17 a perfect smoother
Source: Eilers (2003)
Noisy series \(y\) of length \(m\).
The smoothed series is called \(z\).
We have conflicting interests:
- we want a \(z\) series “as smooth as possible”.
- however, the smoother \(z\) is, the farthest from \(y\) it will be (low fidelity).
Roughness:
\[ R = \displaystyle\sum_i (z_i - z_{i-1})^2 \]
Fit to data:
\[ S = \displaystyle\sum_i (y_i - z_i)^2 \]
Cost functional to be minimized:
\[ Q = S + \lambda R \]