| lmPE {biogeom} | R Documentation |
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 |
y |
the |
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 |
xlim |
the range of the |
ylim |
the range of the |
unit |
the units of the |
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 |
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 |
RSS.vector |
the vector of residual sum of squares corresponding to |
x.obs |
the observed |
y.obs |
the observed |
y.pred |
the predicted |
x.stand.obs |
the observed |
y.stand.obs |
the observed |
y.stand.pred |
the predicted |
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 |
RSS |
the residual sum of squares between the observed and predicted |
sample.size |
the number of data points used in the numerical calculation. |
RMSE.scaled |
the root-mean-square errors between the observed and predicted |
RMSE |
the root-mean-square errors between the observed and predicted |
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.
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
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
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., 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
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
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
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
See Also
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()