---
title: "Connectivity tutorial"
output:
  rmarkdown::html_vignette:
    toc: true
    fig_caption: true
fontsize: 11pt
documentclass: article
bibliography: references.bib
csl: reference-style.csl
vignette: >
  %\VignetteIndexEntry{Connectivity tutorial}
  %\VignetteEngine{knitr::rmarkdown_notangle}
---

```{r, include = FALSE}
h <- 4.5
w <- 4.5
is_check <- ("CheckExEnv" %in% search()) || any(c("_R_CHECK_TIMINGS_",
             "_R_CHECK_LICENSE_") %in% names(Sys.getenv()))
knitr::opts_knit$set(global.par = TRUE)
knitr::opts_chunk$set(
  fig.align = "center", fig.width = 4.0, fig.height = 3.5,
  eval = !is_check, purl = !is_check
)
```

```{r, include = FALSE}
# wrapper for printing problem objects
print_problem <- function(x) {
  if (interactive()) {
    print(x)
    return(invisible())
  }
  res <- knitr::knit_child(
    "print_problem.txt",
    envir = list2env(
      list(name = rlang::as_name(rlang::enquo(x)), x = x),
      parent = parent.frame()),
    quiet = TRUE,
  )
  cat(res, sep = "\n")
}
```

## Introduction

Connectivity is a key consideration in systematic conservation planning [@r4; @r55]. This is because isolated and fragmented populations are often more vulnerable to extinction [@r60; @r61; @r62]. To promote connectivity in prioritizations, a range of different approaches are available [reviewed in @r56]. These approaches can solely on the spatial configuration of a prioritization to enhance structural connectivity [e.g, reducing the spatial fragmentation of a prioritization\; @r2]. They can also leverage data -- such as environmental, river flow, and telemetry data -- to generate prioritizations that promote functional connectivity [e.g., @r43; @r58; @r59].

The aim of this tutorial is to show how connectivity can be incorporated into prioritizations using the _prioritizr R_ package. Here we will explore various approaches for incorporating connectivity, and see how they alter the spatial configuration of prioritizations. As you will discover, many of these approaches involve setting threshold or penalty values to specify the relative importance of connectivity compared to other criteria (e.g., overall cost). For more information on calibrating these values, please see the [_Calibrating trade-offs tutorial_](calibrating_trade-offs_tutorial.html).

## Data

The dataset used in this tutorial was created for the Coastal Douglas-fir Conservation Partnership [CDFCP\; @r29]. Although the original dataset covers a much larger area; for brevity, here we focus only on Salt Spring Island, British Columbia. Briefly, Salt Spring Island supports a diverse and globally unique mix of dry forest and savanna habitats. Today, these habitats are critically threatened due to land conversion, invasive species, and altered disturbance regimes. For more information on the data, please refer to the [Marxan tool portal](https://arcese.forestry.ubc.ca/marxan-tool/) and the [tool tutorial](https://peter-arcese-lab.sites.olt.ubc.ca/files/2016/09/CDFCP_tutorial_2017_05.pdf).

<center>

![Extent of Coastal Douglas-fir Conservation Partnership Tool area and location of Salt Spring Island](figures/map.jpg)

</center>

Let's begin by loading the packages and data for this tutorial. Since this tutorial requires the _prioritizrdata R_ package, please ensure that it is installed. Specifically, two objects underpin the data for this tutorial. The `salt_pu` object specifies the planning unit data as a single-layer raster (i.e., `terra::rast()` object), and the `salt_features` object contains biodiversity data represented as a multi-layer raster (i.e., `terra::rast()` object).

```{r, message = FALSE}
# load packages
library(prioritizr)
library(prioritizrdata)
library(sf)
library(terra)
library(scales)

# load planning unit data
salt_pu <- get_salt_pu()

# load biodiversity feature data
salt_features <- get_salt_features()

# load connectivity data
salt_con <- get_salt_con()
```

Now we will conduct some preliminary processing. Specifically, we will aggregate from the 100 m resolution to the 300 m resolution. This is to reduce the time needed to generate prioritizations in this tutorial. In practice, we generally recommend consider other criteria too -- such as the spatial scale that is relevant for decision making and the resolution of available datasets -- when deciding on an appropriate scale for planning units.

```{r}
# aggregate data to coarser resolution
salt_pu <- aggregate(salt_pu, fact = 3)
salt_features <- aggregate(salt_features, fact = 3)
salt_con <- aggregate(salt_con, fact = 3)
```

Next, let's have a look at the `salt_pu` object. Here each grid cell represents a planning unit, and the grid cell values denote acquisition costs [@r28]. To aid with visualization, we will log-transform the values when plotting them on a map.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r}
# print planning unit data
print(salt_pu)

# plot map showing the planning units costs on a log-scale
plot(log(salt_pu), main = "Planning unit costs (log)", axes = FALSE)
```

Next, let's look at the `salt_features` object. It has multiple layers. Each layer corresponds to different ecological communities (i.e., _Old Forest_, _Savannah_, _Wetland_, and _Shrub_ communities), and their cell values indicate the probability of encountering a bird species associated a given community.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r}
# print feature data
print(salt_features)

# plot map showing the feature data
plot(salt_features, axes = FALSE)
```

Let's look at the `salt_con` object. It describes the inverse probability of occurrence of human commensal species. Here we will assume that human modified areas impede connectivity for native species, and so cells with higher values will have greater connectivity.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1), oma = c(0, 0, 0, 0.5))
```

```{r}
# print connectivity data
print(salt_con)

# plot map showing the connectivity data
plot(salt_con, axes = FALSE)
```

## Baseline problem

In this tutorial, we will explore a few different ways of incorporating connectivity into prioritizations. To enable comparisons among prioritizations based on different approaches, we will first create a baseline problem formulation that we will subsequently customize to incorporate connectivity. Specifically, we will formulate the baseline problem using the minimum set objective. We will use representation targets of 17% -- based on [Aichi Biodiversity Target 11](https://www.cbd.int/sp/targets/) -- to provide adequate coverage of each ecological community. Additionally, because land properties on Salt Spring Island can either be acquired in their entirety or not at all, we will use binary decision types. This means that planning units are either selected in the solution or not selected in the solution---planning units cannot be partially acquired. Given all these details, let's formulate the baseline problem.

```{r}
# create problem
p0 <-
  problem(salt_pu, salt_features) %>%
  add_min_set_objective() %>%
  add_relative_targets(0.17) %>%
  add_binary_decisions() %>%
  add_default_solver()
```

```{r, echo = FALSE, results = "asis"}
print_problem(p0)
```

After formulating the baseline problem, we can solve it to generate a prioritization.

```{r, results = "hide"}
# solve problem
s0 <- solve(p0)
```

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r}
# print solution
print(s0)

# plot solution
plot(
  s0, main = "Baseline prioritization", axes = FALSE,
  breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

Next, let's explore some options for incorporating connectivity.

## Adding constraints

Let's explore approaches for promoting connectivity in prioritizations by adding constraints to the baseline problem formulation. These approaches ensure that prioritizations exhibit certain characteristics [e.g., ensure prioritizations form a contiguous reserve\; @r57]. This means that, regardless of the optimality gap used to generate a prioritization, the prioritization will always exhibit these characteristics.

### Neighbor constraints

Neighbor constraints can be added to ensure that each selected planning unit has a certain number of neighbors surrounding it (using the `add_neighbor_constraints()` function) [based on @r16]. The `k` parameter can be used to specify the required number of neighbors for each selected planning unit. Let's generate a prioritization by specifying that each planning unit requires at least two neighbors.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with added neighbor constraints and solve it
s1 <-
  p0 %>%
  add_neighbor_constraints(k = 2) %>%
  solve()

# plot solutions
plot(
  c(s0, s1), main = c("baseline", "neighbors constraints"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

### Contiguity constraints

Contiguity constraints can be added to ensure that all planning units form a single contiguous reserve (using the `add_contiguity_constraints()` function) [similar to @r57]. These constraints are extremely complex. As such, they can only be applied to small conservation planning problems and the _Gurobi_ solver is required to solve them in a feasible period of time. Since it would take a long time to generate a near-optimal prioritization for this dataset with contiguity constraints, we will also tell the solver to simply return the first solution that it finds which meets the representation targets and the contiguity constraints.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with added contiguity constraints and solve it
s2 <-
  p0 %>%
  add_contiguity_constraints() %>%
  add_gurobi_solver(first_feasible = TRUE) %>%
  solve()

# plot solutions
plot(
  c(s0, s2), main = c("baseline", "contiguity constraints"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

There is also an even more complex version of the contiguity constraints that is available. These constraints -- termed feature contiguity constraints [similar to @r64] -- can be added to ensure that all of the selected planning units used to the reach representation targets within a prioritization form a contiguous network for each feature (using the `add_feature_contiguity_constraints()` function). In other words, they ensure that each feature can disperse through the prioritization to access a target threshold amount of habitat. However, these constraints are extraordinarily complex, only feasible for small problems, and require preprocessing routines to identify initial solutions. As such, we will not consider them in this tutorial.

### Linear constraints

Linear constraints can be used to specify that the prioritizations must meet an arbitrary set of criteria. As such, they can be used to ensure that prioritizations provide adequate coverage of planning units that have facilitate a high level of connectivity. Recall that the `salt_con` data are used to describe connectivity across the study area. Since higher values denote planning units with greater connectivity, we could use linear constraints to ensure that the total sum of connectivity values -- based on this dataset -- meets a particular threshold (e.g. cover at least 30% of the total amount). This would effectively be treating connectivity as an additional feature [similar to @r52].

```{r}
# compute threshold for constraints
## here we use a threshold of 30% of the total connectivity values
threshold <- global(salt_con, "sum", na.rm = TRUE)[[1]] * 0.3

# print threshold
print(threshold)
```

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with added linear constraints and solve it
s3 <-
  p0 %>%
  add_linear_constraints(
    data = salt_con, threshold = threshold, sense = ">="
  ) %>%
  solve()

# plot solutions
plot(
  c(s0, s3), main = c("baseline", "linear constraints"),
   axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

Although using continuous values has the advantage that the prioritization process can explicitly account for differences in the relative amount of connectivity facilitated by different planning units, the disadvantage is that the prioritization could potentially focus on selecting lots of planning units with low connectivity values. To avoid this result, one strategy is to convert the continuous values into binary values using a threshold limit [similar to @r53]. By applying such a threshold limit, linear constraints can then be used to ensure that the prioritization selects a minimum amount of planning units with high connectivity values (i.e., those with connectivity values that are equal to or greater than the threshold limit).

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r}
# calculate threshold limit
## here we set a threshold limit based on the median
threshold_limit <- global(
  salt_con, fun = quantile, na.rm = TRUE, probs = 0.5
)[[1]]

# convert continuous values to binary values
salt_con_binary <- round(salt_con <= threshold_limit)

# plot binary values
plot(salt_con_binary, main = "salt_con_binary", axes = FALSE)
```

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with added linear constraints and solve it
## note that we use the original threshold computed before,
## to ensure the prioritization covers at least 30% of the total amount
## connectivity values
s4 <-
  p0 %>%
  add_linear_constraints(
    data = salt_con_binary, threshold = threshold, sense = ">="
  ) %>%
  solve()

# plot solutions
plot(
  c(s0, s4), main = c("baseline", "linear constraints (binary)"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

Another strategy is to clamp the continuous values below a threshold limit are assigned a value of zero [similar to @r54]. This strategy has the advantage that (i) the prioritization won't focus on selecting lots of planning units with low connectivity values to meet the constraint threshold, and (ii) the optimization process can use semi-continuous values to distinguish between places that can facilitate a moderate amount and a high amount of connectivity.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, results = "hide"}
# clamp continuous values using the threshold limit we computed before
salt_con_clamp <- salt_con
salt_con_clamp[salt_con <= threshold_limit] <- 0

# plot clamped values
plot(salt_con_clamp, main = "salt_con_clamp", axes = FALSE)
```

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with added linear constraints and solve it
## note that we use the original threshold computed before,
## to ensure the prioritization covers at least 30% of the total amount
## connectivity values
s5 <-
  p0 %>%
  add_linear_constraints(
    data = salt_con_clamp, threshold = threshold, sense = ">="
  ) %>%
  solve()

# plot solutions
plot(
  c(s0, s5), main = c("baseline", "linear constraints (clamped)"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

If we were concerned that the prioritization did not facilitate a high enough level of connectivity, we could increase the `threshold` value or the `threshold_limit` value. For example, let's increase the `threshold_limit` value used to clamp the continuous connectivity values.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, results = "hide"}
# compute threshold limit
threshold_limit2 <- global(
  salt_con, fun = quantile, na.rm = TRUE, probs = 0.7
)[[1]]

# clamp continuous values using the threshold limit we computed before
salt_con_clamp2 <- salt_con
salt_con_clamp2[salt_con <= threshold_limit2] <- 0

# plot clamped values
plot(salt_con_clamp2, main = "salt_con_clamp", axes = FALSE)
```

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with added linear constraints and solve it
## note that we use the original threshold computed before,
## to ensure the prioritization covers at least 30% of the total amount
## connectivity values
s6 <-
  p0 %>%
  add_linear_constraints(
    data = salt_con_clamp2, threshold = threshold, sense = ">="
  ) %>%
  solve()

# plot solutions
plot(
  c(s0, s6), main = c("baseline", "linear constraints (clamped 2)"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

Despite the advantages of clamping the connectivity values, we can see that the prioritization has a relatively high level of spatial fragmentation. In fact, all prioritizations generated using the linear constraints can potentially have this issue. This is because linear constraints do not explicitly account for the spatial arrangement of the planning units. As such, we recommend combining the linear constraints approach with another approach [e.g., the boundary penalties approach discussed below; @r53].

## Adding penalties

Now let's explore approaches for promoting connectivity in prioritizations by adding penalties to the baseline problem formulation. These approaches involve penalizing solutions according to exhibit certain [e.g., penalize spatial fragmentation of prioritizations\; @r2]. Unlike constraint-based methods for incorporating connectivity -- if the optimality gap used to generate a prioritization is too high -- they may not necessarily produce prioritizations that exhibit desirable characteristics.

### Boundary penalties

Boundary penalties can be added to used to reduce the spatial fragmentation of prioritizations (using the `add_boundary_penalties()` function). Specifically, these penalties update the problem formulation to penalize solutions that have a high total amount of exposed boundary length [@r3]. Since boundary data often have large values which can degrade solver performance and result in excessive run times (see the [_Calibrating trade-offs tutorial_](calibrating_trade-offs_tutorial.html) for details), we will first precompute rescale the boundary data.

```{r}
# precompute the boundary data
salt_boundary_data <- boundary_matrix(salt_pu)

# rescale boundary data
salt_boundary_data@x <- rescale(salt_boundary_data@x, to = c(0.01, 100))
```

Next, let's generate a prioritization using boundary penalties. To specify the relative importance reducing spatial fragmentation -- compared with the primary objective of a problem (e.g. minimizing cost) -- we need to a value for the `penalty` parameter is used. Setting a higher value for `penalty` indicates that it is more important to avoid highly fragmented solutions. Let's generate a prioritization with a `penalty` value of 0.001.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with added boundary penalties
s7 <-
  p0 %>%
  add_boundary_penalties(penalty = 0.001, data = salt_boundary_data) %>%
  solve()

# plot solutions
plot(
  c(s0, s7), main = c("baseline", "boundary penalties (0.001)"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

We can see that the resulting prioritization is still relatively fragmented, so let's try generating another prioritization with a higher `penalty` value.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with increased boundary penalties
s8 <-
  p0 %>%
  add_boundary_penalties(penalty = 10, data = salt_boundary_data) %>%
  solve()

# plot solutions
plot(
  c(s0, s8), main = c("baseline", "boundary penalties (10)"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

Although the prioritization is now less fragmented, it has also selected a greater number of planning units. Let's calculate the cost of the prioritizations to see how they vary in overall cost.

```{r}
# calculate cost of baseline prioritization
eval_cost_summary(p0, s0)

# calculate cost of prioritization with low boundary penalties (i.e., 0.001)
eval_cost_summary(p0, s7)

# calculate cost of prioritization high low boundary penalties (i.e., 0.1)
eval_cost_summary(p0, s8)
```

We can see that the cost of the prioritizations increase with when we use higher `penalty` values. This is because there is a trade-off between the cost of a prioritization and the level of spatial fragmentation. Although it can be challenging to find the best balance, there are qualitative and quantitative methods available to help navigate such trade-offs. Please see the [_Calibrating trade-offs tutorial_](calibrating_trade-offs_tutorial.html) for a details on these methods.

### Connectivity penalties

Connectivity penalties can be used to promote connectivity in prioritizations (using the `add_connectivity_penalties()` function). These penalties use  connectivity scores to parametrize the strength of connectivity between pairs of planning units [@r38]. Thus higher scores denote a greater level of connectivity between different planning units. For example, previous studies have parametrized connectivity scores using habitat quality, environmental, and river flow data [e.g. @r59; @r63; @r43]. Although there are many approaches to calculate connectivity scores, one approach involves using conductance data -- data that describe how much each planning unit facilitates movement (opposite of landscape resistance data) -- and calculating scores for each pair of planning units by averaging their conductance values (implemented using the `connectivity_matrix()` function).

Let's compute connectivity scores by treating the `salt_con` object as conductance data. This means that we assume that neighboring planning units with higher values in the  `salt_con` object are capable of facilitating a greater amount of connectivity. **Note that the data used to compute connectivity scores must conform to the same spatial properties as the planning unit data (e.g., resolution, spatial extent, coordinate reference system).** Also, although we are using raster data here, these scores can also be computed for vector data too (e.g., `sf::st_sf()` objects). Similar to the boundary data, we will also rescale the connectivity scores to avoid numerical issues during optimization.

```{r}
# compute connectivity scores
salt_con_scores <- connectivity_matrix(salt_pu, salt_con)

# rescale scores
salt_con_scores@x <- rescale(salt_con_scores@x, to = c(0.01, 100))
```

After computing the connectivity scores, we can use them to generate prioritizations using connectivity penalties. Similar to the boundary penalties, we use the `penalty` parameter to specify the relative importance of promoting connectivity relative to the primary objective of a problem (i.e., minimizing overall cost). Let's generate a prioritization with a `penalty` value of 0.001.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with added connectivity penalties
s9 <-
  p0 %>%
  add_connectivity_penalties(penalty = 0.001, data = salt_con_scores) %>%
  solve()

# plot solutions
plot(
  c(s0, s9), main = c("baseline", "connectivity penalties (0.001)"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

Now let's try generating another prioritization with a higher `penalty` value.

```{r, include = FALSE}
par(mar = c(1.1, 4.1, 4.1, 2.1))
```

```{r, fig.width = 7.0, results = "hide"}
# create problem with increased connectivity penalties
s10 <-
  p0 %>%
  add_connectivity_penalties(penalty = 0.002, data = salt_con_scores) %>%
  solve()

# plot solutions
plot(
  c(s0, s10), main = c("baseline", "connectivity penalties (0.002)"),
  axes = FALSE, breaks = c(0, 0.5, 1), col = c("grey70", "darkgreen")
)
```

We can see that increasing the `penalty` parameter causes the prioritizations to select planning units in regions with greater connectivity values (i.e., per the `salt_con` object). As discussed with the boundary penalties, increasing the `penalty` value tells the optimization process to focus more on promoting connectivity---meaning that it won't focus as much on the primary objective (i.e., because the primary objective is to minimize overall costs). For details on calibrating these trade-offs please see the [_Calibrating trade-offs tutorial_](calibrating_trade-offs_tutorial.html). **Note that you will need to the use `eval_connectivity_summary()` function -- instead of the `eval_boundary_summary()` function -- when adapting the tutorial code for connectivity penalties.**

## Conclusion

Hopefully, this tutorial has provided a helpful introduction for incorporating connectivity into prioritizations. Broadly speaking, we recommend using the boundary penalties or the connectivity penalties to ensure that prioritizations explicitly account for the spatial configuration of selected planning units. Additionally, though not fully explored here, the connectivity penalties are a very flexible approach for promoting connectivity. For instance, in addition to parametrizing pair-wise connectivity scores for neighboring planning units, they can also be used to parametrize pair-wise connectivity scores between more distant planning units. Thus connectivity penalties could be used to parametrize connectivity across both small scales and large spatial scales (e.g., using a scaling procedure wherein connectivity scores between pairs of planning units decline with the distance between them).

## References
