Use Of B-splines Research Articles

Quantile Regression in Space-Time Varying Coefficient Model of Upper Respiratory Tract Infections Data

Space-time varying coefficient models, which are used to identify the effects of covariates that change over time and spatial location, have been widely studied in recent years. One such model, called the quantile regression model, is particularly useful when dealing with outliers or non-standard conditional distributions in the data. However, when the functions of the covariates are not easily specified in a parametric manner, a nonparametric regression technique is often employed. One such technique is the use of B-splines, a nonparametric approach used to estimate the parameters of the unspecified functions in the model. B-splines smoothing has potential to overfit when the number of knots is increased, and thus, a penalty is added to the quantile objective function known as P-splines. The estimation procedure involves minimizing the quantile loss function using an LP-Problem technique. This method was applied to upper respiratory tract infection data in the city of Bandung, Indonesia, which were measured monthly across 30 districts. The results of the study indicate that there are differences in the effect of covariates between quantile levels for both space and time coefficients. The quantile curve estimates also demonstrate robustness with respect to outliers. However, the simultaneous estimation of the quantile curves produced estimates that were relatively close to one another, meaning that some quantile curves did not depict the actual data pattern as precisely. This suggests that each district in Bandung City not only has different categories of incidence rates but also has a heterogeneous incidence rate based on three quantile levels, due to the difference in the effects of covariates over time and space.

Estimation of linear transformation cure models with informatively interval-censored failure time data

Linear transformation models have been one type of models commonly used for regression analysis of failure time data partly due to their flexibility. More recently they have been generalised to the case where there may exist a cured subgroup or the censoring may be informative. In this paper, we consider a more complicated and general situation where both a cured subgroup and informative censoring, or more specifically informative interval censoring, exist. As pointed out in the literature, the analysis that fails to take into account either the cured subgroup or the informative censoring can yield biased estimation or misleading conclusions. For the problem, a three-component mixture cure model is presented and we develop a two-step estimation procedure with the use of B-splines to approximate unknown functions. The proposed approach is quite flexible and can be easily implemented. Also the proposed estimators of regression parameters are shown to be consistent and asymptotically normal. An extensive simulation study is conducted and suggests that the method works well for practical situations. Furthermore a real application is provided to illustrate the proposed methodology.

Real-Time B-Spline Interpolation for Harmonic Phasor Estimation in Power Systems

The fundamental component synchrophasors are widely used nowadays, based on the use of Phasor Measurement Units (PMU), in several applications in electrical power systems. For the advancement of monitoring technology, harmonic synchrophasor estimation is still a challenge. To make it possible, it is necessary to correctly estimate the harmonic phasor (magnitude and phase of each harmonic component), mainly in scenarios with frequency deviations and noise. In this context, the present paper proposes the use of B-Spline based interpolation as a preprocessing algorithm for the Discrete Fourier Transform (DFT). The performance is evaluated in terms of Total Vector Error (TVE) considering the tests proposed by the IEEE C37.118.1 and IEC 60255-118-1 standards and compared with four different harmonic phasor estimation methods, the HPE, SIFE, TFT and FT FIR. The proposed method can provide a TVE below 1% for all harmonic phasors, from the first to the fiftieth order, under significant frequency deviation. Also, for real time validation, the proposed method was implemented in an industrial computer as a software routine capable of processing the signals in real time. The signals are acquired by a commercial Merging Unit (MU) and the samples are published using the IEC 61850 Sampled Values (SV) protocol.

B-spline wavelet discrete-continual finite element method for the local solution to the two-dimensional problem of the theory of elasticity

Introduction. The distinctive article presents a local semi-analytical solution to the problem of the two-dimensional theory of elasticity. The corresponding structures, featuring the regularity (constancy) of physical and geometric parameters (the modulus of elasticity of the material of the structure, the Poisson’s ratio of the material of the structure, dimensions of the cross section of the structure) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction.
 Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial ope­rational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov.
 Results. Some topical issues of construction of normalized basis functions of a B-spline are considered, the approximation technique for corresponding vector functions and operators within DCFEM is described. Along the basic direction, the problem remains continual and an exact analytical solution can be obtained, while along the non-basic direction the finite element approximation is used in combination with a wavelet analysis apparatus. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, a two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method for the solution of such problems was developed, described and verified in numerous papers written by the authors. In particular, we consider the simplest sample analysis of a deep beam, fixed along the side faces and subjected to the load concentrated in the centre of the structure.
 Conclusions. The solution to the verification problem obtained using the proposed version of the wavelet-based DCFEM was in good agreement with the solution obtained using a conventional finite element method (corresponding solutions were constructed with localization and without localization; these solutions coincide almost completely, while the advantages of the numerical-analytical approach are quite obvious). It is shown that the use of B-splines of various degrees within the wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

The use of B-splines to represent the topography of river networks

This work presents a new extension to B-Splines that enables them to model functions on directed tree graphs such as non-braided river networks. The main challenge of the application of B-splines to graphs is their definition in the neighbourhood of nodes with more than two incident edges. Achieving that the B-splines are continuous at these points is non-trivial. For both, simplification reasons and in view of our application, we limit the graphs to directed tree graphs. To fulfil the requirement of continuity, the knots defining the B-Splines need to be located symmetrically along the edges with the same direction. With such defined B-Splines, we approximate the topography of the Mekong River system from scattered height data along the river. To this end, we first test and validate successfully the method with synthetic water level data, with and without added annual signal. The quality of the resulting heights is assessed besides others by means of root mean square errors (RMSE) and mean absolute differences (MAD). The RMSE values are 0.26 m and 1.05 m without and with added annual variation respectively and the MAD values are even lower with 0.11 m and 0.60 m. For the second test, we use real water level observations measured by satellite altimetry. Again, we successfully estimate the river topography, but also discuss the short comings and problems with unevenly distributed data. The unevenly distributed data leads to some very large outliers close to the upstream ends of the rivers tributaries and in regions with rapidly changing topography such as the Mekong Falls. Without the outlier removal the standard deviation of the resulting heights can be as large as 50 m with a mean value of 15.73 m. After the outlier removal the mean standard deviation drops to 8.34 m.

About the B-spline wavelet discrete-continual finite element method of the local plate analysis

Introduction. This distinctive paper addresses the local semi-analytical solution to the problem of plate analysis. Isotropic plates featuring the regularity (constancy) of physical and geometric parameters (modulus of elasticity of the plate material, Poisson’s ratio of the plate material, dimensions of the cross section of the plate) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction.
 Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial operational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov.
 Results. Some relevant issues of construction of normalized basis functions of the B-spline are considered; the technique of approximation of corresponding vector functions and operators within DCFEM is described. The problem remains continual if analyzed along the basic direction, and its exact analytical solution can be obtained, whereas the finite element approximation is used in combination with a wavelet analysis apparatus in respect of the non-basic direction. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method of solving such problems was developed, described and verified in the numerous papers of the co-authors. In particular, we consider the simplest sample analysis of a plate (rectangular in plan) fixed along the side faces exposed to the influence of the load concentrated in the center of the plate.
 Conclusions. The solution to the verification problem obtained using the proposed version of wavelet-based DCFEM was in good agreement with the solution obtained using the conventional finite element method (the corresponding solutions were constructed with and without localization; these solutions almost completely coincided, while the advantages of the numerical-analytical approach were quite obvious). It is shown that the use of B-splines of various degrees within wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

Two- and three-dimensional size-dependent couple stress response using a displacement-based variational method

Based upon the principle of minimum total potential energy for consistent couple stress theory, a Ritz variational approach is developed using tensor product B-splines for both two- and three-dimensional elastostatic analysis. The underlying theory is size-dependent due to incorporation of the energy conjugate skew-symmetric couple-stress and mean curvature tensors, in addition to the force-stress and strain conjugate pair of classical theory. The use of B-splines as the basis functions can assure the required C1 continuity of the displacement field, while also permitting higher order representations. Both displacements and rotations are essential boundary conditions in this theory. Displacements can be enforced in the usual way, but rotations require special treatment to maintain symmetry and positive definiteness of the stiffness matrix for well-posed elastostatic problems. Several computational examples are considered to validate the formulation, illustrate convergence characteristics, and investigate mechanical behavior under consistent couple stress theory. All previous numerical analysis of consistent couple stress theory has been limited to plane strain problems. Thus, the extension here to three-dimensions is of great importance, especially because the behavior in several cases shows significant deviation from the plane strain solutions. Consequently, the variational formulations and computational methodology presented in this paper can play a critical role in assessing the predictive capability of consistent couple stress theory and in understanding size-dependent elastic response.

On the Use of B-Splines as Ritz Variational Basis Functions to Solve the Schrodinger Equation (TISE) for a constrained free Quantum Particle

B-Splines as piecewise adaptation of Bernstein polynomials (aka, B-polys) are widely used as Ritz variational basis functions in solving many problems in the fields of quantum mechanics and atomic physics. In this paper they are used to solve the 1-D stationary Schrodinger equation (TISE) for a free quantum particle subject to a fixed domain length by using the Python software SPLIPY with different sets of computation parameters. In every case it was found that over 60 percent of energy levels had excellent accuracy thereby proving that the use of B-spline collocation is a preferred method.

Spatiotemporal Calibration of Camera and 3D Laser Scanner

The multi-sensory setups consisting of the laser scanners and cameras are popular as the measurements complement each other and provide necessary robustness for applications. Under dynamic conditions or when in motion, a direct transformation (spatial calibration) and time offset between sensors (temporal calibration) is needed to determine the correspondence between measurements. We propose an open-source spatiotemporal calibration framework for a camera and a 3D laser scanner. Our solution is based on commonly available chessboard markers requiring one-minute calibration before the operation that offers accurate and repeatable results. The framework is based on batch optimization of point-to-plane constraints with a time offset calibration possible by a novel continuous representation of the plane equations based on a minimal representation in the Lie algebra and the use of B-splines. The framework's properties are evaluated in simulation while correctness is verified with two distinct sensory setups with Velodyne VLP-16 and SICK MRS6124 3D laser scanners.

Isogeometric analysis of fiber reinforced composites using Kirchhoff–Love shell elements

A second gradient theory for woven fabrics is applied to Kirchhoff–Love shell elements to analyze the mechanics of fiber reinforced composite materials. In particular, we assume a continuous distribution of the fibers embedded into the shell surface, accounting for additional in-plane flexural resistances within the hyperelastic regime. For the finite element discretization we apply isogeometric methods, i.e. we make use of B-splines as basis functions omitting the usage of mixed approaches. The higher gradient formulation of the fabric is verified by a series of numerical examples, followed by suitable validation steps using experimental measurements on organic sheets. A final example using a non-flat reference geometry demonstrates the capabilities of the presented formulation.

A comparison of the B-spline group-based trajectory model with the polynomial group-based trajectory model for identifying trajectories of depressive symptoms around old-age retirement

Objectives: The life event of retirement may be associated with changes in levels of depressive symptoms. The use of polynomial group-based trajectory modelling allows any changes to vary between different groups in a sample. A new approach, estimating these models using B-splines rather than polynomials, may improve modelling of complex changes in depressive symptoms at retirement.Methods: The sample contained 1497 participants from the Swedish Longitudinal Occupational Survey of Health (SLOSH). Polynomial and B-spline approaches to estimating group-based trajectory models were compared.Results: Polynomial group-based trajectory models produced unexpected changes in direction of trajectories unsupported by the data. In contrast, B-splines provided improved insights into trajectory shapes and more homogeneous groups. While retirement was associated with reductions in depressive symptoms in the sample as a whole, the nature of changes at retirement varied between groups.Conclusions: Depressive symptoms trajectories around old age retirement changed in complex ways that were modelled more accurately by the use of B-splines. We recommend estimation of group-based trajectory models with B-splines, particularly where abrupt changes might occur. Improved trajectory modelling may support research into risk factors and consequences of major depressive disorder, ultimately assisting with identification of groups which may benefit from treatment.

FUNÇÕES SPLINES APLICADAS EM DADOS DE CRESCIMENTO

In animal breeding, new methodologies can be applied in statistical analysis to improve the genetic evaluation and, for this reason, they have been the subject in several studies. In the last years, several research works have intended the model development with more adjustable functions to the distinct variables. A set of functions known as Spline functions has called the attention of researches. Then, the purpose of this review is to discuss the use of Spline functions that are applied to growth data in animal breeding. Splines are segmented regression functions that are united by points known as joint points and have the ability to improve the curvature of models and, therefore, the function adjustment. These functions have interesting properties such as the interpolatory nature, less multicolinearity problems, parameter linearity and the ability of increasing the approximation domain, all of which provide estimates in a wide range of possible values. There are three types of Spline functions: natural spline functions, smoothing spline 223 Colloquium Agrariae, vol. 13, n. Especial 2, Jan–Jun, 2017, p. 222-234. ISSN: 1809-8215. DOI: 10.5747/ca.2017.v13.nesp2.000229 functions or nonparametric regression and B-splines functions. These latter functions are more applied to animal breeding, mainly as alternatives to random regression models (RRM) that use the Legendre polynomials. The matrices formed by RRMs with the use of B-spline functions or Legendre polynomials are more scarce and easier to be inverted. Then, the use of Spline functions has been more intensified in the last years because studies have had the purpose of improving the adjustment with less model parameters in functions. New studies will allow improving the methodology and finding out new applications to the Spline functions.

Computer experiments with functional inputs and scalar outputs by a norm-based approach

A framework for designing and analyzing computer experiments is presented, which is constructed for dealing with functional and scalar inputs and scalar outputs. For designing experiments with both functional and scalar inputs, a two-stage approach is suggested. The first stage consists of constructing a candidate set for each functional input. During the second stage, an optimal combination of the found candidate sets and a Latin hypercube for the scalar inputs is sought. The resulting designs can be considered to be generalizations of Latin hypercubes. Gaussian process models are explored as metamodel. The functional inputs are incorporated into the Kriging model by applying norms in order to define distances between two functional inputs. We propose the use of B-splines to make the calculation of these norms computationally feasible.

Multilevel summation with B-spline interpolation for pairwise interactions in molecular dynamics simulations.

The multilevel summation method for calculating electrostatic interactions in molecular dynamics simulations constructs an approximation to a pairwise interaction kernel and its gradient, which can be evaluated at a cost that scales linearly with the number of atoms. The method smoothly splits the kernel into a sum of partial kernels of increasing range and decreasing variability with the longer-range parts interpolated from grids of increasing coarseness. Multilevel summation is especially appropriate in the context of dynamics and minimization, because it can produce continuous gradients. This article explores the use of B-splines to increase the accuracy of the multilevel summation method (for nonperiodic boundaries) without incurring additional computation other than a preprocessing step (whose cost also scales linearly). To obtain accurate results efficiently involves technical difficulties, which are overcome by a novel preprocessing algorithm. Numerical experiments demonstrate that the resulting method offers substantial improvements in accuracy and that its performance is competitive with an implementation of the fast multipole method in general and markedly better for Hamiltonian formulations of molecular dynamics. The improvement is great enough to establish multilevel summation as a serious contender for calculating pairwise interactions in molecular dynamics simulations. In particular, the method appears to be uniquely capable for molecular dynamics in two situations, nonperiodic boundary conditions and massively parallel computation, where the fast Fourier transform employed in the particle-mesh Ewald method falls short.

Smoothing Group-Based Trajectory Models Through B-Splines

This paper investigates the use of B-spline smoothers as an alternative to polynomials when estimating trajectory shape in group-based trajectory models. The use of polynomials in these models can cause undesirable curve shapes, such as uplifts at the end of the trajectory, which may not be present in the data. Moreover, polynomial curves are global, meaning that a data point at one end of the trajectory can affect the shape of the curve at the other end. We use the UK Offenders Index 1963 birth cohort to investigate the use of B-splines. The models are fitted using Latent Gold, and two information criteria (BIC and ICL-BIC) are used to estimate the number of knots of the B-spline, as well as the number of groups. A small simulation study is also presented. A three-group solution was chosen. It is shown that B-splines can provide a better fit to the observed data than cubic polynomials. The offending trajectory groups correspond to the classic groups of adolescent-limited, low-rate chronic and high-rate chronic which were proposed by Moffitt. The shapes of the two chronic trajectory curves from the B-spline fitting are more consistent with the life-course persistent nature of chronic offending than the traditional cubic polynomial curves. The simulation shows improved performance of the B-spline over cubic polynomials. The use of B-splines is recommended when fitting group-based trajectory models. Some software products need further development to include such facilities, and we encourage this development.

Examining the Use of B-Splines in Parking Assist Systems

The main objective of the presented study and simulations conducted was to investigate the prospect of using B-spline curves for the automatic parking, i.e. self-driving, or intelligent vehicles. We consider the problem of parallel parking for a non-holonomic vehicle with a known maximum path curvature. The relationship between the properties of the path and the geometry of corresponding parking spot is revealed. The unique properties of B-splines are exploited to synthesize a path that is smooth and of continuous curvature. The contributions of this project are in the generations of better, smooth continuous paths. This improves passenger comfort during the parallel parking maneuver and allow vehicles to park in tighter spots by increasing the feasible range of the parking manoeuver.

Estimation for the Power-transformed Varying-coefficient Quantile Regression Model

In this article, we consider the varying-coefficient quantile regression model with a power transformation on the dependent variable. The functional coefficients are estimated by the use of B-spline and the number of knots is chosen according to Schwarz-type Information Criterion. The transformation parameter is estimated by minimizing the cusum residual square sum. We show that the estimates of transformation parameter and the functional coefficients are consistent, and also obtain the rates of convergence of the estimation of functional coefficients. The proposed methods are illustrated by some simulated studies and a real data example.

Formula omitted]-spline-decomposition-based output tracking with preview for nonminimum-phase linear systems

This article proposes a B-spline-decomposition-based approach to output tracking with preview for nonminimum-phase (NMP) systems. It has been shown that when there exists a finite (in time) preview of future desired trajectory, precision output tracking of NMP systems can be achieved by using the preview-based stable-inversion technique. The performance of this approach, however, can be sensitive to system dynamics uncertainty, and the computation involved can be demanding. We propose to address these challenges by integrating the notion of trajectory decomposition with the iterative learning control (ILC) technique. Particularly, the B-splines are used to construct a library of output elements and the corresponding input elements a priori, and the ILC techniques are used to obtain the input elements for precision tracking of the output elements. During the tracking with preview, the previewed future desired trajectory is decomposed by using the output elements, and the input is synthesized by using the corresponding input elements with chosen pre- and post-actuation times. The required pre-/post-actuation times are quantified based on the stable-inversion theory. The use of B-splines substantially reduces the number of output elements in the library, and the decomposition-synthesis occurs only at time instants separated by the difference between the preview time and pre-actuation time. The proposed approach is illustrated through a simulation study of nanomanipulation application using a NMP piezo actuator model.

B-Spline interpolation of Kirchhoff-Love space rods

The paper deals with the isogeometric analysis via B-splines of space rods under Kirchhoff-Love hypotheses. The approach was used by Gontier and Vollmer [12] for developing a plane curve element within the framework of the Timoshenko rod model, but they adopted only one patch to represent entirely the geometry of the rod; furthermore the authors developed their theory only for plane elements. In this work we develop an isogeometric approach for the numerical analysis of the 3D Kirchhoff-Love rod theory. We use B-splines and Bezier interpolations and we show that they are able to attain very good accuracy for rod structures, particularly for developing a 3D exact curve element with geometric torsion. The paper presents an original parametrization of the geometric torsion that proves to be very effective. The use of B-splines allows to avoid discontinuities on the geometrical quantities, and particularly on the normal fields, so that even relatively low order interpolation functions are able to yield accurate results.

Formula omitted]-spline surface fitting by iterative geometric interpolation/approximation algorithms

Recently, the use of B-spline curves/surfaces to fit point clouds by iteratively repositioning the B-spline’s control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a B-spline surface that passes through the data points, whereas approximation generates a B-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.