Comparative Analysis for Robust Penalized Spline Smoothing Methods

Algorithm 2

𝑆-estimator iterative algorithm for penalized regression splines.

Step  0. Input an initial regression coefficients 𝜷⁽⁰⁾, the termination tolerance 𝜀, and the maximum

   iteration number Iter_max. Meanwhile, set 𝑘 = 0 and enter into the following loop iterations.

Step  1. Estimate 𝜎_𝜀 based on (24), that is

        𝜎_𝜀𝜷^(𝑘)=MADN+Median𝐫𝜷^(𝑘).

Step  2. Compute the intermediate variables 𝜏(𝜷^(𝑘)) and Λ(𝜷^(𝑘)).

Step  3. Calculate the coefficient vector 𝜷^(𝑘+1) according to (30) as

    𝜷^(𝑘+1)=𝐗^𝑇Λ𝜷^(𝑘)𝜆𝐗+𝜏(𝜷^(𝑘))𝐷⁻¹𝐗^𝑇Λ𝛽^(𝑘)𝐲.

   Hereinto, the penalty parameter 𝜆 is figured out based on the regularized GCV criterion (32).

Step  4. If ‖𝜷^(𝑘+1)−𝜷^(𝑘)‖<𝜀‖𝜷^(𝑘)‖ or 𝑘 = Iter_max, terminate and output the convergent

   coefficient vector for the penalized regression spline of 𝑆-estimator 𝜷^★=𝜷^(𝑘+1); else

   accumulate the counter by setting 𝑘 = 𝑘 + 1 and repeat Step  1.