This paper develops a framework for the estimation of the functional mean and the functional principal components when the functions form a random field. More specifically, the data we study consist of curves $X(\mathbf{s}_k;t),t\in[0,T]$, observed at spatial points $\mathbf{s}_1,\mathbf{s}_2,\ldots,\mathbf{s}_N$. We establish conditions for the sample average (in space) of the $X(\mathbf{s}_k)$ to be a consistent estimator of the population mean function, and for the usual empirical covariance operator to be a consistent estimator of the population covariance operator. These conditions involve an interplay of the assumptions on an appropriately defined dependence between the functions $X(\mathbf{s}_k)$ and the assumptions on the spatial distribution of the points $\mathbf{s}_k$. The rates of convergence may be the same as for i.i.d. functional samples, but generally depend on the strength of dependence and appropriately quantified distances between the points $\mathbf{s}_k$. We also formulate conditions for the lack of consistency.
Unified Principal Component Analysis for Sparse and Dense Functional Data under Spatial Dependency
We consider spatially dependent functional data collected under a geostatistics setting, where locations are sampled from a spatial point process. The functional response is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect. Observations on each function are made on discrete time points and contaminated with measurement errors . Under the assumption of spatial stationarity and isotropy, we propose a tensor product spline estimator for the spatio-temporal covariance function . When a coregionalization covariance structure is further assumed,…
Principal points and elliptical distributions from the multivariate setting to the functional case
随机向量$\mathbf{X}$的$k$主点被定义为使$\mathbf{X}$与集合中最近点之间的期望平方距离最小化的点的集合。Flury（1990，1993）、Tarpey（1995）和Tarpey，Li and Flury（1995）对它们进行了深入研究。对于它们的处理，检查通常限于椭圆分布族。在本文中，我们将以前的结果推广到函数椭圆分布的情况，即，当处理可分希尔伯特空间${\cal H}$上的随机元时。高斯过程的主点在Tarpey和Kinateder（2003）中定义。在本文中，我们推广的概念，主点，自洽点和椭圆分布，使他们适合在这个功能框架。结果连接自洽性和特征向量的协方差算子重新获得在这个新的设置以及一个明确的公式的$k=2$的情况下，以便包括椭圆分布的随机元在${\cal H}$。
