Geometrically designed spline (GeDS) regression is a non-parametric method for fitting spline regression models with variable knots. The GeDS technique is inspired by geometric principles and falls within the domain of generalized non-linear models (GNM), which include generalized linear models (GLM) as a special case. GeDS regression is fitted based on a sample of \(N\) observations of a response variable \(y\), dependent on a set of (currently up to two) covariates, assuming \(y\) has a distribution from the exponential family. In addition, GeDS methodology is implemented both in the context of generalized additive models (GAM) and functional gradient boosting (FGB). On the one hand, GAM consist of an additive modeling technique where the impact of the predictor variables is captured through smooth (GeDS, in this case) functions. On the other hand, GeDS incorporates gradient boosting machine learning technique by implementing functional gradient descent algorithm to optimize general risk functions utilizing component-wise GeDS estimates.
Dimitrina S. Dimitrova <D.Dimitrova@citystgeorges.ac.uk>,
Vladimir K. Kaishev <V.Kaishev@citystgeorges.ac.uk>,
Andrea Lattuada <Andrea.Lattuada@hotmail.com>,
Emilio L. Sáenz Guillén <Emilio.Saenz-Guillen@citystgeorges.ac.uk> and
Richard J. Verrall <R.J.Verrall@citystgeorges.ac.uk>
GeDS provides a novel solution to the spline regression problem and, in particular, to the problem of estimating the number and position of the knots. The GeDS estimation method is based on two stages: first, in stage A, a piecewise linear fit (spline fit of order 2) capturing the underlying functional shape determined by the data is constructed; second, in stage B, the latter fit is approximated through shape preserving (variation diminishing) spline fits of higher orders (\(n = 3\), \(n = 4\),\(\dots\), i.e., degrees 2, 3,\(\dots\)). As a result, GeDS simultaneously produces a linear, a quadratic and a cubic spline fit.
The GeDS method was originally developed by Kaishev et al. (2016) for the univariate case, assuming the response variable \(y\) to be normally distributed and a corresponding Mathematica code was provided.
The GeDS method was extended by Dimitrova et al. (2023) to cover any
distribution from the exponential family. The GeDS R package presented
here provides an enhanced implementation of the original normal GeDS
Mathematica code, through the NGeDS function; it also
includes a generalization, GGeDS, which extends the method to
any distribution in the exponential family.
The GeDS package allows also to fit two dimensional response surfaces
and to construct multivariate (predictor) models with a GeD spline component
and a parametric component (see the functions f,
formula, NGeDS and
GGeDS for details).
Dimitrova et al. (2025) have recently made significant enhancements to the
GeDS methodology, by incorporating generalized additive models (GAM-GeDS)
and functional gradient boosting (FGB-GeDS). On the one hand, generalized additive
models are encompassed by implementing the local-scoring algorithm
using normal GeD splines (i.e., NGeDS) as function smoothers
within the backfitting iterations. This is implemented through the function
NGeDSgam. On the other hand, the GeDS package incorporates
functional gradient descent algorithm by utilizing normal GeD splines (i.e.,
NGeDS) as base learners within the boosting iterations. Unlike
typical boosting methods, the final FGB-GeDS model is expressed as a single
spline model rather than as a sum of base-learner fits. For this,
NGeDSboost leverages the piecewise polynomial representation of
B-splines, and, at each boosting iteration, performs a piecewise update of the
corresponding polynomial coefficients.
The outputs of both NGeDS and GGeDS functions are
"GeDS" class objects, while the outputs of NGeDSgam
and NGeDSboost functions are"GeDSgam" class and
"GeDSboost" class objects, respectively. "GeDS" class,
"GeDSgam" class and "GeDSboost" class objects contain
second, third and fourth order spline fits. As described in
Kaishev et al. (2016), Dimitrova et al. (2023) and Dimitrova et al. (2025),
the "final" GeDS fit is the one minimizing the empirical deviance. Nevertheless,
the user can choose to use any of the available fits.
The GeDS package also includes some datasets where GeDS regression
proves to be very efficient and some user friendly functions that are designed
to easily extract required information. Several methods are also provided to
handle GeDS, GAM-GeDS and FGB-GeDS output results (see NGeDS/GGeDS,
NGeDSgam and NGeDSboost, respectively).
Throughout this document, we use the terms GeDS predictor model, GeDS regression and GeDS fit interchangeably.
Please report any issue arising or bug in the code to Emilio.Saenz-Guillen@citystgeorges.ac.uk.
Kaishev, V.K., Dimitrova, D.S., Haberman, S., & Verrall, R.J. (2016).
Geometrically designed, variable knot regression splines.
Computational Statistics, 31, 1079--1105.
DOI: tools:::Rd_expr_doi("10.1007/s00180-015-0621-7")
Dimitrova, D. S., Kaishev, V. K., Lattuada, A. and Verrall, R. J. (2023).
Geometrically designed variable knot splines in generalized (non-)linear
models.
Applied Mathematics and Computation, 436.
DOI: tools:::Rd_expr_doi("10.1016/j.amc.2022.127493")
Dimitrova, D. S., Kaishev, V. K. and Saenz Guillen, E. L. (2025). GeDS: An R Package for Regression, Generalized Additive Models and Functional Gradient Boosting, based on Geometrically Designed (GeD) Splines. Manuscript submitted for publication.
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