fitsigmoid {biogeom}

Data-Fitting Function for the Sigmoid Growth Equation

Description

fitsigmoid is used to estimate the parameters of a sigmoid growth equation based on the integral of a performance equation or one of its simplified versions.

Usage

fitsigmoid(expr, x, y, ini.val, simpver = 1, 
           control = list(), par.list = FALSE, fig.opt = FALSE,
           xlim = NULL, ylim = NULL, xlab = NULL, ylab = NULL,         
           main = NULL, subdivisions = 100L,
           rel.tol = .Machine$double.eps^0.25, 
           abs.tol = rel.tol, stop.on.error = TRUE, 
           keep.xy = FALSE, aux = NULL)

Arguments

expr

a performance equation or one of its simplified versions that is used to build a sigmoid growth equation.

x

the observed investigation times.

y

the observed y values (i.e., biomass, height, body length, etc.).

ini.val

the initial values of the model parameters.

simpver

an optional argument to use the simplified version of the performance equation.

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.

fig.opt

an optional argument to draw the observations and the predicted sigmoid curve.

xlim

the range of the x-axis over which to plot a sigmoid growth curve.

ylim

the range of the y-axis over which to plot a sigmoid growth curve.

xlab

the label of the x-axis when showing a sigmoid growth curve.

ylab

the label of the y-axis when showing a sigmoid growth curve.

main

the main title of the figure.

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

Here, ini.val only includes the initial values of the model parameters as a list. 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. The performance equations denote MbetaE, MBriereE, MLRFE, MPerformanceE and their simplified versions. The arguments of P and simpver should correspond to expr (i.e., MbetaE or MBriereE or MLRFE or MPerformanceE). The sigmoid equation is the integral of a performance equation or one of its simplified versions.

Value

par

the estimates of the model parameters.

r.sq

the coefficient of determination between the observed and predicted y values.

RSS

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

sample.size

the number of data points used in the data fitting.

x

the observed x values.

y

the observed y values.

y.pred

the predicted y values.

Note

Here, the user can define other performance equations, but new equations or their simplified versions should include the lower and upper thresholds on the x-axis corresponding to y = 0, whose indices should be the same as those in MbetaE or MBriereE or MLRFE or MPerformanceE.

Author(s)

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

References

Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. Plants 11, 1769. doi:10.3390/plants11131769

Lian, M., Shi, P., Zhang, L., Yao, W., Gielis, J., Niklas, K.J. (2023) A generalized performance equation and its application in measuring the Gini index of leaf size inequality. Trees - Structure and Function 37, 1555-1565. doi:10.1007/s00468-023-02448-8

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., Fan, M., Ratkowsky, D.A., Huang, J., Wu, H., Chen, L., Fang, S., Zhang, C. (2017) Comparison of two ontogenetic growth equations for animals and plants. Ecological Modelling 349, 1-10. doi:10.1016/j.ecolmodel.2017.01.012

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

See Also

areaovate, MbetaE, MBriereE, MLRFE, MPerformanceE, sigmoid

Examples

# The shrimp growth data(See the supplementary table in West et al., 2001)
# West, G.B., Brown, J.H., Enquist, B.J. (2001) A general model for ontogenetic growth. 
#     Nature 413, 628-631.
t0 <- c(3, 60, 90, 120, 150, 180, 384)
m0 <- c(0.001, 0.005, 0.018, 0.037, 0.06, 0.067, 0.07)

dev.new()
plot( t0, m0, cex.lab=1.5, cex.axis=1.5, col=4,
      xlab=expression(italic(x)), ylab=expression(italic(y)) )

xopt0   <- seq(100, 150, by=5)
ini.val <- list(0.035, xopt0, 200, 1)
resu1   <- fitsigmoid(MLRFE, x=t0, y=m0, ini.val=ini.val, simpver=1, fig.opt=TRUE, par.list=TRUE)


  xopt0   <- seq(100, 150, by=5)
  ini.val <- list(0.035, xopt0, 200, 1)
  resu1   <- fitsigmoid(MbetaE, x=t0, y=m0, ini.val=ini.val, simpver=1, fig.opt=TRUE)

  m.ini   <- c(0.5, 1, 2, 3, 4, 5, 10, 20)
  ini.val <- list(1e-8, m.ini, 200, 1)
  resu2   <- fitsigmoid(MBriereE, x=t0, y=m0, ini.val=ini.val, simpver=1, 
             fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000, trace=FALSE), 
             subdivisions=100L, rel.tol=.Machine$double.eps^0.25, 
             abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
             keep.xy=FALSE, aux=NULL)

  delta0  <- c(0.5, 1, 2, 5, 10, 20)
  ini.val <- list(0.035, 150, -100, 200, delta0)
  resu3   <- fitsigmoid(MLRFE, x=t0, y=m0, ini.val=ini.val, simpver=NULL, 
               fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000), 
               subdivisions = 100L, rel.tol=.Machine$double.eps^0.25, 
               abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
               keep.xy=FALSE, aux=NULL)

  a.ini   <- c(0.1, 1, 10, 100, 200)
  b.ini   <- 200
  ini.val <- list(0.001, 0.02, 0.15, 0, 200, a.ini, b.ini)
  resu4   <- fitsigmoid(MPerformanceE, x=t0, y=m0, ini.val=ini.val, simpver=NULL, 
             fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000, trace=FALSE), 
             subdivisions=100L, rel.tol=.Machine$double.eps^0.25, 
             abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
             keep.xy=FALSE, aux=NULL)
  resu5   <- fitsigmoid(MPerformanceE, x=t0, y=m0, ini.val=resu4$par, simpver=NULL, 
               fig.opt=TRUE, control=list(reltol=1e-30, maxit=200000, trace=FALSE))

  ini.val <- list(0.001, 0.01, c(0.1, 1, 10), 0, 200)
  resu6   <- fitsigmoid(MPerformanceE, x=t0, y=m0, ini.val=ini.val, simpver=2, 
             fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000, trace=FALSE), 
             subdivisions=100L, rel.tol=.Machine$double.eps^0.25, 
             abs.tol=.Machine$double.eps^0.25, stop.on.error=TRUE, 
             keep.xy=FALSE, aux=NULL)


graphics.off()