fitNRGE {biogeom}

Parameter Estimation for the Narushin-Romanov-Griffin Equation

Description

fitNRGE is used to estimate the parameters of the Narushin-Romanov-Griffin equation.

Usage

fitNRGE(x, y, dev.angle = NULL, ini.C = c(-1, 0.1, 0.5, 1), 
        strip.num = 2000, control = list(), fig.opt = TRUE, np = 2000,
        xlim = NULL, ylim = NULL, unit = NULL, main = NULL)

Arguments

x

the x coordinates of the edge of an egg's boundary.

y

the y coordinates of the edge of an egg's boundary.

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.

ini.C

the initial value(s) of parameter C in the Narushin-Romanov-Griffin equation.

strip.num

the number of equidistant strips intersecting with the egg's boundary that are horizontally placed. See Shi et al. (2018, 2020) for details.

control

the list of control parameters for using the optim function in package stats.

fig.opt

an optional argument to draw the observed and predicted egg's boundaries.

np

the number of data points on the predicted Narushin-Romanov-Griffin curve.

xlim

the range of the x-axis over which to plot the Narushin-Romanov-Griffin curve.

ylim

the range of the y-axis over which to plot the Narushin-Romanov-Griffin curve.

unit

the units of the x-axis and the y-axis when showing the Narushin-Romanov-Griffin curve.

main

the main title of the figure.

Details

The NRGE (see NRGE) has a complex model structure with four parameters (i.e., A, B, C, and D). Because three out of four parameters of NRGE have clear biological and geometric meanings (i.e., A, B, and D), their values could be estimated by means of numerical calculation. After obtaining the numerical values of the three parameters, the Nelder-Mead algorithm (Nelder and Mead, 1965) was used to estimate C. Because of the failure of the optimization method to estimate the major axis (i.e., the mid-line) and model parameters of NRGE, it was difficult to define the egg length axis, although it is essential for calculating A, B, and D. For this reason, two methods were used to obtain the major axis: 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 profile is defined as the major axis (i.e., the mid-line). 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 angle = NULL, the maximum distance method is used; when 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. Once the major axis is established, the distance of the major axis can be calculated as the estimate of A. Using the maximum distance method, A equals the maximum distance. Using the longest axis adjustment method, A may be slightly smaller than the maximum distance. After rotating the major axis to make it overlap with the x-axis, a large number of equidistant strips can be used (e.g., 2000) from the egg base to egg tip to intersect the egg's boundary. This methodology makes it easy to obtain the maximum egg's breadth (i.e., B) and D. The residual sum of squares (RSS) between the observed and predicted y values can be minimized using an optimization method (Nelder and Mead, 1965) to estimate C. Despite the complex structure of NRGE (see NRGE), the optimization method for estimating the remaining parameter C becomes feasible after the other three parameters have been numerically estimated. Please see Shi et al. (2022) for details.

Value

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 x-axis.

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 dev.angle is not NULL.

RSS.vector

the vector of residual sum of squares corresponding to dev.angle, when dev.angle is not NULL.

x.obs

the observed x coordinates.

y.obs

the observed y coordinates.

y.pred

the predicted y coordinates corresponding to the the observed x coordinates.

par

the estimates of the four model parameters in the Narushin-Romanov-Griffin equation.

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

the residual sum of squares between the observed and predicted y values.

sample.size

the number of data points used in the numerical calculation.

RMSE

the root-mean-square errors between the observed and predicted y values.

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. In the outputs, there is no x.pred, because y.obs and y.pred share the same x values (i.e., x.obs).

Author(s)

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

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

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., Niklas, K.J. (2022) Comparison of a universal (but complex) model for avian egg shape with a simpler model. Annals of the New York Academy of Sciences 1514, 34-42. doi:10.1111/nyas.14799

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., 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., Zheng, X., Ratkowsky, D.A., Li, Y., Wang, P., Cheng, L. (2018) A simple method for measuring the bilateral symmetry of leaves. Symmetry 10, 118. doi:10.3390/sym10040118

Tian, F., Wang, Y., Sandhu, H.S., Gielis, J., Shi, P. (2020) Comparison of seed morphology of two ginkgo cultivars. Journal of Forestry Research 31, 751-758. doi:10.1007/s11676-018-0770-y

See Also

curveNRGE, NRGE

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 <- fitNRGE(x1, y1, dev.angle=NULL, ini.C=c(-1, -0.1, seq(0.1, 1, by=0.05)), 
                  strip.num=2000, fig.opt=TRUE) 
         
  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", col=4) 
  lines( Res2$x.obs, Res2$y.pred, col=2)

  Res3 <- fitNRGE(x1, y1, dev.angle=seq(-0.05, 0.05, by=0.01), 
                  ini.C=c(-1, -0.1, seq(0.1, 1, by=0.05)), 
                  strip.num=2000, fig.opt=TRUE)   

  zeta <- Res3$theta + Res3$epsilon
  x2   <- x1*cos(zeta) + y1*sin(zeta)
  y2   <- y1*cos(zeta) - x1*sin(zeta)
  plot( x2-min(x2), y2-min(y2), asp=1, col="grey70", cex=1,
        xlab=expression(italic("x")), ylab=expression(italic("y")) )  
  lines(Res3$x.obs-min(Res3$x.obs), Res3$y.obs-min(Res3$y.obs), col=4)
  lines(Res3$x.obs-min(Res3$x.obs), Res3$y.pred-min(Res3$y.obs), col=2)

  RMSE <- sqrt( Res3$RSS / Res3$sample.size )
  RMSE

  # To calculate the volume of the egg 
  VolumeNRGE(P=Res3$par)

  # To calculate the surface area of the egg
  SurfaceAreaNRGE(P=Res3$par)


graphics.off()