Selecting the Number of Knots for Penalized Splines

By David Ruppert. 2002

816 citations recorded by the source corpus.

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In Computational Mechanics

Citations in the corpus

Smoothing and forecasting mortality rates (2004, background): “Theoretical aspects of P-splines are discussed in Wand (1999), Ruppert and Carroll (2000), Aerts et al. (2002), Ruppert (2002) and Durban and Currie (2003).”
Likelihood ratio tests in linear mixed models with one variance component (2003, methodology): “Crainiceanu, Ruppert and Vogelsang (2002) showed that this mass can be very large for LRT∞ and RLRT∞ and is equal to the null probability that LRT∞(·) and RLRT∞(·) have a global maximum at zero.”
Some asymptotic results on generalized penalized spline smoothing (2009, background): “tical impact if the dimension of the spline basis is chosen in lush and generous manner, see Ruppert (2002) .”
Smoothing and mixed models (2003, background): “French, Kammann and Wand (2001) provide a small amount of evidence to support this claim, as does Ruppert (2002) .”
Knot selection for least-squares and penalized splines (2013, methodology): “If, for example, k = 5 (a value recommended by Ruppert [9] for some situations) then acquiring an extra significant figure of precision will increase computation time by a factor of 100,000! Random search methods can provide a practical alternative to deterministic grid search techniques.”
Splines, knots, and penalties (2010, background): “The number of knots is essentially immaterial, as long as it is large enough (Ruppert, 2002), hence in the limiting case of one knot per bin we still get a nicely smoothed histogram.”
Shape constrained additive models (2015, methodology): “As with any penalized regression spline approaches, the choice of the basis dimension, q, is not crucial but should be generous enough to avoid oversmoothing/underfitting (Ruppert 2002; Li and Ruppert 2008).”
A Note on Penalized Spline Smoothing With Correlated Errors (2007, methodology): “With respect to the dimension K , we follow Ruppert (2002), who showed that the actual choice of K and the location of the knots have little influence on the resulting penalized fit as long as K is large, for example, K = min(n/4,40).”
Spatially Adaptive Bayesian Penalized Splines With Heteroscedastic Errors (2007, methodology): “Following Gray (1994) and Ruppert (2002), we take Km large enough (e.g., 20) to ensure the desired flexibility.”
Density Estimation in the Presence of Heteroscedastic Measurement Error (2008, background): “The number of knots, K − 1, is chosen to be equal to min(n/4,35) (Ruppert 2002).”
Penalized spline models for functional principal component analysis (2006, methodology): “Since Ruppert (2002) showed that the penalized spline estimators are relatively insensitive to the choice of basis functions compared with the choice of λÅ, as long as enough of them are used, here an adequate choice of knots (10th, . . . , 90th percentiles of the pooled observation times) was used…”
On the asymptotics of penalized spline smoothing (2011, background): “1 provide theoretical justification of the observation that the number of knots is not important, as long as it is above some minimal level (Ruppert [21]).”
Restricted Likelihood Ratio Testing for Zero Variance Components in Linear Mixed Models (2008, background): “…σ 2 u1 I K ), with X = [1 xi ]1≤i≤n, β = (β0, β1), Z 1 = Z K −1/2 K , Z K = [|xi − κk | 3] 1≤i≤n 1≤k≤K and K = [|κk − κk′ | 3]1≤k≤K 1≤k′≤K , where the κk, k = 1, . . . , K , are at least five knots placed at every 4th or 5th unique xi , up to a maximum of 35 knots (Ruppert 2002; Ngo and Wand 2004).”
Classification and valuation of urban green spaces—A hedonic house price valuation (2013, background): “Ruppert (2002) recommends setting } 4 nmin{40,=k whereas Wood (2006) argues that the choice of k is context specific.”
Penalized Functional Regression (2011, neutral): “In the smoothing literature, the choice of the number of knots in φ has been shown (Ruppert 2002; Li and Ruppert 2008) to be unimportant as long as it is large enough to capture the maximum complexity of the regression function: in penalized spline regression it is the explicit smoothness…”
Estimation and inference in spatially varying coefficient models (2018, methodology): “In this work, we choose a common λ for all coefficient functions; nevertheless, separate smoothing parameters for coefficient functions can be adopted with heavier computing; see the discussion by Ruppert (2002).”