Probability Ridges and Distortion Flows: Visualizing Multivariate Time Series Using a Variational Bayesian Manifold Learning Method

  • Alessandra Tosi
  • Iván Olier
  • Alfredo Vellido
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 295)


Time-dependent natural phenomena and artificial processes can often be quantitatively expressed as multivariate time series (MTS). As in any other process of knowledge extraction from data, the analyst can benefit from the exploration of the characteristics of MTS through data visualization. This visualization often becomes difficult to interpret when MTS are modelled using nonlinear techniques. Despite their flexibility, nonlinear models can be rendered useless if such interpretability is lacking. In this brief paper, we model MTS using Variational Bayesian Generative Topographic Mapping Through Time (VB-GTM-TT), a variational Bayesian variant of a constrained hidden Markov model of the manifold learning family defined for MTS visualization. We aim to increase its interpretability by taking advantage of two results of the probabilistic definition of the model: the explicit estimation of probabilities of transition between states described in the visualization space and the quantification of the nonlinear mapping distortion.


Multivariate time series Nonlinear dimensionality reduction Mapping distortion Magnification Factors Visualization Generative Topographic Mapping Variational Bayesian methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fu, T.C.: A Review on Time Series Data Mining. Engineering Applications of Artificial Intelligence 24(1), 164–181 (2011)CrossRefGoogle Scholar
  2. 2.
    Vellido, A., Martín-Guerrero, J.D., Lisboa, P.J.G.: Making Machine Learning Models Interpretable. In: ESANN 2012, pp. 163–172. d-Side Pub. (2012)Google Scholar
  3. 3.
    Vellido, A., Martín, J.D., Rossi, F., Lisboa, P.J.G.: Seeing is Believing: The Importance of Visualization in Real-World Machine Learning Applications. In: ESANN 2011, pp. 219–226. d-Side Pub. (2011)Google Scholar
  4. 4.
    Lee, J.A., Verleysen, M.: Nonlinear Dimensionality Reduction. Springer (2007)Google Scholar
  5. 5.
    Van Belle, V.: Lisboa. P.: Research Directions in Interpretable Machine Learning Models. In: ESANN 2013, pp. 533–541. Pub. (2013)Google Scholar
  6. 6.
    Bishop, C.M., Hinton, G.E., Strachan, I.G.D.: GTM Through Time. In: Fifth International Conference on Artificial Neural Networks, pp. 111–116 (1997)Google Scholar
  7. 7.
    Rabiner, L.R.: A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE 77(2), 257–286 (1989)CrossRefGoogle Scholar
  8. 8.
    Olier, I., Vellido: A Variational Formulation for GTM Through Time. In: International Joint Conference on Neural Networks (IJCNN 2008), pp. 517-522 (2008)Google Scholar
  9. 9.
    Bishop, C.M., Svensén, M., Williams, C.K.I.: GTM: The Generative Topographic Mapping. Neural Computation 10, 215–234 (1998)CrossRefGoogle Scholar
  10. 10.
    Bishop, C.M., Svensén, M., Williams, C.K.I.: Developments of the Generative Topographic Mapping. Neurocomputing 21(1), 203–224 (1998)zbMATHCrossRefGoogle Scholar
  11. 11.
    Olier, I., Vellido, A.: Variational Bayesian Generative Topographic Mapping. Journal of Mathematical Modelling and Algorithms 7(4), 371–387 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Olier, I., Amengual, J., Vellido, A.: A Variational Bayesian Approach for the Robust Estimation of Cortical Silent Periods from EMG Time Series of Brain Stroke Patients. Neurocomputing 74(9), 1301–1314 (2011)CrossRefGoogle Scholar
  13. 13.
    Bishop, C.M., Svensén, M., Williams, C.K.I.: Magnification Factors for the SOM and GTM Algorithms. In: Proceedings of the 1997 Workshop on Self-Organizing Maps (WSOM), pp. 333–338 (1997)Google Scholar
  14. 14.
    Tosi, A., Vellido, A.: Robust Cartogram Visualization of Outliers in Manifold Learning. In: ESANN 2013, pp. 555–560. Pub. (2013)Google Scholar
  15. 15.
    Lin, J., Vlachos, M., Keogh, E., Gunopulos, D.: Iterative Incremental Clustering of Time Series. In: Bertino, E., Christodoulakis, S., Plexousakis, D., Christophides, V., Koubarakis, M., Böhm, K. (eds.) EDBT 2004. LNCS, vol. 2992, pp. 106–122. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Lawrence, N.: Probabilistic Non-Linear Principal Component Analysis with Gaussian Process Latent Variable Models. The Journal of Machine Learning Research 6, 1783–1816 (2005)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Damianou, A.C., Titsias, M.K., Lawrence, N.D.: Variational Gaussian Process Dynamical Systems. In: Advances in Neural Information Processing Systems, NIPS (2011)Google Scholar
  18. 18.
    Wang, J.M., Fleet, D.J., Hertzmann, A.: Gaussian Process Dynamical Models for Human Motion. IEEE Transactions on Pattern Analysis and Machine Intelligence 30(2), 283–298 (2008)CrossRefGoogle Scholar
  19. 19.
    Lewandowski, M., Martínez-del-Rincón, J., Makris, D., Nebel, J.C.: Temporal Extension of Laplacian Eigenmaps for Unsupervised Dimensionality Reduction of Time Series. In: 20th International Conference on Pattern Recognition (ICPR), pp. 161–164. IEEE (2013)Google Scholar
  20. 20.
    Tosi, A., Vellido, A.: Cartogram Representation of the Batch-SOM Magnification Factor. In: ESANN 2012, pp. 203–208 (2012)Google Scholar
  21. 21.
    Vellido, A., García, D., Nebot, À.: Cartogram Visualization for Nonlinear Manifold Learning Models. Data Mining and Knowledge Discovery 27(1), 22–54 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gianniotis, N.: Interpretable magnification factors for topographic maps of high dimensional and structured data. In: IEEE CIDM, pp. 238–245 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Alessandra Tosi
    • 1
  • Iván Olier
    • 2
  • Alfredo Vellido
    • 1
  1. 1.Dept. de Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Manchester Institute of BiotechnologyThe University of ManchesterManchesterUK

Personalised recommendations