Free knot spline estimation with two roughness penalty terms for functional data and its application to clustering

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 methods 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. 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.