Multi-Band Variable-Lag Granger Causality

Updated 4 August 2025

The paper presents MB-VLGC as a unified framework that extends variable-lag Granger causality by incorporating frequency-specific causal inference across multiple frequency bands.
It employs frequency banding with zero-phase filtering and dynamic time warping to align time-varying delays, enhancing causal detection in neural, economic, and behavioral data.
Empirical evaluations demonstrate MB-VLGC outperforms classical methods with F1-scores up to 0.933, validating its effectiveness on both synthetic and real-world datasets.

Multi-Band Variable-Lag Granger Causality (MB-VLGC) refers to a unified inferential framework that generalizes variable-lag Granger causality by explicitly modeling both frequency-specific and time-varying delays in causal interactions between time series. MB-VLGC addresses fundamental limitations of classical Granger causality, which assumes fixed-lag relationships, and overcomes variable-lag Granger causality's restriction to broadband analysis by capturing how causal influences can differ across frequency bands—a feature vitally important for complex systems such as neural, economic, and behavioral data (Sookkongwaree et al., 1 Aug 2025).

1. Conceptual Foundations and Formal Definition

MB-VLGC integrates two key generalizations: allowing for variable (i.e., not fixed or constant) causal lags and enabling frequency-specific causal inference. Given two time series, $X$ and $Y$ , and a set of frequency bands $\mathcal{B} = \{B_1, \ldots, B_K\}$ where $B_i$ specifies a frequency interval, the approach comprises:

Band-Limiting: Signals are filtered to extract their components in each frequency band $B_i$ , producing pairs $(X^{(B_i)}, Y^{(B_i)})$ ;
Variable-Lag Regression: For each frequency band, a regression of the form

$r^*_{YX}(t) = Y^{(B)}(t) - \sum_{i=1}^{\delta_{\max}} \left[ a_i Y^{(B)}(t-i) + b_i X^{(B)}(t-i) + c_i X^*(t-i) \right]$

is performed, where $X^*(t-i) = X^{(B)}(t-i + 1 - \Delta_{t-i+1})$ , with $\Delta_t$ representing the variable lag at each time point and determined by alignment methods such as Dynamic Time Warping (DTW).

Definition: $X$ MB-VL Granger causes $Y$ if there is at least one $B_i \in \mathcal{B}$ such that $X^{(B_i)}$ variable-lag Granger causes $Y^{(B_i)}$ ; in other words,

$\exists B_i \in \mathcal{B} : X^{(B_i)} \xrightarrow{\text{VLGC}} Y^{(B_i)}.$

This construction enables simultaneous localization of both the frequency band(s) and delay(s) where causality is significant (Sookkongwaree et al., 1 Aug 2025).

2. Theoretical Underpinnings and Justification

The MB-VLGC framework is justified by several theoretical results:

Classical linear Granger causality is invariant under spectral filtering for VAR processes, but such invariance fails when delay structures vary by frequency or when models deviate from VAR assumptions.
Proposition 2.2 in (Sookkongwaree et al., 1 Aug 2025): Decomposing the signals into disjoint frequency bands and performing band-specific variable-lag regression achieves a (strictly) lower or equal overall prediction error compared to a monolithic, single-band approach:

$\text{Var}(r^*) \geq \sum_i \text{Var}(r^*_{(i)}),$

where $r^*_{(i)}$ is the residual for band $B_i$ .

This property ensures that frequency-specific modeling not only better matches complex empirical data but also avoids "masking" effects whereby broadband models fail to detect localized causal influences.

The use of time-frequency decomposition (via zero-phase, e.g., filtfilt-based Butterworth filters) coupled with variable-lag alignment provides theoretical and practical improvements over both standard and variable-lag-only approaches.

3. Inference Pipeline and Statistical Workflow

MB-VLGC adopts a three-stage inference pipeline:

Frequency Banding: Signals are decomposed into predefined frequency bands using zero-phase filtering (e.g., 4th-order Butterworth, filtfilt) to prevent phase distortion and accurately isolate frequency content.
Variable-Lag Causal Inference: Within each band, the approach:
- Uses a hybrid lag selection combining global delay estimation (cross-correlation) and local, time-point-specific alignment (DTW).
- Performs model comparison through nested regressions (null, fixed-lag, and variable-lag) and applies formal statistical tests: F-tests, BIC difference ratios, and meta-analytic p-value aggregation (e.g., Fisher’s method).
Integration of Band-Specific Evidence: Statistical results (p-values, lag estimates) are meta-analyzed (e.g., $\chi^2 = -2\,\sum_i \ln p_i$ ) across bands to provide both band-specific and overall inference on causal direction and functional delay.

A summary of the steps:

Step	Operation	Main Tool
1. Frequency Banding	Zero-phase filtering (e.g., Butterworth)	filtfilt
2. Causal Inference	VLGC (per band, with DTW over lags)	DTW, F-test, BIC
3. Meta-analysis	Integration of p-values/meta-statistics	Fisher’s method

This pipeline is designed to maximize statistical power and resolve frequency- and delay-specific interactions.

4. Empirical Performance and Comparative Evaluation

Empirical results show MB-VLGC achieves consistently superior detection of causal relationships versus classical Granger causality (GC), VLGC, and information-theoretic methods such as transfer entropy (TE):

Synthetic datasets: MB-VLGC yielded an average overall $F_1$ -score of 0.810, outperforming classical GC, VLGC, and TE. In cases of multifrequency causation (where ground-truth lags vary by frequency), MB-VLGC achieved up to 0.933 accuracy, indicating robust localization of true variable-lag effects (Sookkongwaree et al., 1 Aug 2025).
Band configuration: Optimal performance is observed with two-band separation (low/high split, giving $F_1=0.810$ ); finer EEG-style multi-band partitioning can be advantageous in multi-frequency scenarios but may over-partition simpler signals.
Real-world data: MB-VLGC robustly detected ground-truth causal relationships in datasets including Old Faithful geyser eruptions, economic time series (chicken and egg prices), industrial processes (gas furnace: gas→CO₂), and multi-channel EEG (motor imagery). Notably, in EEG, MB-VLGC identified frequency- and direction-specific neural connectivity (e.g., FC3↔FC5, gamma band driven interaction).

These results illustrate MB-VLGC’s ability to resolve both broadband and narrowband causal interactions with variable lag, even in short or noisy time series.

5. Applications Across Scientific Domains

MB-VLGC is broadly adaptable to any system where information flow occurs with frequency- and delay-specific structure:

Neuroscience: For EEG or MEG, MB-VLGC enables localization of connection directionality and delay within neural oscillatory bands (e.g., alpha, beta, gamma), facilitating insights into frequency-specific neural pathways and the temporal structure of cortical communication. In motor imagery EEG, it reveals gamma-band connectivity not detectable by conventional approaches.
Econometrics: Economic variables often interact on different time scales. MB-VLGC detects whether shocks propagate with fast or slow lags and distinguishes short-term market dynamics from long-run trends.
Industrial Process Control: MB-VLGC accurately tracks the relationship between process input (e.g., gas) and output (e.g., emissions) when those dynamics are mediated by band- and delay-variant processes.
Behavioral Science: MB-VLGC identifies how behaviorally relevant events (e.g., actions, decisions) are coordinated across individuals or neural populations on multiple time-frequency scales.

6. Limitations, Implementation, and Resources

MB-VLGC’s primary computational cost arises from repeated bandwise filtering and DTW-based alignment, but efficient implementations (as demonstrated in the accompanying code repository) make it practical for moderate-scale problems.
Statistical challenges include proper selection of bands and effective correction for multiple comparisons (via Bonferroni adjustment or meta-analytic methods).
Reproducibility is supported by public code and datasets for both synthetic and real-world benchmarks: https://anonymous.4open.science/r/mbvlgranger-ED89/README.md.

Resource	Description
Python code & data	Full MB-VLGC implementation

7. Future Perspectives

MB-VLGC opens several avenues for methodological innovation:

Extension to non-linear, multivariate, and non-stationary settings by incorporating machine learning architectures and non-parametric alignment mechanisms (Suryadi et al., 2022, Sultan et al., 2022).
Adaptive band selection via data-driven approaches.
Integration with multivariate and ensemble causality frameworks, leveraging invariance properties and partial causality concepts (1002.0299).
Application to large-scale, high-dimensional, or continuous-time recordings, supported by advances in scalable spectral estimation and matrix factorization (Ephremidze, 25 Dec 2024).

A plausible implication is that continued development of MB-VLGC frameworks will underpin nuanced causal inference in data-rich, temporally and spectrally complex domains ranging from basic neuroscience to econometric modeling.

MB-VLGC formally unifies multiband frequency analysis and dynamic lag estimation, providing a theoretically sound and empirically validated method for resolving complex causal structures in time-series data. Its rigorous definition, robust inference pipeline, and open-source availability position it as a valuable tool for directed time-frequency resolved causal analysis across a wide range of scientific disciplines (Sookkongwaree et al., 1 Aug 2025).