spectral analysis

This work addresses the statistical modeling and application of spectrogram zeros of noisy signals. Specifically, it investigates the random point process formed by spectrogram zeros in the complex plane—a fundamental object in time-frequency analysis—and establishes, for the first time, a rigorous theoretical connection between these zeros and those of Gaussian analytic functions, thereby bridging time-frequency analysis, random analytic function theory, and spatial point process theory. Building upon this foundation, we develop a statistically principled model for zero-point distributions and design novel signal detection and adaptive denoising algorithms grounded in spatial statistical inference. The proposed methods enjoy strong theoretical guarantees—including consistency and asymptotic optimality—and demonstrate robustness and interpretability even at low signal-to-noise ratios. By recasting time-frequency signal processing through the lens of stochastic geometry and random zero sets, this work introduces a new paradigm for analyzing and processing nonstationary signals in the time-frequency domain.

Traditional time-series analysis has predominantly focused on time- or state-domain approaches, while frequency-domain methods have long lacked a systematic, cross-disciplinary survey. Method: This paper presents the first comprehensive review of Fourier, Laplace, and wavelet transforms in time-series analysis—covering theoretical foundations, applicability boundaries, strengths, and limitations—and evaluates their applications across finance, meteorology, molecular dynamics, and other domains. We propose a unified evaluation framework, a reproducible technical pipeline, and open-source an integrated toolchain (hosted on GitHub). Contribution/Results: The work fills a critical gap in the literature by delivering the first systematic, domain-agnostic survey of frequency-domain techniques for time-series modeling. It provides both authoritative scholarly reference and practical engineering support for cross-domain temporal modeling, enabling rigorous method selection, benchmarking, and deployment.

Traditional discrete Fourier transform (DFT) is constrained by uniform sampling and fixed-length sequences, rendering it inadequate for non-uniformly sampled, missing-data, or ultra-long signals. To address this, we propose the Extended Discrete Fourier Transform (EDFT), which formulates spectral estimation as an optimization problem minimizing the Fourier integral residual. EDFT adaptively constructs frequency-domain basis functions without requiring equispaced time-domain sampling or identical sequence lengths. Our method integrates iterative optimization, explicit Fourier integral constraints, and adaptive inverse DFT-based signal reconstruction. It enables high-resolution spectral estimation, time-domain extrapolation, missing-data imputation, and direct processing of non-uniformly sampled signals. Compared to DFT, EDFT substantially broadens the applicability of Fourier analysis while preserving theoretical rigor and computational feasibility.

Existing spatial spectral analysis methods are constrained by data types (e.g., point processes, lattice fields, irregularly sampled processes) and domain structures (limited to regular grids). To address these limitations, this paper proposes a unified multitaper spectral estimation framework. Methodologically, it introduces, for the first time, a theoretical framework coupling discrete and continuous taper windows, thereby relaxing classical Fourier-based assumptions of Cartesian domains and uniform sampling. It establishes rigorous asymptotic and finite-sample statistical foundations for partial spectral coherence estimation and significance testing. The framework integrates multitaper windowing, tapered discrete Fourier transforms, and efficient computational algorithms. Empirical validation on large-scale ecological datasets demonstrates robust estimation of cross-spectral associations among heterogeneous spatial processes—spanning point patterns, gridded fields, and irregular samples—while delivering interpretable, statistically principled inference.

To address key bottlenecks in high-dimensional frequency-domain analysis of time series—including poor scalability, difficulty handling component heterogeneity, and lack of non-asymptotic theoretical guarantees for estimating sparse spectral precision matrices (i.e., inverses of spectral density matrices)—this paper proposes the Complex Graphical Lasso (CGLASSO) and its adaptive extension (CAGLASSO). We introduce the first real-valued coordinate descent algorithm grounded in ring isomorphism, overcoming the complex-valued optimization challenges arising from the non-i.i.d. structure of the discrete Fourier transform (DFT). Moreover, we establish the first non-asymptotic error decomposition theory tailored to frequency-domain sparse estimation, rigorously characterizing both high-dimensional approximation and estimation errors. The proposed methods achieve superior statistical consistency, computational efficiency, and sparse structure recovery compared to state-of-the-art alternatives. Extensive simulations and applications to real neuroscience data empirically validate their advantages.

This study systematically investigates the frequency-domain encoding capabilities of the Chronos foundation model, addressing a critical gap in understanding how such models represent fundamental signal properties. Through controlled experiments using discrete sinusoidal signals and a lightweight online Minimum Description Length (MDL) probing framework, the work examines the existence, separability, and cross-spectral fidelity of internal frequency representations within the Chronos decoder. The research reveals, for the first time, a degradation in representation quality in high-frequency regions, thereby delineating both the strengths and limitations of Chronos’s frequency encoding mechanism. These findings offer novel insights into the interpretability of time-series foundation models and provide practical guidance for applications in signal processing and multimodal fusion.

This study addresses the challenge of effectively identifying high-frequency dynamic signals in electroencephalography (EEG) that are associated with neurological disorders. For the first time, dynamic mode decomposition (DMD) is applied to analyze high-frequency EEG components. By extracting stable and consistent high-frequency dynamic features from neurologically relevant channels, the authors construct discriminative feature representations through a pipeline integrating principal component analysis (PCA), statistical testing based on random distribution assumptions, and machine learning–based classification. Experimental results reveal significant and stable high-frequency dynamic patterns in approximately 70% of the samples, which successfully differentiate individuals with alcohol dependence from healthy controls, thereby demonstrating the method’s effectiveness and novelty.

This study addresses a fundamental challenge in system identification: distinguishing spurious eigenvalues arising from limited data from those genuinely reflecting the underlying system dynamics. To this end, the paper introduces—for the first time—the probabilistic sampling pseudospectrum \( P(\lambda) \) and its computationally efficient estimator \( \hat{P}(\lambda) \). By leveraging resampling and statistical inference, this framework quantifies the uncertainty of eigenvalues across the complex plane. The proposed approach provides a general and rigorous statistical criterion for data-driven methods such as Dynamic Mode Decomposition and subspace identification, substantially enhancing the reliability of identifying true dynamical modes from noisy, finite-length observations.

This study addresses the problem of quantifying the goodness-of-fit of moving average MA(q) models to the spectral density of stationary processes by proposing a spectral-domain coefficient of determination. This coefficient extends, for the first time, the classical notion of the coefficient of determination into the framework of spectral analysis to measure how closely an MA(q) model approximates the true spectral density. Constructed via periodogram-based estimation, the proposed coefficient is shown to possess asymptotic normality under rigorous derivation, enabling the development of both a model order selection criterion and a goodness-of-fit test specifically tailored for MA(q) models. The approach adaptively identifies the minimal order q that achieves a pre-specified accuracy level, offering a method that is theoretically sound and practically useful.

This study addresses the challenge of identifying an unknown number of periodic components in functional time series by proposing a novel information criterion with theoretical consistency guarantees. The method integrates least squares fitting with residual process analysis and employs an iterative strategy to adaptively estimate the number of periodicities, making it applicable to a broad class of functional time series models. Extensive numerical simulations demonstrate that the proposed criterion performs exceptionally well in finite samples. Furthermore, its practical utility and effectiveness are corroborated through real-data applications to temperature and sunspot records, where it successfully uncovers statistically significant periodic structures.