Clustering, or unsupervised classification, is a task often plagued by outliers. Yet there is a paucity of work on handling outliers in clustering. Outlier identification algorithms tend to fall into three broad categories: outlier inclusion, outlier trimming, and post hoc outlier identification methods, with the former two often requiring pre-specification of the number of outliers. The fact that sample squared Mahalanobis distance is beta-distributed is used to derive an approximate distribution for the log-likelihoods of subset finite Gaussian mixture models. An algorithm is then proposed that removes the least plausible points according to the subset log-likelihoods, which are deemed outliers, until the subset log-likelihoods adhere to the reference distribution. This results in a trimming method, called OCLUST, that inherently estimates the number of outliers.
We consider the problem of clustering datasets in the presence of arbitrary outliers. Traditional clustering algorithms such as k-means and spectral clustering are known to perform poorly for datasets contaminated with even a small number of outliers. In this paper, we develop a provably robust spectral clustering algorithm that applies a simple rounding scheme to denoise a Gaussian kernel matrix built from the data points and uses vanilla spectral clustering to recover the cluster labels of data points. We analyze the performance of our algorithm under the assumption that the "good" data poin…
A weighted likelihood approach for robust fitting of a mixture of multivariate Gaussian components is developed in this work. Two approaches have been proposed that are driven by a suitable modification of the standard EM and CEM algorithms, respectively. In both techniques, the M-step is enhanced by the computation of weights aimed at downweighting outliers. The weights are based on Pearson residuals stemming from robust Mahalanobis-type distances. Formal rules for robust clustering and outlier detection can be also defined based on the fitted mixture model. The behavior of the proposed metho…
Classification of imbalanced data is one of the common problems in the recent field of data mining. Imbalanced data substantially affects the performance of standard classification models. Data-level approaches mainly use the oversampling methods to solve the problem, such as synthetic minority oversampling Technique (SMOTE). However, since the methods such as SMOTE generate instances by linear interpolation, synthetic data space may look like a polygonal. Also, the oversampling methods generate outliers of the minority class. In this paper, we proposed Gaussian based minority oversampling tec…
We give the first outlier-robust efficient algorithm for clustering a mixture of $k$ statistically separated d-dimensional Gaussians (k-GMMs). Concretely, our algorithm takes input an $ε$-corrupted sample from a $k$-GMM and whp in $d^{\text{poly}(k/η)}$ time, outputs an approximate clustering that misclassifies at most $k^{O(k)}(ε+η)$ fraction of the points whenever every pair of mixture components are separated by $1-\exp(-\text{poly}(k/η)^k)$ in total variation (TV) distance. Such a result was not previously known even for $k=2$. TV separation is the statistically weakest possible notion of …
Imperial College London（伦敦帝国理工学院）；University College London（伦敦大学学院）
We propose a parameter efficient Bayesian layer for hierarchical convolutional Gaussian Processes that incorporates Gaussian Processes operating in Wasserstein-2 space to reliably propagate uncertainty. This directly replaces convolving Gaussian Processes with a distance-preserving affine operator on distributions. Our experiments on brain tissue-segmentation show that the resulting architecture approaches the performance of well-established deterministic segmentation algorithms (U-Net), which has never been achieved with previous hierarchical Gaussian Processes. Moreover, by applying the same…
In this paper, a scale mixture of Normal distributions model is developed for classification and clustering of data having outliers and missing values. The classification method, based on a mixture model, focuses on the introduction of latent variables that gives us the possibility to handle sensitivity of model to outliers and to allow a less restrictive modelling of missing data. Inference is processed through a Variational Bayesian Approximation and a Bayesian treatment is adopted for model learning, supervised classification and clustering.
Laboratoire des Signaux et Systèmes (L2S)（信号与系统实验室）；CentraleSupélec-CNRS-Université Paris-Sud（巴黎中央理工-高等电力学院-法国国家科学研究中心-巴黎第十一大学）；Université Paris-Saclay（巴黎萨克雷大学）；Instituto de Cálculo（计算研究所）；Universidad de Buenos Aires（布宜诺斯艾利斯大学）；IMAS-CONICET（阿根廷国家科学技术研究委员会数学应用研究所）
Though very popular, it is well known that the EM for GMM algorithm suffers from non-Gaussian distribution shapes, outliers and high-dimensionality. In this paper, we design a new robust clustering algorithm that can efficiently deal with noise and outliers in diverse data sets. As an EM-like algorithm, it is based on both estimations of clusters centers and covariances. In addition, using a semi-parametric paradigm, the method estimates an unknown scale parameter per data-point. This allows the algorithm to accommodate for heavier tails distributions and outliers without significantly loosing…
Imperial College London（帝国理工学院）；University College London（伦敦大学学院）
Stacking Gaussian Processes severely diminishes the model's ability to detect outliers, which when combined with non-zero mean functions, further extrapolates low non-parametric variance to low training data density regions. We propose a hybrid kernel inspired from Varifold theory, operating in both Euclidean and Wasserstein space. We posit that directly taking into account the variance in the computation of Wasserstein-2 distances is of key importance towards maintaining outlier status throughout the hierarchy. We show improved performance on medium and large scale datasets and enhanced out-o…
Impulsed noise outliers are data points that differs significantly from other observations.They are generally removed from the data set through local regression or Kalman filter algorithm.However, these methods, or their generalizations, are not well suited when the number of outliers is ofthe same order as the number of low-noise data. In this article, we propose a new model for impulsenoised outliers based on simple latent linear Gaussian processes as in the Kalman Filter. We present a fastforward-backward algorithm to filter and smooth sequential data and which also detect these outliers.We…
Dynamic predictions of survival outcomes are of great interest to physicians and patients, since such predictions are useful elements of clinical decision-making. Joint modelling of longitudinal and survival data has been increasingly used to obtain dynamic predictions. A common assumption of joint modelling is that random-effects and error terms in the longitudinal sub-model are Gaussian. However, this assumption may be too restrictive, e.g. in the presence of outliers as commonly encountered in many real-life applications. A natural extension is to robustify the joint models by assuming more…
In this paper, we propose a robust non-Gaussian process emulator based on the Schweppe-type generalized maximum likelihood estimator, which is trained on metered time series of voltage phasors and power injections to perform stochastic power flow. Power system data are often corrupted with outliers caused by fault conditions, power outages, and extreme weather, to name a few. The proposed emulator bounds the influence of the outliers using weights calculated based on projection statistics, which are robust distances of the data points associated with the rows vectors of the factor space. Speci…
Non-Gaussian mixture models are gaining increasing attention for mixture model-based clustering particularly when dealing with data that exhibit features such as skewness and heavy tails. Here, such a mixture distribution is presented, based on the multivariate normal inverse Gaussian (MNIG) distribution. For parameter estimation of the mixture, a Bayesian approach via Gibbs sampler is used; for this, a novel approach to simulate univariate generalized inverse Gaussian random variables and matrix generalized inverse Gaussian random matrices is provided. The proposed algorithm will be applied t…
We present an approach to clustering time series data using a model-based generalization of the K-Means algorithm which we call K-Models. We prove the convergence of this general algorithm and relate it to the hard-EM algorithm for mixture modeling. We then apply our method first with an AR($p$) clustering example and show how the clustering algorithm can be made robust to outliers using a least-absolute deviations criteria. We then build our clustering algorithm up for ARMA($p,q$) models and extend this to ARIMA($p,d,q$) models. We develop a goodness of fit statistic for the models fitted to …
Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid（数学科学学院，马德里complutense大学）；Facultad de Ciencias Económicas y Empresariales, Universidad Autónoma de Madrid（经济与企业科学学院，马德里自治大学）
A Gaussian measurement error assumption, i.e., an assumption that the data are observed up to Gaussian noise, can bias any parameter estimation in the presence of outliers. A heavy tailed error assumption based on Student's t distribution helps reduce the bias. However, it may be less efficient in estimating parameters if the heavy tailed assumption is uniformly applied to all of the data when most of them are normally observed. We propose a mixture error assumption that selectively converts Gaussian errors into Student's t errors according to latent outlier indicators, leveraging the best of …
Universitá della Svizzera Italiana (USI)（瑞士意大利语区大学）
Entropic Outlier Sparsification (EOS) is proposed as a robust computational strategy for the detection of data anomalies in a broad class of learning methods, including the unsupervised problems (like detection of non-Gaussian outliers in mostly-Gaussian data) and in the supervised learning with mislabeled data. EOS dwells on the derived analytic closed-form solution of the (weighted) expected error minimization problem subject to the Shannon entropy regularization. In contrast to common regularization strategies requiring computational costs that scale polynomial with the data dimension, iden…
The conventional approach to data-driven inversion framework is based on Gaussian statistics that presents serious difficulties, especially in the presence of outliers in the measurements. In this work, we present maximum likelihood estimators associated with generalized Gaussian distributions in the context of Rényi, Tsallis and Kaniadakis statistics. In this regard, we analytically analyse the outlier-resistance of each proposal through the so-called influence function. In this way, we formulate inverse problems by constructing objective functions linked to the maximum likelihood estimators.…
