Adaptive kernel regression for constrained route alignment

Route alignment design in surveying and transportation engineering frequently involves fixed waypoint constraints, where a path is required to pass through specific coordinates. While existing literature primarily relies on geometric optimization or control-theoretic spline frameworks, there is a lack of systematic statistical modeling approaches that balance global smoothness with exact point adherence. This thesis proposes an adaptive Nadaraya--Watson (ANW) kernel regression estimator designed to address the fixed waypoint problem. By incorporating waypoint-specific weight tuning parameters, the ANW estimator decouples global smoothing from local constraint satisfaction, avoiding the "jagged" artifacts common in naive local bandwidth-shrinking strategies. To further enhance estimation accuracy, we develop an iterative data sharpening algorithm that systematically reduces bias while maintaining the stability of the kernel framework. We establish the theoretical foundation for the ANW estimator by deriving its asymptotic bias and variance and proving its convergence properties under the internal constraint model. Repeated numerical studies in one- and two-dimensional trajectory settings, including simulations with externally imposed waypoint constraints, show that the proposed method provides a stable balance among global accuracy, waypoint adherence, and geometric regularity, while its data-sharpened variants can further improve RMSE at the cost of slightly weaker waypoint enforcement. Finally, we validate the practical utility of the framework through empirical applications to railway, high-speed rail, and highway route planning. In sum, this work provides a stable, theoretically grounded, and computationally efficient solution for complex, constrained alignment design problems.