LAD Regression Research Articles

Heterogeneous robust estimation with the mixed penalty in high-dimensional regression model

In this paper, we propose a MIXED penalty for the LAD regression model, which can estimate parameters and select important variables efficiently and stably. The proposed method has a good performance in the case of dependent variable with heavy tail and outliers, so this estimator is robust and efficient for tackling the problem of heterogeniety. We show that the proposed estimator possesses the good properties by applying certain assumptions. In the part of numerical simulation, we give several simulation studies to examine the asymptotic results, which shows that the method we proposed behaves better.

PENGENDALIAN KOEFISIEN REGRESI LEAST ABSOLUTE DEVIATION PADA RENTANG BERMAKNA MENGGUNAKAN PROGRAM LINIER

So far, regression analysis is used to model the mean of response variable as a function of some independent variables, using the least squares (LS) method. In general, the LS method is able to describe well the measure of central tendency, however it is not robust against outliers. Therefore, in certain cases, a regression analysis that minimizes the sum of absolute residuals (least absolute deviation - LAD) is required, which is more robust against outliers. So far, the value of the regression coefficient is not modeled and only depends entirely on the data processed. In some cases, the sign and the value of regression coefficients need to be controlled, in order to be in the meaningful range. The results of this study showed that the modification of the constraints on the LAD regression able to control the regression coefficients to be in the meaningful range. The results of bootstrap showed that distribution of controlled regression coefficients were different from distribution of uncontrolled regression coefficients.

Performances of LAD Regression, M-Regression and Quantile Regression Methods in order to Investigate Stock Prices of the Banks in the BIST Bank Index

In this study, performances of LAD regression, M-regression, Q25 and Q75 quantile regression models as robust regression methods alternative to the classical LS method are compared in the case of violations from the normality assumption of the error terms and the presence of an outlier. By using these alternative regression methods, stock prices of the 12 commercial banks and 1 participation bank listed in the Istanbul Stock Exchange (BIST) bank index between 2012 and 2016 are investigated in terms of equity size and equity profitability. As a result of this study, M-regression is the most suitable robust regression model with the smallest value of the mean squared error (MSE) measure and the small values for the standard errors of the parameter estimates belonging to the equity size and equity profitability. The smaller the standard errors of the parameter estimates, the narrower the resulting confidence intervals are obtained in M- regression. The accuracy as a measure of closeness of parameter estimates to the true values of the parameters is also obtained higher in M-regression.

MPS 05-03 RELATIONSHIP BETWEEN INCIDENCE OF OBSTRUCTIVE SLEEP APNOE SYNDROME AND ATRIAL FIBRILLATION AND THE SERUM URIC ACID IN ESSENTIAL HYPERTENSION PATIENTS

Objective: Our aim was to investigate the relationship of SUA with HTN, AF as also the correlation of SUA with sleep apnea and specific diastolic echocardiographic parameters in hypertensive patients. Design and Method: We prospectively enrolled 418 hypertensive patients and 103 controls. The group of hypertensives was divided into: subgroup A (N = 78): with AF and subgroup B (N = 443): in sinus rhythm. We investigated the differences of SUA levels among the groups. In the second part we studied the correlation of SUA with apnea hypopnoea index (AHI), left atrial diameter (LA), left ventricular mass (LVM) and pulse wave velocity. Results: The mean age of patients with AF was (67.5 ± 10.2) yr vs. (61.8 ± 10.1)yr in subgroup B. Comparative analysis showed that SUA levels were significantly increased in patients with hypertension and AF (9.7 ± 6.1 mg/dl vs group B: 6.0 ± 1.4 mg/dl vs control group: 5.4 ± 1,2 mg/dl, p < 0.01). After adjustment for LAD, LVEDD, LVMI, LVEF, TC, SUA and age logistic regression analysis showed that only age (OR: 1.054; 95% CI: 1.019–1.090, p = 0.002), LAD (OR: 1.173; 95% CI: 1.114–1.235; p < 0.001) and LVEF (OR: 0.942; 95% CI: 0.914–0.971; p < 0.001) were independently associated with AF, while SUA was not independently associated with AF in EH patients. Patients in subgroup A presented more often sever OSAS (58% vs 26%, p < 0,05). Adjusted for age, BMI, BP, the patients with hyperuricemia presented greater AHI, mean SpO2, frequency of SpO2 decreased >=4% and >=5%. The UA levels significantly increased with the severity of OSAS. Conclusions: Increased SUA levels are not independently associated with AF in patients with hypertension, while age, LAD and LVEF are independently associated with AF in patients with EH. A strong association was found between UA levels and OSAS in a large number of hypertensive patients with AF.

New Formulae and Uses of Mean Absolute Deviation

Mean absolute deviation about mean and median is extended to define MAD-comoment similarly analogous to classical central moment notation of covariance. Therefore, the MAD-comoment is used to derive the coefficient of determination in LAD regression, to analyze total sum of absolute into the pure between sum of absolute and the pure within sum of absolute that provides the exact decomposition of Pietria index of inequality, to develop a new formula for beta coefficient in financial risk analysis that gives less weight at the extreme ends than least squares method for market return and to define MAD-correlation under first moment assumption.

Penalized LAD Regression for Single-index Models

In this article, we consider the penalized LAD regression which deals with variable selection and estimation simultaneously for single-index models. The proposed estimator is robust and efficient with respect to heavy tailed errors or outliers in the response. Then, we introduce a practical algorithm where the unknown link function is estimated by local linear smoothing combined with LAD regression and the parametric index is estimated through linear LAD regression. Furthermore, we show that, under certain appropriate conditions, the penalized LAD estimator has desired large sample properties including -consistency, and oracle property. Simulation studies and a real data application are carried to illustrate the performances of the proposed method for finite-sample cases. Finally, based on the idea of sure independence screening procedure and the distance correlation (DC-SIS) proposed by Li et al. (2012), a robust two-step estimator is introduced to deal with ultra-high dimensional data.

Compressive Sensing Signal Reconstruction by Weighted Median Regression Estimates

In this paper, we propose a simple and robust algorithm for compressive sensing (CS) signal reconstruction based on the weighted median (WM) operator. The proposed approach addresses the reconstruction problem by solving a l0-regularized least absolute deviation (l0-LAD) regression problem with a tunable regularization parameter, being suitable for applications where the underlying contamination follows a statistical model with heavier-than-Gaussian tails. The solution to this regularized LAD regression problem is efficiently computed, under a coordinate descent framework, by an iterative algorithm that comprises two stages. In the first stage, an estimation of the sparse signal is found by recasting the reconstruction problem as a parameter location estimation for each entry in the sparse vector leading to the minimization of a sum of weighted absolute deviations. The solution to this one-dimensional minimization problem turns out to be the WM operator acting on a shifted-and-scaled version of the measurement samples with weights taken from the entries in the measurement matrix. The resultant estimated value is then passed to a second stage that identifies whether the corresponding entry is relevant or not. This stage is achieved by a hard threshold operator with adaptable thresholding parameter that is suitably tuned as the algorithm progresses. This two-stage operation, WM operator followed by a hard threshold operator, adds the desired robustness to the estimation of the sparse signal and, at the same time, ensures the sparsity of the solution. Extensive simulations demonstrate the reconstruction capability of the proposed approach under different noise models. We compare the performance of the proposed approach to those yielded by state-of-the-art CS reconstruction algorithms showing that our approach achieves a better performance for different noise distributions. In particular, as the distribution tails become heavier the performance gain achieved by the proposed approach increases significantly.

A Comparison of Different Methods for LAD Regression

The least squares method is very sensitive to outliers, one of the simple alternative is the least absolute deviation, i.e. L1 regression, which is less sensitive to outliers, so which is more suitable the small sample and much noise situation. In this paper, the L1 problem of linear model is discussed, the previous work is reviewed systematically, different algorithms is compared, it is proved that the dual forms of different algorithms are the same.

Quantile Uncorrelation and Instrumental Regressions

We introduce a notion of median uncorre- lation that is a natural extension of mean (linear) uncorrelation. A scalar random variable Y is median uncorrelated with a k-dimensional random vector X if and only if the slope from an LAD regression of Y on X is zero. Using this simple definition, we characterize properties of median uncorrelated random variables, and introduce a notion of multivariate median uncor- relation. We provide measures of median uncorrela- tion that are similar to the linear correlation coefficient and the coefficient of determination. We also extend this median uncorrelation to other loss functions. As two stage least squares exploits mean uncorrelation between an instrument vector and the error to derive consistent estimators for parameters in linear regressi- ons with endogenous regressors, the main result of this paper shows how a median uncorrelation assumption between an instrument vector and the error can simi- larly be used to derive consistent estimators in these linear models with endogenous regressors. We also show how median uncorrelation can be used in linear panel models with quantile restrictions and in linear models with measurement errors.

Open areas in a landscape enhance pollen-mediated gene flow of a tree species: evidence from northern Switzerland

Habitat fragmentation often has negative consequences for genetic diversity, and thereby for the viability of populations. However, these negative consequences might be counteracted by gene flow as the latter provides functional connectivity between apparently isolated habitat fragments. Gene flow is itself influenced by landscape structure and composition, and it is therefore important to understand the relationship between gene flow and landscape structure and composition. We used linear LAD regression models to investigate the relationship between contemporary gene flow by pollen in the rare, insect-pollinated forest tree Sorbus domestica and several landscape features. None of the landscape components—which included closed forest, deep valleys, open land and settlements—proved to be an impermeable barrier to gene flow by pollen. We found evidence that settlements, large open areas, and a pronounced topography increased long-distance gene flow in the landscape as compared to a random model including all possible gene flow trajectories. These results are encouraging from a conservation view, as gene flow in species pollinated by generalist insects seems to provide functional connectivity and may help to maintain genetic diversity in rare plant species in fragmented landscapes.

A robust penalized method for the analysis of noisy DNA copy number data

BackgroundDeletions and amplifications of the human genomic DNA copy number are the causes of numerous diseases, such as, various forms of cancer. Therefore, the detection of DNA copy number variations (CNV) is important in understanding the genetic basis of many diseases. Various techniques and platforms have been developed for genome-wide analysis of DNA copy number, such as, array-based comparative genomic hybridization (aCGH) and high-resolution mapping with high-density tiling oligonucleotide arrays. Since complicated biological and experimental processes are often associated with these platforms, data can be potentially contaminated by outliers.ResultsWe propose a penalized LAD regression model with the adaptive fused lasso penalty for detecting CNV. This method contains robust properties and incorporates both the spatial dependence and sparsity of CNV into the analysis. Our simulation studies and real data analysis indicate that the proposed method can correctly detect the numbers and locations of the true breakpoints while appropriately controlling the false positives.ConclusionsThe proposed method has three advantages for detecting CNV change points: it contains robustness properties; incorporates both spatial dependence and sparsity; and estimates the true values at each marker accurately.

A Comparison of Methods for Analyzing Health-Related Quality-of-Life Measures

Objectives Self-reported health status is often measured using psychometric or utility indices that provide a score intended to summarize an individual's health. Measurements of health status can be subject to a ceiling effect. Frequently, researchers want to examine relationships between determinants of health and measures of health status. Regression methods that ignore the censoring in the health status measurement can produce biased coefficient estimates. The authors examine the performance of three different models for assessing the relationship between demographic characteristics and health status. Methods Three methods that allow one to analyze data subject to a ceiling effect are compared. The first model is the classic Tobit model. The second and third models are robust variants of the Tobit model: symmetrically trimmed least squares and censored least absolute deviations (Censored LAD) regression. These models were fit to data from the Canadian National Population Health Survey. The results are compared to three models that ignore the presence of a ceiling effect. Results The Censored LAD model produced coefficient estimates that tended to be shrunk toward 0, compared to the other two models. The three models produced conflicting evidence on the effect of gender on health status. Similarly, the rate of decay in health status with increasing age differed across the three models. The Censored LAD model produced results very similar to median regression. Furthermore, the censored LAD model had the lowest prediction error in an independent validation dataset. Conclusions Our results highlight the need for careful consideration about how best to model variation in health status. Based upon our study, we recommend the use of Censored LAD regression.

Tests of Fit for the Laplace Distribution, With Applications

Tests are given for the Laplace or double exponential distribution. The test statistics are based on the empirical distribution function and include the families of Cramér-von Mises and Kolmogorov-Smirnov. Asymptotic theory is given, and asymptotic points are calculated, for the Cramér-von Mises family, and Monte Carlo points for finite samples are given for all the statistics. Power studies suggest that the Watson statistic is the most powerful for the common problem of testing Laplace against other symmetric distributions. An application of the Laplace distribution is in LAD (or L1) regression. This is also discussed in the article, with two examples.

Tests of Fit for the Laplace Distribution, with Applications

Tests are given for the Laplace or double exponential distribution. The test statistics are based on the empirical distribution function and include the families of Cramér-von Mises and Kolmogorov-Smirnov. Asymptotic theory is given, and asymptotic points are calculated, for the Cramér-von Mises family, and Monte Carlo points for finite samples are given for all the statistics. Power studies suggest that the Watson statistic is the most powerful for the common problem of testing Laplace against other symmetric distributions. An application of the Laplace distribution is in LAD (or L1) regression. This is also discussed in the article, with two examples.

NONPARAMETRIC ESTIMATION OF THE CONDITIONAL MEDIAN FUNCTION FOR LONG-RANGE DEPENDENT PROCESSES

This paper considers the problem of estimating the conditional median of Yi=v(Xi)+εi, given Xi=x. We assume{Xi}and{εi}are independent linear processes with long-range dependence, estimate the conditional median by local LAD regression, and show the asymptotic normality of the estimator. The asymptotic property depends on the degree of dependence. When the dependence is not very strong, the asymptotic behavior is the same as in the case of weakly dependent processes. When the dependence is very strong, the estimator behaves as the estimator of the conditional mean function, and the effect of long-range dependence apoears.

LAD Regression for Detecting Outliers in Response and Explanatory Variables

Least absolute deviations regression resists outliers in the response variable but is relatively sensitive to outlying observations in the explanatory variables. In this paper a simple solution is proposed to overcome this problem. This is achieved by minimizing the absolute values of vertical and horizontal deviations in turn. Two algorithms are proposed: one for the simple and one for the multiple regression case. The methods presented have been tested on a variety of data and have proven to be quite effective.

Artificial Skill and Validation in Meteorological Forecasting

The results of a simulation study of multiple regression prediction models for meteorological forecasting are reported. The effects of sample size, amount, and severity of nonrepresentative data in the population, inclusion of noninformative predictors, and least ( sum of ) absolute deviations ( LAD ) and least ( sum of ) squared deviations ( LSD ) regression models are examined on five populations constructed from meteorological data. Artificial skill is shown to be a product of small sample size, LSD regression, and nonrepresentative data. Validation of sample results is examined, and LAD regression is found to be superior to LSD regression when sample size is small and nonrepresentative data are present.