Choosing among notions of multivariate depth statistics

Abstract

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 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. We discuss the theoretical and practical aspects of this choice, including invariance and uniqueness, robustness and computational feasibility. Complexity and speed of exact algorithms are compared and the use of different depths in classification problems. The accuracy of approximate approaches like the random Tukey depth is discussed as well as the application to large and high-dimensional data. Also, local and functional depths are shortly addressed.

Eikonal depth: an optimal control approach to statistical depths

“…The eikonal depth does not necessarily define a single center, i.e. a point with maximal depth, in contrast to the Tukey depth and other well-studied statistical depths such as the Mahalanobis depth, or the convex peeling depth [39]. In some settings, this is actually a desirable property of the eikonal depth because it enables it to capture the multimodality of probability distributions, a property that is not shared by the aforementioned definitions of depth.…”

“…These include, for example, the projection depth [55], the Oja depth [42], the zonoid depth [17], the Mahalanobis depth [35], the convex peeling depth [3], and the Monge Kantorovich depth [12]. These depths each carry unique advantages and disadvantages, which are compared in [39]. In briefest summary, these depths tend to either be i) robust and intepretable, but difficult to compute (e.g.…”

“…A well-known property of the Tukey depth is its invariance under affine transformations, a property shared by some other statistical depths such as the Mahalanobis, simplicial, zonoid and convex peeling depths [39]. Although the eikonal depth is not invariant under affine transformations, Figure 2: For φ(s) = s α , the figures show the effect of α on the eikonal depth of a Gaussian mixture.…”

“…an affine invariant version of it can be constructed, in the same way as for the Oja, spatial, and lens depth [39]. More details on this procedure are provided after Proposition It is not hard to see that the eikonal depth is not affine invariant: As can be seen from its control interpretation, the value of the depth depends on the length of trajectories.…”

“…This property is important because a result by Serfling [45] guarantees that functions with this invariant property can be made affine invariant using a scatter transform. Following [39], a scatter matrix R X is a positive definite d×d matrix that satisfies R AX+b = λ X,A,b A R X A T for any A of full rank, any b, and some λ X,A,b > 0. In order to make the depth affine invariant, one needs to transform the data as…”