The L 1 Method for Robust Nonparametric Regression

Wang, Ferdinand T.; Scott, David W.

doi:10.2307/2291201

Cited by 54 publications

(65 citation statements)

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“…Up to now, this task is only rudimentarily treated, even for vertical outliers. In the context of local L 1 regression, Wang and Scott (1994) introduced a version of CV with robustness against outlying responses.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…For example, a common approach to robustification is to replace the quadratic loss function lðzÞ ¼ z 2 by functions which are less sensitive to outliers, e.g. the L 1 norm lðzÞ ¼ jzj as proposed by Wang and Scott (1994). More specifically, in the context of local constant fitting, the estimate of a function mðÁÞ at point x, given the data ðX i ; Y i Þ; i ¼ 1; .…”

Section: Introductionmentioning

confidence: 99%

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Local smoothing with robustness against outlying predictors

2004

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Outlying pollutant concentration data are frequently observed in time series studies conducted to investigate the effects of atmospheric pollution on mortality/morbidity. These outliers may severely affect the estimation procedures and even generate unexpected results like a protective effect of pollution. Although robust methods have been proposed to downweight the effect of outliers in the response variable distribution, little has been done to handle outlying explanatory variable values. We consider a robust local polynomial smoothing technique which may be useful for such purposes. It is based on downweighting points with a small design density and may also be used as a diagnostic tool to identify outliers. Using data from a study conducted in São Paulo, Brazil, we show how an unexpected form of the relative risk curve of mortality attributable to pollution by SO2 obtained via nonrobust methods may be completely reversed when the proposed technique is employed. Copyright © 2004 John Wiley & Sons, Ltd.

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Section: Discussion and Outlookmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local smoothing with robustness against outlying predictors

2004

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“…The observational noise is assumed to be rough and the number of subsequent spikes to be small as compared to the durations between the shifts. Fan and Hall (1994) and Wang and Scott (1994) propose local constant weighted L 1 -estimatesĝ(x) based on the minimization (1),…”

Section: Weighted Median Smoothing and Filteringmentioning

confidence: 99%

Weighted Repeated Median Smoothing and Filtering

Fried

Einbeck

Gather

2007

Journal of the American Statistical Association

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We propose weighted repeated median filters and smoothers for robust non-parametric regression in general and for robust signal extraction from time series in particular. The proposed methods allow to remove outlying sequences and to preserve discontinuities (shifts) in the underlying regression function (the signal) in the presence of local linear trends. Suitable weighting of the observations according to their distances in the design space reduces the bias arising from non-linearities. It also allows to improve the efficiency of (unweighted) repeated median filters using larger bandwidths, keeping their properties for distinguishing between outlier sequences and long-term shifts. Robust smoothers based on weighted L 1 -regression are included for the reason of comparison.

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“…For example, LAD regression can be adapted to the nonparametric estimation of smooth regression curves (see [17] for a general discussion of nonparametric regression estimation), by local fitting of WLAD regressions ( [18,19]). Formulations of the type in this section allowed Giloni and Simonoff ([5]) to derive the exact breakdown properties of local LAD regression at any evaluation point and to make recommendations concerning the best choice of weight function.…”

Section: Lad Regression and Breakdownmentioning

confidence: 99%

A mathematical programming approach for improving the robustness of least sum of absolute deviations regression

Giloni

Sengupta

Simonoff

2006

Naval Research Logistics

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This paper discusses a novel application of mathematical programming techniques to a regression problem. While least squares regression techniques have been used for a long time, it is known that their robustness properties are not desirable. Specifically, the estimators are known to be too sensitive to data contamination. In this paper we examine regressions based on Least-sum of Absolute Deviations (LAD) and show that the robustness of the estimator can be improved significantly through a judicious choice of weights. The problem of finding optimum weights is formulated as a nonlinear mixed integer program, which is too difficult to solve exactly in general. We demonstrate that our problem is equivalent to a mathematical program with a single functional constraint resembling the knapsack problem and then solve it for a special case. We then generalize this solution to general regression designs. Furthermore, we provide an efficient algorithm to solve the general nonlinear, mixed integer programming problem when the number of predictors is small. We show the efficacy of the weighted LAD estimator using numerical examples.

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The L 1 Method for Robust Nonparametric Regression

Cited by 54 publications

References 0 publications

Local smoothing with robustness against outlying predictors

Local smoothing with robustness against outlying predictors

Weighted Repeated Median Smoothing and Filtering

A mathematical programming approach for improving the robustness of least sum of absolute deviations regression

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