An improved robust hierarchical registration algorithm

Network Topology Error Identification in Robust State Estimation for Power Systems

Say that a regression method is "unstable" at a data set if a small change in the data can cause a relatively large change in the fitted plane. A well-known example of this is the instability of least squares regression (LS) near (multi)collinear data sets. It is known that least absolute deviation (LAD) and least median of squares (LMS) linear regression can exhibit instability at data sets that are far from collinear. Clear-cut instability occurs at a "singularity"--a data set, arbitrarily small changes to which can substantially change the fit. For example, the collinear data sets are the singularities of LS. One way to measure the extent of instability of a regression method is to measure the size of its "singular set" (set of singularities). The dimension of the singular set is a tractable measure of its size that can be estimated without distributional assumptions or asymptotics. By applying a general theorem on the dimension of singular sets, we find that the singular sets of LAD and LMS are at least as large as that of LS and often much larger. Thus, prima facie, LAD and LMS are frequently unstable. This casts doubt on the trustworthiness of LAD and LMS as exploratory regression tools.

The incorporation of the robust regression methods Least Median Square (LMS) and Least Trimmed Squares (LTS) is proposed in structural equation modeling. Results show that, in situations of high deviations of symmetry, the evaluated methods would be recommended for applications including smaller sample sizes.

In this paper, the computation of the least median of squares (LMS) estimator for regression and the minimum volume ellipsoid estimator for multivariate location and covariance are considered, using the P3R and PAR programs from the widely available statistical package BMDP. Multivariate outliers are detected using robust distances based on the MVE estimators of the explanatory variables. The identification of leverage points in linear regression is based on the combined use of the LMS standardized residuals and the robust distances of the explanatory regression variables. FORTRAN procedures and the required BMDP input to produce the corresponding robust estimates are presented. The Hawkins-Bradu-Kass data is used as a test example.

This paper considers the robustness properties in the time series context of the least median of squares (LMS) estimator. The influence function of the LMS estimator is derived under additive outlier contamination. This influence function is redescending and bounded for fixed values of the AR parameters. The gross-error sensitivity, however, is an unbounded function of the AR parameters. In order to asses the global robustness behavior of the LMS estimator, we consider several notions of breakdown. The breakdown points of the LMS estimator depend on the value of the underlying AR parameter. Generally, the breakdown point is below one half for high values of the AR parameter. The bias curves of the LMS estimator reveal, however, that the magnitude of outliers has to be considerable in order to cause breakdown.

Outlier detection is an important task for fitting a model to a set of data. Two different outlier detection approaches are given as tests for outliers and robust methods. For these approaches, usually outliers are considered as additive bias terms neglected in the original adjustment model. However, there is another approach that outlier is considered as a model error in the Gauss–Markov model. This model error is represented as an unknown parameter. As it cannot be known before which observation includes outlier; this method is applied on the data for each observation separately and tested with t-test or F-test. It is successful if the sample includes only one outlier. To detect multiple outliers more successfully, in this article, a new outlier detection method is introduced. In this method, all the possible combinations of multiple outliers are considered as model errors and it is accepted that the smallest variance of them gives the solution for a certain number of outliers, then the estimated model errors are detected by comparing with a critical value. The critical value is chosen as 3σo. To compare the results of the new method, with those of the Least Median of Squares (LMS) and Huber M-estimators, Monte Carlo simulation technique is used for linear regression. The Mean Success Rate is proposed to measure the reliabilities of the methods. We showed that the new method is robust and includes the property of high breakdown point as LMS; and more efficient than LMS.

Summary In this paper we derive an asymptotic normality result for an adaptive trimmed likelihood estimator of regression starting from initial high breakdownpoint robust regression estimates. The approach leads to quickly and easily computed robust and efficient estimates for regression. A highlight of the method is that it tends automatically in one algorithm to expose the outliers and give least squares estimates with the outliers removed. The idea is to begin with a rapidly computed consistent robust estimator such as the least median of squares (LMS) or least trimmed squares (LTS) or for example the more recent MM estimators of Yohai. Such estimators are now standard in statistics computing packages, for example as in SPLUS or R. In addition to the asymptotics we provide data analyses supporting the new adaptive approach. This approach appears to work well on a number of data sets and is quicker than the related brute force adaptive regression approach described in Clarke (2000). This current approach builds on the work of Bednarski and Clarke (2002) which considered the asymptotics for the location estimator only.

In robust statistics, there are several types of robust estimators, including M-estimator (maximum likelihood type estimator), L-estimator (linear combinations of order statistics), R-estimator (estimator based on rank transformation) [77], RM estimator (repeated median) [141], and LMS estimator (estimator using the least median of squares) [133]. When information theoretic learning is applied to robust statistics, the Gaussian kernel in entropy plays a role of Welsch M-estimator and can be efficiently optimized by half-quadratic minimization. Hence, in this chapter, we introduce some basic concepts of M-estimation and half-quadratic minimization.

Image registration technology has been widely used in many parts of the computer vision system such as the automatic optical inspection system which is used to detect the printed circuit board (PCB) defects. The accuracy of the image registration will deeply influence the system's performance, so this study proposed an accurate image registration algorithm and applied it to the PCB defect detection. Good features to track feature detector and speeded up robust feature descriptor are combined to extract efficient features to achieve the first accurate image registration. Afterwards, cross‐correlation functions were used to compute the shift between the reference image and the first‐registered image for further accurate registration. Experimental results show that the authors’ algorithm performs a much better registration, with a lower root‐mean‐square error value between the reference image and transformed image. What is more, they applied it to detect the defects of PCB with a high accuracy.

Robust Regression for the Detection of Independent 3D Motion by a Binocular Observer

Finding and fitting circles from a set of points is a frequent problem in the data analysis of high-energy physics experiments. In a tracker immersed in a homogeneous magnetic field, tracks are close to perfect circles if projected to the bending plane. In a ring-imaging Cherenkov (RICH) detector, circles of photons around the crossing point of charged particles have to be found and their radii estimated. In both cases, non-negligible background may be present that tends to complicate the pattern recognition and to bias the circle fit. In this contribution we present a robust circle fit based on a modified Riemann fit that removes or significantly reduces the effect of background points. As in the standard Riemann fit, the measured points are projected to the Riemann sphere or paraboloid, and a plane is fitted to the projected points. The fit is made robust by replacing the usual least-squares regression by a least median of squares (LMS) regression. Because of the high breakdown point of the LMS estimator, the fit is insensitive to background points. The LMS plane is used to initialize the weights of an M-estimator that refits the plane in order to suppress eventual remaining outliers and to obtain the final circle parameters. The method is demonstrated on three sets of artificial data: points on a circle plus a comparable number of background points; points on two overlapping circles with additional background; and points obtained by the simulation of tracks in a drift chamber with mirror points and additional background. The results show high circle finding efficiency and small contamination of the final fitted circles.

Natural mortality rate (M) and fishing catchability coefficient (q) were estimated by using the Paloheimo pairwise linear regression model with simulated and real catch–effort data. The four estimation methods used were least squares (LS), least absolute value (LAV), least median of squares (LMS), and LMS-based reweighted least squares (RLS). When catch or effort data or both contained homogeneous error variances, mean squared errors (MSE) of LS and RLS were significantly smaller than those of LAV and LMS. However, there was no significant difference in MSE between LS and RLS. When catch–effort data contained variances inconsistent among years, MSEs of the robust methods were much smaller than those of LS. Reweighted least squares substantially reduced the variances in the LS-estimated q and M, and it had the smallest MSE among the four estimation methods. We suggest using the following procedures in estimating q and M based on the Paloheimo pairwise regression model: (1) applying LMS to the model to identify outliers in the data; and (2) weighting the defined outliers by a factor of 0 and normal data points by a factor of 1 and applying LS to the model to estimate q and M and their associated variances.

We propose a new method to fill missing or invalid values in depth images generated from the Kinect depth sensor. To fill the missing depth values, we use a robust least median of squares (LMedS) approach. We apply our method for telepresence environments, where Kinects are used very often for reconstructing the captured scene in 3D. We introduce a modified 1D LMedS approach for efficient traversal of consecutive image frames. Our approach solves the unstable nature of depth values in static scenes that is perceived as flickering. We obtain very good result both for static and moving objects inside a scene.

Least trimmed squares (LTS) provides a parametric family of high breakdown estimators in regression with better asymptotic properties than least median of squares (LMS) estimators. We adapt the forward search algorithm of Atkinson (1994) to LTS and provide methods for determining the amount of data to be trimmed. We examine the efficiency of different trimming proportions by simulation and demonstrate the increasing efficiency of parameter estimation as larger proportions of data are fitted using the LTS criterion. Some standard data examples are analysed. One shows that LTS provides more stable solutions than LMS.

New technological developments in optics and electronics are rendering laser scanning systems cheaper and more accurate. Such systems can directly capture depth information from objects simplifying range image analysis and enlarging their applications scope. Accurate image registration algorithms represent a pivotal aspect of range image analysis which still require substantial improvements. While a large number of algorithms have been proposed, it is very important to compare the performance of different registration algorithms so that good algorithms can be selected for real applications. Unfortunately, most of these algorithms are iterative with a number of parameters initialised by users. The main disadvantages of these methods are that these parameters are often context dependent. Thus, while aiming to develop automatic registration algorithms instead of comparing the registration algorithms themselves, we compare the structural constraints based on a single registration algorithm(GICP). A large number of experiments using real range images demonstrate their effectiveness on improving the GICP algorithm.