Estimating Instantaneous Traffic Flow from Event-Time Detectors: An Adaptive Kernel-Intensity Observer for Emerging Transport Systems

Estimating instantaneous traffic flow $q(t)$ from discrete event times is a core estimation problem in emerging transport systems, where widely deployed point detectors must operate alongside connected, automated, and data-driven control components. Common approaches, especially the reciprocal of headway ($q=1/h$), can systematically overestimate flow under stationary renewal assumptions because of Jensen's Inequality, and the issue becomes pronounced when traffic is highly variable ($\kappa \le 1$). This paper places several smoothing approaches within a kernel intensity estimation view, and shows that causal exponential smoothing can be interpreted as a steady-state scalar Kalman filter. Building on this link, we develop a second-order observer with an adaptive bandwidth by using a relative process-noise model ($Q \propto q^2$) together with Poisson-like measurement noise ($R \propto q$). The resulting observer responds quickly when flow is high and applies stronger smoothing when observations are sparse. Controlled simulation studies with known intensity profiles validate the bias analysis and demonstrate improved tracking under nonstationary conditions. For event-only field deployment, where instantaneous-flow ground truth is not directly observable, we also formalize a validation framework based on predictive consistency, calibration, and operational indicators. This validation framework supports real-time use under partial observability and provides observable predictive diagnostics for monitoring, adaptive control, and traffic management analytics.