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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., Comment: Under submission at IEEE/ACM TASLP. arXiv admin note: text overlap with arXiv:2202.11479

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.

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. Anomaly 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., Comment: 41 pages, 25 figures, 13 tables

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., Comment: 30 pages

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., Comment: 8 pages, 4 figures, 2 tables, WACV2024 - SCIoT workshop

Data augmentation is an essential building block for learning efficient deep learning models. Among all augmentation techniques proposed so far, linear interpolation of training data points, also called mixup, has found to be effective for a large panel of applications. While the majority of works have focused on selecting the right points to mix, or applying complex non-linear interpolation, we are interested in mixing similar points more frequently and strongly than less similar ones. To this end, we propose to dynamically change the underlying distribution of interpolation coefficients through warping functions, depending on the similarity between data points to combine. We define an efficient and flexible framework to do so without losing in diversity. We provide extensive experiments for classification and regression tasks, showing that our proposed method improves both performance and calibration of models. Code available in https://github.com/ENSTA-U2IS/torch-uncertainty

Classical multivariate statistics measures the outlyingness of a point by its Mahalanobis distance from the mean, which is based on the mean and the covariance matrix of the data. A multivariate depth function is a function which, given a point and a distribution in d-space, measures centrality by a number between 0 and 1, while satisfying certain postulates regarding invariance, monotonicity, convexity and continuity. Accordingly, numerous notions of multivariate depth have been proposed in the literature, some of which are also robust against extremely outlying data. The departure from classical Mahalanobis distance does not come without cost. There is a trade-off between invariance, robustness and computational feasibility. In the last few years, efficient exact algorithms as well as approximate ones have been constructed and made available in R-packages. Consequently, in practical applications the choice of a depth statistic is no more restricted to one or two notions due to computational limits; rather often more notions are feasible, among which the researcher has to decide. The article debates theoretical and practical aspects of this choice, including invariance and uniqueness, robustness and computational feasibility. Complexity and speed of exact algorithms are compared. The accuracy of approximate approaches like the random Tukey depth is discussed as well as the application to large and high-dimensional data. Extensions to local and functional depths and connections to regression depth are shortly addressed.

Anomaly detection is a branch of machine learning and data analysis 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 industry and science. 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.

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.

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., Comment: Accepted at NeurIPS 2022

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 $\Sigma$ of a finite set, $\{1,\; \ldots,\; n\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $\Sigma$'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.

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.

Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in R-d. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional projections of the data, then the depth satisfies the so-called projection property. Such depths form an important class that includes many of the depths that have been proposed in literature. For depths that satisfy the projection property an approximate algorithm can easily be constructed since taking the minimum of the depths with respect to only a finite number of one-dimensional projections yields an upper bound for the depth with respect to the multivariate data. Such an algorithm is particularly useful if no exact algorithm exists or if the exact algorithm has a high computational complexity, as is the case with the halfspace depth or the projection depth. To compute these depths in high dimensions, the use of an approximate algorithm with better complexity is surely preferable. Instead of focusing on a single method we provide a comprehensive and fair comparison of several methods, both already described in the literature and original. (C) 2021 Elsevier B.V. All rights reserved.

Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in R-d. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional projections of the data, then the depth satisfies the so-called projection property. Such depths form an important class that includes many of the depths that have been proposed in literature. For depths that satisfy the projection property an approximate algorithm can easily be constructed since taking the minimum of the depths with respect to only a finite number of one-dimensional projections yields an upper bound for the depth with respect to the multivariate data. Such an algorithm is particularly useful if no exact algorithm exists or if the exact algorithm has a high computational complexity, as is the case with the halfspace depth or the projection depth. To compute these depths in high dimensions, the use of an approximate algorithm with better complexity is surely preferable. Instead of focusing on a single method we provide a comprehensive and fair comparison of several methods, both already described in the literature and original. (C) 2021 Elsevier B.V. All rights reserved.

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.

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.

The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.

The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.

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.

The recent enthusiasm for artificial intelligence (AI) is due principally to advances in deep learning. Deep learning methods are remarkably accurate, but also opaque, which limits their potential use in safety-critical applications. To achieve trust and accountability, designers and operators of machine learning algorithms must be able to explain the inner workings, the results and the causes of failures of algorithms to users, regulators, and citizens. The originality of this paper is to combine technical, legal and economic aspects of explainability to develop a framework for defining the "right" level of explain-ability in a given context. We propose three logical steps: First, define the main contextual factors, such as who the audience of the explanation is, the operational context, the level of harm that the system could cause, and the legal/regulatory framework. This step will help characterize the operational and legal needs for explanation, and the corresponding social benefits. Second, examine the technical tools available, including post hoc approaches (input perturbation, saliency maps...) and hybrid AI approaches. Third, as function of the first two steps, choose the right levels of global and local explanation outputs, taking into the account the costs involved. We identify seven kinds of costs and emphasize that explanations are socially useful only when total social benefits exceed costs.

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.

Following the seminal idea of Tukey (1975), 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 DD alpha-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.

Given data in , a Tukey kappa-trimmed region is the set of all points that have at least Tukey depth kappa w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension p > 2. We construct two novel algorithms to compute a Tukey kappa-trimmed region, a naive one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the naive one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median. for this article are available online.

Given data in , a Tukey kappa-trimmed region is the set of all points that have at least Tukey depth kappa w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension p > 2. We construct two novel algorithms to compute a Tukey kappa-trimmed region, a naive one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the naive one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median. for this article are available online.

Following the seminal idea of Tukey (1975), 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 DD alpha-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.

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.

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.

A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional space. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the DD-plot, which is a subset of the unit square. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification in . The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Chervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as by a benchmark study.

A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional space. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the DD-plot, which is a subset of the unit square. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification in . The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Chervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as by a benchmark study.

In this article, we describe five new Stata commands that fit and provide statistical inference in nonparametric frontier models. The tenonradial and teradial commands fit data envelopment models where nonradial and radial technical efficiency measures are computed (Fare, 1998, Fundamentals of Production Theory; Fare and Lovell, 1978, Journal of Economic Theory 19: 150-162; Fare, Grosskopf, and Lovell, 1994a, Production Frontiers). Technical efficiency measures are obtained by solving linear programming problems. The teradialbc, nptestind, and nptestrts commands provide tools for making statistical inference regarding radial technical efficiency measures (Simar and Wilson, 1998, Management Science 44: 49-61; 2000, Journal of Applied Statistics 27: 779-802; 2002, European Journal of Operational Research 139: 115-132). We provide a brief overview of the nonparametric efficiency measurement, and we describe the syntax and options of the new commands. Additionally, we provide an example showing the capabilities of the new commands. Finally, we perform a small empirical study of productivity growth.

For computing the exact value of the halfspace depth of a point w.r.t. a data cloud of n points in arbitrary dimension, a theoretical framework is suggested. Based on this framework a whole class of algorithms can be derived. In all of these algorithms the depth is calculated as the minimum over a finite number of depth values w.r.t. proper projections of the data cloud. Three variants of this class are studied in more detail. All of these algorithms are capable of dealing with data that are not in general position and even with data that contain ties. As is shown by simulations, all proposed algorithms prove to be very efficient. (C) 2015 Elsevier B.V. All rights reserved.

Composite Marginal Likelihood (CML) has become a popular approach for estimating spatial probit models. However, for spatial autoregressive specifications the existing brute-force implementations are infeasible in large samples as they rely on inverting the high-dimensional precision matrix of the latent state variable. The contribution of this paper is to provide a CML implementation that circumvents inversion of that matrix and therefore can also be applied to very large sample sizes. (C) 2016 Elsevier B.V. All rights reserved.

In this article, we describe five new Stata commands that fit and provide statistical inference in nonparametric frontier models. The tenonradial and teradial commands fit data envelopment models where nonradial and radial technical efficiency measures are computed (Fare, 1998, Fundamentals of Production Theory; Fare and Lovell, 1978, Journal of Economic Theory 19: 150-162; Fare, Grosskopf, and Lovell, 1994a, Production Frontiers). Technical efficiency measures are obtained by solving linear programming problems. The teradialbc, nptestind, and nptestrts commands provide tools for making statistical inference regarding radial technical efficiency measures (Simar and Wilson, 1998, Management Science 44: 49-61; 2000, Journal of Applied Statistics 27: 779-802; 2002, European Journal of Operational Research 139: 115-132). We provide a brief overview of the nonparametric efficiency measurement, and we describe the syntax and options of the new commands. Additionally, we provide an example showing the capabilities of the new commands. Finally, we perform a small empirical study of productivity growth.

For computing the exact value of the halfspace depth of a point w.r.t. a data cloud of n points in arbitrary dimension, a theoretical framework is suggested. Based on this framework a whole class of algorithms can be derived. In all of these algorithms the depth is calculated as the minimum over a finite number of depth values w.r.t. proper projections of the data cloud. Three variants of this class are studied in more detail. All of these algorithms are capable of dealing with data that are not in general position and even with data that contain ties. As is shown by simulations, all proposed algorithms prove to be very efficient. (C) 2015 Elsevier B.V. All rights reserved.

Composite Marginal Likelihood (CML) has become a popular approach for estimating spatial probit models. However, for spatial autoregressive specifications the existing brute-force implementations are infeasible in large samples as they rely on inverting the high-dimensional precision matrix of the latent state variable. The contribution of this paper is to provide a CML implementation that circumvents inversion of that matrix and therefore can also be applied to very large sample sizes. (C) 2016 Elsevier B.V. All rights reserved.

The -classifier, a nonparametric fast and very robust procedure, is described and applied to fifty classification problems regarding a broad spectrum of real-world data. The procedure first transforms the data from their original property space into a depth space, which is a low-dimensional unit cube, and then separates them by a projective invariant procedure, called -procedure. To each data point the transformation assigns its depth values with respect to the given classes. Several alternative depth notions (spatial depth, Mahalanobis depth, projection depth, and Tukey depth, the latter two being approximated by univariate projections) are used in the procedure, and compared regarding their average error rates. With the Tukey depth, which fits the distributions' shape best and is most robust, 'outsiders', that is data points having zero depth in all classes, appear. They need an additional treatment for classification. Evidence is also given about the dimension of the extended feature space needed for linear separation. The -procedure is available as an R-package.

The -classifier, a nonparametric fast and very robust procedure, is described and applied to fifty classification problems regarding a broad spectrum of real-world data. The procedure first transforms the data from their original property space into a depth space, which is a low-dimensional unit cube, and then separates them by a projective invariant procedure, called -procedure. To each data point the transformation assigns its depth values with respect to the given classes. Several alternative depth notions (spatial depth, Mahalanobis depth, projection depth, and Tukey depth, the latter two being approximated by univariate projections) are used in the procedure, and compared regarding their average error rates. With the Tukey depth, which fits the distributions' shape best and is most robust, 'outsiders', that is data points having zero depth in all classes, appear. They need an additional treatment for classification. Evidence is also given about the dimension of the extended feature space needed for linear separation. The -procedure is available as an R-package.

A new procedure, called D D alpha-procedure, is developed to solve the problem of classifying d-dimensional objects into q a parts per thousand yen 2 classes. The procedure is 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 D D alpha-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 support vector machine.

A new procedure, called D D alpha-procedure, is developed to solve the problem of classifying d-dimensional objects into q a parts per thousand yen 2 classes. The procedure is 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 D D alpha-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 support vector machine.

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.