Depth Function Research Articles

Mapping high resolution National Soil Information Grids of China

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.

Computability of finite quotients of finitely generated groups

Abstract We systematically study groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can fail to be recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem but not have computable finite quotients.

Quantifying Depth‐Dependent Seismic Anisotropy in the Critical Zone Enhanced by Weathering of a Piedmont Schist

AbstractWeathering processes weaken and break apart rock, freeing nutrients and enhancing permeability through the subsurface. To better understand these processes, it is useful to constrain physical properties of materials derived from weathering within the critical zone. Foliated rocks exhibit permeability, strength and seismic anisotropy–the former two bear hydrological and geomorphological consequences while the latter is geophysically quantifiable. Each of these types of anisotropy are related to rock fabric (fractures and foliation); thus, characterizing weathering‐dependent changes in rock fabric with depth may have a range of implications (e.g., landslide susceptibility, groundwater modeling, and landscape evolution). To better understand how weathering effects rock fabric, we quantify seismic anisotropy in saprolite and weathered bedrock within two catchments underlain by the Precambrian Loch Raven schist, located in Oregon Ridge Park, MD. Using circular geophone arrays and perpendicular seismic refraction profiles, anisotropy versus depth functions are created for material 0–25 m below ground surface (bgs). We find that anisotropy is relatively low (0%–15%) in the deepest material sampled (12–25 m bgs) but becomes more pronounced (29%–33%) at depths corresponding with saprolite and highly weathered bedrock (5–12 m bgs). At shallow soil depths (0–5 m bgs), material is seismically isotropic, indicating that mixing processes have destroyed parent fabric. Therefore, in situ weathering and anisotropy appear to be correlated, suggesting that in‐place weathering amplifies the intrinsic anisotropy of bedrock.

On depth-based fuzzy trimmed means and a notion of depth specifically defined for fuzzy numbers

Empirical trimmed means have been studied in general spaces and, in particular, they have been applied to the one-dimensional fuzzy case. They provide a competing robust estimation procedure of the central tendency for fuzzy number-valued data, but they are not the only way to define a trimmed mean in this space. The aim is to adapt trimmed means defined on the basis of certain depth function to the framework of fuzzy number-valued data and compare their behaviour with that of empirical fuzzy trimmed means. The first idea for evaluating the depth of a fuzzy number-valued observation consists of applying an existing functional depth to the expression of such an observation as a function. The second alternative introduces a depth function specifically defined for fuzzy numbers. The empirical performance of both proposals is analyzed.

Open Science Practices for Studies of Event-Related Potentials (ERPs): Reporting ERP Reliability and Data Quality in the Spirit of Transparency

Hydrological frequency analysis (HFA) is commonly used for the assessment of the risk associated to hydrological events. HFA is generally based on the assumptions of homogeneity, independence and stationarity of the hydrological data. Hydrological events are often described through a number of dependent characteristics, such as peak, volume and duration for floods. Unfortunately, in this multivariate setting, the verification of the above assumptions is often neglected. When a shift occurs in a data series, it can affect the stationarity and the homogeneity of the data. The objective of this paper is to study tests for shift detection in multivariate hydrological data. The considered shift tests are mainly based on the notion of depth function, except for one test that is considered for comparison purposes. A simulation study is performed to evaluate and compare the power of all these tests with hydrological constraints. A flood analysis application is also carried out to show the practical aspects of the considered tests. The power of the considered tests is influenced by a number of factors, including the sample size, the shift amplitude, the magnitude of the series and the location of the shift in the series.

A sensors-based profile heterogeneity index for soil characterization

The technological and computational development has raised new possibilities for soil scientists to describe and infer about soil-forming processes and functions. Thus, why not to insert the contrasts between soil layers in a profile revealed by proximal sensing? We proposed to study the soil profile heterogeneity by reflectance (Vis-NIR-SWIR), X-ray fluorescence and magnetic susceptibility aiming to quantify it with the proposed “Profile Heterogeneity Index” (PHI) through seven soil sequences in twenty-seven locations over different hillslope positions and parent materials. The study was conducted in a 165-ha farm, at Rio das Pedras municipality, São Paulo State, Brazil. The differences between two consecutive soil layers were calculated by subtracting the data to the bottom and the immediately above for each soil profile. The results are depicted as a depth function with proportional colors to highlight the heterogeneity in the soil profiles. This approach provided a new method to identify where the transitions of soil attributes occur exactly and improve the understanding about soil-formation factors and processes. In this study, greater PHI values were mainly related to erosion and redeposition process through the landscape, differences in soil parent materials and also soil-forming processes. The sensors can furnish important and complementary information to the characterization of soil profile heterogeneity.

Non‐parametric depth‐based tests for the multivariate location problem

SummaryIn 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.

Outlier Detection Using Minimum Vector Variance Algorithm with Depth Function and Mahalanobis Distance

Outliers are observations where the point of observation deviates from the data pattern. The existence of outliers in the data can cause irregularities in the results of data analysis. One solution to this problem is to detect outliers using a statistical approach. The statistical approach method used in this study is the Minimum Vector Variance (MVV) algorithm which has robust characteristics for outliers. The purpose of this research is to detect outliers using the MVV algorithm by changing the data sorting criteria using the Robust Depth Mahalanobis to produce maximum detection. The results obtained from this study are that RDMMVV is superior to the observed value in showing the outliers and the location of the outliers in the data plot compared to DMMVV and MMVV.

Depth classification based on affine-invariant, weighted and kernel-based spatial depth functions

Several multivariate depth functions have been proposed in the literature, of which some satisfy all the conditions for statistical depth functions while some do not. Spatial depth is known to be invariant to spherical and shift transformations. In this paper, the possibility of using different versions of spatial depth in classification is considered. The covariance-adjusted, weighted, and kernel-based versions of spatial depth functions are presented to classify multivariate outcomes. We extend the maximal depth classification notions for the covariance-adjusted, weighted, and kernel-based spatial depth versions. The classifiers' performance is considered and compared with some existing classification methods using simulated and real datasets.

Comparison-based centrality measures

Recently, learning only from ordinal information of the type “item x is closer to item y than to item z” has received increasing attention in the machine learning community. Such triplet comparisons are particularly well suited for learning from crowdsourced human intelligence tasks, in which workers make statements about the relative distances in a triplet of items. In this paper, we systematically investigate comparison-based centrality measures on triplets and theoretically analyze their underlying Euclidean notion of centrality. Two such measures already appear in the literature under opposing approaches, and we propose a third measure, which is a natural compromise between these two. We further discuss their relation to statistical depth functions, which comprise desirable properties for centrality measures, and conclude with experiments on real and synthetic datasets for medoid estimation and outlier detection.

Expectile depth: Theory and computation for bivariate datasets

Expectiles are the solution to an asymmetric least squares minimization problem for univariate data. They resemble the quantiles, and just like them, expectiles are indexed by a level α in the unit interval. In the present paper, we introduce and discuss the main properties of the (multivariate) expectile regions, a nested family of sets, whose instance with level 0<α≤1/2 is built up by all points whose univariate projections lie between the expectiles of levels α and 1−α of the projected dataset. Such level is interpreted as the degree of centrality of a point with respect to a multivariate distribution and therefore serves as a depth function. We propose here algorithms for determining all the extreme points of the bivariate expectile regions as well as for computing the depth of a point in the plane. We also study the convergence of the sample expectile regions to the population ones and the uniform consistency of the sample expectile depth. Finally, we present some real data examples for which the Bivariate Expectile Plot (BExPlot) is introduced.

Depth functions of powers of homogeneous ideals

We settle a conjecture of Herzog and Hibi, which states that the function d e p t h S / Q n \mathrm {depth}\, S/Q^n , n ≥ 1 n \ge 1 , where Q Q is a homogeneous ideal in a polynomial ring S S , can be any convergent numerical function. We also give a positive answer to a longstanding open question of Ratliff on the associated primes of powers of ideals.

Assessment of environmental degradation due to anthropogenic processes based on critical zones: a study in a basin in southern Brazil

Different land uses imposed by human activities have affected biotic and abiotic components, modifying the natural environment according to population growth and increasing the environmental impact over time. As a consequence, the relevance of studies to evaluate the magnitude and aspects of environmental changes is increasing. In this context, the Ribeirao do Pinheirinho Basin (Sao Paulo state, southeastern Brazil) was selected for this study, since it is part of a region that recharges a set of drainage channels that provide water to several municipalities (approximately 2 million people), primarily during dry seasons. Additionally, this basin has been intensively used for agricultural purposes over the last 50 years. To determine the current characteristics and to analyze the environmental changes, this study was based on temporal and spatial analyses of the components of the basin’s critical zone at a scale of 1:10,000. In association, due to water availability issues, the HOST (hydrology of soil types) classification was adopted, as well as previous data, satellite images, aerial photos and results of laboratory and field tests. As a result, anthropogenic activities have changed the depth, areal distribution, and environmental functions of geological materials. These activities have affected the relationship between rainfall, infiltration and surface runoff, resulting in a lower groundwater level and decreasing water availability. In conclusion, the adopted methodology aids in the assessment of the changes in the environmental area of study. This could be useful for environmental recovery projects as well as the establishment of environmental policies.

Multivariate process control charts based on the Lp depth

AbstractEven if large historical dataset could be available for monitoring key quality features of a process via multivariate control charts, previous knowledge may not be enough to reliably identify or adopt a unique model for all the variables. When no specific parametric model turns out to be appropriate, some alternative solutions should be adopted and exploiting non‐parametric methods to build a control chart appears a reasonable choice. Among the possible non‐parametric statistical techniques, data depth functions are gaining a growing interest in multivariate quality control. Within the literature, several notions of depth are effective for this purpose, even in the case of deviation from the normality assumption. However, the use of the Lp depth for constructing non‐parametric multivariate control charts has been surprisingly neglected so far. Hence, the goal of this work is to investigate the behavior the Lp depth in the statistical process control and to compare its performances to those of the Mahalanobis depth, which is often adopted to build depth‐based control charts.

Numerical simulation of the spheroidization process with cored wire injection in liquid iron using finite volume method

We develop a finite volume method for the simulation of the liquid iron cored wire feeding spheroidization technology, which can realize the visualization of the spheroidization process and provide guidance for parameter optimization. In this work, we first define a two-dimensional unsteady heat-transfer mathematical model for single-core structure cored wire melting in liquid iron and then discretize the governing equation numerically. The present solver is validated by comparing it with the data obtained from experimental studies. Interestingly, in the numerical results we found that the absorption velocity of the magnesium element can be improved to the greatest extent to get the most ideal spheroidizing effect. To further show the melting and boiling characteristics of spheroidization, we also explore the relationship between melting and boiling depth, wire feeding speed, and temperature of liquid iron in the feeding process, and the wire feeding speed–depth function was also obtained.

Robust multivariate estimation based on statistical depth filters

In the classical contamination models, such as the gross-error (Huber and Tukey contamination model or case-wise contamination), observations are considered as the units to be identified as outliers or not. This model is very useful when the number of considered variables is moderately small. Alqallaf et al. (Ann Stat 37(1):311–331, 2009) show the limits of this approach for a larger number of variables and introduced the independent contamination model (cell-wise contamination) where now the cells are the units to be identified as outliers or not. One approach to deal, at the same time, with both type of contamination is filter out the contaminated cells from the data set and then apply a robust procedure able to handle case-wise outliers and missing values. Here, we develop a general framework to build filters in any dimension based on statistical data depth functions. We show that previous approaches, e.g., Agostinelli et al. (TEST 24(3):441–461, 2015b) and Leung et al. (Comput Stat Data Anal 111:59–76, 2017), are special cases. We illustrate our method by using the half-space depth.

An exactly solvable problem of wave fronts and applications to the asymptotic theory

A nontrivially solvable 4-dimensional Hamiltonian system is applied to the problem of wave fronts and to the asymptotic theory of partial differential equations. The Hamilton function we consider is H(x,p)=D(x)|p|. Such Hamiltonians arise when describing the fronts of linear waves generated by a localized source in a basin with a variable depth. We consider two realistic types of bottom shape: 1) the depth of the basin is determined, in the polar coordinates, by the function D(ϱ,φ)=(ϱ2+b)∕(ϱ2+a) and 2) the depth function is D(x,y)=(x2+b)∕(x2+a). As an application, we construct the asymptotic solution to the wave equation with localized initial conditions and asymptotic solutions of the Helmholtz equation with a localized right-hand side.

Characteristics of Pluto’s Haze and Surface from an Analytic Radiative Transfer Model

Abstract Observations of Pluto from New Horizons have been combined with previous ground-based observations and fit to a radiative transfer model based on Chandrasekhar’s planetary problem and Hapke theory to simultaneously derive the physical properties of the dwarf planet’s surface and atmosphere. We derive the macroscopic roughness, single-scattering albedo, and directional scattering properties of the surface, and the single-scattering albedo, optical depth, and single-scattering phase function of Pluto’s haze. The haze particles are small, with best-fit sizes in the range of ∼0.41–1.14 μm. We find that Pluto’s haze is more similar to that of Titan, rich in organic compounds and highly forward scattering, than that of Triton. With organic compounds and a likely subsurface water ocean, Pluto may harbor sustainable habitable environments. Our model, which includes the coherent backscatter effect, fits the anomalously large opposition surge recently discovered on Pluto.

Three-Dimensional Mapping of Forest Soil Carbon Stocks Using SCORPAN Modelling and Relative Depth Gradients in the North-Eastern Lowlands of Germany

To cope with the challenges in forest management that are contemporarily caused by climate change, data on current chemical and physical soil properties are more and more necessary. For this purpose, we present a further amalgam of depth functions and SCORPAN modelling to provide data at arbitrary depth layers. In this concept, regionalisation is split up into the modelling of plot totals and the estimation of vertical distributions. The intended benefits by splitting up are: consistency between estimates on plot level and depth layer level, avoidance of artificial depth gradients, straightforward interpretation of covariates in the sense of pedogenetic processes, and circumnavigation of the propagation of uncertainties associated with separation between horizons during field sampling. The methodology was tailored to the circumstances within the north-eastern lowlands and the utilisation of current inventory data of the National Forest Soil Inventory (NFSI) in Brandenburg (Germany). Using the regionalisation of soil organic carbon (SOC) as an example, the application is demonstrated and discussed in detail. The depth to groundwater table and terrain parameters related to the catchment area were the main factors in SOC storage. The use of kriging did not improve the model performance. The relative depth gradients of SOC were especially distinguished by tree species composition and stand age. We suppose that interesting fields of application may be found in scenario-based modelling of SOC and when SOC serves as a basis for hydrological modelling.

Classification of High-resolution Solar Hα Spectra Using t-distributed Stochastic Neighbor Embedding

Abstract The Hα spectral line is a well-studied absorption line revealing properties of the highly structured and dynamic solar chromosphere. Typical features with distinct spectral signatures in Hα include filaments and prominences, bright active-region plages, superpenumbrae around sunspots, surges, flares, Ellerman bombs, filigree, and mottles and rosettes, among others. This study is based on high-spectral resolution Hα spectra obtained with the Echelle spectrograph of the Vacuum Tower Telescope (VTT) located at Observatorio del Teide, Tenerife, Spain. The t-distributed stochastic neighbor embedding (t-SNE) is a machine-learning algorithm, which is used for nonlinear dimensionality reduction. In this application, it projects Hα spectra onto a two-dimensional map, where it becomes possible to classify the spectra according to results of cloud model (CM) inversions. The CM parameters optical depth, Doppler width, line-of-sight velocity, and source function describe properties of the cloud material. Initial results of t-SNE indicate its strong discriminatory power to separate quiet-Sun and plage profiles from those that are suitable for CM inversions. In addition, a detailed study of various t-SNE parameters is conducted, the impact of seeing conditions on the classification is assessed, results for various types of input data are compared, and the identified clusters are linked to chromospheric features. Although t-SNE proves to be efficient in clustering high-dimensional data, human inference is required at each step to interpret the results. This exploratory study provides a framework and ideas on how to tailor a classification scheme toward specific spectral data and science questions.