Least Absolute Deviation Research Articles

Optimum Placement of Phasor Measurement Units in Power Systems

In this paper, we proposed to evaluate the optimal phasor measurement unit (PMU) placement based on the sum of variance (SV) of the robust estimators. Variance is traditionally used as a measure of the quality of estimates. In this paper, after satisfying the requirements of the minimum number of PMUs, maximum measurement redundancy, and observability, the placements obtained are further distinguished on the basis of the variance of the estimated states. Both the weighted least squares (WLS) and robust estimators are considered. Examples on the IEEE 30-bus system using least absolute value (LAV) estimation are given. The covariance of the state estimates from the WLS can be calculated using the existing formulas. However, there is no equivalent formula for the robust LAV estimator. A formula is derived in this paper using influence function to approximately calculate the covariance matrix of the state estimates from the robust LAV estimator. The optimum placement, i.e., the placement with the smallest SV, can be selected.

Application Of L1- Regularization Approach In QSAR Problem. Linear Regression And Artificial Neural Networks

One of the primary tasks of analytical chemistry and QSAR/QSPR researches is building of prognostic regression equations based on descriptors sets. The one of the most important problems here is to decrease the number of descriptors in the initial descriptor set which is usually way too big. In current investigation the descriptor set is proposed to be reduced employing the least absolute shrinkage and selection operator (LASSO) approach. Decreased descriptor sets were used for calculations with application of the following QSAR/QSPR methods: ordinary least squares (OLS), the least absolute deviation (LAD) regressions and artificial neural networks (ANN). Contrary to aforementioned methods principal component regression (PCR) and partial least squares (PLS) approaches can produce solutions containing numerous descriptors. In this article we compared the viability of these two different descriptor handling ideologies in application to molecular chemical and physical properties prediction. From the obtained results it is possible to see that there are tasks for which PCR and PLS approaches can fail to produce accurate regression equations. At the same time, methods OLS and LAD that use small amount of descriptors can provide viable solutions for the same cases. It was shown that these small sets of descriptors selected with LASSO approach can be used in ANN to obtain models with even better internal validation characteristics.

Theoretical study of energy levels and transitions 4s<sup>2</sup>4p<sup>3</sup>−4s4p<sup>4</sup> for ions Rh XIII to Cd XVI

Ions from Rh XIII to Cd XVI belong to the arsenic isoelectronic sequence ions. Their ground configuration is 4s<sup>2</sup>4p<sup>3</sup>, and the lower excited configurations are 4s4p<sup>4</sup>, 4s<sup>2</sup>4p<sup>2</sup>4d and 4s<sup>2</sup>4p<sup>2</sup>5s etc. The present study aims to predict the energy levels and transition data unknown in experiment for configurations 4s<sup>2</sup>4p<sup>3</sup> and 4s4p<sup>4</sup> from Rh XIII to Cd XVI ions, by analyzing the trend of the variation of Slater-Condon parameters along the As-like sequence based on the experimental energy levels available in the literature. So, the theoretical analyses of fine-structure energy levels of these configurations are conducted for the sequence ions from Rb V to Cd XVI by Hartree-Fock with Relativistic correction (HFR) method in Cowan’ code. The Slater-Condon parameter values of energy levels are obtained by least-square-fit (LSF) technique for ions mentioned above with the available experimental data. For the unknown parameters, the generalized-least-square-fit (GLSF) technique is used together with the extra (or inter)-polation method. With these new parameter values, the energy levels of 4s<sup>2</sup>4p<sup>3</sup> and 4s4p<sup>4</sup>, the wavelengths and oscillator strengths of the transition array 4s<sup>2</sup>4p<sup>3</sup>−4s4p<sup>4</sup> are computed. This research shows that for 4s<sup>2</sup>4p<sup>3</sup>, the single-configuration approximation of HFR calculation can present the satisfactory results, however, for 4s4p<sup>4</sup>, the reasonable good results can be achieved only by multi-configuration(4s4p<sup>4</sup> + 4s<sup>2</sup>4p<sup>2</sup>4d) approximation, which can be verified by the obtained data. Comparing the absolute differences between observed and present LSF calculated levels’ values (including multi-configuration interaction) for the 4s4p<sup>4</sup> configuration in ions from Rb V to Mo X with the results computed in a similar Hartree-Fock single-configuration approximate method by Person and Pettersson (Person W, Pettersson S G 1984 <i>Phys. Scr.</i> <b>29</b> 308), we can see that the present LSF energy levels are improved substantially. For example, the LSF minimum and maximum absolute deviation value at present are 1 cm<sup>–1</sup> and 140 cm<sup>–1</sup>, respectively, much more accurate than the results presented by Person et al., which are 45 cm<sup>–1</sup> and 382 cm<sup>–1</sup>. The predicted data are in good agreement with the experimental results. For obtaining more information, the energy levels of 4s<sup>2</sup>4p<sup>3</sup> and 4s4p<sup>4</sup> configurations are computed by grasp2K-DEV package in valence-valence correlation scheme, which is based on the fully relativistic multi-configuration Dirac-Hartree-Fock (MCDHF) theory. The overall MCDHF energy levels are generally in accordance with the experimental results. The data obtained in this research are expected to be used in the future relevant theoretical and experimental investigations.

A Comparative Analysis on Some Estimators of Parameters of Linear Regression Models in Presence of Multicollinearity

Linear regression is the measure of relationship between two or more variables known as dependent and independent variables. Classical least squares method for estimating regression models consist of minimising the sum of the squared residuals. Among the assumptions of Ordinary least squares method (OLS) is that there is no correlations (multicollinearity) between the independent variables. Violation of this assumptions arises most often in regression analysis and can lead to inefficiency of the least square method. This study, therefore, determined the efficient estimator between Least Absolute Deviation (LAD) and Weighted Least Square (WLS) in multiple linear regression models at different levels of multicollinearity in the explanatory variables. Simulation techniques were conducted using R Statistical software, to investigate the performance of the two estimators under violation of assumptions of lack of multicollinearity. Their performances were compared at different sample sizes. Finite properties of estimators’ criteria namely, mean absolute error, absolute bias and mean squared error were used for comparing the methods. The best estimator was selected based on minimum value of these criteria at a specified level of multicollinearity and sample size. The results showed that, LAD was the best at different levels of multicollinearity and was recommended as alternative to OLS under this condition. The performances of the two estimators decreased when the levels of multicollinearity was increased.

Robust semi-parametric multiple change-points detection

This paper is dedicated to define two new multiple change-points detectors in the case of an unknown number of changes in the mean of a signal corrupted by additive noise. Both these methods are based on the Least-Absolute Value (LAV) criterion. Such criterion is well known for improving the robustness of the procedure, especially in the case of outliers or heavy-tailed distributions. The first method is inspired by model selection theory and leads to a data-driven estimator. The second one is an algorithm based on total variation type penalty. These strategies are numerically studied on Monte-Carlo experiments.

Lagrangian Duality with ELM for Word Sense Multiprototype Discovery

ABSTRACT Homonymy and polysemy are major issues in word sense disambiguation. Combining with multilayer neural network, word sense multiprototyping tackles the issues by defining multiple feature embedding representations for each word which are based on the average feature weight of the word’s different context windows called prototypes. The complexity of parameter estimation of neural network regression as well as the fixed context window size are the restrictions on the implementation of word sense multiprototyping. We propose approximating the least absolute deviation (LAD) between pair-wise word frequency covariance and pair-wise word semantic relatedness by Extreme Machine Learning (ELM) with less-constraint parameter estimation. Lagrangian duality proves the method’s feasibility. An in-cluster closeness calculation is performed to extract a variable context window to contextually identify multiprototypes of word senses based on Kmeans clustering. The higher accuracy of the discovered multiprototypes is verified by our experiments.

The Parameter Identification of Least Absolute Deviation Based on an Improved Gravitational Search Algorithm

The least squares (LS) identification algorithm is vulnerable to outliers and has large residual square when the measured data is mixed with impulse noise which obeys symmetrical alpha stable (SaS) distribution, so the least absolute deviation (LAD) is selected as the objective function to get better identification performance when impulse noise exists. And taking the non-difference of least absolute deviation into consideration, we adopt an improved gravitational search algorithm as optimal algorithm to search for optimal solution globally. Then the parameter identification method based on LAD objective function using an improved gravitational search algorithm (LAD-IGSA) is put forward creatively. Simulation results show that the LAD-IGSA method can restrain the influence of impulse noise effectively and achieve higher identification accuracy. Moreover, LAD-IGSA method presents better robustness and bettidentification accuracy than LP method with small data sets.

Polyphenol Liquid-Liquid Extraction Process Development Using NRTL-SAC.

Liquid–liquid extraction (LLE) can be an effective strategy for the purification of polyphenols from a fermentation broth. However, solvents need to be chosen to ensure high extraction capacity and selectivity. For that purpose, a systematic study is here presented, where the partition of different polyphenols—naringin, naringenin, p-coumaric acid, and trans-resveratrol—was measured in different solvents and solvent mixtures and described using the semipredictive NRTL-SAC model. The minimum average absolute deviation obtained, based on predicted activity coefficients, was of 40%. With the exception of naringin, the NRTL-SAC molecular descriptors were estimated using solubility data already available in the literature. The obtained results made it possible to propose suitable LLE-based downstream process schemes for two possible purification scenarios: the recovery of trans-resveratrol and the purification of both naringenin and trans-resveratrol, two similar hydrophobic polyphenols, both from a fermentation broth containing hydrophilic impurities (e.g., sugars, proteins).

ROBUST GEOGRAPHICALLY WEIGHTED REGRESSION DENGAN METODE MUTLAK SIMPANGAN TERKECIL PADA PEMODELAN KEJADIAN DIARE DI KOTA SEMARANG

Diarrhea is one of many health issues in developing country like Indonesia, because the sickness and the death number are still high. According to health profile of Semarang City, the people who suffer from diarrhea from 2010-2015 are decreasing. The lowest point happened at the year 2013 with the total case of 38.001, however there are an increasing number from 2014-2015. The distribution data of diarrhea is a spatial data. The differences between environment and sanitation could cause spatial heterogeneity. The spatial heterogeneity could cause the produced variant value no longer constant, but instead it is different on each region. Therefore, regression model that involves the effects of spatial heterogeneity is needed, which are Geographically Weighted Regression (GWR) that is built by Weighted Least Square (WLS) adjuster. Although, GWR parameter adjuster that used WLS is very sensitive with the existence of outliers. The existence of the outlier in the data will create a huge residual. Thus, more robust method is needed, which is Least Absolute Deviation (LAD) methods in order to estimate the parameter on model GWR. This model is called Robust GWR (RGWR). The result shows that the model events of diarrhea on each region in Semarang City are different. Furthermore, the model events of diarrhea with RGWR model generate MAPE 16,3396% which means the performance of RGWR is formed well. Keyword: Diarrhea, Robust, Geographically Weighted Regression, Least Absolute Deviation

Robust sparse accelerated failure time model for survival analysis.

To identify the bio-mark genes related to disease with high dimension and low sample size gene expression data, various regression approaches with different regularization methods have been proposed to solve this problem. Nevertheless, high-noises in biological data significantly reduce the performances of methods. The accelerated failure time (AFT) modelwas designed for gene selection and survival time estimation in cancer survival analysis. In this article, we proposed a novel robust sparse accelerated failure time model (RS-AFT) through combining the least absolute deviation (LAD) and L regularization. An iterative weighted linear programming algorithm without regularization parameter tuning was proposed to solve this RS-AFT model. The results of the experiments show our method has better performancebothin gene selection and survival time estimationthan some widely used regularization methods such as lasso, elastic net and SCAD. Hence we thought the RS-AFT model may be a competitive regularization method in cancer survival analysis.

Complex correntropy function: Properties, and application to a channel equalization problem

The use of correntropy as a similarity measure has been increasing in different scenarios due to the well-known ability to extract high-order statistic information from data. Recently, a new similarity measure between complex random variables was defined and called complex correntropy. Based on a Gaussian kernel, it extends the benefits of correntropy to complex-valued data. However, its properties have not yet been formalized. This paper studies the properties of this new similarity measure and extends this definition to positive-definite kernels. Complex correntropy is applied to a channel equalization problem as good results are achieved when compared with other algorithms such as the complex least mean square (CLMS), complex recursive least squares (CRLS), and least absolute deviation (LAD).

A Decentralization Method for Hybrid State Estimators

Hybrid state estimation determines the states of systems monitored both by conventional SCADA measurements and PMUs. This paper proposes a decentralization method in order to improve the computational performance of hybrid state estimators with minimum loss of accuracy. The proposed method utilizes sensitivity matrix that relates measurement errors to state estimates in order to form the so-called isolated bus groups and observable subislands. Although the proposed decentralization method can be applied to any state estimator, the paper is developed using a least absolute value (LAV) estimator for demonstration purposes, which is known to be robust against bad data. The high computational burden of LAV estimator compared to the well-known weighted least squares estimator is eliminated, thanks to the proposed decentralization method.

Robust optimisation‐based state estimation considering parameter errors for systems observed by phasor measurement units

This study presents a new robust state estimation (SE) approach for power systems that are monitored by synchronised phasor measurements. The proposed robust SE utilises the least absolute value (LAV) of residuals to determine the system state. Although conventional deterministic LAV-based SE is robust against bad measurements, it is vulnerable with respect to the network parameters’ errors. In other words, the conventional LAV-SE method is not robust against the network parameters’ errors and these errors can make the SE results inaccurate. The proposed robust SE aims at solving this problem of the conventional LAV-SE. Robust optimisation, strong duality theorem, and Big-M linearisation technique have been used to construct the proposed SE approach, which is robust against the network parameters’ errors. Additionally, the proposed robust SE is finally formulated as a tractable mixed-integer linear programming optimisation problem. Comprehensive numerical experiments and a probabilistic evaluation approach are presented to evaluate the effectiveness of the proposed robust SE compared to the conventional deterministic SE in the presence of the uncertainty source of network parameters’ errors.

Robust Interpolation of DEMs From Lidar-Derived Elevation Data

Light detection and ranging (lidar)-derived elevation data are commonly subjected to outliers due to the boundaries of occlusions, physical imperfections of sensors, and surface reflectance. Outliers have a serious negative effect on the accuracy of digital elevation models (DEMs). To decrease the impact of outliers on DEM construction, we propose a robust interpolation algorithm of multiquadric (MQ) based on a regularized least absolute deviation (LAD) technique. The objective function of the proposed method includes a regularization-based smoothing term and an LAD-based fitting term, respectively, used to smooth noisy samples and resist the influence of outliers. To solve the objective function of the proposed method, we develop a simple scheme based on the split-Bregman iteration algorithm. Results from simulated data sets indicate that when sample points are noisy or contaminated by outliers, the proposed method is more accurate than the classical MQ and two recently developed robust algorithms of MQ for surface modeling. Real-world examples of interpolating 1 private and 11 publicly available airborne lidar-derived data sets demonstrate that the proposed method averagely produces better results than two promising interpolation methods including regularized spline with tension (RST) and gridded data-based robust thin plate spline (RTPS). Specifically, the image of RTPS is too smooth to retain terrain details. Although RST can keep subtle terrain features, it is distorted by some misclassified object points (i.e., pseudooutliers). The proposed method obtains a good tradeoff between resisting the effect of outliers and preserving terrain features. Overall, the proposed method can be considered as an alternative for interpolating lidar-derived data sets potentially including outliers.

Analysis of Algorithms for Stable Estimation of Coefficients of Multiple Linear Regression Models

Computational experiments on model data were performed in order to study the effectiveness of the algorithms for realization of the least absolute deviations (LAD) method and the generalized method of the least absolute deviations (GLAD) when estimating the parameters of multiple linear regression models based on descent through the nodal straight lines. In addition, a comparative analysis of the algorithms of descent through nodal straight lines for LAD and GLAD with known exact and approximate methods to solve tasks (2) and (3) was carried out.

Detecting Activation in fMRI Data: An Approach Based on Sparse Representation of BOLD Signal

This paper proposes a simple yet effective approach for detecting activated voxels in fMRI data by exploiting the inherent sparsity property of the BOLD signal in temporal and spatial domains. In the time domain, the approach combines the General Linear Model (GLM) with a Least Absolute Deviation (LAD) based regression method regularized by the pseudonorml0to promote sparsity in the parameter vector of the model. In the spatial domain, detection of activated regions is based on thresholding the spatial map of estimated parameters associated with a particular stimulus. The threshold is calculated by exploiting the sparseness of the BOLD signal in the spatial domain assuming a Laplacian distribution model. The proposed approach is validated using synthetic and real fMRI data. For synthetic data, results show that the proposed approach is able to detect most activated voxels without any false activation. For real data, the method is evaluated through comparison with the SPM software. Results indicate that this approach can effectively find activated regions that are similar to those found by SPM, but using a much simpler approach. This study may lead to the development of robust spatial approaches to further simplifying the complexity of classical schemes.

Measurement and correlation of thermophysical properties of waste lubricant oil

Measurements of viscosity, density and surface tension of waste lubricant oil (WLO) were conducted at different temperatures. The dataset was constituted by nine WLO samples from different producers. The main objective of the work is the investigation of the temperature dependence of viscosity, including the glass transition, Tg, and dynamic crossover, Tx, temperatures. In addition, different correlations were evaluated to predict viscosity and surface tension starting from a property easily obtained such as density. WLO show rapid decrease of viscosity with shear rate, followed by an extensive and well defined Newtonian plateau. The temperature dependence of viscosity at the Newtonian plateau of WLO deviated from the Arrhenius behavior, and was accurately described by four different models: Williams–Landel–Ferry (WLF), MYEGA, power law and Ghatee. The minimum and maximum absolute average relative deviation (AARD) were 0.08% and 0.9%, respectively. Tg predicted by WLF equation varied between T=170.5K and T=198.7K, while MYEGA tends to predict lower Tg (about T=20K). The power law and Ghatee models estimate Tx nearly with the same accuracy. A ratio Tx/Tg of about 1.28±0.03 was obtained for WLO, which agrees with the literature. Waste oil is a moderately fragile glass forming fluid with a fragility parameter, m, ranging from 47.8 and 64.5. New correlations were developed for the prediction of viscosity and surface tension from density at different temperatures. AARD of 7% was observed for viscosity and 2% for surface tension.

The linearized alternating direction method of multipliers for sparse group LAD model

While the least absolute shrinkage and selection operator (LASSO) became a popular method due to its wide applications in high dimensional settings, some generalized LASSO models were developed. The sparse group LASSO is one of the important lasso-type methods, which aims to solve the linear regression problems with grouped covariates and tends to produce a solution with sparse effects both on a group and within group level. At the same time, we know that the least absolute deviation (LAD) is a useful and robust method when the noise distribution may be heavy-tailed or heterogeneous. In this paper, we combine these two classical ideas together to develop sparse group LAD model. We show that the sparse group LAD estimator achieves near oracle performance, i.e., with high probability, the $$L_2$$ norm of the estimation error is of order $$O(\sqrt{k\text{ log }p/n}).$$ Moreover, with the help of the linearization technique we propose the linearized alternating direction method of multipliers to solve the sparse group LAD estimator and establish its convergence. Numerical experiments are reported to illustrate the efficiency of the proposed algorithm.

Ordinary Least Squares and Quantile Regression: An Inquiry-Based Learning Approach to a Comparison of Regression Methods

We present a short, inquiry-based learning course on concepts and methods underlying ordinary least squares (OLS), least absolute deviation (LAD), and quantile regression (QR). Students investigate squared, absolute, and weighted absolute distance functions (metrics) as location measures. Using differential calculus and properties of convex functions, students discover the sample mean, median, and pth quantile by minimizing sums of distances. Students use these metrics to define loss functions for OLS, LAD, and QR, and explore methods to minimize them. We discuss classroom experiences over two semesters. Classroom activities, Maple worksheets, and R demonstration code are available at a companion website.

A descent method for least absolute deviation lasso problems

Variable selection is an important method to analyze large quantity of data and extract useful information. Although least square regression is the most widely used scheme for its flexibility in obtaining explicit solutions, least absolute deviation (LAD) regression combined with lasso penalty becomes popular for its resistance to heavy-tailed errors in response variable, denoted as LAD-LASSO. In this paper, we consider the LAD-LASSO problem for variable selection. Based on a dynamic optimality condition of nonsmooth optimization problem, we develop a descent method to solve the nonsmooth optimization problem. Numerical experiments are conducted to confirm that the proposed method is more efficient than existing methods.