Robust principal component analysis via ES-algorithm

In this paper, a new method for robust principal component analysis (PCA) is proposed. PCA is a widely used tool for dimension reduction without substantial loss of information. However, the classical PCA is vulnerable to outliers due to its dependence on the empirical covariance matrix. To avoid such weakness, several alternative approaches based on robust scatter matrix were suggested. A popular choice is ROBPCA that combines projection pursuit ideas with robust covariance estimation via variance maximization criterion. Our approach is based on the fact that PCA can be formulated as a regression-type optimization problem, which is the main difference from the previous approaches. The proposed robust PCA is derived by substituting square loss function with a robust penalty function, Huber loss function. A practical algorithm is proposed in order to implement an optimization computation, and furthermore, convergence properties of the algorithm are investigated. Results from a simulation study and a real data example demonstrate the promising empirical properties of the proposed method.     

稳健主成分分析方法研究及其在经济管理中的应用

传统的多元统计分析方法,如主成分分析方法和因子分析方法等的共同点是计算样本的均值向量和协方差矩阵,并在这两者的基础上计算其他统计量。当样本数据中没有离群值时,这些方法都能得到优良的结果。但是当样本数据中包括离群值时,计算结果就会很容易受到这些离群值的影响,这是因为传统的均值向量和协方差矩阵都不是稳健的统计量。本文对目前较流行的FAST-MCD方法的算法进行研究,构造了稳健的均值向量和稳健的协方差矩阵,应用到主成分分析中,并针对其不足之处提出改进方法。从模拟和实证的结果来看,改进后的的方法和新的稳健估计量确实能够对离群值起到很好的抵抗作用,大幅度地降低它们对计算结果的影响。     

Asymptotic Distribution of Robust Estimator for Functional Nonparametric Models

We propose a family of robust nonparametric estimators for regression function based on kernel method. We establish the asymptotic normality of the estimator under the concentration properties on small balls of the probability measure of the functional explanatory variables. Useful applications to prediction, discrimination in a semi-metric space, and confidence curves are given. In addition, to highlight the generality of our purpose and to emphasize the role of each of our hypotheses, several special cases of our general conditions are also discussed. Finally, some numerical study in chemiometrical real data are carried out to compare the sensitivity to outliers between the classical and robust regression.     

A comparison of different procedures for principal component analysis in the presence of outliers

Principal component analysis (PCA) is a popular technique that is useful for dimensionality reduction but it is affected by the presence of outliers. The outlier sensitivity of classical PCA (CPCA) has caused the development of new approaches. Effects of using estimates obtained by expectation–maximization – EM and multiple imputation – MI instead of outliers were examined on the artificial and a real data set. Furthermore, robust PCA based on minimum covariance determinant (MCD), PCA based on estimates obtained by EM instead of outliers and PCA based on estimates obtained by MI instead of outliers were compared with the results of CPCA. In this study, we tried to show the effects of using estimates obtained by MI and EM instead of outliers, depending on the ratio of outliers in data set. Finally, when the ratio of outliers exceeds 20%, we suggest the use of estimates obtained by MI and EM instead of outliers as an alternative approach.     

The evaluation of socio-economic development of development agency regions in Turkey using classical and robust principal component analyses

In this study, classical and robust principal component analyses are used to evaluate socioeconomic development of regions of development agencies that give service on the purpose of decreasing development difference among regions in Turkey. Due to the high differences between development levels of regions outlier problem occurs, hence robust statistical methods are used. Also, classical and robust statistical methods are used to investigate if there are any outliers in data set. In classic principal component analyse, the number of observations must be larger than the number of variables. Otherwise determinant of covariance matrix is zero. In Robust method for Principal Component Analysis (ROBPCA), a robust approach to principal component analyse in high-dimensional data, even if the number of variables is larger than the number of observations, principal components are obtained. In this paper, firstly 26 development agencies are evaluated with 19 variables by using principal component analysis based on classical and robust scatter matrices and then these 26 development agencies are evaluated with 46 variables by using the ROBPCA method.     

Fast and robust bootstrap

In this paper we review recent developments on a bootstrap method for robust estimators which is computationally faster and more resistant to outliers than the classical bootstrap. This fast and robust bootstrap method is, under reasonable regularity conditions, asymptotically consistent. We describe the method in general and then consider its application to perform inference based on robust estimators for the linear regression and multivariate location-scatter models. In particular, we study confidence and prediction intervals and tests of hypotheses for linear regression models, inference for location-scatter parameters and principal components, and classification error estimation for discriminant analysis.     

Geometric consistency of principal component scores for high-dimensional mixture models and its application

In this article, we consider clustering based on principal component analysis (PCA) for high-dimensional mixture models. We present theoretical reasons why PCA is effective for clustering high-dimensional data. First, we derive a geometric representation of high-dimension, low-sample-size (HDLSS) data taken from a two-class mixture model. With the help of the geometric representation, we give geometric consistency properties of sample principal component scores in the HDLSS context. We develop ideas of the geometric representation and provide geometric consistency properties for multiclass mixture models. We show that PCA can cluster HDLSS data under certain conditions in a surprisingly explicit way. Finally, we demonstrate the performance of the clustering using gene expression datasets.     

The FastHCS algorithm for robust PCA

Principal component analysis (PCA) is widely used to analyze high-dimensional data, but it is very sensitive to outliers. Robust PCA methods seek fits that are unaffected by the outliers and can therefore be trusted to reveal them. FastHCS (high-dimensional congruent subsets) is a robust PCA algorithm suitable for high-dimensional applications, including cases where the number of variables exceeds the number of observations. After detailing the FastHCS algorithm, we carry out an extensive simulation study and three real data applications, the results of which show that FastHCS is systematically more robust to outliers than state-of-the-art methods.     

A robust functional time series forecasting method

Univariate time series often take the form of a collection of curves observed sequentially over time. Examples of these include hourly ground-level ozone concentration curves. These curves can be viewed as a time series of functions observed at equally spaced intervals over a dense grid. Since functional time series may contain various types of outliers, we introduce a robust functional time series forecasting method to down-weigh the influence of outliers in forecasting. Through a robust principal component analysis based on projection pursuit, a time series of functions can be decomposed into a set of robust dynamic functional principal components and their associated scores. Conditioning on the estimated functional principal components, the crux of the curve-forecasting problem lies in modelling and forecasting principal component scores, through a robust vector autoregressive forecasting method. Via a simulation study and an empirical study on forecasting ground-level ozone concentration, the robust method demonstrates the superior forecast accuracy that dynamic functional principal component regression entails. The robust method also shows the superior estimation accuracy of the parameters in the vector autoregressive models for modelling and forecasting principal component scores, and thus improves curve forecast accuracy.     

Doubly robust weighted composite quantile regression based on SCAD-L2

In this article, a robust variable selection procedure based on the weighted composite quantile regression (WCQR) is proposed. Compared with the composite quantile regression (CQR), WCQR is robust to heavy-tailed errors and outliers in the explanatory variables. For the choice of the weights in the WCQR, we employ a weighting scheme based on the principal component method. To select variables with grouping effect, we consider WCQR with SCAD-L₂ penalization. Furthermore, under some suitable assumptions, the theoretical properties, including the consistency and oracle property of the estimator, are established with a diverging number of parameters. In addition, we study the numerical performance of the proposed method in the case of ultrahigh-dimensional data. Simulation studies and real examples are provided to demonstrate the superiority of our method over the CQR method when there are outliers in the explanatory variables and/or the random error is from a heavy-tailed distribution.     

Robust group non-convex estimations for high-dimensional partially linear models

High-dimensional data with a group structure of variables arise always in many contemporary statistical modelling problems. Heavy-tailed errors or outliers in the response often exist in these data. We consider robust group selection for partially linear models when the number of covariates can be larger than the sample size. The non-convex penalty function is applied to achieve both goals of variable selection and estimation in the linear part simultaneously, and we use polynomial splines to estimate the nonparametric component. Under regular conditions, we show that the robust estimator enjoys the oracle property. Simulation studies demonstrate the performance of the proposed method with samples of moderate size. The analysis of a real example illustrates that our method works well.     

Variable selection in partially linear additive models for modal regression

Based on B-spline basis functions and smoothly clipped absolute deviation (SCAD) penalty, we present a new estimation and variable selection procedure based on modal regression for partially linear additive models. The outstanding merit of the new method is that it is robust against outliers or heavy-tail error distributions and performs no worse than the least-square-based estimation for normal error case. The main difference is that the standard quadratic loss is replaced by a kernel function depending on a bandwidth that can be automatically selected based on the observed data. With appropriate selection of the regularization parameters, the new method possesses the consistency in variable selection and oracle property in estimation. Finally, both simulation study and real data analysis are performed to examine the performance of our approach.     

Multiple imputation for continuous variables using a Bayesian principal component analysis

ABSTRACT

We propose a multiple imputation method based on principal component analysis (PCA) to deal with incomplete continuous data. To reflect the uncertainty of the parameters from one imputation to the next, we use a Bayesian treatment of the PCA model. Using a simulation study and real data sets, the method is compared to two classical approaches: multiple imputation based on joint modelling and on fully conditional modelling. Contrary to the others, the proposed method can be easily used on data sets where the number of individuals is less than the number of variables and when the variables are highly correlated. In addition, it provides unbiased point estimates of quantities of interest, such as an expectation, a regression coefficient or a correlation coefficient, with a smaller mean squared error. Furthermore, the widths of the confidence intervals built for the quantities of interest are often smaller whilst ensuring a valid coverage.     

Robust Detection of Multiple Outliers in Grouped Multivariate Data

Many methods have been developed for detecting multiple outliers in a single multivariate sample, but very few for the case where there may be groups in the data. We propose a method of simultaneously determining groups (as in cluster analysis) and detecting outliers, which are points that are distant from every group. Our method is an adaptation of the BACON algorithm proposed by Billor, Hadi and Velleman for the robust detection of multiple outliers in a single group of multivariate data. There are two versions of our method, depending on whether or not the groups can be assumed to have equal covariance matrices. The effectiveness of the method is illustrated by its application to two real data sets and further shown by a simulation study for different sample sizes and dimensions for 2 and 3 groups, with and without planted outliers in the data. When the number of groups is not known in advance, the algorithm could be used as a robust method of cluster analysis, by running it for various numbers of groups and choosing the best solution.     

Interpreting the Principal Component Analysis of Multivariate Density Functions

Functional principal component analysis (FPCA) as a reduction data technique of a finite number T of functions can be used to identify the dominant modes of variation of numeric three-way data.

We carry out the FPCA on multidimensional probability density functions, relate this method to other standard methods and define its centered or standardized versions. Grounded on the relationship between FPCA of densities, FPCA of their corresponding characteristic functions, PCA of the MacLaurin expansions of these characteristic functions and dual STATIS method applied to their variance matrices, we propose a method for interpreting the results of the FPCA of densities. This method is based on the investigations of the relationships between the scores of the FPCA and the moments associated to the densities.

The method is illustrated using known Gaussian densities. In practice, FPCA of densities deals with observations of multidimensional variables on T occasions. These observations can be used to estimate the T associated densities (i) by estimating the parameters of these densities, assuming that they are Gaussian, or (ii) by using the Gaussian kernel method and choosing the matrix bandwidth by the normal reference rule. Thereafter, FPCA estimate is derived from these estimates and the interpretation method is carried out to explore the dominant modes of variation of the types of three-way data encountered in sensory analysis and archaeology.     

Robust wavelet shrinkage using robust selection of thresholds

This paper considers the problem of selecting a robust threshold of wavelet shrinkage. Previous approaches reported in literature to handle the presence of outliers mainly focus on developing a robust procedure for a given threshold; this is related to solving a nontrivial optimization problem. The drawback of this approach is that the selection of a robust threshold, which is crucial for the resulting fit is ignored. This paper points out that the best fit can be achieved by a robust wavelet shrinkage with a robust threshold. We propose data-driven selection methods for a robust threshold. These approaches are based on a coupling of classical wavelet thresholding rules with pseudo data. The concept of pseudo data has influenced the implementation of the proposed methods, and provides a fast and efficient algorithm. Results from a simulation study and a real example demonstrate the promising empirical properties of the proposed approaches.     

A fuzzy robust regression approach applied to bedload transport data

Fuzzy least-square regression can be very sensitive to unusual data (e.g., outliers). In this article, we describe how to fit an alternative robust-regression estimator in fuzzy environment, which attempts to identify and ignore unusual data. The proposed approach concerns classical robust regression and estimation methods that are insensitive to outliers. In this regard, based on the least trimmed square estimation method, an estimation procedure is proposed for determining the coefficients of the fuzzy regression model for crisp input-fuzzy output data. The investigated fuzzy regression model is applied to bedload transport data forecasting suspended load by discharge based on a real world data. The accuracy of the proposed method is compared with the well-known fuzzy least-square regression model. The comparison results reveal that the fuzzy robust regression model performs better than the other models in suspended load estimation for the particular dataset. This comparison is done based on a similarity measure between fuzzy sets. The proposed model is general and can be used for modeling natural phenomena whose available observations are reported as imprecise rather than crisp.     

An approach to non-linear principal components analysis using radially symmetric kernel functions

An approach to non-linear principal components using radially symmetric kernel basis functions is described. The procedure consists of two steps: a projection of the data set to a reduced dimension using a non-linear transformation whose parameters are determined by the solution of a generalized symmetric eigenvector equation. This is achieved by demanding a maximum variance transformation subject to a normalization condition (Hotelling's approach) and can be related to the homogeneity analysis approach of Gifi through the minimization of a loss function. The transformed variables are the principal components whose values define contours, or more generally hypersurfaces, in the data space. The second stage of the procedure defines the fitting surface, the principal surface, in the data space (again as a weighted sum of kernel basis functions) using the definition of self-consistency of Hastie and Stuetzle. The parameters of this principal surface are determined by a singular value decomposition and crossvalidation is used to obtain the kernel bandwidths. The approach is assessed on four data sets.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

Probabilistic Principal Component Analysis

Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based on a probability model. We demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model that is closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss, with illustrative examples, the advantages conveyed by this probabilistic approach to PCA.