# Gompertz Growth Model (Old Style) Fit a Gompertz growth model to otoliths and/or tags, using a traditional parametrization. ```r gompertzo(par, data, silent = TRUE, ...) gompertzo_curve(t, Linf, k, tau) gompertzo_objfun(par, data) ``` ## Arguments - `par`: a parameter list. - `data`: a data list. - `silent`: passed to `MakeADFun`. - ``...``: passed to `MakeADFun`. - `t`: age (vector). - `Linf`: asymptotic maximum length. - `k`: growth coefficient. - `tau`: location parameter. ## Returns The `gompertzo` function returns a TMB model object, produced by `MakeADFun`. The `gompertzo_curve` function returns a numeric vector of predicted length at age. The `gompertzo_objfun` function returns the negative log-likelihood as a single number, describing the goodness of fit of `par` to the `data`. ## Details The main function `gompertzo` creates a model object, ready for parameter estimation. The auxiliary functions `gompertzo_curve` and `gompertzo_objfun` are called by the main function to calculate the regression curve and objective function value. The user can also call the auxiliary functions directly for plotting and model exploration. The `par` list contains the following elements: * `log_Linf`, asymptotic maximum length * `log_k`, growth coefficient * `tau`, location parameter * `log_sigma_min`, growth variability at the shortest observed length in the data * `log_sigma_max` (*), growth variability at the longest observed length in the data * `log_age` (*), age at release of tagged individuals (vector) *: The parameter `log_sigma_max` can be omitted to estimate growth variability that does not vary with length. The parameter vector `log_age` can be omitted to fit to otoliths only. The `data` list contains the following elements: * `Aoto` (*), age from otoliths (vector) * `Loto` (*), length from otoliths (vector) * `Lrel` (*), length at release of tagged individuals (vector) * `Lrec` (*), length at recapture of tagged individuals (vector) * `liberty` (*), time at liberty of tagged individuals in years (vector) *: The data vectors `Aoto` and `Loto` can be omitted to fit to tagging data only. The data vectors `Lrel`, `Lrec`, and `liberty` can be omitted to fit to otoliths only. ## Note The Schnute parametrization used in `gompertz` reduces parameter correlation and improves convergence reliability compared to the traditional parametrization used in `gompertzo`. Therefore, the `gompertz` parametrization can be recommended for general usage, as both parametrizations produce the same growth curve. However, there can be some use cases where the traditional parametrization (`Linf`, `k`, `tau`) is preferred over the Schnute parametrization (`L1`, `L2`, `k`). Gompertz is a special case of the Richards (1959) model, where $b=0$. If the best model fit of a `richards` model to a particular dataset involves a very small estimated value of $b$, then the `gompertz` model offers a preferable parametrization, as it produces the same curve using fewer parameters. The Gompertz (1825) growth model, as parametrized by Ricker (1979, Eq. 23) predicts length at age as: $$ \hat L_t ~=~ L_\infty\exp\!\big(\!\!-\!e^{-k(t-\tau)}\big)Lhat = Linf * exp(-exp(-k * (t-tau))) $$ The variability of length at age increases linearly with length, $$ \sigma_L ~=~ \alpha \,+\, \beta \hat Lsigma = alpha + beta * Lhat $$ where the intercept is c("$alpha =\n$", "$sigma_min - beta * L_min$"), the slope is c("$beta =\n$", "$(sigma_max-sigma_min) / (L_max-L_min)$"), and $L_min$ and $L_max$ are the shortest and longest observed lengths in the data. Alternatively, growth variability can be modelled as a constant $sigma_L=sigma_min$ that does not vary with length, by omitting `log_sigma_max` from the parameter list (see above). The negative log-likelihood objective function integrates (sums) the likelihood components from the otoliths and tags, $$ f ~=~ \sum_{i=1}^{N_\mathrm{oto}}\, 0.5\log(2\pi) \,+\,\log\sigma_i \,+\, \frac{(L_i-\hat L_i)^2}{2\sigma_i^2}f= sum(0.5*log(2*pi) + log(sigma_i) + ((L_i-Lhat_i)^2)/(2*sigma_i^2)) $$ $$ ~~~~ ~+\,~ \sum_{j=1}^{N_\mathrm{tag}}\, 0.5\log(2\pi) \,+\,\log\sigma_j \,+\, \frac{(L_j-\hat L_j)^2}{2\sigma_j^2}+ sum(0.5*log(2*pi) + log(sigma_j) + ((L_j-Lhat_j)^2)/(2*sigma_j^2)) $$ $$ ~~~~ ~+\,~ \sum_{k=1}^{N_\mathrm{tag}}\, 0.5\log(2\pi) \,+\,\log\sigma_k \,+\, \frac{(L_k-\hat L_k)^2}{2\sigma_k^2}+ sum(0.5*log(2*pi) + log(sigma_k) + ((L_k-Lhat_k)^2)/(2*sigma_k^2)) $$ where $L_i$ are length measurements from the otolith data, $L_j$ are length measurements at tag release, and $L_k$ are length measurements at tag recapture. ## Examples ```r # Model 1: Fit to haddock otoliths # Explore initial parameter values plot(len~age, otoliths_had, xlim=c(0,18), ylim=c(0,105), pch=16, col="#0080a010") x <- seq(1, 18, 0.1) lines(x, gompertzo_curve(x, Linf=73, k=0.4, tau=2), lty=3) # Prepare parameters and data init <- list(log_Linf=log(73), log_k=log(0.4), tau=2, log_sigma_min=log(3), log_sigma_max=log(3)) dat <- list(Aoto=otoliths_had$age, Loto=otoliths_had$len) gompertzo_objfun(init, dat) # Fit model model <- gompertzo(init, dat) fit <- nlminb(model$par, model$fn, model$gr, control=list(eval.max=1e4, iter.max=1e4)) report <- model$report() sdreport <- sdreport(model) # Plot results Lhat <- with(report, gompertzo_curve(x, Linf, k, tau)) lines(x, Lhat, lwd=2, col=2) legend("bottomright", c("initial curve","model fit"), col=c(1,2), lty=c(3,1), lwd=c(1,2), bty="n", inset=0.02, y.intersp=1.25) # Model summary report[c("Linf", "k", "tau", "sigma_min", "sigma_max")] fit[-1] summary(sdreport) # Plot 95% prediction band band <- pred_band(x, model) areaplot::confplot(cbind(lower,upper)~age, band, xlim=c(0,18), ylim=c(0,100), ylab="len", col="mistyrose") points(len~age, otoliths_had, pch=16, col="#0080a010") lines(Lhat~age, band, lwd=2, col=2) lines(lower~age, band, lty=1, lwd=0.5, col=2) lines(upper~age, band, lty=1, lwd=0.5, col=2) ############################################################################# # Model 2: Fit to skipjack otoliths and tags # Explore initial parameter values plot(len~age, otoliths_skj, xlim=c(0,4), ylim=c(0,100)) x <- seq(0, 4, 0.1) points(lenRel~I(lenRel/60), tags_skj, col=4) points(lenRec~I(lenRel/60+liberty), tags_skj, col=3) lines(x, gompertzo_curve(x, Linf=75, k=1, tau=0), lty=2) legend("bottomright", c("otoliths","tag releases","tag recaptures", "initial curve"), col=c(1,4,3,1), pch=c(1,1,1,NA), lty=c(0,0,0,2), lwd=c(1.2,1.2,1.2,1), bty="n", inset=0.02, y.intersp=1.25) # Prepare parameters and data init <- list(log_Linf=log(75), log_k=log(1), tau=0, log_sigma_min=log(3), log_sigma_max=log(3), log_age=log(tags_skj$lenRel/60)) dat <- list(Aoto=otoliths_skj$age, Loto=otoliths_skj$len, Lrel=tags_skj$lenRel, Lrec=tags_skj$lenRec, liberty=tags_skj$liberty) gompertzo_objfun(init, dat) # Fit model model <- gompertzo(init, dat) fit <- nlminb(model$par, model$fn, model$gr, control=list(eval.max=1e4, iter.max=1e4)) report <- model$report() sdreport <- sdreport(model) # Plot results band <- pred_band(x, model) areaplot::confplot(cbind(lower,upper)~age, band, xlim=c(0,4), ylim=c(0,100), ylab="len", col="gray90") points(len~age, otoliths_skj) points(report$age, report$Lrel, col=4) points(report$age+report$liberty, report$Lrec, col=3) Lhat <- with(report, gompertzo_curve(x, Linf, k, tau)) lines(x, Lhat, lwd=2) legend("bottomright", c("otoliths","tag releases","tag recaptures", "model fit"), col=c(1,4,3,1), pch=c(1,1,1,NA), lty=c(0,0,0,1), lwd=c(1.2,1.2,1.2,2), bty="n", inset=0.02, y.intersp=1.25) # Model summary report[c("Linf", "k", "tau", "sigma_min", "sigma_max")] fit[-1] head(summary(sdreport), 5) ############################################################################# # Model 3: Fit to skipjack otoliths only init <- list(log_Linf=log(75), log_k=log(1), tau=0, log_sigma_min=log(3), log_sigma_max=log(3)) dat <- list(Aoto=otoliths_skj$age, Loto=otoliths_skj$len) model <- gompertzo(init, dat) fit <- nlminb(model$par, model$fn, model$gr, control=list(eval.max=1e4, iter.max=1e4)) model$report()[c("Linf", "k", "tau", "sigma_min", "sigma_max")] ############################################################################# # Model 4: Fit to skipjack otoliths only, # but now estimating constant sigma instead of sigma varying by length # We do this by omitting log_sigma_max init <- list(log_Linf=log(75), log_k=log(1), tau=0, log_sigma_min=log(3)) dat <- list(Aoto=otoliths_skj$age, Loto=otoliths_skj$len) model <- gompertzo(init, dat) fit <- nlminb(model$par, model$fn, model$gr, control=list(eval.max=1e4, iter.max=1e4)) model$report()[c("Linf", "k", "tau", "sigma_min")] ############################################################################# # Model 5: Fit to skipjack tags only init <- list(log_Linf=log(75), log_k=log(1), tau=0, log_sigma_min=log(3), log_sigma_max=log(3), log_age=log(tags_skj$lenRel/60)) dat <- list(Lrel=tags_skj$lenRel, Lrec=tags_skj$lenRec, liberty=tags_skj$liberty) model <- gompertzo(init, dat) fit <- nlminb(model$par, model$fn, model$gr, control=list(eval.max=1e4, iter.max=1e4)) model$report()[c("Linf", "k", "tau", "sigma_min", "sigma_max")] ``` ## References Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. **Philosophical Transactions of the Royal Society**, 115 , 513-583. Ricker, W.E. (1979). Growth rates and models. In: W.S. Hoar et al. (eds.) **Fish physiology 8: Bioenergetics and growth**. New York: Academic Press, pp. 677-743. The `fishgrowth-package` help page includes references describing the parameter estimation method. ## See Also `gcm`, `gompertz`, `gompertzo`, `richards`, `richards`, `schnute3`, `vonbert`, and `vonberto` are alternative growth models. `pred_band` calculates a prediction band for a fitted growth model. `otoliths_had`, `otoliths_skj`, and `tags_skj` are example datasets. `fishgrowth-package` gives an overview of the package.

gompertzo function