LOWLAD: a locally weighted L₁ smoothing spline algorithm with cross validated choice of smoothing parameters

The computation of L₁ smoothing splines on large data sets is often desirable, but computationally infeasible. A locally weighted, LAD smoothing spline based smoother is suggested, and preliminary results will be discussed. Specifically, one can seek smoothing splines in the spaces W_m(D), with [0, 1]ⁿ⊆D. We assume data of the form y_i=f(t_i)+ε_i, i=1,..., N with {t_i}_i=1^N⊂D, the ε_i are errors with E(ε_i)=0, and f is assumed to be in W_m. An LAD smoothing spline is the solution, s_λ, of the following optimization problem {Mathematical expression} where J_m(g) is the seminorm consisting of the standard sum of the squared L₂ norms of the mth partial derivatives of g. Such an LAD smoothing spline, s_λ, would be expected to give robust smoothed estimates of f in situations where the ε_i are from a distribution with heavy tails. For fixed λ>0, the solution to such a problem is known to be a thin plate spline on W_m, and hence s_λ is assumed to be of the form {Mathematical expression} where {Mathematical expression} is the reproducing kernel for W_m(D), R₁(t_i, t)=proj W_m⁰R(t_i, t), and the functions {φ_v}_v=1^M span the Kern (proj_Wm⁰)=Kern(J_m). Optimality conditions defining s_λ as the solution to (1) yield an algorithm for its computation. However, this computation becomes unwieldy when N≃O(10³). A possible remedy is to solve "local" problems of the form of (1), on neighborhoods of "size"b, and to blend these locally optimal LAD splines together producing a globally smooth estimator. Two smoothing parameters (the global value of "λ", and the "local neighborhood" size "b") should preferably have data driven, cross validated, choice.