5.4 Smoothing Splines
Derivation of Equation (5.12)
Equal the derivative of Equation (5.11) as zero, we get
Put the terms related to $\theta$ on one side and the others on the other side, we get
Multiply the inverse of $N^TN+\lambda\Omega_N$ on both sides completes the derivation of Equation (5.12)
Explanations on Equation (5.17) and (5.18)
It’s a little confusing to get Equation (5.18) directly from Equation (5.17) and its original form Equation (5.11). In order to give a clear explanation, here we give the proof of the equation,
which are the different terms between Equation (5.11) and Equation (5.18).
We know that
following Equation (5.14) in the book.
From Equation (5.17), we can get
Plug the above two equation into the right side of the equation remains to be proved, we get
which completes the proof.