Content deleted Content added
MewTheEditor (talk | contribs) Moved second half of the lead to a section. |
|||
Line 10:
|chapter=8. Hierarchical Clustering | url=https://www.springer.com/gp/book/9783319219028 |chapter-url=https://www.researchgate.net/publication/314700681 }}</ref> are usually presented in a [[dendrogram]].
Hierarchical clustering has the distinct advantage that any valid measure of distance can be used. In fact, the observations themselves are not required: all that is used is a [[distance matrix|matrix of distances]]. On the other hand, except for the special case of single-linkage distance, none of the algorithms (except exhaustive search in <math>\mathcal{O}(2^n)</math>) can be guaranteed to find the optimum solution.▼
The standard algorithm for '''hierarchical agglomerative clustering''' (HAC) has a [[time complexity]] of <math>\mathcal{O}(n^3)</math> and requires <math>\Omega(n^2)</math> memory, which makes it too slow for even medium data sets. However, for some special cases, optimal efficient agglomerative methods (of complexity <math>\mathcal{O}(n^2)</math>) are known: '''SLINK'''<!--boldface per WP:R#PLA--><ref name="SLINK">{{cite journal | author=R. Sibson | title=SLINK: an optimally efficient algorithm for the single-link cluster method | journal=The Computer Journal | volume=16 | issue=1 | pages=30–34 | year=1973 | publisher=British Computer Society | url=http://www.cs.gsu.edu/~wkim/index_files/papers/sibson.pdf | doi=10.1093/comjnl/16.1.30| doi-access=free }}</ref> for [[Single-linkage clustering|single-linkage]] and CLINK<ref name="CLINK">{{cite journal | author=D. Defays | title=An efficient algorithm for a complete-link method | journal=The Computer Journal | volume=20 | issue=4 | pages=364–6 | year=1977 | publisher=British Computer Society | doi=10.1093/comjnl/20.4.364| doi-access=free }}</ref> for [[complete-linkage clustering]]. With a [[heap (data structure)|heap]], the runtime of the general case can be reduced to <math>\mathcal{O}(n^2 \log n)</math>, an improvement on the aforementioned bound of <math>\mathcal{O}(n^3)</math>, at the cost of further increasing the memory requirements. In many cases, the memory overheads of this approach are too large to make it practically usable.▼
Divisive clustering with an exhaustive search is <math>\mathcal{O}(2^n)</math>, but it is common to use faster heuristics to choose splits, such as [[k-means clustering|''k''-means]].▼
== Complexity ==
▲Hierarchical clustering has the distinct advantage that any valid measure of distance can be used. In fact, the observations themselves are not required: all that is used is a [[distance matrix|matrix of distances]].
▲The standard algorithm for '''hierarchical agglomerative clustering''' (HAC) has a [[time complexity]] of <math>\mathcal{O}(n^3)</math> and requires <math>\Omega(n^2)</math> memory, which makes it too slow for even medium data sets. However, for some special cases, optimal efficient agglomerative methods (of complexity <math>\mathcal{O}(n^2)</math>) are known: '''SLINK'''<!--boldface per WP:R#PLA--><ref name="SLINK">{{cite journal | author=R. Sibson | title=SLINK: an optimally efficient algorithm for the single-link cluster method | journal=The Computer Journal | volume=16 | issue=1 | pages=30–34 | year=1973 | publisher=British Computer Society | url=http://www.cs.gsu.edu/~wkim/index_files/papers/sibson.pdf | doi=10.1093/comjnl/16.1.30| doi-access=free }}</ref> for [[Single-linkage clustering|single-linkage]] and CLINK<ref name="CLINK">{{cite journal | author=D. Defays | title=An efficient algorithm for a complete-link method | journal=The Computer Journal | volume=20 | issue=4 | pages=364–6 | year=1977 | publisher=British Computer Society | doi=10.1093/comjnl/20.4.364| doi-access=free }}</ref> for [[complete-linkage clustering]]. With a [[heap (data structure)|heap]], the runtime of the general case can be reduced to <math>\mathcal{O}(n^2 \log n)</math>, an improvement on the aforementioned bound of <math>\mathcal{O}(n^3)</math>, at the cost of further increasing the memory requirements. In many cases, the memory overheads of this approach are too large to make it practically usable.
▲Divisive clustering with an exhaustive search is <math>\mathcal{O}(2^n)</math>, but it is common to use faster heuristics to choose splits, such as [[k-means clustering|''k''-means]].
== Cluster Linkage ==
| |||