Shape Constrained Additive Smooth Models

Generalized additive smooth models, including those for location scale and shape, would sometimes benefit from the imposition of shape constraints on some smooth components. When these components are represented using spline basis expansions, many constraints, such as monotonicity or convexity, can be expressed as linear inequality constraints on the basis coefficients. Given smoothing parameters, model fitting by Newton's method then becomes a sequential quadratic programming problem, while the equally important task of finding initial coefficients satisfying the constraints is a simple linear programming problem. Smoothing parameters can then be estimated by an adaptation of the extended Fellner-Schall method. This article brings together and reviews the necessary methods required to provide a general purpose framework for shape constrained smooth additive modelling, covering the widely applicable case in which not all smooth terms are constrained. As well as providing examples of shape constrained generalized additive models, it is shown how the same approach facilitates non-parametric density estimation. The article is accompanied by a new function scasm in R package mgcv.