# Data-Fitting Function for the Explicit Troscianko Equation `fitETE` is used to estimate the parameters of the explicit Troscianko equation. UTF-8 ```r 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) ``` ## Arguments - `x`: the $x$ coordinates of an egg's profile. - `y`: the $y$ 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 $x$-axis). - `angle`: the angle between the mid-line and the $x$-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 $x$-axis. - `np`: the number of data points on the predicted explicit Troscianko curve. - `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 unit of the $x$-axis and the $y$-axis when showing the Troscianko curve. - `main`: the main title of the figure. ## Details 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 $y$ 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., $\pi/4$) defined by the user, it indicates that the major axis is rotated by the amount ($\pi/4$) counterclockwise from the $x$-axis. ## Returns - **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 $y$ values on the Troscianko curve. - **RSS**: the residual sum of squares between the observed and predicted $y$ values on the Troscianko curve. - **sample.size**: the number of data points used in the data fitting. - **x.stand.obs**: the observed $x$ coordinates of the points on the Troscianko curve at the standard state. - **y.stand.obs**: the observed $y$ coordinates of the points on the Troscianko curve at the standard state. - **y.stand.pred**: the predicted $y$ coordinates of the points on the Troscianko curve at the standard state. - **x.obs**: the observed $x$ coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user. - **y.obs**: the observed $y$ coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user. - **y.pred**: the predicted $y$ coordinates of the points on the Troscianko curve at the transferred polar angles as defined by the user. ## Note In the outputs, there are no `x.stand.pred` and `x.pred`, because `y.stand.obs` and `y.stand.pred` share the same $x$ values (i.e., `x.stand.obs`), and `y.obs` and `y.pred` share the same $x$ values (i.e., `x.obs`). ## 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. 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$-$9738. tools:::Rd_expr_doi("10.1002/ece3.4412") Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. **Computer Journal** 7, 308$-$313. tools:::Rd_expr_doi("10.1093/comjnl/7.4.308") Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. **Annals of the New York Academy of Sciences** 1516, 123$-$134. tools:::Rd_expr_doi("10.1111/nyas.14862") Shi, P., Wang, L., Quinn, B.K., Gielis, J. (2023) A new program to estimate the parameters of Preston's equation, a general formula for describing the egg shape of birds. **Symmetry** 15, 231. tools:::Rd_expr_doi("10.3390/sym15010231") Troscianko, J. (2014). A simple tool for calculating egg shape, volume and surface area from digital images. **Ibis**, 156, 874$-$878. tools:::Rd_expr_doi("10.1111/ibi.12177") ## See Also `curveETE`, `TE`, `lmTE` ## Examples ```r data(eggs) uni.C <- sort( unique(eggs$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() ```

fitETE function