Local depth

Abstract

Local depth is a generalization of ordinary depth able to reveal local features of the probability distribution. Liu's simplicial depth is primarily used, but results for Tukey's halfspace depth are also derived. It is shown that the maximizers of local depth can help to detect the mode(s) of a probability distribution. This work is devoted to the univariate case, but the main definitions are stated in the general multivariate case. Theoretical results and applications are illustrated with several examples.

Introduction

Data depth is a general method for the analysis of probability distributions and data sets. Of particular interest are geometrical depth functions, like Liu's simplicial depth (Liu, 1990), Oja's simplicial volume depth (Oja, 1983, Zuo and Serfling, 2000) and Tukey's halfspace depth (Tukey, 1975), because no distributional assumptions are required and the results are distribution-free. Depth methods allow several features and properties of univariate distributions to be extended to multivariate distributions. Notable examples are new functionals describing multivariate location and spread (e.g., deepest points and depth regions), depth L-statistics and general purpose graphical displays for data exploration and comparison in any dimension, e.g., the DD-plot and the scale curve (Liu et al., 1999). Inferential applications include multivariate analogues of Wilcoxon rank sum test (Liu and Singh, 1993), diagnostics of non-normality (Liu et al., 1999) and tests of location and scale differences derived from the DD-plot (Li and Liu, 2004). Depth methods are also used in classification and clustering (Ghosh and Chauduri, 2005, Jornsten, 2004) with results which appear competitive with classical methods.

The basic idea underlying all applications of data depth is to rank the points of the space according to their degree of centrality with the location(s) of the deepest point(s) identifying the center(s) of the distribution. Regular depth functions, including halfspace, simplicial and simplicial volume depth, are decreasing from the center and this monotonicity property implies that they admit just one maximizer, whatever the distribution may be, unimodal or multimodal. This fact has long been known (e.g., Zuo and Serfling, 2000, p. 461, see also Rafalin et al., 2006) and is perfectly coherent when the focus is on the global features of the distribution, disregarding local features like the peaks of the density function. However, the existence of multiple centers, their locations and relative importance are of interest in several theoretical and applied domains, e.g., mixture distributions and cluster analysis. In this paper we suggest simple modifications of simplicial and halfspace depth so as to allow them to record the local space geometry, near any given point. For simplicial depth, the proposal is to consider only random simplices with size not greater than a fixed threshold. For halfspace depth, halfspaces are replaced by infinite slabs with finite width. The formal definitions are given in Section 2. When the tuning parameter (simplex size, slab width) tends to infinity, the ordinary definitions are recovered, thus regular depth is a special case of local depth and this could be an advantage over alternative approaches, like Rafalin et al. (2006). Indeed, the role of the tuning parameter is to provide a continuum of degrees of detail of the description, from very narrow neighbourhoods of the points up to the whole space. Hence local depth is a class of center-outward ranking functions serving multiple purposes, according to the value of the tuning parameter: low values describe centralness of the points of the space conditional on a small window around them, higher values force wider windows and therefore produce rankings more and more similar to ordinary depth. The stationary points of local depth, also indexed by the window size, define a (set) parameter of the distribution including the points corresponding to local maxima/minima of local depth, coincident with (or near to) the modes and anti-modes of the distribution. The rationale is different from the familiar descriptions provided by the moments or, in the univariate case, by the quantiles: this new parameter corresponds to the points with comparatively higher/lower probability mass in the surrounding window. Again, as the window size grows higher, the parameter converges to the (unique) overall center of the distribution. In summary, local depth provides a broad framework for nonparametric analysis, including all the issues of ordinary depth as particular instances.

In the univariate case, the local versions of simplicial and halfspace depth are simple enough to allow a detailed investigation, at least for absolutely continuous distributions. The main result is that, unlike their global counterparts, they can indeed register multiple modes. This suggests that in the sample case the maximizers of local depth can be used to estimate the mode(s) of the underlying distribution. The details are given in 3 Maximizers of local depth, 4 Sample data.

Local depth has some contact points with density function and the relationship in the univariate case is described in Section 2.3. However, the method retains some distinctive features. The ranks it produces are probabilities (or estimates of probabilities, in the sample case) based on nonparametric functionals and they are invariant to all affine transformations. We argue that, in the univariate case, for suitable low values of the tuning parameter, sample local depth can be a valid substitute of a density estimate but at the present stage it is still unclear whether this result could extend to the general multivariate case.

To illustrate the topics, examples are provided throughout, both with specific distribution models and real data. The final section is devoted to an overall discussion.