The uncertainty of changepoints in time series

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Nam, Christopher F. H.;
  • Subject: QA
    arxiv: Statistics::Computation | Statistics::Methodology

Analysis concerning time series exhibiting changepoints have predominantly\ud focused on detection and estimation. However, changepoint estimates such as their\ud number and location are subject to uncertainty which is often not captured explicitly,\ud or requires sampl... View more
  • References (76)
    76 references, page 1 of 8

    Chapter 1 Introduction 1 1.1 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    Chapter 2 Literature Review 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 At Most One Change . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Binary Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Penalised Likelihood approaches . . . . . . . . . . . . . . . . . . . . 18 2.6 Global Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.7 Pruned Exact Linear Time algorithm . . . . . . . . . . . . . . . . . . 23 2.8 AutoPARM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.9 Bayesian Model Selection methods . . . . . . . . . . . . . . . . . . . 28 2.10 Reversible Jump Markov Chain Monte Carlo . . . . . . . . . . . . . 31 2.11 Product-Partition models . . . . . . . . . . . . . . . . . . . . . . . . 32 2.12 Hidden Markov Models based methods . . . . . . . . . . . . . . . . . 35 2.12.1 Deterministic State Sequence Inference . . . . . . . . . . . . . 37 2.12.2 Exact CP Distributions . . . . . . . . . . . . . . . . . . . . . 42 2.12.3 Constrained HMMs . . . . . . . . . . . . . . . . . . . . . . . 43 2.13 Exact Sampling of the Posterior via Recursions . . . . . . . . . . . . 47 2.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    Chapter 4 Quantifying the Uncertainty of Brain Activity 98 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.2 A Brief Introduction to Neuroimaging and fMRI . . . . . . . . . . . 99 4.2.1 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2.2 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3 An Anxiety Inducing Experiment . . . . . . . . . . . . . . . . . . . . 107 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 112

    Chapter 5 Model Selection for Hidden Markov Models 116 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3.1 An Information Theoretic Approach . . . . . . . . . . . . . . 119 5.3.2 Parallel Markov Chain Monte Carlo . . . . . . . . . . . . . . 120 5.3.3 Reversible Jump Markov Chain Monte Carlo . . . . . . . . . 121 5.3.4 Sequential Hidden Markov Model . . . . . . . . . . . . . . . . 123 5.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.1 Approximating p(y1:n|H) . . . . . . . . . . . . . . . . . . . . 128 5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.5.1 Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5.2 Hamilton's GNP data . . . . . . . . . . . . . . . . . . . . . . 138 5.6 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 141 8

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    • Nam, C. F. H., Aston, J. A. D., and Johansen, A. M. (2012b). Quantifying the uncertainty in change points. Journal of Time Series Analysis, 33(5):807-823

    • Nam, C. F. H., Aston, J. A. D., and Johansen, A. M. (2012a). Parallel Sequential Monte Carlo samplers and estimation of the number of states in a Hidden Markov model. CRiSM Research Report, 12(23)

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