LOWLAD: a locally weightedL ₁ smoothing spline algorithm with cross validated choice of smoothing parameters

Abstract

The computation ofL ₁ 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 spacesW _m (D), with [0, 1]ⁿ⊆D. We assume data of the formy _i =f(t _i )+ε _i ,i=1,..., N with {t _i } ^N _i=1 ⊂D, the ε _i are errors withE(ε _i )=0, andf is assumed to be inW _m . An LAD smoothing spline is the solution,s _λ, of the following optimization problem

$$\mathop {\min }\limits_{g \in W_m } \frac{1}{N}\sum\limits_{i = 1}^N {\left| {y_i - g(t_i )} \right| + \lambda J_m (f),} $$

whereJ _m (g) is the seminorm consisting of the standard 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 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 onW _m , and hences _λ is assumed to be of the form$s_\lambda = \sum\nolimits_{\nu = 1}^M {d_\nu } \phi _\nu + \sum\nolimits_{i = 1}^N {c_i } \zeta _i$ where$\zeta _i (t) = R_1 (t_i ,t),R(s,t) = R_0 (s,t) + R_1 (s,t)$ is the reproducing kernel forW _m (D), R ₁ (t _i ,t)=projW ⁰ _m R(t _i ,t), and the functions {φ _v } ^M _v=1 span the Kern (proj _W ⁰ _m )=Kern(J _m ). Optimality conditions definings _λ as the solution to (1) yield an algorithm for its computation. However, this computation becomes unwieldy whenN≃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.