The generalized additive model for location, scale and shape (GAMLSS) is a semiparametric regression model in which a parametric statistical distribution is assumed for the response (target) variable but the parameters of this distribution can vary according to explanatory variables.
GAMLSS is a form of supervised machine learning.
GAMLSS enables flexible regression and smoothing models to be fitted to the data.
GAMLSS assumes the response variable follows an arbitrary parametric distribution, which might be heavy or light-tailed, and positively or negatively skewed.
In addition, all the parameters of the distribution – location (e.g., mean), scale (e.g., variance) and shape (skewness and kurtosis) – can be modeled as linear, nonlinear or smooth functions of explanatory variables.
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)[1] and Hastie's and Tibshirani's book.
[2] In GAMLSS the exponential family distribution assumption for the response variable, (
), (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 modeling the scale or shape parameters using explanatory variables.
There are several packages written in R related to GAMLSS models,[3] and tutorials for using and interpreting GAMLSS.
[4] A GAMLSS model assumes independent observations
with probability (density) function
a vector of four distribution parameters, each of which can be a function of the explanatory variables.
The first two population distribution parameters
where μ, σ, ν, τ and
is a parameter vector of length
is a fixed known design matrix of order
is a smooth non-parametric function of explanatory variable
For centile estimation the WHO Multicentre Growth Reference Study Group have recommended GAMLSS and the Box–Cox power exponential (BCPE) distributions[5] for the construction of the WHO Child Growth Standards.
[6][7] The form of the distribution assumed for the response variable y, is very general.
Such implementations also allow use of truncated distributions and censored (or interval) response variables.