Smoothing parameter selection method for multiresponse nonparametric regression model using smoothing spline and Kernel estimators approaches

Abstract

The principle problem in multiresponse nonparametric regression model is how we estimate the regression functions which draw association between some dependent (response) variables and some independent (predictor) variables where there are correlations between responses. There are many techniques used to estimate the regression function. Two of them are spline and kernel smoothing techniques. Speaking about smoothing techniques, not only in uniresponse spline and kernel nonparametric regression models but also in multiresponse spline and kernel nonparametric regression models, the estimations of regression functions depend on smoothing parameters. In the privious researches the covariance matrices were assumed to be known. Matrix of covariance is not assumed known in this research. The goals of this research are selecting of optimal smoothing parameters for the model we consider through spline and kernel smoothing techniques. Optimal smoothing parameters can be obtained by taking the solution to generalized cross validation (GCV) optimization problem. The obtained results of this research are the optimal smoothing parameter for smoothing spline estimator approach and the optimal smoothing parameter namely optimal bandwidth for kernel estimator approach.

Nonparametric Regression Modeling Based on Spline Truncated Estimator on Simulation Data

“…Regresi nonparametrik memiliki fleksibilitas yang tinggi, karena data diharapkan mencari sendiri bentuk estimasi kurva regresinya tanpa dipengaruhi oleh faktor subyektifitas peneliti [6]. Pada pendekatan menggunakan model regresi nonparametrik yang tidak memiliki spesifikasi pola fungsi regresi tertentu, dapat digunakan estimator-estimator untuk memperkirakan fungsi regresi dalam model, estimator-estimator yang biasa digunakan tersebut misalnya spline, kernel, linier lokal, dan lain sebagainya [12]. Estimator-estimator pada analisis regresi nonparametrik juga membuat hasil pendekatan dan interpretasi regresi nonparametrik lebih baik.…”

Analyzing The Effect of BI 7-Days Repo Rate on The Jakarta Composite Index Using Nonparametric Regression Approaches Based on Least Square Spline Estimator

“…Estimator Least Square Spline (LS-Spline) dipilih karena mampu mengikuti pola perilaku data yang berfluktuasi dengan menempatkan titik knot. Pemilihan model terbaik dilakukan dengan membandingkan nilai Generalized Cross Validation (GCV), dimana nilai GCV terendah menunjukkan model yang optimal [11][12]20]. Pendekatan ini telah banyak dipelajari dan diterapkan oleh berbagai peneliti [13].…”

Estimating the confidence interval of the regression coefficient of the blood sugar model through a multivariable linear spline with known variance

“…Non-parametric regression is used when the assumption of the relationship between the predictor and the response is unknown, so we must estimate its function. Some estimators that have been developed include spline truncated (Aprilia, Islamiyati and Anisa, 2019), spline smoothing (Lestari, Budiantara and Chamidah, 2019), spline penalized (Islamiyati, Fatmawati and Chamidah, 2019), local polynomial (Chamidah, Gusti, Tjahjono and Lestari, 2019), Kernel (Chamidah and Saifuddin, 2013), Fourier series (Mardianto, Tjahjono and Rifada, 2020), and Gaussian process (Saegusa, 2020) , and spline principal component analysis (Islamiyati, Kalondeng, Sunusi, Zakir and Amir, 2022). For this article, we use spline truncated to estimate multi-variable non-parametric regression functions.…”