Fit Multi-modal Mallows' models to ranking data.

Description

Fits the Mallows' model to ranking data. Data can be partially or fully-ranked.

Details

Author(s)

Erik Gregory Maintainer: <egregory2007@yahoo.com>

References

"Mixtures of distance-based models for ranking data". Thomas Brendan Murphy & Donal Martin. 1 April 2002. Computational Statistics & Data Analysis 41 (2003) 645-655.

"Estimating a Population Distribution of Sequences of k Items from Cross- Sectional Data". Laurel A. Smith (Beckett) and Denis A. Evans. Journal of the Royal Statistical Society. Series C (Applied Statistics). Vol. 40, No. 1 (1991), pp.31-42. Blackwell Publishing for the Royal Statistical Society. Accessed 16/08/2010. http://www.jstor.org/stable/2347903 .

"A Non-iterative procedure for maximum likelihood estimation of the parameters of Mallows' Model Based on Partial Rankings". Laura Adkins and Michael Flinger. Communication in Statistics - Theory and Methods, 27:9, 2199-2220. 1998, Marchel Dekker, Inc. http://dx.doi.org/10.1080/03610929808832223 .

All Kendall's distances between two sets of rankings.

Description

Calculates all of the Kendall's distances between two different sets of rankings.

Usage

  AllKendall(r, seqs, data.info = NULL)

Arguments

Value

Matrix where output[i, j] represents the distance from sequence "i" in "r" to sequence "j" in "seqs".

Author(s)

Erik Gregory

Examples

data1 <- do.call("rbind", list(1:5, 5:1, c(3, 2, 1, 4, 5)))
data2 <- do.call("rbind", list(1:5, 5:1))
# AllKendall(data1, data2)

Calculate all distances between a set of sequences and a fixed sequence.

Description

Used to calculate the sequence Kendall distance distribution in N! space.

Usage

  AllSeqDists(seqs)

Arguments

Value

Vector of the distances from the sequences to 1:N.

Author(s)

Erik Gregory

Fit Mallows model N times and select most likely model. The EM algorithm to fit Multi-Modal Mallows' models is prone to getting stuck in local maxima, so we run it several times and selec the best one.

Description

Fit Mallows model N times and select most likely model. The EM algorithm to fit Multi-Modal Mallows' models is prone to getting stuck in local maxima, so we run it several times and selec the best one.

Usage

  BestFit(datas, N, iter, G)

Arguments

Value

best fitting model.

Calculate the normalizing coefficient for Mallow's model in a sequence space.

Description

Calculate the normalizing coefficient, as a function of the lambda parameter, and the size of the sequence space.

Usage

  C_lam(lambda, dists = NULL, dists.table = NULL)

Arguments

Value

Normalizing coefficient of Mallows' model in N! space with lambda = lambda.

Author(s)

Erik Gregory

Constructs sequences from Kendall Information matricies.

Description

Sequences in a fully-ordered sequence space have a unique Kendall Information vector associated with them. This function creates the sequence from the Kendall information vector.

Usage

  ConstructSeqs(prefs, n.abils)

Arguments

Value

List of fully-ordered sequences, one for each row of prefs.

Author(s)

Erik Gregory

Examples

ConstructSeqs(matrix(c(1, 1, 1, 0, 0, 0), nrow = 1), 4)
# Should output (4, 1, 2, 3)

Calculate the Kendall distance distribution in N! space.

Description

This function counts the number of fully-ordered vectors at each distance in N! space.

Usage

  DistanceDistribution(N = 3)

Arguments

Value

Table-like structure, where the names represent the distance from the modal sequence of each sequence in N! space, and the values represent the number of sequences at that distance in the sequence space.

Author(s)

Erik Gregory

The Expectation step of the EM algorithm.

Description

Assigns each ranking the probability that it belongs to each cluster, given current parameters.

Usage

  EStep(R, r, p, lambda, G, N, C, all.dists = NULL)

Arguments

Value

Matrix where output[i, j] represents the current probability that subject "i" belongs to cluster "j".

Author(s)

Erik Gregory

References

"Mixtures of distance-based models for ranking data". Thomas Brendan Murphy & Donal Martin. 1 April 2002. Computational Statistics & Data Analysis 41 (2003) 645-655.

Formats the data in the "Solve" function for output.

Description

Data formatting function.

Usage

  FormatOut(R, p, lambda, z, datas, likelihood)

Arguments

Value

Author(s)

Erik Gregory

All information used to calculate Kendall's distance.

Description

Performs each column-wise comparison on a matrix of sequences. A 0 value denotes that there is an increase between the two columns, 1 a decrease, and NA indicates that the column values are identical in the row.

Usage

  KendallInfo(r, inds = NULL)

Arguments

Value

Matrix of 0s, 1s, and NAs representing pairwise comparisons of vector values.

Author(s)

Erik Gregory

References

http://en.wikipedia.org/wiki/Kendall_tau_distance

Objective function to determine lambda.

Description

Objective function to find the root of in calculating the lambda parameters for each cluster.

Usage

  Lambda(lambda, rhs, dists, dists.table = NULL)

Arguments

Value

Output of the objective function to determine the root of. Goal is zero.

Author(s)

Erik Gregory

References

"Mixtures of distance-based models for ranking data". Thomas Brendan Murphy & Donal Martin. 1 April 2002. Computational Statistics & Data Analysis 41 (2003) 645-655.

Likelihood of the data and parameters.

Description

Calculates the log-likelihood of the data with the current parameters and Kendall's distance.

Usage

  Likelihood(z, p, C.lam, lambda, all.dists.data)

Arguments

Value

Current log-likelihood of the data with the current parameters.

Author(s)

Erik Gregory

References

"Mixtures of distance-based models for ranking data". Thomas Brendan Murphy & Donal Martin. 1 April 2002. Computational Statistics & Data Analysis 41 (2003) 645-655.

Fits a Multi-Modal Mallows' model to ranking data.

Description

Fits the Multi-Modal Mallows' model to partial or full ranking data, using Kendall's metric and an EM algorithm. This is essentially metric sequence clustering.

Usage

  Mallows(datas, G, iter = 10, hyp = NULL,
    plot.like = FALSE)

Arguments

Value

See output of FormatOut

Author(s)

Erik Gregory

References

"Mixtures of distance-based models for ranking data". Thomas Brendan Murphy & Donal Martin. 1 April 2002. Computational Statistics & Data Analysis 41 (2003) 645-655.

Calculates the table of Kendall distances in (N+1)! space, given those in N! space.

Description

This is identical to counting the number of fully-ordered vectors at each bubble sort distance in (N+1)! space.

Usage

  NextTable(last.table, N.last)

Arguments

Value

Table of distances in (N+1)! space.

Author(s)

Erik Gregory

Initialize sequence modes for the clustering process.

Description

Initialize sequence modes for the clustering process.

Usage

  Rgen(G, hyp = NULL, abils)

Arguments

Value

A list of G cluster centers, each of length abils

Author(s)

Erik Gregory

Examples

Rgen(3, 1:5, 5)

Calculates distances in N! space.

Description

Calculates Kendall's distances of each sequence in N! space. This is VERY Inefficient for N >= 8. See DistanceDistribution for an astronomical improvement (possibly on the order of 10^10).

Usage

  SeqDistribution(N)

Arguments

Value

Vector of Kendall distances from 1:N to each sequence in N! space.

Author(s)

Erik Gregory

Change the form of ordered sequences.

Description

Simplifies sequences so that each tie group is only of distance 1 to the next tie group. For example, we would simplify (1, 1, 2, 4, 4, 5) to (1, 1, 2, 3, 3, 4).

Usage

  SimplifySequences(loss.time)

Arguments

Value

Simplified sequences, as described in Description.

Author(s)

Erik Gregory

Update the Lambda parameters of clusters.

Description

Updates the Lambda parameters to maximize the likelihood of the data under Mallows' model.

Usage

  UpdateLambda(r, R, z, G, dists.to.Rg, dists.table,
    top.bound = 1000)

Arguments

Value

Vector of new lambda parameters for the clusters.

Author(s)

Erik Gregory

References

"Mixtures of distance-based models for ranking data". Thomas Brendan Murphy & Donal Martin. 1 April 2002. Computational Statistics & Data Analysis 41 (2003) 645-655.

Update Proportion in each cluster.

Description

Updates the proportion of data assigned to each cluster.

Usage

  UpdateP(z)

Arguments

Value

Proportion of data in each cluster.

Author(s)

Erik Gregory

References

"Mixtures of distance-based models for ranking data". Thomas Brendan Murphy & Donal Martin. 1 April 2002. Computational Statistics & Data Analysis 41 (2003) 645-655.

Update modal sequences in each cluster.

Description

Maximizes the likelihood of the data by updating the cluster centers of the model.

Usage

  UpdateR(r, z, infos = NULL)

Arguments

Value

New cluster centers for each cluster.

Author(s)

Erik Gregory

References

"Mixtures of distance-based models for ranking data". Thomas Brendan Murphy & Donal Martin. 1 April 2002. Computational Statistics & Data Analysis 41 (2003) 645-655.

Sample data set.

Description

Simple synthetic data set containing 3 modal sequences in 15! space, with some noise added.

Format

The format is: num [1:1700, 1:15] 1 15 1 15 15 12 10 4 1 15 ...

Examples

data(datas)
head(datas)

1980 APA Presidential Candidate ranking data.

Description

This data is a pre-processed version of the 1980 American Psychological Association Presidential candidate ranking data. It has uninformative rankings removed, and values pre-simplified into partial rankings.

Format

The format is: int [1:1378, 1:3] 1 1 1 1 2 2 1 1 2 2 ... - attr(*, "dimnames")=List of 2 ..$ : chr [1:1378] "1" "2" "3" "6" ... ..$ : chr [1:3] "Carter" "Reagan" "Anderson"

Source

The American Psychological Association, http://www.electionstudies.org/studypages/1980prepost/1980prepost.htm

Examples

data(elect)
head(elect)

Hello, World!

Description

Prints 'Hello, world!'.

Usage

hello()

Examples

hello()

Fitted version of the toy datas data set, with three modal sequences.

Description

The data has 3 modal sequences, and we can compare this to the two.mode data set.

Format

The format is: List of 5 $ R :List of 3 ..$ : int [1:15] 1 2 3 4 5 6 7 8 9 10 ... ..$ : int [1:15] 1 3 5 7 9 2 4 6 8 10 ... ..$ : int [1:15] 15 14 13 12 11 10 9 8 7 6 ... $ p : num [1:3] 0.447 0.118 0.435 $ lambda : num [1:3] 2.01 1000 2.04 $ datas :'data.frame': 1700 obs. of 23 variables: ..$ X1 : num [1:1700] 1 15 1 15 15 12 10 4 1 15 ... ..$ X2 : num [1:1700] 2 14 2 14 14 13 13 12 2 14 ... ..$ X3 : num [1:1700] 3 13 3 13 13 2 4 6 3 13 ... ..$ X4 : num [1:1700] 4 12 4 12 12 8 7 1 4 12 ... ..$ X5 : num [1:1700] 5 11 5 11 11 9 14 5 5 11 ... ..$ X6 : num [1:1700] 6 10 6 10 10 1 8 10 6 10 ... ..$ X7 : num [1:1700] 7 9 7 9 9 15 1 13 7 9 ... ..$ X8 : num [1:1700] 8 8 8 8 8 10 9 9 8 8 ... ..$ X9 : num [1:1700] 9 7 9 7 7 6 5 14 9 7 ... ..$ X10 : num [1:1700] 10 6 10 6 6 11 11 8 10 6 ... ..$ X11 : num [1:1700] 11 5 11 5 5 3 15 2 11 5 ... ..$ X12 : num [1:1700] 12 4 12 4 4 14 12 11 12 4 ... ..$ X13 : num [1:1700] 13 3 13 3 3 7 2 7 13 3 ... ..$ X14 : num [1:1700] 14 2 14 2 2 5 3 15 14 2 ... ..$ X15 : num [1:1700] 15 1 15 1 1 4 6 3 15 1 ... ..$ clust : int [1:1700] 1 3 1 3 3 3 3 1 1 3 ... ..$ pvals.1: num [1:1700] 1.00 1.03e-91 1.00 2.04e-93 1.03e-91 ... ..$ pvals.2: num [1:1700] 0 0 0 0 0 0 0 0 0 0 ... ..$ pvals.3: num [1:1700] 1.02e-92 1.00 1.34e-93 1.00 1.00 ... ..$ seq : Factor w/ 3 levels "1 2 3 4 5 6 7 8 9 10 11 12 13 14 15",..: 1 3 1 3 3 3 3 1 1 3 ... ..$ dists.1: num [1:1700] 0 105 0 105 105 61 58 46 0 105 ... ..$ dists.2: num [1:1700] 10 95 10 95 95 61 54 54 10 95 ... ..$ dists.3: num [1:1700] 105 0 105 0 0 44 47 59 105 0 ... $ min.like: num [1:100] -122710 -51439 -50310 -49976 -49718 ...

Examples

data(three.mode)
head(three.mode[[4]])

Two-mode Mallows' model fit to toy data set "datas"

Description

"datas" has 3 modes, but we observe here what happens when we try to fit it with 2 modal sequences. The most prominent modal sequences are 1:15, 15:1

Format

The format is: List of 5 $ R :List of 2 ..$ : int [1:15] 1 2 3 4 5 6 7 8 9 10 ... ..$ : int [1:15] 15 14 13 12 11 10 9 8 7 6 ... $ p : num [1:2] 0.557 0.443 $ lambda : num [1:2] 2.05 2.02 $ datas :'data.frame': 1700 obs. of 21 variables: ..$ X1 : num [1:1700] 1 15 1 15 15 12 10 4 1 15 ... ..$ X2 : num [1:1700] 2 14 2 14 14 13 13 12 2 14 ... ..$ X3 : num [1:1700] 3 13 3 13 13 2 4 6 3 13 ... ..$ X4 : num [1:1700] 4 12 4 12 12 8 7 1 4 12 ... ..$ X5 : num [1:1700] 5 11 5 11 11 9 14 5 5 11 ... ..$ X6 : num [1:1700] 6 10 6 10 10 1 8 10 6 10 ... ..$ X7 : num [1:1700] 7 9 7 9 9 15 1 13 7 9 ... ..$ X8 : num [1:1700] 8 8 8 8 8 10 9 9 8 8 ... ..$ X9 : num [1:1700] 9 7 9 7 7 6 5 14 9 7 ... ..$ X10 : num [1:1700] 10 6 10 6 6 11 11 8 10 6 ... ..$ X11 : num [1:1700] 11 5 11 5 5 3 15 2 11 5 ... ..$ X12 : num [1:1700] 12 4 12 4 4 14 12 11 12 4 ... ..$ X13 : num [1:1700] 13 3 13 3 3 7 2 7 13 3 ... ..$ X14 : num [1:1700] 14 2 14 2 2 5 3 15 14 2 ... ..$ X15 : num [1:1700] 15 1 15 1 1 4 6 3 15 1 ... ..$ clust : int [1:1700] 1 2 1 2 2 2 2 1 1 2 ... ..$ pvals.1: num [1:1700] 1.00 4.15e-94 1.00 4.15e-94 4.15e-94 ... ..$ pvals.2: num [1:1700] 5.4e-93 1.0 5.4e-93 1.0 1.0 ... ..$ seq : Factor w/ 2 levels "1 2 3 4 5 6 7 8 9 10 11 12 13 14 15",..: 1 2 1 2 2 2 2 1 1 2 ... ..$ dists.1: num [1:1700] 0 105 0 105 105 61 58 46 0 105 ... ..$ dists.2: num [1:1700] 105 0 105 0 0 44 47 59 105 0 ... $ min.like: num [1:100] -178063 -139298 -58290 -54074 -53902 ...

Examples

data(two.mode)
head(two.mode[[4]])

Bi-modal Mallow's model fit to the APA data set.

Description

The two-modes seem to divide well between Democrats and Republicans...

Format

The format is: List of 5 $ R :List of 2 ..$ : int [1:3] 1 3 2 ..$ : int [1:3] 3 1 2 $ p : num [1:2] 0.541 0.459 $ lambda : num [1:2] 2.19 2.32 $ datas :'data.frame': 1378 obs. of 9 variables: ..$ Carter : int [1:1378] 1 1 1 1 2 2 1 1 2 2 ... ..$ Reagan : int [1:1378] 1 2 2 2 1 1 2 3 1 1 ... ..$ Anderson: int [1:1378] 1 2 2 3 3 3 3 2 3 3 ... ..$ clust : int [1:1378] 1 1 1 1 2 2 1 1 2 2 ... ..$ pvals.1 : num [1:1378] 0.541 0.992 0.992 0.932 0.131 ... ..$ pvals.2 : num [1:1378] 0.45893 0.00809 0.00809 0.06802 0.86945 ... ..$ seq : Factor w/ 2 levels "1 3 2","3 1 2": 1 1 1 1 2 2 1 1 2 2 ... ..$ dists.1 : num [1:1378] 0 0 0 1 2 2 1 0 2 2 ... ..$ dists.2 : num [1:1378] 0 2 2 2 1 1 2 3 1 1 ... $ min.like: num [1:100] -6421 -3386 -2916 -2811 -2799 ...

Source

American Psychological Association http://www.electionstudies.org/studypages/anes_mergedfile_1980/anes_mergedfile_1980.htm

Examples

data(two.seq)
head(two.seq[[4]])