OPTICS algorithm
Ordering points to identify the clustering structure is an algorithm for finding density-based clusters in spatial data. It was presented by Mihael Ankerst, Markus M. Breunig, Hans-Peter Kriegel and Jörg Sander.
Its basic idea is similar to DBSCAN, but it addresses one of DBSCAN's major weaknesses: the problem of detecting meaningful clusters in data of varying density. To do so, the points of the database are ordered such that spatially closest points become neighbors in the ordering. Additionally, a special distance is stored for each point that represents the density that must be accepted for a cluster so that both points belong to the same cluster. This is represented as a dendrogram.
Basic idea
Like DBSCAN, OPTICS requires two parameters:, which describes the maximum distance to consider, and, describing the number of points required to form a cluster. A point is a core point if at least points are found within its -neighborhood . In contrast to DBSCAN, OPTICS also considers points that are part of a more densely packed cluster, so each point is assigned a core distance that describes the distance to the th closest point:The reachability-distance of another point from a point is either the distance between and, or the core distance of, whichever is bigger:
If and are nearest neighbors, this is the we need to assume to have and belong to the same cluster.
Both core-distance and reachability-distance are undefined if no sufficiently dense cluster is available. Given a sufficiently large, this never happens, but then every -neighborhood query returns the entire database, resulting in runtime. Hence, the parameter is required to cut off the density of clusters that are no longer interesting, and to speed up the algorithm.
The parameter is, strictly speaking, not necessary. It can simply be set to the maximum possible value. When a spatial index is available, however, it does play a practical role with regards to complexity. OPTICS abstracts from DBSCAN by removing this parameter, at least to the extent of only having to give the maximum value.
Pseudocode
The basic approach of OPTICS is similar to DBSCAN, but instead of maintaining a set of known, but so far unprocessed cluster members, a priority queue is used.function OPTICS is
for each point p of DB do
p.reachability-distance = UNDEFINED
for each unprocessed point p of DB do
N = getNeighbors
mark p as processed
output p to the ordered list
if core-distance != UNDEFINED then
Seeds = empty priority queue
update
for each next q in Seeds do
N' = getNeighbors
mark q as processed
output q to the ordered list
if core-distance != UNDEFINED do
update
In update, the priority queue Seeds is updated with the -neighborhood of and, respectively:
function update is
coredist = core-distance
for each o in N
if o is not processed then
new-reach-dist = max
if o.reachability-distance UNDEFINED then // o is not in Seeds
o.reachability-distance = new-reach-dist
Seeds.insert
else // o in Seeds, check for improvement
if new-reach-dist < o.reachability-distance then
o.reachability-distance = new-reach-dist
Seeds.move-up
OPTICS hence outputs the points in a particular ordering, annotated with their smallest reachability distance.
Extracting the clusters
Using a reachability-plot, the hierarchical structure of the clusters can be obtained easily. It is a 2D plot, with the ordering of the points as processed by OPTICS on the x-axis and the reachability distance on the y-axis. Since points belonging to a cluster have a low reachability distance to their nearest neighbor, the clusters show up as valleys in the reachability plot. The deeper the valley, the denser the cluster.The image above illustrates this concept. In its upper left area, a synthetic example data set is shown. The upper right part visualizes the spanning tree produced by OPTICS, and the lower part shows the reachability plot as computed by OPTICS. Colors in this plot are labels, and not computed by the algorithm; but it is well visible how the valleys in the plot correspond to the clusters in above data set. The yellow points in this image are considered noise, and no valley is found in their reachability plot. They are usually not assigned to clusters, except the omnipresent "all data" cluster in a hierarchical result.
Extracting clusters from this plot can be done manually by selecting a range on the x-axis after visual inspection, by selecting a threshold on the y-axis, or by different algorithms that try to detect the valleys by steepness, knee detection, or local maxima. Clusterings obtained this way usually are hierarchical, and cannot be achieved by a single DBSCAN run.
Complexity
Like DBSCAN, OPTICS processes each point once, and performs one -neighborhood query during this processing. Given a spatial index that grants a neighborhood query in runtime, an overall runtime of is obtained. The authors of the original OPTICS paper report an actual constant slowdown factor of 1.6 compared to DBSCAN. Note that the value of might heavily influence the cost of the algorithm, since a value too large might raise the cost of a neighborhood query to linear complexity.In particular, choosing is possible, but leads to quadratic complexity, since every neighborhood query returns the full data set. Even when no spatial index is available, this comes at additional cost in managing the heap. Therefore, should be chosen appropriately for the data set.
Extensions
OPTICS-OF is an outlier detection algorithm based on OPTICS. The main use is the extraction of outliers from an existing run of OPTICS at low cost compared to using a different outlier detection method. The better known version LOF is based on the same concepts.DeLi-Clu, Density-Link-Clustering combines ideas from single-linkage clustering and OPTICS, eliminating the parameter and offering performance improvements over OPTICS.
HiSC is a hierarchical subspace clustering method based on OPTICS.
HiCO is a hierarchical correlation clustering algorithm based on OPTICS.
DiSH is an improvement over HiSC that can find more complex hierarchies.
FOPTICS is a faster implementation using random projections.
HDBSCAN* is based on a refinement of DBSCAN, excluding border-points from the clusters and thus following more strictly the basic definition of density-levels by Hartigan.
Availability
Java implementations of OPTICS, OPTICS-OF, DeLi-Clu, HiSC, HiCO and DiSH are available in the ELKI data mining framework. Other Java implementations include the Weka extension.The R package "dbscan" includes a C++ implementation of OPTICS using a k-d tree for index acceleration for Euclidean distance only.
Python implementations of OPTICS are available in the library and in scikit-learn. HDBSCAN* is available in the library.