With the rapid development of data collection techniques, complex data objects that are not in the Euclidean space are frequently encountered in new statistical applications. Fr\'echet regression model (Peterson & M\"uller 2019) provides a promising framework for regression analysis with metric space-valued responses. In this paper, we introduce a flexible sufficient dimension reduction (SDR) method for Fr\'echet regression to achieve two purposes: to mitigate the curse of dimensionality caused by high-dimensional predictors, and to provide a tool for data visualization for Fr\'echet regression. Our approach is flexible enough to turn any existing SDR method for Euclidean (X,Y) into one for Euclidean X and metric space-valued Y. The basic idea is to first map the metric-space valued random object $Y$ to a real-valued random variable $f(Y)$ using a class of functions, and then perform classical SDR to the transformed data. If the class of functions is sufficiently rich, then we are guaranteed to uncover the Fr\'echet SDR space. We showed that such a class, which we call an ensemble, can be generated by a universal kernel. We established the consistency and asymptotic convergence rate of the proposed methods. The finite-sample performance of the proposed methods is illustrated through simulation studies for several commonly encountered metric spaces that include Wasserstein space, the space of symmetric positive definite matrices, and the sphere. We illustrated the data visualization aspect of our method by exploring the human mortality distribution data across countries and by studying the distribution of hematoma density.
翻译:随着数据收集技术的迅速发展,在新的统计应用中经常遇到不属于欧洲空间的复杂数据对象。Fr\'echet回归模型(Peterson & M\""uller 2019)为以空间价值衡量的响应进行回归分析提供了一个很有希望的框架。在本文中,我们为Fr\'echet回归引入了一种灵活的足够维度减少法(SDR),以实现两个目的:减轻高维预测器对维度的诅咒,并为Fr\'echet回归提供一种数据可视化的工具。我们的方法足够灵活,可以将Eucliidean (X,Y) 的任何现有SDR方法转换为Euclidean X 和通用空间价值Y的回归分析模型。我们的基本想法是首先将价值为随机对象$Y(SDR)的公尺空间价值随机值变值值值计算法($(Y) ) 实现真正的随机值变数($f(Y),然后对变换数据进行经典特别提法。如果功能的类别足够丰富,那么我们就能保证通过Frchet STD空间空间空间空间空间空间空间空间的可辨化来发现。我们通过探索空间空间空间空间空间空间空间空间空间空间。我们展示的精确度的精确度分布分布分布分布分布分析方法展示的精确度的精确度的分布。我们展示的精确度分布图展示了这种模型的精确度分布。我们用一个用来用来用来解释的精确度分布。我们用平比度分析方法,我们用平比度分析。 我们用平比度分析法,我们用平的计算出一个平比度计算法, 我们用平的计算法的计算法的计算法,我们用平比的计算法的平的平的计算法,我们用平比的平的计算法,我们用平的计算法,我们用平级计算法的计算法,我们用平比。