A Robust Collaborative Recommendation Algorithm Based on k-distance and Tukey M-estimator

The existing collaborative recommendation algorithms have lower robustness against shilling attacks.With this problem in mind,in this paper we propose a robust collaborative recommendation algorithm based on k-distance and Tukey M-estimator.Firstly,we propose a k-distancebased method to compute user suspicion degree(USD).The reliable neighbor model can be constructed through incorporating the user suspicion degree into user neighbor model.The influence of attack profiles on the recommendation results is reduced through adjusting similarities among users.Then,Tukey M-estimator is introduced to construct robust matrix factorization model,which can realize the robust estimation of user feature matrix and item feature matrix and reduce the influence of attack profiles on item feature matrix.Finally,a robust collaborative recommendation algorithm is devised by combining the reliable neighbor model and robust matrix factorization model.Experimental results show that the proposed algorithm outperforms the existing methods in terms of both recommendation accuracy and robustness.

Study of M-estimator Variational Retrieval Using Simulated Feng Yun-3A Data

This paper adopts satellite channel brightness temperature simulation to study M-estimator variational retrieval. This approach combines both the advantages of classical variational inversion and robust M-estimators. Classical variational inversion depends on prior quality control to elim- inate outliers, and its errors follow a Gaussian distribution. We coupled the M-estimators to the framework of classical variational inversion to obtain a M-estimator variational inversion. The cost function contains the M-estimator to guarantee the robustness to outliers and improve the retrieval re- sults. The experimental evaluation adopts Feng Yun-3A （FY-3A） simulated data to add to the Gaussian and Non-Gaussian error. The variational in- version is used to obtain the inversion brightness temperature, and temperature and humidity data are used for validation. The preliminary results demonstrate the potential of M-estimator variational retrieval.

LIMITING BEHAVIOR OF RECURSIVE M-ESTIMATORS IN MULTIVARIATE LINEAR REGRESSION MODELS AND THEIR ASYMPTOTIC EFFICIENCIES

Recursive algorithms are very useful for computing M-estimators of regression coefficients and scatter parameters. In this article, it is shown that for a nondecreasing ul （t）, under some mild conditions the recursive M-estimators of regression coefficients and scatter parameters are strongly consistent and the recursive M-estimator of the regression coefficients is also asymptotically normal distributed. Furthermore, optimal recursive M-estimators, asymptotic efficiencies of recursive M-estimators and asymptotic relative efficiencies between recursive M-estimators of regression coefficients are studied.

Variable bandwidth and one-step local M-estimator

A robust version of local linear regression smoothers augmented with variable bandwidth is studied. The proposed method inherits the advantages of local polynomial regression and overcomes the shortcoming of lack of robustness of leastsquares techniques. The use of variable bandwidth enhances the flexibility of the resulting local M-estimators and makes them possible to cope well with spatially inhomogeneous curves, heteroscedastic errors and nonuniform design densities. Under appropriate regularity conditions, it is shown that the proposed estimators exist and are asymptotically normal. Based on the robust estimation equation, one-step local M-estimators are introduced to reduce computational burden. It is demonstrated that the one-step local M-estimators share the same asymptotic distributions as the fully iterative M-estimators, as long as the initial estimators are good enough. In other words, the onestep local M-estimators reduce significantly the computation cost of the fully iterative M-estimators without deteriorating their performance. This fact is also illustrated via simulations.

Nonlocal-Means Image Denoising Technique Using Robust M-Estimator

Edge preserved smoothing techniques have gained importance for the purpose of image processing applications A good edge preserving filter is given by nonlocal-means filter rather than any other linear model based approaches. This paper explores a different approach of nonlocal-means filter by using robust M-estimator function rather than the exponential function for its weight calculation. Here the filter output at each pixel is the weighted average of pixels with surrounding neighborhoods using the chosen robust M-estimator function. The main direction of this paper is to identify the best robust M-estimator function for nonlocal-means denoising algorithm. In order to speed up the computation, a new patch classification method is followed to eliminate the uncorrelated patches from the weighted averaging process. This patch classification approach compares favorably to existing techniques in respect of quality versus computational time. Validations using standard test images and brain atlas images have been analyzed and the results were compared with the other known methods. It is seen that there is reason to believe that the proposed refined technique has some notable points.

Algorithmic Study of M-Estimators for Multi-Function Sensor Data Reconstruction

This paper describes a data reconstruction technique for a multi-function sensor based on the Mestimator, which uses least squares and weighted least squares method. The algorithm has better robustness than conventional least squares which can amplify the errors of inaccurate data. The M-estimator places particular emphasis on reducing the effects of large data errors, which are further overcome by an iterative regression process which gives small weights to large off-group data errors and large weights to small data errors. Simulation results are consistent with the hypothesis with 81 groups of regression data having an average accuracy of 3.5%, which demonstrates that the M-estimator provides more accurate and reliable data reconstruction.

Moderate Deviations for M-estimators in Linear Models with φ-mixing Errors

In this paper, the moderate deviations for the M-estimators of regression parameter in a linear model are obtained when the errors form a strictly stationary Ф-mixing sequence. The results are applied to study many different types of M-estimators such as Huber＇s estimator, L^P-regression estimator, least squares estimator and least absolute deviation estimator.

Rates of convergence of M-estimators for partly linear models involving time series

The asymptotic behaviour of M-estimalors constructed with B-spline method based on strictly stationary β-mixing observations of a partly linear model is dealt with. Under some regular conditions, it is proved that the M-estimators of the vector of parameters are asymptotically normal and the M-estimators of the nonparametric component achieve the optimal convergence rates for nonparametric regression. Our asymptotic theory includes L1-, L2-, Lp-norm, and Huber estimators as special cases.

ON B-SPLINE M-ESTIMATORS IN A SEMIPARAMETRIC REGRESSION MODEL

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THE RATES OF CONVERGENCE OF M-ESTIMATORS FOR PARTLY LINEAR MODELS IN DEPENDENT CASES

Consider the partly linear model K = X1& + go(Ti) + ei, where {(Ti, Xi)}T is a strictlystationary Sequence of random variable8, the ei’8 are i.i.d. random errorsl the K’s are realvalued responsest fo is a &vector of parameters, X is a &vector of explanatory variables,Ti is another explanatory variable ranging over a nondegenerate compact interval. Bnd ona segmnt of observations (T1, Xi 1 Y1 ),’’’ f (Tn, X;, Yn), this article investigates the rates ofconvrgence of the M-estimators for Po and go obtained from the minimisation problemwhere H is a space of B-spline functions of order m + 1 and p(-) is a function chosen suitablyUnder some regularity conditions, it is shown that the estimator of go achieves the optimalglobal rate of convergence of estimators for nonparametric regression, and the estdriator offo is asymptotically normal. The M-estimators here include regression quantile estimators,Li-estimators, Lp-norm estimators, Huber’s type M-estimators and usual least squares estimators. Applications of the asymptotic theory to testing the hypothesis H0: A’β0 =β are alsodiscussed, where β is a given vector and A is a known d × do matrix with rank d0.

Bahadur Representation of Nonparametric M-Estimators for Spatial Processes

Under some mild conditions, we establish a strong Bahadur representation of a general class of nonparametric local linear M-estimators for mixing processes on a random field. If the socalled optimal bandwidth hn = O（｜n｜^-1/5）, n ∈ Z^d, is chosen, then the remainder rates in the Bahadur representation for the local M-estimators of the regression function and its derivative are of order O（｜n｜^-4/5 log ｜n｜）. Moreover, we derive some asymptotic properties for the nonparametric local linear M-estimators as applications of our result.

HUBER'S M-ESTIMATOR ON UNDERDETERMINED PROBLEMS

After surveying the theoretical aspects of Huber's M-estimator on underdeter-mined problems, two finite algorithms are presented. Both proceed in a construc-tive manner by moving from one partition to an adjacent one. One of the algorithm,which uses the tuning constant as a continuation parameter, also has the facility to simultaneously estimate the tuning constant and scaling factor. Stable and efficient implementation of the algorithms is presented together with numerical results. The L1-norm problem is mentioned as a special case.

SELECTING AN ADAPTIVE SEQUENCE FOR COMPUTING RECURSIVE M-ESTIMATORS IN MULTIVARIATE LINEAR REGRESSION MODELS

In this paper, the authors consider an adaptive recursive algorithm by selecting an adaptive sequence for computing M-estimators in multivariate linear regression models. Its asymptotic property is investigated. The recursive algorithm given by Miao and Wu （1996） is modified accordingly. Simu- lation studies of the Mgorithm is also provided. In addition, the Newton-Raphson iterative algorithm is considered for the purpose of comparison.

An Approximate M-estimation for the Parameters of Mixed Regression Model

In this paper, to keep scale inveriance, we propose an approximate M-estrmation for the mixed regression model and show consistency of the estimation under weaker conditions than that in [1].

The M-estimate of Local Linear Regression with Variable Window Breadth

In this paper, by using the Brouwer fixed point theorem, we consider the existence and uniqueness of the solution for local linear regression with variable window breadth.

A Study on Fast and Robust Vanishing Point Detection System Using Fast M-Estimation Method and Regional Division for In-vehicle Camera

The vanishing point detection technology helps automatic driving. In this paper, the straight lines on the road associated with the vanishing point are extracted efficiently by using the regional division and angle limitation. And, the vanishing point is detected robustly by using the fast M-estimation method. Proposed method could detect straight-line features associated with vanishing point detection efficient on the road. And the vanishing point was detected exactly by the effect of the fast M-estimation method when the straight-line features not associated with vanishing point detection were detected. The processing time of the proposed method was faster than the camera flame rate （30 fps）. Thus, the proposed method is capable of real-time processing.

基于自适应权值的点云三维物体重建算法研究

基于三维扫描点云数据的三维物体重建是计算机图形学中非常重要的课题,在计算机动画、医学图像处理等多方面都有应用。其中基于最小二乘问题的Levenberg-Marquart算法和基于极大似然估计的M-Estimator算法都是不错的方案。但是当点的数量过多过少或者点云中有噪声时,这些方案产生的结果都会有较大的误差,影响重建的效果。为了解决这两个问题,结合Levenberg-Marquart算法和M-Estimator算法,提出了一种新的算法。该算法结合Levenberg-Marquart算法较快的收敛性和M-Estimator算法的抗噪性,能很好地解决点数量较多和噪声点影响结果的问题。通过在M-Estimator的权重函数上进行改进,提出自适应的权值函数,用灵活变动和自适应的值代替原来的固定值,使算法在噪声等级较高时也能表现良好。最后将算法应用在球体和圆柱上,并和最新的研究成果进行对比,数据说明算法无论是在点云数量较多还是在噪声等级较高的情况下都明显优于其他已知算法。

Bayesian Classifier Based on Robust Kernel Density Estimation and Harris Hawks Optimisation

In real-world applications, datasets frequently contain outliers, which can hinder the generalization ability of machine learning models. Bayesian classifiers, a popular supervised learning method, rely on accurate probability density estimation for classifying continuous datasets. However, achieving precise density estimation with datasets containing outliers poses a significant challenge. This paper introduces a Bayesian classifier that utilizes optimized robust kernel density estimation to address this issue. Our proposed method enhances the accuracy of probability density distribution estimation by mitigating the impact of outliers on the training sample’s estimated distribution. Unlike the conventional kernel density estimator, our robust estimator can be seen as a weighted kernel mapping summary for each sample. This kernel mapping performs the inner product in the Hilbert space, allowing the kernel density estimation to be considered the average of the samples’ mapping in the Hilbert space using a reproducing kernel. M-estimation techniques are used to obtain accurate mean values and solve the weights. Meanwhile, complete cross-validation is used as the objective function to search for the optimal bandwidth, which impacts the estimator. The Harris Hawks Optimisation optimizes the objective function to improve the estimation accuracy. The experimental results show that it outperforms other optimization algorithms regarding convergence speed and objective function value during the bandwidth search. The optimal robust kernel density estimator achieves better fitness performance than the traditional kernel density estimator when the training data contains outliers. The Naïve Bayesian with optimal robust kernel density estimation improves the generalization in the classification with outliers.

Analysis of Salaries and Some Non-traditional Measures of Location

The paper deals with an analysis of how to use certain measures of location in analysis of salaries. One of the traditional measures of location, the mean should offer typical value of variable, representing all its values by the best way. Sometimes, the mean is located in the tail of the distribution and gives a very biased idea about the location of the distribution. In these cases, using different measures of location could be useful. Trimmed mean is described. The trimmed mean refers to a situation where a certain proportion of the largest and smallest observations are removed and the remaining observations are averaged. The construction of some measures of location is based on the analysis of outliers. Outliers are characterized. Then the possibilities of the detection of outliers are analyzed. Computing of one-step M-estimator and modified one-step M-estimator of location is described. A comparison of the trimmed means and M-estimators of location is presented. Finally, the paper focuses on the application of the trimmed mean and M-estimators of location in analysis of salaries. The analysis of salaries of employers of the big Slovak companies in second half of the year 2009 is realized. The data from the census are used in the analysis. The median, 20% trimmed mean and the characteristics, based on the one-step M-estimator of location and modified one step M-estimator, are calculated.

A Novel Robust Nonlinear Dynamic Data Reconciliation

Outlier in one variable will smear the estimation of other measurements in data reconciliation （DR）. In this article, a novel robust method is proposed for nonlinear dynamic data reconciliation, to reduce the influence of outliers on the result of DR. This method introduces a penalty function matrix in a conventional least-square objective function, to assign small weights for outliers and large weights for normal measurements. To avoid the loss of data information, element-wise Mahalanobis distance is proposed, as an improvement on vector-wise distance, to construct a penalty function matrix. The correlation of measurement error is also considered in this article. The method introduces the robust statistical theory into conventional least square estimator by constructing the penalty weight matrix and gets not only good robustness but also simple calculation. Simulation of a continuous stirred tank reactor, verifies the effectiveness of the proposed algorithm.