The Use of Spline, Bayesian Spline and Penalized Bayesian Spline Regression for Modeling

Aims: The aim of this study is modeled the ratios of export to imports data in Turkey by using nonparametric regression methods.
Study Design: This was Spline, Bayesian Spline and Penalized Spline Regression modeling study.
Place and Duration of Study: Turkish Statistical Institute. The ratios of export to import data consist of sixty-seven month periods (May 2007 to November 2012).
Methodology: In this study, distribution graph of ratios of export to import between 2007 to 2012 years in Turkey is modeled using spline and Bayesian spline regression methods. The results of these regression models are compared. Then Penalized spline regression is examined with Bayesian approach and models are established for the different values of the smoothing parameter which obtained using prior distributions. We proposed a new smoothing parameter using the information content of normal distribution. Under the assumption of the coefficients of basis functions are normally distributed, the new smoothing parameter (λ*) is defined as the ratio of the information content of normal distribution.
Results: When we compared the spline and Bayesian spline regression models, both models were shown similar characteristics. The coefficients of β and b parameter vectors were very similar and the coefficients of determination of two models were obtained same. But, the standard errors of parameter estimations of Bayesian spline regression were smaller than spline regression models. For this reason, we conclude that Bayesian spline regression model parameter estimation is more reliable then spline regression model. We also compared penalized Bayesian spline models using different penalty terms. The different models on the same data set have been set up using different value of λ. From the results, observe that the absolute value of the coefficients of basis functions decrease as the penalty term 1/λ increase. Also, the coefficient of determination of the model gradually diminishes. In addition, we proposed a new smoothing parameter using the information content of normal distribution. According to results, small changes in λ* have made drastic changes in smoothing of the model. So, we conclude that λ* is more sensitive than traditional smoothing parameter (λ).
Conclusion: We investigated the three most common nonparametric regression models, which are called spline, Bayesian spline and penalized Bayesian spline, discussing advantages and disadvantages of them using real data. We conclude that Bayesian spline regression model parameter estimation is more reliable than other models. In addition, we conclude that λ* is more sensitive than traditional smoothing parameter (λ).