Kernel smoothing has greatly evolved since its inception to become an essential methodology in the data science tool kit for the 21st century. Its widespread adoption is due to its fundamental role for multivariate exploratory data analysis, as well as the crucial role it plays in composite solutions to complex data challenges. Multivariate Kernel Smoothing and Its Applications offers a comprehensive overview of both aspects. It begins with a thorough exposition of the approaches to achieve the two basic goals of estimating probability density functions and their derivatives. The focus then turns to the applications of these approaches to more complex data analysis goals, many with a geometric/topological flavour, such as level set estimation, clustering (unsupervised learning), principal curves, and feature significance. Other topics, while not direct applications of density (derivative) estimation but sharing many commonalities with the previous settings, include classification (supervised learning), nearest neighbour estimation, and deconvolution for data observed with error. For a data scientist, each chapter contains illustrative Open data examples that are analysed by the most appropriate kernel smoothing method. The emphasis is always placed on an intuitive understanding of the data provided by the accompanying statistical visualisations. For a reader wishing to investigate further the details of their underlying statistical reasoning, a graduated exposition to a unified theoretical framework is provided. The algorithms for efficient software implementation are also discussed. José E. Chacón is an associate professor at the Department of Mathematics of the Universidad de Extremadura in Spain. Tarn Duong is a Senior Data Scientist for a start-up which provides short distance carpooling services in France. Both authors have made important contributions to kernel smoothing research over the last couple of decades.

The adjusted Rand index (ARI) is commonly used in cluster analysis to measure the degree of agreement between two data partitions. Since its introduction, exploring the situations of extreme agreement and disagreement under different circumstances has been a subject of interest, in order to achieve a better understanding of this index. Here, an explicit formula for the lowest possible value of the ARI for two clusterings of given sizes is shown, and moreover a specific pair of clusterings achieving such a bound is provided.

Important information concerning a multivariate data set, such as clusters and modal regions, is contained in the derivatives of the probability density function. Despite this importance, nonparametric estimation of higher order derivatives of the density functions have received only relatively scant attention. Kernel estimators of density functions are widely used as they exhibit excellent theoretical and practical properties, though their generalization to density derivatives has progressed more slowly due to the mathematical intractabilities encountered in the crucial problem of bandwidth (or smoothing parameter) selection. This paper presents the first fully automatic, data-based bandwidth selectors for multivariate kernel density derivative estimators. This is achieved by synthesizing recent advances in matrix analytic theory which allow mathematically and computationally tractable representations of higher order derivatives of multivariate vector valued functions. The theoretical asymptotic properties as well as the finite sample behaviour of the proposed selectors are studied. In addition, we explore in detail the applications of the new data-driven methods for two other statistical problems: clustering and bump hunting. The introduced techniques are combined with the mean shift algorithm to develop novel automatic, nonparametric clustering procedures which are shown to outperform mixture-model cluster analysis and other recent nonparametric approaches in practice. Furthermore, the advantage of the use of smoothing parameters designed for density derivative estimation for feature significance analysis for bump hunting is illustrated with a real data example.

Despite its popularity, it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, whereas for other statistical problems the theoretical population goal is clearly defined (as in regression or classification), for some of the clustering methodologies it is difficult to specify the population goal to which the data-based clustering algorithms should try to get close. This paper aims to provide some insight into the theoretical foundations of clustering by focusing on two main objectives: to provide an explicit formulation for the ideal population goal of the modal clustering methodology, which understands clusters as regions of high density; and to present two new loss functions, applicable in fact to any clustering methodology, to evaluate the performance of a data-based clustering algorithm with respect to the ideal population goal. In particular, it is shown that only mild conditions on a sequence of density estimators are needed to ensure that the sequence of modal clusterings that they induce is consistent.