Non-separable Spatio-temporal Graph Kernels via SPDEs

This repository is the official implementation of the methods in the publication

• Alexander Nikitin, ST John, Arno Solin, and Samuel Kaski (2022). Non-separable spatio-temporal graph kernels via SPDEs. In Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS). [arXiv]

We leverage an explicit link between stochastic partial differential equations (SPDEs) and Gaussian processes on graphs and derive non-separable spatio-temporal graph kernels that capture interaction across space and time. We formulate the graph kernels for the stochastic heat equation and wave equation. We show that by providing novel tools for spatio-temporal GP modelling on graphs, we outperform pre-existing graph kernels in real-world applications that feature diffusion, oscillation, and other complicated interactions.

Use

The repo uses git-lfs to store datasets. To fetch the data use:

git lfs fetch

The code was tested with python==3.6 and should work for python>=3.6.

To install the required packages, run:

pip install -r requirements.txt
pip install -e .

Structure

The repository contains two sets of kernels for time-independent and temporal processes on graphs.

• Time-independent kernels are stored in graph_kernels/kernels.py.
• Temporal kernels are stored in graph_kernels/time_kernels.py.
• SHEK and SWEK are implemented in graph_kernels/time_kernels.py:StochasticHeatEquation and graph_kernels/time_kernels.py:StochasticWaveEquationKernel.

Experiments.

We provide an experimental evaluation of the proposed kernels on several datasets.

Heat Transfer Dataset

Interpolation:

python experiments/1d_experiments.py --interpolation --dump_directory=$PATH_TO_RESULTS

Extrapolation:

python experiments/1d_experiments.py --extrapolation --dump_directory=$PATH_TO_RESULTS

Chickenpox experiments

Interpolation (103 + num_test_weeks):

python experiments/run_chicken_pox.py --num_test_weeks=2 --interpolation --dump_directory=$PATH_TO_RESULTS

Extrapolation:

python experiments/run_chicken_pox.py --num_test_weeks=2 --extrapolation --dump_directory=$PATH_TO_RESULTS

Covid19 Experiments

Interpolation (33 + num_test_weeks):

python experiments/run_covid_experiments.py --log_target --no-use_flight_graph \
        --no-use_normalized_target --num_test_weeks=2 --interpolation --dump_directory=$PATH_TO_RESULTS

Extrapolation:

python experiments/run_covid_experiments.py --log_target --no-use_flight_graph \
        --no-use_normalized_target --num_test_weeks=2 --interpolation --dump_directory=$PATH_TO_RESULTS

Wave Experiments:

Open with jupyter-notebook:

./experiments/1d_wave_experiments.ipynb

Citation

If you use the code in this repository for your research, please cite the paper as follows:

@inproceedings{nikitin2022non,
  title={Non-separable spatio-temporal graph kernels via SPDEs},
  author={Nikitin, Alexander V and John, ST and Solin, Arno and Kaski, Samuel},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={10640--10660},
  year={2022},
  organization={PMLR}
}

Contributing

For all correspondence, please contact alexander.nikitin@aalto.fi.

License

This software is provided under the Apache License 2.0.