# k-Prototypes Clustering Computes k-prototypes clustering for mixed-type data. ```r 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.). ## Returns `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. - **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`. ## 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. ## Examples ```r # 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) ``` ## References * Szepannek, G. (2018): clustMixType: User-Friendly Clustering of Mixed-Type Data in R, **The R Journal 10/2**, 200-208, tools:::Rd_expr_doi("10.32614/RJ-2018-048") . * Aschenbruck, R., Szepannek, G., Wilhelm, A. (2022): Imputation Strategies for Clustering Mixed‑Type Data with Missing Values, **Journal of Classification**, tools:::Rd_expr_doi("10.1007/s00357-022-09422-y") . * Szepannek, G., Aschenbruck, R., Wilhelm, A. (2024): Clustering Large Mixed-Type Data with Ordinal Variables, **Advances in Data Analysis and Classification**, tools:::Rd_expr_doi("10.1007/s11634-024-00595-5") . * 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. ## Author(s) gero.szepannek@web.de

kproto function