Exact Interpolation under Noise: A Reproducible Comparison of Clough-Tocher and Multiquadric RBF Surfaces

TL;DR

This study compares cubic and RBF surface interpolants under noise, showing that cubic interpolants are more stable in noisy conditions, with all experiments fully reproducible and highlighting practical implications for environmental engineering.

Contribution

It provides a reproducible comparison of interpolation methods under noise, demonstrating the stability of cubic interpolants and emphasizing the importance of structured interpolation in noisy measurements.

Findings

Both methods perform well in noise-free settings.

Cubic interpolant is more stable under noisy conditions.

Reproducible experimental protocol established.

Abstract

This paper presents a reproducible comparison of cubic and radial basis function (RBF) interpolants for multivariate surface analysis. To eliminate evaluation bias, both methods are assessed under a unified slice-wise train/test protocol on the same synthetic function family. Performance is reported using RMSE, MAE, and $R^2$ in two regimes: (i) noise-free observations and (ii) noisy observations. In the noise-free regime, both interpolants achieve high accuracy with output-dependent advantages. In the noisy regime, exact interpolation overfits noisy nodes and degrades out-of-sample performance for both methods; in our experimental setting, the cubic interpolant is comparatively more stable. All experiments are fully reproducible through a single SciPy/NumPy-based script with a fixed random seed, repeated splits, and bootstrap-based uncertainty summaries. From an environmental engineering perspective, the main practical implication is that noisy or apparently inconsistent measurements in thermodynamic process systems should not be discarded by default; instead, they can be structured and interpolated to recover physically meaningful process behavior.