Elbow method (clustering): Difference between revisions

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| url = http://www3.interscience.wiley.com/cgi-bin/fulltext/17435/PDFSTART

| doi = 10.1002/(SICI)1097-0266(199606)17:6<441::AID-SMJ819>3.0.CO;2-G

| url-access = subscription

}}{{dead link|date=February 2019|bot=medic}}{{cbignore|bot=medic}}</ref>

This can even hold in cases where all other methods for [[determining the number of clusters in a data set]] (as mentioned in that article) agree on the number of clusters.

[[File:Elbow in Inertia on uniform data.png|thumb|alt=Plot of the sum of squared errors (SSE) as k increases, following a typical 1/k shape.|Example of the typical "elbow" pattern used for choosing the number of clusters even emerging on uniform data.]]

Even on uniform random data (with no meaningful clusters) the curve follows approximately the ratio ''1/k'' where ''k'' is the number of clusters parameter, causing users to see an "elbow" to mistakenly choose some "optimal" number of clusters.<ref name=":0" /><!-- This should be trivial to reproduce from the article, choose 2d uniform data, cluster with k=1 to 20 and plot the resulting "elbow". -->

Because the two axes (the number of clusters and the remaining variance) have no semantic relationship, various attempt to capture the elbow by "slope" are ill-defined and sensitive to the parameter range.<ref name=":0">{{Cite journal |last=Schubert |first=Erich |date=2023-07-05 |title=Stop using the elbow criterion for k-means and how to choose the number of clusters instead |url=https://doi.org/10.1145/3606274.3606278 |journal=ACM SIGKDD Explorations Newsletter |volume=25 |issue=1 |pages=36–42 |doi=10.1145/3606274.3606278 |issn=1931-0145|arxiv=2212.12189 }}</ref> Increasing the maximum number of clusters can change the ___location of the perceived "elbow", and in many cases alternate heuristics such as the [[Calinski–Harabasz index|variance-rarioratio-criterion]] or the [[Silhouette (clustering)|average silhouette width]] are considered to be more reliable.<ref name=":0" /> But even with such measures, the results may depend much on the data preprocessing (feature selection and scaling) and users may come to very different clustering results on the same data.

== Measures of variation ==