Statistical analysis of functional data is challenging due to their complex patterns, for which functional depth provides an effective means of reflecting their ordering structure. In this work, we investigate practical aspects of the recently proposed regularized projection depth (RPD), which induces a meaningful ordering of functional data while appropriately accommodating their complex shape features. Specifically, we examine the impact and choice of its tuning parameter, which regulates the degree of effective dimension reduction applied to the data, and propose a random projection-based approach for its efficient computation, supported by theoretical justification. Through comprehensive numerical studies, we explore a wide range of statistical applications of the RPD and demonstrate its particular usefulness in uncovering shape features in functional data analysis. This ability allows the RPD to outperform competing depth-based methods, especially in tasks such as functional outlier detection, classification, and two-sample hypothesis testing.

翻译：函数型数据的统计分析因其复杂的模式而具有挑战性，而函数型深度为反映其排序结构提供了一种有效手段。本文研究了最近提出的正则化投影深度（RPD）的实践方面，该方法能在适当容纳函数型数据复杂形状特征的同时，诱导出有意义的排序。具体而言，我们考察了其调节参数的影响与选择，该参数控制着应用于数据的有效降维程度，并提出了一种基于随机投影的高效计算方法，并提供了理论依据。通过全面的数值研究，我们探索了RPD在广泛的统计应用中的潜力，并证明了其在揭示函数型数据分析中的形状特征方面具有特殊优势。这种能力使得RPD在函数型异常值检测、分类以及双样本假设检验等任务中，尤其优于其他基于深度的方法。