Simplicial Depth Research Articles

Explicit bivariate simplicial depth

The simplicial depth (SD) is a celebrated tool defining elements of nonparametric and robust statistics for multivariate data. While many properties of SD are well-established, and its applications are abundant, explicit expressions for SD are known only for a handful of the simplest multivariate probability distributions. This paper deals with SD in the plane. It (i) develops a one-dimensional integral formula for SD of any properly continuous probability distribution, (ii) gives exact explicit expressions for SD of uniform distributions on (both convex and non-convex) polygons in the plane or on the boundaries of such polygons, and (iii) discusses several implications of these findings to probability and statistics: (a) An upper bound on the maximum SD in the plane, (b) an implication for a test of symmetry of a bivariate distribution, and (c) a connection of SD with the classical Sylvester problem from geometric probability.

Two-sample testing with local community depth

AbstractIn this paper, we investigate nonparametric two-sample hypothesis testing with local community depth, a measure of data depth calculated from pairwise distances between observations. Unlike many common “center-outwards” measures of data depth, local community depth captures local features of a distribution such as multimodality and can adapt to modes with different scales. This gives local community depth greater power to detect scale changes in multimodal distributions, and importantly the method is parameter-free. The local depth employed supplements existing adaptations of depth to local regions and is computationally cheaper in high dimensions than many classic depths such as halfspace depth, regression depth, simplicial depth, and Mahalanobis depth. We test hypotheses by using local community depth in the Liu–Singh test and give conditions under which our method is consistent against fixed alternatives (and has a limiting null distribution). In simulations and an application to diagnosing dengue fever, we show that local community depth outperforms other depths when the underlying distribution is multimodal or asymmetric, and is comparable to other methods when the data is unimodal and symmetric.

Comparing machine learning algorithms by union-free generic depth

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we provide two examples of classifier comparisons on samples of standard benchmark data sets. Our results demonstrate promisingly the wide variety of different analysis approaches based on ufg methods. Furthermore, the examples outline that our approach differs substantially from existing benchmarking approaches, and thus adds a new perspective to the vivid debate on classifier comparison.1

Simplicial depths for fuzzy random variables

The recently defined concept of statistical depth function for fuzzy sets provides a theoretical framework for ordering fuzzy sets with respect to the distribution of a fuzzy random variable. One of the most used and studied statistical depth functions for multivariate data is simplicial depth, based on multivariate simplices. We introduce a notion of pseudosimplices generated by fuzzy sets and propose three generalizations of simplicial depth to fuzzy sets. Their theoretical properties are analyzed and the behavior of the proposals is illustrated through a study of both synthetic and real data.

Simplicial depth: Characterization and reconstruction

AbstractStatistical depth functions have been designed with the intention of extending nonparametric inference toward multivariate setups. As such, the depths should serve as multivariate analogues of the quantile functions known from the analysis of real‐valued data. The so‐called characterization and reconstruction questions are among the fundamental open problems of the contemporary depth research. Roughly speaking, they ask: (a) Is it is possible that two different datasets, or more generally, two different probability distributions, correspond to identical depths, or does the depth function uniquely characterize the underlying distribution? (b) Knowing a depth function, can we reconstruct the corresponding distribution? For any given depth to constitute a fully‐fledged alternative to the quantile function, the depth must characterize wide classes of probability measures, and these measures must be simple to recover from their depths. We investigate these characterization/reconstruction questions for the classical simplicial depth for multivariate data. We show that, under mild conditions, datasets (represented by measures putting equal mass to each datum in a dataset of size ) and atomic measures are characterized by, and can be easily reconstructed from, their simplicial depth.

Simplicial depth and its median: Selected properties and limitations

AbstractDepth functions are important tools of nonparametric statistics that extend orderings, ranks, and quantiles to the setup of multivariate data. We revisit the classical definition of the simplicial depth and explore its theoretical properties when evaluated with respect to datasets or measures that do not necessarily possess a symmetric density. Recent advances from discrete geometry are used to refine the results about the robustness and continuity of the simplicial depth and its induced multivariate median. Further, we compute the exact simplicial depth in several scenarios and point out some undesirable behavior: (i) the simplicial depth does not have to be maximized at the center of symmetry of the distribution, (ii) it is not necessarily unimodal, and can possess local extremes, and (iii) the sets of the induced multivariate medians or other central regions do not have to be connected.

Tropical Carathéodory with Matroids

Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.

Computing Colourful Simplicial Depth and Median in ℝ2

Computing Colourful Simplicial Depth and Median in ℝ2

Computing Simplicial Depth by Using Importance Sampling Algorithm and Its Application

Simplicial depth (SD) plays an important role in discriminant analysis, hypothesis testing, machine learning, and engineering computations. However, the computation of simplicial depth is hugely challenging because the exact algorithm is an NP problem with dimension d and sample size n as input arguments. The approximate algorithm for simplicial depth computation has extremely low efficiency, especially in high-dimensional cases. In this study, we design an importance sampling algorithm for the computation of simplicial depth. As an advanced Monte Carlo method, the proposed algorithm outperforms other approximate and exact algorithms in accuracy and efficiency, as shown by simulated and real data experiments. Furthermore, we illustrate the robustness of simplicial depth in regression analysis through a concrete physical data experiment.

On General Notions of Depth for Regression

Depth notions in location have generated tremendous attention in the literature. In fact, data depth and its applications remain as one of the most active research topics in statistics over the last three decades. Most favored notions of depth in location include Tukey (In Proceedings of the International Congress of Mathematicians $($Vancouver, B.C., 1974$)$, Vol. 2 (1975) 523–531) half-space depth (HD), Liu (Ann. Statist. 18 (1990) 405–414) simplicial depth and projection depth (PD) (Stahel (1981) and Donoho (1982), Liu (In $L_{1}$-Statistical Analysis and Related Methods $($Neuchâtel, 1992$)$ (1992) 279–294 North-Holland), Zuo and Serfling (Ann. Statist. 28 (2000) 461–482) and (ZS00) and Zuo (Ann. Statist. 31 (2003) 1460–1490)), among others. Depth notions in regression have also been proposed sporadically, nevertheless. The regression depth (RD) of Rousseeuw and Hubert (J. Amer. Statist. Assoc. 94 (1999) 388–433) (RH99), the most famous, exemplifies a direct extension of Tukey HD to regression. Other notions include Carrizosa (J. Multivariate Anal. 58 (1996) 21–26) and the ones proposed in this article via modifying a functional in Maronna and Yohai (Ann. Statist. 21 (1993) 965–990) (MY93). Is there any relationship between Carrizosa depth and the RD of RH99? Do these depth notions possess desirable properties? What are the desirable properties? Can existing notions serve well as depth notions in regression? These questions remain open. The major objectives of the article include (i) revealing the connection between Carrizosa depth and RD of RH99; (ii) expanding location depth evaluating criteria in ZS00 for regression depth notions; (iii) examining the existing regression notions with respect to the gauges; and (iv) proposing the regression counterpart of the eminent location projection depth.

A bivariate test for location based on simplicial depth

We translate the simplicial data depth concept of Liu (1990) into a bivariate test of location that is distribution free under the assumption of angular symmetry about zero, without loss of generality. The test statistic is a count of the number of data triangles that contain zero and rejects the null hypothesis for small values of the test statistic. A straightforward method for computing this statistic is provided. The exact null distribution is computed and tabled for n ≤ 15, the asymptotic null distribution is derived and a formula for approximate critical values is provided. Simulations show that the power of the test is comparable to other bivariate sign type tests. The analogous one dimensional distribution free test is found to be equivalent to a two sided one sample sign test. These ideas are then extended to three dimensions, where we show that, surprisingly, the test is not generally distribution free under the null hypothesis.

Parallel computation of bivariate point data depths and display of intrinsic depth segments

This paper presents a new way to compute simplicial and Tukey data depths using Open Multi‐Processing parallelization, which makes it practical to compute point depths for tens of thousands of points. The definition of point depth is the order statistic depth of a single point, here in two dimensions. Using the point depths, the regional depth characteristics of the dataset as a whole can be explored. Using this new methodology, fast parallel computation of both simplicial depth and Tukey depth for a dataset of n points has time complexity O(n2logn) with O(n) space, which is practical for n up to 100,000. Obtaining depths for a large number of points in a faster manner by parallel computation supports identifying the central region quickly, because the points of maximum depth are known. The point depth computation identifies the depths of selected spoke segments around each origin point. These spoke depths are used to create new visualizations of depth characteristics and contour depths without adding virtual points.

From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices

A set \(P = H \cup \{w\}\) of \(n+1\) points in general position in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on such a set P, it suffices to know the frequency vector of P. While there are roughly \(2^n\) distinct order types that correspond to wheel sets, the number of frequency vectors is only about \(2^{n/2}\). We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, triangulations, and many more. Based on that, the corresponding numbers of graphs can be computed efficiently. In particular, we rediscover an already known formula for w-embracing triangles spanned by H. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of w-embracing simplices. While our previous arguments in the plane do not generalize easily, we show how to use similar ideas in \(\mathbb {R}^d\) for any fixed d. The result is an \(O(n^{d-1})\) time algorithm for computing the simplicial depth of a point w in a set H of n points, improving on the previously best bound of \(O(n^d\log n)\). Based on our result about simplicial depth, we can compute the number of facets of the convex hull of \(n=d+k\) points in general position in \(\mathbb {R}^d\) in time \(O(n^{\max \{\omega ,k-2\}})\) where \(\omega \approx 2.373\), even though the asymptotic number of facets may be as large as \(n^k\).

Achieve privacy-preserving simplicial depth query over collaborative cloud servers

The simplicial depth (SD) of a query point $q\in \mathbb {R}^{d}$ with respect to a dataset $S\subset \mathbb {R}^{d}$ is defined based on counting all (d + 1)-dimensional simplices obtained from S that contain q. The simplicial depth is a ranking function which is frequently used in order to sort a multivariate dataset. In the higher dimension d, no better algorithm is known than the brute force method which takes Θ(nd+ 1) time, where |S| = n. Unfortunately, in contrast to the many advantages that have been previously identified by research studies, this depth function requires a massive amount of computation particularly for higher dimensional datasets. This challenge could be overcome by offloading the computation to cloud servers. However, delegating simplicial depth queries to not fully trusted cloud servers would be a source of serious security breaches and privacy issues. Therefore, in this paper, we target the privacy-preserving simplicial depth query over collaborative cloud servers. To this end, two resource-abundant cloud servers will be employed to perform such time consuming computation while maintaining the user’s privacy. Security analysis shows our proposed scheme achieves privacy-preserving requirements. In addition, some experiments based on a dataset generated by normal distribution are conducted, and the results validate the efficiency and practicality of our proposed scheme. Although this work only focuses on the planar case, our proposed scheme can be extended into higher dimension cases without significant alterations.

Integrated rank-weighted depth

We study depth measures for multivariate data defined by integrating univariate depth measures, specifically, integrated dual (ID) depth introduced by Cuevas and Fraiman (2009) which integrates univariate simplicial depth, and integrated rank-weighted (IRW) depth, which integrates univariate Tukey depth. We build on the results of Cuevas and Fraiman (2009) to show that IRW depth shares many depth properties with ID depth. Further, we provide additional results on exact computation, decreasing along rays, continuity and breakdown point that apply to both ID and IRW depth. We also establish asymptotic normality and consistency of the sample IRW depths. Lastly, we demonstrate the use of this depth measure with real and simulated datasets: calculating robust location estimators and dd-plots.

On Combinatorial Depth Measures

Given a set [Formula: see text] of points and a point [Formula: see text] in the plane, we define a function [Formula: see text] that provides a combinatorial characterization of the multiset of values [Formula: see text], where for each [Formula: see text], [Formula: see text] is the open half-plane determined by [Formula: see text] and [Formula: see text]. We introduce two new natural measures of depth, perihedral depth and eutomic depth, and we show how to express these and the well-known simplicial and Tukey depths concisely in terms of [Formula: see text]. The perihedral and eutomic depths of [Formula: see text] with respect to [Formula: see text] correspond respectively to the number of subsets of [Formula: see text] whose convex hull contains [Formula: see text], and the number of combinatorially distinct bisections of [Formula: see text] determined by a line through [Formula: see text]. We present algorithms to compute the depth of an arbitrary query point in [Formula: see text] time and medians (deepest points) with respect to these depth measures in [Formula: see text] and [Formula: see text] time respectively. For comparison, these results match or slightly improve on the corresponding best-known running times for simplicial depth, whose definition involves similar combinatorial complexity.

Jackknife empirical likelihood goodness-of-fit tests for U-statistics based general estimating equations

Motivated by applications to goodness of fit U-statistic testing, the jackknife empirical likelihood (JEL) for vector U-statistics is justified with two approaches and the Wilks theorems are proved. This generalizes empirical likelihood (EL) for general estimating equations (GEE’s) to U-statistics based GEE’s. The results are extended to allow for the use of estimated constraints and for the number of constraints to grow with the sample size. It is demonstrated that the JEL can be used to construct EL tests for moment based distribution characteristics (e.g., skewness, coefficient of variation) with less computational burden and more flexibility than the usual EL. This can be done in the U-statistic representation approach and the vector U-statistic approach which were illustrated with several examples including JEL tests for Pearson’s correlation, Goodman–Kruskal’s Gamma, overdisperson, U-quantiles, variance components, and the simplicial depth function. The JEL tests are asymptotically distribution free. Simulations were run to exhibit power improvement of the JEL tests with incorporation of side information.

Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

Abstract The colorful simplicial depth of a collection of $d+1$ finite sets of points in Euclidean $d$-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. [7]. Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton [6] in the theory of normal surfaces.

On Evaluation of Ensemble Forecast Calibration Using the Concept of Data Depth

Abstract Various generalizations of the univariate rank histogram have been proposed to inspect the reliability of an ensemble forecast or analysis in multidimensional spaces. Multivariate rank histograms provide insightful information about the misspecification of genuinely multivariate features such as the correlation between various variables in a multivariate ensemble. However, the interpretation of patterns in a multivariate rank histogram should be handled with care. The purpose of this paper is to focus on multivariate rank histograms designed based on the concept of data depth and outline some important considerations that should be accounted for when using such multivariate rank histograms. To generate correct multivariate rank histograms using the concept of data depth, the datatype of the ensemble should be taken into account to define a proper preranking function. This paper demonstrates how and why some preranking functions might not be suitable for multivariate or vector-valued ensembles and proposes preranking functions based on the concept of simplicial depth that are applicable to both multivariate points and vector-valued ensembles. In addition, there exists an inherent identifiability issue associated with center-outward preranking functions used to generate multivariate rank histograms. This problem can be alleviated by complementing the multivariate rank histogram with other well-known multivariate statistical inference tools based on rank statistics such as the depth-versus-depth (DD) plot. Using a synthetic example, it is shown that the DD plot is less sensitive to sample size compared to multivariate rank histograms.

On Liu’s simplicial depth and Randles’ interdirections

At about the same time (approximately 1989), R. Liu introduced the notion of simplicial depth and R. Randles the notion of interdirections. These completely independent and seemingly unrelated initiatives, serving different purposes in nonparametric multivariate analysis, have spawned significant activity within their quite different respective domains. A surprising and fruitful connection between the two notions is shown. Exploiting the connection, statistical procedures based on interdirections can be modified to use simplicial depth instead, at considerable reduction of computational burden in the case of dimensions 2, 3, and 4. Implications regarding multivariate sign test statistics are discussed in detail, and several other potential applications are noted.