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==The Monti consensus clustering algorithm==
The Monti consensus clustering algorithm<ref>{{Cite journal|last=Monti|first=Stefano|last2=Tamayo|first2=Pablo|last3=Mesirov|first3=Jill|last4=Golub|first4=Todd|date=2003-07-01|title=Consensus Clustering: A Resampling-Based Method for Class Discovery and Visualization of Gene Expression Microarray Data|journal=Machine Learning|language=en|volume=52|issue=1|pages=91–118|doi=10.1023/A:1023949509487|issn=1573-0565|doi-access=free}}</ref> is one of the most popular consensus clustering algorithms and is used to determine the number of clusters, <math>K</math>. Given a dataset of <math>N</math> total number of points to cluster, this algorithm works by resampling and clustering the data, for each <math>K</math> and a <math>NXN</math> consensus matrix is calculated, where each element represents the fraction of times two samples clustered together. A perfectly stable matrix would consist entirely of zeros and ones, representing all sample pairs always clustering together or not together over all resampling iterations. The relative stability of the consensus matrices can be used to infer the optimal <math>K</math>.
More specifically, given a set of points to cluster, <math>D=\{e_1,e_2,...e_N\}</math>, let <math>D^1,D^2,...,D^H</math> be the list of <math>H</math> pertubed (resampled) datasets of the original dataset <math>D</math>, and let <math>M^h</math> denote the <math>NXN</math> connectivity matrix resulting from applying a clustering algorithm to the dataset <math>D^h</math>. The entries of <math>M^h</math> are defined as follows:
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==Over-interpretation potential of the Monti consensus clustering algorithm==
[[File:PACexplained.png|400px|thumb|PAC measure (proportion of ambiguous clustering) explained. Optimal K is the K with lowest PAC value.]]
Monti consensus clustering can be a powerful tool for identifying clusters, but it needs to be applied with caution as shown by Şenbabaoğlu ''et al.'' <ref name="SenbabaogluSREP" /> It has been shown that the Monti consensus clustering algorithm is able to claim apparent stability of chance partitioning of null datasets drawn from a unimodal distribution, and thus has the potential to lead to over-interpretation of cluster stability in a real study.<ref name=SenbabaogluSREP>{{cite journal|last=Şenbabaoğlu|first=Y.|author2=Michailidis, G. |author3=Li, J. Z. |title=Critical limitations of consensus clustering in class discovery|journal=Scientific Reports|date=2014|doi=10.1038/srep06207|volume=4|pages=6207|pmid=25158761|pmc=4145288|bibcode=2014NatSR...4E6207.}}</ref><ref name=SenbabaogluRXV>{{cite biorxiv|last=Şenbabaoğlu|first=Y.|author2=Michailidis, G. |author3=Li, J. Z. |title=A reassessment of consensus clustering for class discovery|date=Feb 2014|biorxiv=10.1101/002642}}</ref> If clusters are not well separated, consensus clustering could lead one to conclude apparent structure when there is none, or declare cluster stability when it is subtle. Identifying false positive clusters is a common problem throughout cluster research,<ref name=":0">{{Cite journal|last=Liu|first=Yufeng|last2=Hayes|first2=David Neil|last3=Nobel|first3=Andrew|last4=Marron|first4=J. S.|date=2008-09-01|title=Statistical Significance of Clustering for High-Dimension, Low–Sample Size Data|journal=Journal of the American Statistical Association|volume=103|issue=483|pages=1281–1293|doi=10.1198/016214508000000454|issn=0162-1459}}</ref> and has been addressed by methods such as SigClust<ref name=":0" /> and the GAP-statistic.<ref>{{Cite journal|last=Tibshirani|first=Robert|last2=Walther|first2=Guenther|last3=Hastie|first3=Trevor|date=2001|title=Estimating the number of clusters in a data set via the gap statistic|journal=Journal of the Royal Statistical Society: Series B (Statistical Methodology)|language=en|volume=63|issue=2|pages=411–423|doi=10.1111/1467-9868.00293|issn=1467-9868}}</ref> However, these methods rely on certain assumptions for the null model that may not always be appropiate.
Şenbabaoğlu ''et al'' <ref name="SenbabaogluSREP" /> demonstrated the original delta K metric to decide <math>K</math> in the Monti algorithm performed poorly, and proposed a new superior metric for measuring the stability of consensus matrices using their CDF curves. In the CDF curve of a consensus matrix, the lower left portion represents sample pairs rarely clustered together, the upper right portion represents those almost always clustered together, whereas the middle segment represent those with ambiguous assignments in different clustering runs. The proportion of ambiguous clustering (PAC) score measure quantifies this middle segment; and is defined as the fraction of sample pairs with consensus indices falling in the interval (u<sub>1</sub>, u<sub>2</sub>) ∈ [0, 1] where u<sub>1</sub> is a value close to 0 and u<sub>2</sub> is a value close to 1 (for instance u<sub>1</sub>=0.1 and u<sub>2</sub>=0.9). A low value of PAC indicates a flat middle segment, and a low rate of discordant assignments across permuted clustering runs. One can therefore infer the optimal number of clusters by the <math>K</math> value having the lowest PAC.<ref name="SenbabaogluSREP" /><ref name="SenbabaogluRXV" />
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