Page 383 - Contributed Paper Session (CPS) - Volume 2

CPS1891 Tzee-Ming Huang

A knot selection algorithm for regression splines
Tzee-Ming Huang
Department of Statistics, National Chengchi University, Taipei City 11605, Taiwan
(R.O.C.)

Abstract
In nonparametric regression, splines are commonly used to approximate the
unknown regression function since they are known to have good
approximation power. However, to use splines for function approximation, it
is necessary to set the knots and the order of splines, and knot locations are
crucial to the approximation results. A novel knot selection algorithm based
on statistical testing is proposed. Simulation results are also included to
demonstrate the performance of the proposed algorithm.

Keywords
knot screening; unequally spaced knots

1. Introduction
In this study, we consider the estimation problem in nonparametric
regression using spline approximation. Suppose that we have observations
( , ): = 1, … , from the following nonparametric regression model:

= ( ) + = 1, … , , (1)
,

where is the regression function to be estimated and s are errors. There

are different approaches for estimating such as kernel estimation or basis
approximation. In this study, we focus on estimating based on spline
approximation.
A univariate spline function of order m and knots , … , is a piecewise
1

polynomial that can be expressed as a linear combination of the functions
0
, … , −1 , ( − ) −1 , … , ( − ) −1 , where for ℓ = 1, … , ,
1 +
+

( − ) −1 ≥
( − ) −1 = { 1 ℓ
1 +
0 otherwise.

0
Let (), … , + () denote the basis functions , … , −1 , ( −
1
) −1 , … , ( − ) −1 respectively. When is approximated by the spline
+
1 +
∑ , the coefficients , … , can be estimated using their least square
=1

1
̂
estimators ̂ , … , ̂ . Then, the resulting estimator for is ≝ ∑ ̂ .

1

=1
To approximate using one needs to specify the order and knots for the
̂
spline, and the knot choice is known to be crucial. A simple approach for
determining the knots is to use equally spaced knots and determine the
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