Silhouette (clustering)
Silhouette refers to a method of interpretation and validation of consistency within clusters of data. The technique provides a succinct graphical representation of how well each object has been classified. It was proposed by Belgian statistician Peter Rousseeuw in 1987. The silhouette value is a measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation). The silhouette ranges from −1 to +1, where a high value indicates that the object is well matched to its own cluster and poorly matched to neighboring clusters. If most objects have a high value, then  the clustering configuration is appropriate. If many points have a low or negative value, then the clustering configuration may have too many or too few clusters. The silhouette can be calculated with any distance metric, such as the Euclidean distance or the Manhattan distance. Definition. Assume the data have been clustered via any technique, such as k-means, into formula_1 clusters. For data point formula_2 (data point formula_3 in the cluster formula_4), let to be the "smallest" (hence the formula_6 operator in the formula) mean distance of formula_3 to all points in any other cluster, of which formula_3 is not  a member. The cluster with this smallest mean dissimilarity is said to be the "neighboring cluster" of formula_3 because it is the next best fit cluster for point formula_3. We now define a "silhouette" (value) of one data point formula_3 and Which can be also written as: From the above definition it is clear that Note that formula_18 is not clearly defined for clusters with size = 1, in which case we set formula_19. This choice is arbitrary, but neutral in the sense that it is at the midpoint of the bounds, -1 and 1. For formula_20 to be close  to 1 we require formula_21. As formula_18 is a measure of how dissimilar formula_3 is to its own cluster, a small value means it is well matched. Furthermore, a large formula_24 implies that formula_3 is badly matched to its neighbouring cluster. Thus an formula_20 close to 1 means that the data is appropriately clustered. If formula_20 is close to -1, then by the same logic we see that formula_3 would be more appropriate if it was clustered in its neighbouring cluster. An formula_20 near zero means that the datum is on the border of two natural clusters. The mean formula_20  over all points of a cluster is a measure of how tightly grouped all the points in the cluster are. Thus the mean formula_20 over all data of the entire dataset is a measure of how appropriately the data have been clustered. If there are too many or too few clusters, as may occur when a poor choice of formula_1 is used in the clustering algorithm (e.g. k-means), some of the clusters will typically display much narrower silhouettes than the rest. Thus silhouette plots and means may be used to determine the natural number of clusters within a dataset . One can also increase the likelihood of the silhouette being maximized at the correct number of clusters by re-scaling the data using feature weights that are cluster specific. Kaufman et al. introduced the term "silhouette coefficient" for the maximum value of the mean formula_20 over all data of the entire dataset. Where formula_35 represents the mean formula_20 over all data of the entire dataset for a specific number of clusters formula_1.