# Graph-based Virtual Sensing from Sparse and Partial Multivariate Observations

arXiv: https://arxiv.org/abs/2402.12598 · HTML: https://mlsanity.com/paper/2402.12598/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 61 references that are in the map.
- First posted on arXiv 19 Feb 2024.

## Builds on (most cited first, 28 of 28)

- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Adaptive Graph Convolutional Recurrent Network for Traffic Forecasting](https://mlsanity.com/paper/2007.02842/) (2020)
- [GAIN: Missing Data Imputation using Generative Adversarial Nets](https://mlsanity.com/paper/1806.02920/) (2018)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [BRITS: Bidirectional Recurrent Imputation for Time Series](https://mlsanity.com/paper/1805.10572/) (2018)
- [Multi-Time Attention Networks for Irregularly Sampled Time Series](https://mlsanity.com/paper/2101.10318/) (2021)
- [Deep Learning for Spatio-Temporal Data Mining: A Survey](https://mlsanity.com/paper/1906.04928/) (2019)
- [SAITS: Self-Attention-based Imputation for Time Series](https://mlsanity.com/paper/2202.08516/) (2022)
- [A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection](https://mlsanity.com/paper/2307.03759/) (2023)
- [A Gentle Introduction to Deep Learning for Graphs](https://mlsanity.com/paper/1912.12693/) (2019)
- [Filling the G_ap_s: Multivariate Time Series Imputation by Graph Neural Networks](https://mlsanity.com/paper/2108.00298/) (2021)
- [Inductive Graph Neural Networks for Spatiotemporal Kriging](https://mlsanity.com/paper/2006.07527/) (2020)
- [Estimating Missing Data in Temporal Data Streams Using Multi-directional Recurrent Neural Networks](https://mlsanity.com/paper/1711.08742/) (2017)
- [Principles and Algorithms for Forecasting Groups of Time Series: Locality and Globality](https://mlsanity.com/paper/2008.00444/) (2020)
- [Learning to Reconstruct Missing Data from Spatiotemporal Graphs with Sparse Observations](https://mlsanity.com/paper/2205.13479/) (2022)
- [Taming Local Effects in Graph-based Spatiotemporal Forecasting](https://mlsanity.com/paper/2302.04071/) (2023)
- [Kriging Convolutional Networks](https://mlsanity.com/paper/2306.09463/) (2023)
- [Spatial Aggregation and Temporal Convolution Networks for Real-time Kriging](https://mlsanity.com/paper/2109.12144/) (2021)
- [INCREASE: Inductive Graph Representation Learning for Spatio-Temporal Kriging](https://mlsanity.com/paper/2302.02738/) (2023)
- [CDSA: Cross-Dimensional Self-Attention for Multivariate, Geo-tagged Time Series Imputation](https://mlsanity.com/paper/1905.09904/) (2019)
- [Sparse Graph Learning from Spatiotemporal Time Series](https://mlsanity.com/paper/2205.13492/) (2022)
- [Graph Neural Processes for Spatio-Temporal Extrapolation](https://mlsanity.com/paper/2305.18719/) (2023)
- [Time Series Cluster Kernel for Learning Similarities between Multivariate Time Series with Missing Data](https://mlsanity.com/paper/1704.00794/) (2017)
- [Forecasting Unobserved Node States with spatio-temporal Graph Neural Networks](https://mlsanity.com/paper/2211.11596/) (2022)
- [Discrete Latent Structure in Neural Networks](https://mlsanity.com/paper/2301.07473/) (2023)

## Built on by (most cited first, 8 of 8)

- [Graph Neural Networks for Virtual Sensing in Complex Systems: Addressing Heterogeneous Temporal Dynamics](https://mlsanity.com/paper/2407.18691/) (2024)
- [Physics-Enhanced Graph Neural Networks For Soft Sensing in Industrial Internet of Things](https://mlsanity.com/paper/2404.08061/) (2024)
- [Joint Estimation and Prediction of City-wide Delivery Demand: A Large Language Model Empowered Graph-based Learning Approach](https://mlsanity.com/paper/2408.17258/) (2024)
- [Learning Latent Graph Structures and their Uncertainty](https://mlsanity.com/paper/2405.19933/) (2024)
- [Why Do Time Series Models Need Long Context Windows?](https://mlsanity.com/paper/2606.01999/) (2026)
- [On the Regularization of Learnable Embeddings for Time Series Forecasting](https://mlsanity.com/paper/2410.14630/) (2024)
- [Interpretability in Deep Time Series Models Demands Semantic Alignment](https://mlsanity.com/paper/2602.02239/) (2026)
- [A Dataset and Model for Imputing Water Surface Elevation on a Large and Extremely Sparse Spatiotemporal Graph](https://mlsanity.com/paper/2609.11580/) (2026)

## Abstract (arXiv)

Virtual sensing techniques allow for inferring signals at new unmonitored locations by exploiting spatio-temporal measurements coming from physical sensors at different locations. However, as the sensor coverage becomes sparse due to costs or other constraints, physical proximity cannot be used to support interpolation. In this paper, we overcome this challenge by leveraging dependencies between the target variable and a set of correlated variables (covariates) that can frequently be associated with each location of interest. From this viewpoint, covariates provide partial observability, and the problem consists of inferring values for unobserved channels by exploiting observations at other locations to learn how such variables can correlate. We introduce a novel graph-based methodology to exploit such relationships and design a graph deep learning architecture, named GgNet, implementing the framework. The proposed approach relies on propagating information over a nested graph structure that is used to learn dependencies between variables as well as locations. GgNet is extensively evaluated under different virtual sensing scenarios, demonstrating higher reconstruction accuracy compared to the state-of-the-art.