Projection Depth Research Articles

Projection depth and Lr-type depths for fuzzy random variables

Statistical depth functions are a standard tool in nonparametric statistics to extend order-based univariate methods to the multivariate setting. Since there is no universally accepted total order for fuzzy data (even in the univariate case) and there is a lack of parametric models, a fuzzy extension of depth-based methods is very interesting. In this paper, we adapt the multivariate depths projection depth and Lr-type depth functions to the fuzzy setting, proposing different generalizations for the Lr-type depths. We prove that the proposed fuzzy depth functions have very good properties, obtaining that the fuzzy projection depth is the second example in the literature to satisfy simultaneously the notion of semilinear and of geometric depth. This implies that the fuzzy projection depth is extremely well behave, to order fuzzy sets with respect to fuzzy random variables. Furthermore, we illustrate the good empirical behavior of the proposed fuzzy depth functions with a real data example of trapezoidal fuzzy sets and the used of fuzzy depths in depth-based classification procedures. Finally, as trapezoidal fuzzy sets can be represented by elements of R4, we justify our proposals by also showing empirically the superiority of the fuzzy depths over the multivariate projection depth applied to fuzzy sets.

Bahadur representations for the bootstrap median absolute deviation and the application to projection depth weighted mean

Bahadur representations for the bootstrap median absolute deviation and the application to projection depth weighted mean

Permutation tests of multivariate location using data depth

ABSTRACT We provide two classes of affine invariant statistics based on data depth to test the equality of mean vectors in multivariate paired data. The proposed tests are defined based on the depth values of the deepest point of the sample relative to the negative of the multivariate sample and the expected median under the null hypothesis. The tests are implemented through the idea of the permutation procedure. No distributional assumption is imposed on the data, except that the permutation test assumes a centrally symmetric distribution of the paired data. A simulation study compares the new tests to some competitors. The results show that the new tests are highly competitive for a wide variety of distributional models. More specifically, the results show that the tests based on the halfspace, simplicial, and projection depth functions perform well compared to other methods and are the most robust. A real data example illustrating the use of the tests is also presented.

The Shortest Route is Not Always the Fastest: Probability-Modeled Stereoscopic Eye Movement Completion Time in VR

Speed and consistency of target-shifting play a crucial role in human ability to perform complex tasks. Shifting our gaze between objects of interest quickly and consistently requires changes both in depth and direction. Gaze changes in depth are driven by slow, inconsistent vergence movements which rotate the eyes in opposite directions, while changes in direction are driven by ballistic, consistent movements called saccades , which rotate the eyes in the same direction. In the natural world, most of our eye movements are a combination of both types. While scientific consensus on the nature of saccades exists, vergence and combined movements remain less understood and agreed upon. We eschew the lack of scientific consensus in favor of proposing an operationalized computational model which predicts the completion time of any type of gaze movement during target-shifting in 3D. To this end, we conduct a psychophysical study in a stereo VR environment to collect more than 12,000 gaze movement trials, analyze the temporal distribution of the observed gaze movements, and fit a probabilistic model to the data. We perform a series of objective measurements and user studies to validate the model. The results demonstrate its predictive accuracy, generalization, as well as applications for optimizing visual performance by altering content placement. Lastly, we leverage the model to measure differences in human target-changing time relative to the natural world, as well as suggest scene-aware projection depth. By incorporating the complexities and randomness of human oculomotor control, we hope this research will support new behavior-aware metrics for VR/AR display design, interface layout, and gaze-contingent rendering.

PDI-HFP: An intelligent method for heat flux prediction on hypersonic aircraft based on projection depth images

We propose a novel intelligent method to predict the heat flux on hypersonic aircraft. This method considers the aircraft shape and the inflow conditions as inputs and directly outputs overall surface heat flux values. Specifically, PDI-HFP first projects the aircraft shape onto two-dimensional (2D) images in six directions and then utilizes well-trained neural networks to predict the corresponding heat flux images. Finally, PDI-HFP reconstructs the predicted surface heat flux from 2D space to 3D space, and an interpolation method is then implemented to obtain the heat flux distribution on the surface of the 3D aircraft. To the best of our knowledge, this is the first work to apply deep learning techniques to 3D heat flux prediction on arbitrary types of surface grid. Extensive experimental results demonstrate that the values of the heat flux predicted by our method are very close to those generated by CFD simulation. More importantly, compared with CFD simulation, the use of PDI-HFP effectively shortens the computational time, achieving a speedup by a factor of 200–1000, depending on the aircraft shape.

Fast Robust Location and Scatter Estimation: A Depth-based Method

The minimum covariance determinant (MCD) estimator is ubiquitous in multivariate analysis, the critical step of which is to select a subset of a given size with the lowest sample covariance determinant. The concentration step (C-step) is a common tool for subset-seeking; however, it becomes computationally demanding for high-dimensional data. To alleviate the challenge, we propose a depth-based algorithm, termed as FDB, which replaces the optimal subset with the trimmed region induced by statistical depth. We show that the depth-based region is consistent with the MCD-based subset under a specific class of depth notions, for instance, the projection depth. With the two suggested depths, the FDB estimator is not only computationally more efficient but also reaches the same level of robustness as the MCD estimator. Extensive simulation studies are conducted to assess the empirical performance of our estimators. We also validate the computational efficiency and robustness of our estimators under several typical tasks such as principal component analysis, linear discriminant analysis, image denoise and outlier detection on real-life datasets. An R package FDB and additional results are available in the supplementary materials.

Employing the MCMC technique to compute the projection depth in high dimensions

Projection depth (PD), one of the prevailing location depth notions, is a powerful and favored tool for multivariate nonparametric analysis. It permits the extension of the univariate median and weighted means to a multivariate setting. The multidimensional projection depth median (PM) and depth weighted means, including the Stahel–Donoho (SD) estimator are highly robust and affine equivariant. PM has the highest finite sample breakdown point robustness among affine equivariant location estimators.However, the computation of PD remains a challenge because its exact computation is only feasible for a data set with a dimension that is theoretically no higher than eight but practically no higher than three. Approximate algorithms such as random direction procedure or simulated annealing (SA) algorithm, are time-consuming in high-dimensional cases. Here, we present an efficient SA algorithm and its extension for the computation of PD. Simulated and real data examples indicate that the proposed algorithms outperform their competitors, including the Nelder–Mead method, and the SA algorithm, in high-dimensional cases and can obtain highly accurate results compared with those of the exact algorithm in low-dimensional cases.

Volumetric assessment is superior to axial measurements in estimating the outcome of patients with acute uncomplicated type B aortic intramural haematoma.

Our goal was to investigate the accuracy of the two-dimensional and three-dimensional computed tomography imaging features in predicting the progression of acute uncomplicated type B aortic intramural haematoma (IMH). This study retrospectively screened 140 patients diagnosed with acute uncomplicated type B IMH in our institution from January 2015 to December 2020. Patients were classified as exhibiting progression (aortic dissection, aortic rupture, aneurysm formation, ulcer-like projection depth >10 mm or >10% increase in the initial thickness of the aortic wall) and regression (completely or partially reabsorbed haematoma) based on follow-up computed tomography. During the 11.4-month follow-up [interquartile range (IQR), 2.6-17.8], 55 patients had haematoma progression. The progression group had higher haematoma volume (HV) and total lesion volume [94.8 (IQR, 80.0-108.2) cm3 vs 40.3 (IQR, 30.8-57.9) cm3; 278.0 (IQR, 238.6-369.3) cm3 vs 197.3 (IQR, 152.8-235.9) cm3, both P < 0.001) and longer lesion length [43.2 (IQR, 37.5-46.7) cm vs 30.4 (IQR, 28.1-37.6) cm, P < 0.001)] than the regression group. According to the area under the curve, HV > 66 cm3 is the greatest risk factor for haematoma progression. In multivariable analysis, HV was a powerful independent predictive factor for type B IMH progression, with a hazard ratio of 17.9 (95% confidence interval, 5.5-58.7; P < 0.001). Volumetric parameters may help to predict disease progression more precisely for patients with acute uncomplicated type B IMH compared to standard axial measurements, which might optimize the initial treatment and follow-up protocol.

Real-time, large-depth holographic 3D head-up display: selected aspects

Today's state-of-the-art automotive head-up displays (HUD) possess single- or double layer focal planes that limit the observers' eye focus to these planes when crucial information is shown. Other visual 3D cues such as motion parallax also suffer from this limitation. The resulting viewing experience contradicts the natural way of viewing during driving or interaction, when alerts and hints should appear at the correct projection depth where real objects of interest are located. Here we present a real-time holographic HUD with continuous depth that supports the intuitive and natural way of viewing and interacting with virtual environments outside and inside the car. We demonstrate full-color, 3D real-time encoding within a field-of-view of 5∘×3∘.

Boiling Heat Transfer Characteristics of Rotary Hollow Cylinders With In-Line and Staggered Multiple Arrays of Water Jets

Abstract Transient heat transfer studies of quenching rotary hollow cylinders with in-line and staggered multiple arrays of jets have been carried out experimentally. The study involves three hollow cylinders (Do/d = 12–24) with rotation speed 10–50 rpm, quenched by subcooled water jets (ΔTsub = 50–80 K) with jet flow rate 2.7–10.9 L/min. The increase in area-averaged and maximum heat flux over quenching surface (Af) has been observed in the studied multiple arrays with constant Qtotal compared to previous studies. Investigation of radial temperature distribution at stagnation point of jet reveals that the footprint of configuration of 4-row array is highlighted in radial distances near the outer surface and vanishes further down toward the inner surface. The influence of the main quenching parameters on local average surface heat flux at stagnation point is addressed in all the boiling regimes where the result indicates jet flow rate provides strongest effect in all the boiling regimes. Effectiveness of magnitude of maximum heat flux in the boiling curve for the studied parameters is reported. The result of spatial and temporal heat flux by radial conduction in the solid presents projection depth of cyclic variation of surface heat flux in the radial axis as it disappears near inner surface of hollow cylinder. In addition, correlations are proposed for area-averaged Nusselt number as well as average and maximum local heat flux at stagnation point of jet for the in-line and staggered multiple arrays.

Inferences about a Probabilistic Measure of Effect Size When Dealing with More Than Two Groups

For two independent random variables, X and Y, let p = P(X &gt; Y ) + 0.5P(X = Y ), which is sometimes described as a probabilistic measure of effect size. It has been argued that for various reasons, p represents an important and useful way of characterizing how groups differ. In clinical trials, for example, an issue is the likelihood that one method of treatment will be more effective than another. The paper deals with making inferences about p when three or more groups are to be compared. When tied values can occur, the results suggest using a multiple comparison procedure based on an extension of Cliff’s method used in conjunction with Hochberg’s sequentially rejective technique. If tied values occur with probability zero, an alternative method can be argued to have a practical advantage. As for a global test, extant rank-based methods are unsatisfactory given the goal of comparing groups based on p. The one method that performed well in simulations is based in part on the distribution of the difference between each pair of random variables. A bootstrap method is used where a p-value is based on the projection depth of the null vector relative to the bootstrap cloud. The proposed methods are illustrated using data from an intervention study.

Computation of projection regression depth and its induced median

Notions of depth in regression have been introduced and studied in the literature. The most famous example is Regression Depth (RD), which is a direct extension of location depth to regression. The projection regression depth (PRD) is the extension of another prevailing location depth, the projection depth, to regression. The computation issues of the RD have been discussed in the literature. The computation issues of the PRD and its induced median (maximum depth estimator) in a regression setting, never considered before, are addressed. For a given β∈Rp exact algorithms for the PRD with cost O(n2logn) (p=2) and O(N(n,p)(p3+nlogn+np1.5+npNIter)) (p>2) and approximate algorithms for the PRD and its induced median with cost respectively O(Nvnp) and O(RpNβ(p2+nNvNIter)) are proposed. Here N(n,p) is a number defined based on the total number of (p−1) dimensional hyperplanes formed by points induced from sample points and the β; Nv is the total number of unit directions v utilized; Nβ is the total number of candidate regression parameters β employed; NIter is the total number of iterations carried out in an optimization algorithm; R is the total number of replications. Furthermore, as the second major contribution, three PRD induced estimators, which can be computed up to 30 times faster than that of the PRD induced median while maintaining a similar level of accuracy are introduced. Examples and simulation studies reveal that the depth median induced from the PRD is favorable in terms of robustness and efficiency, compared to the maximum depth estimator induced from the RD, which is the current leading regression median.

Approximate computation of projection depths

Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in Rd. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional projections of the data, then the depth satisfies the so-called projection property. Such depths form an important class that includes many of the depths that have been proposed in literature. For depths that satisfy the projection property an approximate algorithm can easily be constructed since taking the minimum of the depths with respect to only a finite number of one-dimensional projections yields an upper bound for the depth with respect to the multivariate data. Such an algorithm is particularly useful if no exact algorithm exists or if the exact algorithm has a high computational complexity, as is the case with the halfspace depth or the projection depth. To compute these depths in high dimensions, the use of an approximate algorithm with better complexity is surely preferable. Instead of focusing on a single method we provide a comprehensive and fair comparison of several methods, both already described in the literature and original.

NRSfPP: non-rigid structure-from-perspective projection

A state-of-the-art algorithm for perspective projection reconstruction of non-rigid surfaces from single-view and realistic videos is proposed. It overcomes the limitations arising from the usage of orthographic camera model and also the complexity and non-linearity issues of perspective projection equation. Unlike traditional non-rigid structure-from-motion (NRSfM) methods, which have been studied only on synthetic datasets and controlled lab environments that require some prior constraints (such as manually segmented objects, limited rotations and occlusions, and full-length trajectories); the proposed method can be used in realistic video sequences. In addition, contrary to previous methods that use multiple cameras with different relative viewing angles, only a single-view video is required to reconstruct the 3D structures. By only using the 2D frames of incoming video stream, the proposed method extracts the projective depth coefficients of each point in each input frame, rotation matrix, translation vector, varying camera parameters (such as focal lengths for each input frame), and finally reconstructs the 3D deformable shape. Due to the high number of unknowns, the problem has been divided into two parts of projective depth coefficients extraction and 3D shape reconstruction. Perspective reconstruction of non-rigid surfaces has been extended to be certainly converged which leads to a significant increase in execution frequency of the iterated algorithm that has been presented for projective depth coefficients extraction. As such, it produces promising results for perspective projection reconstruction of non-rigid surfaces from single-view and realistic videos. The accuracy and robustness of the proposed method is demonstrated quantitatively on synthetic data and qualitatively on real image sequences. The experimental results show that NRSfPP provides the state-of-the-art results and resolves the failures of previous approaches by eliminating all of the predefined situations and constraints of applying perspective projection as the camera model.

CRATER DETECTION USING TEXTURE FEATURE AND RANDOM PROJECTION DEPTH FUNCTION

Abstract. In this paper, a novel automatic crater detection algorithm (CDA) based on traditional texture feature and random projection depth function has been proposed. By using traditional texture feature, mathematical morphology is used to identify crater initially. To further reduce the false detection rate, random projection depth function is used. For this purpose, firstly, gray level co-occurrence matrix and a novel grade level co-occurrence matrix are both used to further obtain the texture features of these candidate craters. Secondly, based on the above collected features, random projection depth function is used to refine the crater candidate detection results. LRO Narrow Angle Camera (NAC) mosaic images (1 m/pixel) and Wide-angle Camera (WAC) mosaic images (100 m/pixel) are used to test the accuracy of proposed method. The experimental results indicate our proposed method is robust to detect craters located in different terrains.

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.

Uniform convergence rates for the approximated halfspace and projection depth

The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.

Outlier Detection using Nonparametric Depth-Based Techniques in Hydrology

Several issues arise when extending the methods of outlier detection from a single dimension to a higher dimension. These issues include limited methods for visualization, marginal methods inadequacy, lacking a natural order and limitation in parametric modeling. The intension to overcome and address such limitations the nonparametric outlier identifier, based on depth functions, is introduced. These identifiers comprise of four threshold type outlyingness functions for outlier detection that are Mahalanobis distance, Tukey depth, spatial Mahalanobis depth, and projection depth. The object of the present research is the application of the proposed nonparametric technique in hydrology. The study is intended to be executed in two different frameworks that are multivariate hydrological data analysis and functional hydrological data analysis. The event of a flood is graphically represented by hydrograph whose components are used for computing flood characteristics that are peak(p) and volume(v). These characteristics are frequently employed for the various types of analysis in the multivariate study. Whereas, hydrograph is exhaustively employed in the analysis of functional data so that all the important information regarding flood event are not missed while analysis. The proposed technique in a multivariate framework is applied to the bivariate flood characteristics (p,v)while in functional framework proposed approach is applied to the initial two scores of principal components denoted as (z_1,z_2 ), since initial two principal components capture major variation of data employed for analysis.

Automating Spine Curvature Measurement in Volumetric Ultrasound via Adaptive Phase Features

Ultrasound volume projection imaging (VPI) has been recently suggested. This novel imaging method allows a non-radiation assessment of spine deformity with free-hand 3-D ultrasound imaging. This paper presents a fully automatic method to evaluate the spine curve in VPI images corresponding to different projection depth of the volumetric ultrasound, thus making it possible to analyze 3-D spine deformity. The new automatic method is based on prior knowledge about the geometric arrangement of the spinous processes. The frequency bandwidth of log-Gabor filters is adaptively adjusted to calculate the oriented phase congruency, facilitating the segmentation of the spinous column profile. And the spine curvature angle is finally calculated according to the inflection points of the curve over the segmented spinous column profile. The performance of the automatic method is evaluated on spine VPI images among patients with different scoliotic angles. The curvature angles obtained using the proposed method have a high linear correlation with those by the manual method (r = 0.90, p < 0.001) and X-ray Cobb's method (r = 0.87, p < 0.001). The feasibility of 3-D spine deformity assessment is also demonstrated using VPI images corresponding to various projection depth. The results suggest that this method can substantially improve the recognition of the spinous column profile, especially facilitating the applications of 3-D spine deformity assessment.

Covariance‐free non‐homogeneity STAP detector in compound Gaussian clutter based on robust statistics

Space-time adaptive processing (STAP) detects targets by computing adaptive weight vectors for each cell under test using its covariance matrix, as estimated from surrounding secondary cells. In this study, the non-homogeneity detector (NHD) excludes the anomalous secondary cells that adversely affect the detection performance. The existing robust NHDs require estimating the covariance matrix of each secondary cell, which hinders their implementation in modern radars with large-dimensional range cells. In this study, the authors propose a new low-complexity NHD that is suitable for highly correlated clutter environments with both Gaussian and non-Gaussian heavy-tailed distributions. The proposed detector, which is based on the projection depth function from the field of robust statistics, features a non-parametric and covariance-free test statistic. As a result, its computational complexity is much lower than that of current NHDs, such as the widely used normalised adaptive matched filter (NAMF) detector, especially for large-dimensional range cells. In Monte Carlo simulations with different clutter distributions and radar system configurations, the proposed detector shows comparable performance to that of NAMF. The low complexity and robust performance of the new detector make it particularly attractive for real-time applications.