# Fitting occupancy models with flocker

#### Jacob Socolar & Simon Mills

#### 2023-10-20

`flocker` is an R package for fitting occupancy models that incorporate sophisticated effects structures using simple formula-based syntax. `flocker` is built on R package `brms`, which in turn is a front-end for `Stan`.

This vignette is intended as a companion to Socolar & Mills 2023, where we provide details of the models and post-processing functionality available in `flocker` in greater detail. Here, we provide illustrative R code for several types of model, demonstrating data simulation, model fitting, and model post-processing. We also showcase the `brms` syntax that `flocker` can use to fit a variety of sophisticated effect structures.

## Terms and definitions

Socolar & Mills (2023) introduce several terms that figure importantly in this vignette, including:

- closure-unit: The groupings of observations over which closure is assumed. In single-species models, a closure-unit corresponds to a “site” or “point”. In multi-species models, a closure-unit is a species-site combination. In dynamic (multi-season) models, a closure-unit is a site-season combination (or species-site-season in a multi-species dynamic model).

- rep-constant, rep-varying: We refer to models that assume constant detection probabilities across repeat visits within closure-units as rep-constant models, as contrasted with rep-varying models that incorporate event-specific detection covariates. It turns out that rep-constant models enable a more efficient parametrization of the likelihood than rep-varying models.

- unit covariates, event covariates: We refer to any covariate that does not vary across sampling events within closure-units as a “unit covariate”. This includes covariates that are intrinsically properties of single closure-units (e.g. the elevations of sites in a single-species model), covariates that are intrinsically properties of groups of closure units (e.g. elevations of sites in a multi-species model), and covariates that are intrinsically properties of sampling events but happen to be constant within all closure-units (e.g. observer in a sampling design where every site is visited by exactly one observer). We refer to any covariate that varies across sampling events within covariates as an “event covariate”. Note that while unit covariates may appear in either the occupancy or the detection formula, event covariates are restricted to the detection formula. Models that incorporate event covariates are rep-varying (see above); those that do not are rep-constant.

## Installation and feedback

Installation instructions are available here. To request features or report bugs (much appreciated!), please open an issue on GitHub.

To make `flocker` and `brms` functions globally available within an R session run:

    library(flocker)
    library(brms)
    set.seed(1)

## Data simulation

General purpose data simulation is provided via `simulate_flocker_data()`, which by default will simulate a dataset with 30 species sampled at 50 sites using four replicate surveys (i.e. a single-season multi-species dataset). Non-default arguments will simulate example data for other likelihoods, including multi-season and data-augmented occupancy models.

    d <- simulate_flocker_data()

The simulated data `d` are in list form, with elements for the detection/non-detection observations `d$obs`, unit covariates `d$unit_covs`, and event covariates `d$event_covs`. `d$obs` is a matrix where rows are species-site combinations, columns are replicate visits, and entries are `1` (detection), `0` (nondetection), or `NA` (no visit). `d$unit_covs` is a dataframe containing covariates that vary across the rows of obs (i.e. by closure-unit) but not across the columns within any given row (i.e. do not vary across replicate visits). `event_covs` is a named list of matrices, with each matrix having the same dimensions as the observation matrix. Each list element corresponds to a covariate that varies across the columns of `d$obs` (i.e. varies between replicate visits).

## Data formatting

`flock()`, the main function in `flocker` for fitting occupancy models, expects a highly specific data format that we describe more fully here. The function `make_flocker_data()` formats data for use with `flock()` automatically. For single-season models, `make_flocker_data()` takes as input a matrix or dataframe of detection/non-detection data. Rows represent closure-units, columns represent repeated sampling events within closure-units, and entries must be `0` (nondetection), `1` (detection), or `NA` (no corresponding sampling event). The data must be formatted so that all `NA`s are trailing within their rows. For example, if some units were sampled four times and other three times, the three sampling events must be treated as events 1, 2, and 3 (with the fourth event `NA`) rather than as events 1, 3, and 4 (with the second event `NA`) or any other combination.

Many occupancy models also include covariates that influence occupancy or detection probabilities. Unit covariates (see Terms and definitions) can be passed to `make_flocker_data()` as a dataframe with the same number of rows as the observation matrix and data in the same order as the rows of the observation matrix. Columns are covariates, and we recommend using informative column names. Event covariates (see Terms and definitions) can be passed as a named list of matrices whose elements `[i, j]` are the covariate values for the sampling event represented by the corresponding position of the observation matrix. Again, we recommend using informative names for the list elements. If the corresponding observation is `NA`, then the value of the event covariate does not matter.

To pass data to `flocker`, we first pass the output from `simulate_flocker_data()` to `make_flocker_data()`, which will repackage data and apply the necessary formatting:

    fd_rep_varying <- make_flocker_data(
      obs = d$obs,
      unit_covs = d$unit_covs,
      event_covs = d$event_covs
      )
    #> Formatting data for a single-season occupancy model. For details, see make_flocker_data_static. All warnings and error messages should be interpreted in the context of make_flocker_data_static

The function `make_flocker_data()` outputs an object of class `flocker_data` that we can pass to flocker’s model fitting function `flock()`. Note that this is the general workflow users will need to follow with real data. Alternative inputs to `make_flocker_data()` and `flock()` enable the user to readily fit multi-season models as well as multi-species models with data augmentation (see below).

## Model fitting

### The single-season rep-varying model

To fit a model, in this case a single-season multi-species occupancy model, we use the function `flock()`. By supplying different arguments to this function, all flavors of occupancy model available in `flocker` can be fitted. Formulas for the different distributional parameters in the model (occupancy, detection, colonization, extinction, and autologistic terms as applicable) are provided as one-sided formulas to the relevant arguments of `flock()` (`f_occ`, `f_det`, `f_col`, `f_ex`, and `f_auto` as applicable).

    rep_varying <- flock(
      f_occ = ~ uc1 + (1 + uc1 | species),
      f_det = ~ uc1 + ec1 + (1 + uc1 + ec1 | species),
      flocker_data = fd_rep_varying,
      cores = 4,
      silent = 2,
      refresh = 0
      )

Arguments supplied to `flock()` define formulas using `brms` syntax for the occupancy (`f_occ`) and detection (`f_det`) components, and also provide the formatted data. At this stage, the full flexibility and power of `brms` formula syntax are available to the user (see following sections for some examples). `rep_varying` is a `brmsfit` object from package `brms` and also a `flockerfit` object from package `flocker`. Post-processing functions from `brms` will typically not work with this object and are instead replaced by `flocker` equivalents.

### The single-season rep-constant model

`make_flocker_data()` will automatically format the data for a rep-constant model when `event_covs = NULL` and the desired model is a single-season model without data augmentation. To take advantage of the efficiency gains and post-processing functionality of the rep-constant model, it is necessary to supply `event_covs = NULL` to `make_flocker_data()` at the moment of data formatting; it is insufficient to omit event covariates from the detection formula supplied to `flock()` after formatting the data for a rep-varying model.

    fd_rep_constant <- make_flocker_data(
      obs = d$obs,
      unit_covs = d$unit_covs
      )
    #> Formatting data for a single-season occupancy model. For details, see make_flocker_data_static. All warnings and error messages should be interpreted in the context of make_flocker_data_static
    rep_constant <- flock(
      f_occ = ~ uc1 + (1 + uc1 | species),
      f_det = ~ uc1 + (1 + uc1 | species),
      flocker_data = fd_rep_constant,
      save_pars = save_pars(all = TRUE), # for loo with moment matching
      silent = 2,
      refresh = 0,
      cores = 4
      )

Note that within-chain parallelization is available (uniquely so) for the rep-constant mode:

    rep_constant <- flock(
      f_occ = ~ uc1 + (1 + uc1 | species),
      f_det = ~ uc1 + (1 + uc1 | species),
      flocker_data = fd_rep_constant,
      silent = 2,
      refresh = 0,
      chains = 2,
      cores = 2,
      threads = 2
      )

### Multi-season models

Here we provide code examples to complement the companion publication. For a more complete vignette on multi-season models in `flocker`, see the multiseason models vignette.

First, we simulate some data that are valid for use with multi-season models. Here, we will simulate data for three seasons with one unit covariate and one event covariate. The data will be simulated under a colonization-extinction model with explicit inits, but we will be able to fit other models (autologistic, equilibrium inits) to the same data (note that `simulate_flocker_data()` can also simulate directly from these other model types).

    multi_data <- simulate_flocker_data(
      n_season = 3,
      n_pt = 300,
      n_sp = 1,
      multiseason = "colex",
      multi_init = "explicit",
      seed = 1
      )

    fd_multi <- make_flocker_data(
      multi_data$obs,
      multi_data$unit_covs,
      multi_data$event_covs,
      type = "multi",
      quiet = TRUE
      )

Below, we fit the colonization-extinction model with an explicit model for occupancy in the first timestep. Depending on hardware, fitting this model might take several minutes.

    multi_colex <- flock(
      f_occ = ~ uc1,
      f_det = ~ uc1 + ec1,
      f_col = ~ uc1,
      f_ex = ~ uc1,
      flocker_data = fd_multi,
      multiseason = "colex",
      multi_init = "explicit",
      cores = 4,
      silent = 2,
      refresh = 0
    )

Here is the colonization-extinction model using equilibrium occupancy probabilities in the first timestep:

    multi_colex_eq <- flock(
      f_det = ~ uc1 + ec1,
      f_col = ~ uc1,
      f_ex = ~ uc1,
      flocker_data = fd_multi,
      multiseason = "colex",
      multi_init = "equilibrium",
      cores = 4,
      silent = 2,
      refresh = 0
      )

Here is the autologistic model with explicit occupancy probabilities in the first timestep. To reflect the stereotypical autologistic model with a constant logit-scale offset separating colonization and persistence probabilities, we use the formula `f_auto = ~ 1`, but it is fine to relax this constraint and use, e.g. `f_auto = ~ uc1`.

    multi_auto <- flock(
      f_occ = ~ uc1,
      f_det = ~ uc1 + ec1,
      f_col = ~ uc1,
      f_auto = ~ 1,
      flocker_data = fd_multi,
      multiseason = "autologistic",
      multi_init = "explicit",
      cores = 4,
      silent = 2,
      refresh = 0
      )

And the autologistic model with equilibrium occupancy probabilities in the first timestep:

    multi_auto_eq <- flock(
      f_det = ~ uc1 + ec1,
      f_col = ~ uc1,
      f_auto = ~ 1,
      flocker_data = fd_multi,
      multiseason = "autologistic",
      multi_init = "equilibrium",
      cores = 4,
      silent = 2,
      refresh = 0
    )

### Data-augmented multi-species models

Here we provide a simple example of code for a data augmented model. For a more complete unified vignette on data-augmented models in `flocker`, see the data-augmented models vignette.

Fitting the data-augmented model in `flocker` requires passing the observed data as a three-dimensional array with sites along the first dimension, visits along the second, and species along the third. Additionally, we must supply the `n_aug` argument to `make_flocker_data()`, specifying how many all-zero pseudospecies to augment the data with.

    augmented_data <- simulate_flocker_data(
        augmented = TRUE
        )
    fd_augmented <- make_flocker_data(
      augmented_data$obs, augmented_data$unit_covs, augmented_data$event_covs,
      type = "augmented", n_aug = 100,
      quiet = TRUE
      )
    augmented <- flock(
      f_occ = ~ (1 | ff_species),
      f_det = ~ uc1 + ec1 + (1 + uc1 + ec1 | ff_species),
      augmented = TRUE,
      flocker_data = fd_augmented,
      cores = 4,
      silent = 2,
      refresh = 0
      )
    #> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
    #> Running the chains for more iterations may help. See
    #> https://mc-stan.org/misc/warnings.html#bulk-ess

Here, the random effect of species is specified using the special grouping keyword `ff_species` (names beginning with `ff_` are reserved in `flocker` and are not allowed as names for user-supplied covariates).

## Post-processing

`flocker` provides functions for four main types of bespoke post-processing for occupancy models. `fitted_flocker()` computes (and optionally summarizes) posterior distributions of fitted values at the locations of the data used in model fitting or of new data. `get_Z()` provides the posterior distribution for the latent occupancy state. `predict_flocker()` provides posterior predictions at the observed points (e.g. for use in posterior predictive checking) or for new data. `loo_flocker()` and `loo_compare_flocker()` both provide functionality for model comparison. See below for details on all four types of post-processing. Both posterior predictions and model comparison rely on subtle aspects of the occupancy model likelihood that we explain in more detail here.

### brms-native post-processing

All post-processing functions from `brms` work on single-season rep-constant models, but do not work on any other model types. For example:

    predictions_rep_constant <- brms::posterior_predict(rep_constant)
    loo_rep_constant <- brms::loo(rep_constant, moment_match = TRUE)
    brms::conditional_effects(rep_constant)

The following functions work on all model types available in `flocker`.

### Fitted values

Fitted values for any of the distributional parameter (one or more of occupancy, detection, colonization, extinction, autologistic, and/or Omega, the fitted probability that a given (pseudo)species occurs in the metacommunity) are available via `fitted_flocker`. For example:

    fitted_flocker(rep_constant)
    fitted_flocker(rep_varying)
    fitted_flocker(multi_colex)
    fitted_flocker(augmented)

`fitted_flocker` provides a replacement for `brms::posterior_linpred()`. While the `brms`-native function executes on any `flocker` model, it returns in an opaque shape related to the flocker data format. `fitted_flocker()` returns in the shape of the observations passed to `make_flocker_data()`, with posterior iterations stacked along its final dimension.

### The posterior occupancy state

The function `get_Z()` returns the posterior distribution of occupancy probabilities across the closure-units. The shape of the output depends on the class of model, and is an array in the shape of the first visit in `obs` as passed to `make_flocker_data`, with posterior iterations stacked along the final dimension. Thus, for a single-season rep-varying model, the output is a matrix where rows are posterior iterations, columns are closure-units, and values are draws from the posterior distribution of occupancy probabilities:

    get_Z(rep_varying)

For all model types, `get_Z()` accepts an optional `new_data` argument. Leaving the default `new_data = NULL` supplies the posterior for the true occupancy state at the locations of the data used to fit the model. Otherwise, the posterior is computed over the new data. For single-season models, `new_data` can be supplied as a dataframe of unit covariate values or as a `flocker_data` object. For multi-season models, only a `flocker_data` object is allowed. Note that if predictions are desired at sites without observations, it is acceptable to pass an array of dummy observations (e.g. all zeros) to `make_flocker_data()` and then to set `history_condition = FALSE` in the call to `get_Z()`.

`get_Z()` accepts several additional arguments that control the way that posterior is obtained and the values returned. See the companion paper and `?get_Z` for details.

### Posterior prediction

The function `predict_flocker()` provides posterior predictions. By default, predictions are provided for the covariate data to which the model were fit, but predictions to new data are also possible via the `new_data` argument. The output differs by model type. For single-season rep-constant models, the return is a matrix where rows are iterations, columns are units, and values are the number of detections. For single-season rep-varying models, the return is an array whose first dimension is units, second dimension is sampling events, third dimension is iterations, and values are `1`, `0`, or `NA`, representing detection, nondetection, and no corresponding sampling event. For example:

    predict_flocker(rep_varying)

`predict_flocker()` accepts several additional arguments that control the way that posterior is obtained and the values of returned. See the companion paper and `?predict_flocker` for details.

### Model comparison

The most straightforward way to compare models fit with `flocker` is the function `loo_compare_flocker()`. This function takes a list of flocker_fit objects as its argument and returns a model comparison table based on the difference in the expected log predictive density (elpd) between models. This table is a `compare.loo` object from `loo::loo_compare()`. The “leave-one-out” holdouts consist of entire closure-units (single-season models), series (multi-season models), or species (augmented models), not single sampling events (see the companion paper and here for details of why).

`loo_compare_flocker()` accepts as input a list of `flockerfit` objects and outputs a model comparison table. For example, we can compare the rep-constant and rep-varying models that we fit to the same initial data. Recall that the data were simulated with event-covariate effects on detection, and as expected the rep-varying model performs best. Note that we ensure that these comparisons between rep-constant and rep-varying models are valid by omitting the binomial coefficient when computing the log-likelihood for the rep-constant model.

    loo_compare_flocker(
      list(rep_constant, rep_varying)
    )
    #> Warning: Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.

    #> Warning: Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
    #>        elpd_diff se_diff
    #> model2    0.0       0.0
    #> model1 -171.1      17.2

Likewise, we can compare the four flavors of multi-season model that we fit above. Recall that the data were simulated under colonization-extinction dynamics (rather than autologistic) and under explicit initial occupancy probabilities (rather than equilibrium). As expected, the `multi_colex` model performs best:

    loo_compare_flocker(
      list(multi_colex, multi_colex_eq, multi_auto, multi_auto_eq)
    )
    #>        elpd_diff se_diff
    #> model1   0.0       0.0
    #> model3  -7.8       4.1
    #> model2 -27.1       7.2
    #> model4 -32.9       7.9

Flocker also provides the function `loo_flocker()` to return a table of `elpd_loo`, `p_loo`, and `looic` estimates from `loo::loo()` or `brms::loo()` (the latter for single-season rep-constant models only).

## `brms` tips and tricks

Mastering advanced occupancy modeling via `flocker` is mostly a matter of mastering the syntax available in `brms`. Here are some useful pieces of syntax:

### Priors

Priors can be implemented as they would with any `brms` model. Priors can be specified using `set_prior()`, with priors specified for groups of parameters (via `class`) or individual parameters (via `coef`). The priors used for a particular model can be retrieved using `brms::prior_summary()`, and the names of the parameters and their default priors can be displayed prior to model fitting using `get_flocker_prior()` which is a drop-in replacement for `brms::get_prior()`.

    get_flocker_prior(
      f_occ = ~ uc1 + (1 + uc1 | species),
      f_det = ~ uc1 + ec1 + (1 + uc1 + ec1 | species),
      flocker_data = fd_rep_varying
      )
    #>                 prior     class      coef   group resp dpar nlpar lb ub       source
    #>                (flat)         b                                              default
    #>                (flat)         b       ec1                               (vectorized)
    #>                (flat)         b       uc1                               (vectorized)
    #>                lkj(1)       cor                                              default
    #>                lkj(1)       cor           species                       (vectorized)
    #>  student_t(3, 0, 2.5) Intercept                                              default
    #>  student_t(3, 0, 2.5)        sd                                    0         default
    #>  student_t(3, 0, 2.5)        sd           species                  0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd       ec1 species                  0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd Intercept species                  0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd       uc1 species                  0    (vectorized)
    #>                (flat)         b                         occ                  default
    #>                (flat)         b       uc1               occ             (vectorized)
    #>                (flat) Intercept                         occ                  default
    #>  student_t(3, 0, 2.5)        sd                         occ        0         default
    #>  student_t(3, 0, 2.5)        sd           species       occ        0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd Intercept species       occ        0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd       uc1 species       occ        0    (vectorized)
    brms::prior_summary(rep_varying)
    #>                 prior     class      coef   group resp dpar nlpar lb ub       source
    #>                (flat)         b                                              default
    #>                (flat)         b       ec1                               (vectorized)
    #>                (flat)         b       uc1                               (vectorized)
    #>                (flat)         b                         occ                  default
    #>                (flat)         b       uc1               occ             (vectorized)
    #>  student_t(3, 0, 2.5) Intercept                                              default
    #>                (flat) Intercept                         occ                  default
    #>  lkj_corr_cholesky(1)         L                                              default
    #>  lkj_corr_cholesky(1)         L           species                       (vectorized)
    #>  student_t(3, 0, 2.5)        sd                                    0         default
    #>  student_t(3, 0, 2.5)        sd                         occ        0         default
    #>  student_t(3, 0, 2.5)        sd           species                  0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd       ec1 species                  0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd Intercept species                  0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd       uc1 species                  0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd           species       occ        0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd Intercept species       occ        0    (vectorized)
    #>  student_t(3, 0, 2.5)        sd       uc1 species       occ        0    (vectorized)

Note that in examples like the above, with covariates shared between both the occupancy and detection model formulas (`uc1` in this example), then the prior table will contain two entries associated with the covariate, one for the parameter governing occupancy and one for the parameter governing detection. Specifying priors for parameters in formulas other than detection can be done with reference to the `dpar` column, e.g.:

    user_prior <- c(brms::set_prior("normal(0, 1)", coef = "uc1"),
                    brms::set_prior("normal(0, 3)", coef = "uc1", dpar = "occ"))

where the `uc1` parameter in the occupancy component is specified by the addition of the `dpar` argument, and the `uc1` parameter in the detection component is specified without reference to `dpar`.

For more on priors in `brms`, see `?brms::set_prior`.

Users should understand the implications of the default `brms` behavior to internally center the design matrix, which affects how the prior on the intercept gets set (see `?brms::set_prior`). Here is an example, based on a single-season rep-varying model, wherein we set a logistic prior on the value of the intercepts (flat on the probability scale) when all predictors are held at their means and a moderately regularizing prior on the coefficients:

    rep_varying_prior1 <- flock(
      f_occ = ~ uc1,
      f_det = ~ ec1,
      flocker_data = fd_rep_varying,
      prior =
        brms::set_prior("logistic(0,1)", class = "Intercept") +
        brms::set_prior("logistic(0,1)", class = "Intercept", dpar = "occ") +
        brms::set_prior("normal(0,2)", class = "b") +
        brms::set_prior("normal(0,2)", class = "b", dpar = "occ"),
      cores = 4,
      silent = 2,
      refresh = 0
      )
    brms::prior_summary(rep_varying_prior1)
    #>          prior     class coef group resp dpar nlpar lb ub       source
    #>    normal(0,2)         b                                          user
    #>    normal(0,2)         b  ec1                             (vectorized)
    #>    normal(0,2)         b                  occ                     user
    #>    normal(0,2)         b  uc1             occ             (vectorized)
    #>  logistic(0,1) Intercept                                          user
    #>  logistic(0,1) Intercept                  occ                     user

Here is an example where we set informative priors on the intercepts when all covariates are fixed to zero and the same moderately regularizing prior on the coefficients:

    rep_varying_prior2 <- flock(
      f_occ = ~ 0 + Intercept + uc1,
      f_det = ~ 0 + Intercept + ec1,
      flocker_data = fd_rep_varying,
      prior =
        brms::set_prior("normal(0,2)", class = "b") +
        brms::set_prior("normal(0,2)", class = "b", dpar = "occ") +
        brms::set_prior("normal(1, 1)", class = "b", coef = "Intercept") +
        brms::set_prior("normal(-1, 1)", class = "b", coef = "Intercept", dpar = "occ"),
      cores = 4,
      silent = 2,
      refresh = 0
      )
    brms::prior_summary(rep_varying_prior2)
    #>          prior class      coef group resp dpar nlpar lb ub       source
    #>    normal(0,2)     b                                               user
    #>    normal(0,2)     b       ec1                             (vectorized)
    #>   normal(1, 1)     b Intercept                                     user
    #>    normal(0,2)     b                       occ                     user
    #>  normal(-1, 1)     b Intercept             occ                     user
    #>    normal(0,2)     b       uc1             occ             (vectorized)

### Model formulas

Simple formulas follow the same syntax as R’s `lm()` function. For example:

    mod1 <- flock(
      f_occ = ~ uc1 + (1|species),
      f_det = ~ 1,
      flocker_data = fd_rep_constant
      )

### Random effects

Simple random effects follow `lme4` syntax, including advanced `lme4` syntax like `||` for uncorrelated effects and `/` and `:` for expansion of multiple grouping terms. Here’s a simple example:

    mod2 <- flock(
      f_occ = ~ uc1 + (1|species),
      f_det = ~ 1,
      flocker_data = fd_rep_constant
      )

When a model includes multiple random effects with the same grouping term, by default they are modeled as correlated within the occupancy or detection formulas, but as uncorrelated between formulas. For example, the code below estimates a single correlation for the intercept and slope in the occupancy sub-model.

    mod3 <- flock(
      f_occ = ~ uc1 + (1 + uc1 | species),
      f_det = ~ ec1 + (1 | species),
      flocker_data = fd_rep_varying
      )

However, this assumption can easily be relaxed using the `|<ID>|` syntax from `brms`. The `<ID>` is an arbitrary character string representing a group of terms to model as correlated. The below code, for example, models correlated intercepts in the occupancy and detection sub-models, and correlated effects of `sc1` on occupancy and `vc1` on detection, but no correlations between the intercepts and the slopes in either sub-model:

    mod4 <- flock(
      f_occ = ~ uc1 + (1 |g1| species) + (0 + uc1 |g2| species),
      f_det = ~ ec1 + (1 |g1| species) + (0 + ec1 |g2| species),
      flocker_data = fd_rep_varying
      )

For more on `brms` syntax for random effects syntax, see the documentation here.

### Nonlinear models

Via `brms`, `flocker` supports Gaussian processes of arbitrary dimensionality (`brms::gp()`) as well as `mgcv` syntax for thin-plate regression splines (`brms::s()`) and tensor product smooths (`brms::t2()`), and `brms` syntax for monotonic effects of ordinal factors via `brms::mo()` (see here). For example:

    mod5 <- flock(
      f_occ = ~ s(uc1),
      f_det = ~ t2(uc1, ec1),
      flocker_data = fd_rep_varying
      )

    mod6 <- flock(
      f_occ = ~ 1,
      f_det = ~ gp(uc1, ec1),
      flocker_data = fd_rep_varying
      )

In addition, `brms` provides the ability to estimate models wherein the predictors (e.g. for occupancy and detection) are parametric nonlinear functions whose parameters have their own covariate-based linear predictors. For more details and an example, see the nonlinear models vignette.

### Phylogenetic models

Phylogenetic effects can be included by providing a covariance matrix as a `data2` argument and using the `brms::gr()` function to link species identities in `flocker_data` with the supplied covariance matrix. Note that phylogenetic effects can be included in either the occupancy component, the detection component, or both! In our experience, it can be computationally tractable to include multiple phylogenetic effects within a single occupancy model (see Mills et al. 2022).

    # simulate an example phylogeny
    phylogeny <- ape::rtree(30, tip.label = paste0("sp_", 1:30))

    # calculate covariance matrix
    A <- ape::vcv.phylo(phylogeny)

    mod8 <- flock(
      f_occ = ~ 1 + (1|gr(species, cov = A)),
      f_det = ~  1 + ec1 + (1|species),
      flocker_data = fd_rep_varying,
      data2 = list(A = A)
      )

    mod9 <- flock(
      f_occ = ~ 1 + (1|gr(species, cov = A)),
      f_det = ~  1 + ec1 + (1|gr(species, cov = A)),
      flocker_data = fd_rep_varying,
      data2 = list(A = A)
      )

See here for further details about specifying phylogenetic effects in `brms`.

### Spatial and autoregressive structures

In addition to spatial Gaussian processes, `brms` provides a variety of autoregressive structures, both one-dimensional (see `brms::ar()`, `brms::arma()`) and two-dimensional (see `brms::car()`, `brms::sar()`. See here for details about conditional autoregressive (CAR) models in `brms`, and note that `flock()` accepts a `data2` argument that it can pass to `brms` as necessary.

Our principle caution for users is that these autoregressive structures might lead to degenerate models when applied at the visit level (in detection formulas) or at the closure-unit level (in occupancy, colonization, extinction, or autologistic formulas) because observation-level random effects are often degenerate in regressions with Bernoulli responses. Thus we recommend applying autoregressive terms to groupings of multiple visits (detection formula) or multiple closure-units (other formulas). However, we note that `flocker` does provide a well-identified one-dimensional first-order autoregressive structure for occupancy across closure-units in a single-season model. This is achieved by co-opting the autologistic parameterization of the multi-season model and applying it instead to closure-units arranged along a one-dimensional spatial transect, yielding a one-dimensional analog of a spatial autologistic occupancy model.

A second caution is to remind users that in multi-species models, users will likely want to fit separate spatial terms by species (Doser et al 2022). For Gaussian processes, this can be achieved via the `by` argument to `brms::gp()`. For some conditional autoregressive structures (those that allow disconnected islands), this can be achieved by passing a block-diagonal adjacency matrix wherein species are disconnected components.

We note that gaussian process priors for spatially varying coefficients are readily achieved via the nonlinear formula syntax of `brms`, though they may require large volumes of data to successfully fit. For more details and an example, see the nonlinear models vignette.

### Measurement error in covariates

See here for relevant `brms` documentation.

## Additional fitting arguments

`flock` will pass any relevant parameters forward to `brms::brm()`, giving the user important control over the algorithmic details of how the model is fit. See `?brms::brm` for details. To speed up the execution, we recommend supplying the argument `backend = "cmdstanr"`. This requires the `cmdstanr` package and a working installation of `cmdstan`; see here for instructions to get started and further details.