The LLAhclust Package

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1 Verion Date The LLAhclut Package September 1, 2007 Title Hierarchical clutering of variable or object baed on the likelihood linkage analyi method Author Ivan Kojadinovic, Iraël-Céar Lerman, Philippe Peter Maintainer Ivan Kojadinovic Decription The likelihood linkage analyi i a general agglomerative hierarchical clutering method developed in France by Lerman in a long erie of reearch article and book. Initially propoed in the framework of variable clutering, it ha been progreively extended to allow the clutering of very general object decription. The approach mainly conit in replacing the value of the etimated imilarity coefficient by the probability of finding a lower value under the hypothei of abence of link. The package LLAhclut contain routine for computing variou type of probablitic imilarity coefficient between variable or object decription. Once the imilarity value between variable/object are computed, a hierarchical clutering can be performed uing everal probabilitic and non-probabilitic aggregation criteria, and indice meauring the quality of the partition compatible with the reulting hierarchy can be computed. Depend R(>= 2.1.0) Encoding latin1 Licene CeCILL 2 (GNU GPL 2 compatible) URL R topic documented: LLAhclut LLAparteval LLAimobj LLAimvar a.llaim a.matrix.llaim empcopula.imulate Index 14 1

2 2 LLAhclut LLAhclut Likelihood linkage analyi hiearichal clutering Decription Uage Build a hierarchy from imilarity coefficient among object or variable a returned by LLAimvar, LLAimobj or a.llaim. The default aggregation criteria, called lla, can be regarded a a probabilitic verion of the ingle linkage. LLAhclut(, method = "lla", epilon = 1, member = NULL) Argument Value Similarity coefficient a returned by LLAimvar, LLAimobj or a.llaim. method Linkage method (i.e. aggregation criterion). Can be one of lla (default), tippett (Tippett p-value combination method), average, complete, fiher (Fiher p-value combination method), uniform (uniform p-value combination method; can be regarded a a probabilitic verion of the average linkage), normal (normal p-value combination method) or maximum (maximum p-value combination method; can be regarded a a probabilitic verion of the complete linkage). See the lat reference for more detail. epilon member Coefficient ued in the lla linkage. Should lie in [0,1]: epilon=0 correpond to the ingle linkage, epilon=1 (default) yield a probabilitic verion of the ingle linkage. "Weight" of the object to be clutered if not of equal "weight". See hclut for more detail. An object of cla hclut with the correponding attribute. See hclut for more detail. Reference I.C. Lerman (1981), Claification et analye ordinale de donné, Dunod, Pari. I.C. Lerman (1991), Foundation of the likelihood linkage analyi claification method, Applied Stochatic Model and Data Analyi, 7, page I.C. Lerman (1993), Likelihood linkage analyi claification method: An example treated by hand, Biochimie, 75, page I.C. Lerman, Ph. Peter and H. Leredde (1993), Principe et calcul de la méthode implantée dan le programme CHAVL (Claification Hiérarchique par Analye de la Vraiemblance de Lien), Modulad, 12, page I. Kojadinovic (2007), Hierarchical clutering of continuou variable baed on the empirical copula proce, ubmitted.

3 LLAparteval 3 See Alo LLAimvar, LLAimobj, a.llaim, LLAparteval, hclut. Example data(usarret) ## Compute imilaritie between variable baed on ## the LLAnumerical method: <- LLAimvar(USArret) ## Perform the hierarchical clutering of the variable ## uing the default aggregation criterion (lla): h <- LLAhclut() plot(h) ## Compute the quality of the partition compatible ## with the hierarchy in term of the tatitic defined by Lerman: LLAparteval(h,) ## Compute imilaritie between variable uing the claical ## bilateral tet of independence baed on Spearman' rho: <- LLAimvar(USArret, method = "pearman.ab") ## Perform the hierarchical clutering of the variable ## uing Fiher' p-value combination method: h <- LLAhclut(,method="fiher") plot(h) ## NB: the height in the dendrogram i a p-value ## and can be ued to identify mutually independent clae of ## variable, if any. ## Compute the quality of the partition compatible ## with the hierarchy in term of the indice defined in the ## lat reference: LLAparteval(h,) LLAparteval Evalute the quality of each partition compatible with a hierarchy in term of everal indice

4 4 LLAparteval Decription Evalute the quality of each partition compatible with the hierarchy returned by LLAhclut. If the hierarchy i obtained from imilarity coefficient computed uing LLA* method, the global and local tatitic propoed by Lerman are calculated. Otherwie, for imilarity coefficient obtained from independence tet (ee LLAimvar), for each partition, the inter-cla p-value are combined uing Tippett and Fiher rule. Furthermore, the minimum inter-cla p-value and the maximum intra-cla p-value are given. See the lat reference and the example below for more detail. Uage LLAparteval(tree,, m=null) Argument tree m An object of cla hclut a returned by LLAhclut. An object of cla LLAim a returned by LLAimvar, LLAimobj or a.llaim. Integer. If et, the quality of the m coaret partition only i evaluated. Value Return a data.frame whoe column are: global.tat and local.tat if the hierarchy i obtained from imilarity coefficient computed uing LLA* method, and tippett.inter, fiher.inter, min.inter and max.intra in cae of imilarity coefficient obtained from independence tet. Reference I.C. Lerman (1981), Claification et analye ordinale de donné, Dunod, Pari. I.C. Lerman (1991), Foundation of the likelihood linkage analyi claification method, Applied Stochatic Model and Data Analyi, 7, page I.C. Lerman (1993), Likelihood linkage analyi claification method: An example treated by hand, Biochimie, 75, page I.C. Lerman, Ph. Peter and H. Leredde (1993), Principe et calcul de la méthode implantée dan le programme CHAVL (Claification Hiérarchique par Analye de la Vraiemblance de Lien), Modulad, 12, page I. Kojadinovic (2007), Hierarchical clutering of continuou variable baed on the empirical copula proce, ubmitted. See Alo LLAimvar, LLAimobj, a.llaim, LLAhclut.

5 LLAimobj 5 Example data(usarret) ## Compute imilaritie between variable baed on ## the LLAnumerical method: <- LLAimvar(USArret) ## Perform the hierarchical clutering of the variable: h <- LLAhclut() plot(h) ## Compute the quality of the partition compatible ## with the hierarchy in term of the tatitic defined by Lerman: LLAparteval(h,) ## Compute imilaritie between variable uing the claical ## bilateral tet of independence baed on Spearman' rho: <- LLAimvar(USArret, method = "pearman.ab") ## Perform the hierarchical clutering of the variable ## uing Fiher' p-value combination method: h <- LLAhclut(,method="fiher") plot(h) ## NB: the height in the dendrogram i a p-value ## and can be ued to identify mutually independent clae of ## variable, if any. ## Compute the quality of the partition compatible ## with the hierarchy in term of the indice defined in the ## lat reference: LLAparteval(h,) LLAimobj Compute imilaritie among object Decription Compute imilaritie among object uing the likelihood linkage analyi approach propoed by Lerman. The likelihood linkage analyi method mainly conit in replacing the value of the imilarity coefficient between two object by the probability of finding a lower value under the hypothei of abence of link. See the reference below for more detail. Uage LLAimobj(x, method = "LLAeuclidean", upper = FALSE)

6 6 LLAimobj Argument x method upper a numeric matrix or data frame. Can be one of LLAeuclidean, LLAcoinu, LLAcategorical, LLAordinal, or LLAboolean. The two firt method can be ued to compute imiliarty coefficient between object decribed by numerical variable. logical value indicating whether the upper triangle of the imilarity matrix hould be printed by print.llaim. Detail The following function are alo defined for object of cla LLAim: name.llaim, format.llaim, a.matrix.llaim and print.llaim. Value Return an object of cla LLAim whoe attribute are very imilar to thoe of object of cla dit. See dit for more detail. Reference I.C. Lerman (1981), Claification et analye ordinale de donné, Dunod, Pari. I.C. Lerman (1991), Foundation of the likelihood linkage analyi claification method, Applied Stochatic Model and Data Analyi, 7, page I.C. Lerman (1993), Likelihood linkage analyi claification method: An example treated by hand, Biochimie, 75, page I.C. Lerman, Ph. Peter and H. Leredde (1993), Principe et calcul de la méthode implantée dan le programme CHAVL (Claification Hiérarchique par Analye de la Vraiemblance de Lien), Modulad, 12, page See Alo LLAimvar, a.llaim, LLAhclut, LLAparteval, dit. Example data(usarret) ## Compute imilaritie between object baed on ## a local Euclidean ditance (ee reference above): <- LLAimobj(USArret)

7 LLAimvar 7 LLAimvar Compute imilaritie among variable uing the likelihood linkage analyi approach Decription Compute imilaritie among variable uing the likelihood linkage analyi approach propoed by Lerman. The likelihood linkage analyi method mainly conit in replacing the value of the etimated imilarity coefficient between two variable by the probability of finding a lower value under the hypothei of tochatic independence, called abence of link in that context. Nine imilarity coefficient can be computed uing the LLAimvar function. Uage LLAimvar(x, method = "LLAnumerical", upper = FALSE, imulated.ditribution = NULL) Argument x method upper a numeric matrix or data frame. Can be one of LLAnumerical, LLAcategorical, LLAordinal, LLAboolean, chi.quare, pearon.ab, pearman.ab, kendall.ab or empirical.copula. The method LLA* were initially defined by Lerman (ee reference below). The four remaining method compute the imilarity between two variable a one minu the p-value obtained from a tet of independence. See the lat reference and the example ection below for more detail. logical value indicating whether the upper triangle of the imilarity matrix hould be printed by print.llaim. imulated.ditribution Object of cla empcopula.imulation. Should be et only if the method empirical.copula i elected. See function empcopula.imulate and the example ection below for more detail. Detail The following function are alo defined for object of cla LLAim: name.llaim, format.llaim, a.matrix.llaim and print.llaim. Value Return an object of cla LLAim whoe attribute are very imilar to thoe of object of cla dit. See dit for more detail.

8 8 LLAimvar Reference I.C. Lerman (1981), Claification et analye ordinale de donné, Dunod, Pari. I.C. Lerman (1991), Foundation of the likelihood linkage analyi claification method, Applied Stochatic Model and Data Analyi, 7, page I.C. Lerman (1993), Likelihood linkage analyi claification method: An example treated by hand, Biochimie, 75, page I.C. Lerman, Ph. Peter and H. Leredde (1993), Principe et calcul de la méthode implantée dan le programme CHAVL (Claification Hiérarchique par Analye de la Vraiemblance de Lien), Modulad, 12, page P. Deheuvel (1979), La fonction de dépendance empirique et e propriété: un tet non paramétrique d indépendance, Acad. Roy. Belg. Bull. Cl. Sci. 5th Ser. 65, C. Genet and B. Rémillard (2004). Tet of independence and randomne baed on the empirical copula proce. Tet, 13, I. Kojadinovic (2007), Hierarchical clutering of continuou variable baed on the empirical copula proce, ubmitted. See Alo a.llaim, empcopula.imulate, LLAimobj, LLAhclut, LLAparteval, dit. Example data(usarret) ## Compute imilaritie between variable uing the ## LLAnumerical method: <- LLAimvar(USArret) ## Compute imilaritie between variable uing the claical ## bilateral tet of independence baed on Spearman' rho: <- LLAimvar(USArret, method = "pearman.ab") ## Compute imilaritie between variable uing the claical ## bilateral tet of independence baed on Kendall' tau: <- LLAimvar(USArret, method = "kendall.ab") ## Compute imilaritie between variable uing the tet of ## independence à la Deheuvel baed on the empirical copula ## proce recently tudied by Genet and Rémillard: <- LLAimvar(USArret, method = "empirical.copula")

9 a.llaim 9 ## The previou computation could have been done in two tep: d <- empcopula.imulate(n=50,n=2000) <- LLAimvar(USArret, method = "empirical.copula", imulated.ditribution = d) a.llaim Convert the lower triangle of a quare matrix into a LLAim object Decription Uage Convert the lower triangle of a quare matrix into a LLAim object. The LLAim object contain imilarity coefficient among object or variable of interet. a.llaim(m, upper = FALSE, probabilitic = FALSE) Argument m Detail Value input quare imilarity matrix. upper logical value indicating whether the upper triangle of the imilarity matrix hould be printed by print.llaim. probabilitic logical value indicating whether the coefficient in the input imilarity matrix hould be treated a probabilitic imilarity value. If et to FALSE, the input imilarity coefficient are caled. See example below. The following function are alo defined for object of cla LLAim: name.llaim, format.llaim, a.matrix.llaim and print.llaim. Return an object of cla LLAim whoe attribute are very imilar to thoe of object of cla dit. See dit for more detail. Reference I.C. Lerman (1981), Claification et analye ordinale de donné, Dunod, Pari. I.C. Lerman (1991), Foundation of the likelihood linkage analyi claification method, Applied Stochatic Model and Data Analyi, 7, page I.C. Lerman (1993), Likelihood linkage analyi claification method: An example treated by hand, Biochimie, 75, page

10 10 a.matrix.llaim I.C. Lerman, Ph. Peter and H. Leredde (1993), Principe et calcul de la méthode implantée dan le programme CHAVL (Claification Hiérarchique par Analye de la Vraiemblance de Lien), Modulad, 12, page See Alo LLAimvar, LLAimobj, a.matrix.llaim, dit. Example ## Aume that we have at hand a probabilitic imilarity matrix ## between 5 object (lower triangle only): m <- matrix(runif(25), 5, 5) ## The correponding LLAim object i obtained a follow: <- a.llaim(m, probabilitic=true) ## Diplay the initial matrix and the LLAim object: m ## Aume now that we have at hand a non-probabilitic imiliarty ## matrix: m <- matrix(rnorm(25), 5, 5) ## The correponding LLAim object i obtained a follow: <- a.llaim(m, probabilitic=false) ## Diplay the initial matrix and the LLAim object: m ## Notice that the coefficient in are caled: mean() d() a.matrix.llaim Ueful function for dealing with LLAim object Decription Uage The function a.matrix.llaim convert a LLAim object into a quare ymmetrical matrix. The uual R function format, print and name have alo been extended to deal with LLAim object. a.matrix.llaim(x,...)

11 a.matrix.llaim 11 Argument x the LLAim object to be converted.... nothing o far. Value An object of cla matrix. Reference I.C. Lerman (1981), Claification et analye ordinale de donné, Dunod, Pari. I.C. Lerman (1991), Foundation of the likelihood linkage analyi claification method, Applied Stochatic Model and Data Analyi, 7, page I.C. Lerman (1993), Likelihood linkage analyi claification method: An example treated by hand, Biochimie, 75, page I.C. Lerman, Ph. Peter and H. Leredde (1993), Principe et calcul de la méthode implantée dan le programme CHAVL (Claification Hiérarchique par Analye de la Vraiemblance de Lien), Modulad, 12, page See Alo LLAimvar, LLAimobj, a.llaim. Example data(usarret) ## Compute imilaritie between object baed on ## a local Euclidean ditance (ee reference above): <- LLAimobj(USArret) ## Convert to a matrix object: a.matrix() ## Other ueful function: print(, upper=true) name() ## For the format function, ee the R help.

12 12 empcopula.imulate empcopula.imulate Simulation tep ued in the independence tet baed on the empirical copula proce implemented in the LLAimvar function Decription Uage Simulation tep ued in the independence tet baed on the empirical copula proce a propoed by Chritian Genet and Bruno Rémillard. To be ued in conjunction with the LLAimvar function (method="empirical.copula"). The imulation tep conit in imulating the ditribution of the tet tatitic under independence for the ample ize under conideration. More detail can be found in the article cited in the reference ection. empcopula.imulate(n, N = 2000) Argument n N Sample ize when imulating the ditribution of the tet tatitic under independence. Number of repetition when imulating under independence. Detail Value See the reference below for more detail, epecially the third one. The function empcopula.imulate return an object of cla empcop.imulation whoe attribute are: ample.ize, number.repetiton and dit.independence (a vector of length N containing the value of the tet tatitic for each each repetition). Reference P. Deheuvel (1979), La fonction de dépendance empirique et e propriété: un tet non paramétrique d indépendance, Acad. Roy. Belg. Bull. Cl. Sci. 5th Ser. 65, P. Deheuvel (1981), A non parametric tet for independence, Publ. Int. Statit. Univ. Pari 26, C. Genet and B. Rémillard (2004). Tet of independence and randomne baed on the empirical copula proce. Tet, 13, C. Genet, J.-F. Quey and B. Rémillard (2006). Local efficiency of a Cramer-von Mie tet of independence. Journal of Multivariate Analyi, 97, C. Genet, J.-F. Quey and B. Rémillard (2007). Aymptotic local efficiency of Cramer-von Mie tet for multivariate independence. The Annal of Statitic, 35, in pre. I. Kojadinovic (2007), Hierarchical clutering of continuou variable baed on the empirical copula proce, ubmitted.

13 empcopula.imulate 13 See Alo LLAimvar, LLAhclut. Example data(usarret) ## Compute imilaritie between variable uing the tet of ## independence à la Deheuvel baed on the empirical copula ## proce recently tudied by Genet and Rémillard: <- LLAimvar(USArret, method = "empirical.copula") ## The previou computation could have been done in two tep: d <- empcopula.imulate(n=50,n=2000) <- LLAimvar(USArret, method = "empirical.copula", imulated.ditribution = d)

14 Index Topic cluter a.llaim, 9 a.matrix.llaim, 10 empcopula.imulate, 12 LLAhclut, 2 LLAparteval, 3 LLAimobj, 5 LLAimvar, 7 a.llaim, 3, 4, 6, 8, 9, 11 a.matrix.llaim, 10, 10 dit, 6 10 empcopula.imulate, 7, 8, 12 format.llaim (a.matrix.llaim), 10 hclut, 2, 3 LLAhclut, 2, 4, 6, 8, 13 LLAparteval, 3, 3, 6, 8 LLAimobj, 3, 4, 5, 8, 10, 11 LLAimvar, 3, 4, 6, 7, 10, 11, 13 name.llaim (a.matrix.llaim), 10 print.llaim (a.matrix.llaim), 10 14