Notion Of Depth Research Articles

A new approach for the computation of halfspace depth in high dimensions

ABSTRACTHalfspace depth (HD), aka Tukey depth, is one of the most prevailing depth notions among all its competitors. To exactly compute the HD in is a challenging task nevertheless, due to its definition involving infinitely many directional projections. Existing algorithms to compute HD in (d ⩾ 2), more or less involve data projection to the directions perpendicular to hyperplanes (therefore involving polytopes or polyhedral cones) are either dimension d-, or sample size n-limited, or slow. Thanks to the work of Merkle and Bogićević and Merkle, algorithms for the fast computation of HD in ultra high dimension d for large n are proposed and addressed. They are sheer ball-based (or essentially point-wise distances) computations. The worst time complexity to approximately compute the depth m/n of a single point is O((d + m + log (n))n2) which seems to be one of the best known results so far among the competitors for extra large d and n. Unlike most previous algorithms, they do not require the underlying dataset to be in general position. As by-products of the algorithm, some depth regions (or depth level sets), and depth of sample points also could be obtained during the depth calculation of a given point.

Two-Sample Nonparametric Test for Testing Equality of Locations Based on Data Depth

In the recent years, the notion of data depth has been widely used in multivariate data analysis since it measures the centrality or outlyingness of the multivariate data points with respect to the given data cloud and it orders the data from center to outward in any direction called ‘center-outward ordering’. In the present work, we propose a nonparametric test for testing equality of location parameter of two multivariate distributions using notion of data depth. The proposed test is motivated from the concept of correlation. We compare powers of the proposed test with existing tests for multivariate symmetric and skewed distributions through simulation. The proposed test gives an attractive powers against various alternatives. Application to a real life data is also provided.

Nonparametric tests for multivariate locations based on data depth

ABSTRACTThe present paper deals with the problem of testing equality of locations of two multivariate distributions using a notion of data depth. A notion of data depth has been used to measure centrality/outlyingness of a given point in a given data cloud. The paper proposes two nonparametric tests for testing equality of locations of two multivariate populations which are developed by observing the behavior of the depth versus depth plot. Simulation study reveals that the proposed tests are superior to the existing tests based on the data depth with regard to power. Illustrations with real data are provided.

Architecture Of Depth: Gravicells As A Critical Spatial Practice In The 21st Century

This essay provides a comprehensive analysis of Gravicells , by Seiko Mikami and Sota Ichigawa as an architectural experimentation of twenty-first-century media that materializes virtual dimensions of our spatiotemporal experiences. This project revolves around my study of a transdisciplinary field of study directed toward points of overlap and spaces of co-operation between art, architecture and technology that operate as experiential interfaces generating alternative experiences of space. In particular, this essay examines virtual operations behind the perception of space that reveals itself as the primordial spatiality grounding the interconnectedness of bodies and spaces through affective engagement in Gravicells . By inviting participants into a data-saturated environment that visualizes gravitational relations, Gravicells offers an affective experience of the intrinsic interconnectedness of the space and the participant’s body as well as a sense of space that opens up across time and space. To unpack this complex spatiotemporality of the embodied experience of Gravicells , the essay provides a phenomenological reconstruction of the notion of depth. Challenging academic and conventional designations of the notion of depth, the study reconstructs the phenomenological dimension of depth that comprises both distance and voluminosity; distance reveals the immediate link between the subject and the world, while voluminosity describes an atmospheric dimension that indicates the possibility of voluminous bodies involved in the world. As both an origin and a means of enveloping the subject in the world, depth is best understood as the general milieu or primary condition that grounds the continuous inter-implication between body, time, and space; depth supports their envelopment, while they are fundamentally all structured in depth. In my study, thus, depth is no longer treated as an abstract geometrical space that is indirectly inferred from two-dimensional retinal images but as the primary spatial medium that grounds the co-existence of the world. By materializing the structure of depth, Gravicells radically reframes the sense of space, and surfaces our understanding of the ways in which we define ourselves in relation to others and environments.

The Religious Quest As Transformative Journey: Interspiritual Religious Belonging And The Problem Of Religious Depth

Abstract As scholars and the public grope towards understanding emergent forms of religiosity (multiple-religious belonging, spiritual but not religious, interspirituality), notions of discernment, religious depth, and spiritual practice figure prominently in defining and assessing these forms. Some form of commitment to a particular religious tradition is often considered the most important factor in the discernment of religious depth, while “spiritual but not religious” is often seen as the amorphous searching or the drifting whims of an immature ego. I will argue, however, that failing to take into account the most mature forms of emerging religiosity is bound to miss important developments, just as similar methodologies would for traditional religions. Further, I point out problems with correlating religious depth with belonging to a particular religious tradition, and offer an alternate way to conceive of religious depth. In doing so I develop the concept of the religious quest as transformative journey, allowing for a more capacious understanding of religious consciousness. I then introduce interspiritual religious belonging, contrasting it with certain understandings of “multiple-religious” belonging, and providing mature examples of its embodiment. Finally, utilizing new surveys from Pew and PPRI showing accelerating growth among the “spiritual but not religious” and “religiously unaffiliated”-as well as expanding religious and racial diversity within the United States-I briefly reference potential political ramifications the interspiritual movement might have, and address the importance of developing mature theological perspectives from within it. It is my hope that the Theology Without Walls project can provide academic space for the latter.

Local half-region depth for functional data

Data depth has proved successful in the analysis of multivariate data sets, e.g., for deriving an overall center and assigning ranks to the observed units. Two key features are the directions of the ordering, from the center towards the outside, and the recognition of a unique center irrespective of the distribution being unimodal or multimodal. These properties derive from the monotonicity of the ranks that decrease along any ray from the deepest point. Recently, a wider framework allowing for the identification of partial centers was suggested in Agostinelli (2011). The corresponding generalized depth functions, called local depth functions, can record local fluctuations and be used for mode detection, identification of components in mixture models, and cluster analysis. As functional data are becoming more common, López-Pintado and Romo (2011) recently proposed a notion of half-region depth suited for functional data and for high-dimensional data. Here, we propose a local version of this concept, we study its theoretical properties, we define new similarity measures based on it, and we illustrate its behavior with examples based on real data sets.

Metastability of Neuronal Dynamics during General Anesthesia: Time for a Change in Our Assumptions?

There is strong evidence that anesthetics have stereotypical effects on brain state, so that a given anesthetic appears to have a signature in the electroencephalogram (EEG), which may vary with dose. This can be usefully interpreted as the anesthetic determining an attractor in the phase space of the brain. How brain activity shifts between these attractors in time remains understudied, as most studies implicitly assume a one-to-one relationship between drug dose and attractor features by assuming stationarity over the analysis interval and analyzing data segments of several minutes in length. Yet data in rats anesthetized with isoflurane suggests that, at anesthetic levels consistent with surgical anesthesia, brain activity alternates between multiple attractors, often spending on the order of 10 min in one activity pattern before shifting to another. Moreover, the probability of these jumps between attractors changes with anesthetic concentration. This suggests the hypothesis that brain state is metastable during anesthesia: though it appears at equilibrium on short timescales (on the order of seconds to a few minutes), longer intervals show shifting behavior. Compelling evidence for metastability in rats anesthetized with isoflurane is reviewed, but so far only suggestive hints of metastability in brain states exist with other anesthetics or in other species. Explicit testing of metastability during anesthesia will require experiments with longer acquisition intervals and carefully designed analytic approaches; some of the implications of these constraints are reviewed for typical spectral analysis approaches. If metastability exists during anesthesia, it implies degeneracy in the relationship between brain state and effect site concentration, as there is not a one-to-one mapping between the two. This degeneracy could explain some of the reported difficulty in using brain activity monitors to titrate drug dose to prevent awareness during anesthesia and should force a rethinking of the notion of depth of anesthesia as a single dimension. Finally, explicit incorporation of knowledge of the dynamics of the brain during anesthesia could offer better depth of anesthesia monitoring.

On maximum depth classifiers: depth distribution approach

ABSTRACTIn this paper, we consider the notions of data depth for ordering multivariate data and propose a classification rule based on the distribution of some depth functions in . The equivalence of the proposed classification rule to optimal Bayes rule is discussed under suitable conditions. The performance of the proposed classification method is investigated in low- and high-dimensional setting using real datasets. Also, the performance of the proposed classification method is illustrated in comparison to some other depth-based classifiers using simulated data sets.

Generalized Mahalanobis depth in point process and its application in neural coding

In this paper, we propose to generalize the notion of depth in temporal point process observations. The new depth is defined as a weighted product of two probability terms: (1) the number of events in each process, and (2) the center-outward ranking on the event times conditioned on the number of events. In this study, we adopt the Poisson distribution for the first term and the Mahalanobis depth for the second term. We propose an efficient bootstrapping approach to estimate parameters in the defined depth. In the case of Poisson process, the observed events are order statistics where the parameters can be estimated robustly with respect to sample size. We demonstrate the use of the new depth by ranking realizations from a Poisson process. We also test the new method in classification problems using simulations as well as real neural spike train data. It is found that the new framework provides more accurate and robust classifications as compared to commonly used likelihood methods.

Depth‐weighted robust multivariate regression with application to sparse data

AbstractA robust method for multivariate regression is developed based on robust estimators of the joint location and scatter matrix of the explanatory and response variables using the notion of data depth. The multivariate regression estimator possesses desirable affine equivariance properties, achieves the best breakdown point of any affine equivariant estimator, and has an influence function which is bounded in both the response as well as the predictor variable. To increase the efficiency of this estimator, a re‐weighted estimator based on robust Mahalanobis distances of the residual vectors is proposed. In practice, the method is more stable than existing methods that are constructed using subsamples of the data. The resulting multivariate regression technique is computationally feasible, and turns out to perform better than several popular robust multivariate regression methods when applied to various simulated data as well as a real benchmark data set. When the data dimension is quite high compared to the sample size it is still possible to use meaningful notions of data depth along with the corresponding depth values to construct a robust estimator in a sparse setting. The Canadian Journal of Statistics 45: 164–184; 2017 © 2017 Statistical Society of Canada

Robust non‐parametric tests for imaging data based on data depth

Research in many disciplines stands on the analysis of complex high‐dimensional data sets. For example, in clinical neuroscience, large collections of brain images from different subjects are obtained by advanced scanning techniques to study variations in different neurological states. Developing new tools to analyse the main characteristics of these rich data sets is needed. We consider the basic unit of observation to be a general function, which is defined and takes values in spaces of arbitrary dimension. On the basis of a notion of depth for general functions denoted as multivariate volume depth (MVD), images will be ranked from centre to outward and robust estimators can be defined. The theoretical properties of MVD are established, and several non‐parametric depth‐based permutation tests for comparing two groups of images are proposed; in particular, we introduce two‐sample location tests based on MVD. In addition, dispersion measures for a sample of images are introduced and used for testing two sample differences in dispersion. All the proposed tests are calibrated in an extensive simulation study. These statistical tools are applied to detect whether there are differences between the brain images from healthy individuals and patients with major depressive disorders. Copyright © 2017 John Wiley & Sons, Ltd.

På råbeafstand af marxismen. Et bidrag til kritik af kritikken af kritikken (Latour, Foucault, Marx)

In the last couple of decades, critique has come in for a heavy beating: Critique is based on problematic notions of depth and exposure where the critic takes up a position of critical distance that presumes mastery over the material by revealing hidden depths beneath the surface. This article traces significant episodes in the development of this anti-critical critique by embedding it in a longer historical trajectory that has to do with an initially internal Marxist self-critique that paves the way for a downright dismissal of Marxism and the critical analysis of the capitalist mode of production. The article moves backward starting with Bruno Latour and his analysis of critique’s collapse to Michel Foucault and his complicated relationship to Marx and Marxism, ending with a call for a return to an analytic position that is in shouting distance to Marxism.

Power comparison of data depth-based nonparametric tests for testing equality of locations

ABSTRACTIn the recent years, the notion of data depth has been used in nonparametric multivariate data analysis since it gives natural ‘centre-outward’ ordering of multivariate data points with respect to the given data cloud. In the literature, various nonparametric tests are developed for testing equality of location of two multivariate distributions based on data depth. Here, we define two nonparametric tests based on two different test statistic for testing equality of locations of two multivariate distributions. In the present work, we compare the performance of these tests with the tests developed by Li and Liu [New nonparametric tests of multivariate locations and scales using data depth. Statist Sci. 2004;(1):686–696] for testing equality of locations of two multivariate distributions. Comparison in terms of power is done for multivariate symmetric and skewed distributions using simulation for three popular depth functions. Application of tests to real life data is provided. Conclusion and recommendations are also provided.

Classification rules based on distribution functions of functional depth

In ordering multivariate objects, the use of data depth provides a centre-outward ranking. The notion of data depth has been extended to functional data setting and applied in classifying functional data, for example maximal depth classification rules. In this paper, we explore notions of functional depth and propose a classification method based on distribution functions of data depth for functional data. The performance of this method is examined by using simulations and real data sets and the results are compared with the results from existing methods.

Extremal Depth for Functional Data and Applications

ABSTRACTWe propose a new notion called “extremal depth” (ED) for functional data, discuss its properties, and compare its performance with existing concepts. The proposed notion is based on a measure of extreme “outlyingness.” ED has several desirable properties that are not shared by other notions and is especially well suited for obtaining central regions of functional data and function spaces. In particular: (a) the central region achieves the nominal (desired) simultaneous coverage probability; (b) there is a correspondence between ED-based (simultaneous) central regions and appropriate pointwise central regions; and (c) the method is resistant to certain classes of functional outliers. The article examines the performance of ED and compares it with other depth notions. Its usefulness is demonstrated through applications to constructing central regions, functional boxplots, outlier detection, and simultaneous confidence bands in regression problems. Supplementary materials for this article are available online.

DEEP CLASSES

AbstractA set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω.

Subgroup Depth and Twisted Coefficients

Danz computes the depth of certain twisted group algebra extensions in [10], which are less than the values of the depths of the corresponding untwisted group algebra extensions in [8]. In this article, we show that the closely related h-depth of any group crossed product algebra extension is less than or equal to the h-depth of the corresponding (finite rank) group algebra extension. A convenient theoretical underpinning to do so is provided by the entwining structure of a right H-comodule algebra A and a right H-module coalgebra C for a Hopf algebra H. Then A ⊗ C is an A-coring, where corings have a notion of depth extending h-depth. This coring is Galois in certain cases where C is the quotient module Q of a coideal subalgebra R ⊆ H. We note that this applies for the group crossed product algebra extension, so that the depth of this Galois coring is less than the h-depth of H in G. Along the way, we show that the subgroup depth behaves exactly like the combinatorial depth with respect to the core of a subgroup, and extend results in [22] to coideal subalgebras of finite dimension.

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.

The ocean and the urban: Poe's “The Oblong Box”

ABSTRACTAt the end of Poe's oceanic tale “The Oblong Box,” the narrator makes the surprising confession that despite having solved the mystery of what lay inside the mysterious “box,” he rarely “sleeps soundly at night.” Given Poe's propensity for narrating the return of bodies, why might the loss, rather than the return, of a coffin evoke horror from the narrator in this tale at sea? This essay considers the ways that oceanic and urban spaces were constructed as interrelated and codependent, though contrasted, in antebellum literature. As opposed to the city, a space of surfaces and solidified forms, the ocean's fluidity presents a limitless depth, unable to be contained by figures like the city streets and solid geometric forms found in the true crime reporting of antebellum New York and imagined throughout Poe's urban detective fictions. The essay reads Poe's oceanic tale “The Oblong Box” in relation to conventions of detective and gothic fiction and with influence from studies of oceanic and spatial studies of literature before finally turning to city-mysteries to show how the ocean becomes a projected space of depth that refuses mapping in antebellum fiction. This prompts us to explore urban fiction in relation to oceanic space, a developing turn in American literary studies. Specifically, questions of surface and depth open up conversations regarding the epistemological problems of the ocean in an antebellum print world that was already struggling to map and contain the growing spaces of urban locales like Poe's New York and Philadelphia. In a telling inversion of the gothic notions of surface and depth, Poe's narrator discovers that at sea, looking below the surface serves to obfuscate and swallow meaning rather than to reveal it.

Simplified simplicial depth for regression and autoregressive growth processes

We simplify simplicial depth in two directions for regression and autoregressive growth processes. At first we show that simplicial tangent depth often reduces to counting the subsets with alternating signs of the residuals if the regressors are ordered. The second simplification is given by not regarding all subsets of residuals. By consideration of only special subsets of residuals, the asymptotic distributions of the simplified simplicial depth notions are normal distributions so that tests and confidence intervals can be derived easily. We propose two simplifications for the general case and a third simplification for the special case where two parameters are unknown. Additionally, we derive conditions for the consistency of the tests. We show that the simplified depth notions can be used for polynomial regression, for several nonlinear regression models, and for several autoregressive growth processes. We compare the efficiency and robustness of the different simplified versions by a simulation study concerning the Michaelis–Menten model and a nonlinear autoregressive process of order one and provide an application on crack growth.