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Typo and General fixing "a bacteria", replaced: a bacteria → a bacterium, added orphan tag using AWB

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In [[statistics]], the '''generalized additive model location, scale and shape (GAMLSS)''' is a class of [[statistical model]] that provides extended capabilities compared to the simpler [[generalized linear model]]s and [[generalized additive model]]s. These simpler models allow the typical values of a quantity being modelled to be related to whatever [[independent and dependent variables|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 [[independent and dependent variables|dependent variable]]. The GAMLSS approach allows other [[statistical parameter|parameters]] of the distribution to be related to the explanatory variables; where these other parameters might be interpreted as [[scale parameter|scale]] and [[shape parameter]]s of the distribution, although the approach is not limited to such parameters.

==Overview of the model==
The generalized additive model location, scale and shape (GAMLSS) is a statistical model developed by Rigby and Stasinopoulos,{{Citation needed|date=September 2011}} and later expanded<ref>Ahmed Zaheer. Rigby, R. A. and Stasinopoulos D. M. (2005) "Generalized additive models for location, scale and shape, (with discussion)", ''Appl. Statist.'', 54, part 3, pp 507–554.[http://www.gamlss.org/images/stories/papers/gamlss-rss.pdf Link]</ref> to overcome some of the limitations associated with the popular [[generalized linear model]]s (GLMs) and [[generalized additive model]]s (GAMs).<ref>For an overview of these limitations see Nelder and Wedderburn, 1972{{full|date=November 2012}} and Hastie and Tibshirani, 1990.{{full|date=November 2012}}</ref>

In GAMLSS the [[exponential family]] [[probability distribution|distribution]] assumption for the [[Dependent and independent variables#response_variable|response variable]], (<math>y</math>), (essential in [[generalized linear model|GLMs]] and [[generalized additive model|GAMs]]), is relaxed and replaced by a general distribution family, including highly [[Skewness|skew]] and/or [[Kurtosis|kurtotic]] [[continuous distribution|continuous]] and [[discrete distribution]]s.

The systematic part of the model is expanded to allow modeling not only of the [[mean]] (or [[location parameter|location]]) but other parameters of the distribution of ''y'' as linear and/or nonlinear, parametric and/or additive [[non-parametric]] functions of [[explanatory variable]]s and/or [[random effect]]s.

GAMLSS is especially suited for modelling [[leptokurtic]] or [[platykurtic]] and/or positive or negative skew response variable. For [[count data|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 parameter|scale]] or [[shape parameter]]s using explanatory variables. There are several packages written in [[R (programming language)|R]] related to GAMLSS models.<ref>Stasinopoulos D. M.; Rigby R.A. (2007) "Generalized additive models for location scale and shape (GAMLSS) in R". ''Journal of Statistical Software'', 23 (7), Issue 7, Dec 2007. [http://www.jstatsoft.org/v23/i07 Link]</ref>).

A GAMLSS model assumes independent observations <math>y_i</math> for <math>i = 1, 2, \dots , n</math>
with probability (density) function <math>f (y_i | \mu_i , \sigma_i , \nu_i , \tau_i )</math> conditional on <math>(\mu_i , \sigma_i , \nu_i , \tau_i )</math> a vector of four distribution parameters, each of which can be a function to the explanatory variables. The first two population distribution parameters <math>\mu_i</math> and <math>\sigma_i</math> 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.

: <math>

\begin{align}
g_1 (\mu) = \eta_1= X_1 \beta_1 + \sum_{j=1}^{J_1} {h}_{j1}(x_{j1}) \\
g_2(\sigma) = \eta_2= X_2 \beta_2 + \sum_{j=1}^{J_2}{h}_{j2}(x_{j2}) \\
g_3(\nu) = \eta_3 = X_3 \beta_3 + \sum_{j=1}^{J_3}{h}_{j3}(x_{j3}) \\
g_4(\tau)=\eta_4=X_4 \beta_4 + \sum_{j=1}^{J_4}{h}_{j4}(x_{j4})
\end{align}
</math>

where μ, σ, ν, τ and <math>\eta_k</math> are vectors of length <math>n</math>, <math>\beta^{T}_k = (\beta_{1k},\beta_{2k},\ldots,\beta_{J'_{k}
k})</math> is a parameter vector of length <math>J'_k</math>, <math>X_k</math> is a fixed known design matrix of order <math>n \times J'_k</math> and <math>h_{jk}</math> is a smooth non-parametric function of explanatory variable <math>x_{jk}</math>, <math>j=1,2,\ldots, J_{k}</math> and <math>k=1,2,3,4</math>.

For centile estimation the [http://www.who.int/childgrowth/en WHO Multicentre Growth Reference Study Group] have recommended GAMLSS and the Box-Cox power exponential (BCPE) distributions<ref>Robert A. Rigby and D. Mikis Stasinopoulos (2004)[http://www.gamlss.org/images/stories/papers/boxcoxpower23.pdf "Smooth centile curves for skew and kurtotic data modelled using the Box-Cox Power Exponential distribution"]. Preprint </ref> for the construction of the WHO Child Growth Standards.<ref>{{cite doi|10.1002/sim.2227}}</ref><ref>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.</ref>

==What distributions can be used==

The form of the distribution assumed for the response variable y, is very general. For example an implementation of GAMLSS in [[R (programming language)|R]]<ref name=Rp>R packages for GAMLSS can be downloaded from [http://packages.gamlss.org here]</ref> has around 50 different distributions available. Such implementations also allow use of truncated distributions and censored (or interval) response variables.<ref name=Rp/>

==Notes==
{{reflist}}

==Further reading==

* Beyerlein, A., Fahrmeir, L., Mansmann, U., Toschke., A. M. (2001) "Alternative regression models to assess increase in childhood BM". ''IBMC Medical Research Methodology'', 2008, 8(59) {{doi|10.1186/1471-2288-8-59}}
* Cole, T. J., Stanojevic, S., Stocks, J., Coates, A. L., Hankinson, J. L., Wade, A. M. (2009), "Age- and size-related reference ranges: A case study of spirometry through childhood and adulthood", ''Statistics in Medicine'', 28(5), 880-898.[http://www3.interscience.wiley.com/journal/121547617/abstract Link]
* Fenske, N., Fahrmeir, L., Rzehak, P., Hohle, M. (25 September 2008), "Detection of risk factors for obesity in early childhood with quantile regression methods for longitudinal data", ''Department of Statistics: Technical Reports'', No.38 [http://epub.ub.uni-muenchen.de/6260/ Link]
* Hudson, I. L., Kim, S. W., Keatley, M. R. (2010), "Climatic Influences on the Flowering Phenology of Four Eucalypts: A GAMLSS Approach Phenological Research". In ''Phenological Research'', Irene L. Hudson and Marie R. Keatley (eds), Springer Netherlands [http://www.springerlink.com/content/w6231707326g8233/ Link]
* Hudson, I. L., Rea, A., Dalrymple, M. L., Eilers, P. H. C. (2008), "Climate impacts on sudden infant death syndrome: a GAMLSS approach", ''Proceedings of the 23rd international workshop on statistical modelling'' pp. 277–280. [http://arrow.unisa.edu.au:8081/1959.8/62400 Link]
* Nott, D. (2006), "Semiparametric estimation of mean and variance functions for non-Gaussian data", ''Computational Statistics'', 21(3-4), 603-620. [http://www.springerlink.com/content/j02v6t04m5833876/ Link]
* Serinaldi, F. (2011), "Distributional modeling and short-term forecasting of electricity prices by Generalized Additive Models for Location, Scale and Shape", ''Energy Economics'', 33(6), 1216-1226, {{doi|10.1016/j.eneco.2011.05.001}}
* Serinaldi, F., Cuomo, G. (2011) "Characterizing impulsive wave-in-deck loads on coastal bridges by probabilistic models of impact maxima and rise times", ''Coastal Engineering'', 58(9), 908-926, {{doi|10.1016/j.coastaleng.2011.05.010}}
* Serinaldi, F., Villarini, G., Smith, J. A., Krajewski, W. F. (2008), "Change-Point and Trend Analysis on Annual Maximum Discharge in Continental United States", ''American Geophysical Union Fall Meeting 2008'', abstract #H21A-0803*
* van Ogtrop, F. F., Vervoort, R. W., Heller, G. Z., Stasinopoulos, D. M., Rigby, R. A. (2011) "Long-range forecasting of intermittent streamflow", ''Hydrology and Earth System Sciences Discussions'', 8(1), 681-713. {{doi|10.5194/hessd-8-681-2011}}
* Villarini, G., Serinaldi, F. (2011), "Development of statistical models for at-site probabilistic seasonal rainfall forecast", ''International Journal of Climatology''. {{doi|10.1002/joc.3393}}
* Villarini, G., Serinaldi, F., Smith, J. A., Krajewski, W. F. (2009), "On the stationarity of annual flood peaks in the continental United States during the 20th century", ''Water Resources Research'', 45(8). [http://www.agu.org/pubs/crossref/2009/2008WR007645.shtml Link]
* Villarini, G., Smith, J. A., Napolitano, F. (2010), "Nonstationary modeling of a long record of rainfall and temperature over Rome", ''Advances in Water Resources'' {{doi| 10.1016/j.advwatres.2010.03.013}}

==External links==
*[http://www.gamlss.org/ GAMLSS official website gamlss.org]
*[http://manual.gamlss.org/ GAMLSS manual (downloadable)]
*[http://gamlss.org/images/stories/papers/Distributions-2010-onlyThetable.pdf Distribution tables in GAMLSS]
*[http://gamlss.org/images/stories/papers/gamlssreferencecard.pdf The GAMLSS packages reference card (downloadable)]
*[http://www.gamlss.org/images/stories/papers/book-2008-27-6-08.pdf The booklet for the Utrecht short course on GAMLSS (downloadable)]
*[http://packages.gamlss.org/ R packages for GAMLSS on CRAN]
{{Use dmy dates|date=September 2011}}

{{DEFAULTSORT:Generalized Additive Model For Location, Scale And Shape}}
[[Category:Generalized linear models]]
[[Category:Statistical models]]
[[Category:Semi-parametric models]]

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en>John of Reading: Typo and General fixing "a bacteria", replaced: a bacteria → a bacterium, added orphan tag using AWB