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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 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]

In this article we introduce a notion of depth functions for data types that are not given in standard statistical data formats. We focus on data that cannot be represented by one specific data structure, such as normed vector spaces. This covers a wide range of different data types, which we refer to as non-standard data. Depth functions have been studied intensively for normed vector spaces. However, a discussion of depth functions for non-standard data is lacking. In this article, we address this gap by using formal concept analysis to obtain a unified data representation. Building on this representation, we then define depth functions for non-standard data. Furthermore, we provide a systematic basis by introducing structural properties using the data representation provided by formal concept analysis. Finally, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article presents the mathematical formalisation of centrality and outlyingness for non-standard data and increases the number of spaces in which centrality can be discussed. In particular, we provide a basis for defining further depth functions and statistical inference methods for non-standard data. [ABSTRACT FROM AUTHOR]

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]

A new family of depth-based test statistics is proposed for testing the hypothesis of independence between two random vectors. 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. Furthermore, a conditional distribution-free property of the tests is shown. The asymptotic relative efficiency of the tests is discussed under the class of elliptically symmetric distribution. Asymptotic relative efficiencies along with Monte Carlo results suggest that the performance of the proposed class is comparable to the existing ones, and under some circumstances, it has higher power. Finally, we apply the tests to two real data sets and also discuss the robustness of our tests. [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]

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]

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]

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]

Soil pH controls the availability of the majority of plant nutrients, if not all, and determines the growth environment for plant roots. Profile depth functions have been used to represent the vertical distribution of soil attributes and to predict them at continuous depths. This paper proposes a new model to predict pH for a whole soil profile. Soil properties including pH are often similar within the plow layer from mixing during tillage and other agricultural operations. Similarly, soil pH below the root zone tends to be uniform due to low disturbance, leaving a transition zone from the bottom of the tillage layer to the bottom of the root zone due to depth-dependent root density and related soil processes. Keeping this physical description of agricultural field soil profile in mind, a closed form equation (model) was developed similar to a sigmoid curve. The model has 4 parameters including 1) soil pH at the top of a soil profile, 2) soil pH at the bottom of a soil profile, 3) hillslope parameter representing steepness of the curve that is determined by the length of the root zone, and 4) inflection point representing almost the midpoint of the transition zone or root zone. A total of 32 soil cores down to about 1.1 m depths were collected from an agricultural field of Macdonald farm, McGill University. The sub-samples were taken at every 10 cm and analyzed for soil pH in soil: water suspension in the laboratory. The measured pH was used to test the fitting performance of the sigmoid model. Additionally, a global dataset with 432 profiles with various soil classes, drainage types, land use, and altitude was also used to test the generality of the new model. The performance of this model was compared with the results of the commonly used 3rd order polynomial regression function and the equal-area quadratic spline function. Good performance of the sigmoid model with explicit physical explanation showed promise in predicting soil pH at depths. The spline function had the highest accuracy but lacked a general trend in its shape and parameters. The polynomial function had good accuracy and displayed a non-monotonic trend, which can also be used as a substitute for some profiles with complex variability. [ABSTRACT FROM AUTHOR]

A depth-based classification method underlain by a remote concentration measure for processing asymmetric data is developed and investigated. The motivation for the construction of the method was the inefficient use of affine invariant classifiers in combination with depth functions vanishing outside the convex hull of data. The idea of the proposed method is to map a remote space using a remote concentration measure, the Stahel-Donoho remoteness measure, and a corrected remoteness measure. [ABSTRACT FROM AUTHOR]

A nonparametric affine-invariant extrapolation depth-based classifier resistant to spikes and extreme values is proposed and investigated. A multilevel smoothing structure is proposed that makes it possible to obtain global properties of density functions and class boundaries under appropriate regularity conditions. The extrapolation depth-based classifier uses kernel density estimates to efficiently classify multidimensional data at different smoothing levels. [ABSTRACT FROM AUTHOR]

The identification of soil management strategies as well as the evaluation of their effectiveness requires detailed information on the spatial and temporal patterns of soil organic carbon storage. High-resolution SOC profile data are generally not available and traditional methods for collecting these are time consuming and costly. Recent studies use geo-statistical approaches to assess the three-dimensional patterns of SOC storage. However, there is still a large discrepancy between the continuous and high resolution mapping of the horizontal SOC variability on the one hand, and the coarse and discontinuous mapping of the vertical SOC profile on the other. In this study, we combine spectroscopic techniques with spatial modeling in a small, cultivated catchment in Germany and we evaluate the contribution of soil redistribution processes and topographical parameters to the observed spatial and vertical patterns. Using high-resolution data from soil cores, we evaluated the robustness of a third order polynomial function to model the vertical SOC profile. Using a crossvalidation, our results show that this approach results in a robust model (RSME = 0.24%) and performs better than the widely used exponential depth model (RMSE = 0.39%). In a next step, we evaluated the relationship between the parameters of the SOC depth model and co-variables including soil redistribution (inferred from 137 Cs data) and topographical indices using a multiple linear regression model. The performance was calculated by cross-validation and we found a low robustness of the models because of the low number of profiles (i.e. n = 19). A statistical evaluation of the co-variables highlighted two key factors influencing the SOC vertical distribution. Soil redistribution processes mainly influenced the surface SOC content (first centimeters) whereas the shape of the depth distribution was controlled by slope curvature alone. The mapping of polynomial parameters was validated using an external SOC profile dataset and showed a poor prediction of the surface content but a good prediction of the depth distribution once the surface SOC content is known (RMSE = 0.15–0.25%C). This suggests that estimating the vertical SOC profile from topsoil data by applying remote sensing data, in combination with our SOC profile model, is promising and can will result in an accurate mapping of 3D SOC patterns at a very high resolution. [ABSTRACT FROM AUTHOR]

Datasets for modelling and mapping soil properties often consist of samples from many spatial locations, collected from several different soil depth intervals. However, interest may lie in the spatial distribution of the property for a particular target depth interval, which may or may not correspond to the sampled intervals. It is the task of the data analyst to put the data together in such a way that useful and reliable conclusions can be drawn for the soil depths of practical interest. Previous studies to tackle this problem include multi-stage approaches and point-data-based 3-dimensional geostatistical approaches. One disadvantage of a multi-stage approach — for example, first fitting splines to the data for sampled profiles, then imputing new data for the target interval, before considering a spatial analysis with the imputed data — is that the imputation generally ignores any uncertainty in the imputed data, which might give misleading conclusions. Point geostatistical methods, on the other hand, assume that the data represent the value of the target variable at a specific point in the profile, rather than its average over a sampling interval; this too could give misleading estimates. In this work, we present a statistical method that properly deals with the sample support of soil profile data so that all data can be considered in a single geostatistical analysis. The approach is based on the ‘area-to-point’ kriging framework, which can be used to represent the uncertainty from data that are averages over non-negligible sample supports (in our case, the different sampled depth intervals). We combine a covariance model for the increment-averaged data in the vertical domain with another model for the horizontal variation. This enables us to (i). process all data in a single analysis, and (ii). calculate predictions for any target depth and support based on the same statistical model. We test the approach on data from the Murray–Darling basin in eastern Australia, where interest lies in mapping various soil properties that could have an effect on water salinity of the nearby Muttama Creek: we illustrate the methodology for predicting clay content. Finally we discuss a number of possible extensions of the methodology to broaden its applicability, which should provide the basis of further studies. [ABSTRACT FROM AUTHOR]

The use of a physically-based hydrological model for streamflow forecasting is limited by the complexity in the model structure and the data requirements for model calibration. The calibration of such models is a difficult task, and running a complex model for a single simulation can take up to several days, depending on the simulation period and model complexity. The information contained in a time series is not uniformly distributed. Therefore, if we can find the critical events that are important for identification of model parameters, we can facilitate the calibration process. The aim of this study is to test the applicability of the Identification of Critical Events (ICE) algorithm for physically-based models and to test whether ICE algorithm-based calibration depends on any optimization algorithm. The ICE algorithm, which uses the data depth function, was used herein to identify the critical events from a time series. Low depth in multivariate data is an unusual combination and this concept was used to identify the critical events on which the model was then calibrated. The concept is demonstrated by applying the physically-based hydrological model WaSiM-ETH on the Rems catchment, Germany. The model was calibrated on the whole available data, and on critical events selected by the ICE algorithm. In both calibration cases, three different optimization algorithms, shuffled complex evolution (SCE-UA), parameter estimation (PEST) and robust parameter estimation (ROPE), were used. It was found that, for all the optimization algorithms, calibration using only critical events gave very similar performance to that using the whole time series. Hence, the ICE algorithm-based calibration is suitable for physically-based models; it does not depend much on the kind of optimization algorithm. These findings may be useful for calibrating physically-based models on much fewer data. Editor D. Koutsoyiannis; Associate editor A. Montanari Citation Singh, S.K., Liang, J.Y., and Bárdossy, A., 2012. Improving calibration strategy of physically-based model WaSiM-ETH using critical events. Hydrological Sciences Journal, 57 (8), 1487–1505. [ABSTRACT FROM AUTHOR]

Abstract: Consider any kind of parameter for a probability distribution and a fixed distribution. We study the subsets of the parameter space constituted by all the parameters of the probabilities in the -trimming of the fixed distribution. These sets will be referred to as parameter trimmed regions. They are composed of all parameter candidates whose degree of suitability as such a parameter for the distribution is, at least, a specific value . In particular, we analyze location, scale, and location-scale parameters and study the properties of the trimmed regions induced by them. Several specific examples of parameter trimmed regions are studied. Among them, we should mention the zonoid trimmed regions obtained when the chosen parameter is the mean value and the location-scale regions of a univariate distribution obtained when the parameter is the pair given by the mean and the standard deviation. [Copyright &y& Elsevier]

Abstract: It has been commonly admitted that the meaning of a descriptive feature of distributions is given by an ordering and that the measures for this feature are meaningful only if they preserve the ordering. However, while many multivariate kurtosis measures have been introduced, multivariate kurtosis orderings have received relatively little investigation. In this paper, we propose and study a generalized multivariate kurtosis ordering. Under some conditions, this ordering is affine invariant and determines elliptically symmetric distributions within affine equivalence. Some special cases of the generalized ordering provide the kurtosis orderings for various existing multivariate kurtosis measures. Those kurtosis orderings are applied to explore the relationships of the multivariate kurtosis measures. Some other applications of the generalized multivariate kurtosis ordering are also given. [Copyright &y& Elsevier]

Abstract: We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order , given a random sample of size . The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions. [Copyright &y& Elsevier]

Abstract: Parametric and non-parametric depth functions have been used to estimate continuous soil profile properties. However, some soil properties, such as those seen in weathered loess, have anisotropic peak-shaped depth distributions. These distributions are poorly handled by common parametric functions. And while nonparametric functions can handle this data they lack meaningful parameters to describe physical phenomena in the depth distribution of a property such as a peak, an inflection point, or a gradient. The objective of this work is to introduce the use of asymmetric peak functions to model complex and anisotropic soil property depth profiles. These functions have the advantages of providing parameters, which quantify or describe pedogenic processes. We demonstrate the application of the Pearson Type IV (PIV) and the logistic power peak (LPP) functions to high resolution soil property depth profiles measured by diffuse reflectance spectroscopy in a claypan soil landscape of Northeastern Missouri, USA. Both peak functions successfully fit clay, silt, and pH data for an example soil profile from a summit landscape position (R2 =0.90 for pH and 0.98 for silt and clay). The LPP function was further demonstrated to fit clay depth distribution for a shoulder, backslope, footslope, and a depositional landscape position (R2 =0.98, 0.96, 0.96, 0.91). Relationships between the fitted parameters of these profiles were useful to describe landscape trends in their morphological features and show promise to continuously describe pedogenic processes in three dimensions. Peak functions are a useful companion to high-resolution soil profile data collected by sensors and their combined use may allow more intensive mapping and better explanation of soil landscape variability. [Copyright &y& Elsevier]

In this paper, we link the depth-based weighted empirical likelihood (WEL) with general estimating equations to produce a robust estimation of parameters for contaminated data with auxiliary information about the parameters. Such auxiliary information can be expressed through a group of functionally independent general estimating equations. Under general conditions, asymptotic properties of the WEL estimator are established. Furthermore, we prove that the WEL ratio statistic is asymptotically chi-squared distributed. Simulation studies are conducted to test the robustness of the WEL estimator. Finally, we apply the proposed method to analyse the gilgai survey data. [ABSTRACT FROM AUTHOR]

Abstract: Given a probability measure μ on Borel sigma-field of , and a function , the main issue of this work is to establish inequalities of the type , where m is a median (or a deepest point in the sense explained in the paper) of μ and M is a median (or an appropriate quantile) of the measure . For the most popular choice of halfspace depth, we prove that the Jensen''s inequality holds for the class of quasi-convex and lower semi-continuous functions f. To accomplish the task, we give a sequence of results regarding the “type D depth functions” according to classification in [Y. Zuo, R. Serfling, General notions of statistical depth function, Ann. Statist. 28 (2000) 461–482], and prove several structural properties of medians, deepest points and depth functions. We introduce a notion of a median with respect to a partial order in and we present a version of Jensen''s inequality for such medians. Replacing means in classical Jensen''s inequality with medians gives rise to applications in the framework of Pitman''s estimation. [Copyright &y& Elsevier]

Depth functions are increasingly being used in building nonparametric outlier detectors and in constructing useful nonparametric statistics such as depth-weighted L-statistics (DL-statistics). Robustness of a depth function is an essential property for such applications. Here, robustness of three key depth functions, spatial, simplicial, and generalised Tukey, is explored via the influence function (IF) approach. For all three depths, the IFs are derived and found to be bounded, an important robustness property, and are applied to evaluate two other robustness features, gross error sensitivity and local shift sensitivity. These IFs are also used as components of the IFs of associated DL-statistics, for which through a standard approach consistency and asymptotic normality are then derived. In turn, the asymptotic normality is applied to obtain asymptotic relative efficiencies (ARE). For spatial depth, two forms of weight function suggested in the recent literature are considered and AREs in comparison with the mean are obtained. For all three depths and one of these weight functions, finite sample REs are obtained by simulation under normal, contaminated normal, and heavy-tailed t distributions. As a technical tool of general interest, needed here with the simplicial depth, the IF of a general U-statistic is derived. [ABSTRACT FROM AUTHOR]

Suppose y, a d-dimensional ( d ≥ 1) vector, is drawn from a mixture of k ( k ≥ 2) populations, given by ∏1, ∏2,...,∏ k . We wish to identify the population that is the most likely source of the point y. To solve this classification problem many classification rules have been proposed in the literature. In this study, a new nonparametric classifier based on the transvariation probabilities of data depth is proposed. We compare the performance of the newly proposed nonparametric classifier with classical and maximum depth classifiers using some benchmark and simulated data sets. [ABSTRACT FROM AUTHOR]

Abstract: We define a new family of central regions with respect to a probability measure. They are induced by a set or a family of sets of functions and we name them integral trimmed regions. The halfspace trimming and the zonoid trimming are particular cases of integral trimmed regions. We focus our work on the derivation of properties of such integral trimmed regions from conditions satisfied by the generating classes of functions. Further we show that, under mild conditions, the population integral trimmed region of a given depth can be characterized in terms of certain regions based on empirical distributions. [Copyright &y& Elsevier]

• A region of influence approach and data depth is used to estimate flood magnitude. • Performance is affected by statistical outliers having unusual physical features. • More accurate rainfall information will significantly improve method performance. • The method can be used for design in ungauged catchments. A region of influence approach in which statistical depth functions are used to select gauged catchments having similar flood generation behaviour to a given ungauged basin coupled with nonparametric regression on catchment characteristics is employed to estimate mean annual and a quantile flood of 100 year return period in New Zealand basins. This approach may be applied internationally. Relative and root mean square errors assessed by jack-knife resampling are too large to be useful for detailed design in ungauged catchments but are much smaller than those resulting from a conventional multiple nonlinear regression approach. The distribution of errors for samples of sites indicates that large error values result from contributions from a relatively small number of statistical outliers Although these are all natural catchments, the outlier basins have particular physical characteristics such as moraines, karst features and pumice sediments which affect their flood generation processes. Relative errors in estimates for an ungauged basin may be predicted using a relationship involving differences in area, rainfall and statistical depth between the ungauged catchment and gauged similar catchments. Improved analytical techniques and selection and measurement of catchment characteristics should reduce error magnitude but not perhaps sufficiently for use in detailed design. Parallel development based on rainfall-runoff models is desirable but the present lack of requisite information largely concerning rainfalls means that attainment of desired accuracy is a distant prospect. [ABSTRACT FROM AUTHOR]

• Extensions of the increment-averaged kriging approach are presented. • The approach is combined with a machine-learning algorithm, Cubist. • Cubist parameters are fitted while accounting for data support and correlation. • A composite-likelihood approach is presented for large datasets. • The approach is illustrated to map soil electrical conductivity at a regional scale. Prediction and mapping of soil properties for different soil depths can provide important information for effective land management. Such predictions and maps are often built based on soil data from multiple soil surveys, and the sampled depths will rarely align with depths for which the predictions and maps are required. A recently proposed approach to deal with such datasets, termed increment-averaged kriging (IAK), fits a single model using data from all profiles and all soil depths. The approach considered several potential stumbling blocks: the effect that different widths of the sampled depth intervals has on the variances (sample support); potential non-stationarity in the covariances, depending on soil depth; the need to account for interactions between depth and the effect of covariates. In this work, we present some extensions to the original work that (i) extend the covariance model, (ii) integrate the use of a machine learning algorithm, Cubist, into the framework so that its regression parameters can be fitted while accounting for the correlation and sample support of the data, and (iii) apply a composite likelihood approximation to enable estimation and prediction with large datasets. Code to implement the method in R is also made available so that practitioners can test IAK on their own datasets. [ABSTRACT FROM AUTHOR]