CatDynFit {CatDyn} | R Documentation |

## Fit CatDyn Models by Maximum Likelihood

### Description

A wrapper and post-processing tool that checks that the data are passed with proper characteristics, calls optimx() (from package optimx) on any of dozens of possible versions of the generalized depletion models (as internal functions), and then it post-processes optimx() results and join all results in a list of lists.

### Usage

```
CatDynFit(x, p, par, dates, distr, method, control = list(),
hessian = TRUE, itnmax, partial = TRUE)
```

### Arguments

`x` |
A data object of class CatDynData. See as.CatDynData(). |

`p` |
Integer. The process model type, which quantifies the number of perturbations to depletion. In one-fleet cases p is a scalar integer that can take any value between -25 and 25. In two-fleet cases p is a two-components integer vector that quantifies the number of perturbation events of each fleet. It can take values c(0,0), c(0,1), ..., c(0,5), c(1,1), ..., c(1,5), ..., c(4,5), c(5,5), c(6,6), ..., c(25,25). In fisheries with emigration, where in addition to perturbations due to positive pulses of abundance, there are perturbation due to negative pulses of exodus, p should be negative and will take any integer value between -1 and -25, this number fixing both the number of input and exit pulses. |

`par` |
Numeric. Vector of initial parameter values in the log scale. |

`dates` |
Integer. Vector with the time steps of start of season, perturbations (if any), and end of season. In fisheries with emigration, in addition to the timing of entry of perturbations, the timing of exit for each perturbation shall also be provided, right after the time of entry. For example, p=c(1,4,50,10,60,61) would specify a two-perturbations model which starts at time step 1, has the first input perturbation at time step 4, first exit perturbation at time step 50, second input perturbation at time 10, second exit perturbation at time step 60, and season finishing at time step 61. |

`distr` |
Character, either "poisson", "negbin", "normal", "apnormal", "lognormal", "aplnormal", "gamma", "roblognormal", "gumbel", or any pair of these seven (2-fleets systems), corresponding to the likelihood model. |

`method` |
Character. Any method accepted by optimx() can be used, but some may return warnings or errors. |

`control` |
A list of control arguments to be passed to optimx(). |

`hessian` |
Logical. Defaults to TRUE. If set to FALSE all numerical methods tried will fail. |

`itnmax` |
Numeric. Maximum number of iterations, to pass to optimx(). |

`partial` |
Logical, if the model incluyes emigration (p between -1 and -25) partial = TRUE (the default) means that an unknown part of the abundance emigrate at some time step within the season, causing a negative pulse of abundance. Alternatively, partial = FALSE is used for transit stock models where the whole pulse of input yet surviving exits the fishing grounds. |

### Details

Much care should be taken in selecting good initial values to pass in the par argument. To accomplish this CatDyn includes the CatDynExp class, and the catdynexp() and the plot.CatDynExp() functions to graphically fine tune the initial values for all model parameters. In multi-annual applications and monthly time step this might be time consuming but it should be carried out to increase the chance that the optimizers will converge to reasonable parameter space.

Initial parameter values must be passed log-transformed by the user. CatDynFit() will backtransform the maximum likelihood estimates and its numerical Hessian matrix without user intervention using the delta method.

Generally, when p is 5 or lower (one fleet) or c(5,5) (two fleets) or lower, the model is applied to one annual season of data and the time step is "day" or "week". Conversely, when p is 6 (one fleet) or c(6,6) (two fleets) or higher the model is applied to multiannual series and the time step is the month, although it is conceivable that for a higly perturbed fishing system higher p values would be applied to single season cases.

The models set up for fisheries with emigration are single fleet only, so when p is negative, taking any value in the admissible range, its length must be 1.

The discrete Poisson distribution option is recommended for fisheries where the catch is counted in number of fish instead of weight.

The "negbin" value for the distr parameter corresponds to the negative binomial distribution for counts, as an alternative to "poisson" for cases where the assumption of the mean equal to the variance is untenable.

The difference between "normal" and "apnormal", "lognormal" and "aplnormal" is that in the former the dispersion parameters is included in the likelihood function and it is a free parameter to be estimated along with the parameters of the generalized depletion model (and therefore an initial value for the dispersion has to be provided) whereas in the latter the dispersion is eliminated by using the adjusted profile likelihood approximation. Setting distr="roblognormal" will use a robustified version of the lognormal which includes the dispersion parameter. For the "negbin", "gamma" and "gumbel" distributions the dispersion parameter is always estimated along with the model parameters. In two-fleets models any pair of the nine available likelihood models can be specified.

In models with emigration, the logical parameter partial should be TRUE (the default) when the magnitude of the exit pulse is an unknown parameter that needs to be estimated. This could be the case of females exiting the fishing grounds to spawn, for instance. On the other hand partial should be FALSE in transit stock fisheries, and in this case the magnitude of the exit pulse need not be estimated, since all survivors leave. Thus setting partial to TRUE and p negative estimates more parameters than non-emigration and transit stock models.

At this point there is no allowance for models with emigration that include a different number of entry and exit pulses. Transit stock fisheries (partial=FALSE) must have the same number of entry and exit pulses but other emigration models (partial=TRUE) need not be restricted. Next versions of the function will include options for different number of entry and exit pulses.

### Value

A list of length 3.

`Data` |
A list of length 2. Properties is a list of length 3. Units is a dataframe with the units of time step, catch, body weight, and the numbers multiplier. Fleets is a dataframe with the fleets names and the units of nominal effort for each fleet. Dates is a dataframe with the start and end dates of the season in the ISO 8601 format. Data is a list of length equal to the number of fleets (1 or 2). Each component is a dataframe with the raw data, time step, observed effort, observed catch, observed mean body weight, observed catch in numbers, and the catch spike statistic. |

`Initial` |
A dataframe with named initial values of all free parameters in the model. |

`Model` |
A list with length equal to the number of numerical methods. Each component has the perturbation type model, the dates of events, the chosen distribution for the observation of catch, the integer code describing the success or not of covergence returned by the method, the Karush Kuhn Tucker conditions, hopefully TRUE and TRUE, the value of the Akaike Information Criterion, not comparable between different distributions, the back-transformed (from log) maximum likelihood estimates, the numerical gradients at each maximum likelihood estimate, the standard errors of backtransformed (from log) maximum likelihood estimates, and the correlation matrix of the back-transformed (from log) maximum likelihood estimates. |

### Note

Complex models may take several hours to converge on a PC. As an example, a two fleet model with 18 perturbations each fleet, p=c(18,18), and the aplnormal likelihood model, totalling 44 parameters to estimate from 216 monthly time steps, coverged successfully in 16 hours on a Windows 7, 64 bit, 3 GHz processor, 8 GB RAM.

Some effort has been made to avoid being kicked out of numerical optimization by just one numerical method that fails, so that optimization continues with other methods, but there may remain some cases when the whole optimization process is aborted by failure in just one method. Try taking out some suspicious methods and optimize again. Experience shows that methods spg and CG are robust for this kind of model so both should be considered as the baseline case for numerical optimization. When using the option of modeling transit fisheries with the Poisson distribution it has been observed that methods bobyqa and newuoa also perform well, so keep an open mind and take advantage of optimx by trying several numerical methods.

In resident stock models without emigration partial must be left at its default value in order not to have any effect.

### Author(s)

Ruben H. Roa-Ureta (ORCID ID 0000-0002-9620-5224)

### References

Roa-Ureta, R. H. 2012. ICES Journal of Marine Science 69(8), 1403-1415.

Roa-Ureta, R. H. et al. 2015. Fisheries Research 171 (Special Issue), 59-67.

Roa-Ureta, R. H. 2015. Fisheries Research 171 (Special Issue), 68-77.

Lin, Y-J. et al. 2017. Fisheries Research 195, 130-140.

### Examples

```
#NOTE: These examples are run with very few maximum number of iterations for the
#optimization methods passed to the CatDynFit function. Real applications should
#run many more, setting the itnmax parameter in the order of thousands for models
#with dozens of free parameters to estimate.
#
#Falkland Islands one-fleet squid fishery in 1990.
#Create the data object
lgahi <- as.CatDynData(x=lolgahi,
step="day",
fleet.name="Fk",
coleff=2,
colcat=1,
colmbw=3,
unitseff="nboats",
unitscat="kg",
unitsmbw="kg",
nmult="bill",
season.dates=c(as.Date("1990-01-31"),as.Date("1990-05-30")))
#Not run
#plot(lgahi,mark=TRUE,offset=c(NA,NA,.75),hem="S")
#
#1) Fit a 1-fleet 1P model with lognormal observation error and the adjusted
#profile approximation to the likelihood to eliminate the dispersion parameter
M <- 0.011 #1/Time step
N0.ini <- 3.8 #billions
P1.ini <- 1.3 #billions
k.ini <- 5.0e-05 #1/n of boats
alpha.ini <- 1.7 #adimensional
beta.ini <- 0.6 #adimensional
pars.ini <- log(c(M,
N0.ini,
P1.ini,
k.ini,
alpha.ini,
beta.ini))
#Dates
P1 <- 70 #Selected by visual inspection of standard plot
dates <- c(head(lgahi$Data$Fk$time.step,1),
P1,
tail(lgahi$Data$Fk$time.step,1))
lgahi.apln.1P.ini <- catdynexp(x=lgahi,
p=1,
par=pars.ini,
dates=dates,
distr="aplnormal")
plot(x=lgahi.apln.1P.ini,
leg.pos="topright",
Biom.tstep=7,
Cat.tstep=120,
Biom.xpos=0.4,
Biom.ypos=0,
Cat.xpos=0.4,
Cat.ypos=0.1)
#fit
lgahi.apln.1P.fit <- CatDynFit(x=lgahi,
p=1,
par=pars.ini,
dates=dates,
distr="aplnormal",
method="spg",
itnmax=10)
#examine results
lgahi.apln.1P.pred.spg <- CatDynPred(lgahi.apln.1P.fit,"spg")
plot(x=lgahi.apln.1P.pred.spg,
leg.pos="topright",
Biom.tstep=7,
Cat.tstep=120,
Biom.xpos=0.18,
Biom.ypos=0.1,
Cat.xpos=0.18,
Cat.ypos=0.2)
#
#2) Fit a 1-fleet 2P model with lognormal observation error and full exact
#likelihood including the dispersion parameter
M <- 0.011 #1/Time step
N0.ini <- 3.8 #billions
P1.ini <- 1.3 #billions
P2.ini <- 0.5 #billions
k.ini <- 4.0e-05 #1/n of boats
alpha.ini <- 1.7 #adimensional
beta.ini <- 0.6 #adimensional
#Note how to get reasonable initial value for dispersion parameter
psi.ini <- 0.33*sd(log(lgahi$Data$Fk$obscat.bill))^2
pars.ini <- log(c(M,
N0.ini,
P1.ini,
P2.ini,
k.ini,
alpha.ini,
beta.ini,
psi.ini))
#Dates
P1 <- 70 #Selected by visual inspection of standard plot
P2 <- 135 #Selected by visual inspection of standard plot
dates <- c(head(lgahi$Data$Fk$time.step,1),
P1,
P2,
tail(lgahi$Data$Fk$time.step,1))
lgahi.ln.2P.ini <- catdynexp(x=lgahi,
p=2,
par=pars.ini,
dates=dates,
distr="lognormal")
plot(x=lgahi.ln.2P.ini,
leg.pos="topright",
Biom.tstep=7,
Cat.tstep=120,
Biom.xpos=0.4,
Biom.ypos=0,
Cat.xpos=0.18,
Cat.ypos=0.2)
#fit lognormal
lgahi.ln.2P.fit <- CatDynFit(x=lgahi,
p=2,
par=pars.ini,
dates=dates,
distr="lognormal",
method="spg",
itnmax=10)
#examine results
lgahi.ln.2P.pred.spg <- CatDynPred(lgahi.ln.2P.fit,"spg")
plot(x=lgahi.ln.2P.pred.spg,
leg.pos="topright",
Biom.tstep=7,
Cat.tstep=120,
Biom.xpos=0.18,
Biom.ypos=0.1,
Cat.xpos=0.18,
Cat.ypos=0.2)
#
#Summary table for model selection
lgahi.sum <- CatDynSum(x=list(lgahi.apln.1P.fit,
lgahi.ln.2P.fit),
season=1990,
method=c("spg","spg"))
#Plot for correlations among parameter estimates
CatDynCor(x=list(lgahi.apln.1P.fit,
lgahi.ln.2P.fit),
ttl=c("Adjusted Profile Lognormal 1P","Lognormal 2P"),
method=c("spg","spg"),
arr=c(2,1))
#Create neat table with optimization results
CatDynPar(x=lgahi.ln.2P.fit,method="spg")
#
```

*CatDyn*version 1.1-1 Index]