A Structure-Preserving Assessment of VBPBB for Time Series Imputation Under Periodic Trends, Noise, and Missingness Mechanisms

Incomplete time-series data compromise statistical inference, particularly when the underlying process exhibits periodic structure (e.g., annual or monthly cycles). Conventional imputation procedures rarely account for such temporal dependence, leading to attenuation of seasonal signals and biased estimates. This study proposes and evaluates a structure-preserving multiple imputation framework that augments imputation models with frequency-specific covariates derived via the Variable Bandpass Periodic Block Bootstrap (VBPBB). In controlled simulations, we generate series with annual and monthly components, impose Gaussian noise across low, moderate, and high signal-to-noise regimes, and introduce Missing Completely at Random (MCAR) patterns from 5% to 70% missingness. Dominant periodic components are extracted with VBPBB, resampled to stabilize uncertainty, and incorporated as covariates in Amelia II. Compared with baseline methods that do not model temporal structure, the VBPBB-enhanced approach consistently yields lower imputation error and superior retention of periodic features, with the largest gains observed under high noise and when multiple components are included. These findings demonstrate that explicitly modeling periodic content during imputation improves reconstruction accuracy and preserves time-series structure in the presence of substantial missingness.

💡 Research Summary

The paper addresses a critical gap in time‑series imputation: most standard methods ignore inherent periodic structure, leading to attenuation of seasonal amplitudes and distortion of phase, especially when the series contains strong annual, semi‑annual, or monthly cycles. To remedy this, the authors develop a structure‑preserving multiple‑imputation framework that integrates frequency‑specific covariates derived from the Variable Bandpass Periodic Block Bootstrap (VBPBB) into the Amelia II imputation engine.

VBPBB works by first applying a variable‑bandpass filter, based on the Kolmogorov‑Zurbenko Fourier Transform (KZFT), to isolate targeted frequency bands (1/365, 2/365, and 1/30). For each band, the method resamples the series in blocks whose length matches the period of the component, thereby preserving autocorrelation and phase relationships. One thousand bootstrap replicates are generated per component; the point‑wise median across replicates yields a stable, denoised covariate that captures the dominant seasonal signal while reducing the influence of random noise.

These covariates are then fed to Amelia II as auxiliary variables. Amelia II employs an Expectation–Maximization with Bootstrapping (EMB) algorithm to draw multiple imputations under the Missing Completely at Random (MCAR) assumption. By conditioning on the periodic covariates, the imputation model can reconstruct missing observations in a way that respects the underlying frequency structure.

The authors evaluate the approach using a comprehensive simulation study. Synthetic daily series of 6,000 points (≈16.4 years) are generated with three levels of signal complexity: (1) a single annual sinusoid, (2) annual plus its first harmonic, and (3) annual, harmonic, and a monthly component. Gaussian noise is added at five variance levels (σ² = 0.1, 4, 10, 25, 100) to span low to very high signal‑to‑noise ratios. Missingness is imposed under MCAR at rates of 5 %, 10 %, 15 %, 20 %, and 70 %. Each configuration is replicated 30 times for statistical reliability.

Performance is measured by root‑mean‑square error (RMSE), mean absolute error (MAE), and structure‑preservation diagnostics such as band‑limited power retention and seasonal amplitude/phase fidelity. Across all scenarios, the VBPBB‑enhanced method consistently outperforms the baseline Amelia II that lacks periodic covariates. RMSE reductions range from 15 % to 35 %, with the greatest gains observed in high‑noise settings (σ² = 100) and when multiple periodic components are included (scenario 3). The method also maintains a higher proportion of the original seasonal power and yields more accurate estimates of amplitude and phase, demonstrating that explicit modeling of periodic content mitigates noise‑induced attenuation.

The paper highlights the policy relevance of preserving temporal structure: in public‑health surveillance, environmental monitoring, and other domains that rely on continuous reporting, more accurate imputation translates into better trend estimation, resource allocation, and decision‑making. The authors acknowledge limitations, notably the exclusive focus on MCAR and stationary processes, and outline future work to extend the framework to Missing at Random (MAR) or Not‑Missing‑At‑Random (MNAR) mechanisms, non‑stationary series, and formal interval‑coverage assessments.

In summary, this study provides a robust, generalizable framework for periodic‑aware imputation, demonstrates its superiority through rigorous simulation, and offers practical guidance for analysts seeking to improve data quality in seasonally driven time‑series contexts.