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Curve fitting: a method of joining piecewise functions to produce models of complex data

COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2024年第4期53卷 2071-2077页

作者： Orszulik, Stefan T. 6 Kestrels Wantage OX12 0QA Oxon England

This article describes a method of curve-fitting that involves splitting the curve into two (or more) and creating separate functions for each part. These functions are then concatenated via a linking function to create one overall function that better describes the original data than is otherwise achievable. The linking function is able to do this by separating the original two functions so that they are each active in only the relevant portion of the overall curve without the use of dummy variables. The final result is a continuous function in which it is straightforward to smooth the transition at the knot between the piecewise subfunctions. In addition, the piecewise subfunctions do not need to align at the knot since the degree of smoothing is readily controlled. All types of functions may be concatenated so that the method is flexible and relatively simple to apply.

关键词： Continuous function Curve fitting Hinge Knot modeling piecewise functions Spline regression

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piecewise SOS-convex moment optimization and applications via exact semi-definite programs

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EURO JOURNAL ON COMPUTATIONAL OPTIMIZATION 2024年 12卷

作者： Huang, Q. Y. Jeyakumar, V. Li, G. Univ New South Wales Dept Appl Math Sydney NSW 2052 Australia

This paper presents exact Semi-Definite Program (SDP) reformulations for infinite-dimensional moment optimization problems involving a new class of piecewise Sum-of-Squares (SOS)-convex functions and projected spectrahedral support sets. These reformulations show that solving a single SDP finds the optimal value and an optimal probability measure of the original moment problem. This is done by establishing an SOS representation for the non-negativity of a piecewise SOS-convex function over a projected spectrahedron. Finally, as an application and a proof-of-concept illustration, the paper presents numerical results for the Newsvendor and revenue maximization problems with higher-order moments by solving their equivalent SDP reformulations. These reformulations promise a flexible and efficient approach to solving these models. The main novelty of the present work in relation to the recent research lies in finding the solution to moment problems, for the first time, with piecewise SOS-convex functions from their numerically tractable exact SDP reformulations.

关键词： Moment optimization Sum-of-squares convex polynomials piecewise functions Generalized moment problems Semi-definite programming

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A novel approach to egg and math: Improved geometrical standardization of any avian egg profile

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ANNALS OF THE NEW YORK ACADEMY OF SCIENCES 2023年第1期1529卷 61-71页

作者： Narushin, Valeriy G. Orszulik, Stefan T. Romanov, Michael N. Griffin, Darren K. Res Inst Environm Treatment Zaporozhe Ukraine Vita Market Ltd Zaporozhe Ukraine Univ Kent Sch Biosci Canterbury England L K Ernst Fed Res Ctr Anim Husb Podolsk Moscow Oblast Russia

Developing a geometric formulation of any biological object has a number of justifications and applications. Recently, we developed a universal geometric figure for describing a bird's egg in any of the possible basic shapes: spherical, ellipsoidal, ovoid, and pyriform. The formulation proved widely applicable but had a number of drawbacks, including a very obvious join between two parts of the egg. To correct this, we developed the Main Axiom of the universal mathematical formula. This essentially involved making the ordinate of the extremum of the function correspond to half the maximum egg breadth (B), and the abscissa to the reciprocal of the parameter w that reflects the shift of the vertical axis to its coincidence with B. This, in turn, helped us develop a new, simplified mathematical model without a nonbiological join. Experimental verification was performed to confirm the adequacy of the new geometric figure. It accurately described actual avian eggs of various shapes more closely than our previous model. To the best of our knowledge, our new, simplified equation can be applied as a standard for any bird egg that exists in nature. As a rather simple equation, it can be used in a broad range of applications. We had previously developed a universal geometric figure for describing a bird's egg in any of the possible basic shapes: spherical, ellipsoidal, ovoid, and pyriform. However, the formulation had a number of drawbacks, including a very obvious join between two parts of the egg. To correct this, we developed a new, simplified mathematical model without a non-biological join. Experimental verification was performed to confirm the adequacy of the new geometric ***

关键词： avian eggs egg geometry Hugelschaffer model piecewise functions standard egg shape

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Chaotic-quasi-oppositional-phasor based multi populations gorilla troop optimizer for optimal power flow solution

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ENERGY 2024年 301卷

作者： Jamal, Raheela Zhang, Junzhe Men, Baohui Khan, Noor Habib Ebeed, Mohamed Jamal, Tanzeela Mohamed, Emad A. North China Elect Power Univ Sch Water Resources & Hydropower Engn Beijing 102206 Peoples R China Beijing Municipal Engn Consulting Co Ltd Beijing 100124 Peoples R China North China Elect Power Univ Dept New Energy State Key Labortory Wind Power Beijing 102206 Peoples R China Sohag Univ Fac Engn Dept Elect Engn Sohag 82524 Egypt Univ Jaen Dept Elect Engn Jaen 23700 Spain Beijing Technol & Business Univ Beijing 100048 Peoples R China Prince Sattam Bin Abdulaziz Univ Coll Engn Dept Elect Engn Al Kharaj 16278 Saudi Arabia

This paper proposes an efficient modified Gorilla Troops Optimizer (MGTO) to address the problem of prone to local optima and stagnation possibility in the standard Gorilla Troops Optimizer (GTO). The proposed MGTO is based on multiple strategies including chaotic initialization, quasi-oppositional based learning, phasor motion, and multi-populations strategy. The proposed MGTO is applied to resolve the non-convex single objectives optimal power flow (OPF) problem to reduce the fuel cost without and with vple, the emissions, the fuel cost considering prohibted zones (POZs) and piecewise functions, while for multiple objective cases the fuel cost is optimized with the power losses, the voltage deviation and the emissions. To validate the effectiveness of the proposed optimizer, it is tested on IEEE 30-bus and 57-bus systems, as well as 23 standard benchmark functions and the obtained outcomes are compared with the other seven well-known optimization techniques such as SCSO, GWO, WOA, SCA, PF, BWO and GTO. The simulation results verified that the proposed MGTO provides a remarkable and robust solution for the OPF problem solution compared to the other state-of-the-arts algorithms.

关键词： Optimal power flow Valve point loading effect Prohibted zones piecewise functions Gorilla troops optimizer

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Functional Level Abstraction and Simulation of Verilog-AMS piecewise Linear Models 23

Functional Level Abstraction and Simulation of Verilog-AMS P...

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23rd International Symposium on Quality Electronic Design (ISQED)

作者： Azam, Sadia Dall'Ora, Nicola Fraccaroli, Enrico Fummi, Franco Univ Verona Dept Comp Sci Verona Italy

ISBN: (纸本)9781665494663

In electronic design and testing, the simulation speed of analog components is crucial. Moreover, the simulation of heterogeneous components embedded in a Virtual Platforms (VP) needs to be fast and accurate. Often, the analog components are non-linear, and simulating them is not easy to ensure the model's convergence. In this context, techniques for simulating linear circuits are stable and efficient, but there are still many research gaps for non-linear circuits. There are no systematic methods available to solve non-linear equations efficiently. One standard method is to solve these non-linear equations by describing them as a piecewise linear (PWL) models. PWL techniques approximate non-linear functions with a set of linear functions. This is common to most solver methods: they linearize to compute an inverse matrix, finding which direction to move to satisfy the equations. In this article, an abstraction methodology for PWL models is proposed. By using this methodology, it is possible to abstract a piecewise model described with the Verilog-AMS language to the C++ language. These C++ models can be integrated into VPs. A half-wave rectifier and memristor model are selected to explain and validate the methodology. Furthermore, to show the effectiveness of the proposed technique, the abstracted model of the half-wave rectifier is integrated into a MEMS accelerometer. Moreover, the accelerometer is integrated into a VP to show the effectiveness of the functional simulation.

关键词： Functional simulation piecewise functions Verilog-AMS language Analog models

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Approximate expression of the Prandtl membrane analogy in linear elastic pure torsion of open thin-walled cross sections and regular polygons

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INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 2021年 210卷 109-118页

作者： Gil-Martin, L. M. Palomares, A. Hernandez-Montes, E. Univ Granada UGR Dept Struct Mech Campus Univ Fuentenueva S-N Granada 18072 Spain Univ Granada UGR Appl Math Campus Univ Fuentenueva S-N Granada 18072 Spain

This paper presents a new general methodology to obtain an approximate analytical expression of the Saint-Venant's torsion. The shear stress in each of the principal Cartesian directions is obtained by the derivation of the stress function, whose analytical expression is obtained from the Prandtl analogy. The proposed methodology uses two variables quadratic piecewise functions to define the Prandtl membrane. This document shows that the approximate procedure is especially suitable for steel shapes, giving values very close to those obtained with more precise methods. The main advantage of the presented methodology is its simplicity, which makes it useful for both pedagogic and practical applications. Several examples are developed. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Saint-Venant's torsion Linear elasticity Prandtl analogy piecewise functions

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Bayesian regularization for flexible baseline hazard functions in Cox survival models

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BIOMETRICAL JOURNAL 2021年第1期63卷 7-26页

作者： Lazaro, Elena Armero, Carmen Alvares, Danilo Univ Valencia Dept Stat & Operat Res Burjassot 46100 Spain Pontificia Univ Catolica Chile Dept Stat Macul Chile

Fully Bayesian methods for Cox models specify a model for the baseline hazard function. Parametric approaches generally provide monotone estimations. Semi-parametric choices allow for more flexible patterns but they can suffer from overfitting and instability. Regularization methods through prior distributions with correlated structures usually give reasonable answers to these types of situations. We discuss Bayesian regularization for Cox survival models defined via flexible baseline hazards specified by a mixture of piecewise constant functions and by a cubic B-spline function. For those "semi-parametric" proposals, different prior scenarios ranging from prior independence to particular correlated structures are discussed in a real study with microvirulence data and in an extensive simulation scenario that includes different data sample and time axis partition sizes in order to capture risk variations. The posterior distribution of the parameters was approximated using Markov chain Monte Carlo methods. Model selection was performed in accordance with the deviance information criteria and the log pseudo-marginal likelihood. The results obtained reveal that, in general, Cox models present great robustness in covariate effects and survival estimates independent of the baseline hazard specification. In relation to the "semi-parametric" baseline hazard specification, the B-splines hazard function is less dependent on the regularization process than the piecewise specification because it demands a smaller time axis partition to estimate a similar behavior of the risk.

关键词： correlated prior process cubic B-splines piecewise functions survival analysis Weibull distribution

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New Bounds for Truthful Scheduling on Two Unrelated Selfish Machines

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THEORY OF COMPUTING SYSTEMS 2020年第2期64卷 199-226页

作者： Kuryatnikova, Olga Vera, Juan C. Tilburg Univ Dept Econometr & Operat Res NL-5037 AB Tilburg Netherlands

We consider the minimum makespan problem for n tasks and two unrelated parallel selfish machines. Let R-n be the best approximation ratio of randomized monotone scale-free algorithms. This class contains the most efficient algorithms known for truthful scheduling on two machines. We propose a new Min - Max formulation for R-n, as well as upper and lower bounds on R-n based on this formulation. For the lower bound, we exploit pointwise approximations of cumulative distribution functions (CDFs). For the upper bound, we construct randomized algorithms using distributions with piecewise rational CDFs. Our method improves upon the existing bounds on R-n for small n. In particular, we obtain almost tight bounds for n = 2 showing that |R-2 - 1.505996| < 10(- 6).

关键词： Minimax optimization Truthful scheduling Approximation piecewise functions Algorithmic mechanism design

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A Review of Algorithms for Symbolic Domains 21

A Review of Algorithms for Symbolic Domains

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21st International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)

作者： Watt, Stephen Univ Waterloo David R Cheriton Sch Comp Sci Waterloo ON N2L 3G1 Canada

ISBN: (纸本)9781728157245

This talk is about computing with mathematical quantities where the sizes or shapes are not known in advance. We consider polynomials where the exponents can be given by symbolic expressions, matrices with blocks or other internal structure of symbolic size, and piece-wise functions where the shapes of the domains are given by symbolic expressions. © 2019 IEEE.

关键词： computer algebra symbolic computation polynomial algebra matrix algebra piecewise functions

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Monte Carlo Simulation of the Young Measures - Comparison of Random-Number Generators 13

Monte Carlo Simulation of the Young Measures - Comparison of...

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IEEE 13th International Scientific Conference on Informatics

作者： Grzybowski, Andrzej Z. Puchala, Piotr Czestochowa Tech Univ Inst Math PL-42201 Czestochowa Poland

ISBN: (纸本)9781467398688

"Young measure" is an abstract notion from mathematical measure theory growing up from the analysis of some variational problems. From the formal point of view it may be regarded as a continuous linear functional defined on the space of Caratheodory integrands. However, calculating an explicit form of specific Young measure based on its formal theoretical definition involves so-called "weak" limits of the sequences of functions satisfying some general conditions. From a strictly mathematical standpoint, it is quite difficult task which can be solved only in the most simple cases. However in many real-world applications it would be enough to learn only some of the most important probabilistic characteristics of the Young distribution. In such a case the possible solution is to adopt Monte Carlo techniques. In the presentation we propose three different algorithms designed for simulating random variables distributed according to the Young measures associated with piecewise functions. Next with the help of computer simulations we compare their statistical performance via some benchmarking problems. In this study we focus on the accurateness of the distribution of the generated sample.

关键词： Young measure random numbers piecewise functions simulations

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