# Validating k Prototypes Clustering 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`. ```r 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. ## Returns 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 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 i**th** 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 i**th** 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. ## Examples ```r ## 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) ``` ## References * Aschenbruck, R., Szepannek, G. (2020): Cluster Validation for Mixed-Type Data. **Archives of Data Science, Series A, Vol 6, Issue 1**. tools:::Rd_expr_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**. tools:::Rd_expr_doi("10.18637/jss.v061.i06") . ## Author(s) Rabea Aschenbruck

validation_kproto function