• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.233-241
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• 2006

Classical principal component analysis (PCA) can be formulated as finding the linear subspace that best accommodates multidimensional data points in the sense that the sum of squared residual distances is minimized. As alternatives to such LS (least squares) fitting approach, we produce LMS (least median of squares) and LTS (least trimmed squares)-type PCA by minimizing the median of squared residual distances and the trimmed sum of squares, in a similar fashion to Rousseeuw (1984)'s alternative approaches to LS linear regression. Proposed methods adopt the data-driven optimization algorithm of Croux and Ruiz-Gazen (1996, 2005) that is conceptually simple and computationally practical. Numerical examples are given.

• Wind and Structures
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• v.26 no.6
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• pp.383-395
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• 2018

An accurate determination of wind speed distribution is the basis for an evaluation of the wind energy potential required to design a wind turbine, so it is important to estimate unknown parameters of wind speed distribution. In this paper, Gumbel distribution is used in modelling wind speed data, and alternative robust estimation methods to estimate its parameters are considered. The methodologies used to obtain the estimators of the parameters are least absolute deviation, weighted least absolute deviation, median/MAD and least median of squares. The performances of the estimators are compared with traditional estimation methods (i.e., maximum likelihood and least squares) according to bias, mean square deviation and total mean square deviation criteria using a Monte-Carlo simulation study for the data with and without outliers. The simulation results show that least median of squares and median/MAD estimators are more efficient than others for data with outliers in many cases. However, median/MAD estimator is not consistent for location parameter of Gumbel distribution in all cases. In real data application, it is firstly demonstrated that Gumbel distribution fits the daily mean wind speed data well and is also better one to model the data than Weibull distribution with respect to the root mean square error and coefficient of determination criteria. Next, the wind data modified by outliers is analysed to show the performance of the proposed estimators by using numerical and graphical methods.

• Communications for Statistical Applications and Methods
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• v.25 no.6
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• pp.673-685
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• 2018

Procrustes analysis is a useful technique useful to measure, compare shape differences and estimate a mean shape for objects; however it is based on a least squares criterion and is affected by some outliers. Therefore, we propose two generalized Procrustes analysis methods based on M-estimation and least median of squares estimation that are resistant to object outliers. In addition, two algorithms are given for practical implementation. A simulation study and some examples are used to examine and compared the performances of the algorithms with the least square method. Moreover since these resistant GPA methods are available for higher dimensions, we need some methods to visualize the objects and mean shape effectively. Also since we have concentrated on resistant fitting methods without considering shape distributions, we wish to shape analysis not be sensitive to particular model.

• Journal of the Korean Data and Information Science Society
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• v.13 no.2
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• pp.271-284
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• 2002

In this paper we consider the robust inference for the parameter of linear regression model based on weighted least squares. First we consider the sequential test of multiple outliers. Next we suggest the way to assign a weight to each observation $(x_i,\;y_i)$ and recommend the robust inference for linear model. Finally, to check the performance of confidence interval for the slope using proposed method, we conducted a Monte Carlo simulation and presented some numerical results and examples.

• Communications for Statistical Applications and Methods
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• v.3 no.2
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• pp.299-307
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• 1996

This article is concerned with the algorithms for the least median of squares estimator. An algorithm based on the $L{\infty}$ .inf.-estimation procedure is proposed in an attempt to improve the optimality of the estimate. And it is shown that the proposed algorithm yields more optimal estimate than the traditional resampling algorithms. The proposed algorithm employs a linear scaling transformation at each iteration of the$L{\infty}$-algorithm to deal with its computational inefficiency problem.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.903-908
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• 2009

The shrink parameter in ridge regression may be contaminated by outlying points. We propose robust cross validation scores in ridge regression instead of classical cross validation. We use robust location estimators such as median, least trimmed squares, absolute mean for robust cross validation scores. The robust scores have global robustness. Simulations are performed to show the effectiveness of the proposed estimators.

• Journal of Korea Water Resources Association
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• v.31 no.6
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• pp.675-684
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• 1998

This paper presents the empirical formulas for determining the design-width for medium rivers in the Han river basin. The design flood, the watershed ares, and the channel slope of 216 medium rivers in the Han river basin are collected. the design width formulas are then determined by 1) the least squares (LS) method, 2)the least median squares (LMS) method, and 3) the reweighted least squares method based on the LMS (RLS). The six types of formulas are considered to determine the acceptable type for medium streams in the Han river basin. The root mean squared errors (RMSE), the absolute mean (AME) errors, and the mean errors (ME) are computed to test the formulas derived by three regression methods. It si found that the equation related stream width to the watershed area and the channel slope is acceptable for determining the design width for medium streams in the Han river basin. It is expected that the equations proposed by this study be used an index for determining the design-width for medium streams in the Han river basin.

• Communications for Statistical Applications and Methods
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• v.28 no.1
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• pp.81-88
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• 2021

k-least quantile of squares (k-LQS) estimates are a generalization of least median of squares (LMS) estimates. They have not been used as much as LMS because their breakdown points become small as k increases. But if the size of outliers is assumed to be fixed LQS estimates yield a good fit to the majority of data and residuals calculated from LQS estimates can be a reliable tool to detect outliers. We propose to use LQS estimates for separating a clean set from the data in the context of outlyingness of the cases. Three procedures are suggested for the identification of outliers using LQS estimates. Examples are provided to illustrate the methods. A Monte Carlo study show that proposed methods are effective.

• Journal of the Korean Statistical Society
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• v.10
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• pp.105-121
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• 1981

The problem of selection of a subset containing the largest of several slope parameters of regression equations is considered. The proposed selection procedure is based on the weighted median estimators for regression parameters and the median of rescaled absolute residuals for scale parameters. Those estimators are compared with the classical least squares estimators by a simulation study. A Monte Carlo comparison is also made between the new procedure based on the weighted median estiamtors and the procedure based on the least squares estimators. The results show that the proposed procedure is quite robust with respect to the heaviness of distribution tails.

• Journal of Korean Society of Water and Wastewater
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• v.12 no.3
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• pp.99-106
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• 1998

This paper presents a procedure for determining the design rainfall depth and the design rainfall intensity at Incheon city area in Korea. In this study the eight probability distributions are considered to estimate the probable rainfall depths for 11 different durations. The Kolmogorov - Smirnov test and the Chi-square test are adopted to test each distribution. The probable rainfall intensity formulas are then determined by i) the least squares (LS) method, ii) the least median squares (LMS) method, iii) the reweighted least squares method based on the LMS (RLS), and iv) the constrained regression (CR) model. The Talbot, the Sherman, the Japanese, and the Unified type are considered to determine the best type for the Incheon station. The root mean squared (RMS) errors are computed to test the formulas derived by four methods. It is found that the Unified type is the most reliable and that all methods presented herein are acceptable for determining the coefficients of rainfall intensity formulas from an engineering point of view.