通过变换到希尔伯特空间的密度函数的泛函数据分析
Functional data analysis for density functions by transformation to a Hilbert space
摘要：
非负且具有约束积分的函数数据可以被认为是一维密度函数的样本。这样的数据无处不在。由于固有的约束，密度不存在于向量空间中，因此，常用的基于希尔伯特空间的函数数据分析方法是不适用的。为了解决这个问题，我们引入了一种变换方法，通过连续且可逆的映射将概率密度映射到希尔伯特空间的函数。函数数据分析的基本方法，如构造函数的变化模式，函数回归或分类，然后通过使用该线性空间中的密度的表示来实现。通过应用从线性泛函空间到密度空间的逆映射来获得密度本身的表示。感兴趣的变换包括对数分位数密度和对数危险变换等。收敛速度推导出的表示，得到了一般类的变换下的某些结构性质。如果需要根据数据估计特定于对象的密度，则这些速率对应于密度估计的最佳收敛速率。所提出的方法说明通过模拟和脑成像中的应用。
英文摘要：
Functional data that are nonnegative and have a constrained integral can be considered as samples of one-dimensional density functions. Such data are ubiquitous. Due to the inherent constraints, densities do not live in a vector space and, therefore, commonly used Hilbert space based methods of functional data analysis are not applicable. To address this problem, we introduce a transformation approach, mapping probability densities to a Hilbert space of functions through a continuous and invertible map. Basic methods of functional data analysis, such as the construction of functional modes of variation, functional regression or classification, are then implemented by using representations of the densities in this linear space. Representations of the densities themselves are obtained by applying the inverse map from the linear functional space to the density space. Transformations of interest include log quantile density and log hazard transformations, among others. Rates of convergence are derived for the representations that are obtained for a general class of transformations under certain structural properties. If the subject-specific densities need to be estimated from data, these rates correspond to the optimal rates of convergence for density estimation. The proposed methods are illustrated through simulations and applications in brain imaging.
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