Independence Testing in Time-series data

Jaewon Chung

(he/him) - NeuroData lab
Johns Hopkins University - Biomedical Engineering

icon j1c@jhu.edu
icon @j1c (Github)
icon @j1c (Twitter)

Motivation

  • Observe several time series evolving simultaneously
  • Understand the relationship between them

Independence Testing

Let

We want to test:

Indepdence Testing in Time-series data

  • Let

  • Maximum lag .

  • Assymetric test

Stationary Process

A time series is strictly stationary if the joint distribution of is the same as that of .

  • ,

Consquently,

  • is constant, does not depend on
  • If , only depends on

Above two conditions form the weakly stationary.

Auto-covariance/correlation (ACVF/ACF)

Let , be stationary time-series.

  • Auto-covariance function (ACVF)
    • Time dependent covariance with itself

  • Auto-correlation function (ACF)
    • Time dependent correlation with itself

Cross-covariance/correlation (CCVF/CCF)

Let , stationary time-series.

  • Cross-covariance function (CCVF)
    • Time dependent covariance between two time series

  • Cross-correlation function (CCF)
    • Time dependent correlation between two time series

Ljung-Box Test (pronounced Yoong)

  • Max lag

  • Originally for autocorrelated time-series data.

  • Modified for cross-correlated time-series data.

Distance Based Test

  • Max lag

  • Distance cross correlation

  • Multiscale graph cross correlation

Optimal Lag

  • Lag that exhibits the strongest dependence.

Block Permutation Test

  • Dependent data → cannot use standard permutation.

Procedure: Given number of blocks size ,

  1. For each block , produce block:

  2. Choose blocks from with replacement.

  3. Concatenate the chosen blocks to form a new time series .

  4. Compute on the series . Repeat times.

  5. Compute -value as:

Block Permutation Visualization

  • ,

center

Simulations

  1. Independent data

  2. Linear dependence

  3. Non-linear dependence

Visualizations

center

Results



center

Optimal lag estimation

  • Data dependent on lag 3.

center

Real world data

  • Resting state(?) fMRI data

  • n=1200 frames, 0.75 second per frame.

  • 22 brain regions, each part of some large-scale brain network.

    - Default Mode Network (DMN)
    - Dorsal Attention Network (dAtt)
    - Ventral Attention Network (vAtt)
    - Visual Network (Visual)
    - FrontoParietal Network (FP)
    - Limbic Network (Limbic)
    - Somatomotor Network (SM)
    
Network Shorthand Activation?
Default Mode DMN When resting
Dorsal Attention dAtt Selecting stimuli relevant to a goal
Ventral Attention vAtt Detecting and redirecting attention to relevant stimuli
Visual Visual Analyzing the various components of the visual scene
FrontoParietal FP Making decisions in the context of goal-driven behaviour
Limbic Limbic Emotion, consciousness, motivation and long-term memory
Somatomotor SM Detecting somatosensory stimuli, movement

center

Network Shorthand Activation?
Default Mode DMN When resting
Dorsal Attention dAtt Selecting stimuli relevant to a goal
Ventral Attention vAtt Detecting and redirecting attention to relevant stimuli
Visual Visual Analyzing the various components of the visual scene
FrontoParietal FP Making decisions in the context of goal-driven behaviour
Limbic Limbic Emotion, consciousness, motivation and long-term memory
Somatomotor SM Detecting somatosensory stimuli, movement

center

TODO

  • Multivariate simulations

- Analyzing each component individually can lead to wrong inference.