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Canonical Tail Dependence for Soft Extremal Clustering of Multichannel Brain Signals

Mara Sherlin Talento, Jordan Richards, Raphael Huser, Hernando Ombao

TL;DR

This paper develops a canonical tail dependence (CTD) framework based on the tail pairwise dependence matrix (TPDM) to characterize extremal connectivity between brain regions. It derives analytic solutions via TPDM, enabling tail-topology extraction and a fuzzy clustering algorithm for multivariate extremes, and validates the approach with simulations. Applied to neonatal EEG, CTD-based clustering distinguishes seizure-related activity from non-seizure states and reveals interpretable tail-topology driven by specific channels in the gamma band. The work advances representation learning for rare extreme brain events and provides a scalable, interpretable tool for tail-dependent neurophysiological analysis.

Abstract

We develop a novel characterization of extremal dependence between two cortical regions of the brain when its signals display extremely large amplitudes. We show that connectivity in the tails of the distribution reveals unique features of extreme events (e.g., seizures) that can help to identify their occurrence. Numerous studies have established that connectivity-based features are effective for discriminating brain states. Here, we demonstrate the advantage of the proposed approach: that tail connectivity provides additional discriminatory power, enabling more accurate identification of extreme-related events and improved seizure risk management. Common approaches in tail dependence modeling use pairwise summary measures or parametric models. However, these approaches do not identify channels that drive the maximal tail dependence between two groups of signals -- an information that is useful when analyzing electroencephalography of epileptic patients where specific channels are responsible for seizure occurrences. A familiar approach in traditional signal processing is canonical correlation, which we extend to the tails to develop a visualization of extremal channel-contributions. Through the tail pairwise dependence matrix (TPDM), we develop a computationally-efficient estimator for our canonical tail dependence measure. Our method is then used for accurate frequency-based soft clustering of neonates, distinguishing those with seizures from those without.

Canonical Tail Dependence for Soft Extremal Clustering of Multichannel Brain Signals

TL;DR

This paper develops a canonical tail dependence (CTD) framework based on the tail pairwise dependence matrix (TPDM) to characterize extremal connectivity between brain regions. It derives analytic solutions via TPDM, enabling tail-topology extraction and a fuzzy clustering algorithm for multivariate extremes, and validates the approach with simulations. Applied to neonatal EEG, CTD-based clustering distinguishes seizure-related activity from non-seizure states and reveals interpretable tail-topology driven by specific channels in the gamma band. The work advances representation learning for rare extreme brain events and provides a scalable, interpretable tool for tail-dependent neurophysiological analysis.

Abstract

We develop a novel characterization of extremal dependence between two cortical regions of the brain when its signals display extremely large amplitudes. We show that connectivity in the tails of the distribution reveals unique features of extreme events (e.g., seizures) that can help to identify their occurrence. Numerous studies have established that connectivity-based features are effective for discriminating brain states. Here, we demonstrate the advantage of the proposed approach: that tail connectivity provides additional discriminatory power, enabling more accurate identification of extreme-related events and improved seizure risk management. Common approaches in tail dependence modeling use pairwise summary measures or parametric models. However, these approaches do not identify channels that drive the maximal tail dependence between two groups of signals -- an information that is useful when analyzing electroencephalography of epileptic patients where specific channels are responsible for seizure occurrences. A familiar approach in traditional signal processing is canonical correlation, which we extend to the tails to develop a visualization of extremal channel-contributions. Through the tail pairwise dependence matrix (TPDM), we develop a computationally-efficient estimator for our canonical tail dependence measure. Our method is then used for accurate frequency-based soft clustering of neonates, distinguishing those with seizures from those without.

Paper Structure

This paper contains 14 sections, 1 theorem, 28 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Neonatal Electroencephalography Data
  3. Data
  4. Spectral Domain Features
  5. Canonical correlation based soft clustering
  6. The Proposed Methodology
  7. Background: Multivariate Regular Variation
  8. Canonical Tail Dependence
  9. Fuzzy Clustering of Multivariate Extremes
  10. Simulation
  11. Simulation Settings
  12. Simulation Results
  13. Extremal clustering of neonatal EEG data
  14. Conclusion

Key Result

Proposition 3.1

Let $\boldsymbol{Z} = (\boldsymbol{X}^\top, \boldsymbol{Y}^\top)^\top$ where $\boldsymbol{X} \in \text{RV}^P(2)$ and $\boldsymbol{Y} \in \text{RV}^Q(2)$. Define $\boldsymbol{\Gamma}_{XX} \in \mathbb{R}^{P \times P}$ and $\boldsymbol{\Gamma}_{YY} \in \mathbb{R}^{Q \times Q}$ to be the TPDMs of $\bold The CTD in Equation eq:CTD is equivalent to

Figures (10)

  • Figure 1: EEG channel locations on the scalp of different regions of interests: FrontoTemporal (light blue nodes) vs OcciParietal (dark blue nodes), Frontal (light pink nodes) vs TempoOcciParietal (dark pink nodes), and left (brown nodes) vs right hemisphere (orange nodes).
  • Figure 2: Detrended EEG signals from FrontoTemporal lobe (1st and 3rd panels) and OcciParietal lobe (2nd and 4th panels) of non-epileptic patient (1st and 2nd panels) and epileptic patient (3rd and 4th panels).
  • Figure 3: Tomomaps Gursoy2014Tomopy of $|\hat{\boldsymbol{\Lambda}}^{(n)}_0|$, at the gamma band for FrontoTemporal-vs-OcciParietal connectivity. Canonical vectors of six non-epileptic neonates (first row) and eight epileptic neonates (second and third row) with primary localization on right hemisphere. Darker color represents higher eigenvector magnitude for channels $j = 1, \dots, 12$, labeled F3, F7, F4, F8, T3, T4, T5, T6, P3, P4, O1, O2.
  • Figure 4: Accuracy of CCA-based fuzzy clustering evaluated for different values of fuzzy parameter $m$ and for all frequency bands (i.e., delta, theta, alpha, beta and gamma). Colors differ with frequency band and the regions of interest differ across panels.
  • Figure 5: Comparison of $\boldsymbol{\Lambda}_1$ estimates from a single simulated $\{\boldsymbol{Z}^{(n)}_b\}_{b = 1}^{2000}$ obtained using Proposition 1 (middle panel) and direct numerical optimization (right panel) with the true value of $\boldsymbol{\Lambda}_1$ (left panel).
  • ...and 5 more figures

Theorems & Definitions (5)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • proof
  • Definition 3.3