Abstract
Smoothing noisy data is commonly encountered in engineering domain, and currently robust penalized regression spline models are perceived to be the most promising methods for coping with this issue, due to their flexibilities in capturing the nonlinear trends in the data and effectively alleviating the disturbance from the outliers. Against such a background, this paper conducts a thoroughly comparative analysis of two popular robust smoothing techniques, theM-type estimator andS-estimation for penalized regression splines, both of which are reelaborated starting from their origins, with their derivation process reformulated and the corresponding algorithms reorganized under a unified framework. Performances of these two estimators are thoroughly evaluated from the aspects of fitting accuracy, robustness, and execution time upon the MATLAB platform. Elaborately comparative experiments demonstrate that robust penalized spline smoothing methods possess the capability of resistance to the noise effect compared with the nonrobust penalized LS spline regression method. Furthermore, theM-estimator exerts stable performance only for the observations with moderate perturbation error, whereas theS-estimator behaves fairly well even for heavily contaminated observations, but consuming more execution time. These findings can be served as guidance to the selection of appropriate approach for smoothing the noisy data.
Highlights
Penalized spline smoothing technique has already been perceived as a popular nonparametric smoothing approach for smoothing noisy data in engineering domain [1,2,3] during the past 20 years due to its ease of fitting, flexible choice of the inner knots, and smoothing parameter
This paper conducts a thoroughly comparative analysis of two popular robust smoothing techniques, the M-type estimator and S-estimation for penalized regression splines, both of which are reelaborated starting from their origins, with their derivation process reformulated and the corresponding algorithms reorganized under a unified framework
And Rice [5] and Besse et al [6] approximated the smoothing splines by the hybrid splines equipped with the inner knots equal to the data samples and a penalty parameter for determining the smoothing amount of regression curve, which was usually calculated according to the cross-validation criterion
Summary
Penalized spline smoothing technique has already been perceived as a popular nonparametric smoothing approach for smoothing noisy data in engineering domain [1,2,3] during the past 20 years due to its ease of fitting, flexible choice of the inner knots, and smoothing parameter. Research regarding the M-type smoothing technique started from Huber et al [21] and Cox’s [22] work on cubic regression splines Another robust model can be the Sestimation [23] which possesses high-breakdown point and minimizes the residual scale in an extremely robust manner. By replacing the least squares estimation with a suitable Sestimator, Tharmaratnam et al [29] put forward an Sestimation for penalized regression spline as well, which proves to be equivalent to a weighted penalized least squares regression and behaves well even for heavily contaminated observations Against such a background, this paper performs a thoroughly comparative analysis mainly based on these two popular robust smoothing techniques, the M-type estimator and S-estimation for penalized regression splines, both of which are reelaborated starting from their origins, with their derivation process reformulated and the corresponding algorithms reorganized under a unified framework. Concluding remarks are included in the end of Section 4
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