fitGE {biogeom}

Data-Fitting Function for the Gielis Equation

Description

fitGE is used to estimate the parameters of the original (or twin) Gielis equation or one of its simplified versions.

Usage

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)

Arguments

expr

the original (or twin) Gielis equation or one of its simplified versions.

x

the x coordinates of a polygon's boundary.

y

the y coordinates of a polygon's boundary.

ini.val

the list of initial values for the model parameters.

m

the given m value that determines the number of angles of the Gielis curve within [0, 2\pi).

simpver

an optional argument to use the simplified version of the original (or twin) Gielis equation.

nval

the specified value for n_{1} or n_{2} or n_{3} 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 x-axis).

angle

the angle between the major axis and the x-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 x-axis.

np

the number of data points on the predicted Gielis curve.

xlim

the range of the x-axis over which to plot the Gielis curve.

ylim

the range of the y-axis over which to plot the Gielis curve.

unit

the unit of the x-axis and the y-axis when showing the Gielis curve.

main

the main title of the figure.

Details

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., \pi/4) defined by the user, it indicates that the major axis is rotated by the amount (\pi/4) counterclockwise from the x-axis.

Value

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 x coordinates at the standard state.

x.stand.pred

the predicted x coordinates at the standard state.

y.stand.obs

the observed y coordinates at the standard state.

y.stand.pred

the predicted y 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 x coordinates at the transferred polar angles as defined by the user.

x.pred

the predicted x coordinates at the transferred polar angles as defined by the user.

y.obs

the observed y coordinates at the transferred polar angles as defined by the user.

y.pred

the predicted y coordinates at the transferred polar angles as defined by the user.

Note

simpver in GE is different from that in TGE.

Author(s)

Peijian Shi [e-mail], Johan Gielis [e-mail], Brady K. Quinn [e-mail].

References

Gielis, J. (2003) A generic geometric transformation that unifies a wide range of natural and abstract shapes. American Journal of Botany 90, 333-338. 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. doi:10.3390/sym14010023

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308-313. 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. doi:10.1111/nyas.14862

Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. 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. doi:10.1002/ece3.1728

See Also

areaGE, curveGE, DSGE, GE, SurfaceAreaSGE, TGE, VolumeSGE

Examples

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()