constraints: Sampling constraints

In redist: Simulation Methods for Legislative Redistricting

Sampling constraints

Description

The redist_smc() and redist_mergesplit() algorithms in this package allow for additional constraints on the redistricting process to be encoded in the target distribution for sampling. These functions are provided to specify these constraints. All arguments are quoted and evaluated in the context of the data frame provided to redist_constr().

Usage

add_constr_status_quo(constr, strength, current)

add_constr_grp_pow(
  constr,
  strength,
  group_pop,
  total_pop = NULL,
  tgt_group = 0.5,
  tgt_other = 0.5,
  pow = 1
)

add_constr_grp_hinge(
  constr,
  strength,
  group_pop,
  total_pop = NULL,
  tgts_group = c(0.55)
)

add_constr_grp_inv_hinge(
  constr,
  strength,
  group_pop,
  total_pop = NULL,
  tgts_group = c(0.55)
)

add_constr_compet(constr, strength, dvote, rvote, pow = 0.5)

add_constr_incumbency(constr, strength, incumbents)

add_constr_splits(constr, strength, admin)

add_constr_multisplits(constr, strength, admin)

add_constr_total_splits(constr, strength, admin)

add_constr_pop_dev(constr, strength)

add_constr_segregation(constr, strength, group_pop, total_pop = NULL)

add_constr_polsby(constr, strength, perim_df = NULL)

add_constr_fry_hold(
  constr,
  strength,
  total_pop = NULL,
  ssdmat = NULL,
  denominator = 1
)

add_constr_log_st(constr, strength, admin = NULL)

add_constr_edges_rem(constr, strength)

add_constr_custom(constr, strength, fn)

Arguments

constr	A redist_constr() object
strength	The strength of the constraint. Higher values mean a more restrictive constraint.
current	The reference map for the status quo constraint.
group_pop	A vector of group population
total_pop	A vector of total population. Defaults to the population vector used for sampling.
tgt_group, tgt_other	Target group shares for the power-type constraint.
pow	The exponent for the power-type constraint.
tgts_group	A vector of target group shares for the hinge-type constraint.
dvote, rvote	A vector of Democratic or Republican vote counts
incumbents	A vector of unit indices for incumbents. For example, if three incumbents live in the precincts that correspond to rows 1, 2, and 100 of your redist_map, entering incumbents = c(1, 2, 100) would avoid having two or more incumbents be in the same district.
admin	A vector indicating administrative unit membership
perim_df	A dataframe output from redistmetrics::prep_perims
ssdmat	Squared distance matrix for Fryer Holden constraint
denominator	Fryer Holden minimum value to normalize by. Default is 1 (no normalization).
fn	A function

Details

All constraints are fed into a Gibbs measure, with coefficients on each constraint set by the corresponding strength parameter. The strength can be any real number, with zero corresponding to no constraint. Higher and higher strength values will eventually cause the algorithm's accuracy and efficiency to suffer. Whenever you use constraints, be sure to check all sampling diagnostics.

The status_quo constraint adds a term measuring the variation of information distance between the plan and the reference, rescaled to [0, 1].

The grp_hinge constraint takes a list of target group percentages. It matches each district to its nearest target percentage, and then applies a penalty of the form \sqrt{max(0, tgt - grouppct)}, summing across districts. This penalizes districts which are below their target percentage. Use plot.redist_constr() to visualize the effect of this constraint and calibrate strength appropriately.

The grp_inv_hinge constraint takes a list of target group percentages. It matches each district to its nearest target percentage, and then applies a penalty of the form \sqrt{max(0, grouppct - tgt)}, summing across districts. This penalizes districts which are above their target percentage. Use plot.redist_constr() to visualize the effect of this constraint and calibrate strength appropriately.

The grp_pow constraint (for expert use) adds a term of the form (|tgtgroup-grouppct||tgtother-grouppct|)^{pow}), which encourages districts to have group shares near either tgt_group or tgt_other. Values of strength depend heavily on the values of these parameters and especially the pow parameter. Use plot.redist_constr() to visualize the effect of this constraint and calibrate strength appropriately.

The compet constraint encourages competitiveness by applying the grp_pow constraint with target percentages set to 50%. For convenience, it is specified with Democratic and Republican vote shares.

The incumbency constraint adds a term counting the number of districts containing paired-up incumbents. Values of strength should generally be small, given that the underlying values are counts.

The splits constraint adds a term counting the number of counties which are split once or more. Values of strength should generally be small, given that the underlying values are counts.

The multisplits constraint adds a term counting the number of counties which are split twice or more. Values of strength should generally be small, given that the underlying values are counts.

The total_splits constraint adds a term counting the total number of times each county is split, summed across counties (i.e., counting the number of excess district-county pairs). Values of strength should generally be small, given that the underlying values are counts.

The edges_rem constraint adds a term counting the number of edges removed from the adjacency graph. This is only usable with redist_flip(), as other algorithms implicitly use this via the compactness parameter. Values of strength should generally be small, given that the underlying values are counts.

The log_st constraint constraint adds a term counting the log number of spanning trees. This is only usable with redist_flip(), as other algorithms implicitly use this via the compactness parameter.

The polsby constraint adds a term encouraging compactness as defined by the Polsby Popper metric. Values of strength may be of moderate size.

The fry_hold constraint adds a term encouraging compactness as defined by the Fryer Holden metric. Values of strength should be extremely small, as the underlying values are massive when the true minimum Fryer Holden denominator is not known.

The segregation constraint adds a term encouraging segregation among minority groups, as measured by the dissimilarity index.

The pop_dev constraint adds a term encouraging plans to have smaller population deviations from the target population.

The custom constraint allows the user to specify their own constraint using a function which evaluates districts one at a time. The provided function fn should take two arguments: a vector describing the current plan assignment for each unit as its first argument, and an integer describing the district which to evaluate in the second argument. ⁠which([plans == distr])⁠ would give the indices of the units that are assigned to a district distr in any iteration. The function must return a single scalar for each plan - district combination, where a value of 0 indicates no penalty is applied. If users want to penalize an entire plan, they can have the penalty function return a scalar that does not depend on the district. It is important that fn not use information from precincts not included in distr, since in the case of SMC these precincts may not be assigned any district at all (plan will take the value of 0 for these precincts). The flexibility of this constraint comes with an additional computational cost, since the other constraints are written in C++ and so are more performant.

Examples

data(iowa)
iowa_map <- redist_map(iowa, existing_plan = cd_2010, pop_tol = 0.05)
constr <- redist_constr(iowa_map)
constr <- add_constr_splits(constr, strength = 1.5, admin = name)
constr <- add_constr_grp_hinge(constr, strength = 100,
    dem_08, tot_08, tgts_group = c(0.5, 0.6))
# encourage districts to have the same number of counties
constr <- add_constr_custom(constr, strength = 1000, fn = function(plan, distr) {
    # notice that we only use information on precincts in `distr`
    abs(sum(plan == distr) - 99/4)
})
print(constr)

redist documentation built on June 24, 2024, 9:10 a.m.