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By a classical result of Brodmann, the function depthR/I^t is asymptotically a constant, i.e. there is a number s such that depthR/I^t = depthR/I^s for t > s. One calls the smallest number s with this property the index of depth stability of I and denotes it by dstab(I). This invariant remains mysterious till now. The main result of this paper gives an explicit formula for dstab(I) when I is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize dstab(I) explicitly. The formula expresses dstab(I) in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals. [ABSTRACT FROM AUTHOR]

We define morphological operators and filters for directional images whose pixel values are unit vectors. This requires an ordering relation for unit vectors which is obtained by using depth functions. They provide a centre-outward ordering with respect to a specified centre vector. We apply our operators on synthetic directional images and compare them with classical morphological operators for grey-scale images. As application examples, we enhance the fault region in a compressed glass foam and segment misaligned fibre regions of glass fibre-reinforced polymers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The depth-based scale curves have broad applications in nonparametric multivariate statistics. Many nonparametric multivariate measures and procedures are based on the scale curves. In this article, we study the finite-sample performance of the sample scale curve as an estimator of its population counterpart by simulation. It is found that the sample scale curve tends to underestimate its population counterpart and that the discrepancy increases quickly as dimension increases. The effect of data shape characterized by kurtosis on the performance is also studied. Compared with the effect of dimension, the effect of kurtosis is minor. The root cause of the issue is explored and some suggestions for improvement are provided. [ABSTRACT FROM AUTHOR]

The prediction problem of univariate records, though not addressed in multivariate records, has been discussed by many authors based on records values. There are various definitions for multivariate records among which depth-based records have been selected for the aim of this paper. In this paper, by means of the maximum likelihood and conditional median methods, point and interval predictions of depth values which are related to the future depth-based multivariate records are considered on the basis of the observed ones. The observations derived from some elements of the elliptical distributions are the main reason of studying this problem. Finally, the satisfactory performance of the prediction methods is illustrated via some simulation studies and a real dataset about Kermanshah city drought. [ABSTRACT FROM AUTHOR]

This paper aims to propose a new class of permutation-invariant tests for diagonal symmetry around a known point based on the center-outward depth ranking. The asymptotic behavior of the proposed tests under the null distribution is derived. The performance of the proposed tests is assessed through a Monte Carlo study. The results show that the tests perform well comparing other procedures in terms of empirical sizes and empirical powers. We demonstrated that the proposed class includes the celebrated Wilcoxon signed-rank test as a special case in the univariate setting. Finally, we apply the tests to a well-known data set to illustrate the method developed in this paper. [ABSTRACT FROM AUTHOR]

Hydrological models are used for various water resources application. To represent hydrological processes, it need parameters that achieve a discharge simulation as close to the observed series as possible. The simulation result depends on how accurately the models parameters are calibrated. The calibration of model parameters depend on various factors, such as calibration methods and selected objective functions. In this study, some of the automatic calibration methods were investigated and a comparison was made to give better prediction. Different optimization algorithms like SCE-UA, SA, and ROPE were used to illustrate and calibrate the conceptual model HBV-IWS. The study was conducted on the Upper Neckar catchment, Germany. The results show that almost all optimization algorithms gave a very similar result, but the ROPE algorithm seems to be more robust. This is due to ROPE giving a space of parameter values after calibration, instead of a single parameter set as in other optimizations.

In this paper, a class of affine-invariant tests is presented for the multi-sample multivariate location problem. In the procedure to derive the asymptotic distribution of the tests under the null hypothesis, we do not require any symmetric assumption of the distribution functions. The asymptotic relative efficiency of the tests is discussed under the class of elliptically symmetric distributions. Further comparisons are made among several statistics using Monte Carlo results. Asymptotic relative efficiencies along with Monte Carlo results indicate that selected members of the proposed class perform very well for a broad class of distributions. Finally, we apply our proposed tests to Egyptian skulls data for multivariate five different periods comparisons. [ABSTRACT FROM AUTHOR]

[Display omitted] Soil spatial information has traditionally been presented as polygon maps at coarse scales. Solving global and local issues, including food security, water regulation, land degradation, and climate change requires higher quality, more consistent and detailed soil information. Accurate prediction of soil variation over large and complex areas with limited samples remains a challenge, which is especially significant for China due to its vast land area which contains the most diverse soil landscapes in the world. Here, we integrated predictive soil mapping paradigm with adaptive depth function fitting, state-of-the-art ensemble machine learning and high-resolution soil-forming environment characterization in a high-performance parallel computing environment to generate 90-m resolution national gridded maps of nine soil properties (pH, organic carbon, nitrogen, phosphorus, potassium, cation exchange capacity, bulk density, coarse fragments, and thickness) at multiple depths across China. This was based on approximately 5000 representative soil profiles collected in a recent national soil survey and a suite of detailed covariates to characterize soil-forming environments. The predictive accuracy ranged from very good to moderate (Model Efficiency Coefficients from 0.71 to 0.36) at 0–5 cm. The predictive accuracy for most soil properties declined with depth. Compared with previous soil maps, we achieved significantly more detailed and accurate predictions which could well represent soil variations across the territory and are a significant contribution to the GlobalSoilMap.net project. The relative importance of soil-forming factors in the predictions varied by specific soil property and depth, suggesting the complexity and non-stationarity of comprehensive multi-factor interactions in the process of soil development. [ABSTRACT FROM AUTHOR]

Summary: In this paper, using the notion of data depth, we describe two classes of affine invariant test statistics for the one‐sample location problem. The tests are implemented through the idea of permutation tests. The performance of the test against some competitors is investigated with an extensive simulation study. It is observed that the tests perform well when compared to their competitors for a wide spectrum of alternatives. If the proposed test is defined based on a moment‐free depth function, then it is not inherently required to have finite moments of any order and the tests have broader applicability than some of the existing tests. The robustness property of the proposed tests is considered with a simulation study. Finally, we apply the tests to a real data example. Highlights Two classes of affine invariant depth‐based test statistics for the one-sample location problem have been presented.The tests are implemented through the idea of permutation tests.The performance of the test against some competitors is investigated with an extensive simulation study.The tests perform well when compared to their competitors for a wide spectrum of alternatives. The proposed tests based on the simplicial, spatial, and exponential power depth functions have a good performance over a wide spectrum of distributions.The robustness property of the proposed tests is considered with a simulation study. The proposed tests based on the simplicial, halfspace and projection&‐based depth functions are most robust.The tests have been applied to a real data example. [ABSTRACT FROM AUTHOR]

There are different definitions for multivariate records which are based on component-wise approach. In this paper, a new approach based on the notion of depth is employed for introducing a depth-based record as a different multivariate record. This record is more in line with univariate records, has Markovian property, ascertained marginal and joint distributions, and some other exclusive features compared to other records. The depth-based procedure can recognize records satisfactorily. The performance of which is evaluated through studying the drought in Kermanshah, Iran, and the air pollution in Leeds, England. The outputs confirm the suitable potency of depth-based record procedure. [ABSTRACT FROM AUTHOR]

ABSTRACT Soil maps are important to evaluate soil functions and support decision-making process, particularly for soil properties such as pH, carbon content (C), and cation exchange capacity (CEC), but the spatial resolution and soil depth should meet the needs of users. On another hand, the efficiency of statistical models to create soil maps, with an acceptable level of accuracy, often require a large number of samples with an appropriate distribution across the area of interest. However, accessibility for sampling can be a trouble in remote areas, such as the Itatiaia National Park (INP). The hypothesis of this work is that it is possible to obtain a viable result in soil mapping of areas with limited access by using DSM tools. The general objective of this paper was to create 2- and 3-D maps of the soil properties pH, carbon content, and CEC, with the correspondent spatial uncertainty, in the INP plateau. The sampling strategy was designed using conditioned Latin Hypercube Sample (cLHS), and different methods were tested to produce the soil properties maps. For calibration of the models: linear (MLR, multiple linear regression) and nonlinear (GAM, Generalised Additive Models). The results showed differences in predictive performance for all statistical methods and covariate selection approaches. The GAM, with covariates selection based on soil formation factors, was the best method for the limited number of soil samples. The greatest uncertainty was associated with areas with the lowest accessibility and, consequently, with low sampling density and/or noises in covariates. Even though the 2- and 3-D maps of soil properties, each associated with explicit uncertainty, can contribute to the INP decision makers/managers by providing information not available before.

In this paper, distribution-free, affine invariant, signed-rank test statistics are proposed for the hypothesis that a bivariate distribution is centrally symmetric about an arbitrary specified point. The proposed tests are based on the concept of data depth. However, our tests are inherently orthogonal invariant, an affine invariant version of them is provided by using Tyler’s estimator of scatter. The limiting null distribution of proposed tests is derived and the performance of the proposed tests is evaluated through a Monte Carlo study. This study demonstrates that the tests always detect asymmetry and they are convenient to determine small departures from the null hypothesis with high power. Also it shows that the tests perform well comparing other procedures in the literature.

Since the representation ability of statistical global feature for functional data classification algorithm is limited, and the salient point feature is susceptible to noise disturbance, this paper proposed a segmental feature extraction algorithm based on statistical depth notion. Firstly, it used the smoothing technique to pre-smooth the discrete observed data, and introduced the first and second derivatives of the functional data . Then, it calculated depths of Mahalanobis integral of the functions and its low-order derivatives in segments, and thus constructed feature vectors of function curves based on the depth measures. Finally, it selected the optimal number of segments for classification by data-driven, and studied the binary classification of function data. Compared with the other curve feature extraction algorithms, experiments on UCR datasets show that the proposed algorithm performs well in extracting the feature of curve, and improves the classification accuracy effectively. [ABSTRACT FROM AUTHOR]

In many applications, parameters of interest are estimated by solving some non-smooth estimating equations with U-statistic structure. Jackknife empirical likelihood (JEL) approach can solve this problem efficiently by reducing the computation complexity of the empirical likelihood (EL) method. However, as EL, JEL suffers the sensitivity problem to outliers. In this paper, we propose a weighted jackknife empirical likelihood (WJEL) to tackle the above limitation of JEL. The proposed WJEL tilts the JEL function by assigning smaller weights to outliers. The asymptotic of the WJEL ratio statistic is derived. It converges in distribution to a multiple of a chi-square random variable. The multiplying constant depends on the weighting scheme. The self-normalized version of WJEL ratio does not require to know the constant and hence yields the standard chi-square distribution in the limit. Robustness of the proposed method is illustrated by simulation studies and one real data application. [ABSTRACT FROM AUTHOR]

Let I and J be nonzero ideals in two Noetherian algebras A and B over a field k. Let I + J denote the ideal generated by I and J in A ⊗ k B . We prove the following expansion for the symbolic powers: (I + J) (n) = ∑ i + j = n I (i) J (j). If A and B are polynomial rings and if char (k) = 0 or if I and J are monomial ideals, we give exact formulas for the depth and the Castelnuovo–Mumford regularity of (I + J) (n) , which depend on the interplay between the symbolic powers of I and J. The proof involves a result of independent interest which states that the induced map Tor i A (k , I (n)) → Tor i R (k , I (n - 1)) is zero for any homogeneous ideal I and i ≥ 0 , n ≥ 0 . We also investigate other properties and invariants of (I + J) (n) such as the equality between ordinary and symbolic powers, the Waldschmidt constant and the Cohen–Macaulayness. [ABSTRACT FROM AUTHOR]

We prove that the depth functions of cover ideals of balanced hypergraph have the nonincreasing property. Furthermore, we also give a bound for the index of depth stability of these ideals. [ABSTRACT FROM AUTHOR]

Based on the minimal spanning tree (MST) of the observed data set, the paper introduces new notions of data depth and medians for multivariate data. The MST of a data set of size n is the MST of the complete weighted undirected graph on n vertices, where the edge weights are the pairwise distances of the data points. We study several properties of the MST-based depth functions. We consider the corresponding multidimensional medians, investigate their robustness and computational complexity. An example illustrates the use of the MST-based depth functions. [ABSTRACT FROM AUTHOR]

Soil moisture is a key state variable in terrestrial water cycles, which links various land surface and hydrological processes. Owing to the significant spatiotemporal variability in soil moisture, collecting sufficient soil moisture data for relevant studies can be an astonishingly difficult task. Here, a statistical method based on the concept of depth functions was used to define Critical Events (CE) of soil moisture, which was hypothesized to contain disproportionally more system information on soil moisture dynamics. To test the feasibility of applying the CE concept for quantifying soil moisture dynamics, a long-term soil moisture dataset was retrieved from the Automated Weather Data Network (AWDN) located in the continental United States. The method of Temporal Stability Analysis (TSA) was adopted to examine the information embedded in soil moisture data that were collected under different conditions. The results showed that similar information on the relative as well as absolute wetness conditions at the AWDN sites was extracted from the entire time series and CEs of soil moisture, the latter of which contained much less soil moisture observations. Finally, the field data revealed that the occurrence of CEs of soil moisture in the study area was mainly affected by soil depth and interannual variability in precipitation. The number of CEs of soil moisture tended to be larger at deeper soil depths as well as in either dry or wet years than in normal years, suggesting that more soil moisture measurements under those conditions were needed to provide adequate information on soil moisture systems. [ABSTRACT FROM AUTHOR]

In this study, a hybrid method combining 3D kriging and depth function considering heterogeneity (3DK_DF) was proposed to enhance the accuracy and reliability of 3D spatial interpolation for soil heavy metals. Soil samples collected at varied depth intervals in a mining city in China were used as a case dataset. First, the parameters of a logarithmic depth function model for every horizontal soil sample site were fitted on the basis of the observed values of soil Cd collected at varied depth intervals. Second, the 3D trend of soil Cd was obtained on the basis of the spatial distributions of the parameters of the logarithmic depth function model. Third, 3D kriging was used to generate the 3D spatial distribution of residual Cd. Finally, the 3D spatial distribution of the soil Cd was obtained by combining the 3D trend and residual results. The interpolation accuracy of 3DK_DF improved by 29.71% and 48.9% compared with those of the 3D kriging without a trend analysis and the 3D kriging with a polynomial trend model, respectively. The proposed hybrid 3D interpolation method could be of great significance for the comprehensive assessment of soil heavy metal pollution. [Display omitted] • Model a 3D trend of soil Cd using depth function with spatial heterogeneity. • A hybrid technique improved the 3D spatial interpolation accuracy. • 3D distribution generated by the hybrid method best preserve the spatial variation. [ABSTRACT FROM AUTHOR]

Supervised and unsupervised classification is crucial in many areas where different types of data sets are common, such as biology, medicine, or industry, among others. A key consideration is that some units are more typical of the group they belong to than others. For this reason, fuzzy classification approaches are necessary. In this paper, a fuzzy supervised classification method, which is based on the construction of prototypes, is proposed. The method obtains the prototypes from an objective function that includes label information and a distance-based depth function. It works with any distance and it can deal with data sets of a wide nature variety. It can further be applied to data sets where the use of Euclidean distance is not suitable and to high-dimensional data (data sets in which the number of features p is larger than the number of observations n , often written as p > > n). In addition, the model can also cope with unsupervised classification, thus becoming an interesting alternative to other fuzzy clustering methods. With synthetic data sets along with high-dimensional real biomedical and industrial data sets, we demonstrate the good performance of the supervised and unsupervised fuzzy proposed procedures. • New fuzzy classification methodology based on the construction of prototypes • Distance-based, it overcomes the curse of dimensionality • It can be applied to a large spectrum of data, when the Euclidean distance is not suitable • It identifies K observations, selected between the deepest observations • Supervised and unsupervised approaches integrated in the objective function [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the asymptotic behavior of generalized depth-based spread processes, which include the scale curve of Liu et al. (1999) as a special case. Both uniform strong and weak convergences of the generalized depth-based spread processes are established. As applications, we obtain the asymptotic distributions of some nonparametric multivariate kurtosis measures. Applications to compare spread and kurtosis of two multivariate data sets, and to assess multivariate normality, are also discussed. [ABSTRACT FROM AUTHOR]