We derive an estimator of the spectral density of a functional time series that is the output of a multilayer perceptron neural network. The estimator is motivated by difficulties with the computation of existing spectral density estimators for time series of functions defined on very large grids that arise, for example, in climate compute models and medical scans. Existing estimators use autocovariance kernels represented as large $G \times G$ matrices, where $G$ is the number of grid points on which the functions are evaluated. In many recent applications, functions are defined on 2D and 3D domains, and $G$ can be of the order $G \sim 10^5$, making the evaluation of the autocovariance kernels computationally intensive or even impossible. We use the theory of spectral functional principal components to derive our deep learning estimator and prove that it is a universal approximator to the spectral density under general assumptions. Our estimator can be trained without computing the autocovariance kernels and it can be parallelized to provide the estimates much faster than existing approaches. We validate its performance by simulations and an application to fMRI images.

翻译：我们推导了一种用于多层感知机神经网络输出的函数时间序列谱密度估计器。该估计器的提出源于现有谱密度估计方法在计算定义于超大网格上的函数时间序列时遇到的困难，这类问题常见于气候计算模型和医学扫描等领域。现有估计器使用以大型$G \times G$矩阵表示的自协方差核，其中$G$是函数评估所用的网格点数。在近年许多应用中，函数定义于二维和三维空间域，$G$可达$G \sim 10^5$量级，导致自协方差核的计算在计算上极为密集甚至不可行。我们运用谱函数主成分理论推导出该深度学习估计器，并证明其在一般假设条件下是谱密度的通用逼近器。该估计器无需计算自协方差核即可训练，且可通过并行化处理显著加快估计速度。我们通过仿真实验及功能磁共振成像应用验证了其性能。