# Signal Processing on Graphs: Causal Modeling of Unstructured Data

arXiv: https://arxiv.org/abs/1503.00173 · HTML: https://mlsanity.com/paper/1503.00173/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2015.

## Built on by (most cited first, 27 of 27)

- [Graph Learning: A Survey](https://mlsanity.com/paper/2105.00696/) (2021)
- [Sparse Graph Learning from Spatiotemporal Time Series](https://mlsanity.com/paper/2205.13492/) (2022)
- [Graph filtering over expanding graphs](https://mlsanity.com/paper/2203.08058/) (2022)
- [Learning Product Graphs from Spectral Templates](https://mlsanity.com/paper/2211.02893/) (2022)
- [PSO-Convolutional Neural Networks with Heterogeneous Learning Rate](https://mlsanity.com/paper/2205.10456/) (2022)
- [Recovering the Graph Underlying Networked Dynamical Systems under Partial Observability: A Deep Learning Approach](https://mlsanity.com/paper/2208.04405/) (2022)
- [Time-Varying Graph Learning for Data with Heavy-Tailed Distribution](https://mlsanity.com/paper/2501.00606/) (2024)
- [Online Graph Topology Learning from Matrix-valued Time Series](https://mlsanity.com/paper/2107.08020/) (2021)
- [Online Inference for Mixture Model of Streaming Graph Signals with Non-White Excitation](https://mlsanity.com/paper/2207.14019/) (2022)
- [Learning Graph ARMA Processes from Time-Vertex Spectra](https://mlsanity.com/paper/2302.06887/) (2023)
- [Learning to Identify Graphs from Node Trajectories in Multi-Robot Networks](https://mlsanity.com/paper/2307.04374/) (2023)
- [A Review and Roadmap of Deep Causal Model from Different Causal Structures and Representations](https://mlsanity.com/paper/2311.00923/) (2023)
- [Adaptive Least Mean Squares Graph Neural Networks and Online Graph Signal Estimation](https://mlsanity.com/paper/2401.15304/) (2024)
- [Spatiotemporal Forecasting Meets Efficiency: Causal Graph Process Neural Networks](https://mlsanity.com/paper/2405.18879/) (2024)
- [Online Graph Topology Learning via Time-Vertex Adaptive Filters: From Theory to Cardiac Fibrillation](https://mlsanity.com/paper/2411.01567/) (2024)
- [GSA-Forecaster: Forecasting Graph-Based Time-Dependent Data with Graph Sequence Attention](https://mlsanity.com/paper/2104.05914/) (2021)
- [Forecasting Graph Signals with Recursive MIMO Graph Filters](https://mlsanity.com/paper/2210.15258/) (2022)
- [Estimation of a Causal Directed Acyclic Graph Process using Non-Gaussianity](https://mlsanity.com/paper/2211.13800/) (2022)
- [Locally Stationary Graph Processes](https://mlsanity.com/paper/2309.01657/) (2023)
- [Adaptive Least Mean pth Power Graph Neural Networks](https://mlsanity.com/paper/2405.04111/) (2024)
- [A Review of Graph-Powered Data Quality Applications for IoT Monitoring Sensor Networks](https://mlsanity.com/paper/2410.21006/) (2024)
- [A Novel Unified Extended Matrix for Graph Signal Processing: Theory and Application](https://mlsanity.com/paper/2508.16633/) (2025)
- [TorchQuantumDistributed](https://mlsanity.com/paper/2511.19291/) (2025)
- [Graph Embedding with Mel-spectrograms for Underwater Acoustic Target Recognition](https://mlsanity.com/paper/2512.11545/) (2025)
- [Directed Graph Topology Inference via Graph Filter Identification](https://mlsanity.com/paper/2606.27455/) (2026)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)
- [Role-Specific Predictive Geometries for Nonstationary Multivariate Graph-Signal Forecasting](https://mlsanity.com/paper/2609.06519/) (2026)

## Abstract (arXiv)

Many applications collect a large number of time series, for example, the financial data of companies quoted in a stock exchange, the health care data of all patients that visit the emergency room of a hospital, or the temperature sequences continuously measured by weather stations across the US. These data are often referred to as unstructured. A first task in its analytics is to derive a low dimensional representation, a graph or discrete manifold, that describes well the interrelations among the time series and their intrarelations across time. This paper presents a computationally tractable algorithm for estimating this graph that structures the data. The resulting graph is directed and weighted, possibly capturing causal relations, not just reciprocal correlations as in many existing approaches in the literature. A convergence analysis is carried out. The algorithm is demonstrated on random graph datasets and real network time series datasets, and its performance is compared to that of related methods. The adjacency matrices estimated with the new method are close to the true graph in the simulated data and consistent with prior physical knowledge in the real dataset tested.