Recent developments in PROGRESS

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.

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.

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.

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.

Robust principal component regression is development of principal component regression that applies robust method at principal component analysis and principal component regression analysis. Robust principal component regression does not only overcome multicollinearity problems, but also overcomes outlier problems. The robust methods used in this research are Minimum Covariance Determinant (MCD) that is applied when doing principal component analysis and Least Trimmed Squares (LTS) that is applied when doing principal component regression analysis. The case study in this research is Human Development Index (HDI) in Central Java in 2017 which is influenced by labor force participation rates, school enrollment rates, percentage of poor population, population aged 15 years and over who are employed, health facilities, gross enrollment rates, and net enrollment rates. The model of HDI in Central Java in 2017 using robust principal component regression MCD-LTS provides the most effective result for handling multicollinearity and outliers with Adjusted R2 value of 0.6913 and RSE value of 0.469. Keywords: Robust Principal Component Regression, Multicollinearity, Outliers, Minimum Covariance Determinant (MCD), Least Trimmed Squares (LTS), Human Development Index (HDI).

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

The Least Squares Method (MKT) is the best linear regression estimator if the classical assumption tests are met. However, if there are outliers in the data, this approach can provide inaccurate prediction results. Therefore, the purpose of this research is to overcome the inappropriate model due to outliers in variables that affect mutual fund performance in Indonesia. The variables in this study are risk level, inflation, fund size, turnover ratio, and cash flow. Robust regression is an approach designed to provide more accurate estimates without discarding observational data that indicates outliers. One of the estimation approaches is Least Trimmed Squares (LTS). This estimation minimizes the sum of squared residuals from h observations that are not considered outliers. The results showed that there were 10 outliers in the data, and the risk level variable had no effect on mutual fund performance, while the inflation, fund size, turnover ratio, and cash flow variables had a significant effect. So by comparing the value and residual standard error of the two methods, it is found that the value in the LTS method is greater than the MKT method, namely 0.589 0.273, and the residual standard error in the LTS method is smaller than the MKT method, namely 9.73 48.59. Therefore, it can be concluded that the Least Trimmed Squares method provides better estimation results and is more effective for handling outliers than the MKT method.

Least trimmed squares (LTS) is a statistical technique for estimation of unknown parameters of a linear regression model and provides a “robust” alternative to the classical regression method based on minimizing the sum of squared residuals.

The paper by Alfons, Croux and Gelper (2013), Sparse least trimmed squares regression for analyzing high-dimensional large data sets, considered a combination of least trimmed squares (LTS) and lasso penalty for robust and sparse high-dimensional regression. In a recent paper [She and Owen (2011)], a method for outlier detection based on a sparsity penalty on the mean shift parameter was proposed (designated by "SO" in the following). This work is mentioned in Alfons et al. as being an "entirely different approach." Certainly the problem studied by Alfons et al. is novel and interesting.

In the multiple linear regression analysis, the ridge regression estimator and the Liu estimator are often used to address multicollinearity. Besides multicollinearity, outliers are also a problem in the multiple linear regression analysis. We propose new biased estimators based on the least trimmed squares (LTS) ridge estimator and the LTS Liu estimator in the case of the presence of both outliers and multicollinearity. For this purpose, a simulation study is conducted in order to see the difference between the robust ridge estimator and the robust Liu estimator in terms of their effectiveness; the mean square error. In our simulations, the behavior of the new biased estimators is examined for types of outliers: X-space outlier, Y-space outlier, and X-and Y-space outlier. The results for a number of different illustrative cases are presented. This paper also provides the results for the robust ridge regression and robust Liu estimators based on a real-life data set combining the problem of multicollinearity and outliers.

This paper is focusing on the application of robust method in multiple linear regression (MLR) model towards diabetes data. The objectives of this study are to identify the significant variables that affect diabetes by using MLR model and using MLR model with robust method, and to measure the performance of MLR model with/without robust method. Robust method is used in order to overcome the outlier problem of the data. There are three robust methods used in this study which are least quartile difference (LQD), median absolute deviation (MAD) and least-trimmed squares (LTS) estimator. The result shows that multiple linear regression with application of LTS estimator is the best model since it has the lowest value of mean square error (MSE) and mean absolute error (MAE). In conclusion, plasma glucose concentration in an oral glucose tolerance test is positively affected by body mass index, diastolic blood pressure, triceps skin fold thickness, diabetes pedigree function, age and yes/no for diabetes according to WHO criteria while negatively affected by the number of pregnancies. This finding can be used as a guideline for medical doctors as an early prevention of stage 2 of diabetes.

Algorithms such as Least Median of Squares (LMedS) and Random Sample Consensus (RANSAC) have been very successful for low-dimensional robust regression problems. However, the combinatorial nature of these algorithms makes them practically unusable for high-dimensional applications. In this paper, we introduce algorithms that have cubic time complexity in the dimension of the problem, which make them computationally efficient for high-dimensional problems. We formulate the robust regression problem by projecting the dependent variable onto the null space of the independent variables which receives significant contributions only from the outliers. We then identify the outliers using sparse representation/learning based algorithms. Under certain conditions, that follow from the theory of sparse representation, these polynomial algorithms can accurately solve the robust regression problem which is, in general, a combinatorial problem. We present experimental results that demonstrate the efficacy of the proposed algorithms. We also analyze the intrinsic parameter space of robust regression and identify an efficient and accurate class of algorithms for different operating conditions. An application to facial age estimation is presented.

Regression with outlier shrinkage

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.

An optimization procedure for determination of indomethacin and acemethacin by differential pulse adsorptive stripping voltammetry. Application on urine samples