Journal articles: 'Spline Function'

1

YOSHIMOTO, Fujiichi. "Spline Function." Journal of Japan Society for Fuzzy Theory and Systems 6, no. 5 (1994): 895–98. http://dx.doi.org/10.3156/jfuzzy.6.5_895.

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Toraichi, Kazuo, and Takahiko Horiuchi. "Spline Function." IEEJ Transactions on Electronics, Information and Systems 114, no. 2 (1994): 209–16. http://dx.doi.org/10.1541/ieejeiss1987.114.2_209.

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K.H., Faraidun, Karwan H. F. Jwamer, and Sabah Ali Mohammed. "An Algorithm for Computing Spline Function." Journal of Zankoy Sulaimani - Part A 18, no. 3 (April 21, 2016): 251–58. http://dx.doi.org/10.17656/jzs.10554.

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4

MacCarthy, B. L., and N. D. Burns. "An Evaluation of Spline Functions for use in Cam Design." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 199, no. 3 (July 1985): 239–48. http://dx.doi.org/10.1243/pime_proc_1985_199_118_02.

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This paper shows how spline functions can be employed for kinematic motion specification in cam design. The polynomial spline is introduced as a special case of a continuous piecewise function. Cubic and quintic splines are derived and their properties are discussed in the cam design context. It is shown how standard cam laws can be approximated extremely accurately with a small number of points and appropriate boundary conditions. The modified sinusoidal acceleration cam law is used as an example. The application of quintic splines to non-standard and special motions is discussed. The algebraic and B-spline representations of spline functions are compared. The former is considered preferable in this context and a list of useful algorithms is given. The real power of the spline function, in particular the algebraic quintic spline, is its simplicity, ease of computation and adaptability to non-standard design problems. The use of parametrized, deficient and exponential splines is proposed for specific applications.

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Budakçı, Gülter, and Halil Oruç. "Further Properties of Quantum Spline Spaces." Mathematics 8, no. 10 (October 14, 2020): 1770. http://dx.doi.org/10.3390/math8101770.

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We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for 1/q-integration and 1/q-differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function.

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Toraichi, Kazuo, and Masaru Kamada. "Spline Function II." IEEJ Transactions on Electronics, Information and Systems 114, no. 7-8 (1994): 773–82. http://dx.doi.org/10.1541/ieejeiss1987.114.7-8_773.

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K. S., Rostam, and Karwan H.J. "Lacunary Interpolation by Spline function (0,3) Case." Journal of Zankoy Sulaimani - Part A 6, no. 1 (September 16, 2003): 43–49. http://dx.doi.org/10.17656/jzs.10111.

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Strelkovskaya, Irina, Irina Solovskaya, and Juliya Strelkovska. "Application of real and complex splines in infocommunication problems." Problemi telekomunìkacìj, no. 1(28) (December 22, 2021): 3–19. http://dx.doi.org/10.30837/pt.2021.1.01.

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The work offers the solution to problems of analysis and synthesis of infocommunication systems with the help of real and complex spline functions. The use of the spline approximation method for solving problems of recovery of random signals and self-similar traffic, management of network objects and network as a whole, and procedures of infocommunication objects and networks functioning is offered. To solve the problems of forecasting, in particular, forecasting the characteristics of network traffic and maintaining the QoS characteristics in its service and formation of requirements for network buffer devices, developed spline extrapolation based on different types of real spline functions, namely: linear, quadratic, quadratic B-splines, cubic, cubic B-splines, cubic Hermite splines. As a criterion for choosing the type of spline function, the prediction error is selected, the accuracy of which can be increased by using a particular kind of spline, depending on the object being predicted. The use of complex flat spline functions is considered to solve the class of user positioning problems in the radio access network. In general, the use of real and complex spline functions allows obtaining the results of improving the Quality of Service in the infocommunication network and ensuring the scalability of the obtained solutions.

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Toraichi, Kazuo, Ryoichi Mori, and Masaru Kamada. "Design of window function using spline functions." Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 72, no. 11 (1989): 10–19. http://dx.doi.org/10.1002/ecjc.4430721102.

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10

Sivak, Roman. "DEFINITION OF KINEMATICS OF DEFORMATION BASED ON SPLINE-APPROXIMATIONS." Vibrations in engineering and technology, no. 2(97) (August 27, 2020): 101–7. http://dx.doi.org/10.37128/2306-8744-2020-2-11.

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It is suggested to use smoothing cubic spline functions to approximate the current functions. The structure of the selected functional provides the minimum curvature of the spline and the smallest deviation of the spline from the function smoothed at the nodes. The necessary correlation between these requirements is provided by the choice of weights. The algorithm for approximating the current function is implemented by moving from a grid created by current lines and auxiliary lines to a rectangular grid. To reduce the effect of random errors of experimental information on the current line, a smoothing algorithm with a given accuracy was used. The current function is interpolated from the physical grid to the calculated one. The flow velocities were approximated to calculate the strain rates. To estimate the accuracy of the calculations, it is proposed to use the deviation from the incompatibility condition. The main disadvantage is that their behavior around a point determines their behavior as a whole. In this regard, other approaches to the approximation free of this deficiency have been developed in recent years. One such approach that has proven itself well in both theoretical and practical applications is the use of so-called splines. Splines are called functions that are glued together from different pieces of polynomials by a fixed system. The simplest example is broken. Splines naturally occur in many mechanical problems. For example, the spline shape has an elastic beam with point loads. Among the splines, polynomial splines glued together from pieces of polynomials play the most important role. The development of the theory of such splines and their popularization were facilitated by I. Schoenberg's work. Polynomial splines are beginning to be used in many application problems related to function approximation.

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11

Toraichi, Kazuo, Masaru Kamada, and Ryoichi Mori. "A Spline Function Generator." IEEJ Transactions on Electronics, Information and Systems 107, no. 4 (1987): 373–80. http://dx.doi.org/10.1541/ieejeiss1987.107.4_373.

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12

Xie, Jin, and Xiaoyan Liu. "The EH Interpolation Spline and Its Approximation." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/745765.

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A new interpolation spline with two parameters, called EH interpolation spline, is presented in this paper, which is the extension of the standard cubic Hermite interpolation spline, and inherits the same properties of the standard cubic Hermite interpolation spline. Given the fixed interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm.

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13

Zhang, Dai Yuan, and Hai Nan Yang. "Passenger Flow Analysis in Subway Using a Kind of Neural Network." Applied Mechanics and Materials 713-715 (January 2015): 2284–87. http://dx.doi.org/10.4028/www.scientific.net/amm.713-715.2284.

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This paper aims to analyze passenger flow in subway based on a kind of rational spline weight function neural network, in which the numerator of the spline is a cubic polynomial and the denominator of the spline is a quadratic polynomial, and this kind spline is denoted by 3/2 rational splines. There are many factors affecting the passenger flow. Combined the main influential factors with the self-learning method of neural network, we establish the neural network model of passenger flow in subway. This paper introduces the spline weight function neural network and the passenger flow model based on this neural network. Finally MATLAB simulation verifies that the 3/2 rational spline weight function neural network can be applied to analyze the passenger flow in subway with high accuracy.

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14

Dmitriev, V. I., and J. G. Ingtem. "The Regularized Spline (R-Spline) Method for Function Approximation." Computational Mathematics and Modeling 30, no. 3 (July 2019): 198–206. http://dx.doi.org/10.1007/s10598-019-09447-w.

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15

Kumar, Arun, and L. K. Govil. "Interpolation of natural cubic spline." International Journal of Mathematics and Mathematical Sciences 15, no. 2 (1992): 229–34. http://dx.doi.org/10.1155/s0161171292000292.

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From the result in [1] it follows that there is a unique quadratic spline which bounds the same area as that of the function. The matching of the area for the cubic spline does not follow from the corresponding result proved in [2]. We obtain cubic splines which preserve the area of the function.

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HUNT, K. J., R. HAAS, and M. BROWN. "ON THE FUNCTIONAL EQUIVALENCE OF FUZZY INFERENCE SYSTEMS AND SPLINE-BASED NETWORKS." International Journal of Neural Systems 06, no. 02 (June 1995): 171–84. http://dx.doi.org/10.1142/s0129065795000135.

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The conditions under which spline-based networks are functionally equivalent to the Takagi-Sugeno-model of fuzzy inference are formally established. We consider a generalized form of basis function network whose basis functions are splines. The result admits a wide range of fuzzy membership functions which are commonly encountered in fuzzy systems design. We use the theoretical background of functional equivalence to develop a hybrid fuzzy-spline net for inverse dynamic modeling of a hydraulically driven robot manipulator.

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17

Ezhov, Nikolaj, Frank Neitzel, and Svetozar Petrovic. "Spline approximation, Part 1: Basic methodology." Journal of Applied Geodesy 12, no. 2 (April 25, 2018): 139–55. http://dx.doi.org/10.1515/jag-2017-0029.

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Abstract In engineering geodesy point clouds derived from terrestrial laser scanning or from photogrammetric approaches are almost never used as final results. For further processing and analysis a curve or surface approximation with a continuous mathematical function is required. In this paper the approximation of 2D curves by means of splines is treated. Splines offer quite flexible and elegant solutions for interpolation or approximation of “irregularly” distributed data. Depending on the problem they can be expressed as a function or as a set of equations that depend on some parameter. Many different types of splines can be used for spline approximation and all of them have certain advantages and disadvantages depending on the approximation problem. In a series of three articles spline approximation is presented from a geodetic point of view. In this paper (Part 1) the basic methodology of spline approximation is demonstrated using splines constructed from ordinary polynomials and splines constructed from truncated polynomials. In the forthcoming Part 2 the notion of B-spline will be explained in a unique way, namely by using the concept of convex combinations. The numerical stability of all spline approximation approaches as well as the utilization of splines for deformation detection will be investigated on numerical examples in Part 3.

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KATIYAR, S. K., and A. K. B. CHAND. "SHAPE PRESERVING RATIONAL QUARTIC FRACTAL FUNCTIONS." Fractals 27, no. 08 (December 2019): 1950141. http://dx.doi.org/10.1142/s0218348x1950141x.

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The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct [Formula: see text]-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the [Formula: see text]-fractal rational quartic spline when the original function is in [Formula: see text]. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the [Formula: see text]-fractal rational quartic spline to [Formula: see text]. The elements of the iterated function system are identified befittingly so that the class of [Formula: see text]-fractal function [Formula: see text] incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ [Formula: see text]. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

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19

Gomes, Lorrayne, Milena Vieira Lima, Jeferson Corrêa Ribeiro, Andreia Santos Cezário, Eliandra Maria Bianchini Oliveira, Wallacy Barbacena Rosa dos Santos, Tiago Neves Pereira Valente, Crislaine Messias de Souza, and Aline Sousa Camargos. "FUNÇÕES SPLINES APLICADAS EM DADOS DE CRESCIMENTO." COLLOQUIUM AGRARIAE 13, Especial 2 (June 1, 2017): 222–34. http://dx.doi.org/10.5747/ca.2017.v13.nesp2.000229.

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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.

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Vijayakumar, Hannah. "Two-piece Cubic Spline Functions." Mapana - Journal of Sciences 2, no. 1 (October 2, 2003): 25–33. http://dx.doi.org/10.12723/mjs.3.2.

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.M Prenter defines a cubic Spline function in an interval [a, b] as a piecewise cubic polynomial which is twice continuously differentiable in the entire interval [a, b]. The smooth cubic spline functions fitting the given data are the most popular spline functions and when used for interpolation, they do not have the oscillatory behavior which characterized high-degree polynomials. The natural spline has been shown to be unique function possessing the minimum curvature property of all functions interpolating the data and having square integrable second derivative. In this sense, the natural cubic spline is the smoothest function which interpolates the data. Here Two-piece Natural Cubic Spline functions have been defined. An approximation with no indication of its accuracy is utterly valueless. Where an approximation is intended for the general use, one must , of course, go for the trouble of estimating the error as precisely as possible. In this section, an attempt has been made to derive closed form expressions for the error-functions in the case of Two-piece Spline Functions.

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El-Salam, F. A. Abd. "On a Parametric Spline function." IOSR Journal of Mathematics 9, no. 2 (2013): 19–22. http://dx.doi.org/10.9790/5728-0921922.

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22

Toraichi, K., M. Kamada, and R. Mori. "A quadratic spline function generator." IEEE Transactions on Acoustics, Speech, and Signal Processing 37, no. 4 (April 1989): 534–44. http://dx.doi.org/10.1109/29.17534.

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Wang, Ren-Hong, and Zhi-Qiang Xu. "Multivariate weak spline function space." Journal of Computational and Applied Mathematics 144, no. 1-2 (July 2002): 291–99. http://dx.doi.org/10.1016/s0377-0427(01)00568-4.

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24

Yılmaz, Ersin, Syed Ejaz Ahmed, and Dursun Aydın. "A-Spline Regression for Fitting a Nonparametric Regression Function with Censored Data." Stats 3, no. 2 (May 29, 2020): 120–36. http://dx.doi.org/10.3390/stats3020011.

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This paper aims to solve the problem of fitting a nonparametric regression function with right-censored data. In general, issues of censorship in the response variable are solved by synthetic data transformation based on the Kaplan–Meier estimator in the literature. In the context of synthetic data, there have been different studies on the estimation of right-censored nonparametric regression models based on smoothing splines, regression splines, kernel smoothing, local polynomials, and so on. It should be emphasized that synthetic data transformation manipulates the observations because it assigns zero values to censored data points and increases the size of the observations. Thus, an irregularly distributed dataset is obtained. We claim that adaptive spline (A-spline) regression has the potential to deal with this irregular dataset more easily than the smoothing techniques mentioned here, due to the freedom to determine the degree of the spline, as well as the number and location of the knots. The theoretical properties of A-splines with synthetic data are detailed in this paper. Additionally, we support our claim with numerical studies, including a simulation study and a real-world data example.

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25

Howell, Gary W. "Error bounds for two even degree tridiagonal splines." Journal of Applied Mathematics and Stochastic Analysis 3, no. 2 (January 1, 1990): 117–33. http://dx.doi.org/10.1155/s1048953390000107.

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We study a C(1) parabolic and a C(2) quartic spline which are determined by solution of a tridiagonal matrix and which interpolate subinterval midpoints. In contrast to the cubic C(2) spline, both of these algorithms converge to any continuous function as the length of the largest subinterval goes to zero, regardless of mesh ratios. For parabolic splines, this convergence property was discovered by Marsden [1974]. The quartic spline introduced here achieves this convergence by choosing the second derivative zero at the breakpoints. Many of Marsden's bounds are substantially tightened here. We show that for functions of two or fewer coninuous derivatives the quartic spline is shown to give yet better bounds. Several of the bounds given here are optimal.

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26

Karpov, D. A., and V. I. Struchenkov. "Spline approximation of multivalued functions in linear structures routing." Russian Technological Journal 10, no. 4 (July 30, 2022): 65–74. http://dx.doi.org/10.32362/2500-316x-2022-10-4-65-74.

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Objectives. The theory and methods of spline approximation of plane curves given by a sequence of points are currently undergoing rapid development. Despite fundamental differences between used splines and those considered in the theory and its applications, results published earlier demonstrate the possibility of using spline approximation when designing routes of linear structures. The main difference here consists in the impossibility of assuming in advance the number of spline elements when designing the routes. Here, in contrast to widely use polynomial splines, the repeating element is the link “segment of a straight line + arc of a circle” or “segment of a straight line + arc of a clothoid + arc of a circle + arc of a clothoid.” Previously, a two-stage scheme consisting of a determination of the number of elements of the desired spline and subsequent optimization of its parameters was proposed. Although an algorithm for solving the problem in relation to the design of a longitudinal profile has been implemented and published, this is not suitable for designing a route plan, since, unlike a profile, a route plan is generally a multivalued function. The present paper aims to generalize the algorithm for the case of spline approximation of multivalued functions making allowance for the design features of the routes of linear structures.Methods. At the first stage, a novel mathematical model is developed to apply the dynamic programming method taking into account the constraints on the desired spline parameters. At the second stage, nonlinear programming is used. In this case, it is possible to analytically calculate the derivatives of the objective function with respect to the spline parameters in the absence of its analytical expression through these parameters.Results. An algorithm developed for approximating multivalued functions given by a discrete series of points using a spline consisting of arcs of circles conjugated by line segments for solving the first stage of the problem is presented. An additional nonlinear programming algorithm was also used to optimize the parameters of the resulting spline as an initial approximation. However, in the present paper, the first stage is considered only, since the complex algorithm of the second stage and its justification require separate consideration.Conclusions. The presented two-stage spline approximation scheme with an unknown number of spline elements is also suitable for approximating multivalued functions given by a sequence of points on a plane, in particular, for designing a route plan for linear structures.

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Kamada, Masaru, Kazuo Toraichi, and Norio Nakamura. "A function generator based on quadratic B-spline functions." Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 76, no. 1 (1993): 40–53. http://dx.doi.org/10.1002/ecjc.4430760105.

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28

Tsay, Der Min, and Guan Shyong Hwang. "The Synthesis of Follower Motions of Camoids Using Nonparametric B-Splines." Journal of Mechanical Design 118, no. 1 (March 1, 1996): 138–43. http://dx.doi.org/10.1115/1.2826845.

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This paper proposes a tool to synthesize the motion functions of the camoid-follower mechanisms. The characteristics of these kinds of motion functions are that they possess two independent parameters. To implement the work, this study applies the nonparametric B-spline surface interpolation, whose spline functions are constructed by the closed periodic B-splines and the de Boor’s knot sequences in the two parametric directions of the motion function, respectively. The rules and the restrictions needed to be noticed in the process of synthesis are established. Numerical examples are also given to verify the feasibility of the proposed method.

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Meyer, K. "Advances in methodology for random regression analyses." Australian Journal of Experimental Agriculture 45, no. 8 (2005): 847. http://dx.doi.org/10.1071/ea05040.

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Random regression analyses have become standard methodology for the analysis of traits with repeated records that are thought of as representing points on a trajectory. Modelling curves as a regression on functions of a continuous covariable, such as time, for each individual, random regression models are readily implemented in standard, linear mixed model analyses. Early applications have made extensive use of regressions on orthogonal polynomials. Recently, spline functions have been considered as an alternative. The use of a particular type of spline function, the so-called B-splines, as basis functions for random regression analyses is outlined, emphasising the local influence of individual observations and low degree of polynomials employed. While such analyses are likely to involve more regression coefficients than polynomial models, it is demonstrated that reduced rank estimation via the leading principal components is feasible and likely to yield more parsimonious models and more stable estimates than full rank analyses. The combined application of B-spline basis function and reduced rank estimation is illustrated for a small set of data for beef cattle.

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Afzal, Arfan Raheen, Cheng Dong, and Xuewen Lu. "Estimation of partly linear additive hazards model with left-truncated and right-censored data." Statistical Modelling 17, no. 6 (June 30, 2017): 423–48. http://dx.doi.org/10.1177/1471082x17705993.

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In this article, we consider an additive hazards semiparametric model for left-truncated and right-censored data where the risk function has a partly linear structure, we call it the partly linear additive hazards model. The nonlinear components are assumed to be B-splines functions, so the model can be viewed as a semiparametric model with an unknown baseline hazard function and a partly linear parametric risk function, which can model both linear and nonlinear covariate effects, hence is more flexible than a purely linear or nonlinear model. We construct a pseudo-score function to estimate the coefficients of the linear covariates and the B-spline basis functions. The proposed estimators are asymptotically normal under the assumption that the true nonlinear functions are B-spline functions whose knot locations and number of knots are held fixed. On the other hand, when the risk functions are unknown non-parametric functions, the proposed method provides a practical solution to the underlying inference problems. We conduct simulation studies to empirically examine the finite-sample performance of the proposed method and analyze a real dataset for illustration.

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Li, Jie, Yaoyao Tu, and Shilong Fei. "C˜2 Continuous Cubic Hermite Interpolation Splines with Second-Order Elliptic Variation." Tobacco Regulatory Science 7, no. 6 (November 3, 2021): 6317–31. http://dx.doi.org/10.18001/trs.7.6.106.

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In order to solve the deficiency of Hermite interpolation spline with second-order elliptic variation in shape control and continuity, c-2 continuous cubic Hermite interpolation spline with second-order elliptic variation was designed. A set of cubic Hermite basis functions with two parameters was constructed. According to this set of basis functions, the three-order Hermite interpolation spline curves were defined in segments 02, and the parameter selection scheme was discussed. The corresponding cubic Hermite interpolation spline function was studied, and the method to determine the residual term and the best interpolation function was given. The results of an example show that when the interpolation conditions remain unchanged, the cubic Hermite interpolation spline curves not only reach 02 continuity, but also can use the parameters to control the shape of the curves locally or globally. By determining the best values of the parameters, the cubic Hermite interpolation spline function can get a better interpolation effect, and the smoothness of the interpolation spline curve is the best.

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WANG, GANG, XIAN-YAO CHEN, FANG-LI QIAO, ZHAOHUA WU, and NORDEN E. HUANG. "ON INTRINSIC MODE FUNCTION." Advances in Adaptive Data Analysis 02, no. 03 (July 2010): 277–93. http://dx.doi.org/10.1142/s1793536910000549.

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Empirical Mode Decomposition (EMD) has been widely used to analyze non-stationary and nonlinear signal by decomposing data into a series of intrinsic mode functions (IMFs) and a trend function through sifting processes. For lack of a firm mathematical foundation, the implementation of EMD is still empirical and ad hoc. In this paper, we prove mathematically that EMD, as practiced now, only gives an approximation to the true envelope. As a result, there is a potential conflict between the strict definition of IMF and its empirical implementation through natural cubic spline. It is found that the amplitude of IMF is closely connected with the interpolation function defining the upper and lower envelopes: adopting the cubic spline function, the upper (lower) envelope of the resulting IMF is proved to be a unitary cubic spline line as long as the extrema are sparsely distributed compared with the sampling data. Furthermore, when natural spline boundary condition is adopted, the unitary cubic spline line degenerates into a straight line. Unless the amplitude of the IMF is a strictly monotonic function, the slope of the straight line will be zero. It explains why the amplitude of IMF tends to be a constant with the number of sifting increasing ad infinitum. Therefore, to get physically meaningful IMFs the sifting times for each IMF should be kept low as in the practice of EMD. Strictly speaking, the resolution of these difficulties should be either to change the EMD implementation method and eschew the spline, or to define the stoppage criterion more objectively and leniently. Short of the full resolution of the conflict, we should realize that the EMD as implemented now yields an approximation with respect to cubic spline basis. We further concluded that a fixed low number of iterations would be the best option at this time, for it delivers the best approximation.

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Lytvyn, Oleg, Oleg Lytvyn, and Oleksandra Lytvyn. "Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections." Physico-mathematical modelling and informational technologies, no. 33 (September 2, 2021): 12–17. http://dx.doi.org/10.15407/fmmit2021.33.012.

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This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

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Ahmed, Abdarrhim Mohammed, and Saleh Y. Barony. "Plate Bending Spline Finite Strip Models Using Mixed Formulation and Combined Spline Series." European Journal of Engineering Research and Science 4, no. 10 (October 17, 2019): 42–51. http://dx.doi.org/10.24018/ejers.2019.4.10.1489.

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As Classical and Spline finite strip method based on stiffness and mixed variational formulation principle become important tool for continuum structural analysis , especially in the field of plate bending problems , a lot of researches has been focused on interpolation functions in order to improve the efficiency and increase the reliability of the method. The main objective of this paper is to introduce and propose a new spline interpolation function in the light of combination techniques of basic splines through introduction and brief review of previous studies in this field. This work which uses abbreviated form of augmented matrix proposed by authors published in past time , reveals a very good accordance results compared with the analytical and published solutions of different plate bending problems.

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Ahmed, Abdarrhim Mohammed, and Saleh Y. Barony. "Plate Bending Spline Finite Strip Models Using Mixed Formulation and Combined Spline Series." European Journal of Engineering and Technology Research 4, no. 10 (October 17, 2019): 42–51. http://dx.doi.org/10.24018/ejeng.2019.4.10.1489.

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As Classical and Spline finite strip method based on stiffness and mixed variational formulation principle become important tool for continuum structural analysis , especially in the field of plate bending problems , a lot of researches has been focused on interpolation functions in order to improve the efficiency and increase the reliability of the method. The main objective of this paper is to introduce and propose a new spline interpolation function in the light of combination techniques of basic splines through introduction and brief review of previous studies in this field. This work which uses abbreviated form of augmented matrix proposed by authors published in past time , reveals a very good accordance results compared with the analytical and published solutions of different plate bending problems.

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Lixandru, Ion. "Algorithm for the Calculation of the Two Variables Cubic Spline Function." Annals of the Alexandru Ioan Cuza University - Mathematics 59, no. 1 (January 1, 2013): 149–61. http://dx.doi.org/10.2478/v10157-012-0022-y.

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Abstract When having just one variable, the existence and uniqueness of the interpolation spline function reduces to studying the solutions of an algebrical system of equations. This allows us to find a practical way of calculating the interpolation spline function. Also in the case of two variables spline functions, we can construct a linear system of equations determined by the continuity conditions of the spline function and of its partial derivatives on the edge of each division rectangle. The existence and uniqueness of the solution of the obtained system ensure the existence and uniqueness of the two variables interpolation spline function and offers a practical calculation method. This can be used to determine approximate global solutions, of some partial differential equations, solutions whose values can be determined at any point of their domain of definition and can provide information on derivatives approximate of solutions. After calculating the two variable cubic spline function, we must assess the rest of the approximation.

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Lyudmila, Gadasina, and Vyunenko Lyudmila. "Applying spline-based phase analysis to macroeconomic dynamics." Dependence Modeling 10, no. 1 (January 1, 2022): 207–14. http://dx.doi.org/10.1515/demo-2022-0113.

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Abstract The article uses spline-based phase analysis to study the dynamics of a time series of low-frequency data on the values of a certain economic indicator. The approach includes two stages. At the first stage, the original series is approximated by a smooth twice-differentiable function. Natural cubic splines are used as an approximating function y y . Such splines have the smallest curvature over the observation interval compared to other possible functions that satisfy the choice criterion. At the second stage, a phase trajectory is constructed in ( t , y , y ′ ) \left(t,y,y^{\prime} ) -space, corresponding to the original time series, and a phase shadow as a projection of the phase trajectory onto the ( y , y ′ ) (y,y^{\prime} ) -plane. The approach is applied to the values of GDP indicators for the G7 countries. The interrelation between phase shadow loops and cycles of economic indicators evolution is shown. The study also discusses the features, limitations and prospects for the use of spline-based phase analysis.

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Janecki, Dariusz, Leszek Cedro, and Jarosław Zwierzchowski. "Separation Of Non-Periodic And Periodic 2D Profile Features Using B-Spline Functions." Metrology and Measurement Systems 22, no. 2 (June 1, 2015): 289–302. http://dx.doi.org/10.1515/mms-2015-0016.

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Abstract The form, waviness and roughness components of a measured profile are separated by means of digital filters. The aim of analysis was to develop an algorithm for one-dimensional filtering of profiles using approximation by means of B-splines. The theory of B-spline functions introduced by Schoenberg and extended by Unser et al. was used. Unlike the spline filter proposed by Krystek, which is described in ISO standards, the algorithm does not take into account the bending energy of a filtered profile in the functional whose minimization is the principle of the filter. Appropriate smoothness of a filtered profile is achieved by selecting an appropriate distance between nodes of the spline function. In this paper, we determine the Fourier transforms of the filter impulse response at different impulse positions, with respect to the nodes. We show that the filter cutoff length is equal to half of the node-to-node distance. The inclination of the filter frequency characteristic in the transition band can be adjusted by selecting an appropriate degree of the B-spline function. The paper includes examples of separation of 2D roughness, as well as separation of form and waviness of roundness profiles.

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SHIMOJIMA, Koji, Toshio FUKUDA, and Fumihito ARAI. "Fuzzy Inference based on Spline Function." Journal of Japan Society for Fuzzy Theory and Systems 6, no. 4 (1994): 679–89. http://dx.doi.org/10.3156/jfuzzy.6.4_679.

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Grandine, T. A. "Minimal Function Spaces for Spline Interpolation." Constructive Approximation 17, no. 2 (January 1, 2001): 157–68. http://dx.doi.org/10.1007/s002450010005.

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UDDIN, Md KAMAL, and U. V. NAIK-NIMBALKAR. "Regression Function Estimation Using Spline Wavelets." Communications in Statistics - Theory and Methods 34, no. 4 (April 2005): 823–32. http://dx.doi.org/10.1081/sta-200054413.

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Sarfraz, Muhammad, Malik Zawwar Hussain, and Asfar Nisar. "Positive data modeling using spline function." Applied Mathematics and Computation 216, no. 7 (June 2010): 2036–49. http://dx.doi.org/10.1016/j.amc.2010.03.034.

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JENA, M. K. "CONSTRUCTION OF COMPACTLY SUPPORTED WAVELETS FROM TRIGONOMETRIC B-SPLINES." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 05 (September 2011): 843–65. http://dx.doi.org/10.1142/s021969131100433x.

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We construct a class of semiorthogonal wavelets by taking a normalized trigonometric B-spline of any order as the scaling function. The construction is based on generalized Euler–Frobenius polynomial and generalized autocorrelation function. We also show that the odd order normalized trigonometric B-spline satisfies convex hull property as well as partition of unity property. Moreover, we also present a subdivision algorithm for the convolution of normalized trigonometric B-splines. Several examples of wavelet are also provided.

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Cano, G., M. Blanco, I. Casasús, X. Cortés-Lacruz, and D. Villalba. "Comparison of B-splines and non-linear functions to describe growth patterns and predict mature weight of female beef cattle." Animal Production Science 56, no. 11 (2016): 1787. http://dx.doi.org/10.1071/an15089.

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The objective of this study was to compare the ability of Basis spline (B-spline) models and five non-linear functions (Richards, Brody, Von Bertalanffy, Gompertz and Logistic) to describe the growth of females of a beef cattle breed and predict cow mature weight (A). Random regression models that included animal variation within function parameters were fitted using mixed model procedures. Comparisons were made among these functions for goodness of fit, standardised residuals and biological interpretability of the growth curve parameters. The B-spline function showed the best goodness of fit and within non-linear functions, the Richards and Von Bertalanffy functions estimated bodyweight at different periods accurately. The method of fitting the residual variance that provided the best goodness of fit in the model was the constant plus power variance function. The Richards function was found to be the best non-linear function and was compared with the B-spline function to predict mature weight. When the A parameter was estimated using fixed effects, it had a low correlation with the actual mature weight of the cow and the use of this estimate yielded no more gain in predictive accuracy of mature weight than the use of average breed mature weight. When A was estimated using fixed and random effects, it had a moderate correlation with actual mature weight for the B-spline and Richards functions. The use of both types of effects to estimate the maturity index reduced the error compared with the use of average mature weight, especially for the B-spline function, which is recommended as the best function to describe animal growth and predict mature weight.

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Cano, G., M. Blanco, I. Casasús, X. Cortés-Lacruz, and D. Villalba. "Corrigendum to: Comparison of B-splines and non-linear functions to describe growth patterns and predict mature weight of female beef cattle." Animal Production Science 56, no. 12 (2016): 2161. http://dx.doi.org/10.1071/an15089_co.

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The objective of this study was to compare the ability of Basis spline (B-spline) models and five non-linear functions (Richards, Brody, Von Bertalanffy, Gompertz and Logistic) to describe the growth of females of a beef cattle breed and predict cow mature weight (A). Random regression models that included animal variation within function parameters were fitted using mixed model procedures. Comparisons were made among these functions for goodness of fit, standardised residuals and biological interpretability of the growth curve parameters. The B-spline function showed the best goodness of fit and within non-linear functions, the Richards and Von Bertalanffy functions estimated bodyweight at different periods accurately. The method of fitting the residual variance that provided the best goodness of fit in the model was the constant plus power variance function. The Richards function was found to be the best non-linear function and was compared with the B-spline function to predict mature weight. When the A parameter was estimated using fixed effects, it had a low correlation with the actual mature weight of the cow and the use of this estimate yielded no more gain in predictive accuracy of mature weight than the use of average breed mature weight. When A was estimated using fixed and random effects, it had a moderate correlation with actual mature weight for the B-spline and Richards functions. The use of both types of effects to estimate the maturity index reduced the error compared with the use of average mature weight, especially for the B-spline function, which is recommended as the best function to describe animal growth and predict mature weight.

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Andronov, I. L. "Spline Fits: Modelling the Observations." Symposium - International Astronomical Union 155 (1993): 325. http://dx.doi.org/10.1017/s0074180900171141.

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The main results on the elaboration of the algorithms and programs for the search of the ‘hidden periodicities’ in the ‘unevenly spaced’ data, are discussed. They are mainly based on the application of the cubic spline-functions practically for all purposed of the variable star research — period determination, approximation of the phase curve, search for the possible period changes and/or the shape of the phase curve, detection of the periodic components in the case of the multi-harmonic oscillations, restoration of the smoothed function by removing the ‘apparat function’, numerical integration etc. Contrary to the commonly used splines with the arguments of the basic points coinciding with that of the real observations, we use the smoothing by the spline with the number of the basic points m, which is smaller than the number of the observations n. The main expressions for the corresponding fitting curves and their statistical properties were published by Andronov (1988, 1989), as well as the FORTRAN programs for the fits and the integration (Andronov, 1986, 1987, respectively).

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Pepin, A., S. S. Beauchemin, S. Léger, and N. Beaudoin. "A New Method for High-Degree Spline Interpolation: Proof of Continuity for Piecewise Polynomials." Canadian Mathematical Bulletin 63, no. 3 (December 9, 2019): 655–69. http://dx.doi.org/10.4153/s0008439519000742.

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AbstractEffective and accurate high-degree spline interpolation is still a challenging task in today’s applications. Higher degree spline interpolation is not so commonly used, because it requires the knowledge of higher order derivatives at the nodes of a function on a given mesh.In this article, our goal is to demonstrate the continuity of the piecewise polynomials and their derivatives at the connecting points, obtained with a method initially developed by Beaudoin (1998, 2003) and Beauchemin (2003). This new method, involving the discrete Fourier transform (DFT/FFT), leads to higher degree spline interpolation for equally spaced data on an interval $[0,T]$. To do this, we analyze the singularities that may occur when solving the system of equations that enables the construction of splines of any degree. We also note an important difference between the odd-degree splines and even-degree splines. These results prove that Beaudoin and Beauchemin’s method leads to spline interpolation of any degree and that this new method could eventually be used to improve the accuracy of spline interpolation in traditional problems.

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Journal, Baghdad Science. "B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations." Baghdad Science Journal 1, no. 2 (June 6, 2004): 340–46. http://dx.doi.org/10.21123/bsj.1.2.340-346.

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Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.

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Duan, Jun-Sheng, Ming Li, Yan Wang, and Yu-Lian An. "Approximate Solution of Fractional Differential Equation by Quadratic Splines." Fractal and Fractional 6, no. 7 (June 30, 2022): 369. http://dx.doi.org/10.3390/fractalfract6070369.

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In this article, we consider approximate solutions by quadratic splines for a fractional differential equation with two Caputo fractional derivatives, the orders of which satisfy 1<α<2 and 0<β<1. Numerical computing schemes of the two fractional derivatives based on quadratic spline interpolation function are derived. Then, the recursion scheme for numerical solutions and the quadratic spline approximate solution are generated. Two numerical examples are used to check the proposed method. Additionally, comparisons with the L1–L2 numerical solutions are conducted. For the considered fractional differential equation with the leading order α, the involved undetermined parameters in the quadratic spline interpolation function can be exactly resolved.

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Liu, Huan-Wen. "An expansion of bivariate spline functions." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 41, no. 4 (April 2000): 527–41. http://dx.doi.org/10.1017/s0334270000011802.

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AbstractLet Δ denote a triangulation of a planar polygon Ω. For any positive integer 0 ≤ r < k, let denote the vector space of functions in Cr whose restrictions to each triangle of Δ are polynomials of total degree at most k. Such spaces, called bivariate spline spaces, have many applications in surface fitting, scattered data interpolation, function approximation and numerical solutions of partial differential equations. An important problem is to give the function expression. In this paper, we prove that, if (Δ, Ω) is type-X, then any bivariate spline function in can be expressed by a series of univariate polynomials and a special bivariate finite element function in satisfying a so-called integral conformality condition system. We also give a direct sum decomposition of the space . In addition, the dimension of for a kind of triangulation has been determined.

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Journal articles on the topic 'Spline Function'

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