Average diameter & Maximum split of every partition of a given dendrogram

Description

Average diameter & Maximum split of every partition of a given dendrogram

Usage

Adiam.Msplit(X, dend = dend, dist.func = "PD", estim.method = c("copula"))

Arguments

Value

a data frame

Estimate for T(Y,X) based on function codec

Description

Estimate for T(Y,X) based on function codec

Usage

Codec.Tq(X, Y)

Arguments

Value

a value

Estimate for T_bar(Y,X) based on function Codec & a sample of all / all increasing / all decreasing permutations

Description

Estimate for T_bar(Y,X) based on function Codec & a sample of all / all increasing / all decreasing permutations

Usage

Codec.Tq.Perm(X, Y, method = c("sample"))

Arguments

Value

a value

Estimate for \xi(Y,X) using codec function

Description

Estimate for \xi(Y,X) using codec function

Usage

CodecCorr(X, y)

Arguments

Value

a value

Estimate for T(Y,X) based on dimension reduction principle

Description

Estimate for T(Y,X) based on dimension reduction principle

Usage

Copula.Tq(X, Y)

Arguments

Value

a value

Estimate for T_bar(Y,X) based on dimension reduction principle

Description

Estimate for T_bar(Y,X) based on dimension reduction principle

Usage

Copula.Tq.Perm(X, Y, method = c("sample"))

Arguments

Value

a value

Estimate for \xi(Y,X) based on dimension reduction principle

Description

Estimate for \xi(Y,X) based on dimension reduction principle

Usage

CopulaCorr(X, y)

Arguments

Value

a value

Markov product estimate from single (q=1) endogenuous and (p>=1) exogenous variables based on dimension reduction

Description

Markov product estimate from single (q=1) endogenuous and (p>=1) exogenous variables based on dimension reduction

Usage

MPhi(X, y)

Arguments

Value

a value

Silhouette value for the i. variable given variable partition

Description

Silhouette value for the i. variable given variable partition

Usage

Silhouette(i, df, partition, dist.func = "PD", estim.method = c("copula"))

Arguments

Value

a value for Silhouette

Silhouette coefficients given a dendrogram

Description

Silhouette coefficients given a dendrogram

Usage

Silhouette.coefficient(X, dend, dist.func = "PD", estim.method = c("copula"))

Arguments

Value

a data frame

Hierarchical variable clustering and partition.

Description

VarClustPartition is a hierarchical variable clustering algorithm based on the directed dependence coefficient (didec) or a concordance measure (Kendall tau \tau or Spearman's footrule) according to a pre-selected number of clusters or an optimality criterion (Adiam&Msplit or Silhouette coefficient).

Usage

VarClustPartition(
  X,
  trans = FALSE,
  trans.method = c("standardization"),
  dist.method = c("PD"),
  estim.method = c("copula"),
  linkage = FALSE,
  link.method = c("complete"),
  part.method = c("optimal"),
  part.criterion = c("Adiam&Msplit"),
  num.cluster = NULL,
  plot = FALSE
)

Arguments

Details

VarClustPartition performs a hierarchical variable clustering based on the directed dependence coefficient (didec) and provides a partition of the set of variables.

If dist.method =="PD" (perfect dependence) or dist.method =="MPD" (mutual perfect dependence) the clustering is performed using didec either as a directed ("PD") or as a symmetric ("MPD") dependence coefficient. If dist.method =="kendall" or dist.method =="footrule", clustering is performed using either multivariate Kendall's tau ("kendall") or multivariate Spearman's footrule ("footrule"). "kendall" uses the function cor.fk which is provided in the R package pcaPP to calculate bivariate Kendall's tau.

Instead of using one of the above-mentioned four multivariate measures for the clustering, the option linkage == TRUE enables the use of bivariate linkage methods, including complete linkage (link.method == "complete"), average linkage (link.method == "average") and single linkage (link.method == "single"). Note that the multivariate distance methods are computationally demanding because higher-dimensional dependencies are included in the calculation, in contrast to linkage methods which only incorporate pairwise dependencies.

A pre-selected number of clusters num.cluster can be realized with the option part.method == "selected". Otherwise (part.method == "optimal"), the number of clusters is determined by maximizing the intra-cluster similarity (similarity within the same cluster) and minimizing the inter-cluster similarity (similarity among the clusters). Two optimality criteria (Fuchs & Wang 2024) are available:

"Adiam&Msplit": Adiam measures the intra-cluster similarity and Msplit measures the inter-cluster similarity.

"Silhouette": A mixed coefficient incorporating the intra-cluster similarity and the inter-cluster similarity. The optimal number of clusters corresponds to the maximum Silhouette coefficient.

Value

A list containing:

dendrogram

A dendrogram without colored branches;

num.cluster

An integer value determining the number of clusters after partitioning;

clusters

A list containing the clusters after partitioning.

Author(s)

Yuping Wang, Sebastian Fuchs

References

S. Fuchs, Y. Wang, Hierarchical variable clustering based on the predictive strength between random vectors, Int. J. Approx. Reason. 170, Article ID 109185, 2024.

P. Hansen, B. Jaumard, Cluster analysis and mathematical programming, Math. Program. 79 (1) 191–215, 1997.

L. Kaufman, Finding Groups in Data, John Wiley & Sons, 1990.

Examples

library(didec)
n  <- 50
X1 <- rnorm(n,0,1)
X2 <- X1
X3 <- rnorm(n,0,1)
X4 <- X3 + X2
X  <- data.frame(X1=X1,X2=X2,X3=X3,X4=X4)
vcp <- VarClustPartition(X,
                            dist.method = c("PD"),
                            part.method = c("optimal"),
                            part.criterion   = c("Silhouette"),
                            plot        = TRUE)
vcp$clusters

data("bioclimatic")
X   <- bioclimatic[c(2:4,9)]
vcp1 <- VarClustPartition(X,
                          linkage     = TRUE,
                          link.method = c("complete"),
                          dist.method = "PD",
                          part.method = "optimal",
                          part.criterion   = "Silhouette",
                          plot        = TRUE)
vcp1$clusters
vcp2 <- VarClustPartition(X,
                          linkage     = TRUE,
                          link.method = c("complete"),
                          dist.method = "footrule",
                          part.method = "optimal",
                          part.criterion   = "Adiam&Msplit",
                          plot        = TRUE)
vcp2$clusters

Bioclimatic variables

Description

A data set of bioclimatic variables for n=1,862 locations homogeneously distributed over the global landmass from CHELSA ("Climatologies at high resolution for the earth’s land surface areas").

Usage

bioclimatic

Format

An object of class data.frame with 1862 rows and 19 columns.

References

D.N. Karger, O. Conrad, J. Böhner, T. Kawohl, H. Kreft, R.W. Soria-Auza, N.E. Zimmermann, H.P. Linder, M. Kessler, Climatologies at high resolution for the Earth's land surface areas, Sci. Data 4(1), 2017.

Examples

data(bioclimatic)
head(bioclimatic)

Cluster a set of variables using distance function based on predictive measure

Description

Cluster a set of variables using distance function based on predictive measure

Usage

clust.Tq(X, estim.method = c("copula"), mutual = FALSE)

Arguments

Value

a list for hierarchical clustering result

Clustering a set of variables using distance function based on multivariate concordance measures

Description

Clustering a set of variables using distance function based on multivariate concordance measures

Usage

clust.concor.M(X, method = c("footrule"))

Arguments

Value

a list for hierarchical clustering result

Estimation for multivariate concordance measures

Description

Estimation for multivariate concordance measures

Usage

concor.M(X, method = c("footrule"))

Arguments

Value

a value of the estimator for the multivariate concordance measures

Read a dendrogram from a list for hierarchical clustering result

Description

Read a dendrogram from a list for hierarchical clustering result

Usage

dendrogram(clust, step = TRUE)

Arguments

Value

an object of class "dendrogram"

Diameter of a class of variables based on different distance function

Description

Diameter of a class of variables based on different distance function

Usage

diam(X, dist.func = "PD", estim.method = c("copula"))

Arguments

Value

a value

Computes the directed dependence coefficient.

Description

The directed dependence coefficient (didec) estimates the degree of functional dependence of a random vector Y on a random vector X, based on an i.i.d. sample of (X,Y).

Usage

didec(
  X,
  Y,
  trans = FALSE,
  trans.method = c("standardization"),
  estim.method = c("copula"),
  perm = FALSE,
  perm.method = c("decreasing")
)

Arguments

Details

The directed dependence coefficient (didec) is an extension of Azadkia & Chatterjee's measure of functional dependence (Azadkia & Chatterjee, 2021) to a vector of response variables introduced in (Ansari & Fuchs, 2025). estim.method specifies two methods for estimating the directed dependence coefficient. "codec" uses the function codec which estimates Azadkia & Chatterjee’s measure of functional dependence and is provided in the R package FOCI. "copula" estimates the directed dependence coefficient based on a dimension reduction principle; see (Fuchs 2024). The value returned by didec may be positive or negative. In the asymptotic limit, however, it is guaranteed to lie between 0 and 1.

By definition, didec is invariant under permutations of the variables within the predictor vector X. Invariance under permutations within the q-dimensional response vector Y is achieved by computing the arithmetic mean over all possible permutations. In addition to the option "full" of running all q! permutations of (1, ..., q), less computationally intensive options are also available: a random selection of q permutations "sample", cyclic permutations such as (1,2,...,q), (2,...,q,1) either "increasing" or "decreasing". Note that when the number of variables q is large, choosing "full" may result in long computation times.

Value

The degree of functional dependence of the random vector Y on the random vector X.

Author(s)

Yuping Wang, Sebastian Fuchs, Jonathan Ansari

References

J. Ansari, S. Fuchs, A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors, Available at https://arxiv.org/abs/2212.01621, 2025.

M. Azadkia, S. Chatterjee, A simple measure of conditional dependence, Ann. Stat. 49 (6), 2021.

S. Fuchs, Quantifying directed dependence via dimension reduction, J. Multivariate Anal. 201, Article ID 105266, 2024.

distance function based on T

Description

distance function based on T

Usage

dist.Tq(X, Y, estim.method = c("copula"), mutual = FALSE)

Arguments

Value

a value for distance between two vectors

distance function based on multivariate concordance measures

Description

distance function based on multivariate concordance measures

Usage

dist.concor.M(X, Y, method = c("footrule"))

Arguments

Value

a value for distance between two vectors

Distance Matrix Computation using distance function based on T^q

Description

Distance Matrix Computation using distance function based on T^q

Usage

dist.mat.T(X, estim.method = c("copula"), mutual = FALSE)

Arguments

Value

an object of class "dist"

Distance Matrix Computation using distance function based on multivariate concordance measures

Description

Distance Matrix Computation using distance function based on multivariate concordance measures

Usage

dist.mat.concor(X, method = c("footrule"))

Arguments

Value

an object of class "dist"

Multivariate feature ordering by conditional independence.

Description

A variable selection algorithm based on the directed dependence coefficient (didec).

Usage

mfoci(
  X,
  Y,
  trans = FALSE,
  trans.method = c("standardization"),
  estim.method = c("copula"),
  perm = FALSE,
  perm.method = c("decreasing"),
  pre.selected = NULL,
  select.method = c("forward"),
  autostop = TRUE,
  max.num = NULL
)

Arguments

Details

mfoci involves a forward feature selection algorithm for multiple-outcome data that employs the directed dependence coefficient (didec) at each step.

If autostop == TRUE the algorithm stops at the first non-increasing value of didec, thereby selecting a subset of variables. Otherwise, all predictor variables are ranked according to their predictive strength measured by didec.

In addition to the forward feature selection algorithm, this function also provides a best subset selection, which can be accomplished by select.method == "subset". This method selects features by calculating the directed dependence coefficient of all possible feature combinations. Note that the features selected by this method are not ordered.

Value

A list containing:

features

A vector listing all features in X;

pre.selected.features

A vector listing the pre.selected features in X if pre.selected != NULL;

selected.features

A data.frame listing the selected and ranked variables and the corresponding values of the directed dependence coefficient if select.method == "forward"; A vector listing the selected features if select.method == "subset";

valueT

The values of the directed dependence coefficient if select.method == "subset".

Author(s)

Sebastian Fuchs, Jonathan Ansari, Yuping Wang

References

J. Ansari, S. Fuchs, A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors, Available at https://arxiv.org/abs/2212.01621, 2025.

Examples

library(didec)
df <- as.data.frame(bioclimatic)
X <- df[, c(9:12)]
Y <- df[, c(1,8)]
mfoci(X, Y, pre.selected = c(1, 3))

Plot the trade-off of Adiam and Msplit

Description

Plot the trade-off of Adiam and Msplit

Usage

plot_Adiam.Msplit(tradeoff, main = NULL, sub = NULL)

Arguments

Value

ggplot

Plot the Silhouette coefficient

Description

Plot the Silhouette coefficient

Usage

plot_Silhouette.coefficient(Silhouette_Index, main = NULL, sub = NULL)

Arguments

Value

ggplot

plotting a dendrogram with colored branches

Description

plotting a dendrogram with colored branches

Usage

plot_dendrogram(
  dend,
  num.cluster = num.cluster,
  linkage = FALSE,
  ylab = ylab,
  cex.lab = 0.6,
  cex.axis = 0.6
)

Arguments

Value

plot

Powerset without empty set

Description

Powerset without empty set

Usage

powerset(s)

Arguments

Value

a list

Split of two classes of variables based on different distance function

Description

Split of two classes of variables based on different distance function

Usage

split(X, Y, dist.func = "PD", estim.method = c("copula"))

Arguments

Value

a value