PROS Helmina A, Wahyu W, Santi PR Penalized spline fulltext
Proceedings of the IConSSE FSM SWCU (2015), pp. MA.1–4
MA.1
ISBN: 978-602-1047-21-7
Penalized spline estimator in nonparametric regression
Helmina Andriania*, Wahyu Wibowob, Santi Puteri Rahayuc
a
Student, Institut Teknologi Sepuluh Nopember, Jl. Arif Rahman Hakim, Surabaya 60111, Indonesia
Lecturer, Institut Teknologi Sepuluh Nopember, Jl. Arif Rahman Hakim, Surabaya 60111, Indonesia
b,c
Abstract
Regression analysis is a statistical process that is used to investigate patterns of
relationship and to know the influence of the independent variables on the dependent
variables through regression curve estimation. If the shape of regression curve is assumed
unknown then we used nonparametric regression. Nonparametric regression approach
that quite popular is spline. There are two kinds of nonparametric model approach to
spline, spline regression and smoothing spline. Combination of both approaches is known
as penalized spline regression. This paper focuses on how to obtain the penalty matrix
using penalized least square (PLS) method, then used this penalty matrix to estimate
nonparametric regression model based on penalized spline estimator.
Keywords nonparametric regression, penalized spline, penalty matrix
1.
Introduction
Regression analysis is a statistical process that is used to investigate patterns of
relationships and to know the influence of the independent variables on the dependent
variable through regression curve estimation. If the shape of regression curve is assumed
known then we used parametric regression, but if the shape of regression curve is assumed
unknown then we used nonparametric regression (Utami, 2013). Regression curve estimation
using nonparametric regression model can be done by several methods, including the kernel,
spline, and Wavelet and Fourier series expansion (Djuraidah & Aunuddin, 2006).
Nonparametric regression approach that quite popular is spline. Spline is one kind of
piecewise polynomial, which had segmented poperties. The segmented properties provides
more flexibility than ordinary polynomials, thus allowing it to adapt more effectively to the
local characteristics of a function or data (Budiantara et al., 2006). Other advantages owned
by spline is able to describe the change in the pattern of behavior of the function in the subspecified interval and can be used to address the data patterns experienced a sharp increase
or decrease with the help of knots, as well as the resulting curve is relatively (Fathurahman,
2011).
There are two kinds of nonparametric model approach to spline, i.e. spline regression
and smoothing spline. Combination of both approaches known as penalized spline regression
(Djuraidah & Aunuddin, 2006). In applying penalized spline, there are some things that need
to be considered, namely: (a) the location and number of knots, (b) basis spline functions, and
(c) degree of freedom and penalty matrix (Montoya et al., 2014). This paper focuses on how
*
Corresponding author. E-mail address: eena.andriani@gmail.com
SWUP
Penalized spline estimator in nonparametric regression
MA.2
to obtain the penalty matrix using penalized least square (PLS) method, then used this penalty
matrix to estimate nonparametric regression model based on penalized spline estimator.
2.
Materials and methods
To get the penalty matrix, it takes several mathematical steps.
Step 1.
Suppose that n pairs of measurements are observed,
, , = 1,2, … , , satisfying the
model in Eq. (1), where
is an unknown regression function and the errors ε i are
independent with constant variance .
=
(1)
+ , = 1,2, … , .
Step 2.
It is assumed that
can be well modeled by the truncated power basis of degree , the
basis is 1, , … , , −
,…, −
. Hence, the
-degree spline model is
=
+
+⋯+
+ ∑!# !
− ! ",
MA.1
ISBN: 978-602-1047-21-7
Penalized spline estimator in nonparametric regression
Helmina Andriania*, Wahyu Wibowob, Santi Puteri Rahayuc
a
Student, Institut Teknologi Sepuluh Nopember, Jl. Arif Rahman Hakim, Surabaya 60111, Indonesia
Lecturer, Institut Teknologi Sepuluh Nopember, Jl. Arif Rahman Hakim, Surabaya 60111, Indonesia
b,c
Abstract
Regression analysis is a statistical process that is used to investigate patterns of
relationship and to know the influence of the independent variables on the dependent
variables through regression curve estimation. If the shape of regression curve is assumed
unknown then we used nonparametric regression. Nonparametric regression approach
that quite popular is spline. There are two kinds of nonparametric model approach to
spline, spline regression and smoothing spline. Combination of both approaches is known
as penalized spline regression. This paper focuses on how to obtain the penalty matrix
using penalized least square (PLS) method, then used this penalty matrix to estimate
nonparametric regression model based on penalized spline estimator.
Keywords nonparametric regression, penalized spline, penalty matrix
1.
Introduction
Regression analysis is a statistical process that is used to investigate patterns of
relationships and to know the influence of the independent variables on the dependent
variable through regression curve estimation. If the shape of regression curve is assumed
known then we used parametric regression, but if the shape of regression curve is assumed
unknown then we used nonparametric regression (Utami, 2013). Regression curve estimation
using nonparametric regression model can be done by several methods, including the kernel,
spline, and Wavelet and Fourier series expansion (Djuraidah & Aunuddin, 2006).
Nonparametric regression approach that quite popular is spline. Spline is one kind of
piecewise polynomial, which had segmented poperties. The segmented properties provides
more flexibility than ordinary polynomials, thus allowing it to adapt more effectively to the
local characteristics of a function or data (Budiantara et al., 2006). Other advantages owned
by spline is able to describe the change in the pattern of behavior of the function in the subspecified interval and can be used to address the data patterns experienced a sharp increase
or decrease with the help of knots, as well as the resulting curve is relatively (Fathurahman,
2011).
There are two kinds of nonparametric model approach to spline, i.e. spline regression
and smoothing spline. Combination of both approaches known as penalized spline regression
(Djuraidah & Aunuddin, 2006). In applying penalized spline, there are some things that need
to be considered, namely: (a) the location and number of knots, (b) basis spline functions, and
(c) degree of freedom and penalty matrix (Montoya et al., 2014). This paper focuses on how
*
Corresponding author. E-mail address: eena.andriani@gmail.com
SWUP
Penalized spline estimator in nonparametric regression
MA.2
to obtain the penalty matrix using penalized least square (PLS) method, then used this penalty
matrix to estimate nonparametric regression model based on penalized spline estimator.
2.
Materials and methods
To get the penalty matrix, it takes several mathematical steps.
Step 1.
Suppose that n pairs of measurements are observed,
, , = 1,2, … , , satisfying the
model in Eq. (1), where
is an unknown regression function and the errors ε i are
independent with constant variance .
=
(1)
+ , = 1,2, … , .
Step 2.
It is assumed that
can be well modeled by the truncated power basis of degree , the
basis is 1, , … , , −
,…, −
. Hence, the
-degree spline model is
=
+
+⋯+
+ ∑!# !
− ! ",