Natural Cubic Splines

This lecture covers the concept of natural cubic splines, focusing on the integration of u"(x) twice to determine the first derivatives at specific points. The lecture explains how to ensure continuity and differentiability of u(x) by writing it as a linear function outside certain intervals. It also discusses the penalty matrix and the basis matrix used in Bayesian inference and mixed effects models. The instructor emphasizes the importance of smoothness and continuity in constructing natural cubic splines, illustrating the process with matrix forms and vector spaces.