Data-Fitting Function for the Gielis Equation
fitGE is used to estimate the parameters of the original (or twin) Gielis equation or one of its simplified versions.
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fitGE(expr, x, y, ini.val, m = 1, simpver = NULL, nval = nval, control = list(), par.list = FALSE, stand.fig = TRUE, angle = NULL, fig.opt = FALSE, np = 2000, xlim = NULL, ylim = NULL, unit = NULL, main = NULL)
expr: the original (or twin) Gielis equation or one of its simplified versions.x: the coordinates of a polygon's boundary.y: the coordinates of a polygon's boundary.ini.val: the list of initial values for the model parameters.m: the given value that determines the number of angles of the Gielis curve within .simpver: an optional argument to use the simplified version of the original (or twin) Gielis equation.nval: the specified value for or or in the simplified versions.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 polygons at the standard state (i.e., the polar point is located at (0, 0), and the major axis overlaps with the -axis).angle: the angle between the major axis and the -axis, which can be defined by the user.fig.opt: an optional argument of drawing the observed and predicted polygons at arbitrary angle between the major axis and the -axis.np: the number of data points on the predicted Gielis curve.xlim: the range of the -axis over which to plot the Gielis curve.ylim: the range of the -axis over which to plot the Gielis curve.unit: the unit of the -axis and the -axis when showing the Gielis curve.main: the main title of the figure.The arguments of m, simpver, and nval should correspond to expr (i.e., GE or TGE). Please note the differences in the simplified version number and the number of parameters between GE and TGE. 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 radii. The optim function in package stats was used to carry out the Nelder-Mead algorithm. When angle = NULL, the observed polygon 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 polygon.
scan.width: the observed width of the polygon.
scan.area: the observed area of the polygon.
r.sq: the coefficient of determination between the observed and predicted polar radii.
RSS: the residual sum of squares between the observed and predicted polar radii.
sample.size: the number of data points used in the data fitting.
phi.stand.obs: the polar angles at the standard state.
phi.trans: the transferred polar angles rotated as defined by the user.
r.stand.obs: the observed polar radii at the standard state.
r.stand.pred: the predicted polar radii at the standard state.
x.stand.obs: the observed coordinates at the standard state.
x.stand.pred: the predicted coordinates at the standard state.
y.stand.obs: the observed coordinates at the standard state.
y.stand.pred: the predicted coordinates at the standard state.
r.obs: the observed polar radii at the transferred polar angles as defined by the user.
r.pred: the predicted polar radii at the transferred polar angles as defined by the user.
x.obs: the observed coordinates at the transferred polar angles as defined by the user.
x.pred: the predicted coordinates at the transferred polar angles as defined by the user.
y.obs: the observed coordinates at the transferred polar angles as defined by the user.
y.pred: the predicted coordinates at the transferred polar angles as defined by the user.
simpver in GE is different from that in TGE.
Peijian Shi pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca .
Gielis, J. (2003) A generic geometric transformation that unifies a wide range of natural and abstract shapes. American Journal of Botany 90, 333338. tools:::Rd_expr_doi("10.3732/ajb.90.3.333")
Li, Y., Quinn, B.K., Gielis, J., Li, Y., Shi, P. (2022) Evidence that supertriangles exist in nature from the vertical projections of Koelreuteria paniculata fruit. Symmetry
14, 23. tools:::Rd_expr_doi("10.3390/sym14010023")
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., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. tools:::Rd_expr_doi("10.3390/sym12040645")
Shi, P., Xu, Q., Sandhu, H.S., Gielis, J., Ding, Y., Li, H., Dong, X. (2015) Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant. Ecology and Evolution 5, 4578-4589. tools:::Rd_expr_doi("10.1002/ece3.1728")
areaGE, curveGE, DSGE, GE, SurfaceAreaSGE, TGE, VolumeSGE
data(eggs) uni.C <- sort( unique(eggs$Code) ) ind <- 1 Data <- eggs[eggs$Code==uni.C[ind], ] x0 <- Data$x y0 <- Data$y Res1 <- adjdata(x0, y0, ub.np=200, times=1.2, len.pro=1/20) x1 <- Res1$x y1 <- Res1$y Res2 <- adjdata(x0, y0, ub.np=40, times=1, len.pro=1/2, index.sp=20) x2 <- Res2$x y2 <- Res2$y Res3 <- adjdata(x0, y0, ub.np=100, times=1, len.pro=1/2, index.sp=100) x3 <- Res3$x y3 <- Res3$y dev.new() plot( x2, y2, asp=1, cex.lab=1.5, cex.axis=1.5, xlab=expression(italic("x")), ylab=expression(italic("y")), pch=1, col=4 ) points( x3, y3, col=2) x0.ini <- mean( x1 ) y0.ini <- mean( y1 ) theta.ini <- pi a.ini <- sqrt(2) * max( y0.ini-min(y1), x0.ini-min(x1) ) n1.ini <- c(5, 25) n2.ini <- c(15, 25) if(ind == 2){ n1.ini <- c(0.5, 1) n2.ini <- c(6, 12) } ini.val <- list(x0.ini, y0.ini, theta.ini, a.ini, n1.ini, n2.ini) Res4 <- fitGE( GE, x=x1, y=y1, ini.val=ini.val, m=1, simpver=1, nval=1, unit="cm", par.list=FALSE, fig.opt=TRUE, angle=NULL, control=list(reltol=1e-20, maxit=20000), np=2000 ) Res4$par sqrt(sum((Res4$y.stand.obs-Res4$y.stand.pred)^2)/Res4$sample.size) xx <- Res4$x.stand.obs yy <- Res4$y.stand.obs library(spatstat.geom) poly0 <- as.polygonal(owin(poly=list(x=xx, y=yy))) area(poly0) areaGE(GE, P = Res4$par[4:6], m=1, simpver=1) # The following code is used to # calculate the root-mean-square error (RMSE) in the y-coordinates ind1 <- which(yy >= 0) ind2 <- which(yy < 0) xx1 <- xx[ind1] # The upper part of the egg yy1 <- yy[ind1] xx2 <- xx[ind2] # The lower part of the egg yy2 <- yy[ind2] Para <- c(0, 0, 0, Res4$par[4:length(Res4$par)]) PartU <- curveGE(GE, P=Para, phi=seq(0, pi, len=100000), m=1, simpver=1, fig.opt=FALSE) xv1 <- PartU$x yv1 <- PartU$y PartL <- curveGE(GE, P=Para, phi=seq(pi, 2*pi, len=100000), m=1, simpver=1, fig.opt=FALSE) xv2 <- PartL$x yv2 <- PartL$y ind3 <- c() for(q in 1:length(xx1)){ ind.temp <- which.min(abs(xx1[q]-xv1)) ind3 <- c(ind3, ind.temp) } ind4 <- c() for(q in 1:length(xx2)){ ind.temp <- which.min(abs(xx2[q]-xv2)) ind4 <- c(ind4, ind.temp) } RSS <- sum((yy1-yv1[ind3])^2) + sum((yy2-yv2[ind4])^2) RMSE <- sqrt( RSS/length(yy) ) # To calculate the volume of the Gielis egg when simpver=1 & m=1 VolumeSGE(P=Res4$par[4:6]) # To calculate the surface area of the Gielis egg when simpver=1 & m=1 SurfaceAreaSGE(P=Res4$par[4:6]) graphics.off()
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