AnL ₁ smoothing spline algorithm with cross validation

Abstract

We propose an algorithm for the computation ofL ₁ (LAD) smoothing splines in the spacesW _M (D), with$[0, 1]^n \subseteq D$. We assume one is given data of the formy _i =(f(t _i ) +ε _i , i=1,...,N with {itt_i} ^{N ⊂D}_i=1 , theε _i are errors withE(ε _i )=0, andf is assumed to be inW _M . The LAD smoothing spline, for fixed smoothing parameterλ⩾0, is defined as the solution,s _λ, of the optimization problem$\min _{g \in W_M }$ (1/N)∑ ^N _i=1 ¦y _i −g(t _i ¦+λJ _M (g), whereJ _M (g) is the seminorm consisting of the sum of the squaredL ₂ norms of theMth partial derivatives ofg. Such an LAD smoothing spline,s _λ, would be expected to give robust smoothed estimates off in situations where theε _i are from a distribution with heavy tails. The solution to such a problem is a “thin plate spline” of known form. An algorithm for computings _λ is given which is based on considering a sequence of quadratic programming problems whose structure is guided by the optimality conditions for the above convex minimization problem, and which are solved readily, if a good initial point is available. The “data driven” selection of the smoothing parameter is achieved by minimizing aCV(λ) score of the form$(1/N)[\sum\nolimits_{i = 1}^N {\left| {y_i - s_\lambda (t_i )} \right| + \sum\nolimits_{res_i = 0} 1 } ]$.The combined LAD-CV smoothing spline algorithm is a continuation scheme in λ↘0 taken on the above SQPs parametrized inλ, with the optimal smoothing parameter taken to be that value ofλ at which theCV(λ) score first begins to increase. The feasibility of constructing the LAD-CV smoothing spline is illustrated by an application to a problem in environment data interpretation.