Introduction Regression splines (parametric Smoothing splines (nonparametric Controlling smoothness with penalization o Here,we directly solve for the function f that minimizes the following objective function,a penalized version of the least squares objective: ∑-feP+Af"o2a o The first term captures the fit to the data,while the second penalizes curvature-note that for a line,f"(u)=0 for all u o Here,A is the smoothing parameter,and it controls the tradeoff between the two terms: =0 imposes no restrictions and f will therefore interpolate the data A=oo renders curvature impossible,thereby returning us to ordinary linear regression Patrick Breheny STA 621:Nonparametric StatisticsIntroduction Regression splines (parametric) Smoothing splines (nonparametric) Controlling smoothness with penalization Here, we directly solve for the function f that minimizes the following objective function, a penalized version of the least squares objective: Xn i=1 {yi − f(xi)} 2 + λ Z {f 00(u)} 2 du The first term captures the fit to the data, while the second penalizes curvature – note that for a line, f 00(u) = 0 for all u Here, λ is the smoothing parameter, and it controls the tradeoff between the two terms: λ = 0 imposes no restrictions and f will therefore interpolate the data λ = ∞ renders curvature impossible, thereby returning us to ordinary linear regression Patrick Breheny STA 621: Nonparametric Statistics