Polar Depth for Potentially Heavy-Tailed Data

Motivated by the analysis of the behaviour of extremes from multivariate heavy-tailed distributions, we introduce a novel notion of statistical depth, referred to as Polar Depth. The polar depth function is naturally expressed in polar coordinates, as is the limiting distribution of a regularly varying random variable, beyond asymptotically large thresholds, once its marginals have been appropriately normalized. Not only does the polar depth function make it easy to order the extreme values taken by a heavy-tailed random variable X and finds natural applications in anomaly detection, but it is also possible to show, as we prove it under appropriate assumptions in this article, that the polar depth of the largest observations, i.e. observations X which norm is larger than t>0, converges to the polar depth of the limiting distribution as t converges to infinity. Although designed to quantify the depth of multivariate extremes, the polar depth is interesting in its own right, insofar as this notion is more relevant for distributions whose support is included in a halfspace than the alternatives proposed in the literature, the halfspace depth in particular. Here, we demonstrate its properties and analyze statistical issues related to its estimation from both finite-sample and asymptotic points of view. We present numerical results to empirically demonstrate its relevance, particularly for the statistical analysis of extreme observations and more specifically for the identification of anomalies among them.

EMOTHAW: A novel database for emotional state recognition from handwriting

The detection of negative emotions through daily activities such as handwriting is useful for promoting well-being. The spread of human-machine interfaces such as tablets makes the collection of handwriting samples easier. In this context, we present a first publicly available handwriting database which relates emotional states to handwriting, that we call EMOTHAW. This database includes samples of 129 participants whose emotional states, namely anxiety, depression and stress, are assessed by the Depression Anxiety Stress Scales (DASS) questionnaire. Seven tasks are recorded through a digitizing tablet: pentagons and house drawing, words copied in handprint, circles and clock drawing, and one sentence copied in cursive writing. Records consist in pen positions, on-paper and in-air, time stamp, pressure, pen azimuth and altitude. We report our analysis on this database. From collected data, we first compute measurements related to timing and ductus. We compute separate measurements according to the position of the writing device: on paper or in-air. We analyse and classify this set of measurements (referred to as features) using a random forest approach. This latter is a machine learning method [2], based on an ensemble of decision trees, which includes a feature ranking process. We use this ranking process to identify the features which best reveal a targeted emotional state. We then build random forest classifiers associated to each emotional state. Our results, obtained from cross-validation experiments, show that the targeted emotional states can be identified with accuracies ranging from 60% to 71%.