Representation Of Curves Research Articles

Technical note: Statistical generation of climate-perturbed flow duration curves

Abstract. Assessing the robustness of a water resource system's performance under climate change involves exploring a wide range of streamflow conditions. This is often achieved through rainfall–runoff models, but these are commonly validated under historical conditions with no guarantee that calibrated parameters would still be valid in a different climate. In this note, we introduce a new method for the statistical generation of plausible streamflow futures. It flexibly combines changes in average flows with changes in the frequency and magnitude of high and low flows. It relies on a three-parameter analytical representation of the flow duration curve (FDC) that has been proved to perform well across a range of basins in different climates. We rigorously prove that, for common sets of streamflow statistics mirroring average behaviour, variability, and low flows, the parameterisation of the FDC under this representation is unique. We also show that conditions applied to these statistics for a solution to exist are commonly met in practice. These analytical results imply that streamflow futures can be explored by sampling wide ranges of three key flow statistics and by deriving the corresponding FDC in relation to model basin response across the full spectrum of flow conditions. We illustrate this method by exploring in which hydro-climatic futures a proposed run-of-river hydropower plant in eastern Turkey is financially viable. Results show that, contrary to approaches that modify streamflow statistics using multipliers applied uniformly throughout a time series, our approach seamlessly represents a large range of futures with increased frequencies of both high and low flows. This matches expected impacts of climate change in the region and supports analyses of the financial robustness of the proposed infrastructure to climate change. We conclude by highlighting how refinements to the approach could further support rigorous explorations of hydro-climatic futures without the help of rainfall–runoff models.

Piecewise polynomial approximation of spatial curvilinear profiles using the Hough transform

Given a curvilinear profile P represented as a set of points in the space R3 and four families of low-degree polynomial curves that respectively depend on the parameters in the space R4, our goal is to identify the piecewise space polynomial curve best fitting the profile P. We use a parametric representation of the space curves and subdivide the profile into smaller portions that can be fitted with regular curves. We provide theoretical guarantees to the existence of such an approximation and an algorithm for the profile approximation. We take advantage of the implicit function theorem to locally project a space curve on at most two planes and to locally recognise it with a low-degree polynomial curve obtained by applying the Hough transform. Finally, we recombine the curve expressions on the two planes backwards in the space R3. The outcome of the algorithm is thus a piecewise polynomial curve approximating the profile. We validate our approach to approximate curvilinear profiles extracted from 3D point clouds representing real objects and to simplify and resample point clouds.

A New Model for Solving Hydrological Connectivity Inside Soils by Fast Field Cycling NMR Relaxometry

In this paper, a new quantitative approach for estimating the structural and functional connectivity inside soil by Fast Field Cycling (FFC) NMR relaxometry is presented, tested by measurements carried out in three samples with different texture characteristics. Measurements by FFC NMR relaxometry have been carried out using water-suspended samples and Proton Larmor frequencies (νL) ranging in the 0.015–35 MHz interval. Two non-degraded soil samples, with different textural characteristics, and a degraded soil collected in a badland area, were analyzed. For a given soil and any applied Proton Larmor frequency, the distribution of the longitudinal relaxation times, T1, (i.e., relaxogram) measured by FFC NMR has been integrated, and the resulting S-shaped curve (i.e., relaxogram integration curve) was represented, for the first time, by Gumbel’s diagram. This new representation of the relaxogram integration curve, transforming the S-shaped curve into a straight line, allowed for distinguishing three linear components, corresponding to three different relaxation time ranges, characterized by three different slopes. Two points, identified by the abrupt slope changes of the relaxogram integration curve plotted in Gumbel’s diagram, are used to identify two characteristic values of relaxation time, T1A and T1B, which define three well-known pore size classes (T1 < T1A micro-pores, T1A < T1 < T1B meso-pores, and T1 > T1B macro-pores). The relaxogram integration curve allowed for calculating the non-exceeding empirical cumulative frequency, F(T1), corresponding to the characteristic T1A and T1B values. The analysis demonstrated that the relaxogram can be used to determine the pore-size ranges of each investigated sample. Finally, using the slope values of the three components of the relaxogram integration curve, a new definition of the Structural Connectivity Index, SCI, and Functional Connectivity Index, FCI, was proposed.

Comparing overlapping data distributions using visualization

We present results from a preregistered and crowdsourced user study where we asked members of the general population to determine whether two samples represented using different forms of data visualizations are drawn from the same or different populations. Such a task reduces to assessing whether the overlap between the two visualized samples is large enough to suggest similar or different origins. When using idealized normal curves fitted on the samples, it is essentially a graphical formulation of the classic Student’s t-test. However, we speculate that using more sophisticated visual representations, such as bar histograms, Wilkinson dot plots, strip plots, or Tukey boxplots will both allow people to be more accurate at this task as well as better understand its meaning. In other words, the purpose of our study is to explore which visualization best scaffolds novices in making graphical inferences about data. However, our results indicate that the more abstracted idealized bell curve representation of the task yields more accuracy.

Explicit three-piece quadratic model for accurate analytic representation of S-shaped solar cell current–voltage characteristics

Some interest has been recently directed towards analytic mathematical models for reproducing the current–voltage (I–V) characteristics of solar cells, mainly due to their convenient property of expressing current as an explicit function of voltage (which is not true for the traditional one-diode model). However, as is the case with the one-diode model itself, existing analytic models do not yield a good fit in the case of S-shaped I–V characteristics. In this brief paper, an analytic explicit model based on a three-piece quadratic curve is proposed that surpasses such difficulty and enables accurate representation of S-shaped curves. To evaluate the proposed model, validation was carried out against three empirical solar cell I–V characteristics. In all cases, simulation results show that the proposed approach is more accurate than other closed-form models in the literature.

Parametric Generation of Small Ship Hulls with CAD Software

CAD software is a daily tool in ship design offices and shipyards, and every software uses NURBS or B-splines curves and surfaces as common foundations. The CAD tools of today are not static software products and most of them now include parametric design modules, which enable users to change the shape of an object based on its key geometric feature parameters with the use of sliders or equivalent controls. Although B-spline techniques are commonly applied to the representation of the ship hull curves and surfaces, the parametric deformation of the hull surfaces based on geometric parameters is less used. This paper presents a methodology to define the parametric definition of a ship hull with the use of a standard and non-specialized CAD software that is of common use in the ship design offices and universities: Rhinoceros. The presented parametric design methodology will use specific ship hull parameters or feature parameters with a clear geometric meaning, such as displacement, waterplane area, LCB, and LCF, together with the properties of the B-spline curves and the power of Grasshopper, the parametric design tool inside Rhinoceros, to create parametric ship hulls.

Raman spectroscopy and topological machine learning for cancer grading

In the last decade, Raman Spectroscopy is establishing itself as a highly promising technique for the classification of tumour tissues as it allows to obtain the biochemical maps of the tissues under investigation, making it possible to observe changes among different tissues in terms of biochemical constituents (proteins, lipid structures, DNA, vitamins, and so on). In this paper, we aim to show that techniques emerging from the cross-fertilization of persistent homology and machine learning can support the classification of Raman spectra extracted from cancerous tissues for tumour grading. In more detail, topological features of Raman spectra and machine learning classifiers are trained in combination as an automatic classification pipeline in order to select the best-performing pair. The case study is the grading of chondrosarcoma in four classes: cross and leave-one-patient-out validations have been used to assess the classification accuracy of the method. The binary classification achieves a validation accuracy of 81% and a test accuracy of 90%. Moreover, the test dataset has been collected at a different time and with different equipment. Such results are achieved by a support vector classifier trained with the Betti Curve representation of the topological features extracted from the Raman spectra, and are excellent compared with the existing literature. The added value of such results is that the model for the prediction of the chondrosarcoma grading could easily be implemented in clinical practice, possibly integrated into the acquisition system.

Automatic classification of ultrasound thyroids images using vision transformers and generative adversarial networks

Ultrasound has become the main imaging modality for the diagnosis of thyroid nodules. However, opinions issued and decisions taken by doctors are always subjective and depend on the experience of the doctor and his entourage. It is often essential to use a tool to guide doctors during decision-making. In this paper, we develop a new model, based on a deep learning approach, more specifically convolutional neural networks and vision transformers. Our goal was the classification of thyroid ultrasound images which were categorized into two classes (class: Benign and class: Malignant). At the beginning of this work, we implemented a Deep Convolutional Generative Adversarial Network (DCGAN) technique to fix the data shortage and the problem of the imbalanced data set. Then, we used two different methods for the splitting of the data, the first is the classic data split and the second is the cross-validation method. For the modeling techniques, we used two types of models such as: Convolutional Neural Network: (VGG16, EffecientNetB0 and ResNet50) and Vision Transformers: (ViT_B16 and Hybrid ViT). All the models were trained with Softmax and Support Vector Machine (SVM) classifiers. After evaluation of both the CNN and ViT models, we concluded that the SVM classification produces better performance than the Softmax classification for all of the models with the Hybrid ViT being on top in the matter of the metrics evaluation and the curves representation, the model achieved 97.63% Accuracy, resulting in a slight improvement over the existing studies. Our study achieved promising results, hopefully, this will help doctors better diagnose thyroid patients.

Development of an extended Schnute model for more physically realistic representations of soil water retention and moisture capacity curves

Commonly used soil water retention, θ( h), and moisture capacity, C( h), functions implicitly assume that ( i) the θ( h) data curve is sigmoid-shaped with an inflection and ( ii) the C( h) data curve has a value of zero at soil saturation. Desorption measurements on intact soils indicate, however, that the θ( h) data curve is frequently convex-monotonic in shape with no inflection, and C( h) at saturation is often a finite negative value rather than zero. As these model-data mismatches may cause substantial error in simulation or prediction of near-saturated soil hydraulic properties and water flow, a new “Extended Schnute” θ( h)– C( h) function was proposed that can provide θ( h) curve shapes and saturated C( h) values which are consistent with θ( h) and C( h) measurements. The new function and/or its nested Schnute sub-model provided high-quality and physically realistic fits to desorption data collected from intact cores of coarse sand, loamy sand, loam, clay loam, sandy clay loam, clay, and organic clay soils; and it out-performed or equalled the three-parameter van Genuchten θ( h)– C( h) function for every data-set. The new function also provided accurate and physically realistic representations of θ( h) and C( h) data from structured soils containing macropores and strongly graded pore size distributions. It was concluded that the Extended Schnute model is capable of providing accurate and physically realistic representations for a wide range of θ( h) and C( h) data, and it was further recommended that this model be considered over other models when measurements indicate that θ( h) is convex-monotonic in shape and/or C( h) is not zero at soil saturation.

Generalized plane offsets and rational parameterizations

In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties and a study on self-intersection avoidance is performed. The representation of a rational curve as the envelope of its tangent lines, following the approach of Pottmann, is revisited to give the explicit expression of all rational generalized offsets. Other famous shapes, such as constant width curves, bicycle tire-tracks curves and Zindler curves are related to these generalized offsets. This gives rise to the second part of the paper, where the particular case of rational parameterizations by a support function is considered and explicit families of rational constant width curves, rational bicycle tire-track curves and rational Zindler curves are generated and some examples are shown.

Isogeometric-based mapping modeling and buckling analysis of stiffened panels

Modeling and analysis of stiffened panels are two key technologies in the design of aerospace thin-walled structures. For the stiffened panels with complex geometry, classical finite element analysis (FEA) and conventional isogeometric analysis (IGA) based on explicit geometry usually require time-consuming and labor-intensive geometric processing, and additional coupling matrices to be ready for analysis. In this study, a new method for modeling and buckling analysis of stiffened panels is proposed, which provides a more efficient and simpler way. During the modeling process, the stiffeners are treated as curves on surfaces, which is not explicitly defined using the control-point-based representation of curves, but implicitly defined using parameter curves in the parametric space of the surface. Mapping modeling provides more accurate geometric description and transfer the complex modeling problems (three-dimensional space) of stiffeners on free-form surface into simple modeling problems in the regular parametric space (two-dimensional space). During the buckling analysis process, a new mapped stiffener element based on mapping modeling is proposed, which can model the section of the eccentric stiffener without changing the geometry. The precise normal information of the Non-Uniform Rational B-Splines (NURBS) surface can ensure that the stiffeners are perpendicular to the skin. In addition, the coupling of the stiffener and the skin is automatic, without any additional coupling matrix. This buckling analysis framework realizes the complete integration of modeling and analysis. Furthermore, for the stiffened panels with cutouts, the trimmed surface analysis (TSA) method is extended to be used for the numerical integration of the trimmed stiffeners, which means that no additional geometric process is required. Finally, four numerical examples of different types of stiffened panels are constructed, involving metal, trimmed surface, classical grid-stiffener, free-form surface, variable-stiffness composites, and curvilinear grid-stiffener. Several numerical examples of static and buckling analysis of stiffened panels with high fidelity demonstrate the effectiveness of the proposed framework.

Flow Regime-Dependent, Discharge Uncertainty Envelope for Uncertainty Analysis with Ensemble Methods

A discharge uncertainty envelope is presented that provides an observation error model for data assimilation (DA) using discharge observations derived from measurement of stage using a rating curve. It uniquely represents the rating curve representation error, which is due to scale and process incompatibility between the rating curve hydrodynamic model and “true” discharge, within the observation error model. Ensemble methods, specifically, the iterative ensemble smoother (IES) algorithms in PEST++, provide the DA framework for this observation error model. The purpose of the uncertainty envelope is to describe prior observation uncertainty for ensemble methods of DA. Envelope implementation goals are (1) limiting the spread of the envelope to avoid conditioning to extreme parameter values and producing posterior parameter distributions with increased variance, and (2) incorporating a representative degree of observation uncertainty to avoid overfitting, which will introduce bias into posterior parameter estimates and predicted model outcomes. The expected uncertainty envelope is flow regime dependent and is delineated using stochastic, statistical methods before undertaking history matching with IES. Analysis of the goodness-of-fit between stochastically estimated “true” discharge and observed discharge provides criteria for the selection of best-fit parameter ensembles from IES results.

Evolution of growth charts in India

Growth charts are graphical representations of centile or standard deviation curves prepared from population-based studies. Growth charts are derived from large numbers of children either cross-sectionally or longitudinally. The variation in anthropometric measures amongst individuals of different ethnicities requires specific charts also for Indian children and need to be updated to reflect secular trends. Growth charts in India have come a long way from the development of first ones in the mid-1940s by National Institute of Nutrition (NIN) to Paediatrician Friendly IAP growth charts with mid-parental height and BMI tool. Indian BMI charts recommend overweight and obesity cut-offs for children to the adult equivalents of 23 and 27kg/m2 to prevent the under-diagnosis of obesity. Indian growth charts have been published for upper:lower segment ratios, Turner syndrome, waist circumference, height velocity and sitting height.

Piecewise Business Bubble System under Classical and Nonsingular Kernel of Mittag-Leffler Law.

This study aims to investigate the dynamics of three agents in the emerging business bubble model based on the Mittag-Leffler law pertaining to the piecewise classical derivative and non-singular kernel. By generalizing the business bubble dynamics in terms of fractional operators and the piecewise concept, this study presents a new perspective to the field. The entire set of intervals is partitioned into two piecewise intervals to analyse the classical order and conformable order derivatives of an Atangana-Baleanu operator. The subinterval analysis is critical for removing discontinuities in each sub-partition. The existence and uniqueness of the solution based on a piecewise global derivative are tested for the considered model. The approximate root of the system is determined using the piecewise numerically iterative technique of the Newton polynomial. Under the classical order and non-singular law, the approximate root scheme is applied to the piecewise derivative. The curve representation for the piece-wise globalised system is tested by applying the data for the classical and different conformable orders. This establishes the entire density of each compartment and shows a continuous spectrum instead of discrete dynamics. The concept of this study can also be applied to investigate crossover behaviours or abrupt changes in the dynamics of the values of each market.

On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2‐space

In this paper, we mainly investigate (contra)pedals and (anti)orthotomics of frontals in the de Sitter 2‐space from the viewpoint of singularity theory and differential geometry. We utilize the de Sitter Legendrian Frenet frames to provide parametric representations of (contra)pedal curves of spacelike and timelike frontals in the de Sitter 2‐space and to investigate the geometric and singularity properties of these (contra)pedal curves. We then introduce orthotomics of frontals in the de Sitter 2‐space and explain these orthotomics as wavefronts from the viewpoint of Legendrian singularity theory. Furthermore, we generalize these methods to study antiorthotomics of frontals in the de Sitter 2‐space.

COMPUTING LEVEL LINES OF A POLYNOMIAL ON THE PLANE

Application of the method of computing the location of all types of level lines of a real polynomial on the real plane is demonstrated. The theory underlying this method is based on methods of local and global analysis by the means of power geometry and computer algebra. Three nontrivial examples of computing level lines of real polynomials on the real plane are discussed in detail. The following computer algebra algorithms are used: factorization of polynomials, computation of the Gröbner basis, construction of the Newton polygon, and representation of an algebraic curve on a plane. It is shown how computational difficulties can be overcome.

Error estimates for a linear folding model

Abstract An interior penalty discontinuous Galerkin method is devised to approximate minimizers of a linear folding model by discontinuous isoparametric finite element functions that account for an approximation of a folding arc. The numerical analysis of the discrete model includes an a priori error estimate in case of an accurate representation of the folding curve by the isoparametric mesh. Additional estimates show that geometric consistency errors may be controlled separately if the folding arc is approximated by piecewise polynomial curves. Various numerical experiments are carried out to validate the a priori error estimate for the folding model.

FRACTIONAL ORDER DISTRIBUTION ON HEAT FLUX FOR CRYSTALLINE CONCRETE MATERIAL

The present work devoted to analyse the impact of fractional order one dimensional heat equation on heat transfer in concrete substances, during conduction process. Throughout the evolution the thermal properties of substances are considered as constant parameters. Investigations were done through statistical phenomena with linear representation of curve.

Numerical optimization-based design studies on biaxial tensile tests

Biaxial tensile tests are of great significance for the characterization of materials. For ductile materials, they allow the representation of the yield strength curve separating elastic and plastic material behavior. Contrary demands regarding the specimen design arise, if a biaxial stress state with the equivalent stress equal or close to the yield strength shall be obtained. On one hand, the specimen shall be of constant thickness and stiffness, in order to allow an unconstrained deformation without undesired lateral strains. On the other hand, stress concentrations arising from the load introduction make a thickened border region necessary, in order to avoid a premature failure of the specimen. The present paper addresses this problem by a multi-objective structural optimization using Evolutionary Algorithms. Well-defined parametric models of biaxial tensile test specimens allow to systematically investigate the influence of the thickness distribution and the sizing of the border region. Additionally, the influence of actuators of constant force compared to actuators of constant displacement are quantified. In all cases, the minimization of the stress difference as well as the actuation effort are considered as objectives. The results allow a deep insight to the optimal definition of biaxial tensile test specimens and are summed up in well-defined design proposals.

ASTROMER

Taking inspiration from natural language embeddings, we present ASTROMER, a transformer-based model to create representations of light curves. ASTROMER was pre-trained in a self-supervised manner, requiring no human-labeled data. We used millions of R-band light sequences to adjust the ASTROMER weights. The learned representation can be easily adapted to other surveys by re-training ASTROMER on new sources. The power of ASTROMER consists in using the representation to extract light curve embeddings that can enhance the training of other models, such as classifiers or regressors. As an example, we used ASTROMER embeddings to train two neural-based classifiers that use labeled variable stars from MACHO, OGLE-III, and ATLAS. In all experiments, ASTROMER-based classifiers outperformed a baseline recurrent neural network trained on light curves directly when limited labeled data were available. Furthermore, using ASTROMER embeddings decreases the computational resources needed while achieving state-of-the-art results. Finally, we provide a Python library that includes all the functionalities employed in this work.