A Comparison of Different Methods for LAD Regression

2010 ◽  
Vol 143-144 ◽  
pp. 1328-1331
Author(s):  
Hai Jun Chen ◽  
Xiao Ling Liu ◽  
Ling Hui Liu
Keyword(s):  
Least Squares ◽  
Linear Model ◽  
Least Squares Method ◽  
Small Sample ◽  
Least Absolute Deviation ◽  
Absolute Deviation ◽  
Simple Alternative ◽  
Dual Forms

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.

scholarly journals Improved Image Quality for Static BLADE Magnetic Resonance Imaging Using the Total-Variation Regularized Least Absolute Deviation Solver

Tomography ◽  
2021 ◽  
Vol 7 (4) ◽  
pp. 555-572
Author(s):  
Hsin-Chia Chen ◽  
Haw-Chiao Yang ◽  
Chih-Ching Chen ◽  
Seb Harrevelt ◽  
Yu-Chieh Chao ◽  
...  
Keyword(s):  
Magnetic Resonance Imaging ◽  
Magnetic Resonance ◽  
Image Quality ◽  
Least Squares ◽  
Total Variation ◽  
Least Squares Method ◽  
Processing Unit ◽  
Least Absolute Deviation ◽  
Resonance Imaging ◽  
Absolute Deviation

In order to improve the image quality of BLADE magnetic resonance imaging (MRI) using the index tensor solvers and to evaluate MRI image quality in a clinical setting, we implemented BLADE MRI reconstructions using two tensor solvers (the least-squares solver and the L1 total-variation regularized least absolute deviation (L1TV-LAD) solver) on a graphics processing unit (GPU). The BLADE raw data were prospectively acquired and presented in random order before being assessed by two independent radiologists. Evaluation scores were examined for consistency and then by repeated measures analysis of variance (ANOVA) to identify the superior algorithm. The simulation showed the structural similarity index (SSIM) of various tensor solvers ranged between 0.995 and 0.999. Inter-reader reliability was high (Intraclass correlation coefficient (ICC) = 0.845, 95% confidence interval: 0.817, 0.87). The image score of L1TV-LAD was significantly higher than that of vendor-provided image and the least-squares method. The image score of the least-squares method was significantly lower than that of the vendor-provided image. No significance was identified in L1TV-LAD with a regularization strength of λ= 0.4–1.0. The L1TV-LAD with a regularization strength of λ= 0.4–0.7 was found consistently better than least-squares and vendor-provided reconstruction in BLADE MRI with a SENSitivity Encoding (SENSE) factor of 2. This warrants further development of the integrated computing system with the scanner.

Robust Identification of an Exponential Autoregressive Model

2020 ◽  
pp. 42-57
Author(s):  
A.V. Goryainov ◽  
V.B. Goryainov ◽  
W.M. Khing
Keyword(s):  
Least Squares ◽  
Autoregressive Model ◽  
Least Squares Method ◽  
Probability Distributions ◽  
Asymptotic Variance ◽  
Estimation Method ◽  
Nonlinear Effects ◽  
Exponential Model ◽  
Least Absolute Deviation ◽  
Absolute Deviation

One of the most common nonlinear time series (random processes with discrete time) models is the exponential autoregressive model. In particular, it describes such nonlinear effects as limit cycles, resonant jumps, and dependence of the oscillation frequency on amplitude. When identifying this model, the problem arises of estimating its parameters --- the coefficients of the corresponding autoregressive equation. The most common methods for estimating the parameters of an exponential model are the least squares method and the least absolute deviation method. Both of these methods have a number of disadvantages, to eliminate which the paper proposes an estimation method based on the robust Huber approach. The obtained estimates occupy an intermediate position between the least squares and least absolute deviation estimates. It is assumed that the stochastic sequence is described by the autoregressive equation of the first order, is stationary and ergodic, and the probability distribution of the innovations process of the model is unknown. Unbiased, consistency and asymptotic normality of the proposed estimate are established by computer simulation. Its asymptotic variance was found, which allows to obtain an explicit expression for the relative efficiency of the proposed estimate with respect to the least squares estimate and the least absolute deviation estimate and to calculate this efficiency for the most widespread probability distributions of the innovations sequence of the equation of the autoregressive model

scholarly journals Least Squares Estimations for the General Linear Model Parameters with Epsilon Skew Normal Error Term

2020 ◽  
pp. 636-645
Author(s):  
Hussain Karim Nashoor ◽  
Ebtisam Karim Abdulah
Keyword(s):  
Least Squares ◽  
Linear Model ◽  
Least Squares Method ◽  
General Linear Model ◽  
Ordinary Least Squares ◽  
Skewed Distribution ◽  
General Linear ◽  
Average Error ◽  
Model Parameters ◽  
Skew Normal

Examination of skewness makes academics more aware of the importance of accurate statistical analysis. Undoubtedly, most phenomena contain a certain percentage of skewness which resulted to the appearance of what is -called "asymmetry" and, consequently, the importance of the skew normal family . The epsilon skew normal distribution ESN (μ, σ, ε) is one of the probability distributions which provide a more flexible model because the skewness parameter provides the possibility to fluctuate from normal to skewed distribution. Theoretically, the estimation of linear regression model parameters, with an average error value that is not zero, is considered a major challenge due to having difficulties, as no explicit formula to calculate these estimates can be obtained. Practically, values for these estimates can be obtained only by referring to numerical methods. This research paper is dedicated to estimate parameters of the Epsilon Skew Normal General Linear Model (ESNGLM) using an adaptive least squares method, as along with the employment of the ordinary least squares method for estimating parameters of the General Linear Model (GLM). In addition, the coefficient of determination was used as a criterion to compare the models’ preference. These methods were applied to real data represented by dollar exchange rates. The Matlab software was applied in this work and the results showed that the ESNGLM represents a satisfactory model. 

INFLUENCE OF MATRIX ERRORS ON PARAMETER ESTIMATES BY THE LEAST SQUARES METHOD

2021 ◽  
Author(s):  
V. A. Galanina ◽  
◽  
L. A. Reshetov ◽  
M. V. Sokolovskay ◽  
A. E. Farafonova ◽  
...  
Keyword(s):  
Least Squares ◽  
Linear Model ◽  
Least Squares Method ◽  
Statistical Characteristics ◽  
Parameter Estimates ◽  
Model Matrix ◽  
Least Squares Estimates

The paper investigates the effect of distorsions of the linear model matrix on the statistical characteristics of the least squares estimates.

Least absolute deviation estimates in autoregression with infinite variance

1979 ◽  
Vol 16 (1) ◽  
pp. 104-116 ◽  
Author(s):  
S. Gross ◽  
W. L. Steiger
Keyword(s):  
Monte Carlo ◽  
Convergence Rate ◽  
Least Squares ◽  
Finite Order ◽  
Monte Carlo Study ◽  
Least Squares Estimator ◽  
Infinite Variance ◽  
Least Absolute Deviation ◽  
Absolute Deviation ◽  
Strongly Consistent

We consider an L1 analogue of the least squares estimator for the parameters of stationary, finite-order autoregressions. This estimator, the least absolute deviation (LAD), is shown to be strongly consistent via a result that may have independent interest. The striking feature is that the conditions are so mild as to include processes with infinite variance, notably the stationary, finite autoregressions driven by stable increments in Lα, α > 1. Finally, sampling properties of LAD are compared to those of least squares. Together with a known convergence rate result for least squares, the Monte Carlo study provides evidence for a conjecture on the convergence rate of LAD.

ASYMPTOTICALLY EFFICIENT MEDIAN REGRESSION IN THE PRESENCE OF HETEROSKEDASTICITY OF UNKNOWN FORM

2001 ◽  
Vol 17 (4) ◽  
pp. 765-784 ◽  
Author(s):  
Quanshui Zhao
Keyword(s):  
Least Squares Method ◽  
Generalized Least Squares ◽  
Least Absolute Deviation ◽  
Finite Sample ◽  
Median Regression ◽  
Absolute Deviation ◽  
Least Absolute Deviation Regression ◽  
Unknown Form ◽  
The One ◽  
Asymptotically Efficient

We consider a linear model with heteroskedasticity of unknown form. Using Stone's (1977, Annals of Statistics 5, 595–645) k nearest neighbors (k-NN) estimation approach, the optimal weightings for efficient least absolute deviation regression are estimated consistently using residuals from preliminary estimation. The reweighted least absolute deviation or median regression estimator with the estimated weights is shown to be equivalent to the estimator using the true but unknown weights under mild conditions. Asymptotic normality of the estimators is also established. In the finite sample case, the proposed estimators are found to outperform the generalized least squares method of Robinson (1987, Econometrica 55, 875–891) and the one-step estimator of Newey and Powell (1990, Econometric Theory 6, 295–317) based on a Monte Carlo simulation experiment.

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