Least Median Of Squares Method Research Articles

MENGATASI PENCILAN PADA PEMODELAN REGRESI LINEAR BERGANDA DENGAN METODE REGRESI ROBUST PENAKSIR LMS

Ordinary Least Squares (OLS) is frequent used method for estimating parameters. OLS estimator is not a robust regression procedure for the presence of outliers, so the estimate becomes inappropriate. Least Median of Squares (LMS) is one of a robust estimator for the presence of outliers and has a high breakdown value. LMS estimate parameters by minimizing the median of squared residuals. Least Median of Squares (LMS) The purpose of this study is geting a regression equation that better than the regression equation before using OLS for the data that having outlier. For the first step, checking if there is outlier at data and then searching regression equation with LMS method. In this study used data stackloss and from estimation parameter of this data, LMS estimator showed better results compared to the OLS estimator because the regression equation from LMS method have smaller value of Mean Absolute Percentage Error (MAPE).

Multiple Harmonic Source Location Using the Least Median of Squares Method with the Presence of Outliers in High Voltage Electric Power Systems

A static multipoint method to locate harmonic sources in high-voltage electric power systems based on the least median of squares approach is presented. Because of statistical robustness, the least median of squares method is used, even in the presence of multiple erroneous measurements. The special features of the method are described in this article. IEEE-30 bus test system with three harmonic sources and uniformly distributed erroneous measurements is solved with the least median of squares, Heydt’s and Euclidean norm methods. Several harmonic sources are correctly located with the least median of squares method. The least median of squares approach estimated harmonic bus voltages and currents is close to the actual values. With Heydt’s and the Euclidean norm methods the harmonic sources cannot be located in the presence of uniformly distributed erroneous measurements.

Fast and accurate line detection with GPU-based least median of squares

We propose an accurate and efficient 2D line detection technique based on the standard Hough transform (SHT) and least median of squares (LMS). We prove our method to be very accurate and robust to noise and occlusions by comparing it with state-of-the-art line detection methods using both qualitative and quantitative experiments. LMS is known as being very robust but also as having high computation complexity. To make our method practical for real-time applications, we propose a parallel algorithm for LMS computation which is based on point-line duality. We also offer a very efficient implementation of this algorithm for GPU on CUDA architecture. Despite many years since LMS methods have first been described and the widespread use of GPU technology in computer vision and image-processing systems, we are unaware of previous work reporting the use of GPUs for LMS and line detection. We measure the computation time of our GPU-accelerated algorithm and prove it is suitable for real-time applications. Our accelerated LMS algorithm is up to 40 times faster than the fastest single-threaded CPU-based implementation of the state-of-the-art sequential algorithm.

Improved Least Median of Squares Localization for Non-Line-of-Sight Mitigation

Non-line-of-sight (NLOS) propagation is one of the major problems that cause errors in localization systems. There exist many algorithms to reduce NLOS caused errors. Most of these algorithms require ap rioriinformation about the NLOS links, which limits their application. The least median of squares (LMedS) method is an exception and is thus attractive for its simplicity. LMedS selects only a subset of the anchors with the smallest residue for location calculation, but it tends to discard more range measurements than necessary. Its performance can be improved significantly, particularly when the percentage of NLOS links is not high. In this paper, we present an improved LMedS method, which maximizes the subset of reliable anchors to be used. We show via simulation that, compared with the existing LMedS method, the proposed algorithm has significantly higher localization accuracy and is more stable. Index Terms—Least median of squares method (LMedS), local- ization, non-line-of-sight (NLOS) propagation.

Error analysis of robust optical flow estimation by least median of squares methods for the varying illumination model

The apparent pixel motion in an image sequence, called optical flow, is a useful primitive for automatic scene analysis and various other applications of computer vision. In general, however, the optical flow estimation suffers from two significant problems: the problem of illumination that varies with time and the problem of motion discontinuities induced by objects moving with respect to either other objects or with respect to the background. Various integrated approaches for solving these two problems simultaneously have been proposed. Of these, those that are based on the LMedS (Least Median of Squares) appear to be the most robust. The goal of this paper is to carry out an error analysis of two different LMedS-based approaches, one based on the standard LMedS regression and the other using a modification thereof as proposed by us recently. While it is to be expected that the estimation accuracy of any approach would decrease with increasing levels of noise, for LMedS-like methods, it is not always clear as to how much of that decrease in performance can be attributed to the fact that only a small number of randomly selected samples is used for forming temporary solutions. To answer this question, our study here includes a baseline implementation in which all of the image data is used for forming motion estimates. We then compare the estimation errors of the two LMedS-based methods with the baseline implementation. Our error analysis demonstrates that, for the case of Gaussian noise, our modified LMedS approach yields better estimates at moderate levels of noise, but is outperformed by the standard LMedS method as the level of noise increases. For the case of salt-and-pepper noise, the modified LMedS method consistently performs better than the standard LMedS method.

Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness

The findings of this paper are summarized as follows: (1) We propose a sign-constrained robust estimation method, which can tolerate 50% of data contamination and meanwhile achieve high, least-squares-comparable efficiency. Since the objective function is identical with least squares, the method may also be called sign-constrained robust least squares. An iterative version of the method has been implemented and shown to be capable of resisting against more than 50% of contamination. As a by-product, a robust estimate of scale parameter can also be obtained. Unlike the least median of squares method and repeated medians, which use a least possible number of data to derive the solution, the sign-constrained robust least squares method attempts to employ a maximum possible number of good data to derive the robust solution, and thus will not be affected by partial near multi-collinearity among part of the data or if some of the data are clustered together; (2) although M-estimates have been reported to have a breakdown point of 1/(t+1), we have shown that the weights of observations can readily deteriorate such results and bring the breakdown point of M-estimates of Huber’s type to zero. The same zero breakdown point of the L 1-norm method is also derived, again due to the weights of observations; (3) by assuming a prior distribution for the signs of outliers, we have developed the concept of subjective breakdown point, which may be thought of as an extension of stochastic breakdown by Donoho and Huber but can be important in explaining real-life problems in Earth Sciences and image reconstruction; and finally, (4) We have shown that the least median of squares method can still break down with a single outlier, even if no highly concentrated good data nor highly concentrated outliers exist.

The asymptotic variance–covariance matrix, Baarda test and the reliability of L1-norm estimates

This paper derives the complete Bahadur-type linear representation of the basic vector including the residual vector and the adjusted vector of the observations for the L1-norm estimation. The asymptotic variance–covariance matrix of the basic vector is obtained accordingly. The L1-Baarda test statistic is successfully obtained from the residual and its variance. The reliability, understood as the ability to detect as well as resist outliers, of the L1-norm estimate is discussed for the L1-Baarda test statistic, as compared with the reliability of the L2-norm estimate; it is less affected by the variation of the redundancy component. According to the relationship between residuals and true errors for the L1-norm estimate, the outlier is almost completely projected onto the corresponding residual. This is why the L1-Baarda test seems to locate outliers more correctly. For the L2-norm estimates, the outlier is partly projected onto the corresponding residual at the proportion of the corresponding redundancy component, which may easily result in an error of the third kind - the mis-location of the outlier through the L2-Baarda test. Besides, the robustness of the L1–norm estimate is no longer guaranteed if there exist leverage points in observations. This paper suggests the modified L1-norm estimate for robust estimation of leverage points, and derives its Bahadur-type representation and the corresponding variance–covariance matrix. In numerical examples, the L1-norm estimate and modified L1–norm estimate are compared with the L2–norm estimate, Biber-estimate, Koch’s method and the least median of squares method. The result shows that the L1-norm estimation can better identify small outliers, and that the modified L1-norm estimate is valid for robust estimation in the presence of leverage points.

Role of Fuzzy Sets in Power System State Estimation

This paper presents an enhancement to the least median of squares (LMS) estimation in the power system state estimation. A weighted least square estimation may be used in conjunction with LMS, yet LMS could face difficulties for eliminating bad data in specific cases. The proposed method, denoted as fuzzy LMS+LAV estimation, employs the applications of fuzzy sets to improve the accuracy of LMS. The proposed method applies the LMS method to determine the bounds of residuals and then applies fuzzy sets to the least absolute value (LAV) estimator to eliminate bad data. The paper shows that the proposed method produces more accurate estimate of power system states than that of LMS. The complementary advantages of fuzzy LAV and LMS estimators without lengthy computational efforts, which are targeted by this paper include: * less computational effort than the fuzzy LAV. * better accuracy than LMS. * same if not better robustness than both estimators. A qualitative comparison illustrates the superior robustness of the proposed method in the presence of leverage bad data.

Robust estimation of BRDF model parameters

The effect of the Bidirectional Reflectance Distribution Function (BRDF) is one of the most important factors in correcting the reflectance obtained from remotely sensed data. Estimation of BRDF model parameters can be deteriorated by various factors; contamination of the observations by undetected subresolution clouds or snow patches, inconsistent atmospheric correction in multiangular time series due to uncertainties in the atmospheric parameters, slight variations of the surface condition during a period of observation, for example due to soil moisture changes, diurnal effects on vegetation structure, and geolocation errors [Lucht and Roujean, 2000]. In the present paper, parameter estimation robustness is examined using Bidirectional Reflectance Factor (BRF) data measured for paddy fields in Japan. We compare both the M-estimator and the least median of squares (LMedS) methods for robust parameter estimation to the ordinary least squares method (LSM). In experiments, simulated data that were produced by adding noises to the data measured on the ground surface were used. Experimental results demonstrate that if a robust estimation is sought, the LMedS method can be adopted for the robust estimation of a BRDF model parameter.

Robust linear regression taking into account errors in the predictor and response variables.

We developed a robust regression technique that is a generalization of the least median of squares (LMS) technique to the field in which the errors in both the predictor and the response variables are taken into account. This simple generalization is limited in the sense that the resulting straight line is found by using only two points from the initial data set. In this way a simulation step is added by using the Monte Carlo method to generate the best robust regression line. We call this new technique 'bivariate least median of squares' (BLMS), following the notation of the LMS method. We checked the robustness of the new regression technique by calculating its breakdown point, which was 50%. This confirms the robustness of the BLMS regression line. In order to show its applicability to the chemical field we tested it on simulated data sets and real data sets with outliers. The BLMS robust regression line was not affected by many types of outlying points in the data sets.

Frequency analysis of low flows at gaged points of the Ansung stream

The Power and SMEMAX transformation, Gumbel type III, Weibull, log-Pearson, type III, lognormal probability papers based on plotting position formula are considered to estimate 7-day and 30-day low flows for return periods at four gaged points of the Ansung stream in Korea. It is found that the Power transformation is the superior ability to the SMEMAX transformation. A trial is also performed to examine the applicability of the Power transformation by changing the parameters. The Kolmogorov-Smirnov, test (K-S test) and the Chi-square test (x 2 test) are adopted in the trial There may exist a range of acceptable parameters, which implies the high applicability of the Power transformation. The method of moments or the maximum likelihood procedure is adopted to calculate the parameters of the Gumbel type III, the Weibull and the log-Pearson type III distribution. The K-S test is performed to test the goodness of fit for each distribution. The lognormal distribution with the Weibull plotting position consistently performs well and the data separation technique based on the least median of squares (LMS) method improves the coefficients of skewness in the lognormal distribution. The LMS method, easily detects the outliers of the sample and contributes to improve the properties of the normal distribution. The Power transformation, the log-Pearson type III and the lognormal distribution provide good fit to low flows for the Ansung stream. The LMS method may enhance the reliability of low flow analysis. It is also found that the flows at the Ansung stream have been significantly influenced by controlled releases from the upper reservoirs.

Recursive Estimation of Motion Parameters

A robust algroithm that estimates the motion parameters recursively from a sequence of noisy images is presented here. We propose the use of the least median of squares method in conjunction with a computationally efficient recursive scheme. The method works well even when nearly half of the features have been matched very poorly. A recursive constrained least squares method is developed while dealing with a range or stereo data sequence and a recursive total least squares method is proposed for the monocular data sequence.

Concepts and Suggestions for Robust Regression Analysis

Theory: Robust methods for regression yield parameter estimates that are insensitive to small departures in the data from the assumed model. A review of some basic ideas of robust estimation focuses on a class of techniques called M-estimators that discount the impact of outlying observations. Methods: These ideas are extended to three practically important areas: (1) some simple methods for inference for robust estimators are described; (2) a more general class of robust estimators for generalized linear models is then introduced; (3) the high breakdown least median of squares method is presented. Results: Applications from comparative and American politics illustrate ideas in these areas.

Optical fiber geometry by gray-scale analysis with robust regression

We have used least-median-of-squares (LMS) regression to analyze gray-scale images of optical fiber ends. This regression is a form of robust regression that ignores outlying data points. We fitted ellipses to the images of each of two fiber ends by using LMS and least-sum-of-squares regression. The two methods yielded nearly identical results on a pristine fiber end, but the LMS method was far superior on a damaged fiber end, even though we made no effort to filter the outlying data points.

Comparison of Model Misspecification Diagnostics Using Residuals from Least Mean of Squares and Least Median of Squares Fits

Abstract This article explores model misspecification diagnostics based on least squares and least median of squares fits. It shows that in some circumstances, least median of squares methods (or any other estimator with the exact fit property) fail to reveal an incorrectly specified mean function, but least squares methods succeed.

Comparison of robust regression methods based on least-median and adaptive kalman filtering approaches applied to linear calibration data

Five algorithms for data analysis are evaluated for their abilities to discriminate against outliers in small data sets (4–10 points). These methods included least-squares regression, the least absolute -deviation method, the least median of squares method, and two techniques based on an adaptive Kalman filter. For data sets consisting of 4–9 points with one outlier, the average errors in the estimation of the slope were found to be 18.9 % by least-squares, 17.7% by the least absolute deviation method, 0.5% by the least median of squares algorithm, 9.1% by an adaptive Kalman filter algorithm, and 0.9% by a zero-lag adaptive Kalman filter algorithm. Based on these results, the conclusion is that the zero-lag adaptive Kalman filter and the least median of squares approaches are best suited for the detection of outliers in small calibration data sets.