December 02, 2023
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# データ不足に数理モデルで立ち向かう / Japan.R 2023

2023年12月2日に行われたJapan.R 2023での発表資料です
https://japanr.connpass.com/event/302622/

## 森下光之助

December 02, 2023

## Transcript

4. ### • GRP TRP • CM ⾒ • CM ⾒ •

• CM 1 ⾒ • CM 2 1 2 3 4 5 A 1 0 1 0 1 B 0 1 0 1 0 C 1 1 1 0 1 D 0 0 1 0 0 E 0 0 0 0 0 2 (40%) 4 (80%) 7 (140%) 8 (160%) 10 (200%) 2 (40%) 3 (60%) 4 (80%) 4 (80%) 4 (80%)

7. ### • • 0 0 頻 ⾒ 100% lm(y ~ 0

+ x) lm(y ~ 0 + log1p(x))

9. ### l 𝑔 l CM 𝐹 Pr 𝐹 = 𝑓 ∣

𝑔 l CM 1 ⾒ 𝑟 𝑔 = Pr 𝐹 ≥ 1 ∣ 𝑔 = 1 − Pr 𝐹 = 0 ∣ 𝑔 Pr 𝐹 = 𝑓 ∣ 𝑔 𝑟 𝑔 CM
10. ### l Poisson 𝑓 𝜆 = 1 Γ 𝑓 + 1

𝜆!𝑒"# l 𝜆 𝑔 𝜆 = 𝑔 𝑟 𝑔 = 1 − Pr 𝐹 = 0 ∣ 𝑔 = 1 − 1 Γ 0 + 1 𝑔\$𝑒"% = 1 − 𝑒"% dpois(f, lambda) Poisson(𝑓 ∣ 𝜆 = 5) Poisson(𝑓 ∣ 𝜆 = 3) 1 - dpois(0, g) Poisson(𝑓 ∣ 𝜆 = 2)

= 2.06) CM
13. ### l CM CM CM l CM 𝜆 CM 𝜆 CM

⾒ 𝜆 Poisson(𝑓 ∣ 𝜆 = 2) Poisson(𝑓 ∣ 𝜆 = 3) Poisson(𝑓 ∣ 𝜆 = 5)
14. ### l ⾒ ⾒ 𝜆 l 𝜆 頻 𝜆 l 𝜆

Gamma 𝜆 ∣ 𝜈, 𝜈 𝜇 = 𝜈 𝜇 & Γ 𝜈 𝜆&"'𝑒" & (# E 𝜆 = 𝜇 𝜆 dgamma(nu, nu / mu) Gamma 𝜆 ∣ 1, 1 2 Gamma 𝜆 ∣ 4, 4 2 Gamma 𝜆 ∣ 16, 16 2 𝜆 𝜆
15. ### l 𝜆 ⾒ 𝜆 Pr 𝐹 = 𝑓 ∣ 𝜇,

𝜈 = ; \$ ) Pr 𝐹 = 𝑓 ∣ 𝜆 𝑝 𝜆 𝜇, 𝜈 𝑑𝜆 = ; \$ ) Poisson 𝑓 ∣ 𝜆 Gamma 𝜆 𝜈, 𝜈 𝜇 𝑑𝜆 = ; \$ ) 1 Γ 𝑓 + 1 𝜆!𝑒"# 𝜈 𝜇 & Γ 𝜈 𝜆&"'𝑒" & (# 𝑑𝜆 = 𝜈 𝜇 & Γ 𝑓 + 1 Γ 𝜈 ; \$ ) 𝜆&*!"'𝑒" &"( ( # 𝑑𝜆 = 𝜈 𝜇 & Γ 𝑓 + 1 Γ 𝜈 Γ 𝜈 + 𝑓 𝜈 + 𝜇 𝜇 &*! ; \$ ) 𝜈 + 𝜇 𝜇 &*! Γ 𝜈 + 𝑓 𝜆&*!"'𝑒" &*( ( # 𝑑𝜆 = Γ 𝜈 + 𝑓 Γ 𝑓 + 1 Γ 𝜈 𝜈 𝜈 + 𝜇 & 𝜇 𝜈 + 𝜇 ! = , ! " Gamma 𝜆 𝜈 + 𝑓, 𝜈 + 𝜇 𝜇 𝑑𝜆 = 1
16. ### l ⾒ Negative Binomial Distribution; NBD NB 𝑓 𝜇, 𝜈

= Γ 𝜈 + 𝑓 Γ 𝑓 + 1 Γ 𝜈 𝜈 𝜈 + 𝜇 & 𝜇 𝜈 + 𝜇 ! NB 𝑓 2.06,1 NB 𝑓 2.06,3 NB 𝑓 2.06,10 dnbinom(f, mu = mu, size = nu)
17. ### l ⾒ 𝑟 𝑔, 𝜈 = 1 − Pr 𝐹

= 0 ∣ 𝑔, 𝜈 = 1 − Γ 𝜈 + 0 Γ 0 + 1 Γ 𝜈 𝜈 𝜈 + 𝑔 & 𝑔 𝜈 + 𝑔 \$ = 1 − 𝜈 𝜈 + 𝑔 & l 𝜈 1 - dnbinom(0, mu = g, size = nu) 𝑟 𝑔, 1 𝑟 𝑔, 3 𝑟 𝑔, 10
18. ### l 𝑟 𝑔, 𝜈 𝜈 𝜈 l 𝑟+ 𝑔+ ̂

𝜈 ̂ 𝜈 = argmin & 1 − 𝜈 𝜈 + 𝑔+ & − 𝑟′ l ̂ 𝜈 𝑟 𝑔, ̂ 𝜈 = 1 − ̂ 𝜈 ̂ 𝜈 + 𝑔 , & CM CM
19. ### l 1 ⾒ CM 3 CM ⾒ l CM 𝑓

⾒ 𝑓 + l 𝑓 + 𝑟!* 𝑔, 𝜈 = Pr 𝐹 ≥ 𝑓 ∣ 𝑔, 𝑣 = 1 − Pr 𝐹 ≤ 𝑓 − 1 ∣ 𝑔, 𝜈 = 1 − E !!-\$ !"' Γ 𝜈 + 𝑓+ Γ 𝑓+ + 1 Γ 𝜈 𝜈 𝜈 + 𝑔 & 𝑔 𝜈 + 𝑔 !! 𝑓 𝑓 + 𝑟!" 𝑟#" 𝑟\$" 1 - pnbinom(f - 1, mu = g, size = nu)

21. ### • Goerg, Georg M. "Estimating reach curves from one data

point." (2014).