Deleting Outliers in Robust Regression with Mixed Integer Programming

In robust regression we often have to decide how many are the unusualobservations, which should be removed from the sample in order to obtain better fitting for the restof the observations. Generally, we use the basic principle of LTS, which is to fit the majority ofthe data, identifying as outliers those points that cause the biggest damage to the robust fit.However, in the LTS regression method the choice of default values for high break down-point affectsseriously the efficiency of the estimator. In the proposed approach we introduce penalty cost fordiscarding an outlier, consequently, the best fit for the majority of the data is obtained bydiscarding only catastrophic observations. This penalty cost is based on robust design weights andhigh break down-point residual scale taken from the LTS estimator. The robust estimation is obtainedby solving a convex quadratic mixed integer programming problem, where in the objective functionthe sum of the squared residuals and penalties for discarding observations is minimized. Theproposed mathematical programming formula is suitable for small-sample data. Moreover, we conduct asimulation study to compare other robust estimators with our approach in terms of their efficiencyand robustness.

Bayesian regularized quantile regression:A robust alternative for genome-based prediction of skewed data

Genomic prediction(GP)has become a valuable tool for predicting the performance of selection candidates for the next breeding cycle.A vast majority of statistical linear models on which GP is based rely on the assumption of normality of the residuals and therefore on the response variable itself.In this study,we propose to use Bayesian regularized quantile regression(BRQR)in the context of GP;the model has been successfully used in other research areas.We evaluated the prediction ability of the proposed model and compared it with the Bayesian ridge regression(BRR;equivalent to genomic best linear unbiased predictor,GBLUP).In addition,BLUP can be used with pedigree information obtained from the coefficient of coancestry(ABLUP).We have found that the prediction ability of BRQR is comparable to that of BRR and,in some cases,better;it also has the potential to efficiently deal with outliers.A program written in the R statistical package is available as Supplementary material.

A robust sparse representation algorithm based on adaptive joint dictionary

Sparse representation based on dictionary construction and learning methods have aroused interests in the field of face recognition.Aiming at the shortcomings of face feature dictionary not‘clean’and noise interference dictionary not‘representative’in sparse representation classification model,a new method named as robust sparse representation is proposed based on adaptive joint dictionary(RSR-AJD).First,a fast lowrank subspace recovery algorithm based on LogDet function(Fast LRSR-LogDet)is proposed for accurate low-rank facial intrinsic dictionary representing the similar structure of human face and low computational complexity.Then,the Iteratively Reweighted Robust Principal Component Analysis(IRRPCA)algorithm is used to get a more precise occlusion dictionary for depicting the possible discontinuous interference information attached to human face such as glasses occlusion or scarf occlusion etc.Finally,the above Fast LRSR-LogDet algorithm and IRRPCA algorithm are adopted to construct the adaptive joint dictionary,which includes the low-rank facial intrinsic dictionary,the occlusion dictionary and the remaining intra-class variant dictionary for robust sparse coding.Experiments conducted on four popular databases(AR,Extended Yale B,LFW,and Pubfig)verify the robustness and effectiveness of the authors’method.

Probabilistic robust regression with adaptive weights-a case study on face recognition

Robust regression plays an important role in many machine learning problems.A primal approach relies on the use of Huber loss and an iteratively reweightedℓ2 method.However,because the Huber loss is not smooth and its corresponding distribution cannot be represented as a Gaussian scale mixture,such an approach is extremely difficult to handle using a probabilistic framework.To address those limitations,this paper proposes two novel losses and the corresponding probability functions.One is called Soft Huber,which is well suited for modeling non-Gaussian noise.Another is Nonconvex Huber,which can help produce much sparser results when imposed as a prior on regression vector.They can represent anyℓq loss(1/2≤q<2)with tuning parameters,which makes the regression model more robust.We also show that both distributions have an elegant form,which is a Gaussian scale mixture with a generalized inverse Gaussian mixing density.This enables us to devise an expectation maximization(EM)algorithm for solving the regression model.We can obtain an adaptive weight through EM,which is very useful to remove noise data or irrelevant features in regression problems.We apply our model to the face recognition problem and show that it not only reduces the impact of noise pixels but also removes more irrelevant face images.Our experiments demonstrate the promising results on two datasets.

A COMPARISON BETWEEN TWO ROBUST REGRESSION ESTIMATORS BY MEANS OF ROBUST COVARIANCES

Abstract Two classes of Mallows GM estimators with invariance are considered in the stochastic linear regression model. Some of their asymptotic properties are described, and the fitted value influence and variance components are compared by means of robust covariances.

Variable Selection for Robust Mixture Regression Model with Skew Scale Mixtures of Normal Distributions

In this paper, we propose a robust mixture regression model based on the skew scale mixtures of normal distributions (RMR-SSMN) which can accommodate asymmetric, heavy-tailed and contaminated data better. For the variable selection problem, the penalized likelihood approach with a new combined penalty function which balances the SCAD and l2 penalty is proposed. The adjusted EM algorithm is presented to get parameter estimates of RMR-SSMN models at a faster convergence rate. As simulations show, our mixture models are more robust than general FMR models and the new combined penalty function outperforms SCAD for variable selection. Finally, the proposed methodology and algorithm are applied to a real data set and achieve reasonable results.

Industrial Food Quality Analysis Using New k-Nearest-Neighbour methods

The problem of predicting continuous scalar outcomes from functional predictors has received high levels of interest in recent years in many fields,especially in the food industry.The k-nearest neighbor(k-NN)method of Near-Infrared Reflectance(NIR)analysis is practical,relatively easy to implement,and becoming one of the most popular methods for conducting food quality based on NIR data.The k-NN is often named k nearest neighbor classifier when it is used for classifying categorical variables,while it is called k-nearest neighbor regression when it is applied for predicting noncategorical variables.The objective of this paper is to use the functional Near-Infrared Reflectance(NIR)spectroscopy approach to predict some chemical components with some modern statistical models based on the kernel and k-Nearest Neighbour procedures.In this paper,three NIR spectroscopy datasets are used as examples,namely Cookie dough,sugar,and tecator data.Specifically,we propose three models for this kind of data which are Functional Nonparametric Regression,Functional Robust Regression,and Functional Relative Error Regression,with both kernel and k-NN approaches to compare between them.The experimental result shows the higher efficiency of k-NN predictor over the kernel predictor.The predictive power of the k-NN method was compared with that of the kernel method,and several real data sets were used to determine the predictive power of both methods.

Mixed-Species Allometric Equations to Quantify Stem Volume and Tree Biomass in Dry Afromontane Forest of Ethiopia

Volume and biomass equations are essential tools to determine forest productivity and enable forest managers to make informed decisions. However, volume and biomass estimation equations are scarce for Afromontane forests in Africa in general and Ethiopia in particular. This limits our knowledge of the standing volume of wood, biomass, and carbon stock of the forests therein. In this study, we developed a new mixed-species volume and biomass equations for Afromontane forests and compared them with generic pantropical and local models. A total of 193 sampled trees from seven dominant tree species were used to develop the equations. Various volume and biomass equations were fitted using robust linear and nonlinear regression. Model comparison indicated that the best model to estimate stem volume was ln(v)=-9.909+ 0.954*ln(d2h), whereas the best model to estimate biomass was ln(b)=-2.983+ 0.949*ln(ρd2h) . These equations explained over 85% of the variations in the stem volume and biomass measurements. The mean density and basal area of trees in the forest with d ≥ 2 cm was 631.5 stems·ha-1 and 24.4 m2·ha-1. Based on the newly developed equations, the forest has on average 303.0 m3·ha-1 standing volume of wood and 283.8 Mg·ha-1 biomass stock. The newly developed allometric equations derived from this study can be used to accurately determine the stem volume, biomass, and carbon storage in the Afromontane forests in Ethiopia and elsewhere with similar stand characteristics and ecological conditions. By contrast, the generic pan-tropical and other local models appear to provide biased estimates and are not suitable for dry Afromontane forests in Ethiopia. The estimated stem biomass and carbon stock in the Chilimo forest are comparable with the estimates from various tropical forests and woodlands elsewhere in Africa, indicating the importance of dry Afromontane forest for climate change mitigation.

Empirical Analysis of Forest Pest Control Efficiency from 2003 to 2014 in China

Three indexes including forest pest occurrence area,control area and input fund of 31 provinces from 2003 to 2014 were selected from Forestry Statistical Yearbook,to establish dynamic interaction index evaluation system with clustering robust regression model and Stata 13. 0 software. Total forest pest control efficiency in China was determined according to the computing result of entropy method. Suggestions such as improving forest pest control efficiency,increasing service efficiency and input amount of forest pest control input funds were put forward. It will provide empirical basis for target management evaluation of forest pest control work and accountability system.

Linear regression under model uncertainty

We reexamine the classical linear regression model when it is subject to two types of uncertainty:(i)some covariates are either missing or completely inaccessible,and(ii)the variance of the measurement error is undetermined and changing according to a mechanism unknown to the statistician.By following the recent theory of sublinear expectation,we propose to characterize such mean and variance uncertainty in the response variable by two specific nonlinear random variables,which encompass an infinite family of probability distributions for the response variable in the sense of(linear)classical probability theory.The approach enables a family of estimators under various loss functions for the regression parameter and the parameters related to model uncertainty.The consistency of the estimators is established under mild conditions in the data generation process.Three applications are introduced to assess the quality of the approach including a forecasting model for the S&P Index.

The role of green industrial transformation in mitigating carbon emissions:Exploring the channels of technological innovation and environmental regulation

The industrial sector is vital to economic progress,yet industrial pollution poses environmental and economic concerns.The purpose of the study was to investigate the influence of green industrial transformation in re-ducing Pakistan’s carbon intensity between 1975 and 2020.Carbon emissions are considered an endogenous construct,while foreign direct investment(FDI)inflows,technological innovation,green industrial transforma-tion,environmental legislation,and research and development(R&D)investment are possible mediators.The association between variables is assessed using the robust least-squares approach.Green industrial transforma-tion is connected with lower carbon emissions,yet technical innovation,R&D investment,and inbound FDI raise a country’s carbon emissions.The findings support the pollution haven hypothesis in a country.The causality esti-mates indicate that inward FDI contributes to environmental regulations;green industrial transformation directly relates to inbound FDI and R&D expenditures;and technological innovations correspond to inbound FDI,R&D ex-penditures,industrial ecofriendly progression,and environmental standards.According to the impulse response function,environmental policies are anticipated to have a differential effect on carbon emissions in 2023,2024,2028-2030,while they are likely to decrease in the years 2025-2027 and 2031 forward.Additionally,inward FDI and technology advancements would almost certainly result in a rise in carbon emissions over time.Green industrial transitions are projected to result in a ten-year reduction in carbon emissions.The variance decomposi-tion analysis indicates that eco-friendly industrial adaptations would likely have the largest variance error shock on carbon emissions(11.747%),followed by inbound FDI,technological advancements,and regulatory changes,with R&D spending having a minimal impact over time.Pakistan’s economy should foster a green industrial revolution to avoid pollution and increase environmental sustainability to meet its environmental goals.

Robust Filtration Techniques in Geometrical Metrology and Their Comparison

Filtration is one of the core elements of analysis tools in geometrical metrology. Filtration techniques are progressing along with the advancement of manufacturing technology. Modern filtration techniques are required to be robust against outliers, applicable to surfaces with complex geometry and reliable in whole range of measurement data. A comparison study is conducted to evaluate commonly used robust filtration techniques in the field of geometrical metrology, including the two-stage Gaussian filter, the robust Gaussian regression filter, the robust spline filter and morphological filters. They are compared in terms of four aspects： functionality, mathematical computation, capability and characterization parameters. As a result, this study offers metrologists a guideline to choose the appropriate filter for various applications.