Deep learning estimation of the spectral density of functional time series on large domains

TL;DR

This paper introduces a deep learning-based estimator for the spectral density of functional time series on large domains, overcoming computational challenges of traditional methods in high-dimensional settings.

Contribution

The paper develops a novel neural network estimator for spectral density that avoids autocovariance kernel computation and is scalable to large grid functions.

Findings

Performs well in simulations

Faster estimation compared to traditional methods

Successfully applied to fMRI data

Abstract

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.