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Discriminant analysis (DA) is one of the most popular methods for classification due to its conceptual simplicity, low computational cost, and often solid performance. In its standard form, DA uses the arithmetic mean and sample covariance matrix to estimate the center and scatter of each class. We discuss and illustrate how this makes standard DA very sensitive to suspicious data points, such as outliers and mislabeled cases. We then present an overview of techniques for robust DA, which are more reliable in the presence of deviating cases. In particular, we review DA based on robust estimates of location and scatter, along with graphical diagnostic tools for visualizing the results of DA. This article is categorized under:Statistical and Graphical Methods of Data Analysis > Robust MethodsStatistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification [ABSTRACT FROM AUTHOR]

The product moment covariance matrix is a cornerstone of multivariate data analysis, from which one can derive correlations, principal components, Mahalanobis distances and many other results. Unfortunately, the product moment covariance and the corresponding Pearson correlation are very susceptible to outliers (anomalies) in the data. Several robust estimators of covariance matrices have been developed, but few are suitable for the ultrahigh-dimensional data that are becoming more prevalent nowadays. For that one needs methods whose computation scales well with the dimension, are guaranteed to yield a positive semidefinite matrix, and are sufficiently robust to outliers as well as sufficiently accurate in the statistical sense of low variability. We construct such methods using data transformations. The resulting approach is simple, fast, and widely applicable. We study its robustness by deriving influence functions and breakdown values, and computing the mean squared error on contaminated data. Using these results we select a method that performs well overall. This also allows us to construct a faster version of the DetectDeviatingCells method (Rousseeuw and Van den Bossche 2018) to detect cellwise outliers, which can deal with much higher dimensions. The approach is illustrated on genomic data with 12,600 variables and color video data with 920,000 dimensions. for this article are available online. [ABSTRACT FROM AUTHOR]

Quadratic discriminant analysis (QDA) is a widely used classification technique. Based on a training dataset, each class in the data is characterized by an estimate of its center and shape, which can then be used to assign unseen observations to one of the classes. The traditional QDA rule relies on the empirical mean and covariance matrix. Unfortunately, these estimators are sensitive to label and measurement noise which often impairs the model's predictive ability. Robust estimators of location and scatter are resistant to this type of contamination. However, they have a prohibitive computational cost for large scale industrial experiments. We present a novel QDA method based on a recent real-time robust algorithm. We additionally integrate an anomaly detection step to classify the most atypical observations into a separate class of outliers. Finally, we introduce the label bias plot, a graphical display to identify label and measurement noise in the training data. The performance of the proposed approach is illustrated in a simulation study with huge datasets, and on real datasets about diabetes and fruit. • Classical discriminant analysis is sensitive to mislabeled cases and outliers. • The new robust classification method runs much faster than existing ones. • The proposed label bias plot shows label noise and outliers in the training data. • In a classification of spectra, varying illumination setups created subgroups within classes. [ABSTRACT FROM AUTHOR]

Modern industrial machines can generate gigabytes of data in seconds, frequently pushing the boundaries of available computing power. Together with the time criticality of industrial processing this presents a challenging problem for any data analytics procedure. We focus on the deterministic minimum covariance determinant method (DetMCD), which detects outliers by fitting a robust covariance matrix. We construct a much faster version of DetMCD by replacing its initial estimators by two new methods and incorporating update-based concentration steps. The computation time is reduced further by parallel computing, with a novel robust aggregation method to combine the results from the threads. The speed and accuracy of the proposed real-time DetMCD method (RT-DetMCD) are illustrated by simulation and a real industrial application to food sorting. • Multivariate outliers can be detected relative to a robust center and covariance matrix. • Novel algorithmic techniques drastically speed up the computation of robust estimates. • New developments enable the processing of industrial data volumes. • A new aggregation method combines the results of parallel computations. • Real-time automatic food sorting is possible in milliseconds. [ABSTRACT FROM AUTHOR]