biogeom (version 1.4.3)

lmPE: 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")

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

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.

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.

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.

Author

Peijian Shi , Johan Gielis , Brady K. Quinn .

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.

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")

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. tools:::Rd_expr_doi("10.3389/fpls.2015.00856")

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. tools:::Rd_expr_doi("10.3390/f11020246")

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. tools:::Rd_expr_doi("10.3390/f9110714")

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. tools:::Rd_expr_doi("10.3390/sym15010231")

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. tools:::Rd_expr_doi("10.1016/j.gecco.2019.e00666")

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

curveEPE, fitEPE, PE, TSE

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")) ) 

# \donttest{
  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()