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Research project The main objective of the project is to develop scalable methods for fitting shape constrained additive models with the response distribution beyond the standard exponential family.
When analyzing the relationships between a response and predictors, it might be natural to assume that some of the relationships obey certain shape constraints, such as monotonicity and convexity. In particular, such problems are widespread in ecological and environmental studies. The main goal of this project is to develop new methodologies, and open source software for shape constrained generalized additive models with complex data sets.
Any application area that analyzes the relationship between a response and multiple covariates could potentially benefit from using nonparametric and semiparametric regression models.
Generalized additive models (GAM) form a class of models where ever flexible regression is required, with a key idea being to model a random variable as depending nearly linearly on some smooth functions of known predictor variables. In many studies, when analyzing the relationships between a response and predictors, it might be essential to assume that some of the relationships obey certain shape restrictions. For example, the number of protest events in a county is expected to be increasing with population size, the relationship between tree height and altitude is known to be decreasing, but tree height is increasing with tree age. The dose-effect curve in medicine, the relationship between daily mortality and air pollution concentration, body mass index and incidence of heart diseases are other examples where a shape constraint is required. In such studies, unconstrained flexible GAMs might be too flexible and give implausible or uninterpretable results. Shape constrained additive models, SCAM, implemented in an R package scam, have demonstrated their efficiency and practicality in numerous applications. The popularity of SCAMs lies in the appealing balance between applicability and interpretability.
Despite the success of SCAMs, the current methodological and computational framework for the models could be significantly improved. Existing methods do not support models with large data sets that are now increasingly invaluable and important as technological advances across science and industry generate vast quantities of data. In addition to extending the methodology to handle settings with large data sets, there remain outstanding methodological challenges for tackling other standard distributional families than currently covered. Although various extensions of unconstrained GAMs have been developed, significantly new algorithms are required as the GAM methods can not be simply re-used or modified, mainly due to the non-linearity imposed by the shape constrained smooth representations. Currently, efficient and robust computation is possible in the standard single parameter exponential family setting. Still, models for beta regression, ordered categorical models, zero inflated Poisson, and survival models require methodological development beyond the standard likelihoods. The project aims to address the limitations of modelling with SCAM by developing scalable methods for models with response distribution beyond the standard exponential family.
The methods developed will all be implemented for use in R, by inclusion in the R package scam . The aim will be to develop general modelling software readily usable by applied users in diverse practical fields. As SCAM provides a general and powerful framework for flexible nonparametric regression modelling with shape restrictions, further developments are expected to lead to advances in knowledge that have a significant research impact within academia and industry.
