Use of Two Smoothing Parameters in Penalized Spline Estimator for Bi-variate Predictor Non-parametric Regression Model

Penalized spline criteria involve the function of goodness of fit and penalty, which in the penalty function contains smoothing parameters. It serves to control the smoothness of the curve that works simultaneously with point knots and spline degree. The regression function with two predictors in the non-parametric model will have two different non-parametric regression functions. Therefore, we propose the use of two smoothing parameters in the bi-variate predictor non-parametric regression model. We demonstrated its ability through longitudinal data simulation studies with a comparison of one smoothing parameter. It was done on several numbers of subjects with repeated measurements. The generalized cross validation value which is a measure of the model's ability is poured through the box plot. The results show that the use of two smoothing parameters is more optimal than one smoothing parameter. It was seen through a smaller generalized cross validation value on the use of two smoothing parameters. Application of blood sugar level data for patients with two smoothing parameters produced a penalized spline bi-variate predictor regression model with several segments of change patterns. There are five patterns at the time of treatment and blood pressure with the number of smoothing parameters is two, namely 0.39 and 0.73.