Papers › Fast Machine-Precision Spectral Likelihoods for Stationary Time Series
Fast Machine-Precision Spectral Likelihoods for Stationary Time Series
Christopher J. Geoga
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We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in 𝒪(n logn) time with applications to fitting time series models. In particular, for a symmetric Toeplitz matrix Σ with values Σ_(j,k) = h_(|j-k|) = ∫_(-1/2)^(1/2) e^(2 πi |j-k| ω) S(ω) d ω where S(ω) is piecewise smooth, we give an approximation ℱ Σ ℱᴴ ≈𝐃 + 𝐔 𝐕ᴴ, where ℱ is the DFT matrix, 𝐃 is diagonal, and the matrices 𝐔 and 𝐕 are in ℂ^(n ×r) with r ≪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 Σ ∈ℝ^(10⁵ ×10⁵). 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(ω). 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.
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