Cubic smoothing splines minimize a combination of Interpolation cost and Smoothness (roughness) cost:
$\qquad$ min Icost + $\lambda$ Scost
where
$\qquad$ Icost $\equiv \sum (Y_i - \mu(x_i))^2$
$\qquad$ Scost $\equiv \int \mu''(x)^2 dx$
Are there quadratic smoothing splines that minimize something like this, with a different Scost ?
When does it make sense to minimize first not second derivatives, $\int \mu'(x)^2 dx$ ?
(Added: I gather there are different kinds of quadratic splines, with knots at the $x_i$ or between them; I'll take either one.)