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

TL;DR

A reproducible comparison of cubic and radial basis function (RBF) interpolants for multivariate surface analysis is presented, finding that noisy or apparently inconsistent measurements in thermodynamic process systems should not be discarded by default but can be structured and interpolated to recover physically meaningful process behavior.

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.

Figures (4)

• Figure 1: End-to-end workflow of the unified interpolation evaluation pipeline, from controlled data construction to uncertainty-aware performance analysis and reproducible artifact generation.
• Figure 2: Distribution of RMSE values across repeated slice-wise train/test splits for cubic and RBF interpolants in noise-free and noisy regimes.
• Figure 3: Representative slice-wise surface reconstructions obtained using cubic interpolation and multiquadric RBF interpolation under noise-free and noisy regimes. Panels are organized to enable direct geometric comparison of method and noise effects on the same underlying slice.
• Figure 4: Predicted-versus-true scatter plots for a representative noisy slice with failure behavior (left: cubic, right: multiquadric RBF). The red dashed diagonal denotes ideal predictions ($\hat{y}=y$). Deviations from this line visualize bias, variance, and outlier-driven instability under exact interpolation of noisy nodes.