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1 kubostat2017b p.1 agenda I 2017 (b) probabilit distribution and maimum likelihood estimation kubo@ees.hokudai.ac.jp : :43 1 : 2 3? 4 kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 statistaical models appeared in the class The development of linear models Hierarchical Baesian Model Be more fleible Generalized Linear Mied Model (GLMM) Incoporating random effects such as individualit parameter estimation MCMC MLE Generalized Linear Model (GLM) Alwas normal distribution? That's non-sense! MSE Linear model Kubo Doctrine: Learn the evolution of linear-model famil, firstl! kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 suppose that ou have a count data set... 0, 1, 2 the normal distribution... is NOT this one!? ( {0, 1, 2, 3, } ) response variable e.g. egg number () e.g. bod size () eplanator variable? kubostat2017b ( (b) / 42 response variable? 0? NO! eplanator variable kubostat2017b ( (b) / 42

2 kubostat2017b p.2 the Poisson disribution approimates data?! response variable eplanator variable fair distribution non-negative mean YES! Plot our data and observe it Choose proper distributons! the normal distributon is NOT good at everthing be-be, the normal distribution kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 : 2 : : : R kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 : a simplified data set, eas to understand number of seeds taken from 50 imaginar plants : number of seeds per plant individual () individual i i 50 i {1, 2, 3,, 50} seed number of i i { i }! { i } = { 1, 2,, 50 } : { i } R > data [1] [26] kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42

3 kubostat2017b p.3 : : R: a free statistical software : R RStudio OS free software S RStudio kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 : appl table() to categorize data R : start with data plotting, alwas table() > table(data) ( ) > hist(data, breaks = seq(-0.5, 9.5, 1))! kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 : How to evaluate mean value using R? : statistics to represent dispersion > mean(data) [1] 3.56 > abline(v = mean(data))! kubostat2017b ( (b) / 42 : > var(data) [1] sample standard deviation > sd(data) [1] > sqrt(var(data)) [1] sample variance (SD = variance) : : kubostat2017b ( (b) / 42

4 kubostat2017b p.4 Empirical VS Theoretical Distributions 3. kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 empirical distribution > data.table <- table(factor(data, levels = 0:10)) > cbind( = data.table, prob = data.table / 50) prob kubostat2017b ( (b) / 42? = {0, 1, 2, }? {p 0, p 1, p 2,, p 99, p 100, }? kubostat2017b ( (b) / 42? Mathematical epression of the Poisson distribution () ()! probabilit p( λ) = λ ep( λ)! factorial 4! ep( λ) = e λ (e = ) kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42

5 kubostat2017b p.5 the Poisson distribution? R (λ) 3.56 Poisson distribution > <- 0:9 # () > prob <- dpois(, lambda = 3.56) # > plot(, prob, tpe = "b", lt = 2) > # cbind > cbind(, prob) prob kubostat2017b ( (b) / 42 the Poisson distribution represent data? > hist(data, seq(-0.5, 8.5, 0.5)) # > lines(, prob, tpe = "b", lt = 2) # kubostat2017b ( (b) / 42 parameter λ is the mean of the Poisson distribution λ The Poisson distribution is useful if...? λ λ λ 0 : λ = = > # cbind > cbind(, prob) prob {0, 1, 2,, } 1 p( λ) = 1 =0 kubostat2017b ( (b) / 42 : i {0, 1, 2, } count data i mean variance kubostat2017b ( (b) / 42 λ changes the shape of distribution λ? p( λ) = λ ep( λ)! mean λ kubostat2017b ( (b) / 42 4.!? fitting = parameter estimation kubostat2017b ( (b) / 42

6 kubostat2017b p.6 (likelihood)??? likelihood L(λ) depends on the value of mean, λ L(λ) λ maimum likelihood estimation λ goodness of fit λ 3 { 1, 2, 3 } = {2, 2, 4} = (the likelihood definition for the eample): L(λ) = ( 1 2 ) ( 2 2 ) ( 50 3 ) = p( 1 λ) p( 2 λ) p( 3 λ) p( 50 λ) = p( i λ) = λ i ep( λ), i i i! kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42? evaluate not likelihood, but log likelihood!? λ changes the log likelihood, i.e., goodness of fit λ () (!) (log likelihood function) log L(λ) = i ( ) i i log λ λ log k k log L(λ) L(λ) λ kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 seek the maimum likelihood estimate, ˆλ ˆλ? log L(λ) = ( i i log λ λ i k log k) log likelihood * ˆλ = (ML estimator): i i/50 d log L dλ (ML estimate): ˆλ = 3.56 = 0!? no one knows the true λ based on finite size data λ λ ˆλ ˆλ λ 50 kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42

7 kubostat2017b p.7 : random number generation estimation 5. λ = 3.5 ˆλ = 3.56 kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 prediction probabilit distributions appeared in the class λ = 3.5 ˆλ = 3.56 : ( : {0, 1, 2, 3, } : {0, 1, 2,, N} N : < < kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42 GLMM 線形モデルの発展 推定計算方法階層ベイズモデル (HBM) MCMC もっと自由な統計モデリングを! 一般化線形混合モデル (GLMM) 最尤推定法個体差 場所差といった変量効果をあつかいたい一般化線形モデル (GLM) 正規分布以外の確率分布をあつかいたい 最小二乗法線形モデル The net topic YES! : Poisson Regression, a Generalized Linear Model (GLM) kubostat2017b ( (b) / 42 kubostat2017b ( (b) / 42

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