AnL 1 smoothing spline algorithm with cross validation

We propose an algorithm for the computation ofL₁ (LAD) smoothing splines in the spacesW_M(D), with. We assume one is given data of the formy_i=(f(t_i) +_i, i=1,...,N with {itt_i}_i=1^{N D}, the_i are errors withE(_i)=0, andf is assumed to be inW_M. The LAD smoothing spline, for fixed smoothing parameter0, is defined as the solution,s, of the optimization problem (1/N)_i=1^N¦y_i–g(t_i¦+J_M(g), whereJ_M(g) is the seminorm consisting of the sum of the squaredL₂ norms of theMth partial derivatives ofg. Such an LAD smoothing spline,s, would be expected to give robust smoothed estimates off in situations where the_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 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() score of the form.The combined LAD-CV smoothing spline algorithm is a continuation scheme in 0 taken on the above SQPs parametrized in, with the optimal smoothing parameter taken to be that value of at which theCV() 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.