Abstract
This chapter is devoted to the problems of time-series clustering and change-point analysis. Clustering is grouping together samples generated by the same distributions, while change-point problems are concerned with delimiting parts of a sample that are generated by a different process distributions. Building on the results of the previous chapter, here we are trying to solve these more general problems avoiding the need to answer the “same-different” question about process distributions (discrimination), and only using the asymptotically consistent distance estimates that we have. It turns out that this is enough to solve the clustering problem when the number of clusters is known, as well as several versions of the changepoint problem, without the need to make any assumptions beyond stationarity and ergodicity. In particular, the means, variances, or even single-dimensional marginals of all the processes in question may be the same. For the change-point problem, the main focus of this chapter is on the case of a single change; on overview of the more general formulations is given, with the corresponding results presented without proofs.
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M. Basseville and I.V. Nikiforov. Detection of abrupt changes: theory and application. Prentice Hall information and system sciences series. Prentice Hall, 1993.
B.E. Brodsky and B.S. Darkhovsky. Nonparametric methods in change-point problems. Mathematics and its applications. Kluwer Academic Publishers, 1993.
E. Carlstein and S. Lele. Nonparametric change-point estimation for data from an ergodic sequence. Teor. Veroyatnost. i Primenen., 38:910–917, 1993.
L Giraitis, R Leipus, and D Surgailis. The change-point problem for dependent observations. JStat Plan and Infer, pages 1–15, 1995.
Robert M. Gray. Probability, Random Processes, and Ergodic Properties. Springer Verlag, 1988.
I. Katsavounidis, C.-C. Jay Kuo, and Zhen Zhang. A new initialization technique for generalized Lloyd iteration. IEEE Signal Processing Letters, 1:144–146, 1994.
Azadeh Khaleghi and Daniil Ryabko. Locating changes in highly dependent data with unknown number of change points. In P. Bartlett, F.C.N. Pereira, C.J.C. Burges, L. Bottou, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems 25, pages 3095–3103. 2012.
Azadeh Khaleghi and Daniil Ryabko. Asymptotically consistent estimation of the number of change points in highly dependent time series. In ICML, JMLR W&CP, pages 539–547, Beijing, China, 2014.
Azadeh Khaleghi and Daniil Ryabko. Nonparametric multiple change point estimation in highly dependent time series. Theoretical Computer Science, 620:119–133, 2016.
Azadeh Khaleghi, Daniil Ryabko, Jérémie Mary, and Philippe Preux. Consistent algorithms for clustering time series. Journal of Machine Learning Research, 17:1–32, 2016.
J. Kleinberg. An impossibility theorem for clustering. In 15th Conf. Neiral Information Processing Systems (NIPS’02), pages 446–453, Montreal, Canada, 2002. MIT Press.
M. Mahajan, P. Nimbhorkar, and K. Varadarajan. The planar k-means problem is NP-hard. In WALCOM ’09: Proceedings of the 3rd International Workshop on Algorithms and Computation, pages 274–285, Berlin, Heidelberg, 2009. Springer-Verlag.
D. Ryabko and B. Ryabko. Nonparametric statistical inference for ergodic processes. IEEE Transactions on Information Theory, 56(3):1430–1435, 2010.
Daniil Ryabko. Clustering processes. In Proc. the 27th International Conference on Machine Learning (ICML 2010), pages 919–926, Haifa, Israel, 2010.
Daniil Ryabko. Independence clustering (without a matrix). In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 4013–4023. Curran Associates, Inc., 2017.
R. Zadeh and S. Ben-David. A uniqueness theorem for clustering. In A. Ng J. Bilmes, editor, Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence (UAI’09), Montreal, Canada, 2009.
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Ryabko, D. (2019). Clustering and Change-Point Problems. In: Asymptotic Nonparametric Statistical Analysis of Stationary Time Series. SpringerBriefs in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-12564-6_4
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DOI: https://doi.org/10.1007/978-3-030-12564-6_4
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