biogeom (version 1.4.3)

TSE: 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)

Value

The \(y\) values predicted by the Todd-Smart equation.

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.

Author

Peijian Shi , Johan Gielis , Brady K. Quinn .

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.

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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1002/ece3.4412")

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

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

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. tools:::Rd_expr_doi("10.1111/nyas.14862")

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

See Also

lmPE, PE

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()