Electronic Journal of Statistics

Change detection via affine and quadratic detectors

Yang Cao, Arkadi Nemirovski, Yao Xie, Vincent Guigues, and Anatoli Juditsky

Full-text: Open access

Abstract

The goal of the paper is to develop a specific application of the convex optimization based hypothesis testing techniques developed in A. Juditsky, A. Nemirovski, “Hypothesis testing via affine detectors,” Electronic Journal of Statistics 10:2204–2242, 2016. Namely, we consider the Change Detection problem as follows: observing one by one noisy observations of outputs of a discrete-time linear dynamical system, we intend to decide, in a sequential fashion, on the null hypothesis that the input to the system is a nuisance, vs. the alternative that the input is a “nontrivial signal,” with both the nuisances and the nontrivial signals modeled as inputs belonging to finite unions of some given convex sets. Assuming the observation noises are zero mean sub-Gaussian, we develop “computation-friendly” sequential decision rules and demonstrate that in our context these rules are provably near-optimal.

Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 1-57.

Dates
Received: February 2017
First available in Project Euclid: 3 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1514970025

Digital Object Identifier
doi:10.1214/17-EJS1373

Subjects
Primary: 62C20: Minimax procedures
Secondary: 90C22: Semidefinite programming

Keywords
Change-point detection semi-definite program

Rights
Creative Commons Attribution 4.0 International License.

Citation

Cao, Yang; Nemirovski, Arkadi; Xie, Yao; Guigues, Vincent; Juditsky, Anatoli. Change detection via affine and quadratic detectors. Electron. J. Statist. 12 (2018), no. 1, 1--57. doi:10.1214/17-EJS1373. https://projecteuclid.org/euclid.ejs/1514970025


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References

  • [1] M. Basseville and I. Nikiforov., Detection of Abrupt Changes: Theory and Application. Prentice-Hall, Englewood Cliffs, N.J., 1993.
  • [2] E. Brodsky and B. S. Darkhovsky., Nonparametric methods in change point problems, volume 243. Springer Science & Business Media, 2013.
  • [3] Y. Cao, S. Zhu, Y. Xie, J. Key, J. Kacher, R. R. Unocic, and C. M. Rouleau. Sequential adaptive detection for in-situ transmission electron microscopy (tem)., arXiv preprint arXiv:1710.11297, 2017.
  • [4] J. Chen and A. Gupta., Parametric statistical change point analysis: with applications to genetics, medicine, and finance. Boston: Birkhäuser, 2012.
  • [5] F. Enikeeva and Z. Harchaoui. High-dimensional change-point detection with sparse alternatives., arXiv preprint arXiv:1312.1900, 2013.
  • [6] G. Fellouris and G. Sokolov. Second-order asymptotic optimality in multisensor sequential change detection., IEEE Transactions on Information Theory, 62(6) :3662–3675, 2016.
  • [7] N. H. Gholson and R. L. Moose. Maneuvering target tracking using adaptive state estimation., IEEE Transactions on Aerospace and Electronic Systems, 13(3):310–317, 1977.
  • [8] A. Goldenshluger, A. Juditsky, and A. Nemirovski. Hypothesis testing by convex optimization., Electronic Journal of Statistics, 9(2) :1645–1712, 2015.
  • [9] A. Goldenshluger, A. Juditsky, A. Tsybakov, and A. Zeevi. Change–point estimation from indirect observations. 1. minimax complexity., Ann. Inst. Henri Poincare Probab. Stat., 44:787–818, 2008.
  • [10] A. Goldenshluger, A. Juditsky, A. Tsybakov, and A. Zeevi. Change-point estimation from indirect observations. 2. adaptation., Ann. Inst. H. Poincare Probab. Statist, 44(5):819–836, 2008.
  • [11] L. Gordon and M. Pollak. An efficient sequential nonparametric scheme for detecting a change of distribution., The Annals of Statistics, pages 763–804, 1994.
  • [12] V. Guigues. Nonparametric multidimensional breakpoint detection for the mean and correlations of a discrete time stochastic process., Journal of Nonparametric Statistics, 24:857–882, 2012.
  • [13] Y. I. Ingster, C. Pouet, and A. B. Tsybakov. Classification of sparse high-dimensional vectors., Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 367 (1906):4427–4448, 2009.
  • [14] Y. I. Ingster and I. A. Suslina. On detection of a signal of known shape in multi-channel system., Zapiski Nauchnykh Seminarov POMI, 294:88–112, 2002.
  • [15] Y. I. Ingster, A. B. Tsybakov, and N. Verzelen. Detection boundary in sparse regression., Electron. J. Statist., 4 :1476–1526, 2010.
  • [16] A. Juditski and A. Nemirovski. On sequential hypotheses testing via convex optimization., Automation and Remote Control, 76:809–825, 2015.
  • [17] A. Juditsky and A. Nemirovski. Hypothesis testing via affine detectors., Electronic Journal of Statistics, 10 :2204–2242, 2016.
  • [18] A. Korostelev and O. Lepski. On a multi-channel change-point problem., Mathematical Methods of Statistics, 17(3):187–197, 2008.
  • [19] T. L. Lai. Sequential changepoint detection in quality control and dynamical systems., Journal of the Royal Statistical Society. Series B (Methodological), pages 613–658, 1995.
  • [20] T. L. Lai. Information bounds and quick detection of parameter changes in stochastic systems., IEEE Transactions on Information Theory, 44(7) :2917–2929, 1998.
  • [21] A. Lakhina, M. Crovella, and C. Diot. Diagnosing network-wide traffic anomalies. In, ACM SIGCOMM Computer Communication Review, volume 34, pages 219–230. ACM, 2004.
  • [22] C. Lévy-Leduc and F. Roueff. Detection and localization of change-points in high-dimensional network traffic data., The Annals of Applied Statistics, pages 637–662, 2009.
  • [23] K. Liu, R. Zhang, and Y. Mei. Scalable sum-shrinkage schemes for distributed monitoring large-scale data streams., arXiv preprint arXiv:1603.08652, 2016.
  • [24] G. Lorden. Procedures for reacting to a change in distribution., The Annals of Mathematical Statistics, pages 1897–1908, 1971.
  • [25] E. Mazor, A. Averbuch, Y. Bar-Shalom, and J. Dayan. Interacting multiple model methods in target tracking: a survey., IEEE Transactions on Aerospace and Electronic Systems, 34(1):103–123, 1998.
  • [26] Y. Mei. Asymptotic optimality theory for decentralized sequential hypothesis testing in sensor networks., IEEE Transactions on Information Theory, 54(5) :2072–2089, 2008.
  • [27] G. V. Moustakides. Optimal stopping times for detecting changes in distributions., The Annals of Statistics, pages 1379–1387, 1986.
  • [28] M. H. Neumann. Optimal change-point estimation in inverse problems., Scandinavian Journal of Statistics, 24(4):503–521, 1997.
  • [29] E. Page. Continuous inspection schemes., Biometrika, 41(1/2):100–115, 1954.
  • [30] M. Pollak. Optimal detection of a change in distribution., The Annals of Statistics, pages 206–227, 1985.
  • [31] M. Pollak. Average run lengths of an optimal method of detecting a change in distribution., The Annals of Statistics, pages 749–779, 1987.
  • [32] H. V. Poor and O. Hadjiliadis., Quickest detection, volume 40. Cambridge University Press Cambridge, 2009.
  • [33] W. A. Shewhart., Economic control of quality of manufactured product. ASQ Quality Press, 1931.
  • [34] A. N. Shiryaev. On optimum methods in quickest detection problems., Theory of Probability & Its Applications, 8(1):22–46, 1963.
  • [35] D. Siegmund., Sequential Analysis: Tests and Confidence Intervals. Springer Science & Business Media, 1985.
  • [36] D. Siegmund and B. Yakir., The statistics of gene mapping. Springer Science & Business Media, 2007.
  • [37] A. Tartakovsky, I. Nikiforov, and M. Basseville., Sequential analysis: Hypothesis testing and changepoint detection. CRC Press, 2014.
  • [38] A. G. Tartakovsky and V. V. Veeravalli. Change-point detection in multichannel and distributed systems., Applied Sequential Methodologies: Real-World Examples with Data Analysis, 173:339–370, 2004.
  • [39] A. G. Tartakovsky and V. V. Veeravalli. Asymptotically optimal quickest change detection in distributed sensor systems., Sequential Analysis, 27(4):441–475, 2008.
  • [40] V. V. Veeravalli and T. Banerjee. Quickest change detection., Academic press library in signal processing: Array and statistical signal processing, 3:209–256, 2013.
  • [41] A. S. Willsky., Detection of abrupt changes in dynamic systems. Springer, 1985.
  • [42] Y. Xie and D. Siegmund. Sequential multi-sensor change-point detection., Annals of Statistics, 41(2):670–692, 2013.