Chad Milando, PhD, MS

cmilando@bu.edu

Interested in similar spreadsheets? see here:

https://chadmilando.com/teaching/

A walkthrough of B-spline basis functions

Recommended view % is 75%

if downloaded, some graph formatting gets distored

Helpful source

https://www.hsph.harvard.edu/wp-content/uploads/sites/565/2018/08/572Spr07-BasisMethods-4up.pdf

*Others listed in the Appendix

Sheet overview

https://en.wikipedia.org/wiki/B-spline

calculates recursive functions

references I found in creating this

In these functions, "i" is the spline function (e.g., 1 knot can mean 2+ spline functions, but it could be the left side and right side)

and k is the "order" or "degree" (so degree = 3 is cubic polynomila, degree = 2 is quadratic polynomial)

R code to replicate each page is listed in Red font to the right of each page

Problem statement

you have an independent variable x, and a dependent variable y

They have some relationship described graphically ->

(the true relationship is piecewise described on sheet X below)

You don't know the true relationship, and it doesn't seem linear

and it looks like there is a discontinuity at x = 0.50

So you decide to split X up into components so you can

model the relationship between X and Y more flexibly

with a single knot at x = 0.5

Specifically, you want to use B-splines with various degrees (knowing that many other basis functions exist that could split X up differently)

Below, you compare a linear model of Y ~ X, or linear models of Y ~ bs(X, degree = {0, 1, 2, or 3})