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Estimating Dynamic Graphical Models from Multivariate Time-Series Data: Recent Methods and Results

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Advanced Analysis and Learning on Temporal Data (AALTD 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9785))

Abstract

Dynamic graphical models aim to describe the time-varying dependency structure of multiple time-series. In this article we review research focusing on the formulation and estimation of such models. The bulk of work in graphical structurelearning problems has focused in the stationary i.i.d setting, we present a brief overview of this work before introducing some dynamic extensions. In particular we focuson two classes of dynamic graphical model; continuous (smooth) models which are estimated via localised kernels, and piecewise models utilising regularisation based estimation. We give an overview of theoretical and empirical results regarding these models, before demonstrating their qualitative difference in the context of a real-world financial time-series dataset. We conclude with a discussion of the state of the field and future research directions.

D.B. Nelson—This work is funded by the Defence Science Technology Laboratory (Dstl) National PhD Scheme.

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Notes

  1. 1.

    For simplicity, in this paper we assume the mean parameter is zero \(\varvec{\mu }=\varvec{0}\).

  2. 2.

    Note: the model here refers to the sparsity pattern, rather than the fact that the distribution is Gaussian.

  3. 3.

    Note: we set the first changepoint (denoted by \(k=0\)) at \(\tau _{0}=1\) and the last \(K+1\)st changepoint is located at \(\tau _{K+1}=T\).

  4. 4.

    The matrix \(\varvec{\varSigma }^{o}\) is known as an oracle estimator, as it has access to the ground truth \(\varvec{\varSigma }_{*}\) through the risk function \(\mathcal {R}(\cdot )\).

  5. 5.

    We note that Harcharoui et al. perform their changepoint analysis in a reformulated lasso problem rather than considering directly jumps in \(\hat{\varvec{u}}\) as presented here.

  6. 6.

    In the univariate setting for functional approximation this is analogous to the fused lasso of Tibshirani et al. [40].

  7. 7.

    If we choose \(\hat{\varvec{S}}^{t}\) to be estimated through a localised kernel as in (14) we obtain the SINGLE estimator of Monti et al. [32], if we use a Dirac delta kernel i.e. \(\hat{\varvec{S}}^{t}=\varvec{y}^{\top }\varvec{y}/2\) then we recover the IFGL estimator of our previous work [18]. Note: the TESLA approach of Ahmed et al. [1] uses a regulariser term like \(R_{\mathrm {IFGL}}\) but with a logistic loss function for binary variables.

  8. 8.

    The data can be obtained from Ken French’s website http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

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Authors and Affiliations

  1. Department of Statistical Science, University College London, Gower Street, London, WC1E 6BT, UK

    Alex J. Gibberd & James D. B. Nelson

  2. Department of Security and Crime Science, University College London, London, UK

    Alex J. Gibberd

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  1. Alex J. Gibberd
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Correspondence to Alex J. Gibberd .

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Editors and Affiliations

  1. Lab. d'Informatique de Grenoble, Université Grenoble, Grenoble, France

    Ahlame Douzal-Chouakria

  2. Universidade da Coruna, Coruna, Spain

    José A. Vilar

  3. IRISA, Université de Bretagne-Sud, Vannes, France

    Pierre-François Marteau

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Gibberd, A.J., Nelson, J.D.B. (2016). Estimating Dynamic Graphical Models from Multivariate Time-Series Data: Recent Methods and Results. In: Douzal-Chouakria, A., Vilar, J., Marteau, PF. (eds) Advanced Analysis and Learning on Temporal Data. AALTD 2015. Lecture Notes in Computer Science(), vol 9785. Springer, Cham. https://doi.org/10.1007/978-3-319-44412-3_8

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