MAR 580: Models for Marine Ecosystem-Based Management

TMB workshop

Session IV

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(Acknowledgements: Mollie Brooks, Kasper Kristensen, Arni Magnusson, Anders Nielsen, André Punt)

08 September 2022

Functions

• We sometimes like to re-use chunks of our own code that perform a common task
• Solution: put this code in a function!
• Call the function rather than writing the code each time.
• In TMB, we can add functions via separate files (that we point to), or include them before the objective function object definition.

e.g. ‘square()’

template <class Type> Type square(Type x){return x*x;}

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posfun(). (constrain a quantity to be positive)

Type posfun(Type x, Type eps, Type &pen){

pen += CppAD::CondExpLt(x, eps, Type(0.01) * pow(x-eps,2), Type(0));

return CppAD::CondExpGe(x, eps, x, eps/(Type(2)-x/eps));

}

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Using it in the model:

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number<Type> bpen = 0.;

biomass(year) = biomass(year-1) + posfun(biomass(year-1)*r*(1.- biomass(year-1)/K),0.1,bpen);

nll += 1000*bpen;

Likelihood profile

• We obtained estimates of variance this morning by asymptotic methods
• TMB can also do likelihood profiles

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prof <- tmbprofile(model,”par”)

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print confint(prof, level = 0.95)

namespaces, Multivariate normal

• TMB comes with many probability distributions

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• However there are others we can use by loading different namespaces

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e.g. multivariate normal

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using namespace density;

MVNORM_t<Type> neg_log_dmvnorm(Sigma);

nll += neg_log_dmvnorm(residual);

simple MVNORM example

Spatial random effects example

• Extending the Poisson GLMM
• We might expect summer flounder catch rates to be spatially correlated
• e.g. neighboring states have similar catch rates
• Add simple neighborhood-based correlation structure to our Poisson GLMM
• e.g. Corr(i,j) = e^-⍺d(i,j)
• Distance matrix:

coords <- tibble(x = rep(1,9), y = 1:9)

dd <- as.matrix(dist(coords,

upper = TRUE,

diag = TRUE),nrow=9)

Spatial random effects example

• Modifying the GLMM
• .R:
• Need to add the distance matrix to our data object
• Add correlation and spatial variance parameters
• .cpp:
• Add code to calculate the Variance Covariance matrix given the distance matrix and the correlation/variance parameters
• Include the contribution for the spatial random effects (using the multivariate normal)

Exercise: Growth estimation from lengths at release and recapture

• Data set of bluegill recapture lengths
• head(FSAdata::BluegillIL)

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• Laslett et al. (2002, CJFAS)
• ‘LEP’ method of estimating growth
• Fit to Lengths at release/recapture instead of growth increment, time at liberty
• Challenge: Growth models include Age, but these are not known!
• Solution: Integrate across the age at release using random effects!

Parameters:

• VBK�Linf_mu
• sigma_Linf
• Sigma_obs
• A_i (random effect)
• Sigma_A
• Mu_A

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Data

Length 1

Length 2

dt

For each fish:

Mu_L1 (eq 3.5a)

Mu_L2 (eq 3.5b)

Sigma_L1 (eq 3.5c)

Sigma_L2 (eq 3.5d)

Cov_L1L2 (eq 3.5e)

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MVNORM LIKELIHOOD

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Exercise: fitting population model

• Fit a dynamic Schaefer model to the Catch and Survey abundance data for Pleuronectes electronica
• Step 1: Draw the model graph with relationships between variables
• Step 2: Create a TMB model to fit to the available data
• Assume the logs of both the catch data and survey indices are normally distributed with respect to their mean (model-predicted) value.
• 3: Fit the model, plot the model fits, population trajectory
• 4: Compute a likelihood profile for the growth rate r