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.

翻译：在大数据时代，无论是随时间连续记录还是在不同时间点零星采集，信息量都在持续增长。功能数据分析（FDA）处于这场数据革命的前沿，为处理此类复杂数据集并从中提取有意义的见解提供了强大的框架。当前提出的FDA方法在处理形状多变的曲线时常常面临挑战，这主要归因于该方法在分析过程中高度依赖数据近似这一关键环节。本研究提出了一种带有两个惩罚项的自由节点样条估计方法，用于功能数据分析，并通过在模拟数据和真实数据上比较多种聚类方法的结果，验证了该方法的性能。