Direct generalized additive modeling with penalized likelihood

By Brian D. Marx, Paul H. C. Eilers. 1998

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Partially Interpretable Estimators (PIE): Black-Box-Refined Interpretable Machine Learning (2021, background): “Canonical shape functions are based on various forms of splines (Marx & Eilers, 1998; Hastie & Tibshirani, 1990), trees (Lou et al., 2013), or neural networks in a recent work Neural Additive Models (NAM) (Agarwal et al., 2020).”
Some theory for penalized spline generalized additive models (2002, methodology): “ Marx and Eilers (1998) demonstrate how this can be achieved using penalized splines, also known as P-splines.”
PENALIZED STRUCTURED ADDITIVE REGRESSION FOR SPACE-TIME DATA: A BAYESIAN PERSPECTIVE (2004, background): “…(2001b), Lang and Brezger (2003) and Brezger and Lang (2003)), non-linear effects of continuous covariates as well as smooth time trends are modelled through Bayesian versions of penalized splines (P-splines), introduced in a frequentist setting by Eilers and Marx (1996), Marx and Eilers (1998).”
Monotonic Regression Based on Bayesian P–Splines (2008, methodology): “For modeling the unknown functions fj , j = 1, . . . , p, we follow Lang and Brezger (2004), who proposed a Bayesian version of the P-splines approach introduced in a frequentist setting by Eilers and Marx (1996).”
Theory & Methods: Spatially‐adaptive Penalties for Spline Fitting (2000, unknown): “As discussed in Marx and Eilers (1998), one need not use backfitting to fit a additive spline model.”
Comparison of Volatility Measures: A Risk Management Perspective (2010, unknown): “in the GLM framework by Eilers & Marx (1996) and Marx & Eilers (1998) .”
Stable and Efficient Multiple Smoothing Parameter Estimation for Generalized Additive Models (2004, background): “Hastie and Tibshirani (1990), Hastie (1996), Marx and Eilers (1998), and Wood (2000, 2003) have all discussed using penalized regression splines for GAM modeling.”
Generalized Additive Model Selection (2015, methodology): “, 2007), and the method of Meier et al. (2009). Each of these proposals is based on penalized likelihood and the difference comes from the choice of F and the structure of the penalty term J(f). COSSO models the components fj as belonging to a reproducing kernel Hilbert space (RKHS), and operates by”
Geoadditive Survival Models (2006, methodology): “…of time (in particular, the log-baseline hazard rate) and of continuous covariates or further time scales are modeled through Bayesian versions of penalized splines (P-splines) introduced by Eilers and Marx (1996) and Marx and Eilers (1998) for generalized additive models in a frequentist setting.”
Generalized structured additive regression based on Bayesian P-splines (2006, methodology): “For one dimensional P-splines, the band size is max{degree of spline, order of differences}, for two dimensional P-splines the band size is M j ·l+l, and for i.i.d. random effects the posterior precision matrix is diagonal.”
GENERALIZED ADDITIVE PARTIAL LINEAR MODELS WITH HIGH-DIMENSIONAL COVARIATES (2013, background): “As it is well known that the kernel-based backfitting and marginal integration approaches are computationally expensive, Marx and Eilers (1998), Ruppert, Wand, and Carroll (2003), and Wood (2004) suggested penalized regression splines, but no theoretical justifications are available for these…”
Splines, knots, and penalties (2010, unknown): “Bα = B(γ + a) = Xβ + Fb = F̆D̆p+1(γ + a) = Xβ + F̆Sb, (11)”
Parsimonious additive models (2007, methodology): “…for penalized splines, (Ruppert et al., 2003)), Akaike information criteria (Hastie and Tibshirani, 1990; Hurvich et al., 1998; Marx and Eilers, 1998), Bayesian information criteria (Marx and Eilers, 1998), plug in (Opsomer and Ruppert, 1998), or hypothesis testing (Cantoni and Hastie, 2002).”
P-spline ANOVA-type interaction models for spatio-temporal smoothing (2011, methodology): “There are other alternatives to avoid the identifiability problems: (i) add a ridge penalty on the system of equations in (2.3) as in Marx and Eilers (1998) or (ii) identify and impose the constraints numerically (see Wood, 2006a).”
Selecting the Number of Knots for Penalized Splines (2002, methodology): “Moreover, direct tting of additive spline models is straightforward (Marx and Eilers, 1998). The parameter vector = ( T1 ; : : : ; TL)T can be estimated by penalized least-squares by minimizing n Xi=1 yi m(x; ) 2 + L Xl=1 l Kl X k=1 2 l;p+k; (14) where 1; : : : ; L are smoothing parameters. The comp”
Early plant recruitment stages set the template for the development of vegetation patterns along a hydrological gradient (2015, background): “Nomenclature and growth form according to van der Meijden (2005), with (Ch) chamaephyte, (Hc) hemicryptophyte, (He) helophyte and (Th) therophyte.”