Statistics of spatio-temporal processes



The dynamics of spatial processes evolving in time.

Clearly there are many different problems one might wonder about here. I am thinking in particular of the kind of problem whose discretisation might look like this, as a graphical model.

This is highly stylized - I’ve imagined there is one spatial dimension, but usually there would be two or three. The observed notes are where we have sensors that can measure the state of some parameter of interest \(w\) which evolves in time \(t\). I am wondering what we need to control for to simultaneously learn the parameters of the spatial field \(r_i\), the (possibly emulated) process \(p\) and the state of the unobserved \(w\) nodes.

Intros

Cosma Shalizi’s Data Over Space and Time course.

Ensemble Kalman Filters

A classic; happens to work pretty well with spatial fields. See Ensemble Kalman Filters.

Laplace approximation in spatial fields

AFAICT usually the justification we use for applying Gaussian process formalism to inference. See Laplace approximation for a background.

Low rank spatial fields

Fixed-Rank Kriging etc.

Tools

See python spatial and R spatial software.

References

Alzraiee, Ayman H., Jeremy T. White, Matthew J. Knowling, Randall J. Hunt, and Michael N. Fienen. 2022. A Scalable Model-Independent Iterative Data Assimilation Tool for Sequential and Batch Estimation of High Dimensional Model Parameters and States.” Environmental Modelling & Software 150 (April): 105284.
Ayed, Ibrahim, and Emmanuel de Bézenac. 2019. “Learning Dynamical Systems from Partial Observations.” In Advances In Neural Information Processing Systems, 12.
Bakka, Haakon, Håvard Rue, Geir-Arne Fuglstad, Andrea Riebler, David Bolin, Janine Illian, Elias Krainski, Daniel Simpson, and Finn Lindgren. 2018. Spatial Modeling with R-INLA: A Review.” WIREs Computational Statistics 10 (6): e1443.
Banerjee, Sudipto, Alan E. Gelfand, Andrew O. Finley, and Huiyan Sang. 2008. Gaussian Predictive Process Models for Large Spatial Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 70 (4): 825–48.
Bertolacci, Michael. 2019. “Hierarchical Bayesian Mixture Models for Spatiotemporal Data with Nonstandard Features.”
Brix, Anders, and Peter J. Diggle. 2001. Spatiotemporal Prediction for Log-Gaussian Cox Processes.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63 (4): 823–41.
Chen, Wanfang, Marc G. Genton, and Ying Sun. 2021. Space-Time Covariance Structures and Models.” Annual Review of Statistics and Its Application 8 (1): 191–215.
Cressie, Noel, and Hsin-Cheng Huang. 1999. Classes of Nonseparable, Spatio-Temporal Stationary Covariance Functions.” Journal of the American Statistical Association 94 (448): 1330–39.
Cressie, Noel, Tao Shi, and Emily L. Kang. 2010. Fixed Rank Filtering for Spatio-Temporal Data.” Journal of Computational and Graphical Statistics 19 (3): 724–45.
Cressie, Noel, and Christopher K. Wikle. 2011. Statistics for Spatio-Temporal Data. Wiley Series in Probability and Statistics 2.0. John Wiley and Sons.
———. 2014. Space-Time Kalman Filter.” In Wiley StatsRef: Statistics Reference Online. American Cancer Society.
Curtain, Ruth F. 1975. Infinite-Dimensional Filtering.” SIAM Journal on Control 13 (1): 89–104.
Díaz-Avalos, Carlos, P. Juan, and J. Mateu. 2012. Similarity Measures of Conditional Intensity Functions to Test Separability in Multidimensional Point Processes.” Stochastic Environmental Research and Risk Assessment 27 (5): 1193–1205.
Dubrule, Olivier. 2018. Kriging, Splines, Conditional Simulation, Bayesian Inversion and Ensemble Kalman Filtering.” In Handbook of Mathematical Geosciences: Fifty Years of IAMG, edited by B.S. Daya Sagar, Qiuming Cheng, and Frits Agterberg, 3–24. Cham: Springer International Publishing.
Duffin, Connor, Edward Cripps, Thomas Stemler, and Mark Girolami. 2021. Statistical Finite Elements for Misspecified Models.” Proceedings of the National Academy of Sciences 118 (2).
Filippi, Jean-Baptiste, Vivien Mallet, and Bahaa Nader. 2014. Representation and Evaluation of Wildfire Propagation Simulations.” International Journal of Wildland Fire 23 (1): 46.
Finkenstädt, Bärbel., Leonhard. Held, and Valerie. Isham. 2007. Statistical methods for spatio-temporal systems. Boca Raton: Chapman & Hall/CRC.
Forbes, Jeremy, Dianne Cook, and Rob J. Hyndman. 2020. Spatial Modelling of the Two-Party Preferred Vote in Australian Federal Elections: 2001–2016.” Australian & New Zealand Journal of Statistics 62 (2): 168–85.
Giannakis, Dimitrios, and Suddhasattwa Das. 2017. Extraction and Prediction of Coherent Patterns in Incompressible Flows Through Space-Time Koopman Analysis.” arXiv:1706.06450 [Math], June.
Giannakis, Dimitrios, Abbas Ourmazd, Joanna Slawinska, and Zhizhen Zhao. 2017. Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables.” arXiv:1711.02798 [Physics], November.
Giannakis, Dimitrios, Joanna Slawinska, Abbas Ourmazd, and Zhizhen Zhao. 2018. Vector-Valued Spectral Analysis of Space-Time Data.” arXiv:1805.09134 [Physics], May.
Gneiting, Tilmann. 2002. Nonseparable, Stationary Covariance Functions for Space–Time Data.” Journal of the American Statistical Association 97 (458): 590–600.
Godfried, Isaac, Kriti Mahajan, Maggie Wang, Kevin Li, and Pranjalya Tiwari. 2020. FlowDB a Large Scale Precipitation, River, and Flash Flood Dataset.” arXiv:2012.11154 [Cs], December.
Gopalan, G., and C.K. Wikle. 2021. A Higher-Order Singular Value Decomposition Tensor Emulator for Spatiotemporal Simulators.” Journal of Agricultural, Biological and Environmental Statistics.
Goroshin, Ross, Joan Bruna, Jonathan Tompson, David Eigen, and Yann LeCun. 2014. Unsupervised Learning of Spatiotemporally Coherent Metrics.” arXiv:1412.6056 [Cs], December.
Haran, Murali. 2011. Gaussian Random Field Models for Spatial Data.” In Handbook of Markov Chain Monte Carlo, edited by Steve Brooks, Andrew Gelman, Galin Jones, and Xiao-Li Meng. Vol. 20116022. Chapman and Hall/CRC.
Higdon, Dave. 2002. Space and Space-Time Modeling Using Process Convolutions.” In Quantitative Methods for Current Environmental Issues, edited by Clive W. Anderson, Vic Barnett, Philip C. Chatwin, and Abdel H. El-Shaarawi, 37–56. London: Springer.
Hildeman, Anders, David Bolin, and Igor Rychlik. 2019. Joint Spatial Modeling of Significant Wave Height and Wave Period Using the SPDE Approach.” arXiv:1906.00286 [Stat], June.
Hoffimann, Júlio, Maciel Zortea, Breno de Carvalho, and Bianca Zadrozny. 2021. Geostatistical Learning: Challenges and Opportunities.” Frontiers in Applied Mathematics and Statistics 7.
Hooten, Mevin B., and Christopher K. Wikle. 2008. A Hierarchical Bayesian Non-Linear Spatio-Temporal Model for the Spread of Invasive Species with Application to the Eurasian Collared-Dove.” Environmental and Ecological Statistics 15 (1): 59–70.
Hu, Xiangping, and Ingelin Steinsland. 2016. Spatial Modeling with System of Stochastic Partial Differential Equations.” WIREs Computational Statistics 8 (2): 112–25.
Hu, Xiangping, Ingelin Steinsland, Daniel Simpson, Sara Martino, and Håvard Rue. 2013. Spatial Modelling of Temperature and Humidity Using Systems of Stochastic Partial Differential Equations,” July.
Huston, Carolyn. 2015. Thoughts on Spatio-Temporal Uncertainty Metrics Motivated by Input Sensitivity in the Spark Bushfire Spread Model.” In Weber, T., McPhee, M.J. And Anderssen, R.S. (Eds) MODSIM2015, 21st International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand.
Iranzad, Reza, Xiao Liu, W. Art Chaovalitwongse, Daniel S. Hippe, Shouyi Wang, Jie Han, Phawis Thammasorn, Chunyan Duan, Jing Zeng, and Stephen R. Bowen. 2021. Boost-S: Gradient Boosted Trees for Spatial Data and Its Application to FDG-PET Imaging Data,” January.
Koner, Salil, and Ana-Maria Staicu. 2023. Second-Generation Functional Data.” Annual Review of Statistics and Its Application 10 (1): 547–72.
Lienen, Marten, and Stephan Günnemann. 2021. Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks.” In International Conference on Learning Representations.
Lindgren, Finn, and Håvard Rue. 2015. Bayesian Spatial Modelling with R-INLA.” Journal of Statistical Software 63 (i19): 1–25.
Liu, Xiao, Kyongmin Yeo, and Siyuan Lu. 2020. Statistical Modeling for Spatio-Temporal Data From Stochastic Convection-Diffusion Processes.” Journal of the American Statistical Association 0 (0): 1–18.
Nguyen, Tung, Johannes Brandstetter, Ashish Kapoor, Jayesh K. Gupta, and Aditya Grover. 2023. ClimaX: A Foundation Model for Weather and Climate.” arXiv.
Nikitin, Alexander, S. T. John, Arno Solin, and Samuel Kaski. 2022. Non-Separable Spatio-Temporal Graph Kernels via SPDEs.” arXiv:2111.08524 [Cs, Stat], March.
Park, Ji Hwan, Shinjae Yoo, and Balu Nadiga. 2019. “Machine Learning Climate Variability.” In, 5.
Patraucean, Viorica, Ankur Handa, and Roberto Cipolla. 2015. Spatio-Temporal Video Autoencoder with Differentiable Memory.” arXiv:1511.06309 [Cs], November.
Patterson, Denis D., Simon A. Levin, A. Carla Staver, and Jonathan D. Touboul. 2020. Probabilistic Foundations of Spatial Mean-Field Models in Ecology and Applications.” SIAM Journal on Applied Dynamical Systems 19 (4): 2682–2719.
Pewsey, Arthur, and Eduardo García-Portugués. 2020. Recent Advances in Directional Statistics.” arXiv:2005.06889 [Stat], September.
Rackauckas, Chris, Alan Edelman, Keno Fischer, Mike Innes, Elliot Saba, Viral B Shah, and Will Tebbutt. 2020. Generalized Physics-Informed Learning Through Language-Wide Differentiable Programming.” MIT Web Domain, 6.
Rackauckas, Christopher, Yingbo Ma, Vaibhav Dixit, Xingjian Guo, Mike Innes, Jarrett Revels, Joakim Nyberg, and Vijay Ivaturi. 2018. A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions.” arXiv:1812.01892 [Cs], December.
Rackauckas, Christopher, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. 2020. Universal Differential Equations for Scientific Machine Learning.” arXiv:2001.04385 [Cs, Math, q-Bio, Stat], August.
Raissi, Maziar, P. Perdikaris, and George Em Karniadakis. 2019. Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations.” Journal of Computational Physics 378 (February): 686–707.
Rathbun, Stephen L. 1996. Asymptotic Properties of the Maximum Likelihood Estimator for Spatio-Temporal Point Processes.” Journal of Statistical Planning and Inference 51 (1): 55–74.
Reich, Brian J. 2012. Spatiotemporal Quantile Regression for Detecting Distributional Changes in Environmental Processes.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 61 (4): 535–53.
Reich, Brian J., Montserrat Fuentes, and David B. Dunson. 2011. Bayesian Spatial Quantile Regression.” Journal of the American Statistical Association 106 (493): 6–20.
Santos-Fernandez, Edgar, Jay M. Ver Hoef, Erin E. Peterson, James McGree, Daniel Isaak, and Kerrie Mengersen. 2021. Bayesian Spatio-Temporal Models for Stream Networks,” March.
Särkkä, Simo. 2011. Linear Operators and Stochastic Partial Differential Equations in Gaussian Process Regression.” In Artificial Neural Networks and Machine Learning – ICANN 2011, edited by Timo Honkela, Włodzisław Duch, Mark Girolami, and Samuel Kaski, 6792:151–58. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer.
Särkkä, Simo, and Jouni Hartikainen. 2012. Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression.” In Artificial Intelligence and Statistics.
Särkkä, Simo, A. Solin, and J. Hartikainen. 2013. Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.” IEEE Signal Processing Magazine 30 (4): 51–61.
Särkkä, Simo, and Arno Solin. 2019. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks 10. Cambridge ; New York, NY: Cambridge University Press.
Scheider, Simon, Benedikt Gräler, Edzer Pebesma, and Christoph Stasch. 2016. Modeling Spatiotemporal Information Generation.” International Journal of Geographical Information Science 30 (10): 1980–2008.
Shankar, Varun, Gavin D Portwood, Arvind T Mohan, Peetak P Mitra, Christopher Rackauckas, Lucas A Wilson, David P Schmidt, and Venkatasubramanian Viswanathan. 2020. “Learning Non-Linear Spatio-Temporal Dynamics with Convolutional Neural ODEs.” In Third Workshop on Machine Learning and the Physical Sciences (NeurIPS 2020).
Sigrist, Fabio Roman Albert. 2013. Physics Based Dynamic Modeling of Space-Time Data.” Application/pdf. ETH Zurich.
Sigrist, Fabio, Hans R. Künsch, and Werner A. Stahel. 2015a. Spate : An R Package for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process.” Application/pdf. Journal of Statistical Software 63 (14).
———. 2015b. Stochastic Partial Differential Equation Based Modelling of Large Space-Time Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 (1): 3–33.
Solin, Arno. 2016. Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.” Aalto University.
Solin, Arno, and Simo Särkkä. 2013. Infinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.” Physical Review E 88 (5): 052909.
Szehr, Oleg, Dario Azzimonti, and Laura Azzimonti. 2020. An Exact Kernel Framework for Spatio-Temporal Dynamics.” arXiv:2011.06848 [Math, Stat], November.
Tait, Daniel J., and Theodoros Damoulas. 2020. Variational Autoencoding of PDE Inverse Problems.” arXiv:2006.15641 [Cs, Stat], June.
Tsyrulnikov, Michael, and Alexander Rakitko. 2019. Impact of Non-Stationarity on Hybrid Ensemble Filters: A Study with a Doubly Stochastic Advection-Diffusion-Decay Model.” Quarterly Journal of the Royal Meteorological Society 145 (722): 2255–71.
Wikle, Christopher K., L. Mark Berliner, and Noel Cressie. 1998. Hierarchical Bayesian Space-Time Models.” Environmental and Ecological Statistics 5 (2): 117–54.
Wikle, Christopher K., Noel Cressie, and Andrew Zammit-Mangion. 2019. Spatio-Temporal Statistics with R.
Wikle, Christopher K., and Mevin B. Hooten. 2010. A General Science-Based Framework for Dynamical Spatio-Temporal Models.” TEST 19 (3): 417–51.
Wikle, Christopher K., and Andrew Zammit-Mangion. 2023. Statistical Deep Learning for Spatial and Spatiotemporal Data.” Annual Review of Statistics and Its Application 10 (1): 247–70.
Zammit-Mangion, Andrew, and Christopher K. Wikle. 2020. Deep Integro-Difference Equation Models for Spatio-Temporal Forecasting.” Spatial Statistics 37 (June): 100408.
Zhu, Wanchuang, and Yanan Fan. 2022. A Synthetic Likelihood Approach for Intractable Markov Random Fields.” Computational Statistics, July.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.