Package 'clustMixType'

Profiling k-Prototypes Clustering

Description

Visualization of a k-prototypes clustering result for cluster interpretation.

Usage

clprofiles(object, x, vars = NULL, col = NULL)

Arguments

object	Object resulting from a call of resulting kproto. Also other kmeans like objects with object$cluster and object$size are possible.
x	Original data.
vars	Optional vector of either column indices or variable names.
col	Palette of cluster colours to be used for the plots. As a default RColorBrewer's brewer.pal(max(unique(object$cluster)), "Set3") is used for k > 2 clusters and lightblue and orange else.

Details

For numerical variables boxplots and for factor variables barplots of each cluster are generated.

Author(s)

[email protected]

Examples

# generate toy data with factors and numerics

n   <- 100
prb <- 0.9
muk <- 1.5 
clusid <- rep(1:4, each = n)

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)

x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)

x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))

x <- data.frame(x1,x2,x3,x4)

# apply k-prototyps
kpres <- kproto(x, 4)
clprofiles(kpres, x)

# in real world clusters are often not as clear cut
# by variation of lambda the emphasize is shifted towards factor / numeric variables    
kpres <- kproto(x, 2)
clprofiles(kpres, x)

kpres <- kproto(x, 2, lambda = 0.1)
clprofiles(kpres, x)

kpres <- kproto(x, 2, lambda = 25)
clprofiles(kpres, x)

---

Variable Importance

Description

Computes two types of variable importance for an object of class kproto.

Usage

importance_kproto(object, rand = TRUE)

Arguments

object	Object of class kproto.
rand	Logical indicating whether additional variable importance based on rand indices should be computed.

Details

The contribution of the variables to object$tot.withinss is computed as well as those for a null model where all variables are in the same cluster. Variable importance is computed as variable-wise improvement of object$tot.withinss.

As proposed by Henning and Murphy (2023) the importance of a variable can also be assessed by comparing how much the resulting clusters will change if a variable is ignored: For rand == TRUE in addition variable importance is computed by reclustering the data where the variable of interest is replaced by a constant value. Variable importance (VI) is computed by comparing the original partition with the one obtained after reclustering the data using adjusted Rand indices as $VI = 1-Rand_{adj}$.

As reclustering for each variable may be time consuming, it may be descelected.

Note that for na.rm = "imp.onestep" and "imp.internal" the imputed data are used. For na.rm = "yes" observations with missing values are removed from the data and for na.rm = "no" NAs are ignored for distance computation just as in kproto().

Value

List of three elements:

vi_decomp	Data frame where the rows are: the contribution of the variables to object$tot.withinss, the the contribution of the variables in a null model, the differences and releatiove differences
vi_rand	Vector of variable importances based on the adjusted Rand index after reclustering.

Author(s)

[email protected]

References

• Szepannek, G. (2018): clustMixType: User-Friendly Clustering of Mixed-Type Data in R The R Journal 10/2, 200-208, doi:10.32614/RJ-2018-048.

• Hennig, C., and K. Murphy (2023): Quantyfying Variable Importance in Cluster Analysis, CLADAG 2023 Book of Abstracts and Short Papers, 515-518, Pearson Education Resources.

• Aschenbruck, R., Szepannek, G., Wilhelm, A. (2022): Imputation Strategies for Clustering Mixed‑Type Data with Missing Values, Journal of Classification, doi:10.1007/s00357-022-09422-y.

.

Examples

# generate toy data with factors and numerics

n   <- 100
prb <- 0.9
muk <- 1.5 
clusid <- rep(1:4, each = n)

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)

x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)

x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))

x <- data.frame(x1,x2,x3,x4)

# apply k-prototypes
kpres <- kproto(x, 4)

# compute different kinds of variable importance
importance_kproto(kpres, rand = TRUE)

# reducing the influence of the categorical variables (X1 and X2) on the clustering:
kpres <- kproto(x, 4, lambda = 0.01)
importance_kproto(kpres, rand = FALSE)

# reducing the importance of the numeric variables (X3 and X4) on the clustering:
kpres <- kproto(x, 4, lambda = 100)
importance_kproto(kpres, rand = FALSE)

# weighting all variables equally:
kpres <- kproto(x, 4, lambda = 1)
importance_kproto(kpres, rand = FALSE)

---

k-Prototypes Clustering

Description

Computes k-prototypes clustering for mixed-type data.

Usage

kproto(x, ...)

## Default S3 method:
kproto(
  x,
  k,
  lambda = NULL,
  type = "huang",
  iter.max = 100,
  nstart = 1,
  na.rm = "yes",
  keep.data = TRUE,
  verbose = TRUE,
  init = NULL,
  p_nstart.m = 0.9,
  ...
)

Arguments

x	Data frame with both numerics and factors (also ordered factors are possible).
...	Currently not used.
k	Either the number of clusters, a vector specifying indices of initial prototypes, or a data frame of prototypes of the same columns as x.
lambda	Parameter > 0 to trade off between Euclidean distance of numeric variables and simple matching coefficient between categorical variables (if type = "huang"). Also a vector of variable specific factors is possible where the order must correspond to the order of the variables in the data. In this case all variables' distances will be multiplied by their corresponding lambda value.
type	Character, to specify the distance for clustering. Either "huang" or "gower" (cf. details below).
iter.max	Numeric; maximum number of iterations if no convergence before.
nstart	Numeric; If > 1 repetitive computations with random initializations are computed and the result with minimum tot.dist is returned.
na.rm	Character, either "yes" to strip NA values for complete case analysis, "no" to keep and ignore NA values, "imp.internal" to impute the NAs within the algorithm or "imp.onestep" to apply the algorithm ignoring the NAs and impute them after the partition is determined.
keep.data	Logical, whether original should be included in the returned object.
verbose	Logical, whether additional information about process should be printed. Caution: For verbose=FALSE, if the number of clusters is reduced during the iterations it will not mentioned.
init	Character, to specify the initialization strategy. Either "nbh.dens", "sel.cen" or "nstart.m". Default is "NULL", which results in nstart repetitive algorithm computations with random starting prototypes. Otherwise, nstart is not used. Argument k must be a number if a specific initialization strategy is choosen!
p_nstart.m	Numeric, probability(=0.9 is default) for init="nstart.m", where the strategy assures that with a probability of p_nstart.m at least one of the m sets of initial prototypes contains objects of every cluster group (cf. Aschenbruck et al. (2023): Random-based Initialization for clustering mixed-type data with the k-Prototypes algorithm. In: Cladag 2023 Book of abstracts and short spapers, isbn: 9788891935632.).

Details

Like k-means, the k-prototypes algorithm iteratively recomputes cluster prototypes and reassigns clusters, whereby with type = "huang" clusters are assigned using the distance $d(x,y) = d_{euclid}(x,y) + \lambda d_{simple\,matching}(x,y)$. Cluster prototypes are computed as cluster means for numeric variables and modes for factors (cf. Huang, 1998). Ordered factors variables are treated as categorical variables.
For type = "gower" range-normalized absolute distances from the cluster median are computed for the numeric variables (and for the ranks of the ordered factors respectively). For factors simple matching distance is used as in the original k prototypes algorithm. The prototypes are given by the median for numeric variables, the mode for factors and the level with the closest rank to the median rank of the corresponding cluster (cf. Szepannek et al., 2024).
In case of na.rm = FALSE: for each observation variables with missings are ignored (i.e. only the remaining variables are considered for distance computation). In consequence for observations with missings this might result in a change of variable's weighting compared to the one specified by lambda. For these observations distances to the prototypes will typically be smaller as they are based on fewer variables.
The type argument also accepts input "standard", but this naming convention is deprecated and has been renamed to "huang". Please use "huang" instead.

Value

kmeans like object of class kproto:

cluster	Vector of cluster memberships.
centers	Data frame of cluster prototypes.
lambda	Distance parameter lambda.
size	Vector of cluster sizes.
withinss	Vector of within cluster distances for each cluster, i.e. summed distances of all observations belonging to a cluster to their respective prototype.
tot.withinss	Target function: sum of all observations' distances to their corresponding cluster prototype.
dists	Matrix with distances of observations to all cluster prototypes.
iter	Prespecified maximum number of iterations.
trace	List with two elements (vectors) tracing the iteration process: tot.dists and moved number of observations over all iterations.
inits	Initial prototypes determined by specified initialization strategy, if init is either 'nbh.dens' or 'sel.cen'.
nstart.m	only for 'init = nstart_m': determined number of randomly choosen sets.
data	if 'keep.data = TRUE' than the original data will be added to the output list.
type	type argument of the function call.
k.init	k argument of the function call.
iter.max	iter.max argument of the function call.
nstart	nstart argument of the function call.
na.rm	na.rm argument of the function call.
init	init argument of the function call.
p_nstart.m	p_nstart.m argument of the function call.
stdization	Only returned for type = "gower": List of standardized ranks for ordinal variables and an additional element num_ranges with ranges of all numeric variables. Used by predict.kproto.

Author(s)

[email protected]

References

• Szepannek, G. (2018): clustMixType: User-Friendly Clustering of Mixed-Type Data in R, The R Journal 10/2, 200-208, doi:10.32614/RJ-2018-048.

• Z.Huang (1998): Extensions to the k-Means Algorithm for Clustering Large Data Sets with Categorical Variables, Data Mining and Knowledge Discovery 2, 283-304.

• Szepannek, G., Aschenbruck, R., Wilhelm, A. (2024): Clustering Large Mixed-Type Data with Ordinal Variables, Advances in Data Analysis and Classification, doi:10.1007/s11634-024-00595-5.

• Aschenbruck, R., Szepannek, G., Wilhelm, A. (2022): Imputation Strategies for Clustering Mixed‑Type Data with Missing Values, Journal of Classification, doi:10.1007/s00357-022-09422-y.

Examples

# generate toy data with factors and numerics

n   <- 100
prb <- 0.9
muk <- 1.5 
clusid <- rep(1:4, each = n)

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)

x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)

x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))

x <- data.frame(x1,x2,x3,x4)

# apply k-prototypes
kpres <- kproto(x, 4)
clprofiles(kpres, x)

# in real world clusters are often not as clear cut
# by variation of lambda the emphasize is shifted towards factor / numeric variables    
kpres <- kproto(x, 2)
clprofiles(kpres, x)

kpres <- kproto(x, 2, lambda = 0.1)
clprofiles(kpres, x)

kpres <- kproto(x, 2, lambda = 25)
clprofiles(kpres, x)

---

Compute individual Silhouette Widths

Description

Computes an object of class silhouette from k-prototypes clustering for mixed-type data.

Usage

kproto2silhouette(object)

Arguments

object

Object of class kproto.

Details

Information on the individual silhouette width for all observations is computed. The k-prototypes result of class kproto is turned into an object of class silhouette that can be further used with the functions summary.silhouette or plot.silhouette.

Value

Object of class silhouette to be further used with summary.silhouette or plot.silhouette. It consists in a matrix where each row corresponds to one observation and the columns contain the assigned cluster as well as the neighbor cluster as well as the individual silhouette width ob the observation.

Author(s)

[email protected]

References

• Szepannek, G. (2018): clustMixType: User-Friendly Clustering of Mixed-Type Data in R, The R Journal 10/2, 200-208. doi:10.32614/RJ-2018-048.

• Aschenbruck, R., Szepannek, G. (2020): Cluster Validation for Mixed-Type Data. Archives of Data Science, Series A, Vol 6, Issue 1. doi:10.5445/KSP/1000098011/02.

• Kaufman, L., Rousseeuw, P. (1990): Finding Groups in Data: An Introduction to Cluster Analysis. Wiley.

Examples

# generate toy data with factors and numerics

n   <- 50
prb <- 0.9
muk <- 1.5 
clusid <- rep(1:4, each = n)

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)

x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)

x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))

x <- data.frame(x1,x2,x3,x4)

# apply k-prototypes
kpres <- kproto(x, 4)

# compute silhouette widhts
# extract object of class sil
sils <- kproto2silhouette(kpres)

# apply functions from package cluster
library(cluster)
summary(sils)
plot(sils)

# compute average silhouette width  
mean(sils[,3])

# ...alternatively, compute average silhouette width directly using validation_kproto() 
cindex_value <- validation_kproto(method = "silhouette", object = kpres)
cindex_value

---

Compares Variability of Variables

Description

Investigation of the variables' variances/concentrations to support specification of lambda for k-prototypes clustering.

Usage

lambdaest(
  x,
  num.method = 1,
  fac.method = 1,
  outtype = "numeric",
  verbose = TRUE
)

Arguments

x	Data.frame with both numerics and factors.
num.method	Integer 1 or 2. Specifies the heuristic used for numeric variables.
fac.method	Integer 1 or 2. Specifies the heuristic used for factor variables.
outtype	Specifies the desired output: either 'numeric', 'vector' or 'variation'.
verbose	Logical whether additional information about process should be printed.

Details

Variance (num.method = 1) or standard deviation (num.method = 2) of numeric variables and $1-\sum_i p_i^2$ (fac.method = 1) or $1-\max_i p_i$ (fac.method = 2) for factors is computed.

Value

lambda

Ratio of averages over all numeric/factor variables is returned. In case of outtype = "vector" the separate lambda for all variables is returned as the inverse of the single variables' variation as specified by the num.method and fac.method argument. outtype = "variation" directly returns these quantities and is not meant to be passed directly to kproto().

Author(s)

[email protected]

Examples

# generate toy data with factors and numerics

n   <- 100
prb <- 0.9
muk <- 1.5 
clusid <- rep(1:4, each = n)

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)

x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)

x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))

x <- data.frame(x1,x2,x3,x4)

lambdaest(x)
res <- kproto(x, 4, lambda = lambdaest(x))

---

Assign k-Prototypes Clusters

Description

Plot distributions of the clusters across the variables.

Usage

## S3 method for class 'kproto'
plot(x, ...)

Arguments

x	Object resulting from a call of kproto.
...	Additional arguments to be passet to clprofiles such as e.g. vars.

Details

Wrapper around clprofiles. Only works for kproto object created with keep.data = TRUE.

Author(s)

[email protected]

Examples

# generate toy data with factors and numerics

n   <- 100
prb <- 0.9
muk <- 1.5 
clusid <- rep(1:4, each = n)

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)

x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)

x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))

x <- data.frame(x1,x2,x3,x4)

# apply k-prototyps
kpres <- kproto(x, 4)
plot(kpres, vars = c("x1","x3"))

---

Assign k-Prototypes Clusters

Description

Predicts k-prototypes cluster memberships and distances for new data.

Usage

## S3 method for class 'kproto'
predict(object, newdata, ...)

Arguments

object	Object resulting from a call of kproto.
newdata	New data frame (of same structure) where cluster memberships are to be predicted.
...	Currently not used.

Value

kmeans like object of class kproto:

cluster	Vector of cluster memberships.
dists	Matrix with distances of observations to all cluster prototypes.

Author(s)

[email protected]

Examples

# generate toy data with factors and numerics

n   <- 100
prb <- 0.9
muk <- 1.5 
clusid <- rep(1:4, each = n)

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)

x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)

x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))

x <- data.frame(x1,x2,x3,x4)

# apply k-prototyps
kpres <- kproto(x, 4)
predicted.clusters <- predict(kpres, x)

---

Determination of the stability of k Prototypes Clustering

Description

Calculating the stability for a k-Prototypes clustering with k clusters or computing the stability-based optimal number of clusters for k-Prototype clustering. Possible stability indices are: Jaccard, Rand, Fowlkes \& Mallows and Luxburg.

Usage

stability_kproto(
  object,
  method = c("rand", "jaccard", "luxburg", "fowlkesmallows"),
  B = 100,
  verbose = FALSE,
  ...
)

Arguments

object	Object of class kproto resulting from a call with kproto(..., keep.data=TRUE)
method	character specifying the stability, either one or more of luxburg, fowlkesmallows, rand or/and jaccard.
B	numeric, number of bootstrap samples
verbose	Logical whether information about the bootstrap procedure should be given.
...	Further arguments passed to kproto, like: nstart: If > 1 repetitive computations of kproto with random initial prototypes are computed. lambda: Factor to trade off between Euclidean distance of numeric variables and simple matching coefficient between categorical variables.

Value

The output contains the stability for a given k-Prototype clustering in a list with two elements:

kp_stab	stability values for the given clustering
kp_bts_stab	stability values for each bootstrap samples

Author(s)

Rabea Aschenbruck

References

• Aschenbruck, R., Szepannek, G., Wilhelm, A.F.X (2023): Stability of mixed-type cluster partitions for determination of the number of clusters. In: In: Brito, P., Dias, J.G., Lausen, B., Montanari, A., Nugent, R. (eds) Classification and Data Science in the Digital Age. IFCS 2022. Studies in Classification, Data Analysis, and Knowledge Organization, doi:10.1007/978-3-031-09034-9_6.

• von Luxburg, U. (2010): Clustering stability: an overview. Foundations and Trends in Machine Learning, Vol 2, Issue 3. doi:10.1561/2200000008.

• Ben-Hur, A., Elisseeff, A., Guyon, I. (2002): A stability based method for discovering structure in clustered data. Pacific Symposium on Biocomputing. doi:10/bhfxmf.

Examples

## Not run: 
# generate toy data with factors and numerics
n   <- 10
prb <- 0.99
muk <- 2.5 

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)
x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)
x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x <- data.frame(x1,x2,x3,x4)

#' # apply k-prototypes
kpres <- kproto(x, 4, keep.data = TRUE)

# calculate cluster stability
stab <- stability_kproto(method = c("luxburg","fowlkesmallows"), object = kpres)


## End(Not run)

---

Summary Method for kproto Cluster Result

Description

Investigation of variances to specify lambda for k-prototypes clustering.

Usage

## S3 method for class 'kproto'
summary(object, data = NULL, pct.dig = 3, ...)

Arguments

object	Object of class kproto.
data	Optional data set to be analyzed. If !(is.null(data)) clusters for data are assigned by predict(object, data). If not specified the clusters of the original data ara analyzed which is only possible if kproto has been called using keep.data = TRUE.
pct.dig	Number of digits for rounding percentages of factor variables.
...	Further arguments to be passed to internal call of summary() for numeric variables.

Details

For numeric variables statistics are computed for each clusters using summary(). For categorical variables distribution percentages are computed.

Value

List where each element corresponds to one variable. Each row of any element corresponds to one cluster.

Author(s)

[email protected]

Examples

# generate toy data with factors and numerics

n   <- 100
prb <- 0.9
muk <- 1.5 
clusid <- rep(1:4, each = n)

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)

x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)

x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))

x <- data.frame(x1,x2,x3,x4)

res <- kproto(x, 4)
summary(res)

---

Validating k Prototypes Clustering

Description

Calculating the preferred validation index for a k-Prototypes clustering with k clusters or computing the optimal number of clusters based on the choosen index for k-Prototype clustering. Possible validation indices are: cindex, dunn, gamma, gplus, mcclain, ptbiserial, silhouette and tau.

Usage

validation_kproto(
  method = "silhouette",
  object = NULL,
  data = NULL,
  type = "huang",
  k = NULL,
  lambda = NULL,
  kp_obj = "optimal",
  verbose = FALSE,
  ...
)

Arguments

method	Character specifying the validation index: cindex, dunn, gamma, gplus, mcclain, ptbiserial, silhouette (default) or tau.
object	Object of class kproto resulting from a call with kproto(..., keep.data=TRUE).
data	Original data; only required if object == NULL and neglected if object != NULL.
type	Character, to specify the distance for clustering; either "huang" or "gower".
k	Vector specifying the search range for optimum number of clusters; if NULL the range will set as 2:sqrt(n). Only required if object == NULL and neglected if object != NULL.
lambda	Factor to trade off between Euclidean distance of numeric variables and simple matching coefficient between categorical variables.
kp_obj	character either "optimal" or "all": Output of the index-optimal clustering (kp_obj == "optimal") or all computed cluster partitions (kp_obj == "all"); only required if object != NULL.
verbose	Logical, whether additional information about process should be printed.
...	Further arguments passed to kproto, like: nstart: If > 1 repetitive computations of kproto with random initializations are computed. na.rm: Character, either "yes" to strip NA values for complete case analysis, "no" to keep and ignore NA values, "imp.internal" to impute the NAs within the algorithm or "imp.onestep" to apply the algorithm ignoring the NAs and impute them after the partition is determined.

Details

More information about the implemented validation indices:

• cindex

  $Cindex = \frac{S_w-S_{min}}{S_{max}-S_{min}}$

  
  For $S_{min}$ and $S_{max}$ it is necessary to calculate the distances between all pairs of points in the entire data set ($\frac{n(n-1)}{2}$). $S_{min}$ is the sum of the "total number of pairs of objects belonging to the same cluster" smallest distances and $S_{max}$ is the sum of the "total number of pairs of objects belonging to the same cluster" largest distances. $S_w$ is the sum of the within-cluster distances.
  The minimum value of the index is used to indicate the optimal number of clusters.

• dunn

  $Dunn = \frac{\min_{1 \leq i < j \leq q} d(C_i, C_j)}{\max_{1 \leq k \leq q} diam(C_k)}$

  
  The following applies: The dissimilarity between the two clusters $C_i$ and $C_j$ is defined as $d(C_i, C_j)=\min_{x \in C_i, y \in C_j} d(x,y)$ and the diameter of a cluster is defined as $diam(C_k)=\max_{x,y \in C} d(x,y)$.
  The maximum value of the index is used to indicate the optimal number of clusters.

• gamma

  $Gamma = \frac{s(+)-s(-)}{s(+)+s(-)}$

  
  Comparisons are made between all within-cluster dissimilarities and all between-cluster dissimilarities. $s(+)$ is the number of concordant comparisons and $s(-)$ is the number of discordant comparisons. A comparison is named concordant (resp. discordant) if a within-cluster dissimilarity is strictly less (resp. strictly greater) than a between-cluster dissimilarity.
  The maximum value of the index is used to indicate the optimal number of clusters.

• gplus

  $Gplus = \frac{2 \cdot s(-)}{\frac{n(n-1)}{2} \cdot (\frac{n(n-1)}{2}-1)}$

  
  Comparisons are made between all within-cluster dissimilarities and all between-cluster dissimilarities. $s(-)$ is the number of discordant comparisons and a comparison is named discordant if a within-cluster dissimilarity is strictly greater than a between-cluster dissimilarity.
  The minimum value of the index is used to indicate the optimal number of clusters.

• mcclain

  $McClain = \frac{\bar{S}_w}{\bar{S}_b}$

  
  $\bar{S}_w$ is the sum of within-cluster distances divided by the number of within-cluster distances and $\bar{S}_b$ is the sum of between-cluster distances divided by the number of between-cluster distances.
  The minimum value of the index is used to indicate the optimal number of clusters.

• ptbiserial

  $Ptbiserial = \frac{(\bar{S}_b-\bar{S}_w) \cdot (\frac{N_w \cdot N_b}{N_t^2})^{0.5}}{s_d}$

  
  $\bar{S}_w$ is the sum of within-cluster distances divided by the number of within-cluster distances and $\bar{S}_b$ is the sum of between-cluster distances divided by the number of between-cluster distances.
  $N_t$ is the total number of pairs of objects in the data, $N_w$ is the total number of pairs of objects belonging to the same cluster and $N_b$ is the total number of pairs of objects belonging to different clusters. $s_d$ is the standard deviation of all distances.
  The maximum value of the index is used to indicate the optimal number of clusters.

• silhouette

  $Silhouette = \frac{1}{n} \sum_{i=1}^n \frac{b(i)-a(i)}{max(a(i),b(i))}$

  
  $a(i)$ is the average dissimilarity of the ith object to all other objects of the same/own cluster. $b(i)=min(d(i,C))$, where $d(i,C)$ is the average dissimilarity of the ith object to all the other clusters except the own/same cluster.
  The maximum value of the index is used to indicate the optimal number of clusters.

• tau

  $Tau = \frac{s(+) - s(-)}{((\frac{N_t(N_t-1)}{2}-t)\frac{N_t(N_t-1)}{2})^{0.5}}$

  
  Comparisons are made between all within-cluster dissimilarities and all between-cluster dissimilarities. $s(+)$ is the number of concordant comparisons and $s(-)$ is the number of discordant comparisons. A comparison is named concordant (resp. discordant) if a within-cluster dissimilarity is strictly less (resp. strictly greater) than a between-cluster dissimilarity.
  $N_t$ is the total number of distances $\frac{n(n-1)}{2}$ and $t$ is the number of comparisons of two pairs of objects where both pairs represent within-cluster comparisons or both pairs are between-cluster comparisons.
  The maximum value of the index is used to indicate the optimal number of clusters.

Value

For computing the optimal number of clusters based on the choosen validation index for k-Prototype clustering the output contains:

k_opt	optimal number of clusters (sampled in case of ambiguity)
index_opt	index value of the index optimal clustering
indices	calculated indices for $k=2,...,k_{max}$
kp_obj	if(kp_obj == "optimal") the kproto object of the index optimal clustering and if(kp_obj == "all") all kproto which were calculated

For computing the index-value for a given k-Prototype clustering the output contains:

index

calculated index-value

Author(s)

Rabea Aschenbruck

References

• Aschenbruck, R., Szepannek, G. (2020): Cluster Validation for Mixed-Type Data. Archives of Data Science, Series A, Vol 6, Issue 1. doi:10.5445/KSP/1000098011/02.

• Charrad, M., Ghazzali, N., Boiteau, V., Niknafs, A. (2014): NbClust: An R Package for Determining the Relevant Number of Clusters in a Data Set. Journal of Statistical Software, Vol 61, Issue 6. doi:10.18637/jss.v061.i06.

Examples

## Not run: 
# generate toy data with factors and numerics
n   <- 10
prb <- 0.99
muk <- 2.5 

x1 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x1 <- c(x1, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x1 <- as.factor(x1)
x2 <- sample(c("A","B"), 2*n, replace = TRUE, prob = c(prb, 1-prb))
x2 <- c(x2, sample(c("A","B"), 2*n, replace = TRUE, prob = c(1-prb, prb)))
x2 <- as.factor(x2)
x3 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x4 <- c(rnorm(n, mean = -muk), rnorm(n, mean = muk), rnorm(n, mean = -muk), rnorm(n, mean = muk))
x <- data.frame(x1,x2,x3,x4)


# calculate optimal number of cluster, index values and clusterpartition with Silhouette-index
val <- validation_kproto(method = "silhouette", data = x, k = 3:5, nstart = 5)


# apply k-prototypes
kpres <- kproto(x, 4, keep.data = TRUE)

# calculate cindex-value for the given clusterpartition
cindex_value <- validation_kproto(method = "cindex", object = kpres)

## End(Not run)

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