Time Series Analysis in Frequency Domain: A Survey of Open Challenges, Opportunities and Benchmarks

TL;DR

This survey contextualizes time series analysis in the frequency domain, tracing a trajectory from classical transforms (Fourier, Wavelet, Laplace) to modern spectral neural operators and hybrid physics-ML models. It introduces a unified taxonomy, standard benchmarks, and a pipeline that links preprocessing, frequency transformation, and domain-specific evaluation. The work highlights three core frontiers—preserving causal structure, quantifying uncertainty in learned spectra, and topology-aware analysis for non-Euclidean data—and identifies gaps at the intersection with geometric deep learning and quantum-enhanced spectral analysis. By consolidating over 100 studies, the paper provides practitioners with a principled guide for method selection and offers researchers a roadmap for advancing spectral time-series analysis in complex, real-world settings.

Abstract

Frequency-domain analysis has emerged as a powerful paradigm for time series analysis, offering unique advantages over traditional time-domain approaches while introducing new theoretical and practical challenges. This survey provides a comprehensive examination of spectral methods from classical Fourier analysis to modern neural operators, systematically summarizing three open challenges in current research: (1) causal structure preservation during spectral transformations, (2) uncertainty quantification in learned frequency representations, and (3) topology-aware analysis for non-Euclidean data structures. Through rigorous reviewing of over 100 studies, we develop a unified taxonomy that bridges conventional spectral techniques with cutting-edge machine learning approaches, while establishing standardized benchmarks for performance evaluation. Our work identifies key knowledge gaps in the field, particularly in geometric deep learning and quantum-enhanced spectral analysis. The survey offers practitioners a systematic framework for method selection and implementation, while charting promising directions for future research in this rapidly evolving domain.

Figures (7)

• Figure 1: Comparative visualization of spectral decomposition techniques demonstrating the fundamental trade-offs between temporal and frequency resolution
• Figure 2: The pipeline of frequency methods for time series analysis
• Figure 3: Nonstationary dynamics in football trajectory time series. The piecewise parabolic segments (Player 1: clearance, Player 2: header, Player 5: shot) demonstrate abrupt parameter shifts where velocity $v(t)$ and launch angle $\theta(t)$ change discontinuously at each player interaction ($t_1,t_2,t_3$). Traditional temporal models fail to capture these regime shifts, as the system dynamics $\frac{d\mathbf{x}}{dt} = f_t(\mathbf{x})$ vary unpredictably with player interventions.
• Figure 4: Three-dimensional visualization of time series data evolving on a curved Riemannian manifold $\mathcal{M}$. The red trajectory represents the temporal dynamics $\{x_t\}$, with the tangent space $T_x\mathcal{M}$ showing local linearization. The green ellipse depicts the nonlinear embedding $\phi:\mathcal{M}\to\mathbb{R}^d$ preserving temporal relationships as $\{z_t\}$.
• Figure 5: Taxonomy of frequency transform in time series analysis