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Concurrence: A dependence criterion for time series, applied to biological data

Evangelos Sariyanidi, John D. Herrington, Lisa Yankowitz, Pratik Chaudhari, Theodore D. Satterthwaite, Casey J. Zampella, Jeffrey S. Morris, Edward Gunning, Robert T. Schultz, Russell T. Shinohara, Birkan Tunc

TL;DR

The paper introduces concurrence, a contrastive-learning based criterion to quantify statistical dependence between time series without requiring prior knowledge or large datasets. By training a classifier to distinguish concurrently versus non-concurrently cropped segments, it yields a bounded dependence coefficient and a per-segment score, theoretically and empirically linking to true dependence. The approach is validated on synthetic data and real biological signals (fMRI, physiology, behavior), outperforming several baselines and revealing dependencies that traditional linear metrics miss. Its simple implementation, robustness to modest data sizes, and applicability across diverse modalities suggest concurrence could become a standard tool for exploratory and confirmatory dependence analysis in neuroscience and related fields.

Abstract

Measuring the statistical dependence between observed signals is a primary tool for scientific discovery. However, biological systems often exhibit complex non-linear interactions that currently cannot be captured without a priori knowledge or large datasets. We introduce a criterion for dependence, whereby two time series are deemed dependent if one can construct a classifier that distinguishes between temporally aligned vs. misaligned segments extracted from them. We show that this criterion, concurrence, is theoretically linked with dependence, and can become a standard approach for scientific analyses across disciplines, as it can expose relationships across a wide spectrum of signals (fMRI, physiological and behavioral data) without ad-hoc parameter tuning or large amounts of data.

Concurrence: A dependence criterion for time series, applied to biological data

TL;DR

The paper introduces concurrence, a contrastive-learning based criterion to quantify statistical dependence between time series without requiring prior knowledge or large datasets. By training a classifier to distinguish concurrently versus non-concurrently cropped segments, it yields a bounded dependence coefficient and a per-segment score, theoretically and empirically linking to true dependence. The approach is validated on synthetic data and real biological signals (fMRI, physiology, behavior), outperforming several baselines and revealing dependencies that traditional linear metrics miss. Its simple implementation, robustness to modest data sizes, and applicability across diverse modalities suggest concurrence could become a standard tool for exploratory and confirmatory dependence analysis in neuroscience and related fields.

Abstract

Measuring the statistical dependence between observed signals is a primary tool for scientific discovery. However, biological systems often exhibit complex non-linear interactions that currently cannot be captured without a priori knowledge or large datasets. We introduce a criterion for dependence, whereby two time series are deemed dependent if one can construct a classifier that distinguishes between temporally aligned vs. misaligned segments extracted from them. We show that this criterion, concurrence, is theoretically linked with dependence, and can become a standard approach for scientific analyses across disciplines, as it can expose relationships across a wide spectrum of signals (fMRI, physiological and behavioral data) without ad-hoc parameter tuning or large amounts of data.

Paper Structure

This paper contains 18 sections, 1 theorem, 29 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Concurrence
  3. The loss function and the per-segment concurrence score
  4. Theoretical relation between dependence and concurrence
  5. Comparison with Other Contrastive Learning Approaches
  6. Implementation
  7. Experimental Validation
  8. Measuring stochastic dependence, and the effect of segment size
  9. Comparison with alternative methods
  10. Applications to Real Biological Signals
  11. Limitations and Future Work
  12. Conclusion
  13. Network Architecture, Hyperparameters and Implementation
  14. Proof of Theorem \ref{['th:CNC']}
  15. Numerical Simulations
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that $\mathbf z(\tau)$ is an RV defined as a function of a temporal lag parameter $\tau$ as where $\mathbf{h}$, $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$, $\boldsymbol{\epsilon}_x$ and $\boldsymbol{\epsilon}_y$ are Bernoulli processes with respective parameters $p$, $p_\alpha$, $p_\beta$, $p^\epsilon_x$ and $p^\epsilon_y$ such that Further, suppose that $\mathbf z^+$ and $\mathbf z^-$

Figures (12)

  • Figure 1: (a) Dependent signals $x$ and $y$, and independent signals $w$ and $z$. (b) Concurrent segments from $x$ and $y$ have different characteristics from non-concurrent segments, thus one can find functions $f$ and $g$ such that $f(x_i)g(y_j)$ is, on average, larger for concurrent segments (i.e., $i=j$) compared to non-concurrent segments (e.g., $f$ as the identity operator and $g$ the integral operator). (c) Concurrent and non-concurrent segments extracted from independent signals are not statistically distinguishable.
  • Figure 2: (a) Synthesized signals with deterministic dependence ($\xi=1.0$), stochastic dependence ($\xi=0.5$) and no dependence ($\xi=0.0$). (b) Dependent pairs of signals with varying degrees of noise. (c) Concurrence coefficient vs. $\xi$ (d) Concurrence coefficient vs. signal-to-noise ratio (SNR).
  • Figure 3: Pairs of (dependent) signals from six of the 100 synthesized datasets used in experiments. Dependence is typically not easy to visually ascertain.
  • Figure 4: (a) Correlated fMRI signals from two brain regions. (b) Signals from regions that are dependent (concurrence coefficient: 0.25) but uncorrelated (Pearson’s r: 0.02). (c) Connectivity matrix computed with the concurrence coefficient. (d) Connectivity matrix computed with (absolute) Pearson’s $r$ values. (e) The difference between the concurrence- and correlation-based connectivity matrices. (f) The distributions of the difference between the concurrence- and correlation-based connectivity matrices, shown separately for the seven brain networks. (g) Comparison of the Pearson’s r vs. concurrence coefficients computed from all the brain region pairs. (h) The (unclipped) concurrence coefficient between 10,000 pairs of brain regions of mismatched participants.
  • Figure 5: (a) Scatter plot and correlation between the respiration rate (RR) and ECG signal. (b) A sample ECG signal plotted against the synchronized (i.e., time-aligned) RR signal and the temporally misaligned RR. (c) The per-segment concurrence scores (PSCSs) between the temporally aligned ECG and RR are positive, which indicate that the PSCS correctly predicts that the segments are temporally aligned. (d) The PSCSs between the temporally misaligned ECG and RR are generally negative. (e) The PSCS for (temporally aligned) ECG and RR signals against the RMSSD, computed on a dataset of 30 participants for multiple segments per participant.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Remark 1
  • proof
  • Remark 2
  • proof
  • Remark 3
  • proof
  • Remark 4
  • proof
  • proof : Proof of Theorem \ref{['th:CNC']}