# Framed Data Clustering Find a frame of given size, among all possible such frames on the input data, to minimize the minimum within-cluster sum of square distances. ```r FramedClust( X, K, frame.size, first.frame = 1, last.frame = length(X) - frame.size + 1, method = c("linear.polylog", "kmeans", "Ckmeans.1d.dp") ) ``` ## Arguments - `X`: a vector of data points to perform framed clustering - `K`: the number of clusters in each frame - `frame.size`: the number of points from X to be included in each frame. It is not the width of the frame. - `first.frame`: starting index of the first frame to be clustered. The first point in the first frame is `X[first.frame]`. - `last.frame`: starting index of the last frame to be clustered. The first point in the first frame is `X[last.frame]`. - `method`: the framed clustering method. See Details. ## Returns An object of class `"FramedClust"` which has a `plot` method. It is a list with the following components: - **cluster**: a vector of clusters assigned to each element in x. Each cluster is indexed by an integer from 1 to K. NA represents points from X that are outside the optimal frame, thus not part of any cluster. - **centers**: a numeric vector of the means for each cluster in the frame. - **withinss**: a numeric vector of the within-cluster sum of squared distances for each cluster. - **size**: a vector of the number of elements in each cluster. - **totss**: total sum of squared distances between each element and the sample mean. This statistic is not dependent on the clustering result. - **tot.withinss**: total sum of within-cluster squared distances between each element and its cluster mean. This statistic is minimized given the number of clusters. - **betweenss**: sum of squared distances between each cluster mean and sample mean. This statistic is maximized given the number of clusters. - **X_name**: a character string. The actual name of the `X` argument. ## Details The method option `"linear.polylog"` (default) performs fast optimal framed clustering. The runtime is $O(K N log^2 N)$ if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="Debnath21",package="OptCirClust",cached_env=.Rdpack.currefs) . The `"kmeans"` option repeatedly calling the heuristic k-means algorithm in all frames without any guarantee of cluster optimality. The method option `"Ckmeans.1d.dp"` performs optimal framed clustering by repeatedly finding the best clustering within each frame using the `"Ckmeans.1d.dp"` method. At a runtime of $O(K N^2)$, the algorithm is slow but optimal. It is included to provide a baseline. ## Examples ```r N <- 100 X <- rnorm(N) K <- 5 frame.size <- 60 result <- FramedClust(X, K, frame.size) plot(result, main="Example 1. Framed clustering on all frames") frame.size <- 40 first.frame <- 30 last.frame <- 50 method <- "linear.polylog" result <- FramedClust(X, K, frame.size, first.frame, last.frame, method) plot(result, main="Example 2. Framed clustering on a subset of frames") ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs)

FramedClust function