Enhancing Measurement Accuracy: The Impact of Missing Data on Parameter Estimation in Mass–Spring–Damper Systems

In sensor-driven dynamic systems, missing data can severely degrade parameter estimation accuracy. This paper investigates the impact of missing data on phase estimation in a mass–spring–damper system using an information-theoretic framework based on the Cramér–Rao Lower Bound (CRLB). Closed-form CRLB expressions are derived for four scenarios: complete data, Missing Completely At Random (MCAR) deletion, MCAR-based imputation, and Missing At Random (MAR) missingness via a selection-weighted formulation. These bounds are used as theoretical benchmarks to evaluate classical imputation methods (Last Observation Carried Forward (LOCF), linear interpolation) and advanced approaches (Kalman filtering, Rauch-Tung-Striebel (RTS) smoothing, Bayesian inference, and transformer-based imputation) through Monte Carlo simulations. Empirical results show that classical imputation introduces substantial bias and variance, while advanced methods improve stability but do not outperform direct estimation from available samples nor approach the complete-data or MCAR CRLBs, even under realistic model-parameter mismatch. Normality tests confirm approximately Gaussian estimator behavior, supporting the validity of the CRLB-based analysis. Under MAR conditions, estimation from observed samples consistently yields the lowest variance, highlighting the importance of explicitly accounting for the missingness mechanism in parameter estimation.