 𝑦_𝑗)²

𝑚

𝑗=1

= ∥ 𝑥 − 𝑦 ∥²

𝑆𝑆𝐸 = ∑ ∑ 𝑤^(𝑖,𝑗)∥ 𝑥^(𝑖)− 𝜇^(𝑗)∥²

𝑘

𝑗=1 𝑛

𝑖=1

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δj

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