## Statistical Science

- Statist. Sci.
- Volume 11, Number 2 (1996), 89-121.

### Flexible smoothing with *B*-splines and penalties

Paul H. C. Eilers and Brian D. Marx

**Full-text: Open access**

#### Abstract

*B*-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent *B*-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of *B*-splines, of their construction and of penalized likelihood is presented. We discuss properties of penalized *B*-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.

#### Article information

**Source**

Statist. Sci. Volume 11, Number 2 (1996), 89-121.

**Dates**

First available in Project Euclid: 27 November 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.ss/1038425655

**Digital Object Identifier**

doi:10.1214/ss/1038425655

**Mathematical Reviews number (MathSciNet)**

MR1435485

**Zentralblatt MATH identifier**

0955.62562

**Keywords**

Generalized linear models smoothing nonparametric models splines density estimation

#### Citation

Eilers, Paul H. C.; Marx, Brian D. Flexible smoothing with B -splines and penalties. Statist. Sci. 11 (1996), no. 2, 89--121. doi:10.1214/ss/1038425655. https://projecteuclid.org/euclid.ss/1038425655

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- Gijbels, 1996). Conservation of moments seems unimportant. In regression, I do not see the desirability. In density estimation, simple corrections of kernel density estimates for variance inflation exist, but make little difference away from the normal density (Jones,
- 1991). Indeed, getting means and variances right is a normality-based concept, so corrected kernel estimators act in a normal-driven semiparametric manner. Efron and Tibshirani (1996) propose more sophisticated moment conservation, but initial indications are that this is no better nor worse than alternative semiparametric density estimators (Hjort,
- 1996). "The computations, including those for crossvalidation, are relatively inexpensive and easily incorporated into standard software." Again, proponents of the two competing methods I have mentioned would claim the same for the first half of this and advocates of regression splines would claim the lot. The authors make no particularly novel contribution to automatic bandwidth selection. Crossvalidation and AIC are in a class of methods (e.g., H¨ardle, 1990, pages 166-167) which, while not being downright bad, allow scope for improvement.
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- all kernel-ty pe estimators (Fan and Marron, 1994). The local likelihood approach is already deeply understood theoretically. Comparison of P-splines's reasonable boundary performance with local poly nomials's reasonable boundary performance is not yet available through theory or simulations. An interesting point mentioned in the paper is the apparent continuum between few-parameter parametric fits at one end and fully "nonparametric" techniques at the other, with many-parameter par
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- timality and minimax properties (Fan, 1993). For density estimation Engel and Gasser (1995) show a minimax property of the fixed bandwith kernel method within a large class of estimators containing penalized likelihood estimators. The presented paper does not provide any argument, neither theoretical nor by simulations, supporting any superiority of P-splines over their many competitors. In the regression case, the theoretical properties of P-splines might be evaluated by combining arguments of de Boor (1978) on the asy mptotic bias and variance of B-splines in (dependence on m, the spline order k and the smoothness of the underlying function) with the well-known results on smoothing splines. The authors propose to use AIC or crossvalidation to select the smoothing parameter. However, a careful look at their method reveals that there are in fact two free parameters: and the number n of knots. If n m, then we essentially obtain a smoothing spline fit, while results might be very different if n m. Indeed, the estimate might crucially depend on n. Therefore, why not determine and n by cross-validation or a related method? The following theoretical arguments may suggest that such a procedure will work. Note that AIC and cross-validation are very close to unbiased risk estimation which consists of estimating the optimal values of and n by minimizing m
- et al. (1996). Software for the 1992 version, written in C and interfaced to S-PLUS, is publically available from Statlib. (The 1992 version of LOGSPLINE employ s only knot deletion; here, however, we focus on the 1996 version, which uses both knot addition and knot deletion.) LOGSPLINE can provide estimates on both finite and infinite intervals, and it can handle censored data. The results of LOGSPLINE on the Old Faithful data and the suicide data are very similar to the corresponding results of P-splines [the suicide data is an example in Kooperberg and Stone (1992)]. Here we consider a much more challenging data set. The solid line in Figure 1 shows the logspline density estimate based on a random sample of 7,125 annual net incomes in the United Kingdom [Family Expenditure Survey (1968-1983)]. (The data have been rescaled to have mean 1.) The nine knots that were selected by LOGSPLINE are indicated. Note that four of these knots are extremely close to the peak near 0 24. This peak is due to the UK old age pension, which caused many people to have nearly identical incomes. In Kooperberg and Stone (1992) we concluded that the height and location of this peak are accurately estimated by LOGSPLINE. There are several reasons why this data is more challenging than the Old Faithful and suicide data: the data set is much larger, so that it is more of a challenge to computing resources (the LOGSPLINE estimate took 9 seconds on a Sparc 10 workstation); the width of the peak is about 0.02, compared to the range 11.5 of the data; there is a severe outlier (the largest observation is 11.5, the second largest is 7.8); and the rise of the density to the left of the peak is very steep. To get an impression of what the P-splines procedure would yield for this data, I first removed the largest observation so that there would not be any long gaps in the data, reducing the maximum observation to 7.8. The dashed line in Figure 1 is the LOGSPLINE estimate to the data with fixed knots at i/20 × 7 8, for i = 0 1 20 (using 20 intervals, as in most P-spline examples.) The resulting fit should be similar to a P-spline fit with = 0. In this estimate it appears that the narrow peak is completely missed and that, because of the steep rise of the density to the left of the peak and the lack of sufficiently many knots near the peak, two modes are estimated where only one mode exists.
- ors as in Wahba (1978). In particular, the usual priors are specified independently of sample size, whereas one would want to use more B-splines with a larger sample. Furthermore, the integral of the second derivative squared is easier to interpret from a non-Bayesian perspective than the sum of squares of second differences of B-spline coefficients. I take issue with the authors's claim that their method does not have boundary problems. P-splines are approximately equivalent to smoothing splines
- which do have boundary effects (Speckman, 1983). To explain, consider minimizing from equation (5),
- Sheather (1996). We use AIC and get essentially the same amount of smoothing, "a second generation result at a first generation price." Again, we do not wish to imply that AIC is the final answer, but show that it is more useful than sometimes suggested.
- LR, local regression; LRB, local regression with binning; SS, smoothing splines; SSB, smoothing splines with band solver; RSF, regression splines with fixed knots; RSA, regression splines with adaptive knots; PS, P-splines. The row "Adaptive flexibility available" means that a software implementation is readily available
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Digital Object Identifier: doi:10.1111/j.1467-9892.1995.tb00240.x - Speckman, P. L. (1983). Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Statist. 13 970-983. Stone, C. J., Hansen, M., Kooperberg, C. and Truong, Y. K. Mathematical Reviews (MathSciNet): MR803752

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Project Euclid: euclid.aos/1176349650 - (1996). Poly nomial splines and their tensor products in extended linear modeling. Ann. Statist. To appear.
- Wahba, G. (1978). Improper priors, spline smoothing and the problem of guarding against model errors in regression. J. Roy. Statist. Soc. Ser. B 40 364-372.

#### See also

- Includes: S-T. Chiu. Comment.
- Includes: Douglas Nychka, David Cummins. Comment.
- Includes: Chong Gu. Comment.
- Includes: M. C. Jones. Comment.
- Includes: Joachim Engel, Alois Kneip. Comment.
- Includes: Charles Kooperberg. Comment.
- Includes: Dennis D. Cox. Comment.
- Includes: Stephan R. Sain, David W. Scott. Comment.
- Includes: Paul H. C. Eilers, Brian D. Marx. Rejoinder.

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