Roughness regularization for functional data analysis with free knots spline estimation

In the era of big data, an ever-growing volume of information is recorded, either continuously over time or sporadically, at distinct time intervals. Functional Data Analysis (FDA) stands at the cutting edge of this data revolution, offering a powerful framework for handling and extracting meaningful insights from such complex datasets. The currently proposed FDA me\-thods can often encounter challenges, especially when dealing with curves of varying shapes. This can largely be attributed to the method's strong dependence on data approximation as a key aspect of the analysis process. In this work, we propose a free knots spline estimation method for functional data with two penalty terms and demonstrate its performance by comparing the results of several clustering methods on simulated and real data.