/ 55 2 : : (GLM) 1. 1/23 ( )? GLM? (GLM ) 2.! 1/25 ( ) ffset (GLM )

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1 / 55 ( II) : ( ) 2 2 (GLM) ! kub@ees.hkudai.ac.jp

2 / 55 2 : : (GLM) 1. 1/23 ( )? GLM? (GLM ) 2.! 1/25 ( ) ffset (GLM )

3 / : 2. ffset : 3. : x

4 / 55 1.

5 : : ( ) / 55

6 : / 55

7 / 55!

8 / 55 (Pissn distributin)? lambda = 1.4 y i {0, 1, 2,, } (paramter: λ) λ y exp( λ) 0.0 lambda = y! prbability λ λ lambda = : y

9 (generalized linear mdel; GLM) link : link f() = f( ) : β 0 + β 1 x 1 + β 2 x 2 + x i β i x i (cefficient) ({x i } {y i }) {β i } GLM / 55

10 / 55 R : glm() ( ) rbinm() glm(family = binmial) rbinm() glm(family = binmial) rpis() glm(family = pissn) rnbinm() glm.nb() ( ) rgamma() glm(family = gamma) rnrm() glm(family = gaussian) glm() glm.nb() MASS library

11 / 55 R glm() :? ( z):? link : z (y)? family:?

12 glm() family: pissn, link : "lg" ( z): y ~ x z = a + bx a, b λ lg(λ) = z λ = exp(z) = exp(a + bx) λ : y Pis(λ) / 55

13 GLM plant weight (g) number f flwers plant weight (g) number f flwers / 55

14 / 55

15 / 55 i N i k i i p i = k i /N i j p j = k j /N j i j p p i

16 ? / : ? ( ) / 55

17 : specific leaf area (SLA) : ffset : N k : / 55

18 / ffset

19 / 55 :? x {0.1, 0.2,, 1.0} 10 glm(..., family = pissn)

20 ?!! x A = /! glm() ffset / 55

21 R data.frame: Area, x, y > lad("d2.rdata") > head(d, 8) # 8 Area x y / 55

22 / 55 vs plt(d$x, d$y / d$area) d$y/d$area d$x

23 / 55 A vs y plt(d$area, d$y) d$y d$area A y

24 / 55 x ( ) plt(d$area, d$y, cex = d$x * 2) d$y d$area?

25 / 55 x y x

26 = 1. i y i λ i : y i Pis(λ i ) 2. λ i A i x i λ i = A i exp(a + bx i ) λ i = exp(a + bx i + lg(a i )) lg(λ i ) = a + bx i + lg(a i ) lg(a i ) ffset / 55

27 GLM! family: pissn, link : "lg" : y ~ x ffset : lg(area) z = a + b x + lg(area) a, b λ lg(λ) = z λ = exp(z) = exp(a + b x + lg(area)) λ : / 55

28 glm() / 55

29 / 55 R glm() > fit <- glm(y ~ x, family = pissn(link = "lg"), data = d, ffset = lg(area)) > print(summary(fit)) Call: glm(frmula = y ~ x, family = pissn(link = "lg"), data = d, ffset = lg(area)) (......) Cefficients: Estimate Std. Errr z value Pr(> z ) (Intercept) x e-06 Cefficients

30 / 55 d$y d$area x = 0.9, x = 0.1 glm()

31 / 55 : glm() ffset ffset = exp( ) d$y d$area

32 / 55 3.

33 / 55 (1 = ) seed size [ ] ( )

34 / 55 ( ) seed size seed size 1. ( 4 ) 2. ; {0, 1} 3. ( r r & )

35 / 55? 1 / / 200! seed size? 1? (? )

36 / 55 R glm() : seed size (x) r q q = exp( (a + bx) (lgistic ) a b R glm() ( )

37 / 55 (binmial distributin)? y i {0, 1, 2,, N} (paramter: q, N) ( ) N q y (1 q) N y y Nq Nq(1 q) prbability prb = 0.2 prb = 0.5 prb = : N y y

38 (Bernulli distributin) prb = 0.2 y i {0, 1} (paramter: p) q y (1 q) 1 y 0.2 prb = q q(1 q) prbability prb = 0.8 N = : y / 55

39 / 55? q = exp( (a + bx)) (exp(z) = e Z ) 1.0 a 1.0 b x x {a, b} x q 0 q 1

40 / 55 : lgistic lgit lgistic q = exp( (a + bx)) = lgistic(a + bx) lgit lgit(q) = lg q 1 q = a + bx lgit lgistic lgistic lgit

41 / 55 glm() (1) family: binmial, y {0, 1, 2,, N} link : "lgit" family = binmial link seed size ( z): y ~ x family = binmial(link = "lgit")?

42 / 55 glm() (2) family: binmial, 1.0 link : "lgit" ( z): y ~ x z = a + bx a, b seed size q lgit(q) = z 1 q = exp( z) = exp( (a + bx)) q N : y Binm(q, N)

43 / 55 R glm() :? ( z): x link : lgit family: binmial,

44 / 55 ( ) Ending 1.0 germinatin prb seed size!!? :!

45 : / 55

/ 60 : 1. GLM? 2. A: (pwer functin) x y?

2009-03-17 1/ 60 (2009-03-17) GLM 1. GLM :, link,, deviance (20 ) 2. GLM : (60 ) 3. GLM ( ): ffset (40 ) http://hsh.ees.hkudai.ac.jp/ kub/ce/ecsj2009.html 2009-03-17 2/ 60 : 1. GLM? 2. A: (pwer functin)