Depth Function Research Articles

Depth-based support vector classifiers to detect data nests of rare events

The aim of this project is to combine data depth with support vector machines (SVM) for binary classification. To this end, we introduce data depth functions and SVM and discuss why a combination of the two is assumed to work better in some cases than using SVM alone. For two classes X and Y , one investigates whether an individual data point should be assigned to one of these classes. In this context, our focus lies on the detection of rare events, which are structured in data nests: class X contains much more data points than class Y and Y has less dispersion than X. This form of classification problem is akin to finding the proverbial needle in a haystack. Data structures like these are important in churn prediction analyses which will serve as a motivation for possible applications. Beyond the analytical investigations, comprehensive simulation studies will also be carried out.

Depth-based classification of directional data

A non-parametric procedure based on the concept angular depth function is developed for dealing with classification problems of objects in directional statistics. Several notions of depth for directional data are adopted: the angular simplicial, the angular Tukey’s, the arc distance, the cosine distance and the chord distance depths. The proposed method is flexible and can be applied even in high-dimensional cases when a suitable notion of depth is adopted. Performances are investigated and compared by applying methods to different distributional settings through simulated and real data sets.

Prediction of sediment transport capacity using thawed depth deduced from temperature measurement

In cold regions, changes in thawed soil depth induced by temperature fluctuation are commonly observed and can affect sediment transport capacity of meltwater flow and erosion processes. In this paper, a method based on laboratory experiment was proposed to predict sediment transport capacity using the thawed soil depth derived from temperature at different soil depths measured during the experimental time period. A saturated soil column with platinum resistance sensors at depths of 5, 10, 15, 20, and 25 cm from the soil surface was placed in a freezer at -25, -20, and -15 °C until it was totally frozen and then placed in an incubator with a constant temperature of 25, 30, and 35 °C. The thawing processes of the frozen soil column were monitored by the sensors that measured temperature during the experimental time. Based on soil heat conduction mechanism, the relations among soil depth, temperature, and time were deduced as a function to predict thawed soil depth giving the frozen temperature, soil temperature inside the incubator, and time after being placed in the incubator. The sediment transport capacity was then calculated by combining the deduced thawed soil depth function and an existing model. The predicted sediment transport capacity was compared with laboratory measurements in literatures. The results show that the model has the ability of predicting sediment transport capacity ranged from 0.019 to 0.598 kg s−1 m−1 with high R2 of 0.99 and 0.98. It suggests that the proposed model can be used to predict the sediment transport capacity by concentrated meltwater flow on different thawed soil slope. This research contributes to our understanding of soil erosion processes by meltwater on partially thawed soil slope.

Predicting soil properties in 3D: Should depth be a covariate?

Soil is a three-dimensional volume with property variability in all three dimensions. In Digital Soil Mapping (DSM), the variation of soil properties down a profile is usually harmonised by the use of the equal-area spline depth function approach. Soil observations at various depth intervals are harmonised to pre-determined depth intervals. To create maps of soil at the defined depth intervals, 2.5D model produces maps of individual depth intervals separately. Those maps can be reconstructed to produce a continuous depth function for each predicted location. More recently, several studies propose that soil property at any depth can be mapped using a model incorporating depth along with spatial covariates as predictor variables, creating a ‘3D’ model. The aim of this study is to evaluate the proposition that soil properties can be predicted at any depth.This study compares the 2.5D model and 3D model in two areas. The first test is on a 1500 km2 area in Edgeroi, New South Wales (NSW), Australia, mapping soil organic carbon (SOC, %), carbon storage (kg m−2), pH (H2O), clay content (%), and cation exchange capacity (CEC, mg/kg) based on depth-interval observations. The second study area in the Lower Hunter Valley has SOC observations at every 2 cm increment from a 210 km2 area. 2.5D and 3D models were tested in both study areas using four machine learning techniques: Cubist regression tree, Quantile Regression Forest (QRF), Artificial Neural Network (ANN), and 3D Generalised Additive Model (GAM). Results show that, in terms of R2 and RMSE, 2.5D and 3D models using different machine learning models produce comparable results on the validation of depth interval observations. The 3D tree-based models produce “stepped” prediction of properties with depth. Results on the Hunter Valley area with point observations show that the 3D model cannot replicate field point observations. 3D soil mapping on point depth observation has lower accuracy and larger uncertainty compared to the 2.5D model. For future DSM studies, 3D soil mapping with depth as a covariate requires caution with respect to the prediction method and the requirements of the results.

Empirical evidence of orthogonal relationship between directional site effects and fracture azimuths in an active fault zone: The case of the Mt. Etna lower eastern flank

Mt. Etna, in eastern Sicily, is a composite active volcano whose eastern flank is affected by several normal and strike-slip active faults. In this study geological-structural field survey and ambient vibration measurements are performed in the lower eastern flank of Mt. Etna in order to define the role of the tectono-stratigraphic setting in the seismic site response. The sites of measurements are mainly distributed inside the few hundreds of meters long S. Caterina Graben, which is located between two main seismogenic active faults of this volcano sector, the Acireale and the Fiandaca faults. Our study aims at defining the contribution of the fault related active deformation in determining the local ground motion amplification, also providing a subsoil geological model based on the integration of geological and geophysical investigation. Therefore, structural measurements were made in seven sites, revealing coseismic and aseismic extensional fractures mainly striking NNW. The seismic site response was studied processing the ambient vibration measurements through Horizontal-to-vertical spectral ratio. The measured fundamental frequency falls in the range 1.0–4.0 Hz, that we linked to the presence of a velocity contrast at depth of 80–200 m by calibrating a frequency depth function. Moreover, the seismic site effect is due to the stratigraphic setting and is markedly directional, showing maximum amplification along the direction orthogonal to the strike of the observed isooriented extensional fracture field produced by the activity of the S. Caterina Graben. It is noteworthy that, far from the graben on sites of measurements located near the Fiandaca Fault, we observed directional effects parallel to the direction of maximum extension inferred for this tectonic lineament. Our results suggest a linkage between directional site effects and local stress tensors and fractures field orientation.

Digital mapping of soil texture classes using Random Forest classification algorithm

AbstractSoil texture is the most important soil physical property that determines water holding capacity, nutrient availability and crop growth. Spatial distribution of soil texture at a higher spatial resolution at regional and national level is essential for crop planning and management. In the present study, we mapped the soil textural classes over 16.2 M ha area of Andhra Pradesh state, India, at 250 m spatial resolution up to 2 m depth using the digital soil mapping approach. A total of 2,272 profile observations were used for the prediction of soil textural classes using the Random Forest (RF) classification algorithm. To estimate textural classes at six standard soil depth intervals (0–5, 5–15, 15–30, 30–60, 60–100 and 100–200 cm), we used continuous depth function of texture distribution using average sand, silt and clay content of different textural classes. Depth‐wise spline outputs were then transformed into textural classes as per USDA textural classification. Sixteen environmental variables including Landsat‐8 data, digital elevation model attributes and climatic variables were used for modelling. For model building, 75% of data was used and 25% of data was used for validation. Overall classification accuracy index and kappa index were calculated for validation data sets using 100 RF models. We recorded overall accuracy of 50%–65% and kappa index of 35%–47% for various depths. We found that equal‐area quadratic splines of average sand, silt and clay are useful for soil profile depth harmonization of soil textural classes and random forest classification algorithm is a promising tool for spatial prediction of texture classes at the regional level. The present high‐resolution (250 m) maps of soil texture classes are useful for different hydrological studies and preparation of proper land‐use plans.

Wear modelling in wheel–rail contact problems based on energy dissipation

ABSTRACT This work is concerned with the numerical procedure to solve two-dimensional wheel-rail contact problem between a rigid wheel and an elastic-plastic rail lying on a rigid foundation. The contact phenomenon includes Coulomb friction, frictional heat generation as well as the wear of the contacting surfaces. The wear depth function appears as an internal variable in the non-penetration condition updating the gap between the worn surfaces of the bodies. Moreover the dissipated energy due to friction is calculated to evaluate the loss of material and to determine the shape of the contacting surfaces during the wear evolution process. This contact problem is solved numerically using the finite element method as well as the operator splitting approach and semi-smooth Newton method. The distribution of contact surface stresses and temperatures as well as the evolution of the shape of the contact surfaces and the wear depth are reported and discussed.

Desingularization of a Steady Vortex Pair in the Lake Equation

We construct a family of steady solutions of the lake model perturbed by some small Coriolis force, that converge to a singular vortex pair. The desingularized solutions are obtained by maximization of the kinetic energy over a class of rearrangements of sign changing functions. The precise localization of the asymptotic singular vortex pair is proved to depend on the depth function and the Coriolis parameter, and it is independent on the geometry of the lake domain. We apply our result to construct a singular rotating vortex pair in a rotation invariant lake.

Gravity sparse inversion using the interior-point method and a general model weighting function

This study presents an optimized gravity-sparse inversion method. The proposed method minimizes the global objective function using interior-point method for boundary constraints and a general weighting function comprising the depth, compactness, and kernel weighting functions of the density models. For the compactness weighting function, practical experiments demonstrate that the recovered model becomes more compact with an increasing value for the relative exponential factor β. However, if no appropriate boundary-constraint method is applied, the inversion results cannot be controlled within the designated constraint bounds when β needs to be set to a large value to obtain compact inversion results. The interior-point method allows the use of a larger β to obtain more compact inversion results without violating the boundary constraints. Additionally, models in close proximity can more clearly be recognized using this method. To improve the computational efficiency and obtain a more accurate regularization parameter, the preconditioned conjugate gradient and L-curve, or line search methods, were also applied. The proposed method was applied for three synthetic examples: two positive bodies adjacent to each other at different depths inverted using noise-free gravity anomaly data, three bodies (positive or negative) at different depths inverted using noise-free or contaminated gravity anomaly data, and three bodies (positive or negative) characterized by a certain dip angle inverted using contaminated gravity anomaly data. This method was also applied for the inversion of a Woodlawn sulfide body, Missouri iron ore body, and granitoid rock body in the Rio Maria region in the state of Para, Brazil. In all six test cases, larger β values were used and the density models were recovered with sharper boundaries within the designated bounds.

Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem

Abstract We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which this element has a non-trivial image. We also prove that the depth function of this group grows faster than any recursive function.

On the Boundary Control of a Boussinesq System

A boundary control problem is considered for determining a canal depth function optimally for a canal system modeled by a nonlinear Boussinesq equation. By determining the optimal canal depth function, it is aimed to damp out the undesired waves in canal system filled up water. For achieving this aim, the existence and uniqueness of the solutions to system and controllability properties of the system is investigated. Optimal canal depth control function is obtained by means of a maximum principle, which is an elegant tool for transferring the optimal boundary control problem to solving a system of equations including initial-terminal-boundary conditions. The reason making this paper is important that optimal control function is gained without linearization of nonlinear term in the system. In order to show the correctness of the obtained theoretical results, several numerical examples are presented by MATLAB in graphical and table forms. Observing these tables and graphics, it is concluded that introduced boundary control algorithm is effective and has the potential for extending to other nonlinear control systems.

Dynamic capability development in multinational enterprises: Reconciling routine reconfiguration between the headquarters and subsidiaries

AbstractResearch SummaryStarting from the premise that firms need dynamic capabilities to adapt to changing environments, we discuss how multinational enterprises (MNEs) develop dynamic capabilities from internationalization. Unlike domestic firms that develop dynamic capabilities within one organizational system, MNEs are inherently multi‐level systems with the headquarters and subsidiaries. In this paper, we focus on how internationalization depth and breadth function as sources of learning and unlearning in the headquarters and subsidiaries, and how this serves as the antecedent for routine reconfiguration and dynamic capability development in the MNE. We theorize that the headquarters' and subsidiaries' brokering capabilities are critical for reconciling routine reconfiguration at the two levels so that dynamic capability development can occur, and the MNE can adapt to environmental changes.Managerial SummaryOur model of dynamic capability development in multinational enterprises (MNEs) via internationalization makes dynamic capabilities actionable for managers in three main ways. First, we emphasize both learning and unlearning from internationalization as important for both the subsidiaries and headquarters to reconfigure routines to adapt to changing environments. Second, the routine reconfiguration that occurs from the subsidiaries or the headquarters serves as the impetus for dynamic capability development in the MNE. Third, for dynamic capability development to occur in the MNE, both the headquarters and the subsidiaries must be able to reconcile routine reconfigurations via the headquarters' and the subsidiaries' brokering capabilities.

Numerical soil classification supports soil identification by citizen scientists using limited, simple soil observations

AbstractAccurately identifying the soil map unit component at a specific point‐location within a landscape is critical for implementing sustainable soil management. Recent developments in smartphone‐based technologies for characterizing soil profiles, coupled with improved numerical soil classification algorithms, have made it more accessible for non‐soil scientists to sample, characterize, and classify soil profiles. The main objective of this study was to evaluate an operational soil classification framework for identifying the soil component at a sampling‐location based on the numerical similarity of soil property values between the sampled soil profile and the soil components mapped in that area. To evaluate this soil identification framework, we used a subset of the U.S. National Cooperative Soil Survey Soil Characterization Database (NCSS–SCD) as our soil profile test dataset and the U.S. Soil Survey Geographic (SSURGO) database as our reference dataset using profile data of soil components in the area surrounding each test profile. Numerical similarity was tested using soil property data representing different degrees of generalization, both in terms of generalizing depth‐wise variability (i.e., depth‐support) and generalizing across feature space (i.e., soil properties). Three soil property groups (i.e., Novice, Expert, Expert‐Plus) representing different levels of detail and three types of depth‐support (i.e., genetic horizon, depth intervals, and depth functions) were evaluated. Using a simple set of soil property inputs (i.e., Novice: soil texture class, rock fragment volume class, and soil color) resulted in nearly as high identification accuracy (46–53%) as that achieved with an Expert (48–57%) dataset that included more precise determinations (percent sand, silt, clay, and rock fragment volume), and virtually no further improvement with the addition of pH and organic matter in the Expert‐Plus dataset (53–60%). This study also showed minimal effect from the type of depth‐support used to represent depth‐wise variability. Furthermore, we evaluated several measures of soil functional similarity (i.e., ecological sites, land capability, taxonomic distance) which resulted in management relevant accuracies ranging from 65–89%. These findings support the utility of simple soil observations sampled at fixed depths for soil identification.

Approximate containment relation between mode spaces in uncertain shallow water

Mode spaces are the subspaces that describe where the sound of interest impinging on the vertical line array (VLA) lives. It is spanned by the VLA sampled modes (modal depth functions), and is required in the implementation of some underwater source detection and localization algorithms. However, due to environmental uncertainty, especially in shallow water, the mode space is difficult to obtain. How the mode spaces derived from different realizations of uncertain shallow-water environments contain each other is investigated, in terms of how the mode space can contain the signal wavefronts lying in another, in an energy sense. The relative projection deviation (RPD) and the greatest relative signal-energy loss (GRSL) from one mode space to another are defined to quantify the containment degree. The simulation results in a benchmark uncertain shallow-water waveguide suggest that, due to good modal correlations, the mode space derived from one environmental realization is nearly a subset of those derived from others supporting the same or greater number of modes irrespective of the differences between the same-order sampled modes: Projecting the signal wavefronts in a certain mode space to another same or greater dimensional one will generally cause no more than 0.01 energy to be lost. This property holds increasingly well as the array aperture reduces, and is tolerant of environmental uncertainty degree. It is also demonstrated by SWellEx-96 experimental data. Several applications motivated from this property are also presented.

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.

Remarks on the one-dimensional sloshing problem involving thep-Laplacian operator

In this paper, we study the inverse nodal problem and the eigenvalue gap for the one-dimensional sloshing problem with the $p$-Laplacian operator. By applying the Prüfer substitution, we first derive the reconstruction formula of the depth function by using the information of the nodal data. Furthermore, we employ the Tikhonov regularization method to consider how to reconstruct the depth function using only zeros of one eigenfunction. Finally, we investigate the eigenvalue gap under the restriction of symmetric single-well depth functions. We show the gap attains its minimum when the depth function is constant.

Multi-task convolutional neural networks outperformed random forest for mapping soil particle size fractions in central Iran

Knowledge about the spatial distribution of soil particle size fractions (PSF) is critical for sustainable management and resource assessment of the agricultural regions. Although conventional machine learning algorithms, such as random forest (RF) or support vector machine, have been extensively used in digital soil mapping to predict the PSF, less research examined the potential of state-of-the-art deep learning approaches for such processing. Importantly, deep learning approaches such as convolutional neural networks (CNNs) are able to incorporate contextual information about the landscape, which is of great use for DSM analysis. Accordingly, this study addresses this much-needed investigation by using a patch-based, multi-task CNN for predicting PSF of clay, sand, and silt contents at six standard layers given as soil depth increments as recommended by the GlobalSoilMap.net (i.e., 0–5, 5–15, 15–30, 30–60, 60–100, 100–200 cm). The depth functions were derived from equal-area smoothing splines in a region covering large parts (~140,000 km2) of central Iran. The robustness of the proposed architecture is evaluated against RF. Additionally, to allow a fairer comparison between RF and CNN models, we used simple smoothing (mean) filters to effectively reproduce the auxiliary data which are then fed in the RF (RF*). To evaluate the three models, we established a training (75%) and test set (25%). According to the test set, for all soil depths and all PSFs, the results demonstrate that CNN consistently outperforms RF and RF* in terms of root mean square error (RMSE) and coefficient of determination (R2). At the top layer, for example, CNN decreased the RMSE values for clay, sand, and silt contents compared to the RF (22.4%, 18.9%, and 10.7%) and RF* (18.0%, 7.4%, and 9.6%). These findings indicate that even the use of feature-engineered auxiliary data did not enable the RF* models to reach the performance of CNN. The resulting maps can be used as valuable baseline soil information for the effective management of agricultural and environmental resources in the study area and beyond.

Depths of Pluto's and Charon's craters, and their simple-to-complex transition

Impact craters form as point-like explosions on solar system bodies, excavating a cavity within the surface. Most information about the impactor itself is lost, save its energy, such that the target itself controls much of the final crater size and shape given the energy input. For this reason, impact craters are a useful probe of surface differences across a single body and between bodies. Two properties that are commonly used to compare craters from one body to another are the ratio between crater depth and diameter, and the diameter at which craters transition between simple, bowl-shaped morphologies to more complex morphologies that include flat floors, central peaks, wall terraces, and scalloped rims. Both metrics are important as input into theoretical models and for probing the strength of the surface material. In this work, we have measured these properties on Pluto and Charon and report on the results. However, the analysis is not entirely straightforward; a secondary goal of this work is to explore how to measure and quantify the two properties accurately (depth vs diameter and simple-to-complex transitions). We arrive at different results based on the method, and we report on that variation as a way to reasonably estimate the uncertainty in each quantity. For those reasons, it is difficult to place a single, exact value on each, but we do conclude that the morphology-based simple-to-complex transition occurs at approximately 11–12½ km on Pluto, and 13½–16 km on Charon. We also conclude that the existing, public topography data for at least Charon is likely too low in resolution to measure the simple crater depth vs diameter function accurately, such that a morphometry-based simple-to-complex transition diameter is not quantifiable at this time.

Flood frequency estimation in New Zealand using a region of influence approach and statistical depth functions

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.

Block sparse Bayesian learning for broadband mode extraction in shallow water from a vertical array.

The horizontal wavenumbers and modal depth functions are estimated by block sparse Bayesian learning (BSBL) for broadband signals received by a vertical line array in shallow-water waveguides. The dictionary matrix consists of multi-frequency modal depth functions derived from shooting methods given a large set of hypothetical horizontal wavenumbers. The dispersion relation for multi-frequency horizontal wavenumbers is also taken into account to generate the dictionary. In this dictionary, only a few of the entries are used to describe the pressure field. These entries represent the modal depth functions and associated wavenumbers. With the constraint of block sparsity, the BSBL approach is shown to retrieve the horizontal wavenumbers and corresponding modal depth functions with high precision, while a priori knowledge of sea bottom, moving source, and source locations is not needed. The performance is demonstrated by simulations and experimental data.