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 woudl 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 process \(p\) and the state of the unobserved \(w\) nodes.

Tools

geostack

gstat

Arno Solin. 2016. Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.

Ayed, Ibrahim, and Emmanuel de Bézenac. 2019. “Learning Dynamical Systems from Partial Observations.” In Advances in Neural Information Processing Systems, 12.

Banerjee, Sudipto, Alan E. Gelfand, Andrew O. Finley, and Huiyan Sang. n.d. “Gaussian Predictive Process Models for Large Spatial Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 70 (4): 825–48. https://doi.org/10.1111/j.1467-9868.2008.00663.x.

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. https://doi.org/10.1111/1467-9868.00315.

Cressie, Noel, and Hsin-Cheng Huang. 1999. “Classes of Nonseparable, Spatio-Temporal Stationary Covariance Functions.” Journal of the American Statistical Association 94 (448, 448): 1330–9. https://doi.org/10.1080/01621459.1999.10473885.

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. http://gms.gsfc.nasa.gov/vis/a000000/a003800/a003812/2010_Cressie_et_al_JCGS.pdf.

Cressie, Noel, and Christopher K. Wikle. 2014. “Space-Time Kalman Filter.” In Wiley StatsRef: Statistics Reference Online. American Cancer Society. https://doi.org/10.1002/9781118445112.stat07813.

———. 2011. Statistics for Spatio-Temporal Data. Wiley Series in Probability and Statistics 2.0. John Wiley and Sons. http://books.google.com?id=4L_dCgAAQBAJ.

Curtain, Ruth F. 1975. “Infinite-Dimensional Filtering.” SIAM Journal on Control 13 (1): 89–104. https://doi.org/10.1137/0313005.

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. https://doi.org/10.1007/s00477-012-0654-1.

Filippi, Jean-Baptiste, Vivien Mallet, and Bahaa Nader. 2014. “Representation and Evaluation of Wildfire Propagation Simulations.” International Journal of Wildland Fire 23 (1): 46. https://doi.org/10.1071/WF12202.

Finkenstädt, Bärbel., Leonhard. Held, and Valerie. Isham. 2007. Statistical Methods for Spatio-Temporal Systems. Boca Raton: Chapman & Hall/CRC. http://www.crcnetbase.com/isbn/9781584885931.

Giannakis, Dimitrios, and Suddhasattwa Das. 2017. “Extraction and Prediction of Coherent Patterns in Incompressible Flows Through Space-Time Koopman Analysis,” June. http://arxiv.org/abs/1706.06450.

Giannakis, Dimitrios, Abbas Ourmazd, Joanna Slawinska, and Zhizhen Zhao. 2017. “Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables,” November. http://arxiv.org/abs/1711.02798.

Giannakis, Dimitrios, Joanna Slawinska, Abbas Ourmazd, and Zhizhen Zhao. 2018. “Vector-Valued Spectral Analysis of Space-Time Data,” May. http://arxiv.org/abs/1805.09134.

Gneiting, Tilmann. 2002. “Nonseparable, Stationary Covariance Functions for Space–Time Data.” Journal of the American Statistical Association 97 (458): 590–600. https://doi.org/10.1198/016214502760047113.

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,” November. http://arxiv.org/abs/1511.06309.

Raissi, M., P. Perdikaris, and G. E. 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. https://doi.org/10.1016/j.jcp.2018.10.045.

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. https://doi.org/10.1016/0378-3758(95)00070-4.

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. https://doi.org/10.1111/j.1467-9876.2011.01025.x.

Reich, Brian J., Montserrat Fuentes, and David B. Dunson. 2011. “Bayesian Spatial Quantile Regression.” Journal of the American Statistical Association 106 (493): 6–20. https://doi.org/10.1198/jasa.2010.ap09237.

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. https://doi.org/10.1007/978-3-642-21738-8_20.

Särkkä, Simo, and Jouni Hartikainen. 2012. “Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression.” In Artificial Intelligence and Statistics. http://www.jmlr.org/proceedings/papers/v22/sarkka12.html.

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, 4): 51–61. https://doi.org/10.1109/MSP.2013.2246292.

Särkkä, Simo, and Arno Solin. 2019. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks 10. Cambridge ; New York, NY: Cambridge University Press. https://users.aalto.fi/~ssarkka/pub/sde_book.pdf.

Solin, Arno. 2016. “Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.” Aalto University. https://aaltodoc.aalto.fi:443/handle/123456789/19842.

Solin, Arno, and Simo Särkkä. 2013. “Infinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.” Physical Review E 88 (5): 052909. https://doi.org/10.1103/PhysRevE.88.052909.

Tait, Daniel J., and Theodoros Damoulas. 2020. “Variational Autoencoding of PDE Inverse Problems,” June. http://arxiv.org/abs/2006.15641.

“Thoughts on Spatio-Temporal Uncertainty Metrics Motivated by Input Sensitivity in the Spark Bushfire Spread Model.” 2015. 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. https://doi.org/10.36334/MODSIM.2015.F7.huston.

Wikle, Christopher K., Noel Cressie, and Andrew Zammit-Mangion. 2019. Spatio-Temporal Statistics with R.

Zammit-Mangion, Andrew, and Christopher K. Wikle. 2020. “Deep Integro-Difference Equation Models for Spatio-Temporal Forecasting.” Spatial Statistics 37 (June): 100408. https://doi.org/10.1016/j.spasta.2020.100408.