Mild Outliers Research Articles

Multivariate contaminated normal mixture regression modeling of longitudinal data based on jointmean‐covariancemodel

AbstractOutliers are common in longitudinal data analysis, and the multivariate contaminated normal (MCN) distribution in model‐based clustering is often used to detect outliers and provide robust parameter estimates in each subgroup. In this paper, we propose a method, the mixture of MCN (MCNM), based on the joint mean‐covariance model, specifically designed to analyze longitudinal data characterized by mild outliers. Our model can automatically detect outliers in longitudinal data and provide robust parameter estimates in each subgroup. We use iteratively expectation‐conditional maximization (ECM) algorithm and Aitken acceleration to estimate the model parameters, achieving both algorithm acceleration and stable convergence. Our proposed method simultaneously clusters the population, identifies progression patterns of the mean and covariance structures for different subgroups over time, and detects outliers. To demonstrate the effectiveness of our method, we conduct simulation studies under various cases involving different proportions and degrees of contamination. Additionally, we apply our method to real data on the number of people infected with AIDS in 49 countries or regions from 2001 to 2021. Results show that our proposed method effectively clusters the data based on various mean progression trajectories. In summary, our proposed MCNM based on the joint mean‐covariance model and MCD of covariance matrices provides a robust method for clustering longitudinal data with mild outliers. It effectively detects outliers and identifies progression patterns in different groups over time, making it valuable for various applications in longitudinal data analysis.

Extending the Dynamic Range of Galaxy Outflow Scaling Relations: Massive Compact Galaxies with Extreme Outflows

We investigate galactic winds in the HizEA galaxies, a collection of 46 late-stage galaxy mergers at z = 0.4–0.8, with stellar masses of , star formation rates (SFRs) of 20–500 M ⊙ yr−1, and ultra-compact (a few 100 pc) central star-forming regions. We measure their gas kinematics using the Mg ii λ λ 2796,2803 absorption lines in optical spectra from MMT, Magellan, and Keck. We find evidence of outflows in 90% of targets, with maximum outflow velocities of 550–3200 km s−1. We combine these data with ten samples from the literature to construct scaling relations for outflow velocity versus SFR, star formation surface density (ΣSFR), M *, and SFR/M *. The HizEA galaxies extend the dynamic range of the scaling relations by a factor of ∼2–4 in outflow velocity and an order of magnitude in SFR and ΣSFR. The ensemble scaling relations exhibit strong correlations between outflow velocity, SFR, SFR/R, and ΣSFR, and weaker correlations with M * and SFR/M *. The HizEA galaxies are mild outliers on the SFR and M * scaling relations, but they connect smoothly with more typical star-forming galaxies on plots of outflow velocity versus SFR/R and ΣSFR. These results provide further evidence that the HizEA galaxies’ exceptional outflow velocities are a consequence of their extreme star formation conditions rather than hidden black hole activity, and they strengthen previous claims that ΣSFR is one of the most important properties governing the velocities of galactic winds.

Multivariate Contaminated Normal Censored Regression Model: Properties and Maximum Likelihood Inference

The Multivariate Contaminated Normal (MCN) distribution which contains two extra parameters with respect to parameters of the multivariate normal distribution, one for controlling the proportion of mild outliers and the other for specifying the degree of contamination, has been widely applied in robust statistics in the case of elliptically heavy-tailed empirical distributions. This article extends the MCN model to data with possibly censored values due to limits of quantification, referred to as the MCN with censoring (MCN-C) model, and further establishes the censored multivariate linear regression model where the random errors have the MCN distribution, named as the MCN censored regression (MCN-CR) model. Two computationally feasible Expectation Conditional Maximization (ECM) algorithms are developed for maximum likelihood estimation of MCN-C and MCN-CR models. An information-based method is used to approximate the standard errors of location parameters and regression coefficients. The capability and effectiveness of the MCN-C and MCN-CR models are illustrated via two real-data examples. A simulation study is conducted to investigate the superiority of the proposed models in terms of fit, accuracy of parameter estimation and censored data recovery as compared with classical approaches. Supplementary materials for this article are available online.

Seemingly unrelated clusterwise linear regression for contaminated data

Clusterwise regression is an approach to regression analysis based on finite mixtures which is generally employed when sample observations come from a population composed of several unknown sub-populations. Whenever the response is continuous, Gaussian clusterwise linear regression models are usually employed. Such models have been recently robustified with respect to the possible presence of mild outliers in the sub-populations. However, in some fields of research, especially in the modelling of multivariate economic data or data from the social sciences, there may be prior information on the specific covariates to be considered in the linear term employed in the prediction of a certain response. As a consequence, covariates may not be the same for all responses. Thus, a novel class of multivariate Gaussian linear clusterwise regression models is proposed. This class provides an extension to mixture-based regression analysis for modelling multivariate and correlated responses in the presence of mild outliers that let the researcher free to use a different vector of covariates for each response. Details about the model identification and maximum likelihood estimation via an expectation-conditional maximisation algorithm are given. The performance of the new models is studied by simulation in comparison with other clusterwise linear regression models. A comparative evaluation of their effectiveness and usefulness is provided through the analysis of a real dataset.

Unconstrained representation of orthogonal matrices with application to common principal components

Many statistical problems involve the estimation of a left( dtimes dright) orthogonal matrix varvec{Q}. Such an estimation is often challenging due to the orthonormality constraints on varvec{Q}. To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible left( dtimes dright) matrix as the product of a left( dtimes dright) permutation matrix varvec{P}, a left( dtimes dright) unit lower triangular matrix varvec{L}, and a left( dtimes dright) upper triangular matrix varvec{U}. Thanks to the QR decomposition, we find the formulation of varvec{U} when the PLU decomposition is applied to varvec{Q}. We call the result as PLR decomposition; it produces a one-to-one correspondence between varvec{Q} and the dleft( d-1right) /2 entries below the diagonal of varvec{L}, which are advantageously unconstrained real values. Thus, once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to varvec{L}. For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of varvec{Q} in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.

Substitutions at H134 and in the 430-loop region in influenza B neuraminidases can confer reduced susceptibility to multiple neuraminidase inhibitors

With the introduction of the influenza specific neuraminidase inhibitors (NAIs) in 1999, there were concerns about the emergence and spread of resistant viruses in the community setting. Surveillance and testing of community isolates for their susceptibility to the NAIs was initially carried out by the Neuraminidase Inhibitor Susceptibility Network (NISN) and has subsequently been taken on by the global WHO influenza network laboratories. During the NISN surveillance, we identified two Yamagata lineage influenza B viruses with amino acid substitutions of H134Y (B/Auckland/2/2001) or W438R (B/Yokohama/12/2005) which had slightly elevated IC50 values for zanamivir and/or oseltamivir, but not sufficiently to be characterized as mild outliers at the time. As it has now been well demonstrated that mixed populations can mask the true magnitude of resistance of a mutant, we re-examined both of these isolates by plaque purification to see if the true susceptibilities were being masked due to mixed populations. Results confirmed that the B/Auckland isolate contained both wild type and H134Y mutant populations, with mutant IC50 values > 250 nM for both oseltamivir and peramivir in the enzyme inhibition assay. The B/Yokohama isolate also contained both wild type and W438R mutant populations, the latter now demonstrating IC50 values > 400 nM for zanamivir, oseltamivir and peramivir. In addition, plaque purification of the B/Yokohama isolate identified viruses with other single neuraminidase substitutions H134Y, H134R, H431R, or T436P. H134R and H431R viruses had IC50 values > 400 nM and >250 nM respectively against all three NAIs. All changes conferred much greater resistance to peramivir than to zanamivir, and less to oseltamivir, and affected the kinetics of binding and dissociation of the NAIs. Most affected affinity (Km) for the MUNANA substrate, but some had decreased while others had increased affinity. Despite resistance in the enzyme assay, no reduced susceptibility was seen in plaque reduction assays in MDCK cells for any of the mutant viruses. None of these substitutions was in the active site. Modelling suggests that these substitutions affect the 150 and 430-loop regions described for influenza A NAs, suggesting they may also be important for substrate and inhibitor binding for influenza B NAs.

Multiple scaled contaminated normal distribution and its application in clustering

The multivariate contaminated normal (MCN) distribution represents a simple heavy-tailed generalization of the multivariate normal (MN) distribution to model elliptical contoured scatters in the presence of mild outliers (also referred to as ‘bad’ points herein) and automatically detect bad points. The price of these advantages is two additional parameters: proportion of good observations and degree of contamination. However, in a multivariate setting, only one proportion of good observations and only one degree of contamination may be limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution. Among its parameters, we have an orthogonal matrix Γ. In the space spanned by the vectors (principal components) of Γ, there is a proportion of good observations and a degree of contamination for each component. Moreover, each observation has a posterior probability of being good with respect to each principal component. Thanks to this probability, the method provides directional robust estimates of the parameters of the nested MN and automatic directional detection of bad points. The term ‘directional’ is added to specify that the method works separately for each principal component. Mixtures of MSCN distributions are also proposed, and an expectation-maximization algorithm is used for parameter estimation. Real and simulated data are considered to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails.

Robust model-based clustering with mild and gross outliers

We propose a model-based clustering procedure where each component can take into account cluster-specific mild outliers through a flexible distributional assumption, and a proportion of observations is additionally trimmed. We propose a penalized likelihood approach for estimation and selection of the proportions of mild and gross outliers. A theoretically grounded penalty parameter is then obtained. Simulation studies illustrate the advantages of our procedure over flexible mixtures without trimming, and over trimmed normal mixture models (tclust). We conclude with an original real data example on the identification of the source from illicit drug shipments seized in Italy and Spain. The methodology proposed in this paper has been implemented in R functions which can be downloaded from https://github.com/afarcome/cntclust .

A new look at the inverse Gaussian distribution with applications to insurance and economic data

ABSTRACTInsurance and economic data are often positive, and we need to take into account this peculiarity in choosing a statistical model for their distribution. An example is the inverse Gaussian (IG), which is one of the most famous and considered distributions with positive support. With the aim of increasing the use of the IG distribution on insurance and economic data, we propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various branches of statistics, and we give some examples. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and does not allocate probability mass to unrealistic negative values. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers. Finally, for model-based clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain maximum likelihood estimates of the parameters of the mixture and contaminated models. We use insurance data about bodily injury claims, and economic data about incomes of Italian households, to illustrate the models.

The relationship between compressive strength and flexural strength of pavement geopolymer grouting material

To determine the relationship between compressive strength and flexural strength of pavement geopolymer grouting material, 20 groups of geopolymer grouting materials were prepared, the compressive strength and flexural strength were determined by mechanical properties test. On the basis of excluding the abnormal values through boxplot, the results show that, the compressive strength test results were normal, but there were two mild outliers in 7days flexural strength test. The compressive strength and flexural strength were linearly fitted by SPSS, six regression models were obtained by linear fitting of compressive strength and flexural strength. The linear relationship between compressive strength and flexural strength can be better expressed by the cubic curve model, and the correlation coefficient was 0.842.

ContaminatedMixt: An R Package for Fitting Parsimonious Mixtures of Multivariate Contaminated Normal Distributions

We introduce the R package ContaminatedMixt, conceived to disseminate the use of mixtures of multivariate contaminated normal distributions as a tool for robust clustering and classification under the common assumption of elliptically contoured groups. Thirteen variants of the model are also implemented to introduce parsimony. The expectationconditional maximization algorithm is adopted to obtain maximum likelihood parameter estimates, and likelihood-based model selection criteria are used to select the model and the number of groups. Parallel computation can be used on multicore PCs and computer clusters, when several models have to be fitted. Differently from the more popular mixtures of multivariate normal and t distributions, this approach also allows for automatic detection of mild outliers via the maximum a posteriori probabilities procedure. To exemplify the use of the package, applications to artificial and real data are presented.

Mixtures of multivariate contaminated normal regression models

Mixtures of regression models (MRMs) are widely used to investigate the relationship between variables coming from several unknown latent homogeneous groups. Usually, the conditional distribution of the response in each mixture component is assumed to be (multivariate) normal (MN-MRM). To robustify the approach with respect to possible elliptical heavy-tailed departures from normality, due to the presence of mild outliers, the multivariate contaminated normal MRM is here introduced. In addition to the parameters of the MN-MRM, each mixture component has a parameter controlling the proportion of outliers and one specifying the degree of contamination with respect to the response variable(s). Crucially, these parameters do not have to be specified a priori, adding flexibility to our approach. Furthermore, once the model is estimated and the observations are assigned to the groups, a finer intra-group classification in typical points and (mild) outliers, can be directly obtained. Identifiability conditions are provided, an expectation-conditional maximization algorithm is outlined for parameter estimation, and various implementation and operational issues are discussed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo experiments and compared with other procedures. The performance of this novel family of models is also illustrated on artificial and real data, with particular emphasis to the application in allometric studies.

Motion-Based Temporal Alignment of Independently Moving Cameras

This paper presents a method to establish a nonlinear temporal correspondence between two video sequences captured by cameras independently moving in a dynamic 3D scene. We assume that the 3D spatial poses of the cameras are known for each frame. With predefined trajectory basis, the coefficients of the reconstructed trajectory of a moving scene point reflect the rhythm in motion. A robust rank constraint from the coefficient matrices is exploited to measure the spatiotemporal alignment quality for every feasible pair of video fragments. Point correspondences across sequences are not required or even it is possible that different points are tracked in different sequences, only if they satisfy the assumption that every 3D point tracked in the observed sequence can be described as a linear combination of a subset of the 3D points tracked in the reference sequence. Synchronization is then performed using a graph-based search algorithm to find the globally optimal path that minimizes both spatial and temporal misalignments. Our algorithm can use both complete and incomplete feature trajectories along time, and is robust to mild outliers. We verify the robustness and performance of the proposed approach on synthetic data as well as on challenging real video sequences.

Parsimonious mixtures of multivariate contaminated normal distributions.

A mixture of multivariate contaminated normal distributions is developed for model-based clustering. In addition to the parameters of the classical normal mixture, our contaminated mixture has, for each cluster, a parameter controlling the proportion of mild outliers and one specifying the degree of contamination. Crucially, these parameters do not have to be specified apriori, adding a flexibility to our approach. Parsimony is introduced via eigen-decomposition of the component covariance matrices, and sufficient conditions for the identifiability of all the members of the resulting family are provided. An expectation-conditional maximization algorithm is outlined for parameter estimation and various implementation issues are discussed. Using a large-scale simulation study, the behavior of the proposed approach is investigated and comparison with well-established finite mixtures is provided. The performance of this novel family of models is also illustrated on artificial and real data.

Application of component-wise rank method to bivariate bioequivalence case

AbstractOne of the most common analyses in pharmaceutical research is bioequivalence test of two drugs. The FDA has endorsed the usage of Schuirmann's two one-sided hypotheses for the analyses in such studies. Generally, however, several measures on the drugs are taken simultaneously such that the data are multivariate. Nandakumar and McKean generalized Schuirmann's procedure to this multivariate setting. For bivariate data, the results can be summarized in a graphical display. These procedures are least-squares-type procedures and hence, are quite sensitive to mild outliers. To counter this sensitivity, Nandakumar and McKean also developed a simple highly efficient and robust analogue to their multivariate least-squares procedure. The robust results can also be displayed graphically, overlaid with the least-squares graphical results. In this paper, a SAS algorithm is presented, which implements these least-squares and robust multivariate tests for bioequivalence, including their graphical summaries.

Additional value of FDG-PET to contrast enhanced-computed tomography (CT) for the diagnosis of mediastinal lymph node metastasis in non-small cell lung cancer: a Japanese multicenter clinical study

This study was a controlled multicenter clinical study to verify the diagnostic effects of additional FDG-PET to contrast-enhanced CT for mediastinal lymph node metastasis in patients with operable non-small cell lung cancer (NSCLC). NSCLC patients with enlarged mediastinal lymph nodes (short diameter, 7-20 mm), confirmed using contrast-enhanced CT, were examined using FDG-PET to detect metastases prior to surgery. The primary endpoint was the accuracy for concomitantly used CT and FDG-PET showing the additional effects of FDG, compared with CT alone. The secondary endpoints were the clinical impact of FDG-PET on therapeutic decisions and adverse reaction from FDG administration. The images were interpreted by investigators at each institution. Moreover, blinded readings were performed by an image interpretation committee independent of the institutions. The gold standard was the pathological diagnosis determined by surgery or biopsy after PET, and patients in whom a pathological diagnosis was not obtained were excluded from the analysis. Among 99 subjects, the results for 81 subjects eligible for analysis showed that the accuracy improved from 69.1% (56/81) for CT alone to 75.3% (61/81) for CT + PET (p = 0.404). These findings contributed to treatment decisions in 63.0% (51/81) of the cases, mainly with regard to the selection of the operative procedure. The results of the image interpretation committee showed that the accuracy improved from 64.2% (52/81) (95% CI 52.8-74.6) for CT to 75.3% (61/81) (95% CI 64.5-84.2) for CT + PET. The accuracy for 106 mediastinal lymph nodes improved significantly from 62.3% (66/106) (95% CI 52.3-71.5) for CT to 79.2% (84/106) (95% CI 70.3-86.5) for CT + PET (p < 0.05). We found that no serious adverse drug reactions appeared in any of the 99 patients who received FDG, except for transient mild outliers in the laboratory data for two patients. The addition of FDG-PET to contrast-enhanced CT imaging for the staging of NSCLC improved the diagnostic accuracy for mediastinal lymph node metastasis. FDG-PET improved the precision of the staging of NSCLC and contributed to the surgical decisions.

A note on the behavior of the stahel-donoho estimator for outlier detection

The paper presents a number of empirical results concerning the detection of outliers via the Stahel-Donoho robust multivariate estimator (SDE). Specifically, the strategy put forward by Maronna and Yohai (1995) is deployed in various artificial data sets of relatively large size and dimensionality. It is found that (i) mild outliers tend to pose the most serious problems to the SDE with Huber weights; (ii) these problems are not due to the approximation method, they are intrinsic to the SDE; (iii) outliers can manifest at the lower tail of the distribution of scaled Mahalanobis distances (all outliers detected by Maronna and Yohai are located in the upper[ILL]); and (iv) the detection of outliers located at one corner of the data set may be complicated by the existence of a second outlying patch in the cloud (that is, the SDE is prone to suffer from the problem of masking). Overall, these results suggest that use of the SDE for outlier detection must be complemented with some visual analysis of the data cloud.