Introductory stats books begin with the coin flip to introduce the binomial distribution. In R we can easily simulate an outcome from such a random variable \(Y \sim Binomial(1, p)\) doing something like this:

But a coin flip in reality is a lot more complicated: we might consider the initial force, the height of the toss, the spin, and the weight of the coin.

Bird behavior combined with the observation process presents a more complicated system, that is often treated as a mixture of a count distribution and a detection/nondetection process, e.g.:

```
D <- 2 # individuals / unit area
A <- 1 # area
p <- 0.8 # probability of availability given presence
q <- 0.5 # probability of detection given availability
N <- rpois(1, lambda = A * D)
Y <- rbinom(1, size = N, prob = p * q)
```

This looks not too complicated, corresponding to the true abundance
being a random variables \(N \sim
Poisson(DA)\), while the observed count being \(Y \sim Binomial(N, pq)\). This is the exact
simulation that we need when we want to make sure that an
*estimator* can estimate the *model* parameters
(`lambda`

and `prob`

here). But such probabilistic
simulations are not very useful when we are interested how well the
*model* captures important aspects of *reality*.

Going back to the Poisson–Binomial example, `N`

would be a
result of all the factors influencing bird abundance, such as
geographical location, season, habitat suitability, number of
conspecifics, competitors, or predators. `Y`

however would
largely depend on how the birds behave depending on timing, or how an
observer might detect or miss the different individuals, or count the
same individual twice, etc.

Therefore the package has layers, that by default are
*conditionally independent* of each other. This design decision
is meant to facilitate the comparison of certain settings while keeping
all the underlying realizations identical, thus helping to pinpoint
effects without the extra variability introduced by all the other
effects.

The conditionally independent *layers* of a
**bSims** realization are the following, with the
corresponding function:

- landscape (
`bsims_init`

), - population (
`bsims_populate`

), - behavior with movement and vocalization events
(
`bsims_animate`

), - the physical side of the observation process
(
`bsims_detect`

), and - the human aspect of the observation process
(
`bsims_transcribe`

).

See this example as a sneak peek that we’ll explain in the following subsections:

```
library(bSims)
phi <- 0.5 # singing rate
tau <- 1:3 # detection distances by strata
tbr <- c(3, 5, 10) # time intervals
rbr <- c(0.5, 1, 1.5) # count radii
l <- bsims_init(extent=10, # landscape
road=0.25, edge=0.5)
p <- bsims_populate(l, # population
density=c(1, 1, 0))
e <- bsims_animate(p, # events
vocal_rate=phi,
move_rate=1, movement=0.2)
d <- bsims_detect(e, # detections
tau=tau)
x <- bsims_transcribe(d, # transcription
tint=tbr, rint=rbr)
get_table(x) # removal table
#> 0-3min 3-5min 5-10min
#> 0-50m 0 0 0
#> 50-100m 0 1 0
#> 100-150m 3 0 1
```

```
op <- par(mfrow=c(2,3), cex.main=2)
plot(l, main="Initialize")
plot(p, main="Populate")
plot(e, main="Animate")
plot(d, main="Detect")
plot(x, main="Transcribe")
par(op)
```

The `bsims_ini`

function sets up the geometry of a local
landscape. The `extent`

of the landscape determines the edge
lengths of a square shaped area. With no argument values passed, the
function assumes a homogeneous *habitat* (H) in a 10 units x 10
units landscape, 1 unit is 100 meters. Having units this way allows
easier conversion to ha as area unit that is often used in the North
American bird literature. As a result, our landscape has an area of 1
km\(^2\).

The `road`

argument defines the half-width of the road
that is placed in a vertical position. The `edge`

argument
defines the width of the edge stratum on both sides of the road. Habitat
(H), edge (E), and road (R) defines the 3 strata that we refer to by
their initials (H for no stratification, HER for all 3 strata
present).

The origin of the Cartesian coordinate system inside the landscape is
centered at the middle of the square. The `offset`

argument
allows the road and edge strata to be shifted to the left (negative
values) or to the right (positive values) of the horizontal axis. This
makes it possible to create landscapes with only two strata. The
`bsims_init`

function returns a landscape object (with class
‘bsims_landscape’).

```
(l1 <- bsims_init(extent = 10, road = 0, edge = 0, offset = 0))
#> bSims landscape
#> 1 km x 1 km
#> stratification: H
```

```
(l2 <- bsims_init(extent = 10, road = 1, edge = 0, offset = 0))
#> bSims landscape
#> 1 km x 1 km
#> stratification: HR
```

```
(l3 <- bsims_init(extent = 10, road = 0.5, edge = 1, offset = 2))
#> bSims landscape
#> 1 km x 1 km
#> stratification: HER
```

```
(l4 <- bsims_init(extent = 10, road = 0, edge = 5, offset = 5))
#> bSims landscape
#> 1 km x 1 km
#> stratification: HE
```

```
op <- par(mfrow = c(2, 2))
plot(l1, main = "Habitat")
points(0, 0, pch=3)
plot(l2, main = "Habitat & road")
lines(c(0, 0), c(-5, 5), lty=2)
plot(l3, main = "Habitat, edge, road + offset")
arrows(0, 0, 2, 0, 0.1, 20)
lines(c(2, 2), c(-5, 5), lty=2)
points(0, 0, pch=3)
plot(l4, main = "2 habitats")
arrows(0, 0, 5, 0, 0.1, 20)
lines(c(5, 5), c(-5, 5), lty=2)
points(0, 0, pch=3)
```

The `bsims_populate`

function *populates* the
landscape we created by the `bsims_init`

function, which is
the first argument we have to pass to `bsims_populate`

. The
function returns a population object (with class ‘bsims_population’).
The most important argument that controls how many individuals will
inhabit our landscape is `density`

that defines the expected
value of individuals per unit area (1 ha). By default,
`density = 1`

(\(D=1\)) and
we have 100 ha in the landscape (\(A=100\)) which translates into 100
individuals on average (\(E[N]=\lambda=AD\)). The actual number of
individuals in the landscape might deviate from this expectation,
because \(N\) is a random variable
(\(N \sim f(\lambda)\)). The
`abund_fun`

argument controls this relationship between the
expected (\(\lambda\)) and realized
abundance (\(N\)). The default is a
Poisson distribution:

Changing `abund_fun`

can be useful to make abundance
constant or allow under or overdispersion, e.g.:

```
summary(rpois(100, 100)) # Poisson variation
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 72.0 94.0 100.0 100.7 107.0 130.0
```

```
summary(MASS::rnegbin(100, 100, 0.8)) # NegBin variation
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.00 15.75 69.00 106.59 147.25 563.00
```

```
negbin <- function(lambda, ...) MASS::rnegbin(1, lambda, ...)
bsims_populate(l1, abund_fun = negbin, theta = 0.8)
#> bSims population
#> 1 km x 1 km
#> stratification: H
#> total abundance: 109
```

```
## constant abundance
bsims_populate(l1, abund_fun = function(lambda, ...) lambda)
#> bSims population
#> 1 km x 1 km
#> stratification: H
#> total abundance: 100
```

Once we determine how many individuals will populate the landscape,
we have control over the spatial arrangement of the nest location for
each individual. The default is a homogeneous Poisson point process
(complete spatial randomness). Deviations from this can be controlled by
the `xy_fun`

. This function takes distance as its only
argument and returns a numeric value between 0 and 1. A function
`function(d) reurn(1)`

would be equivalent with the Poisson
process, meaning that every new random location is accepted with
probability 1 irrespective of the distance between the new location and
the previously generated point locations in the landscape. When this
function varies with distance, it leads to a non-homogeneous point
process via this accept-reject algorithm. The other arguments
(`margin`

, `maxit`

, `fail`

) are passed
to the underlying `accepreject`

function to remove edge
effects and handle high rejection rates.

In the next example, we fix the abundance to be constant (i.e. not a random variable, \(N=\lambda\)) and different spatial point processes:

```
D <- 0.5
f_abund <- function(lambda, ...) lambda
## systematic
f_syst <- function(d)
(1-exp(-d^2/1^2) + dlnorm(d, 2)/dlnorm(exp(2-1),2)) / 2
## clustered
f_clust <- function(d)
exp(-d^2/1^2) + 0.5*(1-exp(-d^2/4^2))
p1 <- bsims_populate(l1, density = D, abund_fun = f_abund)
p2 <- bsims_populate(l1, density = D, abund_fun = f_abund, xy_fun = f_syst)
p3 <- bsims_populate(l1, density = D, abund_fun = f_abund, xy_fun = f_clust)
distance <- seq(0,10,0.01)
op <- par(mfrow = c(3, 2))
plot(distance, rep(1, length(distance)), type="l", ylim = c(0, 1),
main = "random", ylab=expression(f(d)), col=2)
plot(p1)
plot(distance, f_syst(distance), type="l", ylim = c(0, 1),
main = "systematic", ylab=expression(f(d)), col=2)
plot(p2)
plot(distance, f_clust(distance), type="l", ylim = c(0, 1),
main = "clustered", ylab=expression(f(d)), col=2)
plot(p3)
```

The `get_nests`

function extracts the nest locations.
`get_abundance`

and `get_density`

gives the total
abundance (\(N\)) and density (\(D=N/A\), where \(A\) is `extent^2`

) in the
landscape, respectively.

If the landscape is stratified, that has no effect on density unless
we specify different values through the `density`

argument as
a vector of length 3 referring to the HER strata:

```
D <- c(H = 2, E = 0.5, R = 0)
op <- par(mfrow = c(2, 2))
plot(bsims_populate(l1, density = D), main = "Habitat")
plot(bsims_populate(l2, density = D), main = "Habitat & road")
plot(bsims_populate(l3, density = D), main = "Habitat, edge, road + offset")
plot(bsims_populate(l4, density = D), main = "2 habitats")
```

The `bsims_animate`

function *animates* the
population created by the `bsims_populate`

function.
`bsims_animate`

returns an events object (with class
‘bsims_events’). The most important arguments are governing the
`duration`

of the simulation in minutes, the vocalization
(`vocal_rate`

), and the movement (`move_rate`

)
rates as average number of events per minute.

We can describe these behavioral events using survival modeling terminology. Event time (\(T\)) is a continuous random variable. In the simplest case, its probability density function is the Exponential distribution: \(f(t)=\phi e^{-t\phi}\). The corresponding cumulative distribution function is: \(F(t)=\int_{0}^{t} f(t)dt=1-e^{-t\phi}=p_t\), giving the probability that the event has occurred by duration \(t\). The parameter \(\phi\) is the rate of the Exponential distribution with mean \(1/\phi\) and variance \(1/\phi^2\).

In survival models, the complement of \(F(t)\) is called the *survival
function* (\(S(t)=1-F(t)\), \(S(0)=1\)), which gives the probability that
the event has not occurred by duration \(t\). The *hazard function* (\(\lambda(t)=f(t)/S(t)\)) defines the
instantaneous rate of occurrence of the event (risk, the density of
events at \(t\) divided by the
probability of surviving). The cumulative hazard (cumulative risk) is
the sum of the risks between duration 0 and \(t\) (\(\Lambda(t)=\int_{0}^{t}
\lambda(t)dt\)).

The simplest survival distribution assumes constant risk over time (\(\lambda(t)=\phi\)), which corresponds to the Exponential distribution. The Exponential distribution also happens to describe the lengths of the inter-event times in a homogeneous Poisson process (events are independent, it is a ‘memory-less’ process).

`bsims_animate`

uses independent Exponential distributions
with rates `vocal_rate`

and `move_rate`

to
simulate vocalization and movement events, respectively. The
`get_events`

function extracts the events as a data frame
with columns describing the location (`x`

, `y`

)
and time (`t`

) of the events (`v`

is 1 for
vocalizations and 0 otherwise) for each individual (`i`

gives
the individual identifier that links individuals to the nest
locations)

```
l <- bsims_init()
p <- bsims_populate(l, density = 0.5)
e1 <- bsims_animate(p, vocal_rate = 1)
head(get_events(e1))
#> x y t v a i
#> 1 -2.508184 2.6764407 0.004043933 1 155 4
#> 2 3.876671 4.4885743 0.022987967 1 314 34
#> 3 -1.840156 0.3421961 0.031965954 1 108 1
#> 4 -1.416870 -2.0850311 0.044264103 1 276 19
#> 5 4.112376 -3.7872979 0.105670752 1 223 53
#> 6 2.000265 2.9710303 0.125082360 1 216 43
```

There are no movement related events when `move_rate = 0`

,
the individuals are always located at the nest, i.e. there is no within
territory movement. If we increase the movement rate, we also have to
increase the value of `movement`

, that is the standard
deviation of bivariate Normal kernels centered around each nest
location. This kernel is used to simulate new locations for the movement
events.

```
e2 <- bsims_animate(p, move_rate = 1, movement = 0.25)
op <- par(mfrow = c(1, 2))
plot(e1, main = "Closure")
plot(e2, main = "Movement")
```

Individuals in the landscape might have different vocalization rates
depending on, e.g., breeding status. Such heterogeneity can be added to
the simulations as a finite mixture: `vocal_rate`

and
`move_rate`

can be supplied as a vector, each element giving
the rate for the groups. The `mixture`

argument is then used
to specify the mixture proportions.

```
e3 <- bsims_animate(p,
vocal_rate = c(25, 1), mixture = c(0.33, 0.67))
plot(get_events(e3))
curve((1-0.67*exp(-1*x)) * get_abundance(e3), col=2, add=TRUE)
```

Vocal and movement rates (and corresponding kernel standard deviations) can be defined four different ways:

- a single number: constant behavior patterns, no groups,
- a vector of length
`length(mixture)`

: behavior based finite mixture groups, - a vector of length 3 with
`mixture = 1`

: mixtures correspond to HER strata, - or a matrix of dimension 3 \(\times\)
`length(mixture)`

: HER strata \(\times\) number of behavior based groups.

Strata based groups are tracked by column `s`

, behavior
based groups are tracked as the column `g`

in the output of
`get_nests`

.

Here is how different territory sizes can be achieved in a two-habitat landscape:

```
plot(bsims_animate(bsims_populate(l4, density = D),
move_rate = c(0.5, 1, 1), movement = c(0, 0.2, 0.2),
mixture = 1), main="Strata based mixtures")
```

Stratum related behavior groups depend on the nest location.
Sometimes it makes sense to restrict movement even further,
i.e. individuals do not land in certain strata (but can cross a stratum
if `movement`

is large enough). For example, we can restrict
movement into the road stratum (this requires density to be 0 in that
stratum):