Learning Granger Causality for Hawkes Processes (1602.04511v2)

Abstract: Learning Granger causality for general point processes is a very challenging task. In this paper, we propose an effective method, learning Granger causality, for a special but significant type of point processes --- Hawkes process. We reveal the relationship between Hawkes process's impact function and its Granger causality graph. Specifically, our model represents impact functions using a series of basis functions and recovers the Granger causality graph via group sparsity of the impact functions' coefficients. We propose an effective learning algorithm combining a maximum likelihood estimator (MLE) with a sparse-group-lasso (SGL) regularizer. Additionally, the flexibility of our model allows to incorporate the clustering structure event types into learning framework. We analyze our learning algorithm and propose an adaptive procedure to select basis functions. Experiments on both synthetic and real-world data show that our method can learn the Granger causality graph and the triggering patterns of the Hawkes processes simultaneously.

• The paper introduces a novel nonparametric Hawkes process model that uses basis functions and group sparsity to infer Granger causality.
• It combines an EM-based algorithm with sparse-group-lasso regularization to efficiently estimate temporal dependencies in complex data.
• Experimental results on synthetic and real datasets demonstrate its superior accuracy in identifying intricate self- and mutual-triggering patterns.

Analysis of "Learning Granger Causality for Hawkes Processes"

The paper "Learning Granger Causality for Hawkes Processes" by Hongteng Xu, Mehrdad Farajtabar, and Hongyuan Zha introduces a novel approach to infer Granger causality within Hawkes processes. Granger causality is a concept used to determine whether one time series can predict another, and its adaptation to the multi-dimensional point processes like Hawkes processes has significant implications in various domains, including bioinformatics, social network analysis, and financial modeling.

Summary of Contributions

The authors address the challenge of learning Granger causality by presenting a nonparametric model of Hawkes processes that utilizes basis functions to represent impact functions. They emphasize the connection between Hawkes processes' impact functions and Granger causality graphs. This insight leads to the proposal that the group sparsity of impact function coefficients can effectively reveal the Granger causality relationships. The paper introduces a learning algorithm that combines a maximum likelihood estimator (MLE) with a sparse-group-lasso (SGL) regularizer to conserve this sparsity. Additionally, the algorithm is adaptable to incorporate event type clustering, enhancing robustness.

Methodology

1. Model Representation:
   • Impact functions are expressed as a linear combination of basis functions, allowing a flexible and tractable representation. The learning algorithm applies convex optimization, facilitating convergence to global optima.
2. Sparse-Group-Lasso (SGL) Regularizer:
   • The SGL regularizer ensures both local independence (through group sparsity) and temporal sparsity of impact functions, directly influencing the estimation of Granger causality.
3. Algorithmic Framework:
   • The paper employs an EM-based strategy, drawing from established methods, to iteratively update model parameters, significantly enhancing computational efficiency.
4. Adaptive Basis Function Selection:
   • A procedure based on sampling theory and Nyquist-Shannon theorem guides the adaptive selection of basis functions, crucial for capturing the essential characteristics of impact functions without excessive model complexity.

Experimental Evaluations

The proposed methodology was validated on both synthetic datasets and real-world IPTV viewing records. Results demonstrated the superiority of their approach in learning Granger causality graphs accurately and efficiently compared to existing state-of-the-art algorithms. The empirical evaluation showed that the algorithm successfully identified temporal dependencies, even in data with complex self- and mutual-triggering patterns.

Implications and Future Directions

The implications of this research extend to multiple application fields. The accurate inference of Granger causality within Hawkes processes offers substantial benefits for dynamic system modeling and prediction tasks, including targeted recommendations in media delivery systems or understanding epidemic spreading patterns in social networks.

Potential future developments could involve extending the model to general point processes beyond Hawkes processes, thereby increasing its applicability. Another avenue could be exploring more efficient computational architectures or distributed frameworks to handle even larger datasets.

In conclusion, this paper contributes significantly to the methodology of learning dependencies in point processes, providing a robust framework for causal inference in domains where irregular, asynchronous event sequences prevail.