From 9354d4b6fc9b1c2688c9e7845dab1a00d3d215bf Mon Sep 17 00:00:00 2001 From: Nicholas Clark Date: Mon, 29 Jan 2024 13:40:49 +1000 Subject: [PATCH] nmixture vignette; few minor bug fixes --- R/add_MACor.R | 4 +- R/ppc.mvgam.R | 4 +- docs/articles/nmixtures.html | 1060 +++++++++++++++++ .../figure-html/unnamed-chunk-10-1.png | Bin 0 -> 145323 bytes .../figure-html/unnamed-chunk-11-1.png | Bin 0 -> 38838 bytes .../figure-html/unnamed-chunk-13-1.png | Bin 0 -> 126158 bytes .../figure-html/unnamed-chunk-13-2.png | Bin 0 -> 113519 bytes .../figure-html/unnamed-chunk-23-1.png | Bin 0 -> 61972 bytes .../figure-html/unnamed-chunk-24-1.png | Bin 0 -> 55844 bytes .../figure-html/unnamed-chunk-26-1.png | Bin 0 -> 83661 bytes .../figure-html/unnamed-chunk-26-2.png | Bin 0 -> 84310 bytes .../figure-html/unnamed-chunk-27-1.png | Bin 0 -> 197809 bytes src/mvgam.dll | Bin 1084416 -> 1084416 bytes tests/testthat/Rplots.pdf | Bin 22917 -> 22937 bytes vignettes/nmixtures.Rmd | 487 ++++++++ 15 files changed, 1551 insertions(+), 4 deletions(-) create mode 100644 docs/articles/nmixtures.html create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-10-1.png create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-11-1.png create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-13-1.png create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-13-2.png create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-23-1.png create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-24-1.png create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-26-1.png create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-26-2.png create mode 100644 docs/articles/nmixtures_files/figure-html/unnamed-chunk-27-1.png create mode 100644 vignettes/nmixtures.Rmd diff --git a/R/add_MACor.R b/R/add_MACor.R index 10ff6a23..9431812d 100644 --- a/R/add_MACor.R +++ b/R/add_MACor.R @@ -19,11 +19,11 @@ add_MaCor = function(model_file, if(any(grepl('transformed data {', model_file, fixed = TRUE))){ model_file[grep('transformed data {', model_file, fixed = TRUE)] <- paste0('transformed data {\n', - 'vector[n_series] trend_zeros = rep_vector(0.0, n_lv);') + 'vector[n_lv] trend_zeros = rep_vector(0.0, n_lv);') } else { model_file[grep('parameters {', model_file, fixed = TRUE)[1]] <- paste0('transformed data {\n', - 'vector[n_series] trend_zeros = rep_vector(0.0, n_lv);\n', + 'vector[n_lv] trend_zeros = rep_vector(0.0, n_lv);\n', '}\nparameters {') } diff --git a/R/ppc.mvgam.R b/R/ppc.mvgam.R index 47ddf108..e32d20d6 100644 --- a/R/ppc.mvgam.R +++ b/R/ppc.mvgam.R @@ -71,7 +71,7 @@ ppc.mvgam = function(object, newdata, data_test, series = 1, type = 'hist', "prop_zero")) if(type == 'rootogram'){ - if(!object$family %in% c('poisson', 'negative binomial', 'tweedie')){ + if(!object$family %in% c('poisson', 'negative binomial', 'tweedie', 'nmix')){ stop('Rootograms not supported for checking non-count data', call. = FALSE) } @@ -235,7 +235,7 @@ ppc.mvgam = function(object, newdata, data_test, series = 1, type = 'hist', preds <- mcmc_chains(object$model_output, 'ypred')[,starts[series]:ends[series]] } - preds <- preds[,1:length(truths)] + preds <- preds[,1:length(truths), drop = FALSE] if(NROW(preds) > 4000){ preds <- preds[sample(1:NROW(preds), 4000, F), ] diff --git a/docs/articles/nmixtures.html b/docs/articles/nmixtures.html new file mode 100644 index 00000000..b94b6883 --- /dev/null +++ b/docs/articles/nmixtures.html @@ -0,0 +1,1060 @@ + + + + + + + + +N-mixtures in mvgam • mvgam + + + + + + + + + + Skip to contents + + +
+ + + + +
+
+ + + +

The purpose of this vignette is to show how the mvgam +package can be used to fit and interrogate N-mixture models for +population abundance counts made with imperfect detection.

+
+

N-mixture models +

+

An N-mixture model is a fairly recent addition to the ecological +modeller’s toolkit that is designed to make inferences about variation +in the abundance of species when observations are imperfect (Royle 2004). Briefly, assume \(\boldsymbol{Y_{i,r}}\) is the number of +individuals recorded at site \(i\) +during replicate sampling observation \(r\) (recorded as a non-negative integer). +If multiple replicate surveys are done within a short enough period to +satisfy the assumption that the population remained closed (i.e. there +was no substantial change in true population size between replicate +surveys), we can account for the fact that observations aren’t perfect. +This is done by assuming that these replicate observations are Binomial +random variables that are parameterized by the true “latent” abundance +\(N\) and a detection probability \(p\):

+

\[\begin{align*} +\boldsymbol{Y_{i,r}} & \sim \text{Binomial}(N_i, p_r) \\ +N_{i} & \sim \text{Poisson}(\lambda_i) \end{align*}\]

+

Using a set of linear predictors, we can estimate effects of +covariates \(\boldsymbol{X}\) on the +expected latent abundance (with a log link for \(\lambda\)) and, jointly, effects of +possibly different covariates (call them \(\boldsymbol{Q}\)) on detection probability +(with a logit link for \(p\)):

+

\[\begin{align*} +log(\lambda) & = \beta \boldsymbol{X} \\ +logit(p) & = \gamma \boldsymbol{Q}\end{align*}\]

+

mvgam can handle this type of model because it is +designed to propagate unobserved temporal processes that evolve +independently of the observation process in a State-space format. This +setup adapts well to N-mixture models because they can be thought of as +State-space models in which the latent state is a discrete variable +representing the “true” but unknown population size. This is very +convenient because we can incorporate any of the package’s diverse +effect types (i.e. multidimensional splines, time-varying effects, +monotonic effects, random effects etc…) into the linear predictors. All +that is required for this to work is a marginalization trick that allows +Stan’s sampling algorithms to handle discrete parameters +(see more about how this method of “integrating out” discrete parameters +works in this nice blog post by Maxwell Joseph).

+

The family nmix() is used to set up N-mixture models in +mvgam, but we still need to do a little bit of data +wrangling to ensure the data are set up in the correct format (this is +especially true when we have more than one replicate survey per time +period). The most important aspects are: (1) how we set up the +observation series and trend_map arguments to +ensure replicate surveys are mapped to the correct latent abundance +model and (2) the inclusion of a cap variable that defines +the maximum possible integer value to use for each observation when +estimating latent abundance. The two examples below give a reasonable +overview of how this can be done.

+
+
+ +

First we will use a simple simulation in which multiple replicate +observations are taken at each timepoint for two different species. The +simulation produces observations at a single site over six years, with +five replicate surveys per year. Each species is simulated to have +different nonlinear temporal trends and different detection +probabilities. For now, detection probability is fixed (i.e. it does not +change over time or in association with any covariates). Notice that we +add the cap variable, which does not need to be static, to +define the maximum possible value that we think the latent abundance +could be for each timepoint. This simply needs to be large enough that +we get a reasonable idea of which latent N values are most likely, +without adding too much computational cost:

+
+set.seed(999)
+# Simulate observations for species 1, which shows a declining trend and 0.7 detection probability
+data.frame(site = 1,
+           # five replicates per year; six years
+           replicate = rep(1:5, 6),
+           time = sort(rep(1:6, 5)),
+           species = 'sp_1',
+           # true abundance declines nonlinearly
+           truth = c(rep(28, 5),
+                     rep(26, 5),
+                     rep(23, 5),
+                     rep(16, 5),
+                     rep(14, 5),
+                     rep(14, 5)),
+           # observations are taken with detection prob = 0.7
+           obs = c(rbinom(5, 28, 0.7),
+                   rbinom(5, 26, 0.7),
+                   rbinom(5, 23, 0.7),
+                   rbinom(5, 15, 0.7),
+                   rbinom(5, 14, 0.7),
+                   rbinom(5, 14, 0.7))) %>%
+  # add 'series' information, which is an identifier of site, replicate and species
+  dplyr::mutate(series = paste0('site_', site,
+                                '_', species,
+                                '_rep_', replicate),
+                time = as.numeric(time),
+                # add a 'cap' variable that defines the maximum latent N to 
+                # marginalize over when estimating latent abundance; in other words
+                # how large do we realistically think the true abundance could be?
+                cap = 100) %>%
+  dplyr::select(- replicate) -> testdat
+
+# Now add another species that has a different temporal trend and a smaller 
+# detection probability (0.45 for this species)
+testdat = testdat %>%
+  dplyr::bind_rows(data.frame(site = 1,
+                              replicate = rep(1:5, 6),
+                              time = sort(rep(1:6, 5)),
+                              species = 'sp_2',
+                              truth = c(rep(4, 5),
+                                        rep(7, 5),
+                                        rep(15, 5),
+                                        rep(16, 5),
+                                        rep(19, 5),
+                                        rep(18, 5)),
+                              obs = c(rbinom(5, 4, 0.45),
+                                      rbinom(5, 7, 0.45),
+                                      rbinom(5, 15, 0.45),
+                                      rbinom(5, 16, 0.45),
+                                      rbinom(5, 19, 0.45),
+                                      rbinom(5, 18, 0.45))) %>%
+                     dplyr::mutate(series = paste0('site_', site,
+                                                   '_', species,
+                                                   '_rep_', replicate),
+                                   time = as.numeric(time),
+                                   cap = 50) %>%
+                     dplyr::select(-replicate))
+

This data format isn’t too difficult to set up, but it does differ +from the traditional multidimensional array setup that is commonly used +for fitting N-mixture models in other software packages. Next we ensure +that species and series IDs are included as factor variables, in case +we’d like to allow certain effects to vary by species

+
+testdat$species <- factor(testdat$species,
+                          levels = unique(testdat$species))
+testdat$series <- factor(testdat$series,
+                         levels = unique(testdat$series))
+

Preview the dataset to get an idea of how it is structured:

+
+dplyr::glimpse(testdat)
+
## Rows: 60
+## Columns: 7
+## $ site    <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
+## $ time    <dbl> 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5,…
+## $ species <fct> sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp…
+## $ truth   <dbl> 28, 28, 28, 28, 28, 26, 26, 26, 26, 26, 23, 23, 23, 23, 23, 16…
+## $ obs     <int> 20, 19, 23, 17, 18, 21, 18, 21, 19, 18, 17, 16, 20, 11, 19, 9,…
+## $ series  <fct> site_1_sp_1_rep_1, site_1_sp_1_rep_2, site_1_sp_1_rep_3, site_…
+## $ cap     <dbl> 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 10…
+
+head(testdat, 12)
+
##    site time species truth obs            series cap
+## 1     1    1    sp_1    28  20 site_1_sp_1_rep_1 100
+## 2     1    1    sp_1    28  19 site_1_sp_1_rep_2 100
+## 3     1    1    sp_1    28  23 site_1_sp_1_rep_3 100
+## 4     1    1    sp_1    28  17 site_1_sp_1_rep_4 100
+## 5     1    1    sp_1    28  18 site_1_sp_1_rep_5 100
+## 6     1    2    sp_1    26  21 site_1_sp_1_rep_1 100
+## 7     1    2    sp_1    26  18 site_1_sp_1_rep_2 100
+## 8     1    2    sp_1    26  21 site_1_sp_1_rep_3 100
+## 9     1    2    sp_1    26  19 site_1_sp_1_rep_4 100
+## 10    1    2    sp_1    26  18 site_1_sp_1_rep_5 100
+## 11    1    3    sp_1    23  17 site_1_sp_1_rep_1 100
+## 12    1    3    sp_1    23  16 site_1_sp_1_rep_2 100
+
+

Setting up the trend_map +

+

Finally, we need to set up the trend_map object. This is +crucial for allowing multiple observations to be linked to the same +latent process model (see more information about this argument in the Shared latent states vignette. In this case, the +mapping operates by species and site to state that each set of replicate +observations from the same time point should all share the exact same +latent abundance model:

+
+testdat %>%
+  # each unique combination of site*species is a separate process
+  dplyr::mutate(trend = as.numeric(factor(paste0(site, species)))) %>%
+  dplyr::select(trend, series) %>%
+  dplyr::distinct() -> trend_map
+trend_map
+
##    trend            series
+## 1      1 site_1_sp_1_rep_1
+## 2      1 site_1_sp_1_rep_2
+## 3      1 site_1_sp_1_rep_3
+## 4      1 site_1_sp_1_rep_4
+## 5      1 site_1_sp_1_rep_5
+## 6      2 site_1_sp_2_rep_1
+## 7      2 site_1_sp_2_rep_2
+## 8      2 site_1_sp_2_rep_3
+## 9      2 site_1_sp_2_rep_4
+## 10     2 site_1_sp_2_rep_5
+

Notice how all of the replicates for species 1 in site 1 share the +same process (i.e. the same trend). This will ensure that +all replicates are Binomial draws of the same latent N.

+
+
+

Modelling with the nmix() family +

+

Now we are ready to fit a model using mvgam(). This +model will allow each species to have different detection probabilities +and different temporal trends. We will use Cmdstan as the +backend, which by default will use Hamiltonian Monte Carlo for full +Bayesian inference

+
+mod <- mvgam(
+  # the observation formula sets up linear predictors for
+  # detection probability on the logit scale
+  formula = obs ~ species - 1,
+  
+  # the trend_formula sets up the linear predictors for 
+  # the latent abundance processes on the log scale
+  trend_formula = ~ s(time, by = trend, k = 4) + species,
+  
+  # the trend_map takes care of the mapping
+  trend_map = trend_map,
+  
+  # nmix() family and data
+  family = nmix(),
+  data = testdat,
+  
+  # priors can be set in the usual way
+  priors = c(prior(std_normal(), class = b),
+             prior(normal(1, 1.5), class = Intercept_trend)))
+

View the automatically-generated Stan code to get a +sense of how the marginalization over latent N works

+
+code(mod)
+
## // Stan model code generated by package mvgam
+## functions {
+##   /* Functions to return the log probability of a Poisson Binomial Mixture */
+##   
+##   /* see Bollen et al 2023 for details (https://doi.org/10.1002/ece3.10595)*/
+##   real poisbin_lpmf(array[] int count, int k, array[] real lambda,
+##                     array[] real p) {
+##     if (max(count) > k) {
+##       return negative_infinity();
+##     }
+##     return poisson_log_lpmf(k | lambda) + binomial_logit_lpmf(count | k, p);
+##   }
+##   vector pb_logp(array[] int count, int max_k, array[] real lambda,
+##                  array[] real p) {
+##     int c_max = max(count);
+##     if (max_k < c_max) {
+##       reject("cap variable max_k must be >= observed counts");
+##     }
+##     vector[max_k + 1] lp;
+##     for (k in 0 : (c_max - 1)) {
+##       lp[k + 1] = negative_infinity();
+##     }
+##     for (k in c_max : max_k) {
+##       lp[k + 1] = poisbin_lpmf(count | k, lambda, p);
+##     }
+##     return lp;
+##   }
+##   real pb_lpmf(array[] int count, array[] int max_k, array[] real lambda,
+##                array[] real p) {
+##     // Take maximum of all supplied caps, in case they vary for some reason
+##     int max_k_max = max(max_k);
+##     vector[max_k_max + 1] lp;
+##     lp = pb_logp(count, max_k_max, lambda, p);
+##     return log_sum_exp(lp);
+##   }
+##   /* Functions to generate truncated Poisson variates */
+##   array[] int nmix_rng(array[] int count, array[] int max_k,
+##                        array[] real lambda, array[] real p) {
+##     // Take maximum of all supplied caps, in case they vary for some reason
+##     int max_k_max = max(max_k);
+##     vector[max_k_max + 1] lp;
+##     lp = pb_logp(count, max_k_max, lambda, p);
+##     return rep_array(categorical_rng(softmax(lp)) - 1, size(count));
+##   }
+##   int trunc_pois_rng(int max_k, real lambda) {
+##     real p_ub = poisson_cdf(max_k | lambda);
+##     if (p_ub < 1e-9) {
+##       return max_k;
+##     }
+##     real u = uniform_rng(0, p_ub);
+##     int i = 0;
+##     int X = 0;
+##     real p = exp(-lambda);
+##     real F = p;
+##     while (1) {
+##       if (u < F) {
+##         X = i;
+##         break;
+##       }
+##       i = i + 1;
+##       p = lambda * p / i;
+##       F = F + p;
+##     }
+##     return X;
+##   }
+## }
+## data {
+##   int<lower=0> total_obs; // total number of observations
+##   int<lower=0> n; // number of timepoints per series
+##   int<lower=0> n_sp_trend; // number of trend smoothing parameters
+##   int<lower=0> n_lv; // number of dynamic factors
+##   int<lower=0> n_series; // number of series
+##   matrix[n_series, n_lv] Z; // matrix mapping series to latent states
+##   int<lower=0> num_basis; // total number of basis coefficients
+##   int<lower=0> num_basis_trend; // number of trend basis coefficients
+##   vector[num_basis_trend] zero_trend; // prior locations for trend basis coefficients
+##   matrix[total_obs, num_basis] X; // mgcv GAM design matrix
+##   matrix[n * n_lv, num_basis_trend] X_trend; // trend model design matrix
+##   array[n, n_series] int<lower=0> ytimes; // time-ordered matrix (which col in X belongs to each [time, series] observation?)
+##   array[n, n_lv] int ytimes_trend;
+##   int<lower=0> n_nonmissing; // number of nonmissing observations
+##   array[total_obs] int<lower=0> cap; // upper limits of latent abundances
+##   array[total_obs] int ytimes_array; // sorted ytimes
+##   array[n, n_series] int<lower=0> ytimes_pred; // time-ordered matrix for prediction
+##   int<lower=0> K_groups; // number of unique replicated observations
+##   int<lower=0> K_reps; // maximum number of replicate observations
+##   array[K_groups] int<lower=0> K_starts; // col of K_inds where each group starts
+##   array[K_groups] int<lower=0> K_stops; // col of K_inds where each group ends
+##   array[K_groups, K_reps] int<lower=0> K_inds; // indices of replicated observations
+##   matrix[3, 6] S_trend1; // mgcv smooth penalty matrix S_trend1
+##   matrix[3, 6] S_trend2; // mgcv smooth penalty matrix S_trend2
+##   array[total_obs] int<lower=0> flat_ys; // flattened observations
+## }
+## transformed data {
+##   
+## }
+## parameters {
+##   // raw basis coefficients
+##   vector[num_basis] b_raw;
+##   vector[num_basis_trend] b_raw_trend;
+##   
+##   // smoothing parameters
+##   vector<lower=0>[n_sp_trend] lambda_trend;
+## }
+## transformed parameters {
+##   // detection probability
+##   vector[total_obs] p;
+##   
+##   // latent states
+##   matrix[n, n_lv] LV;
+##   
+##   // latent states and loading matrix
+##   vector[n * n_lv] trend_mus;
+##   matrix[n, n_series] trend;
+##   matrix[n_series, n_lv] lv_coefs;
+##   
+##   // basis coefficients
+##   vector[num_basis] b;
+##   vector[num_basis_trend] b_trend;
+##   
+##   // observation model basis coefficients
+##   b[1 : num_basis] = b_raw[1 : num_basis];
+##   
+##   // process model basis coefficients
+##   b_trend[1 : num_basis_trend] = b_raw_trend[1 : num_basis_trend];
+##   
+##   // detection probability
+##   p = X[ytimes_array,  : ] * b;
+##   
+##   // latent process linear predictors
+##   trend_mus = X_trend * b_trend;
+##   for (j in 1 : n_lv) {
+##     LV[1 : n, j] = trend_mus[ytimes_trend[1 : n, j]];
+##   }
+##   
+##   // derived latent states
+##   lv_coefs = Z;
+##   for (i in 1 : n) {
+##     for (s in 1 : n_series) {
+##       trend[i, s] = dot_product(lv_coefs[s,  : ], LV[i,  : ]);
+##     }
+##   }
+## }
+## model {
+##   // prior for speciessp_1...
+##   b_raw[1] ~ std_normal();
+##   
+##   // prior for speciessp_2...
+##   b_raw[2] ~ std_normal();
+##   
+##   // dynamic process models
+##   
+##   // prior for (Intercept)_trend...
+##   b_raw_trend[1] ~ normal(1, 1.5);
+##   
+##   // prior for speciessp_2_trend...
+##   b_raw_trend[2] ~ std_normal();
+##   
+##   // prior for s(time):trendtrend1_trend...
+##   b_raw_trend[3 : 5] ~ multi_normal_prec(zero_trend[3 : 5],
+##                                          S_trend1[1 : 3, 1 : 3]
+##                                          * lambda_trend[1]
+##                                          + S_trend1[1 : 3, 4 : 6]
+##                                            * lambda_trend[2]);
+##   
+##   // prior for s(time):trendtrend2_trend...
+##   b_raw_trend[6 : 8] ~ multi_normal_prec(zero_trend[6 : 8],
+##                                          S_trend2[1 : 3, 1 : 3]
+##                                          * lambda_trend[3]
+##                                          + S_trend2[1 : 3, 4 : 6]
+##                                            * lambda_trend[4]);
+##   lambda_trend ~ normal(5, 30);
+##   {
+##     // likelihood functions
+##     vector[total_obs] flat_trends;
+##     flat_trends = to_vector(trend);
+##     for (k in 1 : K_groups) {
+##       target += pb_lpmf(flat_ys[K_inds[k, K_starts[k] : K_stops[k]]] | cap[K_inds[k, K_starts[k] : K_stops[k]]], to_array_1d(flat_trends[K_inds[k, K_starts[k] : K_stops[k]]]), to_array_1d(p[K_inds[k, K_starts[k] : K_stops[k]]]));
+##     }
+##   }
+## }
+## generated quantities {
+##   vector[total_obs] eta;
+##   matrix[n, n_series] mus;
+##   vector[n_sp_trend] rho_trend;
+##   vector[n_lv] penalty;
+##   array[n, n_series] int ypred;
+##   array[n, n_series] int latent_ypred;
+##   array[total_obs] int latent_truncpred;
+##   vector[total_obs] flat_trends;
+##   vector[total_obs] detprob;
+##   detprob = inv_logit(p);
+##   penalty = rep_vector(1e12, n_lv);
+##   rho_trend = log(lambda_trend);
+##   
+##   // posterior predictions
+##   eta = X * b;
+##   {
+##     flat_trends = to_vector(trend);
+##     
+##     // prediction for all timepoints that ignore detection prob
+##     for (i in 1 : total_obs) {
+##       latent_truncpred[i] = trunc_pois_rng(cap[i], exp(flat_trends[i]));
+##     }
+##     
+##     // prediction for the nonmissing timepoints using actual obs
+##     for (k in 1 : K_groups) {
+##       latent_truncpred[K_inds[k, K_starts[k] : K_stops[k]]] = nmix_rng(flat_ys[K_inds[k, K_starts[k] : K_stops[k]]],
+##                                                                     cap[K_inds[k, K_starts[k] : K_stops[k]]],
+##                                                                     to_array_1d(
+##                                                                     flat_trends[K_inds[k, K_starts[k] : K_stops[k]]]),
+##                                                                     to_array_1d(
+##                                                                     p[K_inds[k, K_starts[k] : K_stops[k]]]));
+##     }
+##     for (s in 1 : n_series) {
+##       for (i in 1 : n) {
+##         // true latent abundance
+##         latent_ypred[i, s] = latent_truncpred[ytimes_pred[i, s]];
+##         
+##         // observed abundance
+##         ypred[i, s] = binomial_rng(latent_ypred[i, s],
+##                                    detprob[ytimes_pred[i, s]]);
+##         
+##         // expected values
+##         mus[i, s] = detprob[ytimes[i, s]] * latent_ypred[i, s];
+##       }
+##     }
+##   }
+## }
+

The summary of this model shows that it has converged nicely

+
+summary(mod)
+
## GAM observation formula:
+## obs ~ species - 1
+## 
+## GAM process formula:
+## ~s(time, by = trend, k = 4) + species
+## 
+## Family:
+## nmix
+## 
+## Link function:
+## log
+## 
+## Trend model:
+## None
+## 
+## N process models:
+## 2 
+## 
+## N series:
+## 10 
+## 
+## N timepoints:
+## 6 
+## 
+## Status:
+## Fitted using Stan 
+## 4 chains, each with iter = 1000; warmup = 500; thin = 1 
+## Total post-warmup draws = 2000
+## 
+## 
+## GAM observation model coefficient (beta) estimates:
+##              2.5%  50% 97.5% Rhat n_eff
+## speciessp_1 0.520 1.10   1.6    1   982
+## speciessp_2 0.031 0.71   1.2    1  1302
+## 
+## GAM process model coefficient (beta) estimates:
+##                               2.5%    50%  97.5% Rhat n_eff
+## (Intercept)_trend            2.700  2.900  3.100    1   972
+## speciessp_2_trend           -1.100 -0.820 -0.530    1   881
+## s(time):trendtrend1.1_trend -0.061  0.027  0.220    1   821
+## s(time):trendtrend1.2_trend -0.150  0.028  0.250    1  1523
+## s(time):trendtrend1.3_trend -0.410 -0.280 -0.094    1  1102
+## s(time):trendtrend2.1_trend -0.310 -0.021  0.092    1   481
+## s(time):trendtrend2.2_trend -0.110  0.110  0.750    1   481
+## s(time):trendtrend2.3_trend  0.170  0.410  0.630    1   917
+## 
+## Approximate significance of GAM process smooths:
+##                        edf    F p-value   
+## s(time):seriestrend1 0.596 0.26  0.0013 **
+## s(time):seriestrend2 0.881 0.41  0.0269 * 
+## ---
+## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+## 
+## Stan MCMC diagnostics:
+## n_eff / iter looks reasonable for all parameters
+## Rhat looks reasonable for all parameters
+## 0 of 2000 iterations ended with a divergence (0%)
+## 0 of 2000 iterations saturated the maximum tree depth of 12 (0%)
+## E-FMI indicated no pathological behavior
+## 
+## Samples were drawn using NUTS(diag_e) at Mon Jan 29 1:08:01 PM 2024.
+## For each parameter, n_eff is a crude measure of effective sample size,
+## and Rhat is the potential scale reduction factor on split MCMC chains
+## (at convergence, Rhat = 1)
+

loo() functionality works just as it does for all +mvgam models to aid in model comparison / selection

+
+loo(mod)
+
## Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.
+
## 
+## Computed from 2000 by 60 log-likelihood matrix
+## 
+##          Estimate  SE
+## elpd_loo   -140.0 3.2
+## p_loo         4.2 0.7
+## looic       280.0 6.3
+## ------
+## Monte Carlo SE of elpd_loo is 0.1.
+## 
+## Pareto k diagnostic values:
+##                          Count Pct.    Min. n_eff
+## (-Inf, 0.5]   (good)     57    95.0%   386       
+##  (0.5, 0.7]   (ok)        3     5.0%   567       
+##    (0.7, 1]   (bad)       0     0.0%   <NA>      
+##    (1, Inf)   (very bad)  0     0.0%   <NA>      
+## 
+## All Pareto k estimates are ok (k < 0.7).
+## See help('pareto-k-diagnostic') for details.
+

Plot the estimated smooths of time from each species’ latent +abundance process (on the log scale)

+
+plot(mod, type = 'smooths', trend_effects = TRUE)
+

+

marginaleffects support allows for more useful +prediction-based interrogations on different scales. Objects that use +family nmix() have a few additional prediction scales that +can be used (i.e. link, response, +detection or latent_N). For example, here are +the estimated detection probabilities per species, which shows that the +model has over-estimated detection probability for species 2 (originally +simulated to be 0.45):

+
+plot_predictions(mod, condition = 'species',
+                 type = 'detection') +
+  ylab('Pr(detection)') +
+  ylim(c(0, 1)) +
+  theme_classic() +
+  theme(legend.position = 'none')
+

+

A common goal in N-mixture modelling is to estimate the true latent +abundance. The model has automatically generated predictions of the +latent abundance that are conditional on the observations. We can +extract these and produce decent plots using a small function

+
+hc <- hindcast(mod, type = 'latent_N')
+
+# Function to plot latent abundance estimates vs truth
+plot_latentN = function(hindcasts, data, species = 'sp_1'){
+  all_series <- unique(data %>%
+                         dplyr::filter(species == !!species) %>%
+                         dplyr::pull(series))
+  
+  # Grab the first replicate that represents this series
+  # so we can get the true simulated values
+  series <- as.numeric(all_series[1])
+  truths <- data %>%
+    dplyr::arrange(time, series) %>%
+    dplyr::filter(series == !!levels(data$series)[series]) %>%
+    dplyr::pull(truth)
+  
+  # In case some replicates have missing observations,
+  # pull out predictions for ALL replicates and average over them
+  hcs <- do.call(rbind, lapply(all_series, function(x){
+    ind <- which(names(hindcasts$hindcasts) %in% as.character(x))
+    hindcasts$hindcasts[[ind]]
+  }))
+  
+  # Calculate posterior empirical quantiles of predictions
+  pred_quantiles <- data.frame(t(apply(hcs, 2, function(x) 
+    quantile(x, probs = c(0.05, 0.2, 0.3, 0.4, 
+                          0.5, 0.6, 0.7, 0.8, 0.95)))))
+  pred_quantiles$time <- 1:NROW(pred_quantiles)
+  pred_quantiles$truth <- truths
+  
+  # Grab observations
+  data %>%
+    dplyr::filter(series %in% all_series) %>%
+    dplyr::select(time, obs) -> observations
+  
+  # Plot
+  ggplot(pred_quantiles, aes(x = time, group = 1)) +
+    geom_ribbon(aes(ymin = X5., ymax = X95.), fill = "#DCBCBC") + 
+    geom_ribbon(aes(ymin = X30., ymax = X70.), fill = "#B97C7C") +
+    geom_line(aes(x = time, y = truth),
+              colour = 'black', linewidth = 1) +
+    geom_point(aes(x = time, y = truth),
+               shape = 21, colour = 'white', fill = 'black',
+               size = 2.5) +
+    geom_jitter(data = observations, aes(x = time, y = obs),
+                width = 0.06, 
+                shape = 21, fill = 'darkred', colour = 'white', size = 2.5) +
+    labs(y = 'Latent abundance (N)',
+         x = 'Time',
+         title = species)
+}
+

Latent abundance plots vs the simulated truths for each species are +shown below. Here, the red points show the imperfect observations, the +black line shows the true latent abundance, and the ribbons show +credible intervals of our estimates:

+
+plot_latentN(hc, testdat, species = 'sp_1')
+

+
+plot_latentN(hc, testdat, species = 'sp_2')
+

+

We can see that estimates for both species have correctly captured +the true temporal variation in abundance. However, it is also apparent +that low detection probabilities (like for species 2) make it difficult +to accurately estimate latent abundance. We could likely improve these +estimates if we had some additional information that could inform our +estimates of detection probability, such as covariates that reflect our +ability to take accurate measurements

+
+
+
+ +

Now for another example with a larger dataset. We will use data from +Jeff Doser’s simulation example from the wonderful +spAbundance package. The simulated data include one +continuous site-level covariate, one factor site-level covariate and two +continuous sample-level covariates. This example will allow us to +examine how we can include possibly nonlinear effects in the latent +process and detection probability models.

+

Download the data and grab observations / covariate measurements for +one species

+
+# Date link
+load(url('https://github.com/doserjef/spAbundance/raw/main/data/dataNMixSim.rda'))
+data.one.sp <- dataNMixSim
+
+# Pull out observations for one species
+data.one.sp$y <- data.one.sp$y[1, , ]
+
+# Abundance covariates that don't change across repeat sampling observations
+abund.cov <- dataNMixSim$abund.covs[, 1]
+abund.factor <- as.factor(dataNMixSim$abund.covs[, 2])
+
+# Detection covariates that can change across repeat sampling observations
+# Note that `NA`s are not allowed for covariates in mvgam, so we randomly
+# impute them here
+det.cov <- dataNMixSim$det.covs$det.cov.1[,]
+det.cov[is.na(det.cov)] <- rnorm(length(which(is.na(det.cov))))
+det.cov2 <- dataNMixSim$det.covs$det.cov.2
+det.cov2[is.na(det.cov2)] <- rnorm(length(which(is.na(det.cov2))))
+

Next we wrangle into the appropriate ‘long’ data format, adding +indicators of time and series for working in +mvgam. We also add the cap variable to +represent the maximum latent N to marginalize over for each +observation

+
+mod_data <- do.call(rbind,
+                    lapply(1:NROW(data.one.sp$y), function(x){
+                      data.frame(y = data.one.sp$y[x,],
+                                 abund_cov = abund.cov[x],
+                                 abund_fac = abund.factor[x],
+                                 det_cov = det.cov[x,],
+                                 det_cov2 = det.cov2[x,],
+                                 replicate = 1:NCOL(data.one.sp$y),
+                                 site = paste0('site', x))
+                    })) %>%
+  dplyr::mutate(species = 'sp_1',
+                series = as.factor(paste0(site, '_', species, '_', replicate))) %>%
+  dplyr::mutate(site = factor(site, levels = unique(site)),
+                species = factor(species, levels = unique(species)),
+                time = 1,
+                cap = max(data.one.sp$y, na.rm = TRUE) + 20)
+

The data include observations for 225 sites with three replicates per +site, though some observations are missing

+
+NROW(mod_data)
+
## [1] 675
+
+dplyr::glimpse(mod_data)
+
## Rows: 675
+## Columns: 11
+## $ y         <int> 1, NA, NA, NA, 2, 2, NA, 1, NA, NA, 0, 1, 0, 0, 0, 0, NA, NA…
+## $ abund_cov <dbl> -0.3734384, -0.3734384, -0.3734384, 0.7064305, 0.7064305, 0.…
+## $ abund_fac <fct> 3, 3, 3, 4, 4, 4, 9, 9, 9, 2, 2, 2, 3, 3, 3, 2, 2, 2, 1, 1, …
+## $ det_cov   <dbl> -1.2827999, -0.6412036, 1.7083192, 0.7640157, 0.1954809, 0.9…
+## $ det_cov2  <dbl> 2.030473137, 0.151511085, -0.439251153, -1.481393226, 1.0455…
+## $ replicate <int> 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, …
+## $ site      <fct> site1, site1, site1, site2, site2, site2, site3, site3, site…
+## $ species   <fct> sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, sp_1, …
+## $ series    <fct> site1_sp_1_1, site1_sp_1_2, site1_sp_1_3, site2_sp_1_1, site…
+## $ time      <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
+## $ cap       <dbl> 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, …
+
+head(mod_data)
+
##    y  abund_cov abund_fac    det_cov   det_cov2 replicate  site species
+## 1  1 -0.3734384         3 -1.2827999  2.0304731         1 site1    sp_1
+## 2 NA -0.3734384         3 -0.6412036  0.1515111         2 site1    sp_1
+## 3 NA -0.3734384         3  1.7083192 -0.4392512         3 site1    sp_1
+## 4 NA  0.7064305         4  0.7640157 -1.4813932         1 site2    sp_1
+## 5  2  0.7064305         4  0.1954809  1.0455536         2 site2    sp_1
+## 6  2  0.7064305         4  0.9673034  1.9197118         3 site2    sp_1
+##         series time cap
+## 1 site1_sp_1_1    1  33
+## 2 site1_sp_1_2    1  33
+## 3 site1_sp_1_3    1  33
+## 4 site2_sp_1_1    1  33
+## 5 site2_sp_1_2    1  33
+## 6 site2_sp_1_3    1  33
+

The final step for data preparation is of course the +trend_map, which sets up the mapping between observation +replicates and the latent abundance models. This is done in the same way +as in the example above

+
+mod_data %>%
+  # each unique combination of site*species is a separate process
+  dplyr::mutate(trend = as.numeric(factor(paste0(site, species)))) %>%
+  dplyr::select(trend, series) %>%
+  dplyr::distinct() -> trend_map
+
+trend_map %>%
+  dplyr::arrange(trend) %>%
+  head(12)
+
##    trend         series
+## 1      1 site100_sp_1_1
+## 2      1 site100_sp_1_2
+## 3      1 site100_sp_1_3
+## 4      2 site101_sp_1_1
+## 5      2 site101_sp_1_2
+## 6      2 site101_sp_1_3
+## 7      3 site102_sp_1_1
+## 8      3 site102_sp_1_2
+## 9      3 site102_sp_1_3
+## 10     4 site103_sp_1_1
+## 11     4 site103_sp_1_2
+## 12     4 site103_sp_1_3
+

Now we are ready to fit a model using mvgam(). Here we +will use penalized splines for each of the continuous covariate effects +to detect possible nonlinear associations. We also showcase how +mvgam can make use of the different approximation +algorithms available in Stan by using the meanfield +variational Bayes approximator (this reduces computation time +substantially)

+
+mod <- mvgam(
+  # effects of covariates on detection probability;
+  # here we use penalized splines for both continuous covariates
+  formula = y ~ s(det_cov, k = 3) + s(det_cov2, k = 3),
+  
+  # effects of the covariates on latent abundance;
+  # here we use a penalized spline for the continuous covariate and
+  # hierarchical intercepts for the factor covariate
+  trend_formula = ~ s(abund_cov, k = 3) +
+    s(abund_fac, bs = 're'),
+  
+  # link multiple observations to each site
+  trend_map = trend_map,
+  
+  # nmix() family and supplied data
+  family = nmix(),
+  data = mod_data,
+  
+  # standard normal priors on key regression parameters
+  priors = c(prior(std_normal(), class = 'b'),
+             prior(std_normal(), class = 'Intercept'),
+             prior(std_normal(), class = 'Intercept_trend')),
+  
+  # use Stan's variational inference for quicker results
+  algorithm = 'meanfield',
+  samples = 1000)
+

Inspect the model summary but don’t bother looking at estimates for +all individual spline coefficients. Notice how we no longer receive +information on convergence because we did not use MCMC sampling for this +model

+
+summary(mod, include_betas = FALSE)
+
## GAM observation formula:
+## y ~ s(det_cov, k = 3) + s(det_cov2, k = 3)
+## 
+## GAM process formula:
+## ~s(abund_cov, k = 3) + s(abund_fac, bs = "re")
+## 
+## Family:
+## nmix
+## 
+## Link function:
+## log
+## 
+## Trend model:
+## None
+## 
+## N process models:
+## 225 
+## 
+## N series:
+## 675 
+## 
+## N timepoints:
+## 1 
+## 
+## Status:
+## Fitted using Stan 
+## 1 chains, each with iter = 1000; warmup = ; thin = 1 
+## Total post-warmup draws = 1000
+## 
+## 
+## GAM observation model coefficient (beta) estimates:
+##             2.5%  50% 97.5% Rhat n.eff
+## (Intercept) 0.35 0.75   1.2  NaN   NaN
+## 
+## Approximate significance of GAM observation smooths:
+##              edf Chi.sq p-value    
+## s(det_cov)  1.99   86.7 0.00086 ***
+## s(det_cov2) 2.00  359.2 < 2e-16 ***
+## ---
+## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+## 
+## GAM process model coefficient (beta) estimates:
+##                   2.5% 50% 97.5% Rhat n.eff
+## (Intercept)_trend 0.91 1.2   1.4  NaN   NaN
+## 
+## GAM process model group-level estimates:
+##                           2.5%   50% 97.5% Rhat n.eff
+## mean(s(abund_fac))_trend -1.70 -1.40 -1.20  NaN   NaN
+## sd(s(abund_fac))_trend    0.17  0.28  0.48  NaN   NaN
+## 
+## Approximate significance of GAM process smooths:
+##               edf    F p-value  
+## s(abund_cov) 1.90 2.13   0.978  
+## s(abund_fac) 8.87 1.28   0.039 *
+## ---
+## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+## 
+## Posterior approximation used: no diagnostics to compute
+

Again we can make use of marginaleffects support for +interrogating the model through targeted predictions. First, we can +inspect the estimated average detection probability

+
+avg_predictions(mod, type = 'detection')
+
## 
+##  Estimate 2.5 % 97.5 %
+##     0.647 0.568  0.721
+## 
+## Columns: estimate, conf.low, conf.high 
+## Type:  detection
+

Next investigate estimated effects of covariates on latent abundance +using the conditional_effects() function and specifying +type = 'link'; this will return plots on the expectation +scale

+
+abund_plots <- plot(conditional_effects(mod,
+                                        type = 'link',
+                                        effects = c('abund_cov',
+                                                    'abund_fac')),
+                    plot = FALSE)
+

The effect of the continuous covariate on expected latent +abundance

+
+abund_plots[[1]] +
+  ylab('Expected latent abundance')
+

+

The effect of the factor covariate on expected latent abundance, +estimated as a hierarchical random effect

+
+abund_plots[[2]] +
+  ylab('Expected latent abundance')
+

+

Now we can investigate estimated effects of covariates on detection +probability using type = 'detection'

+
+det_plots <- plot(conditional_effects(mod,
+                                      type = 'detection',
+                                      effects = c('det_cov',
+                                                  'det_cov2')),
+                  plot = FALSE)
+

The covariate smooths were estimated to be somewhat nonlinear on the +logit scale according to the model summary (based on their approximate +significances). But inspecting conditional effects of each covariate on +the probability scale is more intuitive and useful

+
+det_plots[[1]] +
+  ylab('Pr(detection)')
+

+
+det_plots[[2]] +
+  ylab('Pr(detection)')
+

+

More targeted predictions are also easy with +marginaleffects support. For example, we can ask: How does +detection probability change as we change both detection +covariates?

+
+fivenum_round = function(x)round(fivenum(x, na.rm = TRUE), 2)
+
+plot_predictions(mod, 
+                 newdata = datagrid(det_cov = unique,
+                                    det_cov2 = fivenum_round),
+                 by = c('det_cov', 'det_cov2'),
+                 type = 'detection') +
+  theme_classic() +
+  ylab('Pr(detection)')
+

+

The model has found support for some important covariate effects, but +of course we’d want to interrogate how well the model predicts and think +about possible spatial effects to capture unmodelled variation in latent +abundance.

+
+
+

Further reading +

+

The following papers and resources offer useful material about +N-mixture models for ecological population dynamics investigations:

+

Guélat, Jérôme, and Kéry, Marc. “Effects +of Spatial Autocorrelation and Imperfect Detection on Species +Distribution Models.Methods in Ecology and Evolution 9 +(2018): 1614–25.

+

Kéry, Marc, and Royle Andrew J. “Applied +hierarchical modeling in ecology: Analysis of distribution, abundance +and species richness in R and BUGS: Volume 2: Dynamic and advanced +models”. London, UK: Academic Press (2020).

+

Royle, Andrew J. “N‐mixture +models for estimating population size from spatially replicated +counts.Biometrics 60.1 (2004): 108-115.

+
+
+
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