Flexible functional data smoothing and optimization using beta spline

Functional data analysis (FDA) is a method used to analyze data represented in its functional form. The method is particularly useful for exploring both curve and longitudinal data in both exploratory and inferential contexts, with minimal constraints on the parameters. In FDA, the choice of basis function is crucial for the smoothing process. However, traditional basis functions lack flexibility, limiting the ability to modify the shape of curves and accurately represent abnormal details in modern and complex datasets. This study introduced a novel and flexible data smoothing technique for interpreting functional data, employing the beta spline introduced by Barsky in 1981. The beta spline offers flexibility due to the inclusion of two shape parameters. The proposed methodology integrated the roughness penalty approach and generalized cross-validation (GCV) to identify the optimal curve that best fitted the data, ensuring appropriate parameters were considered for transforming data into a functional form. The effectiveness of the approach was assessed by analyzing the GCV color grid chart to determine the optimal curve. In contrast to existing methodologies, the proposed method enhanced flexibility by incorporating the beta spline into the smoothing procedure. This approach was anticipated to effectively handle various forms of time series data, offering improved interpretability and accuracy in data analysis, including forecasting.