基于Carter-Morley Jones蛋形模型的参数估计新方法

doi:10.17520/biods.2024203

生物多样性 ›› 2025, Vol. 33 ›› Issue (1): 24203. DOI: 10.17520/biods.2024203 cstr: 32101.14.biods.2024203

基于Carter-Morley Jones蛋形模型的参数估计新方法

王林¹(), 尹梓杨², 黄慧芳³, 王静⁴^,^*()()

1.四川大学生命科学学院, 成都 610064
2.澳门大学科技学院, 澳门 999078
3.华中科技大学数学与统计学院, 武汉 430074
4.郑州大学生命科学学院, 郑州 450001

收稿日期:2024-05-28 接受日期:2024-08-26 出版日期:2025-01-20 发布日期:2024-12-11
通讯作者: * E-mail: wj0302@zzu.edu.cn
基金资助:
郑州大学科研启动基金;河南省高等学校重点科研项目(24B220007)

A new parameter estimation method based on the Carter-Morley Jones egg- shape model

Lin Wang¹(), Ziyang Yin², Huifang Huang³, Jing Wang⁴^,^*()()

1 College of Life Sciences, Sichuan University, Chengdu 610064, China
2 Faculty of Science and Technology, University of Macau, Macau 999078, China
3 College of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
4 College of Life Sciences, Zhengzhou University, Zhengzhou 450001, China

Received:2024-05-28 Accepted:2024-08-26 Online:2025-01-20 Published:2024-12-11
Contact: * E-mail: wj0302@zzu.edu.cn
Supported by:
Zhengzhou University Scientific Research Start-up Funds;Key Scientific Research Project of Colleges and Universities in Henan Province(24B220007)

1. 附录.zip(5KB)

摘要/Abstract

摘要：

作为鸟类对多样化生存环境适应的一种解释, 鸟蛋的形状长期以来受到生物学家和数学家的关注。Carter-Morley Jones方程是一种描述鸟蛋二维投影形状的极坐标方程, 通常应用多元线性回归方法估计其模型参数。然而, 多元线性回归方法可能因其前提假设不成立而导致拟合较差。本研究提出了一种基于非线性最优化理论的Carter-Morley Jones蛋形模型的参数估计方法, 并使用多元线性回归和非线性最优化两种方法对51个物种共416组实际蛋形数据进行了拟合。应用均方根误差和非线性曲率度量分别评价两种方法的拟合优度和线性近似行为。同时, 本研究还比较了基于Carter-Morley Jones蛋形模型预测的鸟蛋体积和使用排水法测量的实际体积。结果表明: (1)相较于多元线性回归参数估计方法, 非线性最优化方法能够获得更好的拟合优度; (2)在蛋形方程的线性近似行为表现方面, 多元线性回归和非线性最优化两种参数估计方法之间并无显著差异; (3)基于Carter-Morley Jones蛋形模型预测的鸟蛋体积和使用排水法测量的实际体积之间并无显著差异。本研究为蛋形方程的实际应用提供了数学工具。同时, 非线性曲率度量也为将来评估蛋形模型提供了一个新的视角。

关键词: 鸟蛋, Carter-Morley Jones方程, 非线性最优化, 蛋形建模, 生态适应

Abstract

Aim: The shape of eggs is a subject of considerable interest among biologists and mathematicians, particularly regarding avian adaptation to diverse environments. The Carter-Morley Jones equation is a polar coordinate equation that describes the two-dimensional projection shape of bird eggs, typically with its model parameters estimated using multiple linear regression. However, multiple linear regression may result in poor fits due to the failure of its underlying assumptions. This study proposes a novel parameter estimation method for the Carter-Morley Jones equation based on nonlinear optimization theory.

Methods: We employed both multiple linear regression and nonlinear optimization techniques to fit actual egg-shape data obtained from 416 eggs spanning 51 species. The performance of each fitting method was assessed by evaluating the goodness of fit through root-mean-square error and analyzing the linear approximation behavior through nonlinear curvature measures. Additionally, we compared the egg volumes predicted by the Carter-Morley Jones equation with actual volumes measured using a graduated cylinder.

Results: The findings indicated that: (1) The nonlinear optimization method provided a superior goodness of fit compared to the multiple linear regression method; (2) No significant difference was observed in the linear approximation behavior between the two parameter estimation methods; and (3) There was no significant difference between the predicted egg volumes from the Carter-Morley Jones model and the actual volumes measured via the graduated cylinder.

Conclusion: This study offers a robust mathematical tool for applying the egg-shape equation in ecological research. Furthermore, the use of nonlinear curvature measures presents a fresh perspective for evaluating egg shape models in future investigations.

Key words: bird eggs, Carter-Morley Jones equation, nonlinear optimization, egg shape modeling, ecological adaptation

中图分类号:

王林, 尹梓杨, 黄慧芳, 王静 (2025) 基于Carter-Morley Jones蛋形模型的参数估计新方法. 生物多样性, 33, 24203. DOI: 10.17520/biods.2024203.

Lin Wang, Ziyang Yin, Huifang Huang, Jing Wang (2025) A new parameter estimation method based on the Carter-Morley Jones egg- shape model. Biodiversity Science, 33, 24203. DOI: 10.17520/biods.2024203.

导出引用管理器 EndNote|Ris|BibTeX

链接本文: https://www.biodiversity-science.net/CN/10.17520/biods.2024203

https://www.biodiversity-science.net/CN/Y2025/V33/I1/24203

图/表 4

图1 使用Carter-Morley Jones蛋形模型模拟的鸟蛋二维轮廓形状。其中K1 = 2, K2 = 0.1, K4 = 0.05, K5 = 0.01, K3从−0.1变化到−0.5, 每次减少0.1。

Fig. 1 Two-dimensional outline shapes of avian eggs simulated using Carter-Morley Jones egg shape model. Here, K1 = 2, K2 = 0.1, K4 = 0.05, K5 = 0.01, and K3 ranged from −0.1 to −0.5, in 0.1 decrement.

图2 实际蛋形边界数据的拟合示例图。分别使用多元回归方法(A)和非线性最优化方法(B)模拟的一个具有代表性的蛋形的观测边界曲线(灰色)和预测边界曲线(红色)。RMSEadj表示调整的均方根误差。

Fig. 2 Illustration of fitting real egg-shaped boundary data. The observed (gray) and the predicted (red) boundary geometries of a representative egg-shape simulated using linear model based on Carter-Morley Jones equation (A) and optimization method based on Carter-Morley Jones equation (B). RMSEadj represents the adjusted root-mean-square error.

图3 基于lmCMJE和optimCMJE两种方法计算调整后的均方根误差的箱线图。(A)基于366组环颈雉蛋形数据; (B)基于另外50组蛋形数据。每个箱子中的水平实线表示中位数, “*”表示均值。W表示Wilcoxon检验统计量。

Fig. 3 Boxplots of the adjusted root-mean-square error (RMSEadj) calculated using two methods (lmCMJE and optimCMJE). (A) Based on the 366 egg-shape data; (B) Based on the other 50 egg-shape data. The horizontal solid line in each box represents the median, the asterisk within the box represents the mean. W represents the statistic of the Wilcoxon test.

图4 比较lmCMJE和optimCMJE两种方法的非线性曲率度量的箱线图。(A, B)基于366组环颈雉蛋形数据; (C, D)基于另外50组蛋形数据。W表示Wilcoxon检验统计量。

Fig. 4 Boxplots of the nonlinear curvature measures compared between two methods (lmCMJE and optimCMJE). (A, B) Based on the 366 egg-shape data; (C, D) Based on the other 50 egg-shape data. W represents the statistic of the Wilcoxon test.

参考文献 29

[1]	Bates DM, Watts DG (1980) Relative curvature measures of nonlinearity. Journal of the Royal Statistical Society: Series B, 42, 1-16.
[2]	Bates DM, Watts DG (1988) Nonlinear Regression Analysis and its Applications. John Wiley & Sons, New York.
[3]	Biggins JD, Montgomerie R, Thompson JE, Birkhead TR (2022a) Preston’s universal formula for avian egg shape. Ornithology, 139, ukac028.
[4]	Biggins JD, Montgomerie R, Thompson JE, Birkhead TR (2022b) Preston’s universal formula for avian egg shape. https://doi.org/10.5061/dryad.547d7wmbz. (accessed on 2022-06-27)
[5]	Biggins JD, Thompson JE, Birkhead TR (2018) Accurately quantifying the shape of birds’ eggs. Ecology and Evolution, 8, 9728-9738. DOI PMID
[6]	Birkhead TR, Thompson JE, Jackson D, Biggins JD (2017) The point of a Guillemot’s egg. Ibis, 159, 255-265.
[7]	Carter TC, Morley Jones R (1970) The hen’s egg: Shell shape and size parameters and their interrelations. British Poultry Science, 11, 179-188.
[8]	Church SH, Donoughe S, Extavour CG (2019) Insect egg size and shape evolve with ecology but not developmental rate. Nature, 571, 58-62.
[9]	D’alba L, Goldenberg J, Nallapaneni A, Parkinson DY, Zhu CH, Vanthournout B, Shawkey MD (2021) Evolution of eggshell structure in relation to nesting ecology in non-avian reptiles. Journal of Morphology, 282, 1066-1079. DOI PMID
[10]	Gielis J (2003) A generic geometric transformation that unifies a wide range of natural and abstract shapes. American Journal of Botany, 90, 333-338. DOI PMID
[11]	Huang WW, Li YY, Niklas KJ, Gielis J, Ding YY, Cao L, Shi PJ (2020) A superellipse with deformation and its application in describing the cross-sectional shapes of a square bamboo. Symmetry, 12, 2073.
[12]	Maritz MF, Douglas RM (1994) Shape quantization and the estimation of volume and surface area of reptile eggs. Journal of Herpetology, 28, 281-291.
[13]	Narushin VG, Romanov MN, Griffin DK (2021) Egg and math:Introducing a universal formula for egg shape. Annals of the New York Academy of Sciences, 1505, 169-177.
[14]	Nelder JA, Mead R (1965) A simplex method for function minimization. The Computer Journal, 7, 308-313.
[15]	Nishiyama Y (2012) The mathematics of egg shape. International Journal of Pure and Applied Mathematics, 78, 679-689.
[16]	Petrović M, Obradović M, Mijailović R (2011) Suitability analysis of Hügelschäffer’s egg curve application in architectural and structures’ geometry. Bulletin of the Polytechnic Institute of Iasi, 57, 115-122.
[17]	Preston FW (1953) The shapes of birds’ eggs. The Auk, 70, 160-182.
[18]	R Core Team (2022) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/. (accessed on 2022-07-01)
[19]	Shi PJ, Chen L, Quinn BK, Yu KX, Miao QY, Guo XC, Lian M, Gielis J, Niklas KJ (2023a) A simple way to calculate the volume and surface area of avian eggs. Annals of the New York Academy of Sciences, 1524, 118-131.
[20]	Shi PJ, Chen L, Quinn BK, Yu KX, Miao QY, Guo XC, Lian M, Gielis J, Niklas KJ (2023b) Egg-shape data of six species of poultry. https://doi.org/10.5061/dryad.f4qrfj719. (accessed on 2023-04-05)
[21]	Shi PJ, Gielis J, Niklas KJ (2022a) Comparison of a universal (but complex) model for avian egg shape with a simpler model. Annals of the New York Academy of Sciences, 1514, 34-42.
[22]	Shi PJ, Gielis J, Quinn BK, Niklas KJ, Ratkowsky DA, Schrader J, Ruan HH, Wang L, Niinemets Ü (2022b) ‘biogeom’:An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences, 1516, 123-134.
[23]	Shi PJ, Ratkowsky DA, Li Y, Zhang LF, Lin SY, Gielis J (2018) A general leaf area geometric formula exists for plants—Evidence from the simplified Gielis equation. Forests, 9, 714.
[24]	Shi PJ, Ridland P, Ratkowsky DA, Li Y (2024) IPEC: Root mean square curvature calculation. R package version 1.1.0. https://CRAN.R-project.org/package=IPEC. (accessed on 2024-01-14)
[25]	Shi PJ, Wang L, Quinn BK, Gielis J (2023c) A new program to estimate the parameters of Preston’s equation, a general formula for describing the egg shape of birds. Symmetry, 15, 231.
[26]	Stoddard MC, Yong EH, Akkaynak D, Sheard C, Tobias JA, Mahadevan L (2017) Avian egg shape: Form, function, and evolution. Science, 356, 1249-1254. DOI PMID
[27]	Su JL, Niklas KJ, Huang WW, Yu XJ, Yang YY, Shi PJ (2019) Lamina shape does not correlate with lamina surface area: An analysis based on the simplified Gielis equation. Global Ecology and Conservation, 19, e00666.
[28]	Troscianko J (2014) A simple tool for calculating egg shape, volume and surface area from digital images. Ibis, 156, 874-878.
[29]	Wilcoxon F (1945) Individual comparisons by ranking methods. Biometrics Bulletin, 1, 80-83.

基于Carter-Morley Jones蛋形模型的参数估计新方法

A new parameter estimation method based on the Carter-Morley Jones egg- shape model

RichHTML

PDF (PC)

补充材料

可视化

摘要/Abstract

引用本文

使用本文

图/表 4

参考文献 29

相关文章 7

编辑推荐

Metrics

本文评价

[1]	魏雪苹, 张宪春. 蕨类植物不同孢子裂缝类型在中国的分布格局[J]. 生物多样性, 2016, 24(10): 1129-1134.
[2]	钟云芳, 张哲, 宋希强, 周兆德. 海南凤仙花不同海拔种群的传粉生物学[J]. 生物多样性, 2014, 22(4): 467-475.
[3]	徐兆礼. 中国近海浮游动物多样性研究的过去和未来[J]. 生物多样性, 2011, 19(6): 635-645.
[4]	田中平, 庄丽, 李建贵, 程模香. 伊犁河谷北坡野果林木本植物种间关系及环境解释[J]. 生物多样性, 2011, 19(3): 335-342.
[5]	严岳鸿, 何祖霞, 苑虎, 邢福武. 坡向差异对广东古兜山自然保护区蕨类植物多样性的生态影响[J]. 生物多样性, 2011, 19(1): 41-47.
[6]	张洋, 谭敦炎. 地下结实植物白番红花的繁育系统与传粉生物学[J]. 生物多样性, 2009, 17(5): 468-475.
[7]	孙华之, 谭敦炎, 曲荣明. 短命植物小疮菊异形瘦果特性及其对荒漠环境的适应[J]. 生物多样性, 2008, 16(4): 353-361.