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1 60 (W30)? 1. ( ) kubo@ees.hokudai.ac.jp 2. ( ) web site URL ( :41 ) 1/ 77
2 ! : :? ( :41 ) 2/ 77
3 ( :41 ) 3/ 77!!
4 ( :41 ) 4/ 77 ( )!
5 ( :41 ) 5/ 77 : offset : :???
6 ( :41 ) 6/ 77 GLM
7 ( :41 ) 7/ 77 : R OS free software S R
8 ( :41 ) 8/ 77
9 ( :41 ) 9/ 77
10 ( :41 ) 10/ 77
11 ( :41 ) 11/ 77
12 ( :41 ) 12/ 77
13 ( :41 ) 13/ 77
14 ( :41 ) 14/ 77
15 : : ( ) ( :41 ) 15/ 77
16 ( :41 ) 16/ 77 ( : offset
17 何でも割算! , 2, 3! x /x ( :41 ) 17/ 77
18 ? /? : ? ( ) ( :41 ) 18/ 77
19 ( :41 ) 19/ 77 : specific leaf area (SLA) : offset : N k :
20 ( :41 ) 20/ 77 (1) offset
21 ( :41 ) 21/ 77 :? x {0.1, 0.2,, 1.0} 10 glm(..., family = poisson)
22 ?!! x A = /! glm() offset ( :41 ) 22/ 77
23 ( :41 ) 23/ 77 vs plot(d$x, d$y / d$area) d$y/d$area d$x
24 GLM! family: poisson, link : "log" : y ~ x offset : log(area) z = a + b x + log(area) a, b λ log(λ) = z λ = exp(z) = exp(a + b x + log(area)) λ : ( :41 ) 24/ 77
25 ( :41 ) 25/ 77 (2)
26 ( :41 ) 26/ 77 (1 = ) seed size [ ] ( )
27 ( :41 ) 27/ 77 ( ) seed size seed size 1. ( 4 ) 2. ; {0, 1} 3. ( or or & )
28 ( :41 ) 28/ 77? 1 / / 200! seed size? 1? (? )
29 ( :41 ) 29/ 77 R glm() : seed size (x) or q q = exp( (a + bx)) (logistic ) a b R glm() ( )
30 ( :41 ) 30/ 77 (binomial distribution)? y i {0, 1, 2,, N} (paramter: q, N) ( ) N q y (1 q) N y y Nq Nq(1 q) probability prob = 0.2 prob = 0.5 prob = : N y y
31 ( :41 ) 31/ 77 R glm() :? ( z): x link : logit family: binomial,
32 ( :41 ) 32/ 77 ( ) Ending 1.0 germination prob seed size!!? :!
33 ( :41 ) 33/ 77 (ESJ59 : glm()
34 : OK ( :41 ) 34/ 77
35 ( :41 ) 35/ 77 (Poisson distribution)? lambda = 1.4 y {0, 1, 2,, } (paramter: λ) λ y exp( λ) 0.0 lambda = y! probability λ λ lambda = : y
36 ( :41 ) 36/ glm() log-linear model
37 : > d2 <- read.csv("d2.csv") # > (ct2 <- xtabs(y ~ x + Spc, data = d2)) Spc x A B > plot(ct2, col = c("orange", "blue")) ct2 0 1 B Spc A ( :41 ) 37/ 77 x
38 GLM ( ) A ( ) y A,x Pois(λ A,x) log(λ A,x) = α A + β A x > summary(glm(y ~ x, data = d2[d2$spc == "A",], family = poisson) > # SpcA (......) Estimate Std. Error z value Pr(> z ) (Intercept) < 2e-16 x ( :41 ) 38/ 77
39 ( :41 ) 39/ 77 GLM ( ) B y B,x Pois(λ B,x) log(λ B,x) = α B + β B x > summary(glm(y ~ x, data = d2[d2$spc == "B",], family = poisson) > # SpcB (......) Estimate Std. Error z value Pr(> z ) (Intercept) < 2e-16 x e-05?
40 ( :41 ) 40/ 77 log(λ A,x ) = x log(λ B,x ) = x lambda_a lambda_b x x AIC AIC λ A,x = α A + β A x 19.3 λ B,x = α B + β B x 17.1 λ A,x = α A 30.1 λ B,x = α B 32.4!
41 ( :41 ) 41/ 77 GLM GLM GLM: logit(q A,x) = a A + b A x 1 q A,x = 1 + exp[ (a A + b A x)] : log(λ A,x) = α A + β A x λ A,x = exp(α A + β A x) λ B,x = exp(α B + β B x) A? λ A,x λ A,x + λ B,x = = exp(α A + β A x) exp(α A + β A x) + exp(α B + β B x) exp[α B α A + (β B β A )x]
42 : GLM GLM GLM q A,x = exp[ (a A + b A x)] GLM ( ) λ A,x λ A,x + λ B,x = exp[α B α A + (β B β A )x] GLM GLM a A = α A α B b A = β A β B ( :41 ) 42/ 77
43 : GLM GLM GLM a A = = α A α B b A = = β A β B > GLM (A ) > glm(ct2 ~ c(0, 1), data = d2, family = binomial) (Intercept) c(0, 1) > GLM (A ) > glm(y ~ x, data = d2[d2$spc == "A",], family = poisson) (Intercept) x > GLM (B ) > glm(y ~ x, data = d2[d2$spc == "B",], family = poisson) (Intercept) x ( :41 ) 43/ 77
44 : GLM GLM GLM (A ) GLM (B ) lambda_a x lambda_* = lambda_b x 0 1 x = q_a 0 1 GLM (A + B ) 0 1 x ( :41 ) 44/ 77
45 ( :41 ) 45/ glm() GLM R package
46 : 2 3 Spc x A B C > plot(ct3, col = c("orange", "blue", "green")) ct3 0 1 C B Spc A ( :41 ) 46/ 77 x
47 ( :41 ) 47/ 77 GLM ( ) > glm(y ~ x * Spc, data = d3, family = poisson) (......) Coefficients: (Intercept) x SpcB SpcC x:spcb x:spcc GLM y A,x Pois(λ A,x ) log(λ A,x ) = α A + β A x α A = 5.66 α B = α C = β A = β B = β C =
48 GLM GLM lambda_a (A ) (B ) (C ) 0 1 x + + lambda_* lambda_b x = 0 1 x lambda_b = 0 1 x lambda_* 0 1 GLM 0 1 x ( :41 ) 48/ 77
49 ( :41 ) 49/ GLM GLMM!
50 : 3 9! Spc x A B C D E F G H I > plot(ct9, col = c( )) cts 0 1 IHG F E D C Spc B A ( :41 ) 50/ 77 x
51 GLM ( ) > ct9 Spc x A B C D E F G H I > summary(glm(y ~ x * Spc, data = d9, family = poisson)) (Intercept) x SpcB SpcC SpcD SpcE SpcF SpcG SpcH SpcI x:spcb x:spcc x:spcd x:spce x:spcf x:spcg x:spch x:spci H! ( :41 ) 51/ 77
52 ( :41 ) 52/ 77 glm() GLMM!??! Spc?: 9 Spc ( )
53 ( :41 ) 53/ 77 σ best GLMM > (fit.glmm <- glmmml(y ~ x, data = d9, cluster = Spc, family = poisson)) ( ) > fit.glmm$posterior.modes [1] [6] individual small posterior prior σ large σ
54 ( ) データ ( 応答変数 ) 各個体の種子数 Y[i] 個 ポアソン分布平均 lambda[i] 全個体共通 a0 無情報事前分布 b0 データ ( 説明変数 ) X[i] 傾きの差 b[spc[i]] 切片の差 a[spc[i]] 階層事前分布 s[1] 切片差のばらつき 階層事前分布 s[2] 傾き差のばらつき 無情報事前分布 ( 超事前分布 ) WinBUGS (MCMC ) BUGS WinBUGS R ( :41 ) 54/ 77
55 ( :41 ) 55/ 77 ( )??
56 ( :41 ) 56/ 77 : λ = 30 λ = = ! p = {0.60, 0.40}
57 ( :41 ) 57/ 77 : λ = 30 λ = 20 λ = = = ! p = {0.50, 0.33, 0.17}
58 ( :41 ) 58/ 77 (Gamma distribution)? y 0 (paramter: r, s) p(y s, r) = rs Γ(s) ys 1 exp( ry) s/r s/r 2
59 : 1 : (0.6, 0.4) (0.325, 0.675) : (0.5, 0.3, 0.1) (0.28, 0.54, 0.18) 0.6 a = 12, b = 8 a = 6, b = 4 a = 1.2, b = 0.8 a = 0.6, b = y y y y ( :41 ) 59/ 77
60 OK? = = ? p = {0.60, 0.40} ( :41 ) 60/ 77
61 OK? = = = , 0.286? p = {0.50, 0.33, 0.17} ( :41 ) 61/ 77
62 ( :41 ) 62/ 77?? OK : R glm(y ~ x, family = Gamma) ( ): θ = 1/r =
63 > ALake <- ArcticLake > ALake$Y <- DR_data(ALake[,1:3]) > DirichReg(Y ~ depth + I(depth^2), ALake) Coefficients for variable no. 1: sand (Intercept) depth I(depth^2) Coefficients for variable no. 2: silt (Intercept) depth I(depth^2) Coefficients for variable no. 3: clay (Intercept) depth I(depth^2) ( :41 ) 63/ 77 R library(dirichletreg) package GLM
64 ( :41 ) 64/ 77 library(dirichletreg) random effects?
65 ( :41 ) 65/ 77 (ESJ57
66 ( :41 ) 66/ 77 : above ground mass Y 重量 y x below ground mass X
67 ( :41 ) 67/ 77 : q w 観測できない世界観測できる世界 parameter process observation 無情報事前分布 w q 1- q y x Y X 地上部地下部 階層的事前分布 個体差 測定時の誤差 重量 y x
68 BUGS code (process ) for (i in 1:N) { Y[i] ~ dnorm(y[i], Tau.err) # X[i] ~ dnorm(x[i], Tau.err) # y[i] <- q[i] * w[i] x[i] <- (1 - q[i]) * w[i] # logit(q[i]) <- a + b * log.w[i] + re[i] w[i] <- exp(log.w[i]) # log.w[i] + log.w[i] ~ dnorm(0, Tau.noninformative) #!! ( ) ( :41 ) 68/ 77
69 ( :41 ) 69/ 77 : MCMC 1. BUGS code (model1.txt) 2. R (runbus1.r) 3. R runbus1.r (source("runbugs1.r")) 4. R library(r2winbugs) WinBUGS 5. WinBUGS Markov chain Monte Carlo (MCMC) 6. R
70 ( :41 ) 70/ 77 : w b MCMC
71 ( :41 ) 71/ 77 above ground mass Y 重量 y x below ground mass X (median) (95% CI)
72 ( :41 ) 72/ 77
73 ( :41 ) 73/ 77 C-N (coverage) GIS
74 ( :41 ) 74/ 77 : offset : :??
75 ( :41 ) 75/ 77
76 60 (W30)? 1. ( ) kubo@ees.hokudai.ac.jp 2. ( ) web site URL ( :41 ) 76/ 77
77 ( :41 ) 77/ 77
kubostat2017j p.2 CSV CSV (!) d2.csv d2.csv,, 286,0,A 85,0,B 378,1,A 148,1,B ( :27 ) 10/ 51 kubostat2017j (http://goo.gl/76c4i
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