Rights statement: c 2016 Azadeh Khaleghi, Daniil Ryabko, Jeremie Mary and Philippe Preux.
Accepted author manuscript, 477 KB, PDF document
Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License
Final published version, 477 KB, PDF document
Available under license: CC BY: Creative Commons Attribution 4.0 International License
Final published version
Licence: CC BY: Creative Commons Attribution 4.0 International License
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Consistent algorithms for clustering time series
AU - Khaleghi, Azedeh
AU - Ryabko, Daniil
AU - Mari, Jeremie
AU - Preux, Philippe
N1 - c 2016 Azadeh Khaleghi, Daniil Ryabko, Jeremie Mary and Philippe Preux.
PY - 2016/3
Y1 - 2016/3
N2 - The problem of clustering is considered for the case where every point is a time series. The time series are either given in one batch (offline setting), or they are allowed to grow with time and new time series can be added along the way (online setting). We propose a natural notion of consistency for this problem, and show that there are simple, computationally efficient algorithms that are asymptotically consistent under extremely weak assumptions on the distributions that generate the data. The notion of consistency is as follows. A clustering algorithm is called consistent if it places two time series into the same cluster if and only if the distribution that generates them is the same. In the considered framework the time series are allowed to be highly dependent, and the dependence can have arbitrary form. If the number of clusters is known, the only assumption we make is that the (marginal) distribution of each time series is stationary ergodic. No parametric, memory or mixing assumptions are made. When the number of clusters is unknown, stronger assumptions are provably necessary, but it is still possible to devise nonparametric algorithms that are consistent under very general conditions. The theoretical findings of this work are illustrated with experiments on both synthetic and real data.
AB - The problem of clustering is considered for the case where every point is a time series. The time series are either given in one batch (offline setting), or they are allowed to grow with time and new time series can be added along the way (online setting). We propose a natural notion of consistency for this problem, and show that there are simple, computationally efficient algorithms that are asymptotically consistent under extremely weak assumptions on the distributions that generate the data. The notion of consistency is as follows. A clustering algorithm is called consistent if it places two time series into the same cluster if and only if the distribution that generates them is the same. In the considered framework the time series are allowed to be highly dependent, and the dependence can have arbitrary form. If the number of clusters is known, the only assumption we make is that the (marginal) distribution of each time series is stationary ergodic. No parametric, memory or mixing assumptions are made. When the number of clusters is unknown, stronger assumptions are provably necessary, but it is still possible to devise nonparametric algorithms that are consistent under very general conditions. The theoretical findings of this work are illustrated with experiments on both synthetic and real data.
KW - clustering
KW - time series
KW - ergodicity
KW - unsupervised learning
M3 - Journal article
VL - 17
SP - 1
EP - 32
JO - Journal of Machine Learning Research
JF - Journal of Machine Learning Research
SN - 1532-4435
IS - 3
ER -