| Title: | Classification and Clustering of Preference Rankings |
|---|---|
| Description: | Tree-based classification and soft-clustering method for preference rankings, with tools for external validation of fuzzy clustering. It contains the recursive partitioning algorithm for preference rankings, non-parametric tree-based method for a matrix of preference rankings as a response variable. It contains also the distribution-free soft clustering method for preference rankings, namely the K-median cluster component analysis (CCA). The package depends on the 'ConsRank' R package. Options for validate the tree-based method are both test-set procedure and V-fold cross validation. The package contains the routines to compute the adjusted concordance index (a fuzzy version of the adjusted rand index) and the normalized degree of concordance (the corresponding fuzzy version of the rand index). Essential references: D'Ambrosio, A., Amodio, S., Iorio, C., Pandolfo, G., and Siciliano, R. (2021) <doi:10.1007/s00357-020-09367-0> D'Ambrosio, A., and Heiser, W.J. (2019) <doi:10.1007/s41237-018-0069-5>; D'Ambrosio, A., and Heiser W.J. (2016) <doi:10.1007/s11336-016-9505-1>; Hullermeier, E., Rifqi, M., Henzgen, S., and Senge, R. (2012) <doi:10.1109/TFUZZ.2011.2179303>. |
| Authors: | Antonio D'Ambrosio [aut, cre] |
| Maintainer: | Antonio D'Ambrosio <[email protected]> |
| License: | GPL-3 |
| Version: | 1.0.1 |
| Built: | 2024-11-25 06:45:12 UTC |
| Source: | CRAN |
K-Median Cluster Component Analysis, a distribution-free soft-clustering method for preference rankings.
cca(X, k, control = ccacontrol(...), ...)cca(X, k, control = ccacontrol(...), ...)
X |
A n by m data matrix containing preference rankings, in which there are n judges and m objects to be judged. Each row is a ranking of the objects which are represented by the columns. |
k |
The number of cluster components |
control |
a list of options that control details of the |
... |
arguments passed bypassing |
The user can use any algorithm implemented in the consrank function from the ConsRank package. All algorithms allow the user to set the option 'full=TRUE'
if the median ranking(s) must be searched in the restricted space of permutations instead of in the unconstrained universe of rankings of n items including all possible ties.
There are two classification uncertainty measures: Us and Uprods. "Us" is the geometric
mean of the membership probabilities of each individual, normalized in such a way that
in the case of maximum uncertainty Us=1. "Ucca" is the average of all the "Us".
"Uprods" is the product of the membership probabilities of each individual, normalized in such a way that
in the case of maximum uncertainty Uprods=1. "Uprodscca" is the average of all the "Uprods".
An object of the class "cca". It contains:
| pk | the membership probability matrix | |
| clc | cluster centers | |
| oclc | cluster centers in terms of orderings | |
| idc | crisp partition: id of the cluster component associated with the highest membership probability | |
| Hcca | Global homogeneity measure (tau_X rank correlation coefficient) | |
| hk | Homogeneity within cluster | |
| props | estimated proportion of cases within cluster | |
| Us | Uncertainty measure per-individual (see details) | |
| Ucca | Global uncertainty measure | |
| Uprods | Uncertainty measure per-individual (see details) | |
| Uprodscca | Global uncertainty measure | |
| consrankout | complete output of rank aggregation algorithm, containing eventually multiple median rankings |
Antonio D'Ambrosio [email protected]
D'Ambrosio, A. and Heiser, W.J. (2019). A Distribution-free Soft Clustering Method for Preference Rankings. Behaviormetrika , vol. 46(2), pp. 333–351, DOI: 10.1007/s41237-018-0069-5
Heiser W.J., and D'Ambrosio A. (2013). Clustering and Prediction of Rankings within a Kemeny Distance Framework. In Berthold, L., Van den Poel, D, Ultsch, A. (eds). Algorithms from and for Nature and Life.pp-19-31. Springer international. DOI: 10.1007/978-3-319-00035-0_2.
Ben-Israel, A., and Iyigun, C. (2008). Probabilistic d-clustering. Journal of Classification, 25(1), pp.5-26. DOI: 10.1007/s00357-008-9002-z
ccacontrol
ranktree
data(Irish) set.seed(135) #for reproducibility # CCA with four components ccares <- cca(Irish$rankings, 4, itercca=10) summary(ccares)data(Irish) set.seed(135) #for reproducibility # CCA with four components ccares <- cca(Irish$rankings, 4, itercca=10) summary(ccares)
Utility function to use to set the control arguments of cca
ccacontrol( algorithm = "quick", full = FALSE, itercca = 1, consrankitermax = 10, np = 15, gl = 100, ff = 0.4, cr = 0.9, proc = FALSE, ps = FALSE )ccacontrol( algorithm = "quick", full = FALSE, itercca = 1, consrankitermax = 10, np = 15, gl = 100, ff = 0.4, cr = 0.9, proc = FALSE, ps = FALSE )
algorithm |
The algorithm used to compute the median ranking. One among"BB", "quick" (default), "fast" and "decor" |
full |
Specifies if the median ranking must be searched in the universe of rankings including all the possible ties. Default: FALSE |
itercca |
Number of iterations of cca |
consrankitermax |
Number of iterations for "fast" and "decor" algorithms. itermax=10 is the default option. |
np |
(for "decor" only) the number of population individuals. np=15 is the default option. |
gl |
(for"decor" only) generations limit, maximum number of consecutive generations without improvement. gl=100 is the default option. |
ff |
(for"decor" only) the scaling rate for mutation. Must be in [0,1]. ff=0.4 is the default option. |
cr |
(for"decor" only) the crossover range. Must be in [0,1]. cr=0.9 is the default option. |
proc |
(for "BB" only) proc=TRUE allows the branch and bound algorithm to work in difficult cases, i.e. when the number of objects is larger than 15 or 25. proc=FALSE is the default option |
ps |
If PS=TRUE, on the screen some information about how many branches are processed are displayed. Default value: FALSE |
A list containing all the control parameters
Antonio D'Ambrosio [email protected]
Random sub-sample of 3584 cases of the survey conducted in 1999 in 32 countries analyzed by Vermunt (2003).
data("EVS")data("EVS")
The format is: List of 3
$ data:'data.frame': 1911 obs. of 11 variables:
country, gender ,yearbird, mstatus (marital status), eduage (age of education completion), employment (Employment status: ordinal scale 1-8), householdinc (Household income: ordinal scale 1-10), A (Maintain order in Nation),Give people more say in Government decisions, (C) Fight rising prices, (D) Protect freedom of speech.
$ predictors:'data.frame' with all the predictors
$ rankings : matrix with the preferencres for "A" (Maintain order in Nation), "B" (Give people more say in Government decisions), "C" (Fight rising prices), "D" (Protect freedom of speech).
Rankings were obtained by applying the post-materialism scale developed by Inglehart (1977). The scale is based upon an experiment of the type “pick 2 out of 4” most important political goals for your Governments. For this reason, replace the 'NA's with 3 before using the rankings with codes 'ranktree' or 'cca' (see D'Ambrosio and Heiser, 2016). About the predictors, the coding of the Countries are: G1 (Austria, Denmark, Netherlands, Sweden), G2 (Belgium, Croatia, France, Greece, Ireland, Northern Ireland, Spain), G3 (Bulgaria, Czechnia, East, Germany, Finland, Iceland, Luxembourg, Malta, Portugal, Romania, Slovenia, West Germany), G4 (Belarus, Estonia, Hungary, Latvia, Lithuania, Poland, Russia, Slovakia, Ukraine). Coding of predictor "mstatus" are: mar (married), wid (widowed), div (divorced), sep (separated), nevm (never married).
http://statisticalinnovations.com/technicalsupport/choice_datasets.html
Vermunt, J. K. (2003). Multilevel latent class models. Sociological Methodology, 33(1), 213–239.
Inglehart, R. (1977). The silent revolution: Changing values and political styles among Western Publics. Princeton, NJ: Princeton University Press.
D'Ambrosio, A., and Heiser W.J. (2016). A recursive partitioning method for the prediction of preference rankings based upon Kemeny distances. Psychometrika, vol. 81 (3), pp.774-94.
data(EVS) # EVS$rankings[is.na(EVS$rankings)] <- 3 #place unranked objects in a tie to the third position # ccares <- cca(EVS$rankings,4) #solution with 4 componentsdata(EVS) # EVS$rankings[is.na(EVS$rankings)] <- 3 #place unranked objects in a tie to the third position # ccares <- cca(EVS$rankings,4) #solution with 4 components
Given two fuzzy (Ruspini) partitions, it compute the NDC and the ACI. NDC is the fuzzy version of the Rand Index, as well as ACI is the fuzzy version of the Adjusted Rand Index
fuzzyconcordance(P, Q, nperms = 1000)fuzzyconcordance(P, Q, nperms = 1000)
P |
A fuzzy partition. It has to be a matrix with n rows and k columns. Each column is expression of the degree of membership of the i-th row over the k partitions (see details). |
Q |
A fuzzy partition. It has to be a matrix with n rows and h columns. Each column is expression of the degree of membership of the i-th row over the h partitions (see details). |
nperms |
number of permutations necessary to compute ACI. Default: 1000 |
Both P and Q, or only one of those, can be crisp (or hard) partitions. In this case, each row must contain either 0 or 1, and the sum of the i-th row must be 1. In other words, either P or Q (or both) are expressed in terms of dummy coding. If both partitions are crisp, then NDC is equal to Rand Index and ACI is equal to Adjusted Rand Index. This function can be used to externally validate the output of any fuzzy clustering method
A list containing:
| ACI | the Adjusted Concordance Index | |
| NDC | the Normalized Degree of Concordance |
Antonio D'Ambrosio [email protected]
D’Ambrosio, A., Amodio, S., Iorio, C., Pandolfo, G. and Siciliano, R. (2021). Adjusted Concordance Index: an Extension of the Adjusted Rand Index to Fuzzy Partitions. Journal of Classification vol. 38(1), pp. 112–128 (2021). DOI: 10.1007/s00357-020-09367-0
Hullermeier, E., Rifqi, M., Henzgen, S., and Senge, R. (2012). Comparing fuzzy partitions: a generalization of the Rand index and related measures. IEEE Transactions on Fuzzy Systems, 20(3), 546–556. DOI: 10.1109/TFUZZ.2011.2179303
#two random fuzzy partitions P = rbind(c(0.5259, 0.1656, 0.3085), c(0.5623, 0.1036, 0.3341), c(0.2508, 0.1849, 0.5643), c(0.5654, 0.1934, 0.2413), c(0.4529, 0.1679, 0.3792), c(0.2390, 0.1758, 0.5852), c(0.3114, 0.1743, 0.5143), c(0.4188, 0.1392, 0.4420), c(0.5830, 0.1655, 0.2514), c(0.5860, 0.1171, 0.2969), c(0.2630, 0.1706, 0.5664), c(0.5882, 0.1032, 0.3086), c(0.5829, 0.1277, 0.2894), c(0.3942, 0.1046, 0.5012), c(0.5201, 0.1097, 0.3702), c(0.2568, 0.1823, 0.5609), c(0.3687, 0.1695, 0.4618), c(0.5663, 0.1317, 0.3020), c(0.5169, 0.1950, 0.2881), c(0.5838, 0.1034, 0.3128)) Q = rbind(c(0.4494, 0.3755, 0.1751), c(0.5219, 0.3526, 0.1255), c(0.3432, 0.5062, 0.1506), c(0.3120, 0.5181, 0.1699), c(0.5362, 0.2747, 0.1891), c(0.4082, 0.3959, 0.1959), c(0.4670, 0.3782, 0.1547), c(0.4276, 0.4585, 0.1139), c(0.4013, 0.4837, 0.1149), c(0.3724, 0.5019, 0.1258), c(0.5055, 0.3104, 0.1841), c(0.4027, 0.4719, 0.1254), c(0.3565, 0.4620, 0.1814), c(0.6106, 0.2650, 0.1244), c(0.5595, 0.2476, 0.1929), c(0.4657, 0.3993, 0.1350), c(0.2964, 0.5839, 0.1197), c(0.5387, 0.3362, 0.1251), c(0.4043, 0.4341, 0.1616), c(0.5631, 0.2895, 0.1473)) ci <- fuzzyconcordance(P,Q) #generate a random fuzzy partition with two components (clusters) Q2 <- matrix(runif(20),ncol=1) Q2 <- cbind(Q2,1-Q2) ci2 <- fuzzyconcordance(P,Q2) #generate a random crisp partition P2 <- t(rmultinom(20,1,c(0.3,0.3,0.4))) ci3 <- fuzzyconcordance(P2,Q) #-------------------- ## Not run: # install.packages("Rankcluster") library("Rankcluster") # model-based clustering algorithm for # ranking data by Biernacki and Jacques (2013) # <doi:10.1016/j.csda.2012.08.008> data(APA) set.seed(136) #for reproducibility rcres <- rankclust(APA$data,K=3) # solution with 3 centers, it takes about 75 seconds ## ccares <- cca(APA$data,k=3) #solution with 3 components, it takes about 7 seconds ## ci <- fuzzyconcordance(rcres[3]@tik,ccares$pk) ci$ACI # 0.0226 means that the two partitions are similar (see NDC below), # but their similarity is mainly due to chance ci$NDC ## End(Not run)#two random fuzzy partitions P = rbind(c(0.5259, 0.1656, 0.3085), c(0.5623, 0.1036, 0.3341), c(0.2508, 0.1849, 0.5643), c(0.5654, 0.1934, 0.2413), c(0.4529, 0.1679, 0.3792), c(0.2390, 0.1758, 0.5852), c(0.3114, 0.1743, 0.5143), c(0.4188, 0.1392, 0.4420), c(0.5830, 0.1655, 0.2514), c(0.5860, 0.1171, 0.2969), c(0.2630, 0.1706, 0.5664), c(0.5882, 0.1032, 0.3086), c(0.5829, 0.1277, 0.2894), c(0.3942, 0.1046, 0.5012), c(0.5201, 0.1097, 0.3702), c(0.2568, 0.1823, 0.5609), c(0.3687, 0.1695, 0.4618), c(0.5663, 0.1317, 0.3020), c(0.5169, 0.1950, 0.2881), c(0.5838, 0.1034, 0.3128)) Q = rbind(c(0.4494, 0.3755, 0.1751), c(0.5219, 0.3526, 0.1255), c(0.3432, 0.5062, 0.1506), c(0.3120, 0.5181, 0.1699), c(0.5362, 0.2747, 0.1891), c(0.4082, 0.3959, 0.1959), c(0.4670, 0.3782, 0.1547), c(0.4276, 0.4585, 0.1139), c(0.4013, 0.4837, 0.1149), c(0.3724, 0.5019, 0.1258), c(0.5055, 0.3104, 0.1841), c(0.4027, 0.4719, 0.1254), c(0.3565, 0.4620, 0.1814), c(0.6106, 0.2650, 0.1244), c(0.5595, 0.2476, 0.1929), c(0.4657, 0.3993, 0.1350), c(0.2964, 0.5839, 0.1197), c(0.5387, 0.3362, 0.1251), c(0.4043, 0.4341, 0.1616), c(0.5631, 0.2895, 0.1473)) ci <- fuzzyconcordance(P,Q) #generate a random fuzzy partition with two components (clusters) Q2 <- matrix(runif(20),ncol=1) Q2 <- cbind(Q2,1-Q2) ci2 <- fuzzyconcordance(P,Q2) #generate a random crisp partition P2 <- t(rmultinom(20,1,c(0.3,0.3,0.4))) ci3 <- fuzzyconcordance(P2,Q) #-------------------- ## Not run: # install.packages("Rankcluster") library("Rankcluster") # model-based clustering algorithm for # ranking data by Biernacki and Jacques (2013) # <doi:10.1016/j.csda.2012.08.008> data(APA) set.seed(136) #for reproducibility rcres <- rankclust(APA$data,K=3) # solution with 3 centers, it takes about 75 seconds ## ccares <- cca(APA$data,k=3) #solution with 3 components, it takes about 7 seconds ## ci <- fuzzyconcordance(rcres[3]@tik,ccares$pk) ci$ACI # 0.0226 means that the two partitions are similar (see NDC below), # but their similarity is mainly due to chance ci$NDC ## End(Not run)
Given a tree belonging to the class "ranktree", determine a subtree with a given number of terminal nodes
getsubtree(Tree, cut, tokeep = NULL)getsubtree(Tree, cut, tokeep = NULL)
Tree |
An object of the class "ranktree" coming form te function |
cut |
The maximum number of terminal nodes that the Tree must have |
tokeep |
parameter invoked by other internal functions |
If the pruning sequence returns a series of subtrees with, say, 1,2,4,7,9 terminal nodes and the user set cut=8, the function extract the subtree with 7 terminal nodes.
An object of the class "ranktree", containing the same information of the output of the function ranktree
Antonio D'Ambrosio [email protected]
data("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) #see how many terminal nodes have the trees compomimg the nested sequence of subtrees infoprun <- tree$pruneinfo$termnodes #select the tree with, say, 6 terminal nodes tree6 <- getsubtree(tree,6)data("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) #see how many terminal nodes have the trees compomimg the nested sequence of subtrees infoprun <- tree$pruneinfo$termnodes #select the tree with, say, 6 terminal nodes tree6 <- getsubtree(tree,6)
An opinion poll conducted by Irish Marketing Surveys one month prior to the election in 1997. Interviews were conducted on about 1100 respondents, drawn from 100 sampling areas. Interviews took place at randomly located homes, with respondents selected according to a socioeconomic quota. A range of sociological questions was asked of each respondent, as was their voting preference, if any, for each of the candidates.
data("Irish")data("Irish")
The format is: List of 3
$ IrishElection: 'data.frame': 1083 obs. of 11 variables: Gender (male, housewife, nonhousewife), marital status (single, married, separated), age, socialclass (five unordered categories), Area (rural, city, town), government satisfaction (no opinion,m satisfied, dissatisfied), Bano , Roch, McAl, Nall, Scal
$ predictors :'data.frame' with all the predictors
$ rankings : matrix with the preferencres for "Bano" "Roch" "McAl" "Nall"
In the original version of the data, the ranking matrix contains NAs. Here, NAs are replaced with the number 7, to indicate that all the non-stated preferences are in a tie at the last position (see D'Ambrosio and Heiser, 2016). For details about the data set see Gormley and Murphy, 2008.
https://projecteuclid.org/journals/annals-of-applied-statistics/volume-2/issue-4/A-mixture-of-experts-model-for-rank-data-with/10.1214/08-AOAS178.full?tab=ArticleLinkSupplemental
Gormley, I.C., and Murphy, T.B. (2008). A mixture of experts model for rank data withapplications in election studies. Annals of Applied Statistics 2(4): 1452-1477. DOI: 10.1214/08-AOAS178
D'Ambrosio, A., and Heiser W.J. (2016). A recursive partitioning method for the prediction of preference rankings based upon Kemeny distances. Psychometrika, vol. 81 (3), pp.774-94. DOI: 10.1007/s11336-016-9505-1.
data(Irish)data(Irish)
A utility function completing the output of the function ranktree.
layouttree(Tree)layouttree(Tree)
Tree |
an object of the class "ranktree" |
an object of the class "ranktree" completing the output of the function ranktree
Antonio D'Ambrosio [email protected]
Given an object of the class "ranktree", it visualize the path leading to the terminal node
nodepath(termnode, Tree)nodepath(termnode, Tree)
termnode |
The terminal node of which the path has to be extracted |
Tree |
An object of the class "ranktree" |
The path leading to the terminal node
Antonio D'Ambrosio [email protected]
ranktree, treepaths, getsubtree
data(Irish) #build the tree with default options tree <- ranktree(Irish$rankings,Irish$predictors) #get information about all the paths leading to terminal nodes paths <- treepaths(tree) #see the path for terminal node number 8 nodepath(termnode=8,tree)data(Irish) #build the tree with default options tree <- ranktree(Irish$rankings,Irish$predictors) #get information about all the paths leading to terminal nodes paths <- treepaths(tree) #see the path for terminal node number 8 nodepath(termnode=8,tree)
Plot the tree coming from the ranktree or the pruning sequence of the ranktree
## S3 method for class 'ranktree' plot( x, plot.type = "tree", dispclass = FALSE, valtree = NULL, taos = TRUE, ... )## S3 method for class 'ranktree' plot( x, plot.type = "tree", dispclass = FALSE, valtree = NULL, taos = TRUE, ... )
x |
An object of the class "ranktree" |
plot.type |
One among "tree" or "pruningseq" |
dispclass |
Display the median ranking above terminal nodes. Default option: FALSE |
valtree |
If plot.type="pruningseq", it shows the Tau_x rank correlation coefficient or the error along the pruning sequence on the training set. If valtree is the output of the function |
taos |
If plot.type="pruningseq", it plots the Tau_x rank correlation coefficient along the pruning sequence. If taos=FALSE, it plots the error. |
... |
System reserved (No specific usage) |
the plot of either the tree or the pruning sequence
Antonio D'Ambrosio [email protected]
data("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) plot(tree,dispclass=TRUE) data(EVS) EVS$rankings[is.na(EVS$rankings)] <- 3 set.seed(654) training=sample(1911,1434) tree <- ranktree(EVS$rankings[training,],EVS$predictors[training,],decrmin=0.001,num=50) plot(tree,dispclass=TRUE) #test set validation vtreetest <- validatetree(tree,testX=EVS$predictors[-training,],EVS$rankings[-training,]) dtree <- getsubtree(tree,vtreetest$best_tau) plot(dtree,dispclass=TRUE) #see the global weigthted tau_X rank correlation coefficients plot(tree,plot.type="pruningseq",valtree=vtreetest) #see the error rates plot(tree,plot.type="pruningseq",valtree=vtreetest, taos=FALSE)data("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) plot(tree,dispclass=TRUE) data(EVS) EVS$rankings[is.na(EVS$rankings)] <- 3 set.seed(654) training=sample(1911,1434) tree <- ranktree(EVS$rankings[training,],EVS$predictors[training,],decrmin=0.001,num=50) plot(tree,dispclass=TRUE) #test set validation vtreetest <- validatetree(tree,testX=EVS$predictors[-training,],EVS$rankings[-training,]) dtree <- getsubtree(tree,vtreetest$best_tau) plot(dtree,dispclass=TRUE) #see the global weigthted tau_X rank correlation coefficients plot(tree,plot.type="pruningseq",valtree=vtreetest) #see the error rates plot(tree,plot.type="pruningseq",valtree=vtreetest, taos=FALSE)
Predict the median rankings in a tree-based structure built with ranktree for new observations
## S3 method for class 'ranktree' predict(object, newx, ...)## S3 method for class 'ranktree' predict(object, newx, ...)
object |
An object of the class "ranktree" |
newx |
A dataframe of the same nature of the predictor dataframe with which the tree has been built |
... |
System reserved (No specific usage) |
A list containing:
| rankings | the fit in terms of rankings | |
| orderings | the fit in terms of orderings | |
| info | dataframe containing the terminal nodes in which the new x fall down, then the new x and the fit (in terms of rankings) |
Antonio D'Ambrosio [email protected]
data(EVS) EVS$rankings[is.na(EVS$rankings)] <- 3 set.seed(654) training=sample(1911,1434) tree <- ranktree(EVS$rankings[training,],EVS$predictors[training,],decrmin=0.001,num=50) #use the function predict ro predict rankings for new predictors rankfit <- predict(tree,newx=EVS$predictors[-training,]) #fit in terms of rankings rankfit$rankings #fit in terms of orderings rankfit$orderings # information about the fit (terminal node, predictor and fit (in terms of rankings)) rankfit$infodata(EVS) EVS$rankings[is.na(EVS$rankings)] <- 3 set.seed(654) training=sample(1911,1434) tree <- ranktree(EVS$rankings[training,],EVS$predictors[training,],decrmin=0.001,num=50) #use the function predict ro predict rankings for new predictors rankfit <- predict(tree,newx=EVS$predictors[-training,]) #fit in terms of rankings rankfit$rankings #fit in terms of orderings rankfit$orderings # information about the fit (terminal node, predictor and fit (in terms of rankings)) rankfit$info
Print methods for objects of class cca
## S3 method for class 'cca' print(x, ...)## S3 method for class 'cca' print(x, ...)
x |
An object of the class "cca" |
... |
not used |
print a brief summary of the CCA
Print methods for objects of class ranktree
## S3 method for class 'ranktree' print(x, ...)## S3 method for class 'ranktree' print(x, ...)
x |
An object of the class "ranktree" |
... |
not used |
print a brief summary of the prediction tree
data("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) treedata("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) tree
Recursive partitioning method for the prediction of preference rankings based upon Kemeny distances.
ranktree(Y, X, prunplot = FALSE, control = ranktreecontrol(...), ...)ranktree(Y, X, prunplot = FALSE, control = ranktreecontrol(...), ...)
Y |
A n by m data matrix, in which there are n judges and m objects to be judged. Each row is a ranking of the objects which are represented by the columns. |
X |
A dataframe containing the predictor, that must have n rows. |
prunplot |
prunplot=TRUE returns the plot of the pruning sequence. Default value: FALSE |
control |
a list of options that control details of the |
... |
arguments passed bypassing |
The user can use any algorithm implemented in the consrank function from the ConsRank package. All algorithms allow the user to set the option 'full=TRUE'
if the median ranking(s) must be searched in the restricted space of permutations instead of in the unconstrained universe of rankings of n items including all possible ties.
The output consists in a object of the class "ranktree". It contains:
| X | the predictors: it must be a dataframe | ||
| Y | the response variable: the matrix of the rankings | ||
| node | a list containing teh tree-based structure: | ||
| number | node number | ||
| terminal | logical: TRUE is terminal node | ||
| father | father node number of the current node | ||
| idfather | id of the father node of the current node | ||
| size | sample size within node | ||
| impur | impurity at node | ||
| wimpur | weighted impurity at node | ||
| idatnode | id of the observations within node | ||
| class | median ranking within node in terms of orderings | ||
| nclass | median ranking within node in terms of rankings | ||
| mclass | eventual multiple median rankings | ||
| tau | Tau_x rank correlation coefficient at node | ||
| wtau | weighted Tau_x rank correlation coefficient at node | ||
| error | error at node | ||
| werror | weighted error at node | ||
| varsplit | variables generating split | ||
| varsplitid | id of variables generating split | ||
| cutspli | splitting point | ||
| children | children nodes generated by current node | ||
| idchildren | id of children nodes generated by current node | ||
| ... | other info about node | ||
| control | parameters used to build the tree | ||
| numnodes | number of nodes of the tree | ||
| tsynt | list containing the synthesis of the tree: | ||
| children | list containing all information about leaves | ||
| parents | list containing all information about parent nodes | ||
| geneaoly | data frame containing information about all nodes | ||
| idgenealogy | data frame containing information about all nodes in terms of nodes id | ||
| idparents | id of the parents of all the nodes | ||
| goodness | goodness -and badness- of fit measures of the tree: Tau_X, error, impurity | ||
| nomin | information about nature of the predictors | ||
| alpha | alpha parameter for pruning sequence | ||
| pruneinfo | list containing information about the pruning sequence: | ||
| prunelist | information about the pruning | ||
| tau | tau_X rank correlation coefficient of each subtree | ||
| error | error of each subtree | ||
| termnodes | number of terminal nodes of each subtree | ||
| subtrees | list of each subtree created with the cost-complexity pruning procedure |
An object of the class ranktree. See details for detailed information.
Antonio D'Ambrosio [email protected]
D'Ambrosio, A., and Heiser W.J. (2016). A recursive partitioning method for the prediction of preference rankings based upon Kemeny distances. Psychometrika, vol. 81 (3), pp.774-94.
ranktreecontrol, plot.ranktree, summary.ranktree, getsubtree, validatetree, treepaths, nodepath
data("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) data(Irish) #build the tree with default options tree <- ranktree(Irish$rankings,Irish$predictors) #plot the tree plot(tree,dispclass=TRUE) #visualize information summary(tree) #get information about the paths leading to terminal nodes (all the paths) infopaths <- treepaths(tree) #the terminal nodes infopaths$leaves #sample size within each terminal node infopaths$size #visualize the path of the second leave (terminal node number 8) infopaths$paths[[2]] #alternatively nodepath(termnode=8,tree) set.seed(132) #for reproducibility #validation of the tree via v-fold cross-validation (default value of V=5) vtree <- validatetree(tree,method="cv") #extract the "best" tree dtree <- getsubtree(tree,vtree$best_tau) summary(dtree) #plot the validated tree plot(dtree,dispclass=TRUE) #predicted rankings rankfit <- predict(dtree,newx=Irish$predictors) #fit of rankings rankfit$rankings #fit in terms of orderings rankfit$orderings #all info about the fit (id og the leaf, predictor values, and fit) rankfit$orderingsdata("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) data(Irish) #build the tree with default options tree <- ranktree(Irish$rankings,Irish$predictors) #plot the tree plot(tree,dispclass=TRUE) #visualize information summary(tree) #get information about the paths leading to terminal nodes (all the paths) infopaths <- treepaths(tree) #the terminal nodes infopaths$leaves #sample size within each terminal node infopaths$size #visualize the path of the second leave (terminal node number 8) infopaths$paths[[2]] #alternatively nodepath(termnode=8,tree) set.seed(132) #for reproducibility #validation of the tree via v-fold cross-validation (default value of V=5) vtree <- validatetree(tree,method="cv") #extract the "best" tree dtree <- getsubtree(tree,vtree$best_tau) summary(dtree) #plot the validated tree plot(dtree,dispclass=TRUE) #predicted rankings rankfit <- predict(dtree,newx=Irish$predictors) #fit of rankings rankfit$rankings #fit in terms of orderings rankfit$orderings #all info about the fit (id og the leaf, predictor values, and fit) rankfit$orderings
Utility function to use to set the control arguments of ranktree
ranktreecontrol( num = NULL, decrmin = 0.01, algorithm = "quick", full = FALSE, itermax = 10, np = 15, gl = 100, ff = 0.4, cr = 0.9, proc = FALSE, ps = FALSE )ranktreecontrol( num = NULL, decrmin = 0.01, algorithm = "quick", full = FALSE, itermax = 10, np = 15, gl = 100, ff = 0.4, cr = 0.9, proc = FALSE, ps = FALSE )
num |
The maximum number of observations in a node to be split: default, 10% of the sample size |
decrmin |
Minimum decrease in impurity |
algorithm |
The algorithm used to compute the median ranking. One among"BB", "quick" (default), "fast" and "decor" |
full |
Specifies if the median ranking must be searched in the universe of rankings including all the possible ties. Default: FALSE |
itermax |
Number of iterations for "fast" and "decor" algorithms. itermax=10 is the default option. |
np |
(for "decor" only) the number of population individuals. np=15 is the default option. |
gl |
(for"decor" only) generations limit, maximum number of consecutive generations without improvement. gl=100 is the default option. |
ff |
(for"decor" only) the scaling rate for mutation. Must be in [0,1]. ff=0.4 is the default option. |
cr |
(for"decor" only) the crossover range. Must be in [0,1]. cr=0.9 is the default option. |
proc |
(for "BB" only) proc=TRUE allows the branch and bound algorithm to work in difficult cases, i.e. when the number of objects is larger than 15 or 25. proc=FALSE is the default option |
ps |
If PS=TRUE, on the screen some information about how many branches are processed are displayed. Default value: FALSE |
A list containing all the control parameters
Antonio D'Ambrosio [email protected]
Summary methods for objects of class cca
## S3 method for class 'cca' summary(object, ...)## S3 method for class 'cca' summary(object, ...)
object |
An object of the class "cca" |
... |
not used |
it shows the summary of the prediction tree
Summary methods for objects of class ranktree
## S3 method for class 'ranktree' summary(object, ...)## S3 method for class 'ranktree' summary(object, ...)
object |
An object of the class "ranktree" |
... |
not used |
it shows the summary of the prediction tree
data("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) summary(tree)data("Univranks") tree <- ranktree(Univranks$rankings,Univranks$predictors,num=50) summary(tree)
Given an object of the class "ranktree", it extracts the paths of all terminal nodes
treepaths(Tree)treepaths(Tree)
Tree |
An object of the class "ranktree" |
A list containing:
| leaves | the number of the terminal nodes | |
| size | the sample size within each terminal nodes | |
| paths | a list containing all the paths |
Antonio D'Ambrosio [email protected]
ranktree, nodepath, getsubtree
data(Irish) #build the tree with default options tree <- ranktree(Irish$rankings,Irish$predictors) #get information about all the paths leading to terminal nodes paths <- treepaths(tree) # #the terminal nodes paths$leaves # #sample size within each terminal node paths$size # #visualize the path of the second leave (terminal node number 8) paths$paths[[2]]data(Irish) #build the tree with default options tree <- ranktree(Irish$rankings,Irish$predictors) #get information about all the paths leading to terminal nodes paths <- treepaths(tree) # #the terminal nodes paths$leaves # #sample size within each terminal node paths$size # #visualize the path of the second leave (terminal node number 8) paths$paths[[2]]
University rankings dataset was analysed by Dittrich, Hatzinger and Katzenbeisser (1998) to investigate paired comparison data concerning European universities and student's characteristics with the goal to show that university rankings are different for different groups of students. Here both raw data (with paired comparisons) and the version with rankings are preesented (see details). A survey of 303 students studying at the Vienna University of Economics was carried out to examine the student's preference of six universities, namely London, Paris, Milan, St. Gallen, Barcelona and Stockholm. The data set contains 23 variables. The first 15 digits in each row indicate the preferences of a student. For a given comparison, responses were coded by 1 if the first preference was preferred, by 2 if the second university was preferred, and by 3 if universities are tied. All rows containing missing ranked Universities were skipped.
data("Univranks")data("Univranks")
The format is: List of 3
$ rawdata: 'data.frame': 212 obs. of 23 variables: the first 15 are the paired comparisons coded as follows: (1: the first is preferred to the second; 2: the second is preferred to the fisrt; 3 tied)
$ LP : comparison of London to Paris
$ LM : comparison of London to Milan
$ PM : comparison of London to Milan
$ LSg : comparison of London to St. Gallen
$ PSg : comparison of Paris to St. Gallen
$ MSg : comparison of Milan to St. Gallen
$ LB : comparison of London to Barcelona
$ PB : comparison of Paris to Barcelona
$ MB : comparison of Milan to Barcelona
$ SgB : comparison of St. Gallen to Barcelona
$ LSt : comparison of London to Stockholm
$ PSt : comparison of Paris to Stockholm
$ MSt : comparison of Milan to Stockholm
$ SgSt: comparison of St. Gallen to Stockholm
$ BSt : comparison of Barcelona to Stockholm
$ Stud: Factor w/ 2 levels "commerce","other"
$ Eng : Factor w/ 2 levels "good","poor""
$ Fra : Factor w/ 2 levels "good","poor"
$ Spa : Factor w/ 2 levels "good","poor"
$ Ita : Factor w/ 2 levels "good","poor"
$ Wor : Factor w/ 2 levels "no","yes"
$ Deg : Factor w/ 2 levels "no","yes"
$ Sex : Factor w/ 2 levels "female","male"
$ predictors:'data.frame': 212 obs. of 8 variables( the last 8 variables of the "rawdata" dataframe
$ rankings : matrix of preference rankings. The columns are: "L" (London), "P" (Paris), "M" (Milan), "Sg" (St. Gallen), "B" (Barcerlona), "St" (Stockholm)
To obtain the preference rankings from the paired comparisons the procedure has been the following: the first row of the raw data is [1 3 2 1 2 1 1 2 1 1 1 2 1 1 2]. London is preferred to Paris, St. Gallen, Barcelona Stockholm (LP, LM, LSg, LB and LSt are always equal to 1), and there is no preference between London and Milan (they are tied); Milan is preferred to Paris (PM = 2), St. Gallen, Barcelona and Stockholm; and so on. The first ordering is then <L M Sg St B P> corresponding to a ranking [1,5,1,2,4,3], where the columns indicate L P M Sg B St.
http://www.blackwellpublishers.co.uk/rss
Dittrich, R., Hatzinger, R., and Katzenbeisser, W. (1998). Modelling the effect of subject-specific covariates in paired comparison studies with an application to university rankings. Journal of the Royal Statistical Society: Series C (Applied Statistics), 47(4), 511-525. DOI: 10.1111/1467-9876.00125
D'Ambrosio, A. (2008). Tree based methods for data editing and preference rankings. Ph.D. thesis, University of Naples Federico II. https://www.doi.org/10.6092/UNINA/FEDOA/2746
data(Univranks)data(Univranks)
Validation of the tree either with a test set procedure or with v-fold cross validation
validatetree( Tree, testX = NULL, testY = NULL, method = "test", V = 5, plotting = TRUE )validatetree( Tree, testX = NULL, testY = NULL, method = "test", V = 5, plotting = TRUE )
Tree |
An object of the class "ranktree" coming form te function |
testX |
The data frame containing the test set (predictors) |
testY |
The matrix ontaining the test set (response) |
method |
One between "test" (default) or "cv" |
V |
The cross-validation parameter. Default V=5 |
plotting |
With the defaul option plotting=TRUE, the pruning sequence plot is visualized |
A list containing:
| tau | the Tau_x rank correlation coefficient of the sequence of the trees | |
| error | the error of the sequence of the trees | |
| termnodes | the number of terminal nodes of the sequence of the trees | |
| best_tau | the best tree in terms of Tau_x rank correlation coefficient | |
| best_error | the best tree in terms of error (it is the same) | |
| validation | information about the validation procedure |
#'
Antonio D'Ambrosio [email protected]
data(EVS) EVS$rankings[is.na(EVS$rankings)] <- 3 set.seed(654) training=sample(1911,1434) tree <- ranktree(EVS$rankings[training,],EVS$predictors[training,],decrmin=0.001,num=50) #test set validation vtreetest <- validatetree(tree,testX=EVS$predictors[-training,],EVS$rankings[-training,]) #cross-validation vtreecv <- validatetree(tree,method="cv",V=10)data(EVS) EVS$rankings[is.na(EVS$rankings)] <- 3 set.seed(654) training=sample(1911,1434) tree <- ranktree(EVS$rankings[training,],EVS$predictors[training,],decrmin=0.001,num=50) #test set validation vtreetest <- validatetree(tree,testX=EVS$predictors[-training,],EVS$rankings[-training,]) #cross-validation vtreecv <- validatetree(tree,method="cv",V=10)