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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Generalized additive model for location, scale and shape ： ウィキペディア英語版 Generalized additive model for location, scale and shape In statistics, the Generalized Additive Model for Location, Scale and Shape (GAMLSS) is a class of statistical model developed by Rigby and Stasinopoulos. These models provide extended capabilities beyond the simpler generalized linear models and generalized additive models. These simpler models allow the typical values of a quantity being modelled to be related to whatever explanatory variables are available. Here the "typical value" is more formally a location parameter, which only describes a limited aspect of the probability distribution of the dependent variable. The GAMLSS approach allows other parameters of the distribution to be related to the explanatory variables; where these other parameters might be interpreted as scale and shape parameters of the distribution, although the approach is not limited to such parameters. ==Overview of the model== The generalized additive model for location, scale and shape (GAMLSS) is a statistical model developed by Rigby and Stasinopoulos and later expanded to overcome some of the limitations associated with the popular generalized linear models (GLMs) and generalized additive models (GAMs). For an overview of these limitations see Nelder and Wedderburn (1972) and Hastie and Tibshirani's book. In GAMLSS the exponential family distribution assumption for the response variable, ($y$), (essential in GLMs and GAMs), is relaxed and replaced by a general distribution family, including highly skew and/or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modeling not only of the mean (or location) but other parameters of the distribution of ''y'' as linear and/or nonlinear, parametric and/or additive non-parametric functions of explanatory variables and/or random effects. GAMLSS is especially suited for modelling a leptokurtic or platykurtic and/or positively or negatively skewed response variable. For count type response variable data it deals with over-dispersion by using proper over-dispersed discrete distributions. Heterogeneity also is dealt with by modelling the scale or shape parameters using explanatory variables. There are several packages written in R related to GAMLSS models.〔 A GAMLSS model assumes independent observations $y\_i$ for $i\; =\; 1,\; 2,\; \backslash dots\; ,\; n$ with probability (density) function $f\; (y\_i\; |\; \backslash mu\_i\; ,\; \backslash sigma\_i\; ,\; \backslash nu\_i\; ,\; \backslash tau\_i\; )$ conditional on $(\backslash mu\_i\; ,\; \backslash sigma\_i\; ,\; \backslash nu\_i\; ,\; \backslash tau\_i\; )$ a vector of four distribution parameters, each of which can be a function to the explanatory variables. The first two population distribution parameters $\backslash mu\_i$ and $\backslash sigma\_i$ are usually characterized as location and scale parameters, while the remaining parameter(s), if any, are characterized as shape parameters, e.g. skewness and kurtosis parameters, although the model may be applied more generally to the parameters of any population distribution with up to four distribution parameters, and can be generalized to more than four distribution parameters. : $$ \begin g_1 (\mu) = \eta_1= X_1 \beta_1 + \sum_^ _(x_) \\ g_2(\sigma) = \eta_2= X_2 \beta_2 + \sum_^_(x_) \\ g_3(\nu) = \eta_3 = X_3 \beta_3 + \sum_^_(x_) \\ g_4(\tau)=\eta_4=X_4 \beta_4 + \sum_^_(x_) \end where μ, σ, ν, τ and $\backslash eta\_k$ are vectors of length $n$, $\backslash beta^\_k\; =\; (\backslash beta\_,\backslash beta\_,\backslash ldots,\backslash beta\_)$ is a parameter vector of length $J\text{'}\_k$, $X\_k$ is a fixed known design matrix of order $n\; \backslash times\; J\text{'}\_k$ and $h\_$ is a smooth non-parametric function of explanatory variable $x\_$, $j=1,2,\backslash ldots,\; J\_$ and $k=1,2,3,4$. For centile estimation the (WHO Multicentre Growth Reference Study Group ) have recommended GAMLSS and the Box-Cox power exponential (BCPE) distributions〔 for the construction of the WHO Child Growth Standards.〔WHO Multicentre Growth Reference Study Group (2006) WHO Child Growth Standards: Length/height-for-age, weight-for-age, weight-for-length, weight-for-height and body mass index-for-age: Methods and development. Geneva: World Health Organization.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Generalized additive model for location, scale and shape」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.