# Introductory material on splines

I am looking for a basic, step-by-step introduction into modelling with splines.

(I have encountered splines while teaching another topic. The textbook I am using does not cover splines in sufficient detail for the students to be able to replicate things by themselves. I hope there is some good introductory material out there so that I do not have to reinvent the wheel and write my own notes on the topic.)

• Richard, I would urge you to check "Nonparametric Regression and Generalized Linear Models" by Green & Silver; it's not specifically a spline oriented book. But it does extensively and rather lucidly discuss interpolating, smoothing splines introducing the concept of roughness functional using the "wiggly" nature of the curve with the motivation being proportional factor related to strain energy. Oct 3, 2022 at 17:08

Echoing my comment, I would put forth the monograph "Nonparametric Regression and Generalized Linear Models" by Green, Silverman. It is not highly meticulous as the likes of de Boor's "A Practical Guide to Splines" (after all, it's not spline-centric treatise) but it is comprehensive and provides a lucid introductory account of splines: the authors motivate the concept of spline by observing that when a spline is bent in the shape of a curve $$g,$$ the leading term in the strain energy is $$\propto \int {g^{\prime\prime}}^2.$$ In doing that, they basically "quantify" the roughness of a curve.

Now, that is enough of an intuitive opening to a new realm. Again, this book doesn't delve too much in the functional analysis formalism as in "Smoothing Splines: Methods and Applications" by Wang but this must not deter any one to set it aside. The authors cover interpolating, cubic, natural cubic, smoothing splines, their properties, constructions, plotting, existence of minimizing spline and associated algorithms. There is a chapter on partial spline (unfortunately, I didn't cover that, so won't comment).

In all, while this is definitely not a spline centric book, it's worth a try. I am not aware of OP's students' level but as a student myself, I enjoyed the first reading with enough relevant mathematical materials for a first read.

## Recommendation:

Nonparametric Regression and Generalized Linear Models: A roughness penalty approach, P. J. Green, B. W. Silverman, Chapman & Hall, $$1994.$$

• Thank you for the recommendations. Is there anything for dummies (really basic)? Also, could you give a full reference to de Boor, too? Oct 4, 2022 at 6:40
• "A practical guide to splines" by Carl R. de Boor. He maintains the errata in his site. Personally, for me, it's very terse for a beginner even though it was mentioned as a reference in our course. As for books for dummies, unfortunately the said book was instrumental enough for my purpose after a bit dose of ESL. Oct 4, 2022 at 7:29
• Thanks! Green & Silverman's book looks appealing. Oct 4, 2022 at 7:31
• Sure. You can now make a proper blend of rigor and intuition in your notes based on these books for your students. Also, there is a post on Maths.SE (albeit scant): Book for Spline interpolation. Oct 4, 2022 at 7:37

I found the section on splines in Frank Harrell's Regression Modeling Strategies very helpful. Yes, the book is not only about splines, but if your students are learning about them, they may find the rest of this tome helpful, too.

This answer is coming from a biostatistics angle.

I would second Frank Harrell's Regression Modelling Strategies, as per Stephan's answer https://link.springer.com/book/10.1007/978-3-319-19425-7.

Other resources I have used include a 2010 paper on general implementation (1). Per the title, it is focused on restricted cubic splines and includes a SAS macro (I think SAS has a bit more built-in functionality nowadays, but I use R).

I also read a more recent paper (2) focused on discussing different spline types and then considering their implementations in R. This is quite handy since there is some review therein about maturity of different packages for fitting splines. For me this seems reasonably accessible at a conceptual level (for those wanting to consider and use splines in applied settings) while also containing sufficient detail on basis forms for those who prefer a mathematical presentation.

(1) Desquilbet L, Mariotti F. Dose-response analyses using restricted cubic spline functions in public health research. Stat Med 2010;29(9):1037-57. doi: 10.1002/sim.3841 https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.3841

(2) Perperoglou, A., Sauerbrei, W., Abrahamowicz, M. et al. A review of spline function procedures in R. BMC Med Res Methodol 19, 46 (2019). https://doi.org/10.1186/s12874-019-0666-3 https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-019-0666-3

I think Semiparametric Regression with R, in particular Chapter 2 "Penalized Splines", by Harezlak, Ruppert, and Wand (2018) would work as a first introduction. The book is focused on practical implementation in $$\mathsf{R}$$ and accompanied by the $$\mathsf{R}$$ package HRW.
Probably Chapter 3 "Scatterplot Smoothing" in Semiparametric Regression by Ruppert, Wand, and Carroll (2003) would be a good addition.