Research ArticleAPPLIED MATHEMATICS

Improving clustering by imposing network information

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Science Advances  07 Aug 2015:
Vol. 1, no. 7, e1500163
DOI: 10.1126/sciadv.1500163
  • Fig. 1 An example of the imposed network and a cluster model discrimination.

    (A) Imposed (linear) graph: a priori persistency assumption for the underlying dynamics in time. (B) Comparing information content of EEG clusterings: graphs of the AIC values for K = 1 to 3 as a function of the regularization constant ϵ2.

  • Fig. 2 Visualization of the two identified manifolds.
  • Fig. 3 Snapshots of the spatiotemporal dynamics of the most dominant eigenvectors.

    (A, C, E, and G) Several time instances of the extracted dominant wave pattern for the EEG with opened eyes. (B, D, F, and H) Snapshots for the dominant EEG pattern with closed eyes at the same time points. Red color stands for the positive component of the oscillation, and blue color for the negative component. Snapshots are taken in both experiments at time points t = 0.0 s (A and B), 0.018443 s (C and D), 0.043033 s (E and F), and 0.061475 s (G and H).

  • Standard Clustering
    Iteratively repeat until convergence in Lα:
    (Step i) for a current value of Γ, Eq. 2 is minimized to wrt. Θ only (that can be done analytically, for example, in the case of classical k-means);
    (Step ii) for a current value of Θ, Eq. 2 is minimized to wrt. Γ only (that can also be done analytically).
  • Clustering with an imposed graph information
    Iteratively repeat until convergence in Lϵ,α:
    (Step i) for a current value of Γ, Eq. 4 is minimized as an unconstrained convex (for example, quadratic) problem wrt. Θ only (this can be done analytically in many cases);
    (Step ii) for a current value of Θ, Eq. 4 is minimized to wrt. Γ only, as a constrained convex [for example, as a quadratic programming (QP)] problem.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/7/e1500163/DC1

    Text

    Movie S1. Visualization of the most dominant attractor dimension over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with opened eyes.

    Movie S2. Visualization of the most dominant attractor dimension over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with closed eyes.

    Movie S3. Visualization of fourth eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with opened eyes.

    Movie S4. Visualization of the fourth eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with closed eyes.

    Movie S5. Visualization of the seventh eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with opened eyes.

    Movie S6. Visualization of the seventh eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with closed eyes.

    Movie S7. Visualization of the 10th eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with opened eyes.

    Movie S8. Visualization of the 10th eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with closed eyes.

    Fig. S1. Cluster affiliation function for the two identified manifolds.

    Fig. S2. Spectrograms of the EEG data.

    References (4584)

  • Supplementary Materials

    This PDF file includes:

    • Text
    • Legends for movies S1 to S8
    • Fig. S1. Cluster affiliation function for the two identified manifolds.
    • Fig. S2. Spectrograms of the EEG data.
    • References (4584)

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    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.avi format). Visualization of the most dominant attractor dimension over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with opened eyes.
    • Movie S2 (.avi format). Visualization of the most dominant attractor dimension over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with closed eyes.
    • Movie S3 (.avi format). Visualization of fourth eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with opened eyes.
    • Movie S4 (.avi format). Visualization of the fourth eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with closed eyes.
    • Movie S5 (.avi format). Visualization of the seventh eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with opened eyes.
    • Movie S6 (.avi format). Visualization of the seventh eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with closed eyes.
    • Movie S7 (.avi format). Visualization of the 10th eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with opened eyes.
    • Movie S8 (.avi format). Visualization of the 10th eigenvector over a schematic representation of the head (indicating the positions and numbers of electrodes according to the international 10-10 system) for the experiment with closed eyes.

    Files in this Data Supplement:

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