NONLINEAR SUFFICIENT DIMENSION REDUCTION FOR FUNCTIONAL DATA
NONLINEAR SUFFICIENT DIMENSION REDUCTION FOR FUNCTIONAL DATA
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DOI：
10.1214/16-aos1475
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发表时间：
2017-06-01
影响因子：
4.5
通讯作者：
Song, Jun
中科院分区：
文献类型：
--
作者：
Li, Bing;Song, Jun
我们提出了一个一般的理论和估计程序的非线性充分降维的预测和响应都可以是随机函数。反应和预测因子之间的关系可以是任意的，并且观察到的时间点的集合可以因受试者而异。该问题的功能和非线性性质导致两个功能空间的建设：第一个代表的功能数据，假设是一个希尔伯特空间，第二个表征非线性，假设是一个再生核希尔伯特空间。我们的构造的一个特别吸引人的特征是，两个空间是嵌套的，在这个意义上，第二个空间的核是由第一个空间的内积决定的。我们提出了两个估计这个一般的降维问题，并建立了其中之一的一致性和收敛速度。这些渐近结果是足够灵活的，以适应完全和部分观察到的功能数据。我们通过仿真研究了我们的估计器的性能，并将其应用于语音识别和手写符号的数据集。
We propose a general theory and the estimation procedures for nonlinear sufficient dimension reduction where both the predictor and the response may be random functions. The relation between the response and predictor can be arbitrary and the sets of observed time points can vary from subject to subject. The functional and nonlinear nature of the problem leads to construction of two functional spaces: the first representing the functional data, assumed to be a Hilbert space, and the second characterizing nonlinearity, assumed to be a reproducing kernel Hilbert space. A particularly attractive feature of our construction is that the two spaces are nested, in the sense that the kernel for the second space is determined by the inner product of the first. We propose two estimators for this general dimension reduction problem, and establish the consistency and convergence rate for one of them. These asymptotic results are flexible enough to accommodate both fully and partially observed functional data. We investigate the performances of our estimators by simulations, and applied them to data sets about speech recognition and handwritten symbols.