23  Populating the Operating Model

Before the simulation can begin, all user-specified parameter distributions must be expanded into fully-dimensioned arrays covering every simulation replicate, age class, and year. This process is called populating the operating model, and is handled by PopulateOM(), which calls PopulateStock() and PopulateFleet() in sequence, followed by the observation model. For a conceptual overview of the population process and worked examples, see Chapter 17.

23.1 Simulation Dimensions

Every array in a populated OM is indexed over some or all of the following dimensions:

Table 23.1: Core array dimensions.

Index	Symbol	Description
Simulation	\(s\)	Stochastic replicate, \(s = 1, \ldots, n_\text{sim}\)
Age	\(a\)	Age class (years), \(a = a_\text{rec}, \ldots, A_\text{max}\)
Year	\(t\)	Time step (historical + projection), \(t = 1, \ldots, T\)
Area	\(r\)	Spatial area, \(r = 1, \ldots, R\)

The time dimension merges historical and projection years into a single index: \(T = n_\text{year} + p_\text{year}\), where \(n_\text{year}\) is the number of historical years and \(p_\text{year}\) the number of projection years. For seasonal models, each calendar year is divided into \(S\) seasons, so each time step represents \(1/S\) of a year and the total number of time steps is \(T = S \cdot (n_\text{year} + p_\text{year})\).

Age classes are computed from the Ages object. The first age class corresponds to the age of recruitment, \(a_\text{rec}\), which need not be zero. For a non-seasonal model with annual time steps, \(a = a_\text{rec}, a_\text{rec}+1, \ldots, A_\text{max}\) in years. For a seasonal model with \(S\) seasons per year, age advances in increments of \(1/S\), giving \(a = a_\text{rec}, a_\text{rec} + 1/S, \ldots, A_\text{max}\) in years. The Ages@Classes slot stores these values after population.

23.2 Sampling Parameter Distributions

Model parameters are specified as distributions rather than fixed values. The input formats accepted by each subcomponent object are described in detail in Chapter 16. This section focuses on the underlying sampling mechanics.

A common pattern is a uniform distribution bounded by a minimum and maximum value, from which \(n_\text{sim}\) draws are taken:

\[\theta_s \sim \text{Uniform}(\theta_\text{min},\; \theta_\text{max}), \quad s = 1, \ldots, n_\text{sim}\]

When a parameter is fixed (i.e. \(\theta_\text{min} = \theta_\text{max}\)), all simulations share the same value. Some parameters are instead specified with a mean and coefficient of variation (CV), in which case draws are taken from a log-normal distribution:

\[\theta_s \sim \text{LogNormal}\!\left(\log(\bar\theta) - \tfrac{1}{2}\sigma^2,\; \sigma\right), \quad \sigma = \sqrt{\log(1 + \text{CV}^2)}\]

Random draws are seeded deterministically from OM@Seed so that results are reproducible. Each stock and fleet component receives a different offset from the base seed to ensure independence across components while maintaining reproducibility.

23.3 Time-Varying Parameters

There are two broad approaches to specifying a schedule (see Chapter 16 for the full range of accepted input formats):

Direct array. The MeanAtAge (or MeanAtLength, MeanAtWeight) slot of a subcomponent object can be populated directly as a Sim × Age/Class × Year array, bypassing the Pars model-based approach. As with parameter arrays, the Year dimension need only span boundary years (dimnames identify which years are represented), and the Sim dimension must be either 1 or exactly \(n_\text{sim}\).

Model-based via Pars. Each element of a Pars list is processed into a Sim × Year parameter array before being passed to the schedule model. Individual parameters can be:

• Constant (time-invariant): a fixed value or uniform bounds \([\theta_\text{min}, \theta_\text{max}]\) sampled once per simulation (as in Section 23.2) and held constant across years: \(\theta_{s,t} = \theta_s\) for all \(t\).

• Random walk: if a parameter Pars$foo is accompanied by a paired Pars$fooSD entry, a mean-preserving log-normal random walk is applied across years:

\[\theta_{s,t} = \theta_s \cdot \frac{\epsilon_{s,t}}{\bar\epsilon_s}, \quad \log\epsilon_{s,t} \sim \mathcal{N}\!\left(-\tfrac{1}{2}\sigma^2,\; \sigma\right)\]

where \(\sigma\) is the value of Pars$fooSD and \(\bar\epsilon_s = T^{-1}\sum_t \epsilon_{s,t}\) is the within-simulation mean of the deviates (mean-preserving correction). After processing, the *SD entry is removed from the parameter list.

• Pre-built Sim × Year array: supplied directly inside Pars, extended to fill any missing simulation or year indices by repeating year boundary values. The Year dimension only needs to span the boundary years where the values change (intermediate years are filled by extension; see ?Extend). The Sim dimension must be either 1 (the same value is used for all simulations) or exactly \(n_\text{sim}\) (simulation-specific values).

Effort trajectories have an additional input form: a data frame of control points with lower and upper bounds at specified years, between which effort is linearly interpolated (see Chapter 25).

23.4 Populate Stock

PopulateStock() generates all biological arrays for a single stock. It is called once per stock, with the OM’s nYear, pYear, nSim, and Seasons passed as arguments. The sequence of operations is:

1. Calculate age classes (Ages@Classes)
2. Populate length-at-age (PopulateLength())
3. Populate weight-at-age (PopulateWeight())
4. Populate natural mortality (PopulateNaturalMortality())
5. Populate maturity-at-age (PopulateMaturity())
6. Populate fecundity-at-age (PopulateFecundity())
7. Populate stock–recruitment (PopulateSRR())
8. Populate spatial structure (PopulateSpatial())
9. Populate initial depletion (PopulateDepletion())

The equations governing each of these steps are described in Chapter 24. The resulting arrays have dimensions \(n_\text{sim} \times n_\text{age} \times T\) unless noted otherwise.

23.5 Populate Fleet

PopulateFleet() generates all fishery arrays for a single fleet operating on a given stock. It takes the already-populated Stock object as an argument so that fleet schedules can be derived in the same age and size class space. The sequence is:

1. Populate effort across historical years (PopulateEffort())
2. Populate catchability (PopulateCatchability())
3. Populate selectivity-at-age (PopulateSelectivity())
4. Populate retention-at-age (PopulateRetention())
5. Populate discard mortality (PopulateDiscardMortality())
6. Populate spatial closures (PopulateClosure())

The equations for each step are described in Chapter 25.

23.6 Populate Obs

PopulateObs() configures the observation model for each fleet–complex combination. Unlike the biological and fishery arrays, observation parameters do not produce age- or year-dimensioned arrays during population. Instead, they set the sampling properties (e.g., error distributions, bias, sample size) used when simulated data are generated in the historical and projection periods.