biogeom (version 1.4.3)

lmTE: 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)

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

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.

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.

Author

Peijian Shi , Johan Gielis , Brady K. Quinn .

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.

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")

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., 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")

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

See Also

fitETE, TE

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 <- 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()