Help for package biogeom

Type:

Package

Title:

Biological Geometries

Version:

1.4.3

Date:

2024-03-21

Author:

Peijian Shi [aut, cre], Johan Gielis [aut], Brady K. Quinn [aut]

Maintainer:

Peijian Shi <pjshi@njfu.edu.cn>

Imports:

spatstat.geom (≥ 2.4-0)

Description:

Is used to simulate and fit biological geometries. 'biogeom' incorporates several novel universal parametric equations that can generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree-rings (Gielis (2003) <doi:10.3732/ajb.90.3.333>; Shi et al. (2020) <doi:10.3390/sym12040645>), three growth-rate curves representing the ontogenetic growth trajectories of animals and plants against time, and the axially symmetrical and integral forms of all these functions (Shi et al. (2017) <doi:10.1016/j.ecolmodel.2017.01.012>; Shi et al. (2021) <doi:10.3390/sym13081524>). The optimization method proposed by Nelder and Mead (1965) <doi:10.1093/comjnl/7.4.308> was used to estimate model parameters. 'biogeom' includes several real data sets of the boundary coordinates of natural shapes, including avian eggs, fruit, lanceolate and ovate leaves, tree rings, seeds, and sea stars,and can be potentially applied to other natural shapes. 'biogeom' can quantify the conspecific or interspecific similarity of natural outlines, and provides information with important ecological and evolutionary implications for the growth and form of living organisms. Please see Shi et al. (2022) <doi:10.1111/nyas.14862> for details.

Depends:

R (≥ 4.3.0)

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

NeedsCompilation:

Packaged:

2024-03-21 14:14:58 UTC; PEIJIAN SHI

Repository:

CRAN

Date/Publication:

2024-03-29 06:20:02 UTC

Calculation of the First-Order Derivative of the Explicit Preston Equation

Description

DEPE is used to calculate the first-order derivative of the explicit Preston equation at a given x-value.

Usage

DEPE(P, x, simpver = NULL)

Arguments

P

the parameters of the explicit Preston equation or one of its simplified versions.

x

the x-value used in the explicit Preston equation.

simpver

an optional argument to use the simplified version of the explicit Preston equation.

Details

When simpver = NULL, the first-order derivative of the explicit Preston equation at a given x-value is selected:

f(x)=\frac{b\left[a^{4}\,c_{1}+a^{3}\left(2\,c_{2}-1\right)x+ a^2\left(3\,c_{3}-2\,c_{1}\right)x^{2}-3\,a\,c_{2}x^3-4\,c_{3}\,x^{4}\right]}{a^4\sqrt{a^2-x^2}},

where P has five parameters: a, b, c_{1}, c_{2}, and c_{3}.

\quad When simpver = 1, the first-order derivative of the simplified version 1 is selected:

f(x)=\frac{b\left[a^{4}\,c_{1}+a^{3}\left(2\,c_{2}-1\right)x- 2\,a^2\,c_{1}\,x^{2}-3\,a\,c_{2}x^3\right]}{a^4\sqrt{a^2-x^2}},

where P has four parameters: a, b, c_{1}, and c_{2}.

\quad When simpver = 2, the first-order derivative of the simplified version 2 is selected:

f(x)=\frac{b\left[a^{4}\,c_{1}-a^{3}\,x- 2\,a^2\,c_{1}\,x^{2}\right]}{a^4\sqrt{a^2-x^2}},

where P has three parameters: a, b, and c_{1}.

\quad When simpver = 3, the first-order derivative of the simplified version 3 is selected:

f(x)=\frac{b\left[a^{3}\left(2\,c_{2}-1\right)x-3\,a\,c_{2}x^3\right]}{a^4\sqrt{a^2-x^2}},

where P has three parameters: a, b, and c_{2}.

Note

The argument P in the DEPE function has the same parameters, as those in the EPE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Shi, P., Chen, L., Quinn, B.K., Yu, K., Miao, Q., Guo, X., Lian, M., Gielis, J., Niklas, K.J. (2023) A simple way to calculate the volume and surface area of avian eggs. Annals of the New York Academy of Sciences 1524, 118-131. doi:10.1111/nyas.15000

Examples

  Par3 <- c(4.27, 2.90, 0.0868, 0.0224, -0.0287)
  xx1  <- seq(-4.27, 4.27, by=0.001)
  f1   <- DEPE(P=Par3, x=xx1, simpver=NULL)
  f2   <- -DEPE(P=Par3, x=xx1, simpver=NULL)

  dev.new()
  plot(xx1, f1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-5, 5), ylim=c(-35, 35), xlab=expression(italic(x)), 
       ylab=expression(paste(italic(f), "(", italic(x), ")", sep="")))
  lines(xx1, f2, col=2)  

  graphics.off()

Calculation of the First-Order Derivative of the Explicit Troscianko Equation

Description

DETE is used to calculate the first-order derivative of the explicit Troscianko equation at a given x-value.

Usage

DETE(P, x)

Arguments

P

the parameters of the explicit Troscianko equation.

x

the x-value used in the explicit Troscianko equation.

Details

The first-order derivative of the explicit Troscianko equation at a given x-value is:

h(x) = \left\{\alpha_{1}+\frac{2\,\alpha_{2}}{a}\,x-\frac{x}{a}\,\left[1-\left(\frac{x}{a}\right)^2\right]^{-1}\right\}\,\exp\left\{\alpha_{0}+\alpha_{1}\left(\frac{x}{a}\right)+\alpha_{2}\left(\frac{x}{a}\right)^2\right\}\,\sqrt{ 1-\left(\frac{x}{a}\right)^2 },

where P has four parameters: a, \alpha_{0}, \alpha_{1}, and \alpha_{2}.

Note

The argument P in the DETE function has the same parameters, as those in the ETE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Examples

  Par5 <- c(2.25, -0.38, -0.29, -0.16)
  xx2  <- seq(-2.25, 2.25, by=0.001)
  h1   <- DETE(P=Par5, x=xx2)
  h2   <- -DETE(P=Par5, x=xx2)
  ind  <- which(is.na(h1) | is.na(h2))
  xx2  <- xx2[-ind]
  h1   <- h1[-ind]
  h2  <- h2[-ind]

  dev.new()
  plot(xx2, h1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-2.25, 2.25), ylim=c(-30, 30), xlab=expression(italic(x)), 
       ylab=expression(paste(italic(h), "(", italic(x), ")", sep="")))
  lines(xx2, h2, col=2)  

  graphics.off()

Calculation of the First-Order Derivative of the Narushin-Romanov-Griffin Equation

Description

DNRGE is used to calculate the first-order derivative of the Narushin-Romanov-Griffin equation at a given x-value.

Usage

DNRGE(P, x)

Arguments

P

the parameters of the Narushin-Romanov-Griffin equation.

x

the x-value used in the Narushin-Romanov-Griffin equation.

Details

Let us define:

f_{1}(x) = \frac{B}{2}\sqrt{\frac{A^2-4x^2}{A^2+8Cx+4C^2}},

f_{2}(x) = \sqrt{\frac{A\left(A^{2}+8Cx+4C^{2}\right)}{2(A-2C)x^{2}+\left(A^{2}+8AC-4C^{2}\right)x+2AC^{2}+A^{2}C+A^{3}}},

f_{3}(x) = A^2 - 4x,

f_{4}(x) = A^2+8Cx+4C^2,

E = \frac{\sqrt{5.5A^{2}+11AC+4C^{2}} \cdot \left(\sqrt{3}AB-2D\sqrt{A^{2}+2AC+4C^{2 }}\right)}{\sqrt{3}AB\left(\sqrt{5.5A^{2}+11AC+4C^{2}}-2\sqrt{A^{2}+2AC+4C^{2}}\right)},

F = 2\left(A-2C\right),

G = A^{2}+8AC-4C^{2},

H = 2AC^{2}+A^{2}C+A^{3},

and then the first-order derivative of the Narushin-Romanov-Griffin equation at a given x-value is:

J(x) = -\frac{4\,f_{1}(x)\left[C\,f_{3}(x)+x\,f_{4}(x)\right]}{f_{3}(x) \cdot f_{4}(x)}\left\{1-E \cdot \left[1-f_{2}(x)\right]\right\}-\frac{AE}{2}\frac{f_{1}(x)}{f_{2}(x)}\frac{f_{4}(x) \cdot \left(2Fx+G\right)}{\left(Fx^2+Gx+H\right)^2},

where P has four parameters: A, B, C, and D.

Note

The argument P in the DNRGE function has the same parameters, as those in the NRGE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

Narushin, V.G., Romanov, M.N., Mishra, B., Griffin, D.K. (2022) Mathematical progression of avian egg shape with associated area and volume determinations. Annals of the New York Academy of Sciences 1513, 65-78. doi:10.1111/nyas.14771

Examples

  Par6 <- c(4.51, 3.18, 0.1227, 2.2284)
  xx3  <- seq(-4.51/2, 4.51/2, len=2000)
  J1   <- DNRGE(P=Par6, x=xx3)
  J2   <- -DNRGE(P=Par6, x=xx3)
  ind  <- which(is.na(J1) | is.na(J2))
  xx3  <- xx3[-ind]
  J1   <- J1[-ind]
  J2   <- J2[-ind]

  dev.new()
  plot(xx3, J1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-4.51/2, 4.51/2), ylim=c(-20, 20), xlab=expression(italic(x)), 
       ylab=expression(paste(italic(J), "(", italic(x), ")", sep="")))
  lines(xx3, J2, col=2)  

  graphics.off()

Calculation of the First-Order Derivative of the Simplified Gielis Equation

Description

DSGE is used to calculate the first-order derivative of the simplified Gielis equation at a given \varphi-value.

Usage

DSGE(P, phi)

Arguments

P

the parameters of the simplified Gielis equation, including a, n_{1}, and n_{2}.

phi

the \varphi-value used in the simplified Gielis equation.

Details

The first-order derivative of the simplified Gielis equation with arguments simpver = 1 and m = 1 at a given \varphi-value is:

g(x) = \frac{a}{4}\,\frac{n_{2}}{n_{1}}\,\left[\left(\cos{\frac{\varphi}{4}}\right)^{n_{2}-1}\left(\sin{\frac{\varphi}{4}}\right)-\left(\sin{\frac{\varphi}{4}}\right)^{n_{2}-1}\left(\cos{\frac{\varphi}{4}}\right)\right]\ \left[\left(\cos{\frac{\varphi}{4}}\right)^{n_{2}}+\left(\sin{\frac{\varphi}{4}}\right)^{n_{2}}\right]^{-\frac{1}{n_{1}}-1},

where P has three parameters: a, n_{1}, and n_{2}.

Note

The argument P in the DSGE function only has the three parameters: a, n_{1}, and n_{2}.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Chen, Z. (2012) Volume and area of revolution under polar coordinate system. Studies in College Mathematics 15(6), 9-11.

Examples

  Par7 <- c(1.124, 14.86, 49.43)
  phi1 <- seq(0, pi, len=2000)
  g1   <- DSGE(P=Par7, phi=phi1)

  dev.new()
  plot(phi1, g1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlab=expression(italic(varphi)), 
       ylab=expression(paste(italic(g), "(", italic(varphi), ")", sep="")))

  graphics.off()

Calculation of the Ordinate For an Arbitrary Point on the Preston Curve in the Plane

Description

EPE is used to calculate the y-value for an arbitrary point on the Preston curve that was generated by the explicit Preston equation or one of its simplified versions for a given x-value.

Usage

EPE(P, x, simpver = NULL)

Arguments

P

the parameters of the explicit Preston equation or one of its simplified versions.

x

the x-value used in the explicit Preston equation.

simpver

an optional argument to use the simplified version of the explicit Preston equation.

Details

When simpver = NULL, the explicit Preston equation is selected:

y = b\ \sqrt{1-\left(\frac{x}{a}\right)^2}\left(1+c_{1}\ \frac{x}{a}+c_{2}\left(\frac{x}{a}\right)^2+c_{3}\left(\frac{x}{a}\right)^3\right),

where P has five parameters: a, b, c_{1}, c_{2}, and c_{3}.

\quad When simpver = 1, the simplified version 1 is selected:

y = b\ \sqrt{1-\left(\frac{x}{a}\right)^2}\left(1+c_{1}\ \frac{x}{a}+c_{2}\left(\frac{x}{a}\right)^2\right),

where P has four parameters: a, b, c_{1}, and c_{2}.

\quad When simpver = 2, the simplified version 2 is selected:

y = b\ \sqrt{1-\left(\frac{x}{a}\right)^2}\left(1+c_{1}\ \frac{x}{a}\right),

where P has three parameters: a, b, and c_{1}.

\quad When simpver = 3, the simplified version 3 is selected:

y = b\ \sqrt{1-\left(\frac{x}{a}\right)^2}\left(1+c_{2}\left(\frac{x}{a}\right)^2\right),

where P has three parameters: a, b, and c_{2}.

Value

The y values predicted by the explicit Preston equation.

Note

We only considered the upper part of the egg-shape curve in the above expressions because the lower part is symmetrical to the upper part around the x-axis. The mid-line of an egg's profile in EPE is aligned to the x-axis, while the mid-line of an egg's profile in PE is aligned to the y-axis. The EPE function has the same parameters, P, as those in the PE function. The explicit Preston equation is used for calculating an egg's volume and surface area, when the parameters, which are saved in the P vector, are obtained using the fitEPE function or the lmPE function based on the TSE function. In addition, the values in x > a (i.e., the first element in P) are forced to be a, and the values in x < -a will be forced to be -a.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Shi, P., Wang, L., Quinn, B.K., Gielis, J. (2023) A new program to estimate the parameters of Preston's equation, a general formula for describing the egg shape of birds. Symmetry 15, 231. doi:10.3390/sym15010231

Examples

  Par3 <- c(4.27, 2.90, 0.0868, 0.0224, -0.0287)
  xx1  <- seq(-4.27, 4.27, by=0.001)
  yy1  <- EPE(P=Par3, x=xx1, simpver=NULL)
  yy2  <- -EPE(P=Par3, x=xx1, simpver=NULL)

  dev.new()
  plot(xx1, yy1, asp=1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-5, 5), ylim=c(-5, 5), 
       xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(xx1, yy2, col=2) 

  graphics.off()

Calculation of the Ordinate For an Arbitrary Point on the Troscianko Curve in the Plane

Description

ETE is used to calculate the y-value for an arbitrary point on the Troscianko curve that was generated by the explicit Troscianko equation.

Usage

ETE(P, x)

Arguments

P

the parameters of the explicit Troscianko equation, including a, \alpha_{0}, \alpha_{1}, and \alpha_{2}.

x

the x-value used in the explicit Troscianko equation.

Details

The explicit Troscianko equation is recommended as:

y = a\,\exp\left\{\alpha_{0}+\alpha_{1}\,\left(\frac{x}{a}\right)+\alpha_{2}\, \left(\frac{x}{a}\right)^2\right\}\sqrt{1-\left(\frac{x}{a}\right)^2},

where x and y represent the abscissa and ordinate of an arbitrary point on the explicit Troscianko curve; a, \alpha_{0}, \alpha_{1}, and \alpha_{2} are parameters to be estimated.

Value

The y values predicted by the explicit Troscianko equation.

Note

We only considered the upper part of the egg-shape curve in the above expressions because the lower part is symmetrical to the upper part around the x-axis. The mid-line of an egg's profile in ETE is aligned to the x-axis. The argument, P, in the ETE function has the same parameters, \alpha_{0}, \alpha_{1}, and \alpha_{2}, as those in the TE function. However, the former has an additional parameter a than the latter, which represents half the egg's length. The lmTE function is based on the TE function, while the fitETE function is based on the ETE function here. In addition, the values in x > a (i.e., the first element in P) are forced to be a, and the values in x < -a will be forced to be -a.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston’s universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences 1516, 123-134. doi:10.1111/nyas.14862

Troscianko, J. (2014). A simple tool for calculating egg shape, volume and surface area from digital images. Ibis, 156, 874-878. doi:10.1111/ibi.12177

Examples

  Par5 <- c(2.25, -0.38, -0.29, -0.16)
  xx2  <- seq(-2.25, 2.25, len=2000)
  yy3  <- ETE(P=Par5, x=xx2)
  yy4  <- -ETE(P=Par5, x=xx2)

  dev.new()
  plot(xx2, yy3, asp=1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-3, 3), ylim=c(-3, 3), 
       xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(xx2, yy4, col=2)  

  graphics.off()

Calculation of the Polar Radius of the Gielis Curve

Description

GE is used to calculate polar radii of the original Gielis equation or one of its simplified versions at given polar angles.

Usage

GE(P, phi, m = 1, simpver = NULL, nval = 1)

Arguments

P

the parameters of the original Gielis equation or one of its simplified versions.

phi

the polar angle(s).

m

the given m value that determines the number of angles of the Gielis curve within [0, 2\pi).

simpver

an optional argument to use the simplified version of the original Gielis equation.

nval

the specified value for n_{1} or n_{2} or n_{3} in the simplified versions.

Details

When simpver = NULL, the original Gielis equation is selected:

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{3}}\right)^{-\frac{1}{n_{1}}},

where r represents the polar radius at the polar angle \varphi; m determines the number of angles within [0, 2\pi); and a, k, n_{1}, n_{2}, and n_{3} need to be provided in P.

\quad When simpver = 1, the simplified version 1 is selected:

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}\right)^{-\frac{1}{n_{1}}},

where a, n_{1}, and n_{2} need to be provided in P.

\quad When simpver = 2, the simplified version 2 is selected:

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}\right)^{-\frac{1}{n_{1}}},

where a and n_{1} need to be provided in P, and n_{2} should be specified in nval.

\quad When simpver = 3, the simplified version 3 is selected:

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}\right)^{-\frac{1}{n_{1}}},

where a needs to be provided in P, and n_{1} should be specified in nval.

\quad When simpver = 4, the simplified version 4 is selected:

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}\right)^{-\frac{1}{n_{1}}},

where a and n_{1} need to be provided in P.

\quad When simpver = 5, the simplified version 5 is selected:

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{3}}\right)^{-\frac{1}{n_{1}}},

where a, n_{1}, n_{2}, and n_{3} need to be provided in P.

\quad When simpver = 6, the simplified version 6 is selected:

where a, k, n_{1}, and n_{2} need to be provided in P.

\quad When simpver = 7, the simplified version 7 is selected:

where a, k, and n_{1} need to be provided in P, and n_{2} should be specified in nval.

\quad When simpver = 8, the simplified version 8 is selected:

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}+ \left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}\right)^{-\frac{1}{n_{1}}},

where a and k are parameters that need to be provided in P, and n_{1} should be specified in nval.

\quad When simpver = 9, the simplified version 9 is selected:

where a, k, and n_{1} need to be provided in P.

Value

The polar radii predicted by the original Gielis equation or one of its simplified versions.

Note

simpver here is different from that in the TGE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

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:10.3732/ajb.90.3.333

Li, Y., Quinn, B.K., Gielis, J., Li, Y., Shi, P. (2022) Evidence that supertriangles exist in nature from the vertical projections of Koelreuteria paniculata fruit. Symmetry 14, 23. doi:10.3390/sym14010023

Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. doi:10.3390/sym12040645

Shi, P., Xu, Q., Sandhu, H.S., Gielis, J., Ding, Y., Li, H., Dong, X. (2015) Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant. Ecology and Evolution 5, 4578-4589. doi:10.1002/ece3.1728

Examples

GE.par  <- c(2, 1, 4, 6, 3)
varphi.vec <- seq(0, 2*pi, len=2000)
r.theor <- GE(P=GE.par, phi=varphi.vec, m=5)

dev.new()
plot( varphi.vec, r.theor, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic(varphi)), ylab=expression(italic("r")),
      type="l", col=4 ) 

graphics.off()

Leaf size distribution of Shibataea chinensis

Description

The data consist of the leaf size measures of 12 individuad culms of Shibataea chinensis.

Usage

data(LeafSizeDist)

Details

In the data set, there are five columns of variables: Code, L, W, A, and M. Code saves the bamboo number (the number before the hyphen) and the leaf number (the number after the hyphen) on each bamboo. L saves the length of each leaf lamina (cm); W saves the width of each leaf lamina (cm); A saves the area of each leaf lamina (\mbox{cm}^{2}); and M saves the dry mass of each leaf lamina (g).

References

Lian, M., Shi, P., Zhang, L., Yao, W., Gielis, J., Niklas, K.J. (2023) A generalized performance equation and its application in measuring the Gini index of leaf size inequality. Trees - Structure and Function 37, 1555-1565. doi:10.1007/s00468-023-02448-8

Examples

data(LeafSizeDist)

CulmNumber <- c()
for(i in 1:length(LeafSizeDist$Code)){
  temp <- as.numeric( strsplit(LeafSizeDist$Code[i],"-", fixed=TRUE)[[1]][1] )
  CulmNumber <- c(CulmNumber, temp)
}

uni.CN <- sort( unique(CulmNumber) )  
ind    <- CulmNumber==uni.CN[1]

A0 <- LeafSizeDist$A[ind]
A1 <- sort( A0 )
x1 <- 1:length(A1)/length(A1)
y1 <- cumsum(A1)/sum(A1)
x1 <- c(0, x1)
y1 <- c(0, y1)

M0 <- LeafSizeDist$M[ind]
M1 <- sort( M0 )
x2 <- 1:length(M1)/length(M1)
y2 <- cumsum(M1)/sum(M1)
x2 <- c(0, x2)
y2 <- c(0, y2)

dev.new()
plot(x1, y1, cex.lab=1.5, cex.axis=1.5, type="l",
    xlab="Cumulative proportion of the number of leaves", 
    ylab="Cumulative proportion of leaf area") 
lines(c(0,1), c(0,1), col=4) 

dev.new()
plot(x2, y2, cex.lab=1.5, cex.axis=1.5, type="l",
    xlab="Cumulative proportion of the number of leaves", 
    ylab="Cumulative proportion of leaf dry mass")
lines(c(0,1), c(0,1), col=4)   

graphics.off()

Modified Briere Equation

Description

MBriereE is used to calculate y values at given x values using the modified Brière equation or one of its simplified versions.

Usage

MBriereE(P, x, simpver = 1)

Arguments

P

the parameters of the modified Brière equation or one of its simplified versions.

x

the given x values.

simpver

an optional argument to use the simplified version of the modified Brière equation.

Details

When simpver = NULL, the modified Brière equation is selected:

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = a\left|x(x-x_{\mathrm{min}})(x_{\mathrm{max}}-x)^{1/m}\right|^{\delta};

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

Here, x and y represent the independent and dependent variables, respectively; and a, m, x_{\mathrm{min}}, x_{\mathrm{max}}, and \delta are constants to be estimated, where x_{\mathrm{min}} and x_{\mathrm{max}} represents the lower and upper intersections between the curve and the x-axis. y is defined as 0 when x < x_{\mathrm{min}} or x > x_{\mathrm{max}}. There are five elements in P, representing the values of a, m, x_{\mathrm{min}}, x_{\mathrm{max}}, and \delta, respectively.

\quad When simpver = 1, the simplified version 1 is selected:

\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},

y = a\left|x^{2}(x_{\mathrm{max}}-x)^{1/m}\right|^{\delta};

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P, representing the values of a, m, x_{\mathrm{max}}, and \delta, respectively.

\quad When simpver = 2, the simplified version 2 is selected:

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = ax(x-x_{\mathrm{min}})(x_{\mathrm{max}}-x)^{1/m};

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P representing the values of a, m, x_{\mathrm{min}}, and x_{\mathrm{max}}, respectively.

\quad When simpver = 3, the simplified version 3 is selected:

\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},

y = ax^{2}(x_{\mathrm{max}}-x)^{1/m};

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are three elements in P representing the values of a, m, and x_{\mathrm{max}}, respectively.

Value

The y values predicted by the modified Brière equation or one of its simplified versions.

Note

We have added a parameter \delta in the original Brière equation (i.e., simpver = 2) to increase the flexibility for data fitting.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Brière, J.-F., Pracros, P, Le Roux, A.-Y., Pierre, J.-S. (1999) A novel rate model of temperature-dependent development for arthropods. Environmental Entomology 28, 22-29. doi:10.1093/ee/28.1.22

Cao, L., Shi, P., Li, L., Chen, G. (2019) A new flexible sigmoidal growth model. Symmetry 11, 204. doi:10.3390/sym11020204

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. doi:10.3390/plants11131769

Examples

x2   <- seq(-5, 15, len=2000)
Par2 <- c(0.01, 3, 0, 10, 1)
y2   <- MBriereE(P=Par2, x=x2, simpver=NULL)

dev.new()
plot( x2, y2, cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )

graphics.off()

Modified Lobry-Rosso-Flandrois (LRF) Equation

Description

MLRFE is used to calculate y values at given x values using the modified LRF equation or one of its simplified versions.

Usage

MLRFE(P, x, simpver = 1)

Arguments

P

the parameters of the modified LRF equation or one of its simplified versions.

x

the given x values.

simpver

an optional argument to use the simplified version of the modified LRF equation.

Details

When simpver = NULL, the modified LRF equation is selected:

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ \frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}\right)},

y = y_{\mathrm{opt}}\left\{\frac{\left(x-x_{\mathrm{min}}\right)\left(x-x_{\mathrm{max}}\right)^{2}}{\left(x_{\mathrm{max}}-x_{\mathrm{opt}}\right)\left[(x_{\mathrm{max}}-x_{\mathrm{opt}})(x-x_{\mathrm{opt}})-(x_{\mathrm{min}}-x_{\mathrm{opt}})(x_{\mathrm{opt}}+x_{\mathrm{max}}-2x)\right]}\right\}^{\delta};

\mbox{if } x \in{\left[\frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y = y_{\mathrm{opt}}\left\{\frac{\left(x-x_{\mathrm{max}}\right)\left(x-x_{\mathrm{min}}\right)^{2}}{\left(x_{\mathrm{opt}}-x_{\mathrm{min}}\right)\left[(x_{\mathrm{opt}}-x_{\mathrm{min}})(x-x_{\mathrm{opt}})-(x_{\mathrm{opt}}-x_{\mathrm{max}})(x_{\mathrm{opt}}+x_{\mathrm{min}}-2x)\right]}\right\}^{\delta};

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

Here, x and y represent the independent and dependent variables, respectively; y_{\mathrm{opt}}, x_{\mathrm{opt}}, x_{\mathrm{min}}, x_{\mathrm{max}}, and \delta are constants to be estimated; y_{\mathrm{opt}} represents the maximum y, and x_{\mathrm{opt}} is the x value associated with the maximum y (i.e., y_{\mathrm{opt}}); and x_{\mathrm{min}} and x_{\mathrm{max}} represents the lower and upper intersections between the curve and the x-axis. There are five elements in P, representing the values of y_{\mathrm{opt}}, x_{\mathrm{opt}}, x_{\mathrm{min}}, x_{\mathrm{max}}, and \delta, respectively.

\quad When simpver = 1, the simplified version 1 is selected:

\mbox{if } x \in{\left(0, \ \frac{x_{\mathrm{max}}}{2}\right)},

y = y_{\mathrm{opt}}\left\{\frac{x\left(x-x_{\mathrm{max}}\right)^{2}}{\left(x_{\mathrm{max}}-x_{\mathrm{opt}}\right)\left[(x_{\mathrm{max}}-x_{\mathrm{opt}})(x-x_{\mathrm{opt}})+x_{\mathrm{opt}}(x_{\mathrm{opt}}+x_{\mathrm{max}}-2x)\right]}\right\}^{\delta};

\mbox{if } x \in{\left[\frac{x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y = y_{\mathrm{opt}}\left\{\frac{\left(x-x_{\mathrm{max}}\right)x^{2}}{x_{\mathrm{opt}}\left[x_{\mathrm{opt}}(x-x_{\mathrm{opt}})-(x_{\mathrm{opt}}-x_{\mathrm{max}})(x_{\mathrm{opt}}-2x)\right]}\right\}^{\delta};

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P, representing the values of y_{\mathrm{opt}}, x_{\mathrm{opt}}, x_{\mathrm{max}}, and \delta, respectively.

\quad When simpver = 2, the simplified version 2 is selected:

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ \frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}\right)},

y = \frac{y_{\mathrm{opt}}\left(x-x_{\mathrm{min}}\right)\left(x-x_{\mathrm{max}}\right)^{2}}{\left(x_{\mathrm{max}}-x_{\mathrm{opt}}\right)\left[(x_{\mathrm{max}}-x_{\mathrm{opt}})(x-x_{\mathrm{opt}})-(x_{\mathrm{min}}-x_{\mathrm{opt}})(x_{\mathrm{opt}}+x_{\mathrm{max}}-2x)\right]};

\mbox{if } x \in{\left[\frac{x_{\mathrm{min}}+x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y = \frac{y_{\mathrm{opt}}\left(x-x_{\mathrm{max}}\right)\left(x-x_{\mathrm{min}}\right)^{2}}{\left(x_{\mathrm{opt}}-x_{\mathrm{min}}\right)\left[(x_{\mathrm{opt}}-x_{\mathrm{min}})(x-x_{\mathrm{opt}})-(x_{\mathrm{opt}}-x_{\mathrm{max}})(x_{\mathrm{opt}}+x_{\mathrm{min}}-2x)\right]};

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P, representing the values of y_{\mathrm{opt}}, x_{\mathrm{opt}}, x_{\mathrm{min}}, and x_{\mathrm{max}}, respectively.

\quad When simpver = 3, the simplified version 3 is selected:

\mbox{if } x \in{\left(0, \ \frac{x_{\mathrm{max}}}{2}\right)},

y = \frac{y_{\mathrm{opt}}x\left(x-x_{\mathrm{max}}\right)^{2}}{\left(x_{\mathrm{max}}-x_{\mathrm{opt}}\right)\left[(x_{\mathrm{max}}-x_{\mathrm{opt}})(x-x_{\mathrm{opt}})+x_{\mathrm{opt}}(x_{\mathrm{opt}}+x_{\mathrm{max}}-2x)\right]};

\mbox{if } x \in{\left[\frac{x_{\mathrm{max}}}{2}, \ x_{\mathrm{max}}\right)},

y = \frac{y_{\mathrm{opt}}\left(x-x_{\mathrm{max}}\right)x^{2}}{x_{\mathrm{opt}}\left[x_{\mathrm{opt}}(x-x_{\mathrm{opt}})-(x_{\mathrm{opt}}-x_{\mathrm{max}})(x_{\mathrm{opt}}-2x)\right]};

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are three elements in P, representing the values of y_{\mathrm{opt}}, x_{\mathrm{opt}}, and x_{\mathrm{max}}, respectively.

Value

The y values predicted by the modified LRF equation or one of its simplified versions.

Note

We have added n parameter \delta in the original LRF equation (i.e., simpver = 2) to increase the flexibility for data fitting.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Shi, P., Fan, M., Ratkowsky, D.A., Huang, J., Wu, H., Chen, L., Fang, S., Zhang, C. (2017) Comparison of two ontogenetic growth equations for animals and plants. Ecological Modelling 349, 1-10. doi:10.1016/j.ecolmodel.2017.01.012

Examples

x3   <- seq(-5, 15, len=2000)
Par3 <- c(3, 3, 10, 2)
y3   <- MbetaE(P=Par3, x=x3, simpver=1)

dev.new()
plot( x3, y3, cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )
 
graphics.off()

Modified Performance Equation

Description

MPerformanceE is used to calculate y values at given x values using the modified performance equation or one of its simplified versions.

Usage

MPerformanceE(P, x, simpver = 1)

Arguments

P

the parameters of the modified performance equation or one of its simplified versions.

x

the given x values.

simpver

an optional argument to use the simplified version of the modified performance equation.

Details

When simpver = NULL, the modified performance equation is selected:

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = c\left(1-e^{-K_{1}\left(x-x_{\mathrm{min}}\right)}\right)^{a}\left(1-e^{K_{2}\left(x-x_{\mathrm{max}}\right)}\right)^{b};

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

Here, x and y represent the independent and dependent variables, respectively; and c, K_{1}, K_{2}, x_{\mathrm{min}}, x_{\mathrm{max}}, a, and b are constants to be estimated, where x_{\mathrm{min}} and x_{\mathrm{max}} represents the lower and upper intersections between the curve and the x-axis. y is defined as 0 when x < x_{\mathrm{min}} or x > x_{\mathrm{max}}. There are seven elements in P, representing the values of c, K_{1}, K_{2}, x_{\mathrm{min}}, x_{\mathrm{max}}, a, and b, respectively.

\quad When simpver = 1, the simplified version 1 is selected:

\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},

y = c\left(1-e^{-K_{1}x}\right)^{a}\left(1-e^{K_{2}\left(x-x_{\mathrm{max}}\right)}\right)^{b};

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are six elements in P, representing the values of c, K_{1}, K_{2}, x_{\mathrm{max}}, a, and b respectively.

\quad When simpver = 2, the simplified version 2 is selected:

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = c\left(1-e^{-K_{1}\left(x-x_{\mathrm{min}}\right)}\right)\left(1-e^{K_{2}\left(x-x_{\mathrm{max}}\right)}\right);

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

There are five elements in P representing the values of c, K_{1}, K_{2}, x_{\mathrm{min}}, and x_{\mathrm{max}}, respectively.

\quad When simpver = 3, the simplified version 3 is selected:

\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},

y = c\left(1-e^{-K_{1}x}\right)\left(1-e^{K_{2}\left(x-x_{\mathrm{max}}\right)}\right);

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P representing the values of c, K_{1}, K_{2}, and x_{\mathrm{max}}, respectively.

\quad When simpver = 4, the simplified version 4 is selected:

\mbox{if } x \in{\left(0, \ \sqrt{2}\right)},

y = c\left(1-e^{-K_{1}x}\right)^{a}\left(1-e^{K_{2}\left(x-\sqrt{2}\right)}\right)^{b};

\mbox{if } x \notin{\left(0, \ \sqrt{2}\right)},

y = 0.

There are five elements in P, representing the values of c, K_{1}, K_{2}, a, and b, respectively.

\quad When simpver = 5, the simplified version 5 is selected:

\mbox{if } x \in{\left(0, \ \sqrt{2}\right)},

y = c\left(1-e^{-K_{1}x}\right)\left(1-e^{K_{2}\left(x-\sqrt{2}\right)}\right);

\mbox{if } x \notin{\left(0, \ \sqrt{2}\right)},

y = 0.

There are three elements in P, representing the values of c, K_{1}, and K_{2}, respectively.

Value

The y values predicted by the modified performance equation or one of its simplified versions.

Note

We have added two parameters a and b in the original performance equation (i.e., simpver = 2) to increase the flexibility for data fitting. The cases of simpver = 4 and simpver = 5 are used to describe the rotated and right-shifted Lorenz curve (see Lian et al. [2023] for details).

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Huey, R.B., Stevenson, R.D. (1979) Integrating thermal physiology and ecology of ectotherms: a discussion of approaches. American Zoologist 19, 357-366. doi:10.1093/icb/19.1.357

Shi, P., Ge, F., Sun, Y., Chen, C. (2011) A simple model for describing the effect of temperature on insect developmental rate. Journal of Asia-Pacific Entomology 14, 15-20. doi:10.1016/j.aspen.2010.11.008

Examples

x4   <- seq(0, 40, len=2000)
Par4 <- c(0.117, 0.090, 0.255, 5, 35, 1, 1)
y4   <- MPerformanceE(P=Par4, x=x4, simpver=NULL)

dev.new()
plot( x4, y4, cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )

graphics.off()

Modified Beta Equation

Description

MbetaE is used to calculate y values at given x values using the modified beta equation or one of its simplified versions.

Usage

MbetaE(P, x, simpver = 1)

Arguments

P

the parameters of the modified beta equation or one of its simplified versions.

x

the given x values.

simpver

an optional argument to use the simplified version of the modified beta equation.

Details

When simpver = NULL, the modified beta equation is selected:

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = y_{\mathrm{opt}}{ \left[\left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x-x_{\mathrm{min}}}{x_{\mathrm{opt}}-x_{\mathrm{min}}}\right)^{\frac{x_{\mathrm{opt}}-x_{\mathrm{min}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} \right] }^{\delta};

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

Here, x and y represent the independent and dependent variables, respectively; y_{\mathrm{opt}}, x_{\mathrm{opt}}, x_{\mathrm{min}}, x_{\mathrm{max}}, and \delta are constants to be estimated; y_{\mathrm{opt}} represents the maximum y, and x_{\mathrm{opt}} is the x value associated with the maximum y (i.e., y_{\mathrm{opt}}); and x_{\mathrm{min}} and x_{\mathrm{max}} represent the lower and upper intersections between the curve and the x-axis. y is defined as 0 when x < x_{\mathrm{min}} or x > x_{\mathrm{max}}. There are five elements in P, representing the values of y_{\mathrm{opt}}, x_{\mathrm{opt}}, x_{\mathrm{min}}, x_{\mathrm{max}}, and \delta, respectively.

\quad When simpver = 1, the simplified version 1 is selected:

\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},

y = y_{\mathrm{opt}}{ \left[\left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x}{x_{\mathrm{opt}}}\right)^{\frac{x_{\mathrm{opt}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} \right] }^{\delta};

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P, representing the values of y_{\mathrm{opt}}, x_{\mathrm{opt}}, x_{\mathrm{max}}, and \delta, respectively.

\quad When simpver = 2, the simplified version 2 is selected:

\mbox{if } x \in{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = y_{\mathrm{opt}}{ \left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x-x_{\mathrm{min}}}{x_{\mathrm{opt}}-x_{\mathrm{min}}}\right)^{\frac{x_{\mathrm{opt}}-x_{\mathrm{min}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} };

\mbox{if } x \notin{\left(x_{\mathrm{min}}, \ x_{\mathrm{max}}\right)},

y = 0.

There are four elements in P, representing the values of y_{\mathrm{opt}}, x_{\mathrm{opt}}, x_{\mathrm{min}}, and x_{\mathrm{max}}, respectively.

\quad When simpver = 3, the simplified version 3 is selected:

\mbox{if } x \in{\left(0, \ x_{\mathrm{max}}\right)},

y = y_{\mathrm{opt}}{ \left(\frac{x_{\mathrm{max}}-x}{x_{\mathrm{max}}-x_{\mathrm{opt}}}\right)\left(\frac{x}{x_{\mathrm{opt}}}\right)^{\frac{x_{\mathrm{opt}}}{x_{\mathrm{max}}-x_{\mathrm{opt}}}} };

\mbox{if } x \notin{\left(0, \ x_{\mathrm{max}}\right)},

y = 0.

There are three elements in P, representing the values of y_{\mathrm{opt}}, x_{\mathrm{opt}}, and x_{\mathrm{max}}, respectively.

Value

The y values predicted by the modified beta equation or one of its simplified versions.

Note

We have added a parameter \delta in the original beta equation (i.e., simpver = 2) to increase the flexibility for data fitting.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Examples

x1   <- seq(-5, 15, len=2000)
Par1 <- c(3, 3, 10, 2)
y1   <- MbetaE(P=Par1, x=x1, simpver=1)

dev.new()
plot( x1, y1,cex.lab=1.5, cex.axis=1.5, type="l",
      xlab=expression(italic(x)), ylab=expression(italic(y)) )
 
graphics.off()

The Narushin-Romanov-Griffin Equation (NRGE)

Description

NRGE is used to calculate y values at given x values using the Narushin-Romanov-Griffin equation (NRGE).

Usage

NRGE(P, x)

Arguments

P

the four parameters (i.e., A, B, C, and D) of the Narushin-Romanov-Griffin equation.

x

the given x values.

Details

The Narushin-Romanov-Griffin equation (Narushin et al., 2021) has four parameters in total, among which three parameters have clear geometric meanings.

f_{1}(x)=\frac{B}{2}\sqrt{\frac{A^2-4x^2}{A^2+8Cx+4C^2}},

E = \frac{\sqrt{5.5A^{2}+11AC+4C^{2}} \cdot \left(\sqrt{3}AB-2D\sqrt{A^{2}+2AC+4C^{2 }}\right)}{\sqrt{3}AB\left(\sqrt{5.5A^{2}+11AC+4C^{2}}-2\sqrt{A^{2}+2AC+4C^{2}}\right)},

f_{2}(x)=\sqrt{\frac{A\left(A^{2}+8Cx+4C^{2}\right)}{2(A-2C)x^{2}+\left(A^{2}+8AC-4C^{2}\right)x+2AC^{2}+A^{2}C+A^{3}}},

f(x)=\pm f_{1}(x) \cdot \left\{1-E \cdot \left[1-f_{2}(x)\right]\right\}.

Here, f(x) is the Narushin-Romanov-Griffin equation, which is used to predict the y coordinates at the given x coordinates; A represents the egg's length; B represents the egg's maximum breadth; C is a parameter to be estimated, and it can be expressed as \left(A-B\right)/(2q), where q is a parameter to be estimated; D represents the egg's breadth associated with (3/4)L from the egg base (to the egg tip) on the egg length axis (which can be regarded as the major axis of the egg shape).

Value

The y values predicted by the Narushin-Romanov-Griffin equation.

Note

Here, parameter C is a parameter to be estimated, which can be directly calculated numerically based on the egg-shape data.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

Shi, P., Gielis, J., Niklas, K.J. (2022) 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. doi:10.1111/nyas.14799

Tian, F., Wang, Y., Sandhu, H.S., Gielis, J., Shi, P. (2020) Comparison of seed morphology of two ginkgo cultivars. Journal of Forestry Research 31, 751-758. doi:10.1007/s11676-018-0770-y

Examples


P0 <- c(11.5, 7.8, 1.1, 5.6)
x  <- seq(-11.5/2, 11.5/2, len=2000)
y1 <- NRGE(P=P0, x=x)
y2 <- -NRGE(P=P0, x=x)

dev.new()
plot(x, y1, cex.lab=1.5, cex.axis=1.5, type="l", 
     col=4, ylim=c(-4, 4), asp=1, 
     xlab=expression(italic(x)), ylab=expression(italic(y)) )
lines(x, y2, col=2)

graphics.off()

Leaf Boundary Data of Seven Species of Neocinnamomum

Description

The data consist of the leaf boundary data of seven species of Neocinnamomum.

Usage

data(Neocinnamomum)

Details

In the data set, there are four columns of variables: Code, LatinName, x, and y. Code saves the codes of individual leaves; LatinName saves the Latin names of the seven species of Neocinnamomum; x saves the x coordinates of the leaf boundary in the Cartesian coordinate system (cm); and y saves the y coordinates of the leaf boundary in the Cartesian coordinate system (cm).

References

Shi, P., Yu, K., Niklas, K.J., Schrader, J., Song, Y., Zhu, R., Li, Y., Wei, H., Ratkowsky, D.A. (2021) A general model for describing the ovate leaf shape. Symmetry, 13, 1524. doi:10.3390/sym13081524

Examples

data(Neocinnamomum)

uni.C <- sort( unique(Neocinnamomum$Code) )
ind   <- 2
Data  <- Neocinnamomum[Neocinnamomum$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y
length(x0)

Res1  <- adjdata(x0, y0, ub.np=200, len.pro=1/20)
x1    <- Res1$x
y1    <- Res1$y
length(x1)

dev.new()
plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) ) 

graphics.off()

Calculation of the Abscissa, Ordinate and Distance From the Origin For an Arbitrary Point on the Preston Curve

Description

PE is used to calculate the abscissa, ordinate and distance from the origin for an arbitrary point on the Preston curve that was generated by the original Preston equation or one of its simplified versions at a given angle.

Usage

PE(P, zeta, simpver = NULL)

Arguments

P

the parameters of the original Preston equation or one of its simplified versions.

zeta

the angle(s) used in the Preston equation.

simpver

an optional argument to use the simplified version of the original Preston equation.

Details

When simpver = NULL, the original Preston equation is selected:

y = a\ \mathrm{sin}\,\zeta,

x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta+c_{2}\,\mathrm{sin}^{2}\,\zeta+c_{3}\,\mathrm{sin}^{3}\,\zeta\right),

r = \sqrt{x^{2}+y^{2}},

where x and y represent the abscissa and ordinate of an arbitrary point on the Preston curve corresponding to an angle \zeta; r represents the distance of the point from the origin; a, b, c_{1}, c_{2}, and c_{3} are parameters to be estimated.

\quad When simpver = 1, the simplified version 1 is selected:

y = a\ \mathrm{sin}\,\zeta,

x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta+c_{2}\,\mathrm{sin}^{2}\,\zeta\right),

r = \sqrt{x^{2}+y^{2}},

\quad When simpver = 2, the simplified version 2 is selected:

y = a\ \mathrm{sin}\,\zeta,

x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta\right),

r = \sqrt{x^{2}+y^{2}},

\quad When simpver = 3, the simplified version 3 is selected:

y = a\ \mathrm{sin}\,\zeta,

x = b\ \mathrm{cos}\,\zeta\left(1+c_{2}\,\mathrm{sin}^{2}\,\zeta\right),

r = \sqrt{x^{2}+y^{2}},

Value

x

the abscissa(s) of the Preston curve corresponding to the given angle(s).

y

the ordinate(s) of the Preston curve corresponding to the given angle(s).

r

the distance(s) of the Preston curve corresponding to the given angle(s) from the origin.

Note

\zeta is NOT the polar angle corresponding to r, i.e.,

y \neq r\,\mathrm{sin}\,\zeta,

x \neq r\,\mathrm{cos}\,\zeta.

Let \varphi be the polar angle corresponding to r. We have:

\zeta = \mathrm{arc\,sin}\frac{ r\ \mathrm{sin}\,\varphi }{a}.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston's universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Preston, F.W. (1953) The shapes of birds' eggs. The Auk 70, 160-182.

Todd, P.H., Smart, I.H.M. (1984) The shape of birds' eggs. Journal of Theoretical Biology 106, 239-243. doi:10.1016/0022-5193(84)90021-3

Examples

  zeta <- seq(0, 2*pi, len=2000)
  Par1 <- c(10, 6, 0.325, -0.0415)
  Res1 <- PE(P=Par1, zeta=zeta, simpver=1)
  Par2 <- c(10, 6, -0.325, -0.0415)
  Res2 <- PE(P=Par2, zeta=zeta, simpver=1)

  dev.new()
  plot(Res1$x, Res1$y, asp=1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(Res2$x, Res2$y, col=2)

  dev.new()
  plot(Res1$r, Res2$r, asp=1, cex.lab=1.5, cex.axis=1.5,
       xlab=expression(paste(italic(r), ""[1], sep="")), 
       ylab=expression(paste(italic(r), ""[2], sep="")))
  abline(0, 1, col=4)

  graphics.off()

Sarabia-Castillo-Slottje Equation (SCSE)

Description

SCSE is used to calculate y values at given x values using the Sarabia-Castillo-Slottje equation. The equation describes the y coordinates of the Lorenz curve.

Usage

SCSE(P, x)

Arguments

P

the parameters of the Sarabia-Castillo-Slottje equation.

x

the given x values ranging between 0 and 1.

Details

y = x^{\gamma}\left[1-\left(1-x\right)^{\alpha}\right]^{\beta}.

Here, x and y represent the independent and dependent variables, respectively; and \gamma, \alpha and \beta are constants to be estimated, where \gamma \ge 0, 0 < \alpha \le 1, and \beta \ge 1. There are three elements in P, representing the values of \gamma, \alpha and \beta, respectively.

Value

The y values predicted by the Sarabia-Castillo-Slottje equation.

Note

The numerical range of x should range between 0 and 1.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Sarabia, J.-M., Castillo, E., Slottje, D.J. (1999) An ordered family of Lorenz curves. Journal of Econometrics. 91, 43-60. doi:10.1016/S0304-4076(98)00048-7

Sitthiyot, T., Holasut, K. (2023) A universal model for the Lorenz curve with novel applications for datasets containing zeros and/or exhibiting extreme inequality. Scientific Reports 13, 4729. doi:10.1038/s41598-023-31827-x

Examples

X1  <- seq(0, 1, len=2000)
Pa2 <- c(0, 0.790, 1.343)
Y2  <- SCSE(P=Pa2, x=X1)

dev.new()
plot( X1, Y2, cex.lab=1.5, cex.axis=1.5, type="l", asp=1, xaxs="i", 
      yaxs="i", xlim=c(0, 1), ylim=c(0, 1), 
      xlab="Cumulative proportion of the number of infructescences", 
      ylab="Cumulative proportion of the infructescence length" )

graphics.off()

Sitthiyot-Holasut Equation

Description

SHE is used to calculate y values at given x values using the Sitthiyot-Holasut equation. The equation describes the y coordinates of the Lorenz curve.

Usage

SHE(P, x)

Arguments

P

the parameters of the Sitthiyot-Holasut equation.

x

the given x values ranging between 0 and 1.

Details

\mbox{if } x > \delta,

y = \left(1-\rho\right)\,\left[\left(\frac{2}{P+1}\right)\left(\frac{x-\delta}{1-\delta}\right)\right] + \rho\,\left[\left(1-\omega\right)\left(\frac{x-\delta}{1-\delta}\right)^{P}+\omega\,\left\{1-\left[1-\left(\frac{x-\delta}{1-\delta}\right)\right]^{\frac{1}{P}}\right\}\right];

\mbox{if } x \le \delta,

y = 0.

Here, x and y represent the independent and dependent variables, respectively; and \delta, \rho, \omega and P are constants to be estimated, where 0 \le \delta < 1, 0 \le \rho \le 1, 0 \le \omega \le 1, and P \ge 1. There are four elements in P, representing the values of \delta, \rho, \omega and P, respectively.

Value

The y values predicted by the Sitthiyot-Holasut equation.

Note

The numerical range of x should range between 0 and 1. When x < \delta, the x value is assigned to be \delta.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Examples

X1  <- seq(0, 1, len=2000)
Pa3 <- c(0, 1, 0.446, 1.739)
Y3  <- SHE(P=Pa3, x=X1)

dev.new()
plot( X1, Y3, cex.lab=1.5, cex.axis=1.5, type="l", asp=1, xaxs="i", 
      yaxs="i", xlim=c(0, 1), ylim=c(0, 1), 
      xlab="Cumulative proportion of the number of infructescences", 
      ylab="Cumulative proportion of the infructescence length" )

graphics.off()

Sarabia Equation

Description

SarabiaE is used to calculate y values at given x values using the Sarabia equation. The equation describes the y coordinates of the Lorenz curve.

Usage

SarabiaE(P, x)

Arguments

P

the parameters of the Sarabia equation.

x

the given x values ranging between 0 and 1.

Details

y = \left(1-\lambda+\eta\right)x+\lambda x^{a_1 + 1}-\eta \left[1-\left(1-x\right)^{a_2 + 1}\right].

Here, x and y represent the independent and dependent variables, respectively; and \lambda, \eta, a_1 and a_2 are constants to be estimated, where a_1 \ge 0, a_2 + 1 \ge 0, \eta\,a_2 + \lambda \le 1, \lambda \ge 0, and \eta\,a_2 \ge 0. There are four elements in P, representing the values of \lambda, \eta, a_1 and a_2, respectively.

Value

The y values predicted by the Sarabia equation.

Note

The numerical range of x should range between 0 and 1.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Sarabia, J.-M. (1997) A hierarchy of Lorenz curves based on the generalized Tukey's lambda distribution. Econometric Reviews 16, 305-320. doi:10.1080/07474939708800389

Examples

X1  <- seq(0, 1, len=2000)
Pa1 <- c(0.295, 101.485, 0.705, 0.003762)
Y1  <- SarabiaE(P=Pa1, x=X1)

dev.new()
plot( X1, Y1, cex.lab=1.5, cex.axis=1.5, type="l", asp=1, xaxs="i", 
      yaxs="i", xlim=c(0, 1), ylim=c(0, 1), 
      xlab="Cumulative proportion of the number of infructescences", 
      ylab="Cumulative proportion of the infructescence length" )

graphics.off()

Calculation of the Surface Area of An Egg Based on the Explicit Preston Equation

Description

SurfaceAreaEPE is used to calculate the surface area of an egg that follows the explicit Preston equation.

Usage

SurfaceAreaEPE(P, simpver = NULL, subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
          stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

P

the parameters of the explicit Preston equation or one of its simplified versions.

simpver

an optional argument to use the simplified version of the explicit Preston equation.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The formula of the surface area (S) of an egg based on the explicit Preston equation or one of its simplified versions is:

S(x)=2\,\pi\int_{-a}^{a}y\,\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx,

where y denotes the explicit Preston equation (i.e., EPE), and a denotes half the egg's length. When simpver = NULL, P has five parameters: a, b, c_{1}, c_{2}, and c_{3}; when simpver = 1, P has four parameters: a, b, c_{1}, and c_{2}; when simpver = 2, P has three parameters: a, b, and c_{1}; when simpver = 3, P has three parameters: a, b, and c_{2}.

Note

The argument P in the SurfaceAreaEPE function has the same parameters, as those in the EPE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Examples

  Par4 <- c(4.27, 2.90, 0.0868, 0.0224, -0.0287)
  SurfaceAreaEPE(P = Par4, simpver = NULL)

Calculation of the Surface Area of An Egg Based on the Explicit Troscianko Equation

Description

SurfaceAreaETE is used to calculate the surface area of an egg that follows the explicit Troscianko equation.

Usage

SurfaceAreaETE(P, subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
          stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

P

the parameters of the explicit Troscianko equation.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The formula of the surface area (S) of an egg based on the explicit Troscianko equation is:

S(x)=2\,\pi\int_{-a}^{a}y\,\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx,

where y denotes the explicit Troscianko equation (i.e., ETE), and a denotes half the egg's length.

Note

The argument P in the SurfaceAreaETE function has the same parameters, as those in the ETE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Examples

  Par5 <- c(2.25, -0.38, -0.29, -0.16)
  SurfaceAreaETE(P = Par5)

Calculation of the Surface Area of An Egg Based on the Narushin-Romanov-Griffin Equation

Description

SurfaceAreaNRGE is used to calculate the surface area of an egg that follows the Narushin-Romanov-Griffin equation.

Usage

SurfaceAreaNRGE(P, subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
          stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

P

the parameters of the Narushin-Romanov-Griffin equation.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The formula of the surface area (S) of an egg based on the Narushin-Romanov-Griffin equation is:

S(x)=2\,\pi\int_{-A/2}^{A/2}y\,\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx,

where y denotes the Narushin-Romanov-Griffin equation (i.e., NRGE), and A denotes the egg's length, which is the first element in the parameter vector, P.

Note

The argument P in the SurfaceAreaNRGE function has the same parameters, as those in the NRGE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

Examples

  Par6 <- c(4.51, 3.18, 0.1227, 2.2284)
  SurfaceAreaNRGE(P = Par6)

Calculation of the Surface Area of An Egg Based on the Simplified Gielis Equation

Description

SurfaceAreaSGE is used to calculate the surface area of an egg that follows the simplified Gielis equation.

Usage

SurfaceAreaSGE(P, subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
          stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

P

the parameters of the simplified Gielis equation, including a, n_{1}, and n_{2}.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The formula of the surface area (S) of an egg based on the simplified Gielis equation is:

S(\varphi)=2\,\pi\int_{0}^{\pi}\sin{\left(\varphi\right)}\ r\,\sqrt{r^2+\left(\frac{dr}{d\varphi}\right)^2}\,d\varphi,

where the polar raidus (r) is the function of the polar angle (\varphi):

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}\right)^{-\frac{1}{n_{1}}},

namely the simplified Gielis equation (i.e., GE) with arguments simpver = 1 and m = 1.

Note

The argument P in the SurfaceAreaSGE function only has the three parameters: a, n_{1}, and n_{2}.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Chen, Z. (2012) Volume and area of revolution under polar coordinate system. Studies in College Mathematics 15(6), 9-11.

Examples

  Par7 <- c(1.124, 14.86, 49.43)
  SurfaceAreaSGE(P = Par7)

The Troscianko Equation (TE)

Description

TE is used to calculate y values at given x values using the re-expression of Troscianko's egg-shape equation, which was proposed by Biggins et al. (2018, 2022).

Usage

TE(P, x)

Arguments

P

the parameters of the Troscianko equation, including \alpha_{0}, \alpha_{1}, and \alpha_{2}.

x

the given x values ranging from -1 to 1.

Details

The Troscianko equation is recommended as (Biggins et al., 2022):

y = \exp\left(\alpha_{0}+\alpha_{1}\,x+\alpha_{2}\,x^2\right)\sqrt{1-x^2},

where x and y represent the abscissa and ordinate of an arbitrary point on the Troscianko curve; \alpha_{0}, \alpha_{1}, and \alpha_{2} are parameters to be estimated.

Value

The y values predicted by the Troscianko equation.

Note

Here, x and y in the Troscianko equation are actually equal to y/a and x/a, respectively, in the explicit Troscianko equation, where a represents half the egg length (See ETE for details). This means that the egg length is scaled to be 2, and the maximum egg width is correspondingly adjusted to keep the same scale.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston's universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Troscianko, J. (2014). A simple tool for calculating egg shape, volume and surface area from digital images. Ibis, 156, 874-878. doi:10.1111/ibi.12177

Examples

  Par <- c(-0.377, -0.29, -0.16)
  xb1 <- seq(-1, 1, len=20000)
  yb1 <- TE(P=Par, x=xb1)
  xb2 <- seq(1, -1, len=20000)
  yb2 <- -TE(P=Par, x=xb2)

  dev.new()
  plot(xb1, yb1, asp=1, type="l", col=2, ylim=c(-1, 1), cex.lab=1.5, cex.axis=1.5, 
    xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(xb2, yb2, col=4)

  graphics.off()

Calculation of the Polar Radius of the Twin Gielis Curve

Description

TGE is used to calculate the polar radii of the twin Gielis equation or one of its simplified versions at given polar angles.

Usage

TGE(P, phi, m = 1, simpver = NULL, nval = 1)

Arguments

P

the parameters of the twin Gielis equation or one of its simplified versions.

phi

the polar angle(s).

m

the given m value that determines the number of angles of the twin Gielis curve within [0, 2\pi).

simpver

an optional argument to use the simplified version of the twin Gielis equation.

nval

the specified value for n_{2} or n_{3} in the simplified versions.

Details

The general form of the twin Gielis equation can be represented as follows:

r\left(\varphi\right) = \mathrm{exp}\left\{\frac{1}{\alpha+\beta\,\mathrm{ln}\left[r_{e}\left(\varphi\right)\right]}+\gamma\right\},

where r represents the polar radius of the twin Gielis curve at the polar angle \varphi, and r_{e} represents the elementary polar radius at the polar angle \varphi. There is a hyperbolic link function to link their log-transformations, i.e.,

\mathrm{ln}\left[r\left(\varphi\right)\right] = \frac{1}{\alpha+\beta\,\mathrm{ln}\left[r_{e}\left(\varphi\right)\right]}+\gamma.

The first three elements of P are \alpha, \beta, and \gamma, and the remaining element(s) of P are the parameters of the elementary polar function, i.e., r_{e}\left(\varphi\right). See Shi et al. (2020) for details.

\quad When simpver = NULL, the original twin Gielis equation is selected:

r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{3}},

where r_{e} represents the elementary polar radius at the polar angle \varphi; m determines the number of angles of the twin Gielis curve within [0, 2\pi); and k, n_{2}, and n_{3} are the fourth to the sixth elements in P. In total, there are six elements in P.

\quad When simpver = 1, the simplified version 1 is selected:

r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}},

where n_{2} is the fourth element in P. There are four elements in total in P.

\quad When simpver = 2, the simplified version 2 is selected:

r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}},

where n_{2} should be specified in nval, and P only includes three elements, i.e., \alpha, \beta, and \gamma.

\quad When simpver = 3, the simplified version 3 is selected:

r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{3}},

where n_{2} and n_{3} are the fourth and fifth elements in P. There are five elements in total in P.

\quad When simpver = 4, the simplified version 4 is selected:

r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}},

where k and n_{2} are the fourth and fifth elelments in P. There are five elements in total in P.

\quad When simpver = 5, the simplified version 5 is selected:

r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}},

where k is the fourth elelment in P. There are four elements in total in P. n_{2} should be specified in nval.

Value

The polar radii predicted by the twin Gielis equation or one of its simplified versions.

Note

simpver here is different from that in the GE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. doi:10.3390/sym12040645

Examples

TGE.par    <- c(2.88, 0.65, 1.16, 139)
varphi.vec <- seq(0, 2*pi, len=2000)
r2.theor   <- TGE(P=TGE.par, phi=varphi.vec, simpver=1, m=5)

dev.new()
plot( varphi.vec, r2.theor, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic(varphi)), ylab=expression(italic("r")),
      type="l", col=4 ) 

starfish4 <- curveGE(TGE, P=c(0, 0, 0, TGE.par), simpver=1, m=5, fig.opt=TRUE)

graphics.off()

The Todd-Smart Equation (TSE)

Description

TSE is used to calculate y values at given x values using the Todd and Smart's re-expression of Preston's universal egg shape.

Usage

TSE(P, x, simpver = NULL)

Arguments

P

the parameters of the original Todd-Smart equation or one of its simplified versions.

x

the given x values ranging from -1 to 1.

simpver

an optional argument to use the simplified version of the original Todd-Smart equation.

Details

When simpver = NULL, the original Preston equation is selected:

y = d_{0}z_{0} + d_{1}z_{1} + d_{2}z_{2} + d_{3}z_{3},

where

z_{0}=\sqrt{1-x^2},

z_{1}=x\sqrt{1-x^2},

z_{2}=x^{2}\sqrt{1-x^2},

z_{3}=x^{3}\sqrt{1-x^2}.

Here, x and y represent the abscissa and ordinate of an arbitrary point on the Todd-Smart curve; d_{0}, d_{1}, d_{2}, and d_{3} are parameters to be estimated.

\quad When simpver = 1, the simplified version 1 is selected:

y = d_{0}z_{0} + d_{1}z_{1} + d_{2}z_{2},

where x and y represent the abscissa and ordinate of an arbitrary point on the Todd-Smart curve; d_{0}, d_{1}, and d_{2} are parameters to be estimated.

\quad When simpver = 2, the simplified version 2 is selected:

y = d_{0}z_{0} + d_{1}z_{1},

where x and y represent the abscissa and ordinate of an arbitrary point on the Todd-Smart curve; d_{0}, and d_{1} are parameters to be estimated.

\quad When simpver = 3, the simplified version 3 is selected:

y = d_{0}z_{0} + d_{2}z_{2},

where x and y represent the abscissa and ordinate of an arbitrary point on the Todd-Smart curve; d_{0}, and d_{2} are parameters to be estimated.

Value

The y values predicted by the Todd-Smart equation.

Note

Here, x and y in the Todd-Smart equation are actually equal to y/a and x/a, respectively, in the Preston equation (See PE for details). Since a represents half the egg length, this means that the egg length is fixed to be 2, and the maximum egg width is correspondingly adjusted to keep the same scale.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston's universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Preston, F.W. (1953) The shapes of birds' eggs. The Auk 70, 160-182.

Todd, P.H., Smart, I.H.M. (1984) The shape of birds' eggs. Journal of Theoretical Biology 106, 239-243. doi:10.1016/0022-5193(84)90021-3

Examples

  Par <- c(0.695320398, -0.210538656, -0.070373518, 0.116839895)
  xb1 <- seq(-1, 1, len=20000)
  yb1 <- TSE(P=Par, x=xb1)
  xb2 <- seq(1, -1, len=20000)
  yb2 <- -TSE(P=Par, x=xb2)

  dev.new()
  plot(xb1, yb1, asp=1, type="l", col=2, ylim=c(-1, 1), cex.lab=1.5, cex.axis=1.5, 
    xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(xb2, yb2, col=4)

  graphics.off()

Calculation of the Volume of An Egg Based on the Explicit Preston Equation

Description

VolumeEPE is used to calculate the volume of an egg that follows the explicit Preston equation.

Usage

VolumeEPE(P, simpver = NULL)

Arguments

P

the parameters of the explicit Preston equation or one of its simplified versions.

simpver

an optional argument to use the simplified version of the explicit Preston equation.

Details

When simpver = NULL, the volume formula (V) of the explicit Preston equation is selected:

V(x) = \frac{4\,\pi}{315}a\,b^{2}\left(105+21\,c_{1}^{2}+42\,c_{2}+9\,c_{2}^2+18\,c_{1}\,c_{3}+5\,c_{3}^2\right),

where P has five parameters: a, b, c_{1}, c_{2}, and c_{3}.

\quad When simpver = 1, the volume formula of the simplified version 1 is selected:

V(x) = \frac{4\,\pi}{315}a\,b^{2}\left(105+21\,c_{1}^{2}+42\,c_{2}+9\,c_{2}^2\right),

where P has four parameters: a, b, c_{1}, and c_{2}.

\quad When simpver = 2, the volume formula of the simplified version 2 is selected:

V(x) = \frac{4\,\pi}{315}a\,b^{2}\left(105+21\,c_{1}^{2}\right),

where P has three parameters: a, b, and c_{1}.

\quad When simpver = 3, the volume formula of the simplified version 3 is selected:

V(x) = \frac{4\,\pi}{315}a\,b^{2}\left(105+42\,c_{2}+9\,c_{2}^2\right),

where P has three parameters: a, b, and c_{2}.

Note

The argument P in the VolumeEPE function has the same parameters, as those in the EPE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Examples

  Par3 <- c(4.27, 2.90, 0.0868, 0.0224, -0.0287)
  VolumeEPE(P=Par3, simpver=NULL)

  # Test the case when simpver = NULL
  a    <- Par3[1]
  b    <- Par3[2]
  c1   <- Par3[3]
  c2   <- Par3[4]
  c3   <- Par3[5]
  pi*4/315*a*b^2*(105+21*c1^2+42*c2+9*c2^2+18*c1*c3+5*c3^2)

  myfun <- function(x){
    pi*EPE(P=Par3, x=x, simpver=NULL)^2
  }
  integrate(myfun, -4.27, 4.27)$value

Calculation of the Volume of An Egg Based on the Explicit Troscianko Equation

Description

VolumeETE is used to calculate the volume of an egg that follows the explicit Troscianko equation.

Usage

VolumeETE(P, subdivisions = 100L,
         rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
         stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

P

the parameters of the explicit Troscianko equation.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The formula of the volume (V) of an egg based on the explicit Troscianko equation is:

V(x)=\pi\int_{-a}^{a}y^2\,dx,

where y denotes the explicit Troscianko equation (i.e., ETE), and a denotes half the egg's length.

Note

The argument P in the VolumeETE function has the same parameters, as those in the ETE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Examples

  Par5 <- c(2.25, -0.38, -0.29, -0.16)
  VolumeETE(P=Par5)

  myfun <- function(x){
    pi*ETE(P=Par5, x=x)^2
  }
  integrate(myfun, -2.25, 2.25)$value

Calculation of the Volume of An Egg Based on the Narushin-Romanov-Griffin Equation

Description

VolumeNRGE is used to calculate the volume of an egg that follows the Narushin-Romanov-Griffin equation.

Usage

VolumeNRGE(P, subdivisions = 100L,
         rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
         stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

P

the parameters of the Narushin-Romanov-Griffin equation.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The formula of the volume (V) of an egg based on the Narushin-Romanov-Griffin equation is:

V(x)=\pi\int_{-A/2}^{A/2}y^2\,dx,

where y denotes the Narushin-Romanov-Griffin equation (i.e., NRGE), and A denotes the egg's length, which is the first element in the parameter vector, P.

Note

The argument P in the VolumeNRGE function has the same parameters, as those in the NRGE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

Examples

  Par6 <- c(4.51, 3.18, 0.1227, 2.2284)
  VolumeNRGE(P=Par6)

  myfun <- function(x){
    pi*NRGE(P=Par6, x=x)^2
  }
  integrate(myfun, -4.51/2, 4.51/2)$value

Calculation of the Volume of An Egg Based on the Simplified Gielis Equation

Description

VolumeSGE is used to calculate the volume of an egg that follows the simplified Gielis equation.

Usage

VolumeSGE(P, subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
          stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

P

the parameters of the simplified Gielis equation, including a, n_{1}, and n_{2}.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The formula of the volume (V) of an egg based on the simplified Gielis equation is:

V\left(\varphi\right)=\frac{2}{3}\,\pi\int_{0}^{\pi}\sin{\left(\varphi\right)}\ r^3\left(\varphi\right)\,d\varphi,

where the polar raidus (r) is the function of the polar angle (\varphi):

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}\right)^{-\frac{1}{n_{1}}},

namely the simplified Gielis equation (i.e., GE) with arguments simpver = 1 and m = 1.

Note

The argument P in the VolumeSGE function only has the three parameters: a, n_{1}, and n_{2}.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Chen, Z. (2012) Volume and area of revolution under polar coordinate system. Studies in College Mathematics 15(6), 9-11.

Examples

  Par7 <- c(1.124, 14.86, 49.43)
  VolumeSGE(P = Par7)

Boundary Data Adjustment of A Polygon

Description

adjdata adjusts the data points in counterclockwise order based on the shortest distance method.

Usage

adjdata(x, y, ub.np = 2000, times = 1.2, len.pro = 1/20, index.sp = 1)

Arguments

x

the x coordinates of points on a polygon's boundary.

y

the y coordinates of points on a polygon's boundary.

ub.np

the upper bound of the number of points eventually retained on the polygon's boundary.

times

the number of times un.np is initially retained as the number of points.

len.pro

the proportion of the distance between any two points to the maximum distance between the points on the polygon's boundary, which is used to determine whether the second point needs to be deleted.

index.sp

the index of the starting point of a group of indices that regularly divide the number of points on the polygon's boundary into ub.np parts.

Details

When ub.np > length(x), length(x) points on the polygon's boundary are retained. The quantile function in package stats is used to carry out the regular division of data points. From the starting point, the second point is the one that has the shortest distance from the former. When the distance between the two points is larger than len.pro multiplied by the maximum distance between points on the polygon's boundary, the second point is deleted from the coordinates. Then, the third point that has the shortest distance from the first point is defined as the second point. If the distance between the first point and the second point is no more than len.pro multiplied by the maximum distance, the first and second points are recorded in a new matrix for the coordinates of the polygon, and the second point is defined as the first point in the old matrix for the coordinates of the polygon. The shortest distance method is then used to look for a third point that meets the requirement.

Value

x

the x coordinates of points eventually retained on the polygon's boundary.

y

the y coordinates of points eventually retained on the polygon's boundary.

Note

The initial boundary data of a polygon can be obtained by running the M-file based on Matlab (version >= 2009a) developed by Shi et al. (2018) and Su et al. (2019) for a .bmp black and white image of the polygon. See references below.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Shi, P., Ratkowsky, D.A., Li, Y., Zhang, L., Lin, S., Gielis, J. (2018) General leaf-area geometric formula exists for plants - Evidence from the simplified Gielis equation. Forests 9, 714. doi:10.3390/f9110714

Su, J., Niklas, K.J., Huang, W., Yu, X., Yang, Y., Shi, P. (2019) Lamina shape does not correlate with lamina surface area: An analysis based on the simplified Gielis equation. Global Ecology and Conservation 19, e00666. doi:10.1016/j.gecco.2019.e00666

Examples

data(eggs)
uni.C1 <- sort( unique(eggs$Code) )
ind1   <- 2
Data1  <- eggs[eggs$Code==uni.C1[ind1], ]
x0     <- Data1$x
y0     <- Data1$y

Res1   <- adjdata(x0, y0, ub.np=2000, times=1.2, len.pro=1/20)
x1     <- Res1$x
y1     <- Res1$y

dev.new()
plot( x1, y1, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")),
      pch=1, col=1 ) 

Res2   <- adjdata(x0, y0, ub.np=40, times=1, len.pro=1/2, index.sp=20)
x2     <- Res2$x
y2     <- Res2$y

Res3   <- adjdata(x0, y0, ub.np=100, times=1, len.pro=1/2, index.sp=100)
x3     <- Res3$x
y3     <- Res3$y

dev.new()
plot( x2, y2, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")),
      pch=1, col=4 ) 
points( x3, y3, col=2)


data(starfish)

uni.C2 <- sort( unique(starfish$Code) )
ind2   <- 2
Data2  <- starfish[starfish$Code==uni.C2[ind2], ]
x4     <- Data2$x
y4     <- Data2$y

dev.new()
plot( x4, y4, asp=1, type="l", cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) )

Res4 <- adjdata(x4, y4, ub.np=500, times=1.2, len.pro=1/20)
x5   <- Res4$x
y5   <- Res4$y

dev.new()
plot( x5, y5, asp=1, type="l", cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) )

graphics.off()

Area Calculation for the Gielis Curve Within `[0, 2\pi)`

Description

areaGE is used to calculate the area of the polygon generated by the Gielis curve within [0, 2\pi).

Usage

areaGE(expr, P, m = 1, simpver = NULL,  
       nval = 1, subdivisions = 100L,
       rel.tol = .Machine$double.eps^0.25, 
       abs.tol = rel.tol, stop.on.error = TRUE, 
       keep.xy = FALSE, aux = NULL)

Arguments

expr

the original (or twin) Gielis equation or one of its simplified versions.

P

the parameters of the original (or twin) Gielis equation or one of its simplified versions.

m

the given m value that determines the number of angles of the Gielis curve within [0, 2\pi).

simpver

an optional argument to use the simplified version of the original (or twin) Gielis equation.

nval

the specified value for n_{1} or n_{2} or n_{3} in the simplified versions.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The arguments of P, m, simpver, and nval should correspond to expr (i.e., GE or TGE). Please note the differences in the simplified version number and the number of parameters between GE and TGE.

Value

The area of the polygon within [0, 2\pi) generated by the original (or twin) Gielis equation or one of its simplified versions.

Note

simpver in GE is different from that in TGE.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

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:10.3732/ajb.90.3.333

Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. doi:10.3390/sym12040645

Examples

Para1 <- c(1.7170, 5.2258, 7.9802)
areaGE(GE, P = Para1, m=5, simpver=1)

Para2 <- c(2.1066, 3.5449, 0.4619, 10.5697)
areaGE(TGE, P = Para2, m=5, simpver=1)

Area Calculation for an Ovate Polygon

Description

areaovate is used to calculate the area of an ovate polygon made from combing two symmetrical curves generated by a performance equation (e.g., MLRFE).

Usage

areaovate(expr, P, simpver = NULL,  
          subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, 
          abs.tol = rel.tol, stop.on.error = TRUE, 
          keep.xy = FALSE, aux = NULL)

Arguments

expr

a performance equation or one of its simplified versions.

P

the parameters of the performance equation or one of its simplified versions.

simpver

an optional argument to use the simplfied version of the performance equation.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

Value

The area of two symmetrical curves along the x-axis generated by a performance equation or one of its simplified versions.

Note

Here, the user can define other performance equations, but new equations or their simplified versions should include the lower and upper thresholds in the x-axis corresponding to y = 0, whose indices should be the same as those in MbetaE or MBriereE or MLRFE.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. doi:10.3390/plants11131769

Shi, P., Yu, K., Niklas, K.J., Schrader, J., Song, Y., Zhu, R., Li, Y., Wei, H., Ratkowsky, D.A. (2021) A general model for describing the ovate leaf shape. Symmetry 13, 1524. doi:10.3390/sym13081524

Examples

Par1 <- c(1.8175, 2.7795, 7.1557, 1.6030)
areaovate(MbetaE, P = Par1, simpver = 1)

Par2 <- c(0.0550, 0.3192, 7.1965, 0.5226)
areaovate(MBriereE, P = Par2, simpver = 1)

Par3 <- c(1.8168, 2.7967, 7.2623, 0.9662)
areaovate(MLRFE, P = Par3, simpver = 1)

Par4 <- c(2.4, 0.96, 0.64, 7.75, 1.76, 3.68)
areaovate(MPerformanceE, P = Par4, simpver = 1)

Leaf Boundary Data of Phyllostachys incarnata T. H. Wen (Poaceae: Bambusoideae)

Description

The data consist of the boundary data of six leaves of P. incarnata sampled at Nanjing Forestry University campus in early December 2016.

Usage

data(bambooleaves)

Details

In the data set, there are three columns of variables: Code, x, and y. Code saves the codes of individual leaves; x saves the x coordinates of the leaf boundary in the Cartesian coordinate system (cm); and y saves the y coordinates of the leaf boundary in the Cartesian coordinate system (cm).

References

Lin, S., Shao, L., Hui, C., Song, Y., Reddy, G.V.P., Gielis, J., Li, F., Ding, Y., Wei, Q., Shi, P. (2018) Why does not the leaf weight-area allometry of bamboos follow the 3/2-power law? Frontiers in Plant Science 9, 583. doi:10.3389/fpls.2018.00583

Examples

data(bambooleaves)

uni.C <- sort( unique(bambooleaves$Code) )
ind   <- 1
Data  <- bambooleaves[bambooleaves$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

dev.new()
plot( x0, y0, asp=1, type="l", cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) )
length(x0)

Res1 <- adjdata(x0, y0, ub.np=600, len.pro=1/20)
dev.new()
plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, type="l", 
      xlab=expression(italic("x")), ylab=expression(italic("y")) )

graphics.off()

Measure of the Extent of Bilateral Symmetry of A Polygon

Description

bilat is used to measure the extent of bilateral (a)symmetry and other measures for a polygon (e.g., a leaf).

Usage

bilat(x, y, strip.num = 200, peri.np = NULL, n.loop = 60,
      auto.search = TRUE, animation.fig = TRUE, time.interval = 0.001,   
      unit = "cm", main = NULL, diff.fig = TRUE, angle = NULL,  
      ratiox = 0.02, ratioy = 0.08, fd.opt = TRUE, frac.fig = TRUE,
      denomi.range = seq(8, 30, by = 1))

Arguments

x

the x coordinates of a polygon's boundary.

y

the y coordinates of a polygon's boundary.

strip.num

the number of equidistant strips intersecting with the polygon that are horizontally placed. See Shi et al. (2018, 2020) for details.

peri.np

the number of data points on the boundary retained for calculating the perimeter of the polygon.

n.loop

the number of data points to randomly sample for calculating the mean perimeter of the polygon.

auto.search

an optional argument to automatically search the maximum distance between two points on the polygon's boundary.

animation.fig

the option of showing the data points on the polygon's boundary in an animation.

time.interval

the time interval at which to suspend execution, in seconds.

unit

the units of the x-axis and the y-axis when showing the polygon.

main

the main title of the figure.

diff.fig

an optional argument to draw the differences in areas between the intersections of the strips with the upper part of the polygon and the intersections of the strips with the lower part of the polygon. The polygon is divided into the upper and lower parts by the x-axis. See Shi et al. (2018, 2020) for details.

angle

the angle between the major axis (i.e., the leaf length axis) and the x-axis, which can be defined by the user.

fd.opt

An optional argument to use the box-counting method to calculate the fractal dimension of the polygon's boundary on a log-log scale.

ratiox

the x coordinate of the location parameter for positioning the legend in the plot of the linear fitting.

ratioy

the y coordinate of the location parameter for positioning the legend in the plot of the linear fitting.

frac.fig

an optional argument to draw the results of the linear fitting using the box-counting method to calculate the fractal dimension of the polygon's boundary on a log-log scale.

denomi.range

the number of equidistant segments of the maximum range between the range of the x coordinates and that of the y coordinates.

Details

The data of x and y should be the coordinates adjusted using the adjdata function. If peri.np = NULL, the number of length(x) is used to calculate the perimeter of the polygon; if peri.np is a positive integer, the number of data points retained on the polygon's boundary is equal to peri.np and random sampling for retaining peri.np data points is carried out n.loop times for calculating the mean perimeter of the polygon. That is to say, the final output for the perimeter is the mean of the n.loop perimeters (i.e., replicates). If the user wants to get a consistent result for the mean perimeter, the set.seed function can be used. In addition, if length(x) < peri.np, peri.np then becomes length(x) rather than the specified value in Arguments. If the polygon apparently has a major axis (e.g., the leaf length axis for an ovate leaf), auto.search is appropriate. If the major axis of the polygon is not the straight line through two points on the polygon's boundary having the maximum distance, the user can define the major axis using the locator function in graphic by clicking two points on or near the polygon's boundary. The location of the first click should be northeast of the location of the second click. This means that the angle between the straight line through the locations of the two clicks and the x-axis should range from 0 to \pi/2. The locations of the clicks can be on the boundary or be approximate to the boundary. The function will automatically find the nearest data point on the boundary to the location of each click. When angle = NULL, the observed polygon will be shown at its initial angle in the scanned image; when angle is a numerical value (e.g., \pi/4) defined by the user, it indicates that the major axis is rotated \pi/4 counterclockwise from the x-axis.

Value

x

the x coordinates retained on the polygon's boundary.

y

the y coordinates retained on the polygon's boundary.

phi

the angle between the length axis (i.e., the major axis) of the polygon and the x-axis.

n1

the number of data points on the upper boundary of the polygon.

n2

the number of data points on the lower boundary of the polygon.

n

the number of data points on the whole polygon's boundary.

total.poly

an object of class "ppp" representing a point pattern dataset in the two-dimensional plane, representing the polygon's boundary.

upper.poly

an object of class "ppp" representing a point pattern dataset in the two-dimensional plane, representing the upper boundary of the polygon along the x-axis.

lower.poly

an object of class "ppp" representing a point pattern dataset in the two-dimensional plane, representing the lower boundary of the polygon along the x-axis.

D

the differences in areas between the upper and lower boundaries of the polygon.

par.upper.area

the area of the upper boundary of the polygon along the x-axis.

par.lower.area

the area of the lower boundary of the polygon along the x-axis.

SI

the standardized index for bilateral (a)symmetry for the polygon.

AR

the ratio of the areas of the upper to the lower parts of the polygon.

scan.length

the length of the polygon. The default is the maximum distance between two points on the polygon's boundary.

scan.width

the maximum width of the polygon.

scan.area

the area of the polygon.

scan.perimeter

the perimeter of the polygon based on all data points or a mean of n.loop replicates of perimeters using the peri.np data points retained on the polygon's boundary.

x.width

distance from the base to a point on the major axis associated with the maximum width of the polygon.

width.1e

the width associated with 1/8 of scan.length (starting from the base of the polygon).

width.2e

the width associated with 2/8 of scan.length (starting from the base of the polygon).

width.4e

the width associated with 4/8 of scan.length (starting from the base of the polygon).

width.6e

the width associated with 6/8 of scan.length (starting from the base of the polygon).

width.7e

the width associated with 7/8 of scan.length (starting from the base of the polygon).

bi.test

the testing results for D using the Wilcoxon signed rank test with continuity correction. See the wilcox.test function in stats.

a

the estimate of the intercept obtained using the box-counting method to calculate the fractal dimension of the polygon's boundary.

sd.a

the standard deviation of the estimated intercept.

lci.a

the lower bound of the 95% confidence interval of the estimated intercept.

uci.a

the upper bound of the 95% confidence interval of the estimated intercept.

b

the estimate of the slope obtained using the box-counting method to calculate the fractal dimension of the polygon's boundary.

sd.b

the standard deviation of the estimated slope.

lci.a

the lower bound of the 95% confidence interval of the estimated slope.

uci.a

the upper bound of the 95% confidence interval of the estimated slope.

r.sq

the coefficient of determination obtained when using the box-counting method to calculate the fractal dimension of the polygon's boundary.

delta

the vector of box sizes used in the box-counting method to calculate the fractal dimension of the polygon's boundary.

N

the number of boxes that include at least one pixel of the polygon's boundary.

Note

The polygon is expected to have an apparent major axis (e.g., the straight line through two points on the polygon's boundary having the maximum distance or one that can be clearly defined to pass by two landmarks on the polygon's boundary [i.e., the leaf length axis, the egg length axis, etc.]). The polygon is placed with its major axis overlapping the x-axis; the base of the polygon is located at the origin; the apex of the polygon is located to the right of the base. phi is equal to angle when angle is not null. In theory, n1 + n2 = n, but in most cases n1 + n2 is slightly smaller than n. The reason is that very few boundary points fall outside the the lower and upper boundaries of the polygon when using the intersect.owin function in spatstat.geom. However, this does not considerably affect the results. The log-transformed SI and the log-transformed AR are demontrated to have a more symmetrical frequency distribution than their original forms. This is important when performing an analysis of variance between (or among) groups to compared their extents of bilateral (a)symmetry. See Shi et al. (2020) for details. The box-counting approach uses a group of boxes (squares for simplicity) with different sizes (\delta) to divide the leaf vein image into different parts. Let N represent the number of boxes that include at least one pixel of the polygon's boundary. The maximum of the range of the x coordinates and the range of the y coordinates for the pixels of the polygon's boundary is defined as z. Let \delta represent the vector of z/denomi.range. We then used the following equation to calculate the fractal dimension of the polygon's boundary:

\mathrm{ln } N = a + b\, \mathrm{ln} \left(\delta^{-1}\right),

where b is the theoretical value of the fractal dimension. We can use its estimate as the numerical value of the fractal dimension for the polygon's boundary.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Shi, P., Niinemets, Ü., Hui, C., Niklas, K.J., Yu, X., Hölscher, D. (2020) Leaf bilateral symmetry and the scaling of the perimeter vs. the surface area in 15 vine species. Forests 11, 246. doi:10.3390/f11020246

Shi, P., Zheng, X., Ratkowsky, D.A., Li, Y., Wang, P., Cheng, L. (2018) A simple method for measuring the bilateral symmetry of leaves. Symmetry 10, 118. doi:10.3390/sym10040118

Examples

data(bambooleaves)

uni.C <- sort( unique(bambooleaves$Code) )
ind   <- 3
Data  <- bambooleaves[bambooleaves$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

dev.new()
plot( x0, y0, asp=1, type="l", cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic(x)), ylab=expression(italic(y)) )

Res1 <- adjdata(x0, y0, ub.np=2000, len.pro=1/20)
x1   <- Res1$x
y1   <- Res1$y



  Res2 <- bilat( x=x1, y=y1, time.interval=0.00045, 
                 peri.np=NULL, auto.search=TRUE, 
                 fd.opt=TRUE )
  Res2$scan.perimeter

  set.seed(123)
  Res3 <- bilat( x=x1, y=y1, time.interval=0.00045, 
                 peri.np=500, n.loop=30,
                 auto.search=TRUE, fd.opt=FALSE )
  Res3$scan.perimeter

  set.seed(123)
  Res4 <- bilat( x=x1, y=y1, time.interval=0.00045, 
                 peri.np=500, n.loop=30,
                 auto.search=TRUE, fd.opt=FALSE, angle=pi/4 )
  Res4$scan.perimeter

  set.seed(123)  
  Res5 <- bilat( x=x1, y=y1, time.interval=0.00045, 
                 peri.np=500, n.loop=30,
                 auto.search=TRUE, fd.opt=FALSE, angle=0 )
  Res5$scan.perimeter
  
  if(interactive()){
    # The angle between the leaf length axis (namely the straight 
    # line through the leaf apex and base) and the horizontal axis 
    # should be between 0 and pi/2 for a scanned leaf's profile. 
    # Here, the user needs to first click the leaf apex, 
    # and then click the leaf base. 
    set.seed(123)  
    Res6 <- bilat( x=x1, y=y1, time.interval=0.00045, 
                 peri.np=500, n.loop=30,
                 auto.search=FALSE, fd.opt=FALSE, angle=NULL )
    Res6$scan.perimeter
  }

  set.seed(NULL)


graphics.off()

Biological Geometries

Description

Is used to simulate and fit biological geometries. 'biogeom' incorporates several novel universal parametric equations that can generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree-rings (Gielis, 2003; Shi et al., 2020), three growth-rate curves representing the ontogenetic growth trajectories of animals and plants against time, and the axially symmetrical and integral forms of all these functions (Shi et al., 2017, 2021). The optimization method proposed by Nelder and Mead (1965) was used to estimate model parameters. 'biogeom' includes several real data sets of the boundary coordinates of natural shapes, including avian eggs, fruit, lanceolate and ovate leaves, tree rings, seeds, and sea stars,and can be potentially applied to other natural shapes. 'biogeom' can quantify the conspecific or interspecific similarity of natural outlines, and provides information with important ecological and evolutionary implications for the growth and form of living organisms. Please see Shi et al. (2022) for details.

Details

The DESCRIPTION file:

Package:	biogeom
Type:	Package
Title:	Biological Geometries
Version:	1.4.3
Date:	2024-03-21
Authors@R:	c(person(given="Peijian", family="Shi", email="pjshi@njfu.edu.cn", role=c("aut", "cre")), person(given=c("Johan"), family="Gielis", email="johan.gielis@uantwerpen.be", role=c("aut")), person(given=c("Brady K."), family="Quinn", email="brady.quinn@dfo-mpo.gc.ca", role=c("aut")))
Author:	Peijian Shi [aut, cre], Johan Gielis [aut], Brady K. Quinn [aut]
Maintainer:	Peijian Shi <pjshi@njfu.edu.cn>
Imports:	spatstat.geom (>= 2.4-0)
Description:	Is used to simulate and fit biological geometries. 'biogeom' incorporates several novel universal parametric equations that can generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree-rings (Gielis (2003) <doi:10.3732/ajb.90.3.333>; Shi et al. (2020) <doi:10.3390/sym12040645>), three growth-rate curves representing the ontogenetic growth trajectories of animals and plants against time, and the axially symmetrical and integral forms of all these functions (Shi et al. (2017) <doi:10.1016/j.ecolmodel.2017.01.012>; Shi et al. (2021) <doi:10.3390/sym13081524>). The optimization method proposed by Nelder and Mead (1965) <doi:10.1093/comjnl/7.4.308> was used to estimate model parameters. 'biogeom' includes several real data sets of the boundary coordinates of natural shapes, including avian eggs, fruit, lanceolate and ovate leaves, tree rings, seeds, and sea stars,and can be potentially applied to other natural shapes. 'biogeom' can quantify the conspecific or interspecific similarity of natural outlines, and provides information with important ecological and evolutionary implications for the growth and form of living organisms. Please see Shi et al. (2022) <doi:10.1111/nyas.14862> for details.
Depends:	R (>= 4.3.0)
License:	GPL (>= 2)
NeedsCompilation:	no

Index of help topics:

DEPE                    Calculation of the First-Order Derivative of
                        the Explicit Preston Equation
DETE                    Calculation of the First-Order Derivative of
                        the Explicit Troscianko Equation
DNRGE                   Calculation of the First-Order Derivative of
                        the Narushin-Romanov-Griffin Equation
DSGE                    Calculation of the First-Order Derivative of
                        the Simplified Gielis Equation
EPE                     Calculation of the Ordinate For an Arbitrary
                        Point on the Preston Curve in the Plane
ETE                     Calculation of the Ordinate For an Arbitrary
                        Point on the Troscianko Curve in the Plane
GE                      Calculation of the Polar Radius of the Gielis
                        Curve
LeafSizeDist            Leaf size distribution of _Shibataea chinensis_
MBriereE                Modified Briere Equation
MLRFE                   Modified Lobry-Rosso-Flandrois (LRF) Equation
MPerformanceE           Modified Performance Equation
MbetaE                  Modified Beta Equation
NRGE                    The Narushin-Romanov-Griffin Equation (NRGE)
Neocinnamomum           Leaf Boundary Data of Seven Species of
                        _Neocinnamomum_
PE                      Calculation of the Abscissa, Ordinate and
                        Distance From the Origin For an Arbitrary Point
                        on the Preston Curve
SCSE                    Sarabia-Castillo-Slottje Equation (SCSE)
SHE                     Sitthiyot-Holasut Equation
SarabiaE                Sarabia Equation
SurfaceAreaEPE          Calculation of the Surface Area of An Egg Based
                        on the Explicit Preston Equation
SurfaceAreaETE          Calculation of the Surface Area of An Egg Based
                        on the Explicit Troscianko Equation
SurfaceAreaNRGE         Calculation of the Surface Area of An Egg Based
                        on the Narushin-Romanov-Griffin Equation
SurfaceAreaSGE          Calculation of the Surface Area of An Egg Based
                        on the Simplified Gielis Equation
TE                      The Troscianko Equation (TE)
TGE                     Calculation of the Polar Radius of the Twin
                        Gielis Curve
TSE                     The Todd-Smart Equation (TSE)
VolumeEPE               Calculation of the Volume of An Egg Based on
                        the Explicit Preston Equation
VolumeETE               Calculation of the Volume of An Egg Based on
                        the Explicit Troscianko Equation
VolumeNRGE              Calculation of the Volume of An Egg Based on
                        the Narushin-Romanov-Griffin Equation
VolumeSGE               Calculation of the Volume of An Egg Based on
                        the Simplified Gielis Equation
adjdata                 Boundary Data Adjustment of A Polygon
areaGE                  Area Calculation for the Gielis Curve Within
                        [0, 2pi)
areaovate               Area Calculation for an Ovate Polygon
bambooleaves            Leaf Boundary Data of _Phyllostachys incarnata_
                        T. H. Wen (Poaceae: Bambusoideae)
bilat                   Measure of the Extent of Bilateral Symmetry of
                        A Polygon
biogeom                 Biological Geometries
curveEPE                Drawing the Preston Curve Produced by the the
                        Explicit Preston Equation
curveETE                Drawing the Troscianko Curve Produced by the
                        Explicit Troscianko Equation
curveGE                 Drawing the Gielis Curve
curveNRGE               Drawing the Egg Shape Predicted by the
                        Narushin-Romanov-Griffin Equation
curveovate              Drawing the Ovate Leaf-Shape Curve
eggs                    Egg Boundary Data of Nine Species of Birds
fitEPE                  Data-Fitting Function for the Explicit Preston
                        Equation
fitETE                  Data-Fitting Function for the Explicit
                        Troscianko Equation
fitGE                   Data-Fitting Function for the Gielis Equation
fitLorenz               Data-Fitting Function for the Rotated and
                        Right-Shifted Lorenz Curve
fitNRGE                 Parameter Estimation for the
                        Narushin-Romanov-Griffin Equation
fitSuper                Data-Fitting Function for the Superellipse
                        Equation
fitovate                Data-Fitting Function for the Ovate Leaf-Shape
                        Equation
fitsigmoid              Data-Fitting Function for the Sigmoid Growth
                        Equation
fracdim                 Calculation of Fractal Dimension of Lef Veins
                        Based on the Box-Counting Method
ginkgoseed              Boundary Data of the Side Projections of
                        _Ginkgo biloba_ Seeds
kp                      Boundary Data of the Vertical Projections of
                        _Koelreuteria paniculata_ Fruit
lmPE                    Parameter Estimation for the Todd-Smart
                        Equation
lmTE                    Parameter Estimation for the Troscianko
                        Equation
shoots                  Height Growth Data of Bamboo Shoots
sigmoid                 Sigmoid Growth Equation
starfish                Boundary Data of Eight Sea Stars
veins                   Leaf Vein Data of _Michelia compressa_
whitespruce             Planar Coordinates of _Picea glauca_ Tree Rings

Note

We are deeply thankful to Cang Hui, Yang Li, Uwe Ligges, Valeriy G. Narushin, Ülo Niinemets, Karl J. Nikas, Honghua Ruan, David A. Ratkowsky, Julian Schrader, Rolf Turner, Lin Wang, and Victoria Wimmer for their valuable help during creating this package. This work was supported by the National Key Research and Development Program of China (Grant No. 2021YFD02200403) and Simon Stevin Institute for Geometry (Antwerpen, Belguim).

Author(s)

Peijian Shi [aut, cre], Johan Gielis [aut], Brady K. Quinn [aut]

Maintainer: Peijian Shi <pjshi@njfu.edu.cn>

References

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:10.3732/ajb.90.3.333

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. doi:10.3390/sym12040645

Drawing the Preston Curve Produced by the the Explicit Preston Equation

Description

curveEPE is used to draw the Preston curve that is produced by the explicit Preston equation.

Usage

curveEPE(P, np = 5000, simpver = NULL,
        fig.opt = FALSE, deform.fun = NULL, Par = NULL,
        xlim = NULL, ylim = NULL, unit = NULL, main="")

Arguments

P

the three location parameters and the parameters of the explicit Preston equation or one of its simplified versions.

np

the number of data points on the Preston curve.

simpver

an optional argument to use the simplfied version of the explicit Preston equation.

fig.opt

an optional argument to draw the Preston curve.

deform.fun

the deformation function used to describe the deviation from a theoretical Preston curve.

Par

the parameter(s) of the deformation function.

xlim

the range of the x-axis over which to plot the Preston curve.

ylim

the range of the y-axis over which to plot the Preston curve.

unit

the units of the x-axis and the y-axis when showing the Preston curve.

main

the main title of the figure.

Details

The first three elements of P are location parameters. The first two are the planar coordinates of the transferred origin, and the third is the angle between the major axis of the curve and the x-axis. Here, the major axis is a straight line through the two ends of an egg's profile (i.e., the mid-line of the egg's profile). The other arguments in P (except these first three location parameters), and simpver should correspond to those of P in EPE. deform.fun should take the form as: deform.fun <- function(Par, z){...}, where z is a two-dimensional matrix related to the x and y values. And the return value of deform.fun should be a list with two variables x and y.

Value

x

the x coordinates of the Preston curve.

y

the y coordinates of the Preston curve.

Note

When the rotation angle is zero (i.e., the third element in P is zero), np data points are distributed counterclockwise on the Preston curve from the rightmost end of the egg's profile to itself.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Preston, F.W. (1953) The shapes of birds' eggs. The Auk 70, 160-182.

Todd, P.H., Smart, I.H.M. (1984) The shape of birds' eggs. Journal of Theoretical Biology 106, 239-243. doi:10.1016/0022-5193(84)90021-3

Examples

  Para1 <- c(0, 0, 0, 10, 6, 0.325, -0.0415)
  curveEPE(P=Para1, simpver=1, fig.opt=TRUE)
  Para2 <- c(0, 0, pi, 10, 6, -0.325, -0.0415)
  curveEPE(P=Para2, simpver=1, fig.opt=TRUE)

  Para3 <- c(0, 0, 0, 10, 6, 0.325, -0.0415, 0.2)
  curveEPE(P=Para3, simpver=NULL, fig.opt=TRUE)
  Para4 <- c(0, 0, pi, 10, 6, -0.325, -0.0415, 0.2)
  curveEPE(P=Para4, simpver=NULL, fig.opt=TRUE)

  Para5 <- c(0, 0, pi/4, 10, 6, 0.325, -0.0415)
  curveEPE(P=Para5, simpver=1, 
          fig.opt=TRUE, main="A rotated egg shape")

  # There is an example that introduces a deformation function in the egg-shape equation
  myfun <- function(Par, z){
    x  <- z[,1]
    y  <- z[,2]
    k1 <- Par[1]
    k2 <- Par[2]
    y  <- y - k1*(y+k2)^2
    list(x=x, y=y)
  }
  deform.op <- curveEPE(P=Para1, np=5000, simpver=1, 
                       fig.opt=TRUE, deform.fun=myfun, Par=c(0.05, 8))


  graphics.off()

Drawing the Troscianko Curve Produced by the Explicit Troscianko Equation

Description

curveETE is used to draw the Troscianko curve that is produced by the explicit Troscianko equation.

Usage

curveETE(P, np = 5000, fig.opt = FALSE, deform.fun = NULL, Par = NULL,
        xlim = NULL, ylim = NULL, unit = NULL, main="")

Arguments

P

the three location parameters and the parameters of the explicit Troscianko equation.

np

the number of data points on the Troscianko curve.

fig.opt

an optional argument to draw the Troscianko curve.

deform.fun

the deformation function used to describe the deviation from a theoretical Troscianko curve.

Par

the parameter(s) of the deformation function.

xlim

the range of the x-axis over which to plot the Troscianko curve.

ylim

the range of the y-axis over which to plot the Troscianko curve.

unit

the units of the x-axis and the y-axis when showing the Troscianko curve.

main

the main title of the figure.

Details

The first three elements of P are location parameters. The first two are the planar coordinates of the transferred origin, and the third is the angle between the major axis of the curve and the x-axis. Here, the major axis is a straight line through the two ends of an egg's profile (i.e., the mid-line of the egg's profile). The other arguments in P (except these first three location parameters) should correspond to those of P in ETE. deform.fun should take the form as: deform.fun <- function(Par, z){...}, where z is a two-dimensional matrix related to the x and y values. And the return value of deform.fun should be a list with two variables x and y.

Value

x

the x coordinates of the Troscianko curve.

y

the y coordinates of the Troscianko curve.

Note

When the rotation angle is zero (i.e., the third element in P is zero), np data points are distributed counterclockwise on the Troscianko curve from the rightmost end of the egg's profile to itself.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston’s universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Examples

  Para1 <- c(0, 0, 0, 2.25, -0.377, -0.29, -0.16)
  curveETE(P=Para1, fig.opt=TRUE)

  # There is an example that introduces a deformation function in the egg-shape equation
  myfun <- function(Par, z){
    x  <- z[,1]
    y  <- z[,2]
    k1 <- Par[1]
    k2 <- Par[2]
    y  <- y - k1*(y+k2)^2
    list(x=x, y=y)
  }
  deform.op <- curveETE(P=Para1, np=5000, fig.opt=TRUE, deform.fun=myfun, Par=c(0.05, 8))

  graphics.off()

Drawing the Gielis Curve

Description

curveGE is used to draw the Gielis curve.

Usage

curveGE(expr, P, phi = seq(0, 2*pi, len = 2000),
        m = 1, simpver = NULL, nval = 1,
        fig.opt = FALSE, deform.fun = NULL, Par = NULL,
        xlim = NULL, ylim = NULL, unit = NULL, main="")

Arguments

expr

the original (or twin) Gielis equation or one of its simplified versions.

P

the three location parameters and the parameters of the original (or twin) Gielis equation or one of its simplified versions.

phi

the given polar angles at which we want to draw the Gielis curve.

m

the given m value that determines the number of angles of the Gielis curve within [0, 2\pi).

simpver

an optional argument to use the simplfied version of the original (or twin) Gielis equation.

nval

the specified value for n_{1} or n_{2} or n_{3} in the simplified versions.

fig.opt

an optional argument to draw the Gielis curve.

deform.fun

the deformation function used to describe the deviation from a theoretical Gielis curve.

Par

the parameter(s) of the deformation function.

xlim

the range of the x-axis over which to plot the Gielis curve.

ylim

the range of the y-axis over which to plot the Gielis curve.

unit

the units of the x-axis and the y-axis when showing the Gielis curve.

main

the main title of the figure.

Details

The first three elements of P are location parameters. The first two are the planar coordinates of the transferred polar point, and the third is the angle between the major axis of the curve and the x-axis. The other arguments in P (except these first three location parameters), m, simpver, and nval should correspond to expr (i.e., GE or TGE). Please note the differences in the simplified version number and the number of parameters between GE and TGE. deform.fun should take the form as: deform.fun <- function(Par, z){...}, where z is a two-dimensional matrix related to the x and y values. And the return value of deform.fun should be a list with two variables x and y.

Value

x

the x coordinates of the Gielis curve corresponding to the given polar angles phi.

y

the y coordinates of the Gielis curve corresponding to the given polar angles phi.

r

the polar radii of the Gielis curve corresponding to the given polar angles phi.

Note

simpver in GE is different from that in TGE.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

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:10.3732/ajb.90.3.333

Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. doi:10.3390/sym12040645

Examples

GE.par  <- c(2, 1, 4, 6, 3)
phi.vec <- seq(0, 2*pi, len=2000)
r.theor <- GE(P=GE.par, phi=phi.vec, m=5)

dev.new()
plot( phi.vec, r.theor, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic(phi)), ylab=expression(italic("r")),
      type="l", col=4 ) 

curve.par <- c(1, 1, pi/4, GE.par)
GE.res    <- curveGE(GE, P=curve.par, fig.opt=TRUE, deform.fun=NULL, Par=NULL, m=5)
# GE.res$r

GE.res    <- curveGE( GE, P=c(0, 0, 0, 2, 4, 20), m=1, simpver=1, fig.opt=TRUE )
# GE.res$r

GE.res    <- curveGE( GE, P=c(1, 1, pi/4, 2, 1, 3), m=5, simpver=1, fig.opt=TRUE )
# GE.res$r

GE.res    <- curveGE( GE, P=c(1, 1, pi/4, 2, 1, 3), m=2, simpver=1, fig.opt=TRUE )
# GE.res$r

GE.res    <- curveGE( GE, P=c(1, 1, pi/4, 2, 0.05), m=1, simpver=2, fig.opt=TRUE )
# GE.res$r

GE.res    <- curveGE( GE, P=c(1, 1, pi/4, 2), m=4, simpver=3, nval=2, fig.opt=TRUE )
# GE.res$r

GE.res    <- curveGE( GE, P=c(1, 1, pi/4, 2, 0.6), m=4, simpver=8, nval=2, fig.opt=TRUE )
# GE.res$r

graphics.off()

Drawing the Egg Shape Predicted by the Narushin-Romanov-Griffin Equation

Description

curveNRGE is used to draw the egg shape predicted by the Narushin-Romanov-Griffin equation.

Usage

curveNRGE(P, np = 5000, fig.opt = FALSE, deform.fun = NULL, Par = NULL, 
    xlim = NULL, ylim = NULL, unit = NULL, main = "")

Arguments

P

the three location parameters and the four parameters of the Narushin-Romanov-Griffin equation (Narushin et al., 2021).

np

the number of data points on the Narushin-Romanov-Griffin curve.

fig.opt

an optional argument to draw the Narushin-Romanov-Griffin curve.

deform.fun

the deformation function used to describe the deviation from a theoretical Narushin-Romanov-Griffin curve.

Par

the parameter(s) of the deformation function.

xlim

the range of the x-axis over which to plot the Narushin-Romanov-Griffin curve.

ylim

the range of the y-axis over which to plot the Narushin-Romanov-Griffin curve.

unit

the units of the x-axis and the y-axis when showing the Narushin-Romanov-Griffin curve.

main

the main title of the figure.

Details

The first three elements of P are location parameters. The first two are the planar coordinates of the transferred origin, and the third is the angle between the major axis of the curve and the x-axis. The other arguments in P should be the same as those in NRGE. deform.fun should take the form as: deform.fun <- function(Par, z){...}, where z is a two-dimensional matrix related to the x and y values. And the return value of deform.fun should be a list with two variables x and y.

Value

x

the x coordinates of the Narushin-Romanov-Griffin curve.

y

the y coordinates of the Narushin-Romanov-Griffin curve.

Note

When the rotation angle is zero (i.e., the third element in P is zero), np data points are distributed counterclockwise on the Narushin-Romanov-Griffin curve from the rightmost end of the egg's profile to itself.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

Tian, F., Wang, Y., Sandhu, H.S., Gielis, J., Shi, P. (2020) Comparison of seed morphology of two ginkgo cultivars. Journal of Forestry Research 31, 751-758. doi:10.1007/s11676-018-0770-y

Examples

PA   <- c(1, 1, pi/4, 11.5, 7.8, 1.1, 5.6)
resA <- curveNRGE(PA, np=5000, fig.opt=TRUE)
resB <- curveNRGE(PA, np=5000, fig.opt=TRUE, xlim=c(-6, 6), 
                  ylim=c(-6, 6), main="A pear-shaped egg") 
cbind(resB$x, resB$y)

graphics.off()

Drawing the Ovate Leaf-Shape Curve

Description

curveovate is used to draw the ovate leaf-shape curve.

Usage

curveovate(expr, P, x, fig.opt = FALSE, 
           deform.fun = NULL, Par = NULL,
           xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

expr

the simplified version 1 of a performance equation.

P

the three location parameters and the parameters of the simplified version 1 of a performance equation.

x

the given x values to draw the ovate leaf-shape curve.

fig.opt

an optional argument to draw the ovate leaf-shape curve.

deform.fun

the deformation function used to describe the deviation from a theoretical ovate leaf-shape curve.

Par

the parameter(s) of the deformation function.

xlim

the range of the x-axis over which to plot the ovate leaf-shape curve.

ylim

the range of the y-axis over which to plot the ovate leaf-shape curve.

unit

the units of the x-axis and the y-axis when showing the ovate leaf-shape curve.

main

the main title of the figure.

Details

P has two types of elements: three location parameters, and model parameters. This means that expr is limited to be the simplified version 1 (where x_{\mathrm{min}} = 0) in MbetaE, MBriereE, MLRFE, and MPerformanceE. The first three elements of P are location parameters, among which the first two are the planar coordinates of the transferred origin, and the third is the angle between the major axis of the curve and the x-axis. deform.fun should take the form as: deform.fun <- function(Par, z){...}, where z is a two-dimensional matrix related to the x and y values. And the return value of deform.fun should be a list with two variables x and y.

Value

x

the x coordinates of the ovate leaf-shape curve.

y

the y coordinates of the ovate leaf-shape curve.

Note

The number of elements in P here has additional three location parameters than that in MbetaE, MBriereE, MLRFE, MPerformanceE.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. doi:10.3390/plants11131769

Huey, R.B., Stevenson, R.D. (1979) Integrating thermal physiology and ecology of ectotherms: a discussion of approaches. American Zoologist 19, 357-366. doi:10.1093/icb/19.1.357

Li, Y., Zheng, Y., Ratkowsky, D.A., Wei, H., Shi, P. (2022) Application of an ovate leaf shape model to evaluate leaf bilateral asymmetry and calculate lamina centroid location. Frontiers in Plant Science 12, 822907. doi:10.3389/fpls.2021.822907

Lian, M., Shi, P., Zhang, L., Yao, W., Gielis, J., Niklas, K.J. (2023) A generalized performance equation and its application in measuring the Gini index of leaf size inequality. Trees - Structure and Function 37, 1555-1565. doi:10.1007/s00468-023-02448-8

Examples

P1  <- c(1, 1, pi/4, 2, 3, 10, 4)
RE1 <- curveovate(MLRFE, P=P1, x=seq(0, 10, by=0.1), fig.opt=TRUE)
RE2 <- curveovate(MbetaE, P=P1, x=seq(0, 10, by=0.1), fig.opt=TRUE)

dev.new()
plot(RE1$x, RE1$y, cex.lab=1.5, cex.axis=1.5, type="l", 
  xlab=expression(italic(x)), ylab=expression(italic(y))) 
lines(RE2$x, RE2$y, col=4)

P3  <- c(1, 1, pi/4, 2.4, 0.96, 0.64, 7.75, 1.76, 3.68)
RE3 <- curveovate(MPerformanceE, P=P3, x=seq(0, 7.75, by=0.01), fig.opt=TRUE)

dev.new()
plot(RE3$x, RE3$y, cex.lab=1.5, cex.axis=1.5, type="l", 
  xlab=expression(italic(x)), ylab=expression(italic(y))) 

graphics.off()

Egg Boundary Data of Nine Species of Birds

Description

The data consist of the egg boundary data of nine species of birds.

Usage

data(eggs)

Details

In the data set, there are four columns of variables: Code, LatinName, x, and y. Code saves the codes of individual eggs; LatinName saves the Latin names of the nine species of birds; x saves the x coordinates of the egg boundary in the Cartesian coordinate system (cm); and y saves the y coordinates of the egg boundary in the Cartesian coordinate system (cm). In Code, codes 1-9 represent Strix uralensis, Dromaius novaehollandiae, Turdus philomelos, Gallus gallus, Pandion haliaetus, Uria aalge, Uria lomvia, Gallinago media, and Aptenodytes patagonicus, respectively.

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

Tian, F., Wang, Y., Sandhu, H.S., Gielis, J., Shi, P. (2020) Comparison of seed morphology of two ginkgo cultivars. Journal of Forestry Research 31, 751-758. doi:10.1007/s11676-018-0770-y

Examples

data(eggs)

uni.C <- sort( unique(eggs$Code) )
ind   <- 8
Data  <- eggs[eggs$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1 <- adjdata(x0, y0, ub.np=1000, times=1.2, len.pro=1/20)
x1   <- Res1$x
y1   <- Res1$y

dev.new()
plot( x1, y1, asp=1, cex.lab=1.5, cex.axis=1.5, type="l", 
      xlab=expression(italic("x")), ylab=expression(italic("y")),
      pch=1, col=1 ) 

Res2 <- adjdata(x0, y0, ub.np=60, times=1, len.pro=1/2, index.sp=20)
x2   <- Res2$x
y2   <- Res2$y

Res3 <- adjdata(x0, y0, ub.np=60, times=1, len.pro=1/2, index.sp=100)
x3   <- Res3$x
y3   <- Res3$y

dev.new()
plot( x2, y2, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")),
      pch=1, col=4 ) 
points( x3, y3, col=2)

graphics.off()

Data-Fitting Function for the Explicit Preston Equation

Description

fitEPE is used to estimate the parameters of the explicit Preston equation or one of its simplified versions.

Usage

fitEPE(x, y, ini.val, simpver = NULL, 
      control = list(), par.list = FALSE, 
      stand.fig = TRUE, angle = NULL, fig.opt = FALSE, np = 2000,
      xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

x

the x coordinates of an egg's profile.

y

the y coordinates of an egg's profile.

ini.val

the list of initial values for the model parameters.

simpver

an optional argument to use the simplified version of the explicit Preston equation.

control

the list of control parameters for using the optim function in package stats.

par.list

the option of showing the list of parameters on the screen.

stand.fig

the option of drawing the observed and predicted profiles of an egg at the standard state (i.e., the egg's centre is located at (0, 0), and the mid-line is aligned to the x-axis).

angle

the angle between the mid-line and the x-axis, which can be defined by the user.

fig.opt

an optional argument of drawing the observed and predicted profiles of an egg at arbitrary angle between the major axis and the x-axis.

np

the number of data points on the predicted explicit Preston curve.

xlim

the range of the x-axis over which to plot the Preston curve.

ylim

the range of the y-axis over which to plot the Preston curve.

unit

the unit of the x-axis and the y-axis when showing the Preston curve.

main

the main title of the figure.

Details

The simpver argument should correspond to EPE. Here, the major axis (i.e., the mid-line of an egg's profile) is the straight line trhough the two ends of the egg's length. The Nelder-Mead algorithm (Nelder and Mead, 1965) is used to carry out the optimization of minimizing the residual sum of squares (RSS) between the observed and predicted y values. The optim function in package stats was used to carry out the Nelder-Mead algorithm. When angle = NULL, the observed egg's profile will be shown at its initial angle in the scanned image; when angle is a numerical value (e.g., \pi/4) defined by the user, it indicates that the major axis is rotated by the amount (\pi/4) counterclockwise from the x-axis.

Value

par

the estimates of the model parameters.

scan.length

the observed length of the egg's profile.

scan.width

the observed width of the egg's profile.

scan.area

the observed area of the egg's profile.

scan.perimeter

the observed perimeter of the egg's profile.

r.sq

the coefficient of determination between the observed and predicted y values on the Preston curve.

RSS

the residual sum of squares between the observed and predicted y values on the Preston curve.

sample.size

the number of data points used in the data fitting.

x.stand.obs

the observed x coordinates of the points on the Preston curve at the standard state.

y.stand.obs

the observed y coordinates of the points on the Preston curve at the standard state.

y.stand.pred

the predicted y coordinates of the points on the Preston curve at the standard state.

x.obs

the observed x coordinates of the points on the Preston curve at the transferred polar angles as defined by the user.

y.obs

the observed y coordinates of the points on the Preston curve at the transferred polar angles as defined by the user.

y.pred

the predicted y coordinates of the points on the Preston curve at the transferred polar angles as defined by the user.

Note

In the outputs, there are no x.stand.pred and x.pred, because y.stand.obs and y.stand.pred share the same x values (i.e., x.stand.obs), and y.obs and y.pred share the same x values (i.e., x.obs).

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Preston, F.W. (1953) The shapes of birds' eggs. The Auk 70, 160-182.

Todd, P.H., Smart, I.H.M. (1984) The shape of birds' eggs. Journal of Theoretical Biology 106, 239-243. doi:10.1016/0022-5193(84)90021-3

Examples

data(eggs)

uni.C <- sort( unique(eggs$Code) )
ind   <- 8
Data  <- eggs[eggs$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1 <- adjdata(x0, y0, ub.np=2000, times=1.2, len.pro=1/20)
x1   <- Res1$x
y1   <- Res1$y

dev.new()
plot( x1, y1, asp=1, cex.lab=1.5, cex.axis=1.5,  type="l", col=4,  
      xlab=expression(italic("x")), ylab=expression(italic("y")) )


  simpver   <- NULL
  res1      <- lmPE( x1, y1, simpver=simpver, dev.angle=seq(-0.05, 0.05, by=0.0001), 
                     unit="cm", fig.opt=FALSE )
  x0.ini    <- mean( x1 )
  y0.ini    <- mean( y1 )
  theta.ini <- res1$theta
  a.ini     <- res1$scan.length / 2
  b.ini     <- res1$scan.width / 2
  c1.ini    <- res1$par[2] / res1$par[1]
  c2.ini    <- res1$par[3] / res1$par[1]
  c3.ini    <- res1$par[4] / res1$par[1]

  ini.val <- list(x0.ini, y0.ini, theta.ini, a.ini, b.ini, c1.ini, c2.ini, c3.ini)

  res0 <- fitEPE( x=x1, y=y1, ini.val=ini.val, 
                 simpver=simpver, unit="cm", par.list=FALSE, 
                 stand.fig=FALSE, angle=NULL, fig.opt=FALSE, 
                 control=list(reltol=1e-30, maxit=50000), 
                 np=2000 ) 

  n.loop <- 12
  Show   <- FALSE
  for(i in 1:n.loop){
    ini.val <- res0$par
    if(i==n.loop) Show <- TRUE
    print(paste(i, "/", n.loop, sep=""))
    res0 <- fitEPE( x=x1, y=y1, ini.val=ini.val, 
                   simpver=simpver, unit="cm", par.list=FALSE, 
                   stand.fig=Show, angle=pi/4, fig.opt=Show,  
                   control=list(reltol=1e-30, maxit=50000), 
                   np=2000 )    
  }

  # The numerical values of the location and model parameters
  res0$par

  # The root-mean-square error (RMSE) between 
  #   the observed and predicted y values
  sqrt(res0$RSS/res0$sample.size)

  sqrt(sum((res0$y.stand.obs-res0$y.stand.pred)^2)/length(res0$y.stand.obs))

  # To calculate the volume of the egg
  VolumeEPE(P=res0$par[4:8])

  # To calculate the surface area of the egg
  SurfaceAreaEPE(P=res0$par[4:8])


graphics.off()

Data-Fitting Function for the Explicit Troscianko Equation

Description

fitETE is used to estimate the parameters of the explicit Troscianko equation.

Usage

fitETE(x, y, ini.val, control = list(), par.list = FALSE, 
      stand.fig = TRUE, angle = NULL, fig.opt = FALSE, np = 2000,
      xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

x

the x coordinates of an egg's profile.

y

the y coordinates of an egg's profile.

ini.val

the list of initial values for the model parameters.

control

the list of control parameters for using the optim function in package stats.

par.list

the option of showing the list of parameters on the screen.

stand.fig

the option of drawing the observed and predicted profiles of an egg at the standard state (i.e., the egg's centre is located at (0, 0), and the mid-line is aligned to the x-axis).

angle

the angle between the mid-line and the x-axis, which can be defined by the user.

fig.opt

an optional argument of drawing the observed and predicted profiles of an egg at arbitrary angle between the major axis and the x-axis.

np

the number of data points on the predicted explicit Troscianko curve.

xlim

the range of the x-axis over which to plot the Troscianko curve.

ylim

the range of the y-axis over which to plot the Troscianko curve.

unit

the unit of the x-axis and the y-axis when showing the Troscianko curve.

main

the main title of the figure.

Details

Here, the major axis (i.e., the mid-line of an egg's profile) is the straight line trhough the two ends of the egg's length. The Nelder-Mead algorithm (Nelder and Mead, 1965) is used to carry out the optimization of minimizing the residual sum of squares (RSS) between the observed and predicted y values. The optim function in package stats was used to carry out the Nelder-Mead algorithm. When angle = NULL, the observed egg's profile will be shown at its initial angle in the scanned image; when angle is a numerical value (e.g., \pi/4) defined by the user, it indicates that the major axis is rotated by the amount (\pi/4) counterclockwise from the x-axis.

Value

par

the estimates of the model parameters.

scan.length

the observed length of the egg's profile.

scan.width

the observed width of the egg's profile.

scan.area

the observed area of the egg's profile.

scan.perimeter

the observed perimeter of the egg's profile.

r.sq

the coefficient of determination between the observed and predicted y values on the Troscianko curve.

RSS

the residual sum of squares between the observed and predicted y values on the Troscianko curve.

sample.size

the number of data points used in the data fitting.

x.stand.obs

the observed x coordinates of the points on the Troscianko curve at the standard state.

y.stand.obs

the observed y coordinates of the points on the Troscianko curve at the standard state.

y.stand.pred

the predicted y coordinates of the points on the Troscianko curve at the standard state.

x.obs

the observed x coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user.

y.obs

the observed y coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user.

y.pred

the predicted y coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user.

Note

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston’s universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Troscianko, J. (2014). A simple tool for calculating egg shape, volume and surface area from digital images. Ibis, 156, 874-878. doi:10.1111/ibi.12177

Examples

data(eggs)

uni.C <- sort( unique(eggs$Code) )
ind   <- 8
Data  <- eggs[eggs$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1 <- adjdata(x0, y0, ub.np=2000, times=1.2, len.pro=1/20)
x1   <- Res1$x
y1   <- Res1$y

dev.new()
plot( x1, y1, asp=1, cex.lab=1.5, cex.axis=1.5,  type="l", col=4,  
      xlab=expression(italic("x")), ylab=expression(italic("y")) )


  res1      <- lmTE( x1, y1, unit="cm", fig.opt=FALSE )
  
  if(FALSE){
    P0  <- c(res1$scan.length/2, res1$par)
    xx  <- seq(-res1$scan.length/2, res1$scan.length/2, len=2000)
    yy1 <- ETE(P0, xx)
    yy2 <- -ETE(P0, xx)
    dev.new()
    plot( xx, yy1, cex.lab=1.5, cex.axis=1.5, asp=1, col=2, 
          ylim=c(-res1$scan.length/2, res1$scan.length/2),
          type="l", xlab=expression(x), ylab=expression(y) )
    lines( xx, yy2, col=4 )
  }

  x0.ini    <- mean( x1 )
  y0.ini    <- mean( y1 )
  theta.ini <- res1$theta
  a.ini     <- res1$scan.length / 2
  alpha0.ini <- res1$par[1] 
  alpha1.ini <- res1$par[2]
  alpha2.ini <- res1$par[3]

  ini.val <- list(x0.ini, y0.ini, theta.ini, a.ini, alpha0.ini, alpha1.ini, alpha2.ini)

  res0 <- fitETE( x=x1, y=y1, ini.val=ini.val, 
                 unit="cm", par.list=FALSE, 
                 stand.fig=FALSE, angle=NULL, fig.opt=FALSE, 
                 control=list(reltol=1e-30, maxit=50000), 
                 np=2000 ) 

  n.loop <- 12
  Show   <- FALSE
  for(i in 1:n.loop){
    ini.val <- res0$par
    if(i==n.loop) Show <- TRUE
    print(paste(i, "/", n.loop, sep=""))
    res0 <- fitETE( x=x1, y=y1, ini.val=ini.val, 
                   unit="cm", par.list=FALSE, 
                   stand.fig=Show, angle=pi/4, fig.opt=Show,  
                   control=list(reltol=1e-30, maxit=50000), 
                   np=2000 )    
  }

  # The numerical values of the location and model parameters
  res0$par

  # The root-mean-square error (RMSE) between 
  #   the observed and predicted y values
  sqrt(res0$RSS/res0$sample.size)

  sqrt(sum((res0$y.stand.obs-res0$y.stand.pred)^2)/length(res0$y.stand.obs))

  # To calculate the volume of the egg
  VolumeETE(P=res0$par[4:7])

  # To calculate the surface area of the egg
  SurfaceAreaETE(P=res0$par[4:7])


graphics.off()

Data-Fitting Function for the Gielis Equation

Description

fitGE is used to estimate the parameters of the original (or twin) Gielis equation or one of its simplified versions.

Usage

fitGE(expr, x, y, ini.val, m = 1, simpver = NULL, 
      nval = nval, control = list(), par.list = FALSE, 
      stand.fig = TRUE, angle = NULL, fig.opt = FALSE, np = 2000,
      xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

expr

the original (or twin) Gielis equation or one of its simplified versions.

x

the x coordinates of a polygon's boundary.

y

the y coordinates of a polygon's boundary.

ini.val

the list of initial values for the model parameters.

m

the given m value that determines the number of angles of the Gielis curve within [0, 2\pi).

simpver

an optional argument to use the simplified version of the original (or twin) Gielis equation.

nval

the specified value for n_{1} or n_{2} or n_{3} in the simplified versions.

control

the list of control parameters for using the optim function in package stats.

par.list

the option of showing the list of parameters on the screen.

stand.fig

the option of drawing the observed and predicted polygons at the standard state (i.e., the polar point is located at (0, 0), and the major axis overlaps with the x-axis).

angle

the angle between the major axis and the x-axis, which can be defined by the user.

fig.opt

an optional argument of drawing the observed and predicted polygons at arbitrary angle between the major axis and the x-axis.

np

the number of data points on the predicted Gielis curve.

xlim

the range of the x-axis over which to plot the Gielis curve.

ylim

the range of the y-axis over which to plot the Gielis curve.

unit

the unit of the x-axis and the y-axis when showing the Gielis curve.

main

the main title of the figure.

Details

The arguments of m, simpver, and nval should correspond to expr (i.e., GE or TGE). Please note the differences in the simplified version number and the number of parameters between GE and TGE. The Nelder-Mead algorithm (Nelder and Mead, 1965) is used to carry out the optimization of minimizing the residual sum of squares (RSS) between the observed and predicted radii. The optim function in package stats was used to carry out the Nelder-Mead algorithm. When angle = NULL, the observed polygon will be shown at its initial angle in the scanned image; when angle is a numerical value (e.g., \pi/4) defined by the user, it indicates that the major axis is rotated by the amount (\pi/4) counterclockwise from the x-axis.

Value

par

the estimates of the model parameters.

scan.length

the observed length of the polygon.

scan.width

the observed width of the polygon.

scan.area

the observed area of the polygon.

r.sq

the coefficient of determination between the observed and predicted polar radii.

RSS

the residual sum of squares between the observed and predicted polar radii.

sample.size

the number of data points used in the data fitting.

phi.stand.obs

the polar angles at the standard state.

phi.trans

the transferred polar angles rotated as defined by the user.

r.stand.obs

the observed polar radii at the standard state.

r.stand.pred

the predicted polar radii at the standard state.

x.stand.obs

the observed x coordinates at the standard state.

x.stand.pred

the predicted x coordinates at the standard state.

y.stand.obs

the observed y coordinates at the standard state.

y.stand.pred

the predicted y coordinates at the standard state.

r.obs

the observed polar radii at the transferred polar angles as defined by the user.

r.pred

the predicted polar radii at the transferred polar angles as defined by the user.

x.obs

the observed x coordinates at the transferred polar angles as defined by the user.

x.pred

the predicted x coordinates at the transferred polar angles as defined by the user.

y.obs

the observed y coordinates at the transferred polar angles as defined by the user.

y.pred

the predicted y coordinates at the transferred polar angles as defined by the user.

Note

simpver in GE is different from that in TGE.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

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:10.3732/ajb.90.3.333

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. doi:10.3390/sym12040645

Examples

data(eggs)

uni.C <- sort( unique(eggs$Code) )
ind   <- 1
Data  <- eggs[eggs$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1 <- adjdata(x0, y0, ub.np=200, times=1.2, len.pro=1/20)
x1   <- Res1$x
y1   <- Res1$y
Res2 <- adjdata(x0, y0, ub.np=40, times=1, len.pro=1/2, index.sp=20)
x2   <- Res2$x
y2   <- Res2$y
Res3 <- adjdata(x0, y0, ub.np=100, times=1, len.pro=1/2, index.sp=100)
x3   <- Res3$x
y3   <- Res3$y

dev.new()
plot( x2, y2, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")),
      pch=1, col=4 ) 
points( x3, y3, col=2)


  x0.ini    <- mean( x1 )
  y0.ini    <- mean( y1 )
  theta.ini <- pi
  a.ini     <- sqrt(2) * max( y0.ini-min(y1), x0.ini-min(x1) ) 
  n1.ini <- c(5, 25)
  n2.ini <- c(15, 25)
  if(ind == 2){
    n1.ini <- c(0.5, 1)
    n2.ini <- c(6, 12)
  }
  ini.val <- list(x0.ini, y0.ini, theta.ini, a.ini, n1.ini, n2.ini)

  Res4 <- fitGE( GE, x=x1, y=y1, ini.val=ini.val, 
                 m=1, simpver=1, nval=1, unit="cm",  
                 par.list=FALSE, fig.opt=TRUE, angle=NULL, 
                 control=list(reltol=1e-20, maxit=20000), 
                 np=2000 )
  Res4$par
  sqrt(sum((Res4$y.stand.obs-Res4$y.stand.pred)^2)/Res4$sample.size)

  xx    <- Res4$x.stand.obs
  yy    <- Res4$y.stand.obs

  library(spatstat.geom)
  poly0 <- as.polygonal(owin(poly=list(x=xx, y=yy)))
  area(poly0)

  areaGE(GE, P = Res4$par[4:6], 
         m=1, simpver=1)


  # The following code is used to 
  #   calculate the root-mean-square error (RMSE) in the y-coordinates
  ind1  <- which(yy >= 0)
  ind2  <- which(yy < 0)
  xx1   <- xx[ind1] # The upper part of the egg
  yy1   <- yy[ind1]
  xx2   <- xx[ind2] # The lower part of the egg
  yy2   <- yy[ind2]
  Para  <- c(0, 0, 0, Res4$par[4:length(Res4$par)])
  PartU <- curveGE(GE, P=Para, phi=seq(0, pi, len=100000), m=1, simpver=1, fig.opt=FALSE)
  xv1   <- PartU$x
  yv1   <- PartU$y
  PartL <- curveGE(GE, P=Para, phi=seq(pi, 2*pi, len=100000), m=1, simpver=1, fig.opt=FALSE)
  xv2   <- PartL$x
  yv2   <- PartL$y  
  ind3  <- c()
  for(q in 1:length(xx1)){
    ind.temp <- which.min(abs(xx1[q]-xv1))
    ind3     <- c(ind3, ind.temp)
  }  
  ind4  <- c()
  for(q in 1:length(xx2)){
    ind.temp <- which.min(abs(xx2[q]-xv2))
    ind4     <- c(ind4, ind.temp)
  }
  RSS   <- sum((yy1-yv1[ind3])^2) + sum((yy2-yv2[ind4])^2)
  RMSE  <- sqrt( RSS/length(yy) )


  # To calculate the volume of the Gielis egg when simpver=1 & m=1
  VolumeSGE(P=Res4$par[4:6])

  # To calculate the surface area of the Gielis egg when simpver=1 & m=1
  SurfaceAreaSGE(P=Res4$par[4:6])


graphics.off()

Data-Fitting Function for the Rotated and Right-Shifted Lorenz Curve

Description

fitLorenz is used to estimate the parameters of the rotated and right-shifted Lorenz curve using version 4 or 5 of MPerformanceE, or the Lorenz equations including SarabiaE, SCSE, and SHE.

Usage

fitLorenz(expr, z, ini.val, simpver = 4, 
          control = list(), par.list = FALSE, 
          fig.opt = FALSE, np = 2000, 
          xlab=NULL, ylab=NULL, main = NULL, subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, 
          abs.tol = rel.tol, stop.on.error = TRUE, 
          keep.xy = FALSE, aux = NULL, par.limit = TRUE)

Arguments

expr

version 4 or 5 of MPerformanceE, or the Lorenz equations including SarabiaE, SCSE, and SHE.

z

the observations of size distribution (i.e., the household income distribution, the leaf size distribution).

ini.val

the initial values of the model parameters.

simpver

an optional argument to use version 4 or 5 of MPerformanceE.

control

the list of control parameters for using the optim function in package stats.

par.list

the option of showing the list of parameters on the screen.

fig.opt

an optional argument to draw the original and rotated Lorenz curves.

np

the number of data points to draw the predicted original and rotated Lorenz curves.

xlab

the label of the x-axis when showing the original Lorenz curve.

ylab

the label of the y-axis when showing the original Lorenz curve.

main

the main title of the figure.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

par.limit

an optional argument to limit the numerical ranges of model parameters of the three Lorenz equations including SarabiaE, SCSE, and SHE.

Details

Here, ini.val only includes the initial values of the model parameters as a list. The Nelder-Mead algorithm (Nelder and Mead, 1965) is used to carry out the optimization of minimizing the residual sum of squares (RSS) between the observed and predicted y values. The optim function in package stats was used to carry out the Nelder-Mead algorithm. Here, versions 4 and 5 of MPerformanceE and the Lorenz equations including SarabiaE, SCSE, and SHE can be used to fit the rotated and right-shifted Lorenz curve.

\quad When simpver = 4, the simplified version 4 of MPerformanceE is selected:

\mbox{if } x \in{\left(0, \ \sqrt{2}\right)},

y = c\left(1-e^{-K_{1}x}\right)^{a}\left(1-e^{K_{2}\left(x-\sqrt{2}\right)}\right)^{b};

\mbox{if } x \notin{\left(0, \ \sqrt{2}\right)},

y = 0.

There are five elements in P, representing the values of c, K_{1}, K_{2}, a, and b, respectively.

\quad When simpver = 5, the simplified version 5 of MPerformanceE is selected:

\mbox{if } x \in{\left(0, \ \sqrt{2}\right)},

y = c\left(1-e^{-K_{1}x}\right)\left(1-e^{K_{2}\left(x-\sqrt{2}\right)}\right);

\mbox{if } x \notin{\left(0, \ \sqrt{2}\right)},

y = 0.

There are three elements in P, representing the values of c, K_{1}, and K_{2}, respectively.

\quad For the Lorenz functions, the user can define any formulae that follow the below form: Lorenz.fun <- function(P, x){...}, where P is the vector of parameter(s), x is the preditor that ranges between 0 and 1 representing the cumulative proportion of the number of individuals in a statistical unit, and Lorenz.fun is the name of a Lorenz function defined by the user, which also ranges between 0 and 1 representing the cumulative proportion of the income or size in a statistical unit. This package provides three representative Lorenz functions: SarabiaE, SCSE, and SHE.

\quad Here, the Gini coefficient (GC) is calculated as follows when MPerformanceE is selected:

\mbox{GC} = 2\int_{0}^{\sqrt{2}}y\,dx,

where x and y are the independent and dependent variables in version 4 or 5 of MPerformanceE, respectively.

\quad However, the Gini coefficient (GC) is calculated as follows when a Lorenz function, e.g., SCSE, is selected:

\mbox{GC} = 2\int_{0}^{1}y\,dx,

where x and y are the independent and dependent variables in the Lorenz function, respectively.

\quad For SarabiaE and SHE, there are explicit formulae for GC (Sarabia, 1997; Sitthiyot and Holasut, 2023).

Value

x1

the cumulative proportion of the number of an entity of interest, i.e., the number of households of a city, the number of leaves of a plant.

y1

the cumulative proportion of the size of an entity of interest.

x

the x coordinates of the rotated and right-shifted y1 versus x1.

y

the y coordinates of the rotated and right-shifted y1 versus x1.

par

the estimates of the model parameters.

r.sq

the coefficient of determination between the observed and predicted y values.

RSS

the residual sum of squares between the observed and predicted y values.

sample.size

the number of data points used in the data fitting.

GC

the calculated Gini coefficient.

Note

When MPerformanceE is selected, the estimates of the model parameters denote those in MPerformanceE rather than being obtained by directly fitting the y1 versus x1 data; when a Lorenz function is selected, the estimates of the model parameters denote those in the Lorenz function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Huey, R.B., Stevenson, R.D. (1979) Integrating thermal physiology and ecology of ectotherms: a discussion of approaches. American Zoologist 19, 357-366. doi:10.1093/icb/19.1.357

Lorenz, M.O. (1905) Methods of measuring the concentration of wealth. Journal of the American Statistical Association 9(70), 209-219. doi:10.2307/2276207

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Sarabia, J.-M. (1997) A hierarchy of Lorenz curves based on the generalized Tukey's lambda distribution. Econometric Reviews 16, 305-320. doi:10.1080/07474939708800389

Examples

  data(LeafSizeDist)

  CulmNumber <- c()
  for(i in 1:length(LeafSizeDist$Code)){
    temp <- as.numeric( strsplit(LeafSizeDist$Code[i], "-", fixed=TRUE)[[1]][1] )
    CulmNumber <- c(CulmNumber, temp)
  }
  uni.CN <- sort( unique(CulmNumber) )  
  ind    <- CulmNumber==uni.CN[1]
  A0     <- LeafSizeDist$A[ind]

  ini.val1 <- list(0.5, 0.1, c(0.01, 0.1, 1, 5, 10), 1, 1)
  ini.val2 <- list(0.5, 0.1, c(0.01, 0.1, 1, 5, 10))
  resu1 <- fitLorenz(MPerformanceE, z=A0, ini.val=ini.val1, simpver=4, fig.opt=TRUE)
  resu2 <- fitLorenz(MPerformanceE, z=A0, ini.val=ini.val2, simpver=5, fig.opt=TRUE)
  resu1$par
  resu2$par

  ini.val3 <- list(0.9, c(10, 50, 100, 500), 1, 0)   
  resu3 <- fitLorenz( SarabiaE, z=A0, ini.val=ini.val3, par.limit=TRUE, 
                      fig.opt=TRUE, control=list(reltol=1e-20, maxit=10000) )
  resu3$par
  resu3$GC

  graphics.off()

Parameter Estimation for the Narushin-Romanov-Griffin Equation

Description

fitNRGE is used to estimate the parameters of the Narushin-Romanov-Griffin equation.

Usage

fitNRGE(x, y, dev.angle = NULL, ini.C = c(-1, 0.1, 0.5, 1), 
        strip.num = 2000, control = list(), fig.opt = TRUE, np = 2000,
        xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

x

the x coordinates of the edge of an egg's boundary.

y

the y coordinates of the edge of an egg's boundary.

dev.angle

the angle of deviation for the axis associated with the maximum distance between two points on an egg's profile from the mid-line of the egg's profile.

ini.C

the initial value(s) of parameter C in the Narushin-Romanov-Griffin equation.

strip.num

the number of equidistant strips intersecting with the egg's boundary that are horizontally placed. See Shi et al. (2018, 2020) for details.

control

the list of control parameters for using the optim function in package stats.

fig.opt

an optional argument to draw the observed and predicted egg's boundaries.

np

the number of data points on the predicted Narushin-Romanov-Griffin curve.

xlim

the range of the x-axis over which to plot the Narushin-Romanov-Griffin curve.

ylim

the range of the y-axis over which to plot the Narushin-Romanov-Griffin curve.

unit

the units of the x-axis and the y-axis when showing the Narushin-Romanov-Griffin curve.

main

the main title of the figure.

Details

The NRGE (see NRGE) has a complex model structure with four parameters (i.e., A, B, C, and D). Because three out of four parameters of NRGE have clear biological and geometric meanings (i.e., A, B, and D), their values could be estimated by means of numerical calculation. After obtaining the numerical values of the three parameters, the Nelder-Mead algorithm (Nelder and Mead, 1965) was used to estimate C. Because of the failure of the optimization method to estimate the major axis (i.e., the mid-line) and model parameters of NRGE, it was difficult to define the egg length axis, although it is essential for calculating A, B, and D. For this reason, two methods were used to obtain the major axis: the maximum distance method, and the longest axis adjustment method. In the first method, the straight line through two points forming the maximum distance on the egg's profile is defined as the major axis (i.e., the mid-line). In the second method, we assume that there is an angle of deviation for the longest axis (i.e., the axis associated with the maximum distance between two points on an egg's profile) from the mid-line of the egg's profile. That is to say, the mid-line of an egg's profile is not the axis associated with the maximum distance between two points on the egg's profile. When angle = NULL, the maximum distance method is used; when angle is a numerical value or a numerical vector, the longest axis adjustment method is used. Here, the numerical value of dev.angle is not the angle of deviation for the major axis of an egg's profile from the x-axis, and instead it is the angle of deviation for the longest axis (associated with the maximum distance between two points on the egg's profile) from the mid-line of the egg's profile. Once the major axis is established, the distance of the major axis can be calculated as the estimate of A. Using the maximum distance method, A equals the maximum distance. Using the longest axis adjustment method, A may be slightly smaller than the maximum distance. After rotating the major axis to make it overlap with the x-axis, a large number of equidistant strips can be used (e.g., 2000) from the egg base to egg tip to intersect the egg's boundary. This methodology makes it easy to obtain the maximum egg's breadth (i.e., B) and D. The residual sum of squares (RSS) between the observed and predicted y values can be minimized using an optimization method (Nelder and Mead, 1965) to estimate C. Despite the complex structure of NRGE (see NRGE), the optimization method for estimating the remaining parameter C becomes feasible after the other three parameters have been numerically estimated. Please see Shi et al. (2022) for details.

Value

theta

the angle between the longest axis of an egg's profile (i.e., the axis associated with the maximum distance between two points on the egg's profile) and the x-axis.

epsilon

the optimal angle of deviation for the longest axis (associated with the maximum distance between two points on an egg's profile) from the mid-line of the egg's profile, when dev.angle is not NULL.

RSS.vector

the vector of residual sum of squares corresponding to dev.angle, when dev.angle is not NULL.

x.obs

the observed x coordinates.

y.obs

the observed y coordinates.

y.pred

the predicted y coordinates corresponding to the the observed x coordinates.

par

the estimates of the four model parameters in the Narushin-Romanov-Griffin equation.

scan.length

the length of the egg's boundary. The default is the maximum distance between two points on the egg's boundary.

scan.width

the maximum width of the egg's boundary.

scan.area

the area of the egg's boundary.

scan.perimeter

the perimeter of the egg's boundary based on all data points on the egg's boundary.

RSS

the residual sum of squares between the observed and predicted y values.

sample.size

the number of data points used in the numerical calculation.

RMSE

the root-mean-square errors between the observed and predicted y values.

Note

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Shi, P., Zheng, X., Ratkowsky, D.A., Li, Y., Wang, P., Cheng, L. (2018) A simple method for measuring the bilateral symmetry of leaves. Symmetry 10, 118. doi:10.3390/sym10040118

Tian, F., Wang, Y., Sandhu, H.S., Gielis, J., Shi, P. (2020) Comparison of seed morphology of two ginkgo cultivars. Journal of Forestry Research 31, 751-758. doi:10.1007/s11676-018-0770-y

Examples

data(eggs)

uni.C <- sort( unique(eggs$Code) )
ind   <- 8
Data  <- eggs[eggs$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1  <- adjdata(x0, y0, ub.np=3000, len.pro=1/20)
x1    <- Res1$x
y1    <- Res1$y

dev.new()
plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) ) 


  Res2 <- fitNRGE(x1, y1, dev.angle=NULL, ini.C=c(-1, -0.1, seq(0.1, 1, by=0.05)), 
                  strip.num=2000, fig.opt=TRUE) 
         
  dev.new()
  plot(Res2$x.obs, Res2$y.obs, asp=1, cex.lab=1.5, cex.axis=1.5, 
       xlab=expression(italic("x")), ylab=expression(italic("y")),
       type="l", col=4) 
  lines( Res2$x.obs, Res2$y.pred, col=2)

  Res3 <- fitNRGE(x1, y1, dev.angle=seq(-0.05, 0.05, by=0.01), 
                  ini.C=c(-1, -0.1, seq(0.1, 1, by=0.05)), 
                  strip.num=2000, fig.opt=TRUE)   

  zeta <- Res3$theta + Res3$epsilon
  x2   <- x1*cos(zeta) + y1*sin(zeta)
  y2   <- y1*cos(zeta) - x1*sin(zeta)
  plot( x2-min(x2), y2-min(y2), asp=1, col="grey70", cex=1,
        xlab=expression(italic("x")), ylab=expression(italic("y")) )  
  lines(Res3$x.obs-min(Res3$x.obs), Res3$y.obs-min(Res3$y.obs), col=4)
  lines(Res3$x.obs-min(Res3$x.obs), Res3$y.pred-min(Res3$y.obs), col=2)

  RMSE <- sqrt( Res3$RSS / Res3$sample.size )
  RMSE

  # To calculate the volume of the egg 
  VolumeNRGE(P=Res3$par)

  # To calculate the surface area of the egg
  SurfaceAreaNRGE(P=Res3$par)


graphics.off()

Data-Fitting Function for the Superellipse Equation

Description

fitSuper is used to estimate the parameters of the superellipse equation.

Usage

fitSuper(x, y, ini.val, control = list(), par.list = FALSE, 
         stand.fig = TRUE, angle = NULL, fig.opt = FALSE, np = 2000,
         xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

x

the x coordinates of a polygon's boundary.

y

the y coordinates of a polygon's boundary.

ini.val

the list of initial values for the model parameters.

control

the list of control parameters for using the optim function in package stats.

par.list

the option of showing the list of parameters on the screen.

stand.fig

the option of drawing the observed and predicted polygons at the standard state (i.e., the polar point is located at (0, 0), and the major axis overlaps with the x-axis).

angle

the angle between the major axis and the x-axis, which can be defined by the user.

fig.opt

an optional argument of drawing the observed and predicted polygons at arbitrary angle between the major axis and the x-axis.

np

the number of data points on the predicted superellipse curve.

xlim

the range of the x-axis over which to plot the superellipse curve.

ylim

the range of the y-axis over which to plot the superellipse curve.

unit

the unit of the x-axis and the y-axis when showing the superellipse curve.

main

the main title of the figure.

Details

The superellipse equation has its mathematical expression:

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\varphi\right)\right|^{n}+ \left|\frac{1}{k}\,\mathrm{sin}\left(\varphi\right)\right|^{n}\right)^{-\frac{1}{n}},

where r represents the polar radius at the polar angle \varphi. a represents the semi-major axis of the superellipse; k is the ratio of the semi-minor axis to the semi-major axis of the superellipse; and n determines the curvature of the superellipse curve. This function is basically equal to the fitGE function with the arguments m = 4 and simpver = 9. However, this function can make the estimated value of the parameter k be smaller than or equal to 1. Apart from the above three model parameters, there are three additional location parameters, i.e., the planar coordinates of the polar point (x_{0} and y_{0}), and the angle between the major axis of the superellipse and the x-axis (\theta). The input order of ini.val is x_{0}, y_{0}, \theta, a, k, and n. The fitted parameters will be shown after running this function in the same order. The Nelder-Mead algorithm (Nelder and Mead, 1965) is used to carry out the optimization of minimizing the residual sum of squares (RSS) between the observed and predicted radii. The optim function in package stats was used to carry out the Nelder-Mead algorithm. When angle = NULL, the observed polygon will be shown at its initial angle in the scanned image; when angle is a numerical value (e.g., \pi/4) defined by the user, it indicates that the major axis is rotated by the amount (\pi/4) counterclockwise from the x-axis.

Value

par

the estimated values of the parameters in the superellipse equation.

scan.length

the observed length of the polygon.

scan.width

the observed width of the polygon.

scan.area

the observed area of the polygon.

r.sq

the coefficient of determination between the observed and predicted polar radii.

RSS

the residual sum of squares between the observed and predicted polar radii.

sample.size

the number of data points used in the data fitting.

phi.stand.obs

the polar angles at the standard state.

phi.trans

the transferred polar angles rotated as defined by the user.

r.stand.obs

the observed polar radii at the standard state.

r.stand.pred

the predicted polar radii at the standard state.

x.stand.obs

the observed x coordinates at the standard state.

x.stand.pred

the predicted x coordinates at the standard state.

y.stand.obs

the observed y coordinates at the standard state.

y.stand.pred

the predicted y coordinates at the standard state.

r.obs

the observed polar radii at the transferred polar angles as defined by the user.

r.pred

the predicted polar radii at the transferred polar angles as defined by the user.

x.obs

the observed x coordinates at the transferred polar angles as defined by the user.

x.pred

the predicted x coordinates at the transferred polar angles as defined by the user.

y.obs

the observed y coordinates at the transferred polar angles as defined by the user.

y.pred

the predicted y coordinates at the transferred polar angles as defined by the user.

Note

The output values of running this function can be used as those of running the fitGE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Shi, P., Huang, J., Hui, C., Grissino-Mayer, H.D., Tardif, J., Zhai, L., Wang, F., Li, B. (2015) Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape. Frontiers in Plant Science 6, 856. doi:10.3389/fpls.2015.00856

Examples

data(whitespruce)

uni.C <- sort( unique(whitespruce$Code) )  
Data  <- whitespruce[whitespruce$Code==uni.C[12], ]
x0    <- Data$x
y0    <- Data$y
Res1  <- adjdata(x0, y0, ub.np=200, len.pro=1/20)
x1    <- Res1$x
y1    <- Res1$y

plot( x1, y1, asp=1, cex.lab=1.5, cex.axis=1.5, type="l", 
      col="grey73", lwd=2,
      xlab=expression(italic("x")), ylab=expression(italic("y")) )


  x0.ini    <- mean( x1 )
  y0.ini    <- mean( y1 )
  theta.ini <- c(0, pi/4, pi/2)
  a.ini     <- mean(c(max(x1)-min(x1), max(y1)-min(y1)))/2
  k.ini     <- 1
  n.ini     <- c(1.5, 2, 2.5)
  ini.val   <- list( x0.ini, y0.ini, theta.ini, a.ini, k.ini, n.ini )
  Res2      <- fitSuper(x=x1, y=y1, ini.val=ini.val, unit="cm", par.list=FALSE,
                        stand.fig=FALSE, angle=NULL, fig.opt=TRUE,  
                        control=list(reltol=1e-20, maxit=20000), np=2000)
  Res2$par
  Res2$r.sq


graphics.off()

Data-Fitting Function for the Ovate Leaf-Shape Equation

Description

fitovate is used to estimate the parameters of a simplified performance equation.

Usage

fitovate(expr, x, y, ini.val, 
         par.list = FALSE, stand.fig = TRUE, control = list(), 
         angle = NULL, fig.opt = FALSE, index.xmax = 3, np = 2000, 
         xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

expr

the simplified version 1 of the performance equations.

x

the x coordinates of a polygon's boundary.

y

the y coordinates of a polygon's boundary.

ini.val

the initial values of the simplified version 1 of a performance equation.

par.list

an optional argument to show the list of parameters on the screen.

stand.fig

an optional argument to draw the observed and predicted polygons' boundaries at the standard state (i.e., the origin is located at (0, 0), and the major axis overlaps with the x-axis).

control

the list of control parameters for using the optim function in package stats.

angle

the angle between the major axis of the polygon and the x-axis, which can be defined by the user.

fig.opt

an optional argument to draw the observed and predicted polygons at an arbitrary angle between the major axis and the x-axis.

index.xmax

the specified index in parameters representing x_{\mathrm{max}}.

np

the number of data points on the predicted ovate leaf-shape curve.

xlim

the range of the x-axis over which to plot the ovate leaf-shape curve.

ylim

the range of the y-axis over which to plot the ovate leaf-shape curve.

unit

the units of the x-axis and the y-axis when showing the ovate leaf-shape curve.

main

the main title of the figure.

Details

ini.val is a list for two types of parameters: three location parameters, and model parameters. This means that expr is limited to being the simplified version 1 (where x_{\mathrm{min}} = 0) in MbetaE, MBriereE, MLRFE, and MPerformanceE. The initial values for the first three parameters in ini.val are location parameters, among which the first two are the planar coordinates of the transferred origin, and the third is the angle between the major axis of the polygon and the x-axis. The Nelder-Mead algorithm (Nelder and Mead, 1965) is used to carry out the optimization of minimizing the residual sum of squares (RSS) between the observed and predicted y-axis. The optim function in package stats was used to carry out the Nelder-Mead algorithm. When angle = NULL, the observed polygon will be shown at its initial angle in the scanned image; when angle is a numerical value (e.g., \pi/4) defined by the user, it indicates that the major axis is rotated by the amount (\pi/4) counterclockwise from the x-axis.

Value

par

the estimates of the model parameters.

r.sq

the coefficient of determination between the observed and predicted y values.

RSS

the residual sum of squares between the observed and predicted y values.

sample.size

the number of data points on the polygon's boundary in the data fitting.

scan.length

the observed length of the polygon's boundary.

scan.width

the observed width of the polygon's boundary.

scan.perimeter

the observed perimeter of the polygon's boundary.

scan.area

the observed area of the polygon's boundary.

pred.length

the predicted length of the polygon's boundary.

pred.width

the predicted width of the polygon's boundary.

pred.perimeter

the predicted perimeter of the polygon's boundary.

pred.area

the predicted area of the polygon's boundary.

x.stand.obs

the observed x coordinates at the standard state.

x.stand.pred

the predicted x coordinates at the standard state.

y.stand.obs

the observed y coordinates at the standard state.

y.stand.pred

the predicted y coordinates at the standard state.

x.obs

the observed x coordinates at the transferred angles defined by the user.

x.pred

the predicted x coordinates at the transferred angles defined by the user.

y.obs

the observed y coordinates at the transferred angles defined by the user.

y.pred

the predicted y coordinates at the transferred angles defined by the user.

Note

There are two types of parameters (i.e., three location parameters and model parameters) for the value of par. The transferred angle denotes the angle between the major axis and the x-axis. For the argument index.xmax, the default is 3, which corresponds to the order of the model parameter of x_{\mathrm{max}} in MbetaE, MBriereE, and MLRFE. However, in MPerformanceE, index.xmax should be 4 rather than 3.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. doi:10.3390/plants11131769

Huey, R.B., Stevenson, R.D. (1979) Integrating thermal physiology and ecology of ectotherms: a discussion of approaches. American Zoologist 19, 357-366. doi:10.1093/icb/19.1.357

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Examples

data(Neocinnamomum)

uni.C <- sort( unique(Neocinnamomum$Code) )
ind   <- 2
Data  <- Neocinnamomum[Neocinnamomum$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1 <- adjdata(x0, y0, ub.np=200, len.pro=1/20)
x1   <- Res1$x
y1   <- Res1$y

dev.new()
plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) ) 


  x0.ini    <- min( x1 )
  y0.ini    <- min( y1 )
  theta.ini <- pi/4
  len.max   <- max( max(y1)-min(y1), max(x1)-min(x1) ) *2/sqrt(2)
  a.ini     <- c(0.1, 0.01, 0.001, 0.0001)
  m.ini     <- c(0.1, 0.5, 1, 2)
  x2.ini    <- len.max
  delta.ini <- c(0.5, 1)
  ini.val   <- list(x0.ini, y0.ini, theta.ini, a.ini, m.ini, x2.ini, delta.ini)

  Res1 <- fitovate(MBriereE, x=x1, y=y1, ini.val=ini.val,            
            par.list=FALSE, fig.opt=TRUE, angle=pi/6, 
            control=list(reltol=1e-20, maxit=20000), 
            np=2000, unit=NULL)
  Res1$RSS

  x0.ini    <- min( x1 )
  y0.ini    <- min( y1 )
  theta.ini <- pi/4
  len.max   <- max( max(y1)-min(y1), max(x1)-min(x1) ) *2/sqrt(2)
  yc.ini    <- len.max/3 
  xc.ini    <- 1/4*len.max
  x2.ini    <- len.max
  delta.ini <- c(0.5, seq(1, 5, by=5))
  ini.val   <- list(x0.ini, y0.ini, theta.ini, yc.ini, xc.ini, x2.ini, delta.ini)

  Res2 <- fitovate( MbetaE, x=x1, y=y1, ini.val=ini.val,            
                    par.list=TRUE, fig.opt=TRUE, angle=pi/3, 
                    control=list(reltol=1e-20, maxit=20000), 
                    np=2000, unit=NULL )
  Res2$RSS

  Res3 <- fitovate( MLRFE, x=x1, y=y1, ini.val=ini.val, 
                    unit=NULL, par.list=FALSE, fig.opt=TRUE, 
                    angle=NULL, control=list(reltol=1e-20, 
                    maxit=20000), np=2000)
  Res3$RSS


  x0.ini    <- min( x1 )
  y0.ini    <- min( y1 )
  theta.ini <- pi/4
  len.max   <- max( max(y1)-min(y1), max(x1)-min(x1) ) *2/sqrt(2)
  c.ini     <- 1/5*len.max
  K1.ini    <- c(0.1, 1, 5, 10)
  K2.ini    <- 1
  x2.ini    <- len.max
  a.ini     <- 1
  b.ini     <- 1
  ini.val   <- list(x0.ini, y0.ini, theta.ini, c.ini, K1.ini, K2.ini, x2.ini, a.ini, b.ini)

  Res4 <- fitovate( MPerformanceE, x=x1, y=y1, ini.val=ini.val,            
                    par.list=TRUE, fig.opt=TRUE, index.xmax=4, angle=pi/3, 
                    control=list(reltol=1e-20, maxit=20000), 
                    np=2000, unit=NULL )
  Res4$RSS


graphics.off()

Data-Fitting Function for the Sigmoid Growth Equation

Description

fitsigmoid is used to estimate the parameters of a sigmoid growth equation based on the integral of a performance equation or one of its simplified versions.

Usage

fitsigmoid(expr, x, y, ini.val, simpver = 1, 
           control = list(), par.list = FALSE, fig.opt = FALSE,
           xlim = NULL, ylim = NULL, xlab = NULL, ylab = NULL,         
           main = NULL, subdivisions = 100L,
           rel.tol = .Machine$double.eps^0.25, 
           abs.tol = rel.tol, stop.on.error = TRUE, 
           keep.xy = FALSE, aux = NULL)

Arguments

expr

a performance equation or one of its simplified versions that is used to build a sigmoid growth equation.

x

the observed investigation times.

y

the observed y values (i.e., biomass, height, body length, etc.).

ini.val

the initial values of the model parameters.

simpver

an optional argument to use the simplified version of the performance equation.

control

the list of control parameters for using the optim function in package stats.

par.list

the option of showing the list of parameters on the screen.

fig.opt

an optional argument to draw the observations and the predicted sigmoid curve.

xlim

the range of the x-axis over which to plot a sigmoid growth curve.

ylim

the range of the y-axis over which to plot a sigmoid growth curve.

xlab

the label of the x-axis when showing a sigmoid growth curve.

ylab

the label of the y-axis when showing a sigmoid growth curve.

main

the main title of the figure.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

Here, ini.val only includes the initial values of the model parameters as a list. The Nelder-Mead algorithm (Nelder and Mead, 1965) is used to carry out the optimization of minimizing the residual sum of squares (RSS) between the observed and predicted y values. The optim function in package stats was used to carry out the Nelder-Mead algorithm. The performance equations denote MbetaE, MBriereE, MLRFE, MPerformanceE and their simplified versions. The arguments of P and simpver should correspond to expr (i.e., MbetaE or MBriereE or MLRFE or MPerformanceE). The sigmoid equation is the integral of a performance equation or one of its simplified versions.

Value

par

the estimates of the model parameters.

r.sq

the coefficient of determination between the observed and predicted y values.

RSS

the residual sum of squares between the observed and predicted y values.

sample.size

the number of data points used in the data fitting.

x

the observed x values.

y

the observed y values.

y.pred

the predicted y values.

Note

Here, the user can define other performance equations, but new equations or their simplified versions should include the lower and upper thresholds on the x-axis corresponding to y = 0, whose indices should be the same as those in MbetaE or MBriereE or MLRFE or MPerformanceE.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. doi:10.3390/plants11131769

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Examples


# The shrimp growth data(See the supplementary table in West et al., 2001)
# West, G.B., Brown, J.H., Enquist, B.J. (2001) A general model for ontogenetic growth. 
#     Nature 413, 628-631.
t0 <- c(3, 60, 90, 120, 150, 180, 384)
m0 <- c(0.001, 0.005, 0.018, 0.037, 0.06, 0.067, 0.07)

dev.new()
plot( t0, m0, cex.lab=1.5, cex.axis=1.5, col=4,
      xlab=expression(italic(x)), ylab=expression(italic(y)) )

xopt0   <- seq(100, 150, by=5)
ini.val <- list(0.035, xopt0, 200, 1)
resu1   <- fitsigmoid(MLRFE, x=t0, y=m0, ini.val=ini.val, simpver=1, fig.opt=TRUE, par.list=TRUE)


  xopt0   <- seq(100, 150, by=5)
  ini.val <- list(0.035, xopt0, 200, 1)
  resu1   <- fitsigmoid(MbetaE, x=t0, y=m0, ini.val=ini.val, simpver=1, fig.opt=TRUE)

  m.ini   <- c(0.5, 1, 2, 3, 4, 5, 10, 20)
  ini.val <- list(1e-8, m.ini, 200, 1)
  resu2   <- fitsigmoid(MBriereE, x=t0, y=m0, ini.val=ini.val, simpver=1, 
             fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000, trace=FALSE), 
             subdivisions=100L, rel.tol=.Machine$double.eps^0.25, 
             abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
             keep.xy=FALSE, aux=NULL)

  delta0  <- c(0.5, 1, 2, 5, 10, 20)
  ini.val <- list(0.035, 150, -100, 200, delta0)
  resu3   <- fitsigmoid(MLRFE, x=t0, y=m0, ini.val=ini.val, simpver=NULL, 
               fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000), 
               subdivisions = 100L, rel.tol=.Machine$double.eps^0.25, 
               abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
               keep.xy=FALSE, aux=NULL)

  a.ini   <- c(0.1, 1, 10, 100, 200)
  b.ini   <- 200
  ini.val <- list(0.001, 0.02, 0.15, 0, 200, a.ini, b.ini)
  resu4   <- fitsigmoid(MPerformanceE, x=t0, y=m0, ini.val=ini.val, simpver=NULL, 
             fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000, trace=FALSE), 
             subdivisions=100L, rel.tol=.Machine$double.eps^0.25, 
             abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
             keep.xy=FALSE, aux=NULL)
  resu5   <- fitsigmoid(MPerformanceE, x=t0, y=m0, ini.val=resu4$par, simpver=NULL, 
               fig.opt=TRUE, control=list(reltol=1e-30, maxit=200000, trace=FALSE))

  ini.val <- list(0.001, 0.01, c(0.1, 1, 10), 0, 200)
  resu6   <- fitsigmoid(MPerformanceE, x=t0, y=m0, ini.val=ini.val, simpver=2, 
             fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000, trace=FALSE), 
             subdivisions=100L, rel.tol=.Machine$double.eps^0.25, 
             abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
             keep.xy=FALSE, aux=NULL)


graphics.off()

Calculation of Fractal Dimension of Lef Veins Based on the Box-Counting Method

Description

fracdim is used to calculate the fractal dimension of leaf veins based on the box-counting method.

Usage

fracdim(x, y, frac.fig = TRUE, denomi.range = seq(8, 30, by=1), 
        ratiox = 0.02, ratioy = 0.08, main = NULL)

Arguments

x

the x coordinates of leaf-vein pixels.

y

the y coordinates of leaf-vein pixels.

frac.fig

the option of drawing the results of the linear fitting.

denomi.range

the number of equidistant segments of the maximum range between the range of the x coordinates and that of the y coordinates.

ratiox

the the x coordinate of the location parameter for positioning the legend.

ratioy

the the y coordinate of the location parameter for positioning the legend.

main

the main title of the figure.

Details

The box-counting approach uses a group of boxes (squares for simplicity) with different sizes (\delta) to divide the leaf vein image into different parts. Let N represent the number of boxes that include at least one pixel of leaf vein. The maximum of the range of the x coordinates and the range of the y coordinates for leaf-vein pixels is defined as z. Let \delta represent the vector of z/denomi.range. Then, we used the following equation to calculate the fractal dimension of leaf veins:

\mathrm{ln } N = a + b\,\mathrm{ ln} \left({\delta}^{-1}\right),

where b is the theoretical value of the fractal dimension. We can use its estimate as the numerical value of the fractal dimension for a leaf venation network.

Value

a

the estimate of the intercept.

sd.a

the standard deviation of the estimated intercept.

lci.a

the lower bound of the 95% confidence interval of the estimated intercept.

uci.a

the upper bound of the 95% confidence interval of the estimated intercept.

b

the estimate of the slope.

sd.b

the standard deviation of the estimated slope.

lci.a

the lower bound of the 95% confidence interval of the estimated slope.

uci.a

the upper bound of the 95% confidence interval of the estimated slope.

r.sq

the coefficient of determination.

delta

the vector of box sizes.

N

the number of boxes that include at least one pixel of leaf vein.

Note

Here, x and y cannot be adjusted by the adjdata function because the leaf veins are not the leaf's boundary data.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Shi, P., Yu, K., Niinemets, Ü., Gielis, J. (2021) Can leaf shape be represented by the ratio of leaf width to length? Evidence from nine species of Magnolia and Michelia (Magnoliaceae). Forests 12, 41. doi:10.3390/f12010041

Vico, P.G., Kyriacos, S., Heymans, O., Louryan, S., Cartilier, L. (1998) Dynamic study of the extraembryonic vascular network of the chick embryo by fractal analysis. Journal of Theoretical Biology 195, 525-532. doi:10.1006/jtbi.1998.0810

Examples

data(veins)

dev.new()
plot(veins$x, veins$y, cex=0.01, asp=1, cex.lab=1.5, cex.axis=1.5, 
     xlab=expression(italic("x")), ylab=expression(italic("y")))

fracdim(veins$x, veins$y)

graphics.off()

Boundary Data of the Side Projections of Ginkgo biloba Seeds

Description

The data consist of the boundary data of four side projections of G. biloba (Cultivar 'Fozhi') seeds sampled at Nanjing Forestry University campus on September 23, 2021.

Usage

data(ginkgoseed)

Details

In the data set, there are three columns of variables: Code, x, and y. Code saves the codes of individual fruit; x saves the x coordinates of the side projections of seeds in the Cartesian coordinate system (cm); and y saves the y coordinates of the side projections of seeds in the Cartesian coordinate system (cm).

References

Tian, F., Wang, Y., Sandhu, H.S., Gielis, J., Shi, P. (2020) Comparison of seed morphology of two ginkgo cultivars. Journal of Forestry Research 31, 751-758. doi:10.1007/s11676-018-0770-y

Examples

data(ginkgoseed)

uni.C <- sort( unique(ginkgoseed$Code) )
ind   <- 1
Data  <- ginkgoseed[ginkgoseed$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1 <- adjdata(x0, y0, ub.np=2000, len.pro=1/20)
dev.new()
plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, type="l", 
      xlab=expression(italic("x")), ylab=expression(italic("y")) )


  x1        <- Res1$x
  y1        <- Res1$y
  x0.ini    <- mean( x1 )
  y0.ini    <- mean( y1 )
  theta.ini <- pi/4
  a.ini     <- 1
  n1.ini    <- seq(0.6, 1, by=0.1)
  n2.ini    <- 1
  n3.ini    <- 1
  ini.val   <- list(x0.ini, y0.ini, theta.ini, 
                    a.ini, n1.ini, n2.ini, n3.ini)

  Res2 <- fitGE( GE, x=x1, y=y1, ini.val=ini.val, 
                 m=2, simpver=5, nval=1, unit="cm",  
                 par.list=FALSE, fig.opt=TRUE, angle=NULL, 
                 control=list(reltol=1e-20, maxit=20000), 
                 np=2000 )


graphics.off()

Boundary Data of the Vertical Projections of Koelreuteria paniculata Fruit

Description

The data consist of the boundary data of four vertical projections of K. paniculata fruit sampled at Nanjing Forestry University campus in early October 2021.

Usage

data(kp)

Details

In the data set, there are three columns of variables: Code, x, and y. Code saves the codes of individual fruit; x saves the x coordinates of the vertical projections of fruit in the Cartesian coordinate system (cm); and y saves the y coordinates of the vertical projections of fruit in the Cartesian coordinate system (cm).

References

Examples

data(kp)

uni.C <- sort( unique(kp$Code) )
ind   <- 1
Data  <- kp[kp$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1 <- adjdata(x0, y0, ub.np=200, len.pro=1/20)
dev.new()
plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, type="l", 
      xlab=expression(italic("x")), ylab=expression(italic("y")) )


  x1        <- Res1$x
  y1        <- Res1$y
  x0.ini    <- mean( x1 )
  y0.ini    <- mean( y1 )
  theta.ini <- pi
  a.ini     <- 0.9
  n1.ini    <- c(1, 4)
  n2.ini    <- 5
  n3.ini    <- c(5, 10, 15)
  ini.val   <- list(x0.ini, y0.ini, theta.ini, 
                    a.ini, n1.ini, n2.ini, n3.ini)

  Res2 <- fitGE( GE, x=x1, y=y1, ini.val=ini.val, 
                 m=3, simpver=5, nval=1, unit="cm",  
                 par.list=FALSE, fig.opt=TRUE, angle=NULL, 
                 control=list(reltol=1e-20, maxit=20000), 
                 np=2000 )


graphics.off()

Parameter Estimation for the Todd-Smart Equation

Description

lmPE is used to estimate the parameters of the Todd-Smart equation using the multiple linear regression.

Usage

lmPE(x, y, simpver = NULL, dev.angle = NULL, weights = NULL, fig.opt = TRUE, 
     prog.opt = TRUE, xlim = NULL, ylim = NULL, unit = NULL, main = NULL, 
     extr.method = "Shi")

Arguments

x

the x coordinates of the edge of an egg's boundary.

y

the y coordinates of the edge of an egg's boundary.

simpver

an optional argument to use the simplified version of the original Todd-Smart equation.

dev.angle

the angle of deviation for the axis associated with the maximum distance between two points on an egg's profile from the mid-line of the egg's profile.

weights

the weights for the multiple linear regression.

fig.opt

an optional argument to draw the observed and predicted egg's boundaries.

prog.opt

an optional argument to show the running progress for different values of dev.angle, when dev.angle is not NULL.

xlim

the range of the x-axis over which to plot the Todd-Smart curve.

ylim

the range of the y-axis over which to plot the Todd-Smart curve.

unit

the units of the x-axis and the y-axis when showing the Todd-Smart curve.

main

the main title of the figure.

extr.method

an optional argument to fit the planar coordinate data of an egg's profile extracted using different methods.

Details

There are two methods to obtain the major axis (i.e., the mid-line) of an egg's profile: the maximum distance method, and the longest axis adjustment method. In the first method, the straight line through two points forming the maximum distance on the egg's boundary is defined as the major axis. In the second method, we assume that there is an angle of deviation for the longest axis (i.e., the axis associated with the maximum distance between two points on an egg's profile) from the mid-line of the egg's profile. That is to say, the mid-line of an egg's profile is not the axis associated with the maximum distance between two points on the egg's profile. When dev.angle = NULL, the maximum distance method is used; when dev.angle is a numerical value or a numerical vector, the longest axis adjustment method is used. Here, the numerical value of dev.angle is not the angle of deviation for the major axis of an egg's profile from the x-axis, and instead it is the angle of deviation for the longest axis (associated with the maximum distance between two points on the egg's profile) from the mid-line of the egg's profile. It is better to take the option of extr.method = "Shi" for correctly fitting the planar coordinate data of an egg's profile extracted using the protocols proposed by Shi et al. (2015, 2018) (and also see Su et al. (2019)), while it is better to take the option of extr.method = "Biggins" for correctly fitting the planar coordinate data of an egg's profile extracted using the protocols proposed by Biggins et al. (2018). For the planar coordinate data extracted using the protocols of Biggins et al. (2018), there are fewer data points on the two ends of the mid-line than other parts of the egg's profile, which means that the range of the observed x values might be smaller than the actual egg's length. A group of equidistant x values are set along the mid-line, and each x value corresponds to two y values that are respectively located at the upper and lower sides of the egg's profile. Because of the difference in the curvature for differnt parts of the egg's profile, the equidistant x values cannot render the extracted data points on the egg's profile to be regular. For the planar coordinate data extracted using the protocols of Shi et al. (2015, 2018), the data points are more regularly distributed on the egg's profile (perimeter) than those of Biggins et al. (2018), although the x values of the data points along the mid-line are not equidistant.

Value

lm.tse

the fitted results of the multiple linear regression.

par

the estimates of the four model parameters in the Todd-Smart equation.

theta

the angle between the longest axis of an egg's profile (i.e., the axis associated with the maximum distance between two points on the egg's profile) and the x-axis.

epsilon

RSS.vector

the vector of residual sum of squares corresponding to dev.angle, when dev.angle is not NULL.

x.obs

the observed x coordinates.

y.obs

the observed y coordinates.

y.pred

the predicted y coordinates corresponding to the the observed x coordinates.

x.stand.obs

the observed x coordinates when the egg length is fixed to be 2 ranging from -1 to 1.

y.stand.obs

the observed y coordinates when the egg length is fixed to be 2 ranging from -1 to 1.

y.stand.pred

the predicted y coordinates corresponding to the the observed x coordinates, when the egg length is fixed to be 2 ranging from -1 to 1.

scan.length

the length of the egg's boundary. The default is the maximum distance between two points on the egg's boundary.

scan.width

the maximum width of the egg's boundary.

scan.area

the area of the egg's boundary.

scan.perimeter

the perimeter of the egg's boundary based on all data points on the egg's boundary.

RSS.scaled

the residual sum of squares between the observed and predicted y values for a scaled egg's profile whose length equals 2.

RSS

the residual sum of squares between the observed and predicted y values.

sample.size

the number of data points used in the numerical calculation.

RMSE.scaled

the root-mean-square errors between the observed and predicted y values for a scaled egg's profile whose length equals 2.

RMSE

the root-mean-square errors between the observed and predicted y values.

Note

theta is the calculated angle between the longest axis (i.e., the axis associated with the maximum distance between two points on an egg's profile) and the x-axis, and epsilon is the calculated angle of deviation for the longest axis from the mid-line of the egg's profile. This means that the angle between the mid-line and the x-axis is equal to theta + epsilon. Here, RSS, and RMSE are for the observed and predicted y coordinates of the egg's profile, not for those when the egg's length is scaled to 2. There are two figures when fig.opt = TRUE: (i) the observed and predicted egg's boundaries when the egg's length is scaled to 2, and (ii) the observed and predicted egg's boundaries at their actual scales.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston’s universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Preston, F.W. (1953) The shapes of birds' eggs. The Auk 70, 160-182.

Todd, P.H., Smart, I.H.M. (1984) The shape of birds' eggs. Journal of Theoretical Biology 106, 239-243. doi:10.1016/0022-5193(84)90021-3

Examples

data(eggs)

uni.C <- sort( unique(eggs$Code) )
ind   <- 8
Data  <- eggs[eggs$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1  <- adjdata(x0, y0, ub.np=3000, len.pro=1/20)
x1    <- Res1$x
y1    <- Res1$y

dev.new()
plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) ) 


  Res2 <- lmPE(x1, y1, simpver=NULL, dev.angle=NULL, unit="cm")
  summary( Res2$lm.tse )
  Res2$RMSE.scaled / 2

  if(FALSE){
    dev.new()
    xg1 <- seq(-1, 1, len=1000)
    yg1 <- TSE(P=Res2$par, x=xg1, simpver=NULL)
    xg2 <- seq(1, -1, len=1000)
    yg2 <- -TSE(P=Res2$par, x=xg2, simpver=NULL)
    plot(xg1, yg1, asp=1, type="l", col=2, ylim=c(-1,1), cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic(x)), ylab=expression(italic(y)))
    lines(xg2, yg2, col=4)

    dev.new()
    plot(Res2$x.obs, Res2$y.obs, asp=1, cex.lab=1.5, cex.axis=1.5,  
      xlab=expression(italic(x)), ylab=expression(italic(y)), type="l")
    lines(Res2$x.obs, Res2$y.pred, col=2)

    dev.new()
    plot(Res2$x.stand.obs, Res2$y.stand.obs, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic(x)), ylab=expression(italic(y)), type="l")
    lines(Res2$x.stand.obs, Res2$y.stand.pred, col=2)
  }

  Res3 <- lmPE(x1, y1, simpver=NULL, dev.angle=seq(-0.05, 0.05, by=0.0001), unit="cm")
  summary( Res3$lm.tse )
  Res3$epsilon
  Res3$RMSE.scaled / 2

  Res4 <- lmPE(x1, y1, simpver=1, dev.angle=NULL, unit="cm")
  summary( Res4$lm.tse )



graphics.off()

Parameter Estimation for the Troscianko Equation

Description

lmTE is used to estimate the parameters of the Troscianko equation using the multiple linear regression, and the estimated values of the parameters are only used as the initial values for using the fitETE function

Usage

lmTE(x, y, dev.angle = NULL, weights = NULL, fig.opt = TRUE, 
     prog.opt = TRUE, xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

x

the x coordinates of the edge of an egg's boundary.

y

the y coordinates of the edge of an egg's boundary.

dev.angle

the angle of deviation for the axis associated with the maximum distance between two points on an egg's profile from the mid-line of the egg's profile.

weights

the weights for the multiple linear regression.

fig.opt

an optional argument to draw the observed and predicted egg's boundaries.

prog.opt

an optional argument to show the running progress for different values of dev.angle, when dev.angle is not NULL.

xlim

the range of the x-axis over which to plot the Troscianko curve.

ylim

the range of the y-axis over which to plot the Troscianko curve.

unit

the units of the x-axis and the y-axis when showing the Troscianko curve.

main

the main title of the figure.

Details

The estimated values of the parameters using the lmTE function tend to be NOT globally optimal, and the values are only used as the initial values for using the fitETE function. There are two methods to obtain the major axis (i.e., the mid-line) of an egg's profile: the maximum distance method, and the longest axis adjustment method. In the first method, the straight line through two points forming the maximum distance on the egg's boundary is defined as the major axis. In the second method, we assume that there is an angle of deviation for the longest axis (i.e., the axis associated with the maximum distance between two points on an egg's profile) from the mid-line of the egg's profile. That is to say, the mid-line of an egg's profile is not the axis associated with the maximum distance between two points on the egg's profile. When dev.angle = NULL, the maximum distance method is used; when dev.angle is a numerical value or a numerical vector, the longest axis adjustment method is used. Here, the numerical value of dev.angle is not the angle of deviation for the major axis of an egg's profile from the x-axis, and instead it is the angle of deviation for the longest axis (associated with the maximum distance between two points on the egg's profile) from the mid-line of the egg's profile. The planar coordinate data of an egg's profile are extracted using the protocols proposed by Shi et al. (2015, 2018) (and also see Su et al. (2019)). For the planar coordinate data extracted using the protocols of Shi et al. (2015, 2018), the data points are more regularly distributed on the egg's profile (perimeter), although the x values of the data points along the mid-line are not equidistant.

Value

lm.te

the fitted results of the multiple linear regression.

par

the estimates of the four model parameters in the Troscianko equation.

theta

the angle between the longest axis of an egg's profile (i.e., the axis associated with the maximum distance between two points on the egg's profile) and the x-axis.

epsilon

RSS.vector

the vector of residual sum of squares corresponding to dev.angle, when dev.angle is not NULL.

x.obs

the observed x coordinates.

y.obs

the observed y coordinates.

y.pred

the predicted y coordinates corresponding to the the observed x coordinates.

x.stand.obs

the observed x coordinates when the egg length is fixed to be 2 ranging from -1 to 1.

y.stand.obs

the observed y coordinates when the egg length is fixed to be 2 ranging from -1 to 1.

y.stand.pred

the predicted y coordinates corresponding to the the observed x coordinates, when the egg length is fixed to be 2 ranging from -1 to 1.

scan.length

the length of the egg's boundary. The default is the maximum distance between two points on the egg's boundary.

scan.width

the maximum width of the egg's boundary.

scan.area

the area of the egg's boundary.

scan.perimeter

the perimeter of the egg's boundary based on all data points on the egg's boundary.

RSS.scaled

the residual sum of squares between the observed and predicted y values for a scaled egg's profile whose length equals 2.

RSS

the residual sum of squares between the observed and predicted y values.

sample.size

the number of data points used in the numerical calculation.

RMSE.scaled

the root-mean-square errors between the observed and predicted y values for a scaled egg's profile whose length equals 2.

RMSE

the root-mean-square errors between the observed and predicted y values.

Note

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston’s universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Troscianko, J. (2014). A simple tool for calculating egg shape, volume and surface area from digital images. Ibis, 156, 874-878. doi:10.1111/ibi.12177

Examples

data(eggs)

uni.C <- sort( unique(eggs$Code) )
ind   <- 8
Data  <- eggs[eggs$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

Res1  <- adjdata(x0, y0, ub.np=3000, len.pro=1/20)
x1    <- Res1$x
y1    <- Res1$y

dev.new()
plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) ) 


  Res2 <- lmTE(x1, y1, dev.angle=NULL, unit="cm")
  summary( Res2$lm.te )
  Res2$RMSE.scaled / 2

  if(FALSE){
    dev.new()
    xg1 <- seq(-1, 1, len=1000)
    yg1 <- TE(P=Res2$par, x=xg1)
    xg2 <- seq(1, -1, len=1000)
    yg2 <- -TE(P=Res2$par, x=xg2)
    plot(xg1, yg1, asp=1, type="l", col=2, ylim=c(-1,1), cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic(x)), ylab=expression(italic(y)))
    lines(xg2, yg2, col=4)

    dev.new()
    plot(Res2$x.obs, Res2$y.obs, asp=1, cex.lab=1.5, cex.axis=1.5,  
      xlab=expression(italic(x)), ylab=expression(italic(y)), type="l")
    lines(Res2$x.obs, Res2$y.pred, col=2)

    dev.new()
    plot(Res2$x.stand.obs, Res2$y.stand.obs, asp=1, cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic(x)), ylab=expression(italic(y)), type="l")
    lines(Res2$x.stand.obs, Res2$y.stand.pred, col=2)
  }

  Res3 <- lmTE(x1, y1, dev.angle=seq(-0.05, 0.05, by=0.0001), unit="cm")
  summary( Res3$lm.te )
  Res3$epsilon
  Res3$RMSE.scaled / 2



graphics.off()

Height Growth Data of Bamboo Shoots

Description

The height data of four species of bamboo at Nanjing Forestry University campus in 2016.

Usage

data(shoots)

Details

In the data set, there are four columns of variables: Code, LatinName, x, and y. Code saves the number codes of different bamboo species; LatinName saves the Latin names of different bamboo species; x saves the investigation times (days from a specific starting time of growth, and where every bamboo has a different starting time of growth); and y saves the measured aboveground height values (cm).

Code = 1 represents Phyllostachys iridescens, and the starting time (namely time = 0) was defined as 12:00, 3rd April, 2016;

Code = 2 represents Phyllostachys mannii, and the starting time (namely time = 0) was defined as 12:00, 4th April, 2016;

Code = 3 represents Pleioblastus maculatus, and the starting time (namely time = 0) was defined as 12:00, 29th April, 2016;

Code = 4 represents Sinobambusa tootsik, and the starting time (namely time = 0) was defined as 12:00, 18th April, 2016.

References

Examples

data(shoots)
attach(shoots)
# Choose a species
# 1: Phyllostachys iridescens; 2: Phyllostachys mannii; 
# 3: Pleioblastus maculatus; 4: Sinobambusa tootsik
ind <- 4
x1  <- x[Code == ind]
y1  <- y[Code == ind] 

dev.new()
plot(x1, y1, cex=1.5, cex.lab=1.5, cex.axis=1.5, xlab="Time (days)", ylab="Height (cm)")


  delta0  <- c(0.5, 1, 2, 5, 10, 20)
  ini.val <- list(600, 25, 0, 40, delta0)
  resu1   <- fitsigmoid(MLRFE, x=x1, y=y1, ini.val=ini.val, simpver=NULL, 
               fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000), 
               subdivisions = 100L, rel.tol=.Machine$double.eps^0.25,                
               abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
               keep.xy=FALSE, aux=NULL)


graphics.off()

Sigmoid Growth Equation

Description

sigmoid is used to calculate the y values (e.g., biomass, height, body length, and so on) at given investigation times.

Usage

sigmoid(expr, P, x, simpver = 1, subdivisions = 100L,
        rel.tol = .Machine$double.eps^0.25, 
        abs.tol = rel.tol, stop.on.error = TRUE, 
        keep.xy = FALSE, aux = NULL)

Arguments

expr

a performance equation or one of its simplified versions.

P

the parameters of the performance equation or one of its simplified versions.

x

the given investigation times.

simpver

an optional argument to use the simplfied version of the performance equation.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The performance equations denote MbetaE, MBriereE, MLRFE, and their simplified versions. The arguments of P and simpver should correspond to expr (i.e., MbetaE, MBriereE, MLRFE, and MPerformanceE). The sigmoid curve is the integral of the performance equation or one of its simplified versions.

Value

The y values (i.e., biomass, height, body length, and so on) at given investigation times. The growth euqation is actually an integral of the performance equation or one of its simplified versions.

Note

Here, the user can define other performance equations, but new equations or their simplified versions should include the lower and upper thresholds in the x-axis corresponding to y = 0, whose indices of the parameters in P should be the same as those in MbetaE, MBriereE, MLRFE, and MPerformanceE).

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

References

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. doi:10.3390/plants11131769

Examples


Pa1 <- c(3, 3, 10, 1)
xv1 <- seq(-5, 15, len=2000)
yv1 <- sigmoid(MBriereE, P=Pa1, x=xv1, simpver=1)
Pa2 <- c(3, 3, 2, 12, 1)
yv2 <- sigmoid(MBriereE, P=Pa2, x=xv1, simpver=NULL)
Pa3 <- c(200, 1.32, 0.85, 12, 3.7, 17)
yv3 <- sigmoid(MPerformanceE, P=Pa3, x=xv1, simpver=1)

dev.new()
plot( xv1, yv2, cex.lab=1.5, cex.axis=1.5, type="l", col=4,
      xlab=expression(italic(x)), ylab=expression(italic(y)) )
lines( xv1, yv1, col=2 )
lines( xv1, yv3, col=3 )

graphics.off()

Boundary Data of Eight Sea Stars

Description

The data consist of the boundary data of eight sea stars from five species.

Usage

data(starfish)

Details

In the data set, there are four columns of variables: Code, LatinName, x, and y. Code saves the codes of individual sea stars; LatinName saves the Latin names of the eight sea stars; x saves the x coordinates of the eight sea stars in the Cartesian coordinate system (cm); and y saves the y coordinates of the eight sea stars in the Cartesian coordinate system (cm). In Code, codes 1-9 represent Anthenoides tenuis, Culcita schmideliana sample 1, Culcita schmideliana sample 2, Culcita schmideliana sample 3, Stellaster equestris, Tosia australis, Tosia magnifica sample 1, and Tosia magnifica sample 2, respectively. See Table A1 published in Shi et al. (2020).

References

Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. doi:10.3390/sym12040645

Examples

data(starfish)

uni.C <- sort( unique(starfish$Code) )
ind   <- 2
Data  <- starfish[starfish$Code==uni.C[ind], ]
x0    <- Data$x
y0    <- Data$y

dev.new()
plot( x0, y0, asp=1, type="l", cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) )
length(x0)

Res1 <- adjdata(x0, y0, ub.np=400, times=1.2, len.pro=1/20)
x1   <- Res1$x
y1   <- Res1$y

dev.new()
plot( x1, y1, asp=1, type="l", cex.lab=1.5, cex.axis=1.5, 
      xlab=expression(italic("x")), ylab=expression(italic("y")) )
length(x1)

graphics.off()

Leaf Vein Data of Michelia compressa

Description

The data consist of the leaf vein data of a leaf of M. compressa sampled at Nanjing Forestry University campus in late July 2019.

Usage

data(veins)

Details

In the data set, there are two columns of variables: x and y. x saves the x coordinates of the leaf veins in the Cartesian coordinate system (cm); y saves the y coordinates of the leaf veins in the Cartesian coordinate system (cm).

Note

The data cannot be adjusted by the adjdata function.

References

Examples

data(veins)

dev.new()
plot(veins$x, veins$y, cex=0.01, asp=1, cex.lab=1.5, cex.axis=1.5, 
     xlab=expression(italic("x")), ylab=expression(italic("y")))

graphics.off()

Planar Coordinates of Picea glauca Tree Rings

Description

The data consist of the planar coordinates of Picea glauca tree rings.

Usage

data(whitespruce)

Details

In the data set, there are three columns of variables: Code, x, and y. Code saves the age codes of tree rings from the 2nd year to the 44th year; x saves the x coordinates of the tree rings in the Cartesian coordinate system (cm); and y saves the y coordinates of the tree rings in the Cartesian coordinate system (cm).

References

Examples

data(whitespruce)

uni.C <- sort( unique(whitespruce$Code) )  
Data  <- whitespruce[whitespruce$Code==uni.C[10], ]
x0    <- Data$x
y0    <- Data$y
Res1  <- adjdata(x0, y0, ub.np=2000, len.pro=1/20)

plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, type="l", 
      xlim=c(3, 13), ylim=c(3, 13), col="grey73", lwd=2,
      xlab=expression(italic(x)), ylab=expression(italic(y)) )



  uni.C <- sort( unique(whitespruce$Code) )  
  for(i in 1:length(uni.C)){
    Data  <- whitespruce[whitespruce$Code==uni.C[i], ]
    x0    <- Data$x
    y0    <- Data$y

    Res1  <- adjdata(x0, y0, ub.np=200, len.pro=1/10)
  
    if(i == 1){
      plot( Res1$x, Res1$y, asp=1, cex.lab=1.5, cex.axis=1.5, type="l", 
            xlim=c(3, 13), ylim=c(3, 13), col=1, lwd=1,
            xlab=expression(italic(x)), ylab=expression(italic(y)) )
    }
    if(i > 1) lines(Res1$x, Res1$y, col=1, lwd=1)
   
  }

  uni.C   <- sort( unique(whitespruce$Code) ) 
  uni.C   <- uni.C[1:12]
  Length  <- c()
  results <- data.frame(Code=c(), x0=c(), y0=c(), theta=c(),
                   a=c(), k=c(), n1=c(), r.sq=c(), RSS=c(), N=c())
  for(i in 1:length(uni.C)){
    Data      <- whitespruce[whitespruce$Code==uni.C[i], ]
    x0        <- Data$x
    y0        <- Data$y
    Res1      <- adjdata(x0, y0, ub.np=200, len.pro=1/10)
    x1        <- Res1$x
    y1        <- Res1$y
    x0.ini    <- mean( x1 )
    y0.ini    <- mean( y1 )
    theta.ini <- c(0, pi/4, pi/2)
    a.ini     <- 0.9
    k.ini     <- 1
    n1.ini    <- c(1.5, 2, 2.5)
    ini.val   <- list(x0.ini, y0.ini, theta.ini, 
                      a.ini, k.ini, n1.ini)

    print(paste("Progress: ", i, "/", length(uni.C), sep=""))
    H <- NULL
    try( H <- fitGE(GE, x=x1, y=y1, ini.val=ini.val, 
                    m=4, simpver=9, unit="cm", par.list=FALSE,
                    stand.fig=FALSE, angle=NULL, fig.opt=FALSE,  
                    control=list(reltol=1e-20, maxit=20000), 
                    np=2000), silent=TRUE )
    if(is.null(H)){
      RE <- data.frame(Code=uni.C[i], x0=NA, y0=NA, theta=NA,
               a=NA, k=NA, n1=NA, r.sq=NA, RSS=NA, N=NA)
    }
    if(!is.null(H)){
      RE     <- data.frame(Code=uni.C[i], x0=H$par[1], y0=H$par[2], 
                  theta=H$par[3], a=H$par[4], k=H$par[5], n1=H$par[6],
                  r.sq=H$r.sq, RSS=H$RSS, N=H$sample.size)
      Length <- c(Length, max(max(H$y.stand.pred)[1]-min(H$y.stand.pred)[1], 
                  max(H$x.stand.pred)[1]-min(H$x.stand.pred)[1])[1])
      if(i == 1){
        plot(H$x.obs, H$y.obs, asp=1, xlim=c(7.4, 8.6), ylim=c(7.4, 8.6),
             cex.lab=1.5, cex.axis=1.5, type="l", lwd=2, col="grey70",
             xlab=expression(italic(x)), ylab=expression(italic(y)))
        lines(H$x.pred, H$y.pred, col=2)         
      } 
      if(i > 1){
        lines(H$x.obs, H$y.obs, lwd=2, col="grey70")
        lines(H$x.pred, H$y.pred, col=2)   
      }
      
    }
    results <- rbind(results, RE)
  }

  # To adjust the estimates of partial parameters to ensure k <= 1
  results2            <- results
  Ind                 <- results$k > 1
  results2$theta[Ind] <- results$theta[Ind] + pi/2
  results2$a[Ind]     <- results$a[Ind] * results$k[Ind]^(1/results$n1[Ind])
  results2$k[Ind]     <- 1/results$k[Ind] 
  results2
  Length/2

Calculation of the First-Order Derivative of the Explicit Preston Equation

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

Calculation of the First-Order Derivative of the Explicit Troscianko Equation

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

Calculation of the First-Order Derivative of the Narushin-Romanov-Griffin Equation

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

Calculation of the First-Order Derivative of the Simplified Gielis Equation

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

Calculation of the Ordinate For an Arbitrary Point on the Preston Curve in the Plane

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Calculation of the Ordinate For an Arbitrary Point on the Troscianko Curve in the Plane

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Calculation of the Polar Radius of the Gielis Curve

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Leaf size distribution of Shibataea chinensis

Description

Usage

Details

References

Examples

Modified Briere Equation