TY - JOUR

T1 - Reproducing Kernel Hilbert Space Approach to Multiresponse Smoothing Spline Regression Function

AU - Lestari, Budi

AU - Chamidah, Nur

AU - Aydin, Dursun

AU - Yilmaz, Ersin

N1 - Funding Information:
The authors thank Airlangga University for technical support and DRTPM of the Ministry of Education, Culture, Research, and Technology, the Republic of Indonesia for financial support. The authors are grateful to the editors and anonymous peer reviewers of the Symmetry journal, who provided comments, corrections, criticisms, and suggestions that were useful for improving the quality of this article.
Funding Information:
This research was funded by the Directorate of Research, Technology, and Community Service (Direktorat Riset, Teknologi, dan Pengabdian kepada Masyarakat–DRTPM), the Ministry of Education, Culture, Research, and Technology, the Republic of Indonesia through the Featured Basic Research of Higher Education Grant (Hibah Penelitian Dasar Unggulan Perguruan Tinggi–PDUPT, Tahun Ketiga dari Tiga Tahun) with master contract number 010/E5/PG.02.00.PT/2022 and derivative contract number 781/UN3.15/PT/2022.
Publisher Copyright:
© 2022 by the authors.

PY - 2022/11

Y1 - 2022/11

N2 - In statistical analyses, especially those using a multiresponse regression model approach, a mathematical model that describes a functional relationship between more than one response variables and one or more predictor variables is often involved. The relationship between these variables is expressed by a regression function. In the multiresponse nonparametric regression (MNR) model that is part of the multiresponse regression model, estimating the regression function becomes the main problem, as there is a correlation between the responses such that it is necessary to include a symmetric weight matrix into a penalized weighted least square (PWLS) optimization during the estimation process. This is, of course, very complicated mathematically. In this study, to estimate the regression function of the MNR model, we developed a PWLS optimization method for the MNR model proposed by a previous researcher, and used a reproducing kernel Hilbert space (RKHS) approach based on a smoothing spline to obtain the solution to the developed PWLS optimization. Additionally, we determined the symmetric weight matrix and optimal smoothing parameter, and investigated the consistency of the regression function estimator. We provide an illustration of the effects of the smoothing parameters for the estimation results using simulation data. In the future, the theory generated from this study can be developed within the scope of statistical inference, especially for the purpose of testing hypotheses involving multiresponse nonparametric regression models and multiresponse semiparametric regression models, and can be used to estimate the nonparametric component of a multiresponse semiparametric regression model used to model Indonesian toddlers’ standard growth charts.

AB - In statistical analyses, especially those using a multiresponse regression model approach, a mathematical model that describes a functional relationship between more than one response variables and one or more predictor variables is often involved. The relationship between these variables is expressed by a regression function. In the multiresponse nonparametric regression (MNR) model that is part of the multiresponse regression model, estimating the regression function becomes the main problem, as there is a correlation between the responses such that it is necessary to include a symmetric weight matrix into a penalized weighted least square (PWLS) optimization during the estimation process. This is, of course, very complicated mathematically. In this study, to estimate the regression function of the MNR model, we developed a PWLS optimization method for the MNR model proposed by a previous researcher, and used a reproducing kernel Hilbert space (RKHS) approach based on a smoothing spline to obtain the solution to the developed PWLS optimization. Additionally, we determined the symmetric weight matrix and optimal smoothing parameter, and investigated the consistency of the regression function estimator. We provide an illustration of the effects of the smoothing parameters for the estimation results using simulation data. In the future, the theory generated from this study can be developed within the scope of statistical inference, especially for the purpose of testing hypotheses involving multiresponse nonparametric regression models and multiresponse semiparametric regression models, and can be used to estimate the nonparametric component of a multiresponse semiparametric regression model used to model Indonesian toddlers’ standard growth charts.

KW - MNR

KW - RKHS

KW - consistency

KW - smoothing spline regression function

KW - standard growth charts

UR - http://www.scopus.com/inward/record.url?scp=85141792075&partnerID=8YFLogxK

U2 - 10.3390/sym14112227

DO - 10.3390/sym14112227

M3 - Article

AN - SCOPUS:85141792075

SN - 2073-8994

VL - 14

JO - Symmetry

JF - Symmetry

IS - 11

M1 - 2227

ER -