---
title: 'Overview of ubms'
subtitle: 'Unmarked Bayesian Models with Stan'
author: Ken Kellner
date: 2020-09-07
output:
    html_document:
      number_sections: true
      toc: true
      toc_float: true
      css: style.css
      highlight: pygments
bibliography: references.bib
link-citations: yes
csl: plos-one.csl
---

```{r, echo=FALSE}
knitr::opts_chunk$set(message=FALSE, warning=FALSE)
```

# Introduction

## What is ubms?

`ubms` is an R package for fitting models of wildlife occurrence and abundance to with datasets where animals are not individually marked.
It provides a nearly identical interface to the popular `unmarked` package [@Fiske_2011].
Instead of using maximum likelihood to fit models (as with `unmarked`), 
models are fit in a Bayesian framework using [Stan](https://mc-stan.org/) [@Carpenter_2017].
It is generally expected that you are already familiar with `unmarked` when using `ubms`.
You can download `ubms`, report issues, or help with development on [Github](https://github.com/kenkellner/ubms).

## Why should you use it?

There are several advantages to using `ubms` over `unmarked`.
First, it is possible to include random effects in `ubms` models, which is not currently possible in `unmarked`.
These are specified using the familiar syntax of [lme4](https://cran.r-project.org/web/packages/lme4/index.html) [@Bates_2015].
Second, `ubms` generates posterior distributions for model parameters, including latent occupancy and abundance parameters.
These can be useful for post-hoc analyses and diagnostics.
Finally, fitting models with Stan gives you access to the large ecosystem of Stan-related tools, such as [LOO](https://cran.r-project.org/web/packages/loo/index.html) (leave-one-out-cross validation; [@Vehtari_2017]).

Another alternative to `ubms` would be to fit models in an existing modeling language such as [BUGS](http://www.openbugs.net/), [JAGS](http://mcmc-jags.sourceforge.net/), or directly in Stan.
`ubms` abstracts away the complex process of defining and writing custom models in these languages which need to be updated each time you make changes to your model.
It also provides many useful helper functions (e.g. `predict`) which would otherwise require custom code.
Finally, because of Stan's efficient sampler [@Hoffman_2014] and because the underlying likelihoods in `ubms` are marginalized, `ubms` will often fit models much faster than equivalent models in BUGS or JAGS [@Yackulic_2020].

## What are the drawbacks?

Relative to `unmarked`, `ubms` has fewer types of models available.
For example, models that incorporate temporary emigration (like `gdistsamp`) [@Chandler2011] are currently not available in `ubms`.
Models should run much faster in `unmarked` as well.
If you do not need one of the specific benefits of `ubms` described above, it makes sense to stick with `unmarked`.
Even if you do plan to use `ubms`, it makes sense to test the models in `unmarked` first.
The similar interface between the two packages makes this very easy, as you will see in the next section.

Relative to BUGS/JAGS/Stan, `ubms` is less flexible because you cannot customize your model structure.
You are limited to the provided model types.
Furthermore, you cannot currently customize prior distributions (although I plan to add this in the future in some form).
Finally, writing your own BUGS/JAGS model can be valuable for gaining a deeper understanding of how a model works; `ubms`, like `unmarked`, is essentially a black box.

To summarize the advantages and disadvantages: I see `ubms` as an intermediate step along the continuum from `unmarked` to custom models in BUGS/JAGS.
It is not meant to replace either approach but rather to supplement them, for situations when a Bayesian framework is needed and "off-the-shelf" model structures are adequate.

# Example: Fitting a single-season occupancy model

Occupancy models estimate the probability $\psi$ that a species occupies a site, while accounting for detection probability $p < 1$ [@MacKenzie_2002].
In order to estimate both $p$ and $\psi$, repeated observations (detection/non-detection data) at each site are required.

## Set up the data

First, load the dataset we'll be using, which comes with `unmarked`:

```{r}
library(unmarked)
data(crossbill)
```

The `crossbill` dataset is a `data.frame` with many columns.
It contains detection/non-detection data for the European crossbill (*Loxia
curvirostra*) in Switzerland [@Schmid_2004].


```{r}
dim(crossbill)
names(crossbill)
```

Check `?crossbill` for details about each column.
The first three columns `id`, `ele`, and `forest` are site covariates.

```{r}
site_covs <- crossbill[,c("id", "ele", "forest")]
```

The following 27 columns beginning with `det` are the binary detection/non-detection data; 9 years with 3 observations per year.
For this example we want to fit a single-season occupancy model; thus we will use only the first three columns (year 1) of `det` as our response variable `y`.

```{r}
y <- crossbill[,c("det991","det992","det993")]
head(y)
```

Note that missing values are possible.
The final 27 columns beginning with `date` are the Julian dates for each observation.
As with `y` we want only the first three columns corresponding to year 1.

```{r}
date <- crossbill[,c("date991","date992","date993")]
```

Finally, we build our `unmarkedFrame` object holding our detection/non-detection data, site covariates, and observation covariates.
Since we will conduct a single-season occupancy analysis, we need to use `unmarkedFrameOccu` specifically.
The resulting `unmarkedFrame` can be used by both `unmarked` and `ubms`.

```{r}
umf <- unmarkedFrameOccu(y=y, siteCovs=site_covs, obsCovs=list(date=date))
head(umf)
```

## Fit a model

### Fit the model in unmarked

First, we fit a null model (no covariates) in `unmarked` using the `occu` function.
The `occu` function requires as input a double formula (for detection and occupancy, respectively) along with our `unmarkedFrame`.

```{r}
(fit_unm <- occu(~1~1, data=umf))
```

### Fit the model in ubms

Next, we fit the same model in `ubms`.
The equivalent to `occu` in `ubms` is `stan_occu`.
Functions in `ubms` generally use this `stan_` prefix, based on the approach used in package [rstanarm](https://cran.r-project.org/web/packages/rstanarm/index.html) for GLMs.
We need to provide the same arguments to `stan_occu`.
In addition, we will specify that we want 3 MCMC chains (`chains=3`), with 500 iterations per chain (`iter=500`) of which the first half will be warmup iterations.
It is beyond the scope of this vignette to discuss the appropriate number or length of chains; see [the Stan user's guide](https://mc-stan.org/docs/2_24/stan-users-guide/index.html) for more details.
Generally 4 chains of 2000 iterations each is recommended (of which 1000 per chain are warmups).
Thus, 500 iterations per chain is probably not enough, but to keep things running quickly it is sufficient for this vignette.
Note that if you are more familiar with BUGS or JAGS, Stan generally requires a smaller number of iterations to reach convergence thanks to its default NUTS sampler [@Hoffman_2014].
If you have a good multi-core CPU, you can run chains in parallel.
Tell Stan how many cores you want to use by setting the `mc.cores` option.

```{r, eval=FALSE}
library(ubms)
options(mc.cores=3)

(fit_stan <- stan_occu(~1~1, data=umf, chains=3, iter=500))
```

```{r, echo=FALSE}
library(ubms)
(fit_stan <- stan_occu(~1~1, data=umf, chains=3, iter=500, refresh=0))
```

### Compare results

The structure of the output from `unmarked` and `ubms` is intentionally similar.
Estimates of the occupancy and detection parameters are also similar, but not identical.
For a more direct comparison, call the `coef` function on both model objects:

```{r}
cbind(unmarked=coef(fit_unm), stan=coef(fit_stan))
```

### Understanding the ubms output summary

Let's look at the output from our `fit_stan` model again:

```{r}
fit_stan
```

The first part (under `Call:`) is the command we used to get this model output.
Underneath are two tables, one per submodel, corresponding to the occupancy and detection parts of the model.
Within each table there is one row per parameter in the submodel.
Since `fit_stan` had no covariates, there is only an intercept term for each submodel.
Model parameters in this summary table are always shown on the appropriate transformed scale, in this case logit.
To get the corresponding probabilities, you can use the `predict` function, which we will demonstrate later.

For each parameter, the mean and standard deviation of the posterior distribution are given.
Unlike `unmarked` output, there is no $Z$ or $p$-value.
Instead, there is a 95% [uncertainty interval](https://statmodeling.stat.columbia.edu/2016/11/26/reminder-instead-confidence-interval-lets-say-uncertainty-interval/) provided.

The final two columns in each summary table `n_eff` and `Rhat` are MCMC diagnostics.
We will discuss their meaning later.

### Extracting individual parameters

To extract summary values into an R table for easy manipulation, use the `summary` method.
Note that you have to specify which submodel you want (`"state"` for occupancy or `"det"` for detection).

```{r}
sum_tab <- summary(fit_stan, "state")
sum_tab$mean[1]
```

To extract the entire posterior for a parameter, use the `extract` method.
To avoid name collisions you need to use the full name of the parameter (which contains both the submodel and the parameter name) when extracting.
To see a list of the available full parameter names, use the `names` method.

```{r}
names(fit_stan)
occ_intercept <- extract(fit_stan, "beta_state[(Intercept)]")[[1]]
hist(occ_intercept, freq=FALSE)
lines(density(occ_intercept), col='red', lwd=2)
```

## Compare candidate models

Now we'll fit a series of candidate models to the `crossbill` data in `ubms` and compare them.

### Fit the models

Along with our previous null model, we'll fit a model with site covariates (elevation and forest), a model with an observation covariate (date), and a "global" model with both site and observation covariates.
This is just an example; perhaps other models should also be considered if we were preparing this analysis for publication.
In our model formulas, we have normalized all covariates with `scale` so they have a mean of 0 and a standard deviation of 1.
This can help improve model convergence and is generally a good idea.

```{r, eval=FALSE}
fit_null <- fit_stan

fit_sc <- stan_occu(~1~scale(forest)+scale(ele), data=umf,
                    chains=3, iter=500)

fit_oc <- stan_occu(~scale(date)~1, data=umf, chains=3, iter=500)

fit_global <- stan_occu(~scale(date)~scale(forest)+scale(ele), data=umf,
                        chains=3, iter=500)
```

```{r, echo=FALSE}
fit_null <- fit_stan
fit_sc <- stan_occu(~1~scale(forest)+scale(ele), data=umf,
                    chains=3, iter=500, refresh=0)
fit_oc <- stan_occu(~scale(date)~1, data=umf, chains=3, iter=500, refresh=0)
```

```{r, warning=TRUE, echo=FALSE}
fit_global <- stan_occu(~scale(date)~scale(forest)+scale(ele), data=umf,
                        chains=3, iter=500, refresh=0)
```

The `fit_global` model gave us some warnings about the effective sample size (`n_eff`) along with a suggested solution.
We will ignore this warning for now but normally it is a good idea to pay close attention to these warnings.

### Compare the models

First we combine the models into a `fitList`:

```{r}
mods <- fitList(fit_null, fit_sc, fit_oc, fit_global)
```

Then we generate a model selection table:

```{r}
round(modSel(mods), 3)
```

Instead of AIC, models are compared using leave-one-out cross-validation (LOO) [@Vehtari_2017] via the `loo` package.
Based on this cross-validation, the expected predictive accuracy (`elpd`) for each model is calculated.
The model with the largest `elpd` (`fit_global`) performed best.
The `elpd_diff` column shows the difference in `elpd` between a model and the top model; if this difference is several times larger than the standard error of the difference (`se_diff`), we are confident the model with the larger `elpd` performed better.
LOO model weights, analogous to AIC weights, are also calculated.
We can see that the `fit_global` model is clearly the best performing model.

You can obtain LOO information for a single model using the `loo` method:

```{r}
loo(fit_global)
```

The `looic` value is analogous to AIC.

You can also obtain the WAIC (Widely Applicable Information Criterion) if you prefer [@Vehtari_2017]:

```{r}
waic(fit_global)
```

## Diagnostics and model fit

We'll define the global model as our top model:

```{r}
fit_top <- fit_global
fit_top
```

### MCMC diagnostics

Again looking at the summary of `fit_top`, it appears all chains have adequately converged based on $\hat{R}$ values.
Ideally, you want all $\hat{R} > 1.05$.
To visualize convergence, look at the traceplots:

```{r}
traceplot(fit_top)
```

Our effective sample size `n_eff` is lacking for some parameters as we were previously warned.
The rule of thumb is to have `n_eff` > 100 * number of chains (300).
The easy solution would be to re-run this model with more iterations.

### Model fit

Calculating residuals is tricky for occupancy models.
There isn't one widely accepted way of doing it.
`ubms` implements the approach of Wright [@Wright_2019] in which residuals are calculated separately for the state and observation processes.
To quickly plot residuals against fitted values, use the `plot` method:

```{r}
plot(fit_top)
```

Note that the residuals are automatically binned, which is appropriate for a binomial response [@Gelman2007].
If the model fits the data well, you would expect 95% of the binned residual points to fall within the shaded area.
You can also plot residuals against covariate values using the `plot_residuals` function:

```{r}
plot_residuals(fit_top, submodel="state", covariate="forest")
```

`ubms` also supports goodness-of-fit tests (posterior predictive checks) for some models.
In the case of occupancy models, there is support for the MacKenzie-Bailey (M-B) chi-square test [@MacKenzie_2004a] via the `gof` function.
For each posterior draw, the M-B statistic is calculated for the actual data and for a simulated dataset.
The proportion of draws for which the simulated statistic is larger than the actual statistic should be near 0.5 if the model fits well.

```{r}
(fit_top_gof <- gof(fit_top, quiet=TRUE))
plot(fit_top_gof)
```

Our model appears to have adequate fit based on this posterior predictive check.

You can use the `posterior_predict` function to simulate new datasets, which you can use to calculate your own fit statistic.
The following command generates one simulated dataset per posterior draw.

```{r}
sim_y <- posterior_predict(fit_top, "y")
dim(sim_y)
```

The output is a matrix with dimensions draws x observations (in site-major order).
As an example, we can calculate the proportion of zeros in each simulated dataset 

```{r}
prop0 <- apply(sim_y, 1, function(x) mean(x==0, na.rm=TRUE))
```

and compare that to the proportion of zeros in the actual dataset.

```{r}
actual_prop0 <- mean(getY(fit_top) == 0, na.rm=TRUE)

#Compare
hist(prop0, col='gray')
abline(v=actual_prop0, col='red', lwd=2)
```

## Model inference

### Marginal covariate effects

Based on the 95% uncertainty intervals, both forest and elevation have a positive effect on occupancy probability (both intervals do not contain 0).
Similarly, Julian date has a positive impact on detection probability.
We can quickly visualize these marginal covariate effects with the `plot_marginal` function:

```{r}
plot_marginal(fit_top, "state")
plot_marginal(fit_top, "det")
```

### Predict parameter values

As with `unmarked`, we can get the predicted $psi$ or $p$ for each site or observation using the `predict` function.
For example, to get occupancy:

```{r}
head(predict(fit_top, submodel="state"))
```

You can also supply newdata as a `data.frame`.

```{r}
nd <- data.frame(ele=100, forest=25)
predict(fit_top, submodel="state", newdata=nd)
```

### Generate predictive map

You can also supply a `stack` of `raster` objects in order to use your fitted model to generate a map of a parameter.
The `Switzerland` dataset supplied with `unmarked` contains spatial data which can be used with the `crossbill` dataset.
This example was adapted from the `unmarked` "Species Distributions" [vignette](https://cran.r-project.org/web/packages/unmarked/vignettes/spp-dist.pdf).
First, we need to convert the necessary covariates in the dataset into `raster` format.

```{r}
library(raster)
data(Switzerland)

elevation <- rasterFromXYZ(Switzerland[,c("x","y","elevation")],
    crs="+proj=somerc +lat_0=46.95240555555556 +lon_0=7.439583333333333 +k_0=1
    +x_0=600000 +y_0=200000 +ellps=bessel +towgs84=674.374,15.056,405.346,0,0,0,0
    +units=m +no_defs")

forest <- rasterFromXYZ(Switzerland[,c("x","y","forest")],
    crs="+proj=somerc +lat_0=46.95240555555556 +lon_0=7.439583333333333 +k_0=1
    +x_0=600000 +y_0=200000 +ellps=bessel +towgs84=674.374,15.056,405.346,0,0,0,0
    +units=m +no_defs")

```

Next we combine the two covariates into a `raster` `stack`.
We also need to name the layers of this stack to exactly match the names of the covariates as they appear in our model.
If you don't do this, `predict` will not work.

```{r}
ef_raster <- stack(elevation, forest)
names(ef_raster) <- c("ele", "forest")
```

Here's what our covariate maps look like:

```{r}
plot(ef_raster, col=terrain.colors(100))
```

Generating a map of predicted occupancy probability is as simple as calling `predict` with our covariate raster stack as the newdata.

```{r}
pr_raster <- predict(fit_top, "state", newdata=ef_raster)
plot(pr_raster)
```

### Simulate latent occupancy states

One of the advantages of BUGS/JAGS is that you can directly model latent parameters, such as the true unobserved occupancy state of a site $z$.
Using the `posterior_predict` function in `ubms`, you can generate an equivalent posterior distribution of $z$.

```{r}
zpost <- posterior_predict(fit_top, "z")
dim(zpost)
```

The output has one row per posterior draw, and one column per site.
The posterior of $z$ can be useful for post-hoc analyses.
For example, suppose you wanted to test for a difference in mean occupancy probability between sites 1-50 and sites 51-100:

```{r}
group1 <- rowMeans(zpost[,1:50], na.rm=TRUE)
group2 <- rowMeans(zpost[,51:100], na.rm=TRUE)

plot_dat <- rbind(data.frame(group="group1", occ=mean(group1),
                             lower=quantile(group1, 0.025),
                             upper=quantile(group2, 0.975)),
                  data.frame(group="group2", occ=mean(group2),
                             lower=quantile(group2, 0.025),
                             upper=quantile(group2, 0.975)))

```

Now plot the posterior distributions of the two means:

```{r}
library(ggplot2)

ggplot(plot_dat, aes(x=group, y=occ)) +
  geom_errorbar(aes(ymin=lower, ymax=upper), width=0.2) +
  geom_point(size=3) +
  ylim(0,1) +
  labs(x="Group", y="Occupancy + 95% UI")
```

# Example: "Stacked" model with random effect 

The `crossbill` data contains 9 years of detection/non-detection data.
A logical model choice for the full dataset would be the dynamic occupancy model, which estimates initial occupancy along with colonization and extinction probabilities.
This model is implemented in `unmarked` as `colext` and in `ubms` as `stan_colext`.
However, in some cases you might not want to fit a dynamic model - for example if you don't have enough data or you aren't interested in the transition probabilities.
Also, other model types may not have dynamic versions available.
In these cases you can fit multiple years of data into a single-season model using the "stacked" approach.
Essentially, you treat unique site-year combinations as sites.
For a helpful discussion on the topic, see [this](https://groups.google.com/forum/#!topic/unmarked/OHkk98y09Zo) thread on the `unmarked` forums.

Ideally you want to control for the pseudoreplication this creates in some form.
In `unmarked` you are limited to approaches such as including a dummy variable for site and/or year.
In `ubms` you can instead include, for example, random site intercepts to account for this pseudoreplication.
The following example demonstrates how to do this.

## Format the Data

We will use the first 3 years of `crossbill` data (instead of all 9), simply to keep the analysis run time down.
Converting the `crossbill` data to stacked format is a bit complex.
The dataset contains 267 unique sites; thus after stacking we should end up with a response variable that contains `267 * 3` "sites" (site-years).
First, we replicate the site covariates 3 times:

```{r}
sc_stack <- site_covs[rep(1:nrow(site_covs), 3),]
names(sc_stack)[1] <- "site"
dim(sc_stack)
head(sc_stack)
```

Notice the inclusion of the site ID variable in the site covariates.
Next, we write a function that splits a "wide" dataset into pieces and stacks them on top of each other.

```{r}
wide_to_stacked <- function(input_df, surveys_per_year){
  #nyears <- ncol(input_df) / surveys_per_year
  nyears <- 3
  inds <- split(1:(nyears*surveys_per_year), rep(1:nyears, each=surveys_per_year))
  split_df <- lapply(1:nyears, function(i){
                      out <- input_df[,inds[[i]]]
                      out$site <- 1:nrow(input_df)
                      out$year <- i
                      names(out)[1:3] <- paste0("obs",1:3)
                      out
              })
  stack_df <- do.call("rbind", split_df)
  stack_df$site <- as.factor(stack_df$site)
  stack_df$year <- as.factor(stack_df$year)
  stack_df
}
```

Next we use the function to convert our original response data into a stacked form.

```{r}
y_wide <- crossbill[, grep("det", names(crossbill), value=TRUE)]
y_stack <- wide_to_stacked(y_wide, 3)
dim(y_stack)
head(y_stack)
```

Finally, we do the same with the `date` observation covariate.

```{r}
date_wide <- crossbill[,grep("date", names(crossbill), value=TRUE)]
date_stack <- wide_to_stacked(date_wide, 3)
dim(date_stack)
```

With our stacked datasets constructed, we build our `unmarkedFrame`:

```{r}
umf_stack <- unmarkedFrameOccu(y=y_stack[,1:3], siteCovs=sc_stack,
                         obsCovs=list(date=date_stack[,1:3]))
```

## Fit the Stacked Model

We'll now fit our top-ranked model from before, but this time with a random effect of site included in the formula for occupancy.
Random effects are specified using the approach used in with the `lme4` package.
For example, a random intercept for each level of the covariate `site` is specified with the formula component `(1|site)`.
Random slopes, or a combination of random slopes and intercepts, are also possible.
This model will take significantly longer to run.

```{r, eval=FALSE}
fit_stack <- stan_occu(~scale(date) ~scale(ele) + scale(forest) + (1|site), 
                       data=umf_stack, chains=3, iter=500, refresh=0)
fit_stack
```

```{r, echo=FALSE}
fit_stack <- stan_occu(~scale(date) ~scale(ele) + scale(forest) + (1|site), 
                       data=umf_stack, chains=3, iter=300, refresh=0)
fit_stack
```

We get more warnings; again these should be fixed by increasing the iterations.
Our output looks similar to our top model from before, except that we now have an estimate for the site-level variance (`sigma [1|site]`) in our summary table.

## Accessing the random intercepts

In order to get the actual random intercept values, we use the `ranef` function.
Note that this function behaves like the `lme4` version, not like the `unmarked` version.

```{r}
ran <- ranef(fit_stack, submodel="state")
head(ran$site[[1]])
```

You can also generate summary statistics for each random term:

```{r}
ran <- ranef(fit_stack, submodel="state", summary=TRUE)
head(ran$site[[1]])
```

# References

<div id="refs"></div>
