Data-Fitting Function for the Explicit Troscianko Equation
fitETE is used to estimate the parameters of the explicit Troscianko equation.
UTF-8
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)
x: the coordinates of an egg's profile.y: the 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 -axis).angle: the angle between the mid-line and the -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 -axis.np: the number of data points on the predicted explicit Troscianko curve.xlim: the range of the -axis over which to plot the Troscianko curve.ylim: the range of the -axis over which to plot the Troscianko curve.unit: the unit of the -axis and the -axis when showing the Troscianko curve.main: the main title of the figure.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 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., ) defined by the user, it indicates that the major axis is rotated by the amount () counterclockwise from the -axis.
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 values on the Troscianko curve.
RSS: the residual sum of squares between the observed and predicted values on the Troscianko curve.
sample.size: the number of data points used in the data fitting.
x.stand.obs: the observed coordinates of the points on the Troscianko curve at the standard state.
y.stand.obs: the observed coordinates of the points on the Troscianko curve at the standard state.
y.stand.pred: the predicted coordinates of the points on the Troscianko curve at the standard state.
x.obs: the observed coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user.
y.obs: the observed coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user.
y.pred: the predicted coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user.
In the outputs, there are no x.stand.pred and x.pred, because y.stand.obs and y.stand.pred share the same values (i.e., x.stand.obs), and y.obs and y.pred share the same values (i.e., x.obs).
Peijian Shi pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca .
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, 97289738. tools:::Rd_expr_doi("10.1002/ece3.4412")
Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308313. tools:::Rd_expr_doi("10.1093/comjnl/7.4.308")
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, 123134. tools:::Rd_expr_doi("10.1111/nyas.14862")
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")
Troscianko, J. (2014). A simple tool for calculating egg shape, volume and surface area from digital images. Ibis, 156, 874878. tools:::Rd_expr_doi("10.1111/ibi.12177")
curveETE, TE, lmTE
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()
Useful links