Volume 5 • Issue 2 • PP: 09–14 • 2025
Anisotropic Periodic Tikhonov Reconstruction for Structured Gaps in Hourly Urban Demand
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© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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Contiguous observation gaps are difficult to reconstruct when a count process is simultaneously periodic, weathersensitive, and subject to abrupt operating-state changes. A single isotropic penalty treats these components as if they had the same spectral roughness; weak regularization then leaves high-frequency coefficients unstable, whereas strong regularization suppresses legitimate diurnal peaks. This paper formulates hourly reconstruction as an anisotropic periodic inverse problem. The response is represented on a square-root scale by exogenous covariates, nested daily, weekly, and annual Fourier systems, and modulated daily harmonics. A block-diagonal Tikhonov operator assigns separate regularization levels to these mechanisms and weights the kth periodic mode by k4, the Fourier form of a squared-curvature penalty. The estimator is available in closed form. Conditions for uniqueness, a perturbation bound for reconstructed gaps, a generalized spectral-filter representation, and monotonicity of the effective dimension are derived. Numerical reconstruction on one year of hourly bicycle demand was evaluated under independent missingness, eight-hour blocks, and complete-day outages using 24 repeated masks per regime. The proposed estimator reduced RMSE relative to a periodic-profile reconstruction by 29.75%, 26.85%, and 29.24%, respectively. Against an isotropic ridge model using the same 139-dimensional basis, the anisotropic model was slightly better for random and full-day gaps, statistically indistinguishable for eight-hour blocks, and attained an effective dimension of 113.57, illustrating that the principal gain is controlled multiscale regularization rather than an increase in model size.
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