The von Bertalanffy Growth Function (VBGF) was introduced and methods for fitting the function were illustrated in Chapter 12 of the *Introductory Fisheries Analyses with R* (IFAR) book. The `nls()`

function demonstrated in the **IFAR** book uses the Gauss-Newton algorithm by default. However, there are many other algorithms for fitting nonlinear functions. Two of the several algorithms coded in R are demonstrated in this supplement. Other supplements demonstrate how to fit other parameterizations of the VBGF and other growth functions.

### Required Packages for this Supplement

Functions used in this supplement require the packages shown below.

```
> library(FSA)
> library(dplyr)
> library(nlstools)
> library(minpack.lm)
```

### Data Used in this Supplement

The male Black Drum data (view, download, meta-data) used in the **IFAR** book will also be used in this supplement.

```
> bdmf <- read.csv("BlackDrum2001.csv") %>%
select(-c(spname,day,weight)) %>%
filterD(sex %in% c("male","female"),otoage<50)
> bdm <- filterD(bdmf,sex=="male")
> headtail(bdm)
```

```
year agid month tl sex otoage
1 2001 1 4 787.5 male 6
2 2001 2 5 700.0 male 5
3 2001 8 5 1140.0 male 23
72 2001 122 5 1175.0 male 39
73 2001 125 6 590.0 male 4
74 2001 127 6 530.0 male 3
```

## Levenberg-Marquardt Algorithm

The Levenberg-Marquardt (L-M) algorithm is a powerful and common method that switches between two other algorithms depending on when those algorithms perform most optimally (Motulsky and Ransnas 1987). Most practically, the L-M algorithm appears to be quite robust to “poor” starting values.

The L-M algorithm is implemented in `nlsLM()`

from `minpack.lm`

(Elzhov et al. 2013) and uses the same main arguments as `nls()`

. For example, the “Typical” VBGF is fit (and parameter estimates, likelihood profile confidence intervals, and predictions are extracted) to the male Black Drum data below using the L-M algorithm.

```
> vbTyp <- vbFuns()
> svTyp <- list(Linf=1193,K=0.13,t0=-2.0)
> fitLM <- nlsLM(tl~vbTyp(otoage,Linf,K,t0),data=bdm,start=svTyp)
> bootLM <- nlsBoot(fitLM)
> cbind(Ests=coef(fitLM),confint(bootLM))
```

```
Ests 95% LCI 95% UCI
Linf 1196.7193456 1179.8720413 1214.7444331
K 0.1418266 0.1241443 0.1622749
t0 -1.5943403 -2.5034835 -0.8176289
```

`> predict(bootLM,vbTyp,t=3)`

```
prediction 95% LCI 95% UCI
572.8113 535.0526 606.2789
```

## Algorithms that Allow Parameter Constraints

In some instances, the user may want to constrain the model fitting algorithms to only consider parameter values within a certain range. For example, the user may want to constrain the \(L_{\infty}\) and \(K\) parameters of the “Typical” VBGF to be positive. Parameter constraints may substantively effect the results of the model fitting. Thus, they should be used rarely and, when used, set liberally.

Parameter constraints can be used with at least two algorithms in R. In either case, the lower and upper bounds for each parameter are entered into separate named **vectors** in the same order as the list used for starting values. Infinite bounds are the default ,but may be specifically defined for some parameters with `Inf`

and `-Inf`

(where `Inf`

represents infinity). For example, vectors that define constraints for the parameters in the “Typical” VBGF are defined below, with \(L_\infty\) and \(K\) constrained to be positive values and \(t_{0}\) left unconstrained.

```
> clwr <- c(Linf=1, K=0.0001,t0=-Inf)
> cupr <- c(Linf=Inf,K=Inf, t0=Inf)
```

These constraints may be provided to `lower=`

and `upper=`

, respectively, of `nlsLM()`

. In this instance, these bounds had no noticeable effect until bootstapping, where there were fewer instances of lack of convergence.

```
> fitLM1 <- nlsLM(tl~vbTyp(otoage,Linf,K,t0),data=bdm,start=svTyp,lower=clwr,upper=cupr)
> bootLM1 <- nlsBoot(fitLM1)
> cbind(Ests=coef(fitLM1),confint(bootLM1))
```

```
Ests 95% LCI 95% UCI
Linf 1196.7193456 1177.7663812 1214.8739079
K 0.1418266 0.1225476 0.1633342
t0 -1.5943403 -2.6332131 -0.7804033
```

Parameter constraints may be used similarly with `nls()`

, but the optimization algorithm must be changed to “Port” with `algorithm="port"`

.

```
> fitP <- nls(tl~vbTyp(otoage,Linf,K,t0),data=bdm,start=svTyp,
algorithm="port",lower=clwr,upper=cupr)
> bootP <- nlsBoot(fitP)
> cbind(Ests=coef(fitP),confint(bootP))
```

```
Ests 95% LCI 95% UCI
Linf 1196.7188609 1177.616313 1215.1724867
K 0.1418273 0.123604 0.1633974
t0 -1.5943118 -2.578606 -0.8138321
```

## Other Algorithms

Still other algorithms are found in `nlxb()`

from `nlmrt`

(Nash 2014) and `nls2()`

from `nls2`

(Grothendieck 2013).

## Reproducibility Information

**Compiled Date:**Fri Nov 06 2015**Compiled Time:**10:04:38 AM**R Version:**R version 3.2.2 (2015-08-14)**System:**Windows, i386-w64-mingw32/i386 (32-bit)**Base Packages:**base, datasets, graphics, grDevices, methods, stats, utils**Required Packages:**FSA, dplyr, nlstools, minpack.lm, captioner, knitr and their dependencies (assertthat, car, DBI, digest, evaluate, formatR, gdata, gplots, graphics, grDevices, highr, Hmisc, lazyeval, magrittr, markdown, methods, plotrix, plyr, R6, Rcpp, sciplot, stats, stringr, tools, utils, yaml)**Other Packages:**captioner_2.2.3, dplyr_0.4.3, extrafont_0.17, FSA_0.8.4, knitr_1.11, minpack.lm_1.1-9, nlstools_1.0-2**Loaded-Only Packages:**assertthat_0.1, DBI_0.3.1, digest_0.6.8, evaluate_0.8, extrafontdb_1.0, formatR_1.2.1, gdata_2.17.0, gtools_3.5.0, htmltools_0.2.6, lazyeval_0.1.10, magrittr_1.5, parallel_3.2.2, plyr_1.8.3, R6_2.1.1, Rcpp_0.12.1, rmarkdown_0.8.1, Rttf2pt1_1.3.3, stringi_1.0-1, stringr_1.0.0, tools_3.2.2, yaml_2.1.13**Links:**Script / RMarkdown

## References

Elzhov, T. V., K. M. Mullen, A.-N. Spiess, and B. M. Bolker. 2013. minpack.lm: R interface to the Levenberg-Marquardt nonlinear least-squares algorithm found in MINPACK, plus support for bounds.

Grothendieck, G. 2013. nls2: Non-linear regression with brute force.

Motulsky, H. J., and L. A. Ransnas. 1987. Fitting curves to data using nonlinear regression: A practical and nonmathematical review. The Federation of American Societies for Experimental Biology Journal 1:365–374.

Nash, J. C. 2014. nlmrt: Functions for nonlinear least squares solutions.