---
title: "Introduction to redist"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction to redist}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
resource_files:
  - images/smc-schematic.png
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
set.seed(5118)
```

The `redist` package provides algorithms and tools for scalable and replicable
redistricting analyses. This vignette introduces the package by way of an
analysis of redistricting in the state of Iowa, which can broken down into four
distinct steps:

1. [Compiling, cleaning, and preparing the data](#prepare)
2. [Defining the redistricting problem](#define)
3. [Simulating redistricting plans](#simulate)
4. [Analyzing the simulated plans](#analyze)

First, however, a brief overview of the package itself.
```{r setup, message=FALSE}
library(redist)
library(dplyr)
library(ggplot2)
```

# The `redist` package {#package}
To install `redist`, follow the instructions in the
[README](../index.html#installation-instructions).

For more information on package components, check out the 
[full documentation](../reference/index.html).

## Algorithms
The package contains a variety of redistricting simulation and enumeration
algorithms.  Generally you will use one of the following three algorithms:

- `redist_smc()`, the recommended algorithm for most purposes.^[from 
[Sequential Monte Carlo for Sampling Balanced and Compact Redistricting Plans](https://arxiv.org/pdf/2008.06131.pdf)]
- `redist_mergesplit()`, a MCMC version of the SMC proposal.^[based on Carter,
D., Herschlag, G., Hunter, Z., and Mattingly, J. (2019). A merge-split proposal
for reversible Monte Carlo Markov chain sampling of redistricting plans. arXiv
preprint arXiv:1911.01503.]
- `redist_flip()`, another MCMC algorithm which uses more local proposals.^[from 
[Automated Redistricting Simulation Using Markov Chain Monte Carlo](https://doi.org/10.1080/10618600.2020.1739532) *Journal of Computational and Graphical Statistics*]

The other algorithms are

- `redist.enumpart()` for efficient enumeration of small maps.^[from
[The Essential Role of Empirical Validation in Legislative Redistricting Simulation](https://imai.fas.harvard.edu/research/files/enumerate.pdf)]
- `redist_shortburst()` for optimizing a plan according to a user-provided
criterion.^[from Cannon, S., Goldbloom-Helzner, A., Gupta, V., Matthews, J. N.,
& Suwal, B. (2020). Voting Rights, Markov Chains, and Optimization by Short
Bursts. arXiv preprint arXiv:2011.02288.]
- `redist.rsg()` and `redist.crsg()`, which do not sample from a known target
distribution.^[from Jowei Chen and Jonathan Rodden (2013) 
“Unintentional Gerrymandering: Political Geography and Electoral Bias in
Legislatures.” Quarterly Journal of Political Science. 8(3): 239-269.]

## Data
The package comes with several built-in datasets, which may be useful in 
exploring the package's functionality and in becoming familiar with algorithmic
redistricting:

- [`iowa`](../reference/iowa.html) (used in this vignette).
- [`fl25`](../reference/fl25.html), a 25-precinct subset of the state of Florida.
- [`fl25_enum`](../reference/fl25_enum.html), containing all possible sets of three districts drawn on the
25-precinct Florida map.
- [`fl70`](../reference/fl70.html), a 70-precinct subset of the state of Florida.
- [`fl250`](../reference/fl250.html), a 250-precinct subset of the state of Florida.


# Compiling, cleaning, and preparing the data {#prepare}
The most time-consuming part of a redistricting analysis, like most data
analyses, is collecting and cleaning the necessary data. For redistricting,
this data includes geographic shapefiles for precincts and existing legislative 
district plans, precinct- or block-level demographic information from the Census,
and precinct-level political data. These data generally come from different 
sources, and may not fully overlap with each other, further complicating the
problem.

`redist` is not focused on this data collection process.  The 
[`geomander`](https://christophertkenny.com/geomander/) package contains
many helpful functions for compiling these data, and fixing problems in
geographic data.

The ALARM project provides 
[pre-cleaned redistricting data files](https://alarm-redist.org/posts/2021-08-10-census-2020/) 
consisting of VEST election data joined 2020 Census data at the precinct level.
Other sources for precinct-level geographic and political information include 
the [MIT Election Lab](https://electionlab.mit.edu/data), 
the [Census](https://www.census.gov/programs-surveys/decennial-census/about/rdo.html),
the [Redistricting Data Hub](https://redistrictingdatahub.org/),
the [Voting and Election Science Team](https://dataverse.harvard.edu/dataverse/electionscience),
the [Harvard Election Data Archive](https://dataverse.harvard.edu/dataverse/eda),
the [Metric Geometry and Gerrymandering Group](https://github.com/mggg-states),
and some state websites. 

## Iowa

For this analysis of Iowa, we'll use the data included in the package, which
was compiled from the Census and the Harvard Election Data Archive. It contains,
for each county, the total population and voting-age population by race, as
well as the number of votes for President in 2008. The `geometry` column 
contains the geographic [shapefile information](https://r-spatial.github.io/sf/).
```{r}
data(iowa)
print(iowa)
```

# Defining the redistricting problem {#define}
A redistricting problem is defined by the map of the precincts, the number
of contiguous districts to divide the precincts into, the level of population 
parity to enforce, and any other necessary constraints that must be imposed.

## Determining the relevant constraints
In Iowa, congressional districts are constructed not out of precincts but out
of the state's 99 counties, and in the 2010 redistricting cycle, Iowa was
apportioned four congressional districts, down one from the 2000 cycle.
Chapter 42 of the Iowa Code provides guidance on the other constraints imposed on
the redistricting process (our emphasis added):

> **42.4 Redistricting standards.**
>
> ...
>
> 1.b. Congressional districts shall each have a population *as nearly equal as
> practicable* to the ideal district population, derived as prescribed in paragraph
> “a” of this subsection. No congressional district shall have a population which
> varies by more than *one percent from the applicable ideal district population*,
> except as necessary to comply with Article III, section 37 of the Constitution
> of the State of Iowa.
>
> ...
>
> 3\. Districts shall be composed of convenient *contiguous territory*. Areas which
> meet only at the points of adjoining corners are not contiguous.
>
> 4\. Districts shall be *reasonably compact* in form, to the extent consistent
> with the standards established by subsections 1, 2, and 3. In general,
> reasonably compact districts are those which are square, rectangular, or
> hexagonal in shape, and not irregularly shaped, to the extent permitted by
> natural or political boundaries....
>
> 5\. No district shall be drawn for the purpose of favoring a political party,
> incumbent legislator or member of Congress, or other person or group, or for the
> purpose of augmenting or diluting the voting strength of a language or racial
> minority group. In establishing districts, *no use shall be made* of any of the
> following data:
>
> a. Addresses of incumbent legislators or members of Congress.
> b. Political affiliations of registered voters.
> c. Previous election results.
> d. Demographic information, other than population head counts, except as required by
> the Constitution and the laws of the United States.

The section goes on to provide two specific measures of compactness that should be
used to compare districts, one of which is the total perimeter of all districts. 
If the total perimeter is small, then the districts relatively compact.

Contiguity of districts and no reliance on partisan or demographic data 
are built-in to `redist`.  We'll look at how to specify the desired population
deviation (no more than 1% by law) in the next section, and discuss compactness
in the [simulation section](#simulate).

## Setting up the problem in `redist`
In `redist`, a basic redistricting problem is defined by an object of type
`redist_map`, which can be constructed using the eponymous function.  The user
must provide a vector of population counts (defaults to the `pop` column, if
one exists) and the desired population parity, and the number of districts.
The latter can be inferred if a reference redistricting plan exists.  For Iowa,
we'll use the adopted 2010 plan as a reference.

```{r}
iowa_map = redist_map(iowa, existing_plan=cd_2010, pop_tol=0.01, total_pop = pop)
print(iowa_map)
```

This looks much the same as `iowa` itself, but metadata has been added,
and there's a new column, `adj`.

## Adjacency-based redistricting
All redistricting algorithms operate on an _adjacency graph_, which is
constructed from the actual precinct or county geography. In the adjacency graph,
every precinct or county is a node, and two nodes are connected by an edge
if the corresponding precincts are geographically adjacent.^[for `redist`'s
purposes, adjacency requires that two regions touch at more than just one point
or corner.] Creating a contiguous set of districts as part of a redistricting
plan then corresponds to creating a
[_partition_](https://en.wikipedia.org/wiki/Graph_partition) of the adjacency
graph.

The `redist_map()` function automatically computes the adjacency graph from the
provided shapefile (though one can be provided directly as well), and stores it
in the `adj` column as an _adjacency list_, which is, for each precinct, a list
of neighboring precincts. As part of this process, the adjacency graph is 
checked for global contiguity (no islands), which is necessary for the
redistricting algorithms to function properly.

We can visualize the adjacency graph by plotting the `redist_map` object.
```{r iowa-adj, fig.width=8}
plot(iowa_map, adj=T) + plot(iowa_map)
```

## Pre-processing
Often, we want to only analyze a portion of a map, or hold some districts fixed
while others are re-simulated.  We may also want to implement a status-quo-type
constraint that encourages simulated districts to be close to a reference plan.
This can be accomplished by freezing the "cores" of each district.

All of these operations fall under the umbrella of map pre-processing, and
`redist` is well-equipped to handle them.  You can use familiar `dplyr` verbs
like `filter()` and `summarize()`, along with new `redist` operations like
`freeze()`, `make_cores()`, and `merge_by()`, to operate on `redist_map` objects. 
The package will make the appropriate modifications to the geometry and
adjacency graph in the background.  

The [map pre-processing vignette](map-preproc.html) contains more information
and examples about these operations.

## Exploring the geography
To get a feel for the demographic and political geography of the state, we'll
make some plots from the `iowa_map` object. We see that the state is mostly
rural and white, with Polk county (Des Moines) the largest and densest.
Politically, most counties are relatively balanced between Democrats and
Republicans (at least in the '08 election), though there is a rough east-west
gradient.

```{r iowa-chloro, message=FALSE}
areas = as.numeric(units::set_units(sf::st_area(iowa_map$geometry), mi^2))
plot(iowa_map, fill = pop / areas) + 
    scale_fill_viridis_c(name="Population density (people / sq. mi)", 
                         trans="sqrt")
plot(iowa_map, fill = dem_08 / tot_08) +
    scale_fill_gradient2(name="Pct. Democratic '08",  midpoint=0.5)
plot(iowa_map, fill = wvap / vap, by_distr = TRUE)
```


# Simulating redistricting plans {#simulate}
The crux of a redistricting analysis is actually simulating new redistricting
plans.  As discussed above, `redist` provides several algorithms for performing
this simulation, and each has its own advantages and incorporates a different
set of constraints. Here, we'll demonstrate use of the `redist_smc()` algorithm,
a Sequential Monte Carlo (SMC)-based procedure which is the recommended choice for
most redistricting analyses.

SMC creates plans directly, by drawing district boundaries one at a time, as
illustrated below.

![](images/smc-schematic.png)

Because of the way districts are drawn in SMC, the generated districts are
relatively compact by default.  This can be further controlled by the 
`compactness` parameter (although `compactness=1` is particularly
computationally convenient).

To simulate, we call `redist_smc()` on our `redist_map` object. We provide
the `runs=2` parameter, which will run the SMC algorithm twice, in parallel.
This doubles the total number of sampled plans, but more importantly, it
provides extremely valuable diagnostic information about whether the algorithm
is sampling properly.

```{r message=FALSE}
iowa_plans = redist_smc(iowa_map, 500, compactness=1, runs=2)
```

The output from the algorithm is a `redist_plans` object, which stores a matrix
of district assignments for each precinct and simulated plans, and a table of 
summary statistics for each district and simulated plan. The existing 2010
plan has also been automatically added as a reference plan.  Additional
reference or comparison plans may be added using `add_reference()`.

```{r}
print(iowa_plans)
```

We can explore specific simulated plans with `redist.plot.plans()`.

```{r ia-sim-plans}
redist.plot.plans(iowa_plans, draws=1:6, shp=iowa_map)
```

# Analyzing the simulated plans {#analyze}
A `redist_plans` object, the output of a sampling algorithm, links a matrix
of precinct assignments to a table of district statistics, and this linkage 
makes analyzing the output a breeze.

Sometimes it may be useful to renumber the simulated districts (which have
random numbers in general) to match the reference plan as closely as possible.
This adds a `pop_overlap` column which measures how much of the population is in
the same district in both a given plan and the reference plan.
```{r}
iowa_plans = match_numbers(iowa_plans, iowa_map$cd_2010)
print(iowa_plans)
```

Then we can add summary statistics by district, using `redist`'s analysis
functions.  Here, we'll compute the population deviation, the perimeter-based
compactness measure related to the Iowa Code's redistricting requirements, and
the fraction of minority voters and two-party Democratic vote share by district.

```{r message=F}
county_perims = prep_perims(iowa_map, iowa_map$adj)

iowa_plans = iowa_plans %>%
    mutate(pop_dev = abs(total_pop / get_target(iowa_map) - 1),
           comp = comp_polsby(pl(), iowa_map, perim_df=county_perims),
           pct_min = group_frac(iowa_map, vap - wvap, vap),
           pct_dem = group_frac(iowa_map, dem_08, dem_08 + rep_08))
print(iowa_plans)
```


Once summary statistics of interest have been calculated, it's very important
to check the algorithm's diagnostics. As with any complex sampling algorithm,
things can go wrong. Diagnostics, while not flawless, can help catch problems
and stop you from making conclusions that are actually the fault of a broken
sampling process. The `summary()` function is `redist`'s one-stop-shop for
algorithm diagnostics.

```{r}
summary(iowa_plans)
```

There's a lot of diagnostic output there, which you should read more about with
`?summary.redist_plans`. If anything is obviously wrong, the function will
alert you and provide instructions on how to try to fix it. But these warnings
aren't flawless, and it's important to check the numbers yourself. 

If you've used 2 or more `runs`, as we have, `summary()` will calculate R-hat values.
These should be as close to 1 as possible, and generally less than 1.05. If 
they are bigger than that, it means that multiple independent runs of the 
algorithm gave different results.  More samples (higher `nsims`) are usually 
called for.  The other number to keep an eye on is the plan diversity 
(top of the output), whose 80% range should generally cover the range from 
0.5--0.8. 

Since our diagnostics look good, we can return to our analysis.
It's quick to turn district-level statistics from a `redist_plans` object 
into plan-level summary statistics.

```{r}
plan_sum = group_by(iowa_plans, draw) %>%
    summarize(max_dev = max(pop_dev),
              avg_comp = mean(comp),
              max_pct_min = max(pct_min),
              dem_distr = sum(pct_dem > 0.5))
print(plan_sum)
```


These tables of statistics are easily plotted using existing libraries like
`ggplot2`, but `redist` provides a number of helpful plotting functions that
automate some common tasks, like adding a reference line for the reference plan.
The output of these functions is a `ggplot` object, allowing for further 
customization.

```{r dev-comp-plot}
library(patchwork)

hist(plan_sum, max_dev) + hist(iowa_plans, comp) +
    plot_layout(guides="collect")
```

We can see that the adopted plan has nearly complete population parity, and
that its districts are roughly as compact on average as those simulated by
the SMC algorithm.

One of the most common, and useful, plots, for studying the partisan
characteristics of a plan, is to plot the fraction of a group (or party) within
each district, and compare to the reference plan. Generally, we would first 
sort the districts by this quantity first, to make the numbers line up, but
here we've already renumbered the districts to match the reference plan as
closely as possible.

```{r signature}
plot(iowa_plans, pct_dem, sort=FALSE, size=0.5)
```

We see that districts 1 and 2 look normal, but it appears that, relative to
our ensemble, district 4 (NW Iowa) is more Democratic, and district 3 (SW Iowa,
Des Moines) is less Democratic. However, the reference plan does not appear to
be a complete outlier. 

We might also want to look at how the Democratic fraction in each district
compares to the fraction of minority voters. We can make a scatterplot of
districts, and overlay the reference districts, using `redist.plot.scatter`.
We'll also color by the district number (higher numbers are in lighter colors).

Once again, we see that while district 1 and 2 of the reference plan look
normal, district has a lower number of Democrats and minority voters than would
otherwise be expected. 

```{r scatter}
pal = scales::viridis_pal()(5)[-1]
redist.plot.scatter(iowa_plans, pct_min, pct_dem, 
                    color=pal[subset_sampled(iowa_plans)$district]) +
    scale_color_manual(values="black")
```

From here, it is easy to keep exploring, using the functionality of
`redist_plans` and the built-in plotting functions. More complex model-based
analyses could also be performed using the district-level or plan-level
statistics.
