# Data-Fitting Function for the Sigmoid Growth Equation `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. UTF-8 ```r 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. ## Returns - **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 pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca . ## References Jin, J., Quinn, B.K., Shi, P. (2022) The modified Brière equation and its applications. **Plants** 11, 1769. tools:::Rd_expr_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$-$1565. tools:::Rd_expr_doi("10.1007/s00468-023-02448-8") Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. **Computer Journal** 7, 308$-$313. tools:::Rd_expr_doi("10.1093/comjnl/7.4.308") Shi, P., Fan, M., Ratkowsky, D.A., Huang, J., Wu, H., Chen, L., Fang, S., Zhang, C. (2017) Comparison of two ontogenetic growth equations for animals and plants. **Ecological Modelling** 349, 1$-$10. tools:::Rd_expr_doi("10.1016/j.ecolmodel.2017.01.012") Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. **Annals of the New York Academy of Sciences** 1516, 123$-$134. tools:::Rd_expr_doi("10.1111/nyas.14862") ## See Also `areaovate`, `MbetaE`, `MBriereE`, `MLRFE`, `MPerformanceE`, `sigmoid` ## Examples ```r # The shrimp growth data(See the supplementary table in West et al., 2001) # West, G.B., Brown, J.H., Enquist, B.J. (2001) A general model for ontogenetic growth. # Nature 413, 628-631. t0 <- c(3, 60, 90, 120, 150, 180, 384) m0 <- c(0.001, 0.005, 0.018, 0.037, 0.06, 0.067, 0.07) dev.new() plot( t0, m0, cex.lab=1.5, cex.axis=1.5, col=4, xlab=expression(italic(x)), ylab=expression(italic(y)) ) xopt0 <- seq(100, 150, by=5) ini.val <- list(0.035, xopt0, 200, 1) resu1 <- fitsigmoid(MLRFE, x=t0, y=m0, ini.val=ini.val, simpver=1, fig.opt=TRUE, par.list=TRUE) xopt0 <- seq(100, 150, by=5) ini.val <- list(0.035, xopt0, 200, 1) resu1 <- fitsigmoid(MbetaE, x=t0, y=m0, ini.val=ini.val, simpver=1, fig.opt=TRUE) m.ini <- c(0.5, 1, 2, 3, 4, 5, 10, 20) ini.val <- list(1e-8, m.ini, 200, 1) resu2 <- fitsigmoid(MBriereE, x=t0, y=m0, ini.val=ini.val, simpver=1, fig.opt=TRUE, control=list(reltol=1e-20, maxit=20000, trace=FALSE), subdivisions=100L, rel.tol=.Machine$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() ```

fitsigmoid function