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clustMixType (version 0.2-5)

validation_kproto: Validating k Prototypes Clustering

Description

Calculating the prefered 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 = NULL,
  object = NULL,
  data = NULL,
  k = NULL,
  kp_obj = "optimal",
  ...
)

Arguments

method

character specifying the validation index: cindex, dunn, gamma, gplus, mcclain, ptbiserial, silhouette 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.

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.

kp_obj

character either "optimal" or "all": Output of the index-optimal clustering (kp_obj == "optimal") or all computed clusterpartitions (kp_obj == "all"); only required if object != NULL.

...

Further arguments passed to kproto, like:

  • nstart: If > 1 repetetive computations of kproto with random initializations are computed.

  • lambda: Factor to trade off between Euclidean distance of numeric variables and simple matching coefficient between categorical variables.

  • verbose: Logical whether information about the cluster procedure should be given. Caution: If verbose=FALSE, the reduction of the number of clusters is not mentioned.

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

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 nessesary 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 samecluster 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.

References

  • 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. www.jstatsoft.org.

Examples

Run this code
# 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)

# }

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