Data-driven hands-on project on interpolation and curve fitting: approaches to reconstruction and hypothesis testing

This project document defines a technical framework for data reconstruction using first-principles numerical approaches. It mandates the manual assembly of governing matrices, specifically prohibiting "black-box" library functions to ensure a deep mathematical understanding of the interpolation process. Decision-making on data pre-processing, transformations, scaling and centring, validation of hypotheses are integrated to the process. Track A (Spatial Fitting) focuses on global surface reconstruction for images or satellite gas concentrations. It requires manual construction of a Design Matrix for Bilinear or Quadratic models, solved via Normal Equations. Track B (Temporal Interpolation) applies Cubic Splines to time-series data like air quality or kinematics. This requires a manually built Tridiagonal Matrix to ensure C2 continuity across data knots. Originality lies in the requirement for manual matrix synthesis and the physical validation of engineering hypotheses using derived gradients rather than automated software outputs.