# Cluster analysis: disjoint regions Formation of disjoint regions for Regional Frequency Analysis. ```r traceWminim (X, centers) sumtraceW (clusters, X) nearest (clusters, X) ``` ## Arguments - `X`: a numeric matrix of characteristics, or an object that can be coerced to such a matrix (such as a numeric vector or a data frame with all numeric columns) - `centers`: the number of clusters - `clusters`: a numeric vector containing the subdivision of `X` in clusters ## Details The Euclidean distance is used. Given $p$ different classification variables, the distance between two elements $i$ and $j$ is: $$ d_{i j} = \sqrt{\frac{1}{p} \sum_{h=1}^{p} (x_{h i} - x_{h j})^2}d_ij = \sqrt{1/p sum[h from 1 to p](x_hi - x_hj)^2} $$ where $x_hi$ is the value of the $h$-th variable of the $i$-th element. The function `traceWminim` is a composition of a jerarchical algorithm, the Ward (1963) one, and an optimisation procedure consisting in the minimisation of: $$ W = \sum_{i=1}^k \left( \sum_{j=1}^{n_i} \delta_{i j}^2 \right)W = sum[i from 1 to k](sum[j from 1 to ni]delta_ij^2) $$ where $k$ is the number of clusters (obtained initially with Ward's algorithm), $ni$ is the number of sites in the $i$-th cluster and $delta_ij$ is the Euclidean distance between the $j$-th element of the $i$-th group and the center of mass of the $i$-th cluster. $W$ is calculated with `sumtraceW`. The algorithm consist in moving a site from one cluster to another if this makes $W$ decrease. ## Returns `traceWminim` gives a vector defining the subdivision of elements characterized by `X` in n=`centers` clusters. `sumtraceW` gives $W$ (it is used by `traceWminim`). `nearest` gives the nearest site to the centers of mass of clusters (it is used by `traceWminim`). ## Note For information on the package and the Author, and for all the references, see `nsRFA`. ## See Also `roi`, `AD.dist`. ## Examples ```r data(hydroSIMN) parameters summary(parameters) # traceWminim param <- parameters[c("Hm","Ybar")] n <- dim(param)[1]; k <- dim(param)[2] param.norm <- (param - matrix(apply(param,2,mean),nrow=n,ncol=k, byrow=TRUE))/matrix(apply(param,2,sd), nrow=n,ncol=k,byrow=TRUE) clusters <- traceWminim(param.norm,4); names(clusters) <- parameters["cod"][,] clusters annualflows summary(annualflows) x <- annualflows["dato"][,] cod <- annualflows["cod"][,] fac <- factor(annualflows["cod"][,], levels=names(clusters[clusters==1])) x1 <- annualflows[!is.na(fac),"dato"] cod1 <- annualflows[!is.na(fac),"cod"] #HW.tests(x1,cod1) # it takes some time fac <- factor(annualflows["cod"][,], levels=names(clusters[clusters==3])) x3 <- annualflows[!is.na(fac),"dato"] cod3 <- annualflows[!is.na(fac),"cod"] #HW.tests(x3,cod3) # it takes some time ```

traceWminim function