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A polynomial regression model for predicting knuckleball movements in soccer free-kick

PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART P-JOURNAL OF SPORTS ENGINEERING AND TECHNOLOGY 2025年

作者： Liu, Jiahang Liang, Dongfang Cho, Hiroshi Jilin Univ Coll Math Changchun 130015 Peoples R China Univ Cambridge Dept Engn Cambridge England Kumamoto Univ Int Res Org Adv Sci & Technol Kumamoto Japan Kumamoto Univ Fac Adv Sci & Technol FAST Kumamoto Japan

This paper develops a polynomial regression model with physical constraints to comprehensively analyze the complex motion of knuckleballs in soccer free-kick. The predictions match accurately with the ball trajectories extracted from recorded soccer matches. The model uses a polynomial function to account for the ball's projectile-like motion. It further incorporates a critical physical constraint that is related to the aerodynamic drag force on the ball. The regression model is implemented using a relaxed iterative method. The lsqlin solver of MATLAB is used in each iteration. The model is verified through a prominent case study - Cristiano Ronaldo's representative knuckleball free-kick executed in April 2012. The ball trajectory is meticulously derived from image processing of video footage. The paper examines the ball position, velocity, and drag coefficient over the entire flight of the ball. Notably, a significant increase in the drag coefficient is observed during the flight, which is a critical feature of the knuckling effect. The study extensively explores the influence of physical constraints and various model parameters to ensure the regression model's credibility, shedding light on the intricate dynamics of knuckleball free-kicks. With its simplicity, efficiency, and effectiveness, this polynomial regression model can be a powerful tool for studying knuckleball free-kicks in real time and thus improving players' techniques in training and in the match. Besides, it can be used to create authentic flight paths of the ball in computer games.

关键词： Knuckleball free-kick polynomial regression model physical constraint drag coefficient soccer football

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Skin Reflectance Reconstruction Based on the polynomial regression model 16th

Skin Reflectance Reconstruction Based on the Polynomial Regr...

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16th Chinese Conference on Image and Graphics Technologies and Applications (IGTA)

作者： Ma, Long Zhu, Yingying Shenyang Jianzhu Univ Sch Sci Shenyang 110168 Liaoning Peoples R China

ISBN: (纸本)9789811671890;9789811671883

Skin spectral reflectance has applications in numerous medical fields including the diagnosis and treatment of cutaneous disorders and the provision of maxillofacial soft tissue prostheses. This paper describes the polynomial model based on the least square (LS) method for skin spectral reflectance from RGB. Furthermore, this paper uses the real human skin data, which makes our results more practical. The performance is evaluated by the mean, maximum and standard deviation of color difference values under other sets of light sources. The values of standard deviation of root mean square (RMS) errors and goodness of fit coefficient (GFC) between the reproduced and the actual spectrawere also calculated. Results are compared with the Xiao's method. All metrics show that the proposed method leads to considerable improvements in comparison with the Xiao's method.

关键词： Skin spectral reflectance reconstruction polynomial regression model Color difference Root mean square error Goodness of fit coefficient

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Wind speed frequency distribution modeling and wind energy resource assessment based on polynomial regression model

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INTERNATIONAL JOURNAL OF ELECTRICAL POWER & ENERGY SYSTEMS 2021年 130卷 106964-106964页

作者： Wang, Lingzhi Liu, Jun Qian, Fucai Xian Univ Posts & Telecommun Sch Automat Xian 710121 Peoples R China Xian Univ Technol Sch Automat & Informat Xian 710048 Peoples R China Shaanxi Key Lab Complex Syst Control & Intelligen Xian 710048 Peoples R China Xian Technol Univ Autonomous Syst & Intelligent Control Int Joint R Xian 710021 Shaanxi Peoples R China

The single Weibull distribution, which is the most frequently used in evaluating wind energy feasibility, fits poorly with zero and low wind speed values and is ineffectual for multimodal wind speed data. Although some other existing models are effective for multimodal wind speed distribution, the gap remains large at zero and low wind speeds. To overcome the shortcomings of these distribution models, we propose a polynomial regression model as the probability density function for wind speed frequency. The model parameters are obtained through linear least-square method. Comparison experiments were conducted between the single Weibull distribution model and the proposed approach. Then the optimal polynomial regression model, the mixture of two truncated normal distributions, and the tri-Weibull mixed model are compared from fitting performance. Meanwhile, the annual energy production and the cost of wind power generation are calculated based on the wind speed frequency distribution model. The results show that the proposed polynomial regression model provides excellent approximation for different wind speed frequency distributions. The approach can be used to fit zero and low wind speed data, and fit multimodal distribution data. When estimating the annual energy production and the wind power cost, our proposed approach has smaller modelling errors and lower cost of power generation than those from another methods. The proposed wind frequency distribution probability density function model would be very useful for wind energy resource assessment and development.

关键词： Wind speed frequency distribution Probability density function polynomial regression model Linear least-square method Annual energy production Wind power cost

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Ambivalent perception of cause-related marketing: investigating the (in)congruence of opposite motivational perceptions on consumer evaluations

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INDUSTRIAL MANAGEMENT & DATA SYSTEMS 2025年第1期125卷 192-213页

作者： Wang, Yuting Chen, Yao Fang, Jie Xiong, Bingqing Shanghai Univ SILC Business Sch Shanghai Peoples R China Jiangsu Univ Sci & Technol Sch Econ & Management Zhenjiang Peoples R China Univ Nottingham Ningbo China Nottingham Univ Business Sch China Ningbo Peoples R China Deakin Univ Deakin Business Sch Melbourne Australia

PurposeDespite the popularity of leveraging cause-related marketing (CRM) to make societal contributions and bolster business profits, sellers face a profound dilemma when conducting CRM due to consumers' ambivalent understanding of sellers' motivation for the initiative. Therefore, it is imperative to unravel consumers' ambivalent understanding of CRM and determine how sellers can effectively employ CRM to elicit positive evaluations from ***/methodology/approachThis study gathered survey data from 217 participants and applied a polynomial regression model and response surface analysis for disentangling ambivalent perception of CRM by investigating the influence of (in)congruence between perceived egoistic and altruistic *** incongruence between perceived egoistic and altruistic motivation can positively influence consumers' evaluations of sellers. Moreover, when perceived egoistic and altruistic motivations are congruent, increasing their absolute level also enhances consumers' evaluation of sellers. Moreover, sellers' platform function usage behavior can amplify the positive effect of incongruence but has no salient moderating role on the congruence ***/valueDiffering from prior literature that predominantly focused on either the positive or negative interpretation of CRM, this study reveals the coexistence of both positive and negative viewpoints and disentangles the congruence and incongruence effect between the two motivational understandings.

关键词： Ambivalent perception Cause-related marketing Platform function usage polynomial regression model Response surface analysis Stereotype content model

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MINIMIZING THE CONDITION NUMBER TO CONSTRUCT DESIGN POINTS FOR polynomial regression modelS

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SIAM JOURNAL ON OPTIMIZATION 2013年第1期23卷 666-686页

作者： Ye, Jane J. Zhou, Julie Univ Victoria Dept Math & Stat Victoria BC V8W 3R4 Canada

In this paper we study a new optimality criterion, the K-optimality criterion, for constructing optimal experimental designs for polynomial regression models. We focus on the pth order polynomial regression model with symmetric design space [-1, 1]. For this model, we show that there is always a symmetric K-optimal design with exactly p + 1 support points including the boundary points -1 and 1. It is well known that the condition number for a positive definite matrix as the ratio of the maximum eigenvalue to the minimum eigenvalue is usually nonsmooth. We show that for our model, the condition number of the information matrix is continuously differentiable. Theoretical K-optimal designs are derived for p = 1 and 2. Numerical results are presented for 3 <= p <= 10.

关键词： condition number information matrix K-optimal design polynomial regression model semidefinite programming symmetric design

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Asymptotic efficiency of the OLSE for polynomial regression models with spatially correlated errors

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STATISTICS & PROBABILITY LETTERS 2000年第1期47卷 1-10页

作者： Shin, DW Song, SH Ewha Univ Dept Stat Seodaemun Ku Seoul 120750 South Korea Duksung Univ Dept Stat Tobong Ku Seoul 132714 South Korea

For polynomial regression models with spatially correlated errors, the covariance matrix of the ordinary least-squares estimator (OLSE) is shown to have the same limiting value as that of the generalized least-squares estimator (GLSE) under the same normalization. This implies that the OLSE is asymptotically efficient. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： polynomial regression model spatial correlation efficiency OLSE GLSE

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Quantitative study of the relationships among proteolysis, lipid oxidation, structure and texture throughout the dry-cured ham process

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FOOD CHEMISTRY 2015年 166卷 522-530页

作者： Harkouss, Rami Astruc, Thierry Lebert, Andre Gatellier, Philippe Loison, Olivier Safa, Hassan Portanguen, Stephane Parafita, Emilie Mirade, Pierre-Sylvain INRA Qual Prod Anim UR370 F-63122 St Genes Champanelle France UBP CNRS IFMA UMR6602 Inst Pascal F-63171 Aubiere France ADIV ZAC Gravanches F-63039 Clermont Ferrand 2 France

Temperature, salt and water contents are key processing factors in dry-cured ham production. They affect how proteolysis, lipid oxidation, structure and texture evolve, and thus determine the sensory properties and final quality of dry-cured ham. The aim of this study was to quantify the interrelationships and the time course of (i) proteolysis, (ii) lipid oxidation, (iii) five textural parameters: hardness, fragility, cohesiveness, springiness and adhesiveness and (iv) four structural parameters: fibre numbers, extracellular spaces, cross section area, and connective tissue area, during the dry-cured ham process. Applying multiple polynomial regression enabled us to build phenomenological models relating proteolysis, salt and water contents to certain textural and structural parameters investigated. A linear relationship between lipid oxidation and proteolysis was also established. All of these models and relationships, once combined with salt penetration, water migration and heat transfer models, can be used to dynamically simulate all of these phenomena throughout dry-cured ham manufacturing. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Dry-cured ham Proteolysis Structure Texture polynomial regression model

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A novel non-intrusive method using design of experiments and smooth approximation to speed up multi-period load-flows in distribution network planning

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ELECTRIC POWER SYSTEMS RESEARCH 2018年 154卷 444-451页

作者： Baraffe, Heloise Dutrieux Cosson, Marjorie Bect, Julien Delille, Gauthier Francois, Bruno EDF R&D 7 Blvd Gaspard Monge F-91120 Palaiseau France Univ Paris Saclay Univ Paris Sud L2S Cent SupelecCNRS 3 Rue Joliot Curie F-91192 Gif Sur Yvette France Univ Lille Cent Lille Arts & Metiers Paris Tech HEIEA L2EP 2697Lab Elect Engn & Power Elect F-59000 Lille France

Alternative solutions to network reinforcement are now being investigated in distribution network planning studies to reduce the costs and periods for integrating renewable energy sources. However, a thorough techno-economic analysis of these solutions requires a large number of multi-period load flow calculations, which makes it hard to implement in planning tools. A non-intrusive approximation method is therefore proposed to obtain fast and accurate multi-period load-flows. This method builds a surrogate model of the load-flow solver using polynomial regression and kriging, combined with Latin hypercube sampling. Case studies based on real distribution networks show that the proposed method is more efficient for distribution network planning in presence of renewable energy sources than time subsampling, line model simplification, and voltage linearization. In particular, accurate 10-min profiles of voltages, currents, and network power losses are obtained in a satisfactory computation time. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Approximation method Distribution network planning Kriging model Multi-period load-flow polynomial regression model Principal Component Analysis

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Discriminant analysis for regression models with stationary long-memory disturbances

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JOURNAL OF MULTIVARIATE ANALYSIS 1997年第2期60卷 177-187页

作者： Zhang, GQ Komazawa University Tokyo Japan

We shah consider the problems of classifying an observation From regression model with stationary long-memory or short-memory disturbances into one of two populations described by the mean Functions of the model. We use the log-likelihood ratio as a discrimant statistic which is optimal in the sense of its minimizing the misclassification probabilities. Then we confirm the theoretical results by some simple polynomial regression models. (C) 1997 Academic Press.

关键词： discriminant analysis misclassification probability polynomial regression model regression model stationary long-memory disturbances

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modelling of daily maximum temperatures over Australia from 1970 to 2012

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METEOROLOGICAL APPLICATIONS 2016年第1期23卷 115-122页

作者： Wanishsakpong, Wandee McNeil, Nittaya Prince Songkla Univ Fac Sci & Technol Dept Math & Comp Sci Pattani Campus Muang 94000 Pattani Thailand

The variations of daily maximum temperatures over consecutive 5 day periods from 1970 to 2012 in Australia are described. The data were obtained from the Australia Bureau of Meteorology, comprising measurements from 85 weather stations. Linear regression was used initially to model the seasonally adjusted daily maximum temperatures. Factor analysis was used to classify the temperatures from the 85 stations into seven factors which correspond to seven geographical regions. The average maximum annual temperature in these seven regions ranged from 23 to 36 degrees C. A 6(th) order polynomial regression model was then fitted to investigate the trend and pattern of the temperatures in these seven regions. Similar increasing temperature trends occurred in the central, eastern, southern and southeastern parts of the country.

关键词： Australia climate change time series analysis linear regression factor analysis polynomial regression model

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