Anisotropic Periodic Tikhonov Reconstruction for

Structured Gaps in Hourly Urban Demand

Agnes Osagie1,* Mohammad Abobala2

1 Cape Peninsula University of Technology, Faculty of Applied Science, South Africa

2 Department of Mathematics, Faculty of Science, Tishreen University, Syria

Received: May 12, 2025 Revised: August 10, 2025 Accepted: October 07, 2025 ⋆ Corresponding author

ABSTRACT

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.

Keywords: Tikhonov regularization Inverse problems Fourier basis Periodic reconstruction Missing time-series

data Urban mobility Structured gaps

1. OBSERVATION GAPS AS AN INVERSE PROBLEM

Hourly demand curves are not smooth in a single sense. They

contain a strong within-day cycle, a weaker weekly component,

slow seasonal movement, and local changes induced

by temperature, precipitation, holidays, or service availability.

When observations disappear in a contiguous interval, a

reconstruction procedure must preserve these scales without

allowing the least-observed frequencies to dominate the fit.

This is different from ordinary one-step forecasting: values

before and after a gap may both be observed, and the task is

to recover the missing segment from an incomplete annual

field.

Recent bike-sharing work has concentrated mainly on predictive

architectures. Weather-based rule systems remain

useful as interpretable baselines [1], while data-mining models

have been used for trip-duration and demand analysis [2].

System configuration also affects observed demand and op-