Two frameworks for multivariate functional depth based on multivariate depths are introduced in this paper. The first framework is multivariate functional integrated depth, and the second framework involves multivariate functional extremal depth, which is an extension of the extremal depth for univariate functional data. In each framework, global and local multivariate functional depths are proposed. The properties of population multivariate functional depths and consistency of finite sample depths to their population versions are established. In addition, finite sample depths under irregularly observed time grids are estimated. As a by-product, the simplified sparse functional boxplot and simplified intensity sparse functional boxplot are proposed for visualization without data reconstruction. A simulation study demonstrates the advantages of global multivariate functional depths over local multivariate functional depths in outlier detection and running time for big functional data. An application of our frameworks to cyclone tracks data demonstrates the excellent performance of our global multivariate functional depths.
翻译:本文介绍了基于多变量深度的多变量功能深度的两个框架。第一个框架是多变量功能集成深度,第二个框架涉及多变量功能极端深度,这是单ivariate功能数据极端深度的延伸。在每个框架中,都提出了全球和地方多变量功能深度。确立了人口多变量功能深度的特性和有限样本深度与其人口版本的一致性。此外,还估算了在不规则观测的时间网下有限的样本深度。作为副产品,建议不重建数据就可视化使用简化的稀释功能插件和简化的密度稀释功能框。模拟研究展示了全球多变量功能深度相对于本地多变量功能深度的优势,用于外部检测和运行大功能数据的时间。我们应用龙卷风跟踪数据的框架显示了我们全球多变量功能深度的出色表现。