{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/fast-machine-precision-spectral-likelihoods","title":"Fast Machine-Precision Spectral Likelihoods for Stationary Time Series","arxiv_id":"2404.16583","date":"2024-04-25","proceeding":null,"authors":["Christopher J. Geoga"],"abstract":"We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in $\\mathcal{O}(n \\log n)$ time with applications to fitting time series models. In particular, for a symmetric Toeplitz matrix $\\mathbf{\\Sigma}$ with values $\\mathbf{\\Sigma}_{j,k} = h_{|j-k|} = \\int_{-1/2}^{1/2} e^{2 \\pi i |j-k| \\omega} S(\\omega) \\mathrm{d} \\omega$ where $S(\\omega)$ is piecewise smooth, we give an approximation $\\mathbf{\\mathcal{F}} \\mathbf{\\Sigma} \\mathbf{\\mathcal{F}}^H \\approx \\mathbf{D} + \\mathbf{U} \\mathbf{V}^H$, where $\\mathbf{\\mathcal{F}}$ is the DFT matrix, $\\mathbf{D}$ is diagonal, and the matrices $\\mathbf{U}$ and $\\mathbf{V}$ are in $\\mathbb{C}^{n \\times r}$ with $r \\ll n$. Studying these matrices in the context of time series, we offer a theoretical explanation of this structure and connect it to existing spectral-domain approximation frameworks. We then give a complete discussion of the numerical method for assembling the approximation and demonstrate its efficiency for improving Whittle-type likelihood approximations, including dramatic examples where a correction of rank $r = 2$ to the standard Whittle approximation increases the accuracy of the log-likelihood approximation from $3$ to $14$ digits for a matrix $\\mathbf{\\Sigma} \\in \\mathbb{R}^{10^5 \\times 10^5}$. The method and analysis of this work applies well beyond time series analysis, providing an algorithm for extremely accurate solutions to linear systems with a wide variety of symmetric Toeplitz matrices whose entries are generated by a piecewise smooth $S(\\omega)$. The analysis employed here largely depends on asymptotic expansions of oscillatory integrals, and also provides a new perspective on when existing spectral-domain approximation methods for Gaussian log-likelihoods can be particularly problematic.","url_abs":"https://arxiv.org/abs/2404.16583v3","url_pdf":"https://arxiv.org/pdf/2404.16583v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"fast-machine-precision-spectral-likelihoods","repo_url":"https://github.com/cgeoga/spectralestimators.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}