S_array_from_inequalities_and_conditions: Converts system of inequalities into an array of simplices...

In aeggers/pivotprobs: Methods for estimating the probability of ties and other important election events.

Converts system of inequalities into an array of simplices for integration

Description

The SimplicialCubature::adaptIntegrateSimplex() function requires an array of vertices S over which to integrate. This function produces this array from a matrix of inequalities and optional arguments specifiying alterations to these inequalities.

Usage

S_array_from_inequalities_and_conditions(
  inequality_mat,
  rows_to_alter = c(NULL),
  drop_dimension = F,
  limits = c(0, NULL),
  epsilon = 1e-10,
  qhull_options = NULL
)

Arguments

inequality_mat	An R x C+1 matrix. Each row is an inequality of the form Ax > b. The coefficients A come from the first C columns, the constant b comes from the last column, and the x's are (in this case) the shares of each ballot. For example, if there were only two ballots and the sole condition is x_1 - x_2 > .1, inequality_mat would be matrix(c(1, -1, .1), nrow = 1).
rows_to_alter	The indices of any rows in inequality_mat that you wish to set to equality or near-equality. To generate the S array for the plurality pivot event in which candidate 1 ties candidate 2, specify the row in inequality_mat that indicates that candidate 1 must win more than candidate 2.
drop_dimension	If F and rows_to_alter are supplied, then the inequality in inequality_mat is converted to a near inequality based on the limits and the result represents a narrow hypervolume over which to integrate. If T and rows_to_alter are supplied, then the inequality in inequality_mat is converted to a near inequality at b=limits[1] and the result represents a hyperplane over which to integrate.
limits	Determine the range within which the inequality in rows_to_alter is required to hold. If the inequality to be altered was v_1 - v_2 > 0 and drop_dimension=F, the new conditions are v_1 - v_2 > limits[1] and v_1 - v_2 < limits[2]. If drop_dimension=T, the new condition is effectively v_1 - v_2 = limits[1].
epsilon	Tolerance for determining which vertices of the convex hull defined by inequality_mat satisfy the altered conditions when drop_dimension is specified.
qhull_options	These are passed to geometry::delaunayn() and geometry::convexhulln(). See QHull documentation.

Details

For estimating pivot event probabilities, the general procedure is to specify a matrix of conditions under which a given candidates wins (inequality_mat) and then specify which of these conditions (rows_to_alter) should be met with near equality (drop_dimension=F) or exact equality (drop_dimension=T). The function could be used for other purposes, however.

Value

An array of matrices, each representing one simplex. Note that each *column* of each matrix is one vertex. (You might expect rows to indicate vertices.)

Examples

# plurality conditions with 3 candidates
n <- 1000
inequality_mat <- rbind(c(1,-1,0,1/n),
                        c(1,0,-1,1/n))
# simplices covering region where candidate 1 wins
S_array_from_inequalities_and_conditions(inequality_mat)
# simplices covering region where candidate 1 finishes just ahead of candidate 2
S_array_from_inequalities_and_conditions(inequality_mat,
       rows_to_alter = c(1), limits = c(0, 1/n))
# simplices covering facet where candidates 1 and 2 tie
S_array_from_inequalities_and_conditions(inequality_mat,
       rows_to_alter = c(1), drop_dimension = T, limits = c(0, 1/n))

aeggers/pivotprobs documentation built on Oct. 28, 2024, 9:46 a.m.