Generalized additive model for location, scale and shape
The generalized additive model for location, scale and shape is a distributional regression model in which a parametric statistical distribution is assumed for the response variable but the parameters of this distribution can vary according to explanatory variables. Therefore the shape of this distribution for the target variable can change with explanatory variables.
GAMLSS is an input output model, i.e. but differs from the classical model in that the input X effects the distribution of the target variable as a whole not just the mean, i.e..
GAMLSS allows flexible regression by using smoothing or machine learning techniques to model the parameters of the target variable. GAMLSS assumes the response variable could follows any theoretical parametric distribution, which might be heavy or light-tailed, and positively or negatively skewed. In addition, all the parameters of the distribution which often are location, scale and shape – can be modelled as linear, nonlinear or using algorithm modelling functions of the explanatory variables. The distributional assumption for the target variables can be checked through diagnostic plots like Q–Q plot or worm plot. GAMLSS is a supervised machine learning model since the target value is always present.
Overview of the model
The generalized additive model for location, scale and shape is a statistical model introduced by Rigby and Stasinopoulos to overcome some of the limitations associated with the popular generalized linear models of Nelder and Wedderburn and generalized additive models of Hastie and Tibshirani. Most of the limitations arise from the limited choice of distributions for the response. Both GLM and GAM assumed that the response comes from the exponential family a family rich enough to allow continuous and discrete responses but not very flexible enough to model other characteristics of the distribution i.e tails.In GAMLSS the exponential family distribution assumption for the response variable,,, 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 modelling not only of the mean but possibly other parameters of the distribution of response as linear and/or nonlinear, parametric and/or additive non-parametric functions of explanatory variables and/or random effects.
GAMLSS, for continuous responses, is especially suited for modelling a leptokurtic or platykurtic and/or positively or negatively skewed variables therefor modelling the tails of the distribution. For count type response variable data it deals with over-dispersion and zero-inflation by using proper over-dispersed and zero inflated distributions. Heterogeneity also is dealt with by modeling the scale or shape parameters using explanatory variables. GAMLSS allow also mixed distributions, that is, distributions which have discrete and continuous parts, i.e the zero inflated beta is one or them. Note that while the beta distribution allow value in the beta inflated allows values in.
There are several packages written in R related to GAMLSS models, and tutorials for using and interpreting GAMLSS.
A GAMLSS model assumes independent observations for
with probability function conditional on. The parameter often is a vector of four distribution parameters, each of which can be a function of the explanatory variables, for example, The first two distribution parameters and are usually characterised as location and scale parameters, while the remaining parameter, if any, are characterised as shape parameters, e.g. skewness and kurtosis parameters. The model may be applied more generally to the parameters of any population distribution.
The most general formulation of a GAMLSS model is
where is any machine learning model and is the number of parameters in the distribution for the resposnse. The original formulation of GAMLSS in the 2005 RSS paper had only four parameters and it was written as;
where, and are vectors of length, is a parameter vector of length, is a fixed known design matrix of order and is a smooth non-parametric function of explanatory variable, for and.
for are link functions to ensure that the parameter are in the correct range of values.