Alternative Approaches for Movement Parameterization • SPoRC

The SPoRC package offers multiple approaches for modeling fish movement between regions, each with different complexity and flexibility trade-offs. This vignette demonstrates how to configure movement parameterization using the Setup_Mod_Movement function with various structural options. In general, there are two primary movement model types, which include:

Unstructured Markov: Discrete movement matrices with optional blocking structures across population, age, year, season, and sex dimensions
Continuous-Time Markov Chain (CTMC): CTMC-based movement with diffusion and preference parameters estimated using formula-based approaches

The following demonstrations use the three region sablefish dataset as a basis (three_rg_sable_data). The initial setup establishes the general model dimensions. Note that n_pop = 1 and n_seas = 1 are specified here, but movement can be configured for multi-population and multi-season models in an analogous fashion.

# Load in packages
library(SPoRC) 
data("three_rg_sable_data") # load in data

# setup model dimensions
input_list <- Setup_Mod_Dim(years = 1:length(three_rg_sable_data$years),
                            ages = 1:length(three_rg_sable_data$ages),
                            lens = seq(41,99,2),
                            n_regions = three_rg_sable_data$n_regions,
                            n_sexes = three_rg_sable_data$n_sexes,
                            n_fish_fleets = three_rg_sable_data$n_fish_fleets,
                            n_srv_fleets = three_rg_sable_data$n_srv_fleets,
                            n_seas = 1,
                            n_pop = 1,
                            verbose = TRUE
                            )

Like previous vignettes, the Setup_Mod_Dim() function initializes the model structure with 62 years of data (1960–2021), 30 age classes, 3 regions, 2 sexes, 2 fishery fleets (fixed gear and trawl), and multiple survey fleets.

Unstructured Markov

In the simplest case, movement can be parameterized as an unstructured Markov model (move_type = 0), where movement parameters are constant across model partitions and are estimated as discrete transitions between regions. This results in n_regions × (n_regions - 1) parameters estimated.

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0, # recruits don't move
  move_type = 0 # unstructured markov
)
length(unique(input_list$map$move_pars)) # number of parameters estimated

Age Blocks

The unstructured Markov model can be extended to incorporate blocking structures, with parameter sharing within blocks to reduce parameter load. In the following, we specify an unstructured Markov model with 2 age blocks, but constant movement across years and sexes. This results in n_regions × (n_regions - 1) × 2 parameters estimated.

# define age blocks
age_blk <- list(c(1:15), c(16:30))
age_blk

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 0,
  Movement_ageblk_spec = age_blk,
  Movement_yearblk_spec = "constant",
  Movement_sexblk_spec = "constant"
)

length(unique(input_list$map$move_pars)) # number of parameters estimated

Year Blocks

Year blocks can be specified in a similar fashion. Here, 5 year blocks are specified, resulting in n_regions × (n_regions - 1) × 5 parameters estimated.

# define year blocks
yr_blk <- list(c(1:15), c(16:30), c(31:45), c(46:60), c(61:62))
yr_blk

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 0,
  Movement_ageblk_spec = "constant",
  Movement_yearblk_spec = yr_blk,
  Movement_sexblk_spec = "constant"
)

length(unique(input_list$map$move_pars)) # number of parameters estimated

Sex Blocks

Sex blocks are specified similarly. The following example shows sex-specific movement, resulting in n_regions × (n_regions - 1) × n_sexes parameters estimated.

# define sex blocks
sx_blk <- as.list(1:2)
sx_blk

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 0,
  Movement_ageblk_spec = "constant",
  Movement_yearblk_spec = "constant",
  Movement_sexblk_spec = sx_blk
)

Blocks Across All Dimensions

Building on the principles described above, movement blocks can also be defined simultaneously across ages, years, and sexes. The example below specifies two age blocks, five year blocks, and sex-specific movement. This configuration results in n_regions × (n_regions - 1) × 2 × 5 × n_sexes movement parameters being estimated. However, this parameterization is likely excessive and may lead to an unstable model solution. Note that this model uses 1 population and 1 season, but population-specific and season-specific movement rates can be specified in an analogous fashion using Movement_popblk_spec and Movement_seasblk_spec.

# define blocks across all dimensions
age_blk <- list(c(1:15), c(16:30))
yr_blk  <- list(c(1:15), c(16:30), c(31:45), c(46:60), c(61:62))
sx_blk  <- as.list(1:2)

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 0,
  Movement_ageblk_spec = age_blk,
  Movement_yearblk_spec = yr_blk,
  Movement_sexblk_spec = sx_blk
)

Continuous-Time Markov Chain (CTMC)

One potential approach to reduce model parameterization is to represent movement as a continuous-time Markov chain (CTMC) process, in which transitions are governed by a mechanistic framework composed of diffusion (random dispersal) and taxis (directed preference) components. Unlike the unstructured Markov model, CTMC movement processes are defined in continuous time and are converted to annual movement fractions using the matrix exponential, thereby allowing for sequential transitions among regions.

To implement a CTMC-based movement model, an adjacency matrix must first be defined to specify which regions are connected. In the example below, all regions are assumed to be connected, permitting individuals to move among any spatial strata within a given period. An accompanying data frame is then created to define the covariates associated with movement. This data frame must include columns for pop, regions, years, seas, ages, and sexes, plus any additional covariates to be used in the diffusion or preference formulas. The seas column should include all seasons being modelled; projection year covariate values can also be appended to this data frame when n_proj_yrs_devs > 0.

adjacency <- igraph::as_adjacency_matrix(
  igraph::make_graph(
    ~ 1 - 2,
    2 - 3,
    1 - 3
  )
)

# make ctmc data — must include pop, regions, years, seas, ages, sexes columns
ctmc_data <- expand.grid(
  pop     = 1,
  regions = 1:three_rg_sable_data$n_regions,
  years   = 1:length(three_rg_sable_data$years),
  seas    = 1,
  ages    = 1:length(three_rg_sable_data$ages),
  sexes   = 1:three_rg_sable_data$n_sexes
)

Constant Movement

In the code chunk below, movement is defined as arising from a purely diffusive process, which effectively represents constant movement across regions. This formulation results in the estimation of n_regions parameters, providing a more parsimonious alternative to the unstructured Markov approach that estimates n_regions × (n_regions - 1) parameters. Movement among adjacent areas is determined by the defined adjacency matrix, and no directional preference is specified in this case.

The argument area_r = rep(1, 3) specifies how diffusive processes scale with area size. Here, all areas are assumed to be equal. However, when areas differ in size, area_r should be defined as proportional to area, such that smaller areas are associated with higher diffusion rates.

# constant diffusion, no preference
diffusion_formula  <- ~0 + factor(regions)
preference_formula <- ~0

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 1,
  ctmc_move_dat = ctmc_data,
  adjacency_mat = adjacency,
  area_r = rep(1, 3),
  diffusion_formula = diffusion_formula,
  preference_formula = preference_formula
)

length(input_list$par$log_move_diffusion_pars)
length(input_list$par$move_preference_pars)

Age-Varying

Movement can also be specified to vary across model partitions. The examples below illustrate how age-varying movement can be represented within the CTMC framework using both linear and spline-based relationships.

Linear

In this example, diffusion is assumed constant across ages, while preference varies linearly by age within each region. This specification results in the estimation of a single diffusion parameter, along with n_regions additional parameters describing age-specific movement preferences.

# constant diffusion, linear age preference
diffusion_formula  <- ~1
preference_formula <- ~0 + factor(regions):ages

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 1,
  ctmc_move_dat = ctmc_data,
  adjacency_mat = adjacency,
  area_r = rep(1, 3),
  diffusion_formula = diffusion_formula,
  preference_formula = preference_formula
)

length(input_list$par$log_move_diffusion_pars)
length(input_list$par$move_preference_pars)

Spline

In some cases, age-specific movement patterns may be more complex than a simple linear relationship can represent. To capture nonlinear variation, spline-based age-specific movement can be modeled using a spline basis. In the example below, one diffusion parameter is estimated, along with n_regions × 4 parameters describing age-specific movement preferences.

# constant diffusion, spline-based age preference
diffusion_formula  <- ~1
preference_formula <- ~0 + factor(regions):splines2::bSpline(ages, df = 4, intercept = TRUE)

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 1,
  ctmc_move_dat = ctmc_data,
  adjacency_mat = adjacency,
  area_r = rep(1, 3),
  diffusion_formula = diffusion_formula,
  preference_formula = preference_formula
)

length(input_list$par$log_move_diffusion_pars)
length(input_list$par$move_preference_pars)

Movement Across All Dimensions

Movement can also be specified to vary continuously across all model dimensions. In this example, movement is modeled as a function of region, age, year, and sex. For multi-population models, factor(pop) can be added to the formula in an analogous fashion. This specification results in the estimation of n_regions × 4 × 6 × n_sexes preference parameters.

# spline-based age and year preference, sex-specific
diffusion_formula  <- ~1
preference_formula <- ~0 + factor(regions):
  splines2::bSpline(ages, df = 4, intercept = TRUE):
  splines2::bSpline(years, df = 6, intercept = TRUE):
  factor(sexes)

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 1,
  ctmc_move_dat = ctmc_data,
  adjacency_mat = adjacency,
  area_r = rep(1, 3),
  diffusion_formula = diffusion_formula,
  preference_formula = preference_formula
)

length(input_list$par$log_move_diffusion_pars)
length(input_list$par$move_preference_pars)

Process Error

Lastly, process error deviations can also be incorporated into movement estimates. In the example below, movement is modeled using a CTMC framework, although process error can similarly be applied to an unstructured Markov model. Movement is specified to vary smoothly across ages using a spline function, while allowing independent and identically distributed (iid) deviations across years for each source region (cont_vary_movement = 'iid_y'). Process error variance parameters are specified to be shared across populations, source regions, seasons, ages, and sexes (Movement_cont_pe_pars_spec = 'est_shared'). Additional options for cont_vary_movement and Movement_cont_pe_pars_spec are described in the function documentation (?Setup_Mod_Movement).

Users may alternatively treat these variance parameters as random effects (integrated out via the Laplace approximation) or as penalized likelihood terms, depending on how Movement_cont_pe_pars_spec is defined. When Movement_cont_pe_pars_spec = 'fix', users can supply a fixed variance value directly through input_list$par$move_pe_pars. Note that deviations are only estimated for destination regions (i.e., n_regions - 1), as no deviation term is defined for the source region.

# spline age preference with iid year deviations
diffusion_formula  <- ~1
preference_formula <- ~0 + factor(regions):splines2::bSpline(ages, df = 4, intercept = TRUE)

# Setup movement
input_list <- Setup_Mod_Movement(
  input_list = input_list,
  do_recruits_move = 0,
  move_type = 1,
  ctmc_move_dat = ctmc_data,
  adjacency_mat = adjacency,
  area_r = rep(1, 3),
  diffusion_formula = diffusion_formula,
  preference_formula = preference_formula,
  cont_vary_movement = 'iid_y',
  Movement_cont_pe_pars_spec = 'est_shared'
)

length(input_list$par$log_move_diffusion_pars)
length(input_list$par$move_preference_pars)
length(unique(input_list$map$move_devs))