Pavlo Mozharovskyi

Lorenzo Guerra, Linhan Xu, Paolo Bellavista et al.

The integration of digital devices in modern vehicles has revolutionized automotive technology, enhancing safety and the overall driving experience. The Controller Area Network (CAN) bus is a central system for managing in-vehicle communication between the electronic control units (ECUs). However, the CAN protocol poses security challenges due to inherent vulnerabilities, lacking encryption and authentication, which, combined with an expanding attack surface, necessitates robust security measures. In response to this challenge, numerous Intrusion Detection Systems (IDS) have been developed and deployed. Nonetheless, an open, comprehensive, and realistic dataset to test the effectiveness of such IDSs remains absent in the existing literature. This paper addresses this gap by considering the latest ROAD dataset, containing stealthy and sophisticated injections. The methodology involves dataset labelling and the implementation of both state-of-the-art deep learning models and traditional machine learning models to show the discrepancy in performance between the datasets most commonly used in the literature and the ROAD dataset, a more realistic alternative.

Anna Malinovskaya, Pavlo Mozharovskyi, Philipp Otto

The rapid advancement of models based on artificial intelligence demands innovative monitoring techniques which can operate in real time with low computational costs. In machine learning, especially if we consider artificial neural networks (ANNs), the models are often trained in a supervised manner. Consequently, the learned relationship between the input and the output must remain valid during the model's deployment. If this stationarity assumption holds, we can conclude that the ANN provides accurate predictions. Otherwise, the retraining or rebuilding of the model is required. We propose considering the latent feature representation of the data (called "embedding") generated by the ANN to determine the time when the data stream starts being nonstationary. In particular, we monitor embeddings by applying multivariate control charts based on the data depth calculation and normalized ranks. The performance of the introduced method is compared with benchmark approaches for various ANN architectures and different underlying data formats.

Yinghao Wang, Rémi Nahon, Enzo Tartaglione et al.

In this paper, we present a new approach to mental state classification from EEG signals by combining signal processing techniques and machine learning (ML) algorithms. We evaluate the performance of the proposed method on a dataset of EEG recordings collected during a cognitive load task and compared it to other state-of-the-art methods. The results show that the proposed method achieves high accuracy in classifying mental states and outperforms state-of-the-art methods in terms of classification accuracy and computational efficiency.

Pavlo Mozharovskyi, Romain Valla

Anomaly detection is a branch of data analysis and machine learning which aims at identifying observations that exhibit abnormal behaviour. Be it measurement errors, disease development, severe weather, production quality default(s) (items) or failed equipment, financial frauds or crisis events, their on-time identification, isolation and explanation constitute an important task in almost any branch of science and industry. By providing a robust ordering, data depth - statistical function that measures belongingness of any point of the space to a data set - becomes a particularly useful tool for detection of anomalies. Already known for its theoretical properties, data depth has undergone substantial computational developments in the last decade and particularly recent years, which has made it applicable for contemporary-sized problems of data analysis and machine learning. In this article, data depth is studied as an efficient anomaly detection tool, assigning abnormality labels to observations with lower depth values, in a multivariate setting. Practical questions of necessity and reasonability of invariances and shape of the depth function, its robustness and computational complexity, choice of the threshold are discussed. Illustrations include use-cases that underline advantageous behaviour of data depth in various settings.

Quentin Bouniot, Pavlo Mozharovskyi, Florence d'Alché-Buc

Among all data augmentation techniques proposed so far, linear interpolation of training samples, also called Mixup, has found to be effective for a large panel of applications. Along with improved predictive performance, Mixup is also a good technique for improving calibration. However, mixing data carelessly can lead to manifold mismatch, i.e., synthetic data lying outside original class manifolds, which can deteriorate calibration. In this work, we show that the likelihood of assigning a wrong label with mixup increases with the distance between data to mix. To this end, we propose to dynamically change the underlying distributions of interpolation coefficients depending on the similarity between samples to mix, and define a flexible framework to do so without losing in diversity. We provide extensive experiments for classification and regression tasks, showing that our proposed method improves predictive performance and calibration of models, while being much more efficient.

Jayneel Parekh, Quentin Bouniot, Pavlo Mozharovskyi et al.

Developing inherently interpretable models for prediction has gained prominence in recent years. A subclass of these models, wherein the interpretable network relies on learning high-level concepts, are valued because of closeness of concept representations to human communication. However, the visualization and understanding of the learnt unsupervised dictionary of concepts encounters major limitations, especially for large-scale images. We propose here a novel method that relies on mapping the concept features to the latent space of a pretrained generative model. The use of a generative model enables high quality visualization, and lays out an intuitive and interactive procedure for better interpretation of the learnt concepts by imputing concept activations and visualizing generated modifications. Furthermore, leveraging pretrained generative models has the additional advantage of making the training of the system more efficient. We quantitatively ascertain the efficacy of our method in terms of accuracy of the interpretable prediction network, fidelity of reconstruction, as well as faithfulness and consistency of learnt concepts. The experiments are conducted on multiple image recognition benchmarks for large-scale images. Project page available at https://jayneelparekh.github.io/VisCoIN_project_page/

Aël Quélennec, Enzo Tartaglione, Pavlo Mozharovskyi et al.

In the realm of efficient on-device learning under extreme memory and computation constraints, a significant gap in successful approaches persists. Although considerable effort has been devoted to efficient inference, the main obstacle to efficient learning is the prohibitive cost of backpropagation. The resources required to compute gradients and update network parameters often exceed the limits of tightly constrained memory budgets. This paper challenges conventional wisdom and proposes a series of experiments that reveal the existence of superior sub-networks. Furthermore, we hint at the potential for substantial gains through a dynamic neuron selection strategy when fine-tuning a target task. Our efforts extend to the adaptation of a recent dynamic neuron selection strategy pioneered by Bragagnolo et al. (NEq), revealing its effectiveness in the most stringent scenarios. Our experiments demonstrate, in the average case, the superiority of a NEq-inspired approach over a random selection. This observation prompts a compelling avenue for further exploration in the area, highlighting the opportunity to design a new class of algorithms designed to facilitate parameter update selection. Our findings usher in a new era of possibilities in the field of on-device learning under extreme constraints and encourage the pursuit of innovative strategies for efficient, resource-friendly model fine-tuning.

Aël Quélennec, Nour Hezbri, Pavlo Mozharovskyi et al.

Memory-efficient training of deep neural networks has become increasingly important as models grow larger while deployment environments impose strict resource constraints. We propose TraDy, a novel transfer learning scheme leveraging two key insights: layer importance for updates is architecture-dependent and determinable a priori, while dynamic stochastic channel selection provides superior gradient approximation compared to static approaches. We introduce a dynamic channel selection approach that stochastically resamples channels between epochs within preselected layers. Extensive experiments demonstrate TraDy achieves state-of-the-art performance across various downstream tasks and architectures while maintaining strict memory constraints, achieving up to 99% activation sparsity, 95% weight derivative sparsity, and 97% reduction in FLOPs for weight derivative computation.

Arturo Castellanos, Pavlo Mozharovskyi

Data depths are score functions that quantify in an unsupervised fashion how central is a point inside a distribution, with numerous applications such as anomaly detection, multivariate or functional data analysis, arising across various fields. The halfspace depth was the first depth to aim at generalising the notion of quantile beyond the univariate case. Among the existing variety of depth definitions, it remains one of the most used notions of data depth. Taking a different angle from the quantile point of view, we show that the halfspace depth can also be regarded as the minimum loss of a set of classifiers for a specific labelling of the points. By changing the loss or the set of classifiers considered, this new angle naturally leads to a family of "loss depths", extending to well-studied classifiers such as, e.g., SVM or logistic regression, among others. This framework directly inherits computational efficiency of existing machine learning algorithms as well as their fast statistical convergence rates, and opens the data depth realm to the high-dimensional setting. Furthermore, the new loss depths highlight a connection between the dataset and the right amount of complexity or simplicity of the classifiers. The simplicity of classifiers as well as the interpretation as a risk makes our new kind of data depth easy to explain, yet efficient for anomaly detection, as is shown by experiments.

Arturo Castellanos, Pavlo Mozharovskyi, Florence d'Alché-Buc et al.

Data depth is a statistical function that generalizes order and quantiles to the multivariate setting and beyond, with applications spanning over descriptive and visual statistics, anomaly detection, testing, etc. The celebrated halfspace depth exploits data geometry via an optimization program to deliver properties of invariances, robustness, and non-parametricity. Nevertheless, it implicitly assumes convex data supports and requires exponential computational cost. To tackle distribution's multimodality, we extend the halfspace depth in a Reproducing Kernel Hilbert Space (RKHS). We show that the obtained depth is intuitive and establish its consistency with provable concentration bounds that allow for homogeneity testing. The proposed depth can be computed using manifold gradient making faster than halfspace depth by several orders of magnitude. The performance of our depth is demonstrated through numerical simulations as well as applications such as anomaly detection on real data and homogeneity testing.

Aël Quélennec, Pavlo Mozharovskyi, Van-Tam Nguyen et al.

On-device neural network training faces critical memory constraints that limit the adaptation of pre-trained models to downstream tasks. We present MeDyate, a theoretically-grounded framework for memory-constrained dynamic subnetwork adaptation. Our approach introduces two key innovations: LaRa (Layer Ranking), an improved layer importance metric that enables principled layer pre-selection, and a dynamic channel sampling strategy that exploits the temporal stability of channel importance distributions during fine-tuning. MeDyate dynamically resamples channels between epochs according to importance-weighted probabilities, ensuring comprehensive parameter space exploration while respecting strict memory budgets. Extensive evaluation across a large panel of tasks and architectures demonstrates that MeDyate achieves state-of-the-art performance under extreme memory constraints, consistently outperforming existing static and dynamic approaches while maintaining high computational efficiency. Our method represents a significant step towards enabling efficient on-device learning by demonstrating effective fine-tuning with memory budgets as low as a few hundred kB of RAM.

Lorenzo Guerra, Thomas Chapuis, Guillaume Duc et al.

Detecting intrusions in network traffic is a challenging task, particularly under limited supervision and constantly evolving attack patterns. While recent works have leveraged graph neural networks for network intrusion detection, they often decouple representation learning from anomaly detection, limiting the utility of the embeddings for identifying attacks. We propose GraphIDS, a self-supervised intrusion detection model that unifies these two stages by learning local graph representations of normal communication patterns through a masked autoencoder. An inductive graph neural network embeds each flow with its local topological context to capture typical network behavior, while a Transformer-based encoder-decoder reconstructs these embeddings, implicitly learning global co-occurrence patterns via self-attention without requiring explicit positional information. During inference, flows with unusually high reconstruction errors are flagged as potential intrusions. This end-to-end framework ensures that embeddings are directly optimized for the downstream task, facilitating the recognition of malicious traffic. On diverse NetFlow benchmarks, GraphIDS achieves up to 99.98% PR-AUC and 99.61% macro F1-score, outperforming baselines by 5-25 percentage points.

Arturo Castellanos, Anna Korba, Pavlo Mozharovskyi et al.

Distances between probability distributions are a key component of many statistical machine learning tasks, from two-sample testing to generative modeling, among others. We introduce a novel distance between measures that compares them through a Schatten norm of their kernel covariance operators. We show that this new distance is an integral probability metric that can be framed between a Maximum Mean Discrepancy (MMD) and a Wasserstein distance. In particular, we show that it avoids some pitfalls of MMD, by being more discriminative and robust to the choice of hyperparameters. Moreover, it benefits from some compelling properties of kernel methods, that can avoid the curse of dimensionality for their sample complexity. We provide an algorithm to compute the distance in practice by introducing an extension of kernel matrix for difference of distributions that could be of independent interest. Those advantages are illustrated by robust approximate Bayesian computation under contamination as well as particle flow simulations.

Marcos Matabuena, Rahul Ghosal, Pavlo Mozharovskyi et al.

Depth measures are powerful tools for defining level sets in emerging, non--standard, and complex random objects such as high-dimensional multivariate data, functional data, and random graphs. Despite their favorable theoretical properties, the integration of depth measures into regression modeling to provide prediction regions remains a largely underexplored area of research. To address this gap, we propose a novel, model-free uncertainty quantification algorithm based on conditional depth measures--specifically, conditional kernel mean embeddings and an integrated depth measure. These new algorithms can be used to define prediction and tolerance regions when predictors and responses are defined in separable Hilbert spaces. The use of kernel mean embeddings ensures faster convergence rates in prediction region estimation. To enhance the practical utility of the algorithms with finite samples, we also introduce a conformal prediction variant that provides marginal, non-asymptotic guarantees for the derived prediction regions. Additionally, we establish both conditional and unconditional consistency results, as well as fast convergence rates in certain homoscedastic settings. We evaluate the finite--sample performance of our model in extensive simulation studies involving various types of functional data and traditional Euclidean scenarios. Finally, we demonstrate the practical relevance of our approach through a digital health application related to physical activity, aiming to provide personalized recommendations

Romain Valla, Pavlo Mozharovskyi, Florence d'Alché-Buc

At the crossway of machine learning and data analysis, anomaly detection aims at identifying observations that exhibit abnormal behaviour. Be it measurement errors, disease development, severe weather, production quality default(s) (items) or failed equipment, financial frauds or crisis events, their on-time identification and isolation constitute an important task in almost any area of industry and science. While a substantial body of literature is devoted to detection of anomalies, little attention is payed to their explanation. This is the case mostly due to intrinsically non-supervised nature of the task and non-robustness of the exploratory methods like principal component analysis (PCA). We introduce a new statistical tool dedicated for exploratory analysis of abnormal observations using data depth as a score. Abnormal component analysis (shortly ACA) is a method that searches a low-dimensional data representation that best visualises and explains anomalies. This low-dimensional representation not only allows to distinguish groups of anomalies better than the methods of the state of the art, but as well provides a -- linear in variables and thus easily interpretable -- explanation for anomalies. In a comparative simulation and real-data study, ACA also proves advantageous for anomaly analysis with respect to methods present in the literature.

Jayneel Parekh, Sanjeel Parekh, Pavlo Mozharovskyi et al.

This paper tackles two major problem settings for interpretability of audio processing networks, post-hoc and by-design interpretation. For post-hoc interpretation, we aim to interpret decisions of a network in terms of high-level audio objects that are also listenable for the end-user. This is extended to present an inherently interpretable model with high performance. To this end, we propose a novel interpreter design that incorporates non-negative matrix factorization (NMF). In particular, an interpreter is trained to generate a regularized intermediate embedding from hidden layers of a target network, learnt as time-activations of a pre-learnt NMF dictionary. Our methodology allows us to generate intuitive audio-based interpretations that explicitly enhance parts of the input signal most relevant for a network's decision. We demonstrate our method's applicability on a variety of classification tasks, including multi-label data for real-world audio and music.

Jayneel Parekh, Sanjeel Parekh, Pavlo Mozharovskyi et al.

This paper tackles post-hoc interpretability for audio processing networks. Our goal is to interpret decisions of a network in terms of high-level audio objects that are also listenable for the end-user. To this end, we propose a novel interpreter design that incorporates non-negative matrix factorization (NMF). In particular, a carefully regularized interpreter module is trained to take hidden layer representations of the targeted network as input and produce time activations of pre-learnt NMF components as intermediate outputs. Our methodology allows us to generate intuitive audio-based interpretations that explicitly enhance parts of the input signal most relevant for a network's decision. We demonstrate our method's applicability on popular benchmarks, including a real-world multi-label classification task.

Morgane Goibert, Stéphan Clémençon, Ekhine Irurozki et al.

The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation $Σ$ of a finite set, $\{1,\; \ldots,\; n\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $Σ$'s distribution $P$, it may neglect other informative features. It is the purpose of this paper to define analogs of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of depth function on the symmetric group. Overcoming the absence of vector space structure on $\mathfrak{S}_n$, the latter defines a center-outward ordering of the permutations in the support of $P$ and extends the classic metric-based formulation of consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that ranking depths should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments.

Guillaume Staerman, Eric Adjakossa, Pavlo Mozharovskyi et al.

The increasing automation in many areas of the Industry expressly demands to design efficient machine-learning solutions for the detection of abnormal events. With the ubiquitous deployment of sensors monitoring nearly continuously the health of complex infrastructures, anomaly detection can now rely on measurements sampled at a very high frequency, providing a very rich representation of the phenomenon under surveillance. In order to exploit fully the information thus collected, the observations cannot be treated as multivariate data anymore and a functional analysis approach is required. It is the purpose of this paper to investigate the performance of recent techniques for anomaly detection in the functional setup on real datasets. After an overview of the state-of-the-art and a visual-descriptive study, a variety of anomaly detection methods are compared. While taxonomies of abnormalities (e.g. shape, location) in the functional setup are documented in the literature, assigning a specific type to the identified anomalies appears to be a challenging task. Thus, strengths and weaknesses of the existing approaches are benchmarked in view of these highlighted types in a simulation study. Anomaly detection methods are next evaluated on two datasets, related to the monitoring of helicopters in flight and to the spectrometry of construction materials namely. The benchmark analysis is concluded by recommendation guidance for practitioners.

Guillaume Staerman, Pavlo Mozharovskyi, Stéphan Clémençon

Because it determines a center-outward ordering of observations in $\mathbb{R}^d$ with $d\geq 2$, the concept of statistical depth permits to define quantiles and ranks for multivariate data and use them for various statistical tasks (e.g. inference, hypothesis testing). Whereas many depth functions have been proposed \textit{ad-hoc} in the literature since the seminal contribution of \cite{Tukey75}, not all of them possess the properties desirable to emulate the notion of quantile function for univariate probability distributions. In this paper, we propose an extension of the \textit{integrated rank-weighted} statistical depth (IRW depth in abbreviated form) originally introduced in \cite{IRW}, modified in order to satisfy the property of \textit{affine-invariance}, fulfilling thus all the four key axioms listed in the nomenclature elaborated by \cite{ZuoS00a}. The variant we propose, referred to as the Affine-Invariant IRW depth (AI-IRW in short), involves the covariance/precision matrices of the (supposedly square integrable) $d$-dimensional random vector $X$ under study, in order to take into account the directions along which $X$ is most variable to assign a depth value to any point $x\in \mathbb{R}^d$. The accuracy of the sampling version of the AI-IRW depth is investigated from a nonasymptotic perspective. Namely, a concentration result for the statistical counterpart of the AI-IRW depth is proved. Beyond the theoretical analysis carried out, applications to anomaly detection are considered and numerical results are displayed, providing strong empirical evidence of the relevance of the depth function we propose here.

Guillaume Staerman, Pavlo Mozharovskyi, Pierre Colombo et al.

The design of a metric between probability distributions is a longstanding problem motivated by numerous applications in Machine Learning. Focusing on continuous probability distributions on the Euclidean space $\mathbb{R}^d$, we introduce a novel pseudo-metric between probability distributions by leveraging the extension of univariate quantiles to multivariate spaces. Data depth is a nonparametric statistical tool that measures the centrality of any element $x\in\mathbb{R}^d$ with respect to (w.r.t.) a probability distribution or a data set. It is a natural median-oriented extension of the cumulative distribution function (cdf) to the multivariate case. Thus, its upper-level sets -- the depth-trimmed regions -- give rise to a definition of multivariate quantiles. The new pseudo-metric relies on the average of the Hausdorff distance between the depth-based quantile regions w.r.t. each distribution. Its good behavior w.r.t. major transformation groups, as well as its ability to factor out translations, are depicted. Robustness, an appealing feature of this pseudo-metric, is studied through the finite sample breakdown point. Moreover, we propose an efficient approximation method with linear time complexity w.r.t. the size of the data set and its dimension. The quality of this approximation as well as the performance of the proposed approach are illustrated in numerical experiments.

Jayneel Parekh, Pavlo Mozharovskyi, Florence d'Alché-Buc

To tackle interpretability in deep learning, we present a novel framework to jointly learn a predictive model and its associated interpretation model. The interpreter provides both local and global interpretability about the predictive model in terms of human-understandable high level attribute functions, with minimal loss of accuracy. This is achieved by a dedicated architecture and well chosen regularization penalties. We seek for a small-size dictionary of high level attribute functions that take as inputs the outputs of selected hidden layers and whose outputs feed a linear classifier. We impose strong conciseness on the activation of attributes with an entropy-based criterion while enforcing fidelity to both inputs and outputs of the predictive model. A detailed pipeline to visualize the learnt features is also developed. Moreover, besides generating interpretable models by design, our approach can be specialized to provide post-hoc interpretations for a pre-trained neural network. We validate our approach against several state-of-the-art methods on multiple datasets and show its efficacy on both kinds of tasks.

Guillaume Staerman, Pierre Laforgue, Pavlo Mozharovskyi et al.

Issued from Optimal Transport, the Wasserstein distance has gained importance in Machine Learning due to its appealing geometrical properties and the increasing availability of efficient approximations. In this work, we consider the problem of estimating the Wasserstein distance between two probability distributions when observations are polluted by outliers. To that end, we investigate how to leverage Medians of Means (MoM) estimators to robustify the estimation of Wasserstein distance. Exploiting the dual Kantorovitch formulation of Wasserstein distance, we introduce and discuss novel MoM-based robust estimators whose consistency is studied under a data contamination model and for which convergence rates are provided. These MoM estimators enable to make Wasserstein Generative Adversarial Network (WGAN) robust to outliers, as witnessed by an empirical study on two benchmarks CIFAR10 and Fashion MNIST. Eventually, we discuss how to combine MoM with the entropy-regularized approximation of the Wasserstein distance and propose a simple MoM-based re-weighting scheme that could be used in conjunction with the Sinkhorn algorithm.

Guillaume Staerman, Pavlo Mozharovskyi, Stephan Clémençon

With the ubiquity of sensors in the IoT era, statistical observations are becoming increasingly available in the form of massive (multivariate) time-series. Formulated as unsupervised anomaly detection tasks, an abundance of applications like aviation safety management, the health monitoring of complex infrastructures or fraud detection can now rely on such functional data, acquired and stored with an ever finer granularity. The concept of statistical depth, which reflects centrality of an arbitrary observation w.r.t. a statistical population may play a crucial role in this regard, anomalies corresponding to observations with 'small' depth. Supported by sound theoretical and computational developments in the recent decades, it has proven to be extremely useful, in particular in functional spaces. However, most approaches documented in the literature consist in evaluating independently the centrality of each point forming the time series and consequently exhibit a certain insensitivity to possible shape changes. In this paper, we propose a novel notion of functional depth based on the area of the convex hull of sampled curves, capturing gradual departures from centrality, even beyond the envelope of the data, in a natural fashion. We discuss practical relevance of commonly imposed axioms on functional depths and investigate which of them are satisfied by the notion of depth we promote here. Estimation and computational issues are also addressed and various numerical experiments provide empirical evidence of the relevance of the approach proposed.

Guillaume Staerman, Pavlo Mozharovskyi, Stephan Clémençon et al.

For the purpose of monitoring the behavior of complex infrastructures (e.g. aircrafts, transport or energy networks), high-rate sensors are deployed to capture multivariate data, generally unlabeled, in quasi continuous-time to detect quickly the occurrence of anomalies that may jeopardize the smooth operation of the system of interest. The statistical analysis of such massive data of functional nature raises many challenging methodological questions. The primary goal of this paper is to extend the popular Isolation Forest (IF) approach to Anomaly Detection, originally dedicated to finite dimensional observations, to functional data. The major difficulty lies in the wide variety of topological structures that may equip a space of functions and the great variety of patterns that may characterize abnormal curves. We address the issue of (randomly) splitting the functional space in a flexible manner in order to isolate progressively any trajectory from the others, a key ingredient to the efficiency of the algorithm. Beyond a detailed description of the algorithm, computational complexity and stability issues are investigated at length. From the scoring function measuring the degree of abnormality of an observation provided by the proposed variant of the IF algorithm, a Functional Statistical Depth function is defined and discussed as well as a multivariate functional extension. Numerical experiments provide strong empirical evidence of the accuracy of the extension proposed.

Oleksii Pokotylo, Pavlo Mozharovskyi, Rainer Dyckerhoff

Following the seminal idea of Tukey, data depth is a function that measures how close an arbitrary point of the space is located to an implicitly defined center of a data cloud. Having undergone theoretical and computational developments, it is now employed in numerous applications with classification being the most popular one. The R-package ddalpha is a software directed to fuse experience of the applicant with recent achievements in the area of data depth and depth-based classification. ddalpha provides an implementation for exact and approximate computation of most reasonable and widely applied notions of data depth. These can be further used in the depth-based multivariate and functional classifiers implemented in the package, where the $DDα$-procedure is in the main focus. The package is expandable with user-defined custom depth methods and separators. The implemented functions for depth visualization and the built-in benchmark procedures may also serve to provide insights into the geometry of the data and the quality of pattern recognition.

Tatjana Lange, Karl Mosler, Pavlo Mozharovskyi

A new procedure, called DDa-procedure, is developed to solve the problem of classifying d-dimensional objects into q >= 2 classes. The procedure is completely nonparametric; it uses q-dimensional depth plots and a very efficient algorithm for discrimination analysis in the depth space [0,1]^q. Specifically, the depth is the zonoid depth, and the algorithm is the alpha-procedure. In case of more than two classes several binary classifications are performed and a majority rule is applied. Special treatments are discussed for 'outsiders', that is, data having zero depth vector. The DDa-classifier is applied to simulated as well as real data, and the results are compared with those of similar procedures that have been recently proposed. In most cases the new procedure has comparable error rates, but is much faster than other classification approaches, including the SVM.