subit_convergence: Subiterative Convergence

In RankAggSIgFUR: Polynomially Bounded Rank Aggregation under Kemeny's Axiomatic Approach

Subiterative Convergence

Description

Subiterative Convergence finds the consensus ranking by iteratively applying the Modified Kemeny algorithm on smaller number of objects, \eta. Starting with a given seed ranking, the consensus ranking is obtained when the algorithm converges.

Usage

subit_convergence(
  eta,
  seed_rkg,
  input_rkgs,
  universe_rkgs = c(),
  objNames = c(),
  wt = c()
)

Arguments

eta	a subiteration length of number of objects to consider in the smaller subset. Recommended eta values are between 2 and 8. Smaller eta values result in shorter run-time.
seed_rkg	an initial ranking to start the algorithm. An ideal seed ranking for Subiterative Convergence is the mean seed ranking of input rankings.
input_rkgs	a n by k matrix of k rankings of n objects, where each column is a complete ranking.
universe_rkgs	a matrix containing all possible permutations of ranking n objects. Each column in this matrix represents one permuted ranking. An optional parameter.
objNames	a n-length vector containing object names. An optional parameter.
wt	a k-length vector containing weights for each judge or attribute. An optional parameter.

Value

A list containing the consensus ranking (expressed as ordering), total Kemeny distance, and average tau correlation coefficient corresponding to the consensus ranking.

References

Badal, P. S., & Das, A. (2018). Efficient algorithms using subiterative convergence for Kemeny ranking problem. Computers & Operations Research, 98, 198-210. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.cor.2018.06.007")}

See Also

mod_kemeny, fur, sigfur, mean_seed

Examples

## Four input rankings of five objects
eta <- 3
seed_rkg <- c(1, 2, 3, 4, 5)
input_rkgs <- matrix(c(3, 2, 5, 4, 1, 2, 3, 1, 5, 4, 5, 1, 3, 4, 2, 1, 2, 4, 5, 3),
    byrow = FALSE, ncol = 4)
subit_convergence(eta, seed_rkg, input_rkgs) # Determined the consensus ranking, total Kemeny
                                             # distance, and average tau correlation coefficient

## Example with eta=1
eta <- 1
seed_rkg <- c(1, 2, 3, 4, 5)
input_rkgs <- matrix(c(3, 2, 5, 4, 1, 2, 3, 1, 5, 4, 5, 1, 3, 4, 2, 1, 2, 4, 5, 3),
    byrow = FALSE, ncol = 4)
subit_convergence(eta, seed_rkg, input_rkgs) # Shows a warning and returns seed ranking

## Five input rankings with five objects
## 2nd ranking == 3rd ranking, so if a third object is weighted as zero,
## we should get the same answer as the first examples
input_rkgs <- matrix(c(3, 2, 5, 4, 1, 2, 3, 1, 5, 4, 2, 3, 1, 5, 4, 5, 1, 3, 4, 2, 1, 
                       2, 4, 5, 3),byrow = FALSE, ncol = 5)
eta <- 3
seed_rkg <- c(1, 2, 3, 4, 5)
wt = c(1,1,0,1,1)
subit_convergence(eta, seed_rkg, input_rkgs, wt=wt) # Determined the consensus ranking, total Kemeny
                                             # distance, and average tau correlation coefficient

## Using five input rankings with five objects with prepare_data to 
## automatically prepare the weight vector
input_rkgs <- matrix(c(3, 2, 5, 4, 1, 2, 3, 1, 5, 4, 2, 3, 1, 5, 4, 5, 1, 3, 4, 2, 1, 
                       2, 4, 5, 3),byrow = FALSE, ncol = 5)
out = prepare_data(input_rkgs) 
input_rkgs = out$input_rkgs
wt = out$wt
eta <- 3
seed_rkg <- c(1, 2, 3, 4, 5)
subit_convergence(eta, seed_rkg, input_rkgs, wt=wt) # Determined the consensus ranking, total Kemeny
                                             # distance, and average tau correlation coefficient

## Included dataset of 15 input rankings of 50 objects
data(data50x15)
input_rkgs <- as.matrix(data50x15[, -1])
mean_seed_rkg <- mean_seed(t(input_rkgs)) # Use the mean seed ranking as the seed ranking
eta <- 2
subit_convergence(eta, seed_rkg = mean_seed_rkg, input_rkgs)

RankAggSIgFUR documentation built on July 9, 2023, 7:26 p.m.