Spatial Statistics

Last update: 30 Apr 2026 15:45
First version: 25 November 2002 (with random fields/spatial stochastic processes being spun out no later than 14 September 2013)

That is, statistics for random variables spread out in space, i.e., random fields. Includes statistical image processing, which is important but doesn't really grab me as an application.

Things to learn more about:

"Kriging" is the historically-fixed name for using linear least squares regression to predict the value of a field at one point from its values at other points. It's usually presented as relying on the field being Gaussian. How much of the inferential theory really needs the Gaussian assumption? (You can derive the first- and second- order parts of the theory, about conditional expectations and covariances, without it, but beyond...?)
Moran developed an approach to spatial smoothing and regression where one regresses the field on eigenvectors of a matrix related to the adjacency or connectivity matrix of the lattice. How, mathematically, does this relate to kriging? What are the computational and inferential advantages and disadvantages?

The case where the random variables also evolve in time, the spatio-temporal case, is interesting enough to get its own notebook.

Carlo Gaetan and Xavier Guyon, Spatial Statistics and Modeling [Mini-review]
Peter Guttorp, Stochastic Modeling of Scientific Data
Brian D. Ripley
- Spatial Statistics
- Statistical Inference for Spatial Processes

A. C. Davison, S. A. Padoan and M. Ribatet, "Statistical Modeling of Spatial Extremes", Statistical Science 27 (2012): 161--186 [with discussion and response]
Timothy G. Conley and Morgan Kelly, "The Standard Errors of Persistence", Journal of International Economics 153 (2025): 104027 [Superseding a 2019 preprint of the same title by Kelly]
Florence Forbes, Myriam Charras-Garrido, Lamiae Azizi, Senan Doyle, and David Abrial, "Spatial risk mapping for rare disease with hidden Markov fields and variational EM", Annals of Applied Statistics 7 (2013): 1992--1216
M. Ghosh and J. N. K. Rao, "Small Area Estimation: An Appraisal", Statistical Science 9 (1994): 55--76 [Plus discussions and reply]
Xavier Guyon, Random Fields on a Network
Karen Kafadar, "Smoothing Geographical Data, Particularly Rates of Disease", Statistics in Medicine 15 (1996): 2539--2560 [PDF reprint via Prof. Kafadar]
Gary King, A Solution to the Ecological Inference Problem: Reconstructing Individual Behavior from Aggregate Data [Review]
S. N. Lahiri, Resampling Methods for Dependent Data [Mini-review]
Youjin Lee, Elizabeth L. Ogburn, "Testing for Network and Spatial Autocorrelation", arxiv:1710.03296
Elizaveta Levina and Peter J. Bickel, "Texture synthesis and nonparametric resampling of random fields", Annals of Statistics 34 (2006): 1751--1773
Daisuke Murakami, "spmoran: An R package for Moran's eigenvector-based spatial regression analysis", arxiv:1703.04467
John Novembre and Matthew Stephens, "Interpreting principal component analyses of spatial population genetic variation", Nature Genetics 40 (2008): 646--649 [Many PCA patterns commonly taken to be signs of ancestral population movements can also be produced as artifacts from null models. This is distressing, since many of the results based on PCA maps are things which make sense and I'd like to be true, but Novembre and Stephens's arguments check out.]
R. Piasecki, M. T. Martin, and A. Plastino, "Inhomogeneity and complexity measures for spatial patterns," cond-mat/0107471
Peter I. Saparin, Wolfgang Gowin, Jürgen Kurths, and Dieter Felsenber, "Quantification of cancellous bone structure using symbolic dynamics and measures of complexity", Physical Review E 58 (1998): 6449--6459
Jeffrey E. Steif, "Consistent estimation of joint distributions for sufficiently mixing random fields", Annals of Statistics 25 (1997): 293--304
Grace Wahba, Spline Models for Observational Data
Michael E. Wall, Andreas Rechtsteiner and Luis M. Rocha, "Singular Value Decomposition and Principal Component Analysis," physics/0208101
Rongjing Xiang and Jennifer Neville, "Relational Learning with One Network: An Asymptotic Analysis", AI Stats 2011 [PDF reprint]

D. G. Krige, Lognormal-de Wijsian Geostatistics for Ore Evaluation (Johannesburg: South African Institute of Mining and Metallurgy, 1981) [This is not Krige's first publication on what has come to be called "kriging", i.e., least-squares optimal linear prediction of a random field, but it's the only one of his I've read...]
Georges Matheron, Matheron's Theory of Regionalised Variables

CRS, lecture notes for 36-467, Data over Space and Time

Adrian Baddeley et al., Spatial Point Patterns: Methodology and Applications with R/cite>
Roger S. Bivand, Applied Spatial Data Analysis with R
Chris Brunsdon and Lex Comber, An Introduction to R for Spatial Analysis and Mapping
Cressie, Statistics for Spatial Data
David Darmofal, Spatial Analysis for the Social Sciences
Peter J. Diggle, Statistical Analysis of Spatial and Spatio-Temporal Point Patterns
Marie-Josie Fortin and Mark R. Dale, Spatial Analysis: A Guide for Ecologists
P. K. Kitanidis, Introduction to Geostatistics: Applications in Hydrogeology
Jose-Maria Montero, Gema Fernandez-Aviles and Jorge Mateu, Spatial and Spatio-Temporal Geostatistical Modeling and Kriging
Paula Moraga, Spatial Statistics for Data Science: Theory and Practice with R
Michael Sherman, Spatial Statistics and Spatio-Temporal Data: Covariance Functions and Directional Properties
M. N. M. van Lieshout, Theory of Spatial Statistics: A Concise Introduction

Stéphanie Allassonniere, Estelle Kuhn, "Convergent Stochastic Expectation Maximization algorithm with efficient sampling in high dimension. Application to deformable template model estimation", arxiv:1207.5938
Renato M. Assuncao and Pablo A. Ferrari, "Detection of spatial pattern through independence of thinned processes," math.PR/0103104
Yves F. Atchade, "Estimation of Network structures from partially observed Markov random fields", arxiv:1108.2835
Claus Beisbart, Thomas Buchert and Herbert Wagner, "Morphometry of Spatial Patterns," astro-ph/0007459
Claus Beisbart, Martin Kerscher and Klaus Mecke, "Mark correlations: relating physical properties to spatial distributions," physics/0201069
Claus Beisbart, Robert Dahlke, Klaus Mecke, and Herbert Wagner, "Vector- and tensor-valued descriptors for spatial patterns," physics/0203072
A. Brezger, L. Fahrmeir, A. Hennerfeind, "Adaptive Gaussian Markov random fields with applications in human brain mapping", Journal of the Royal Statistical Society C 56 (2007): 327--345
E. Ceyhan, C. E. Priebe, D. J. Marchette, "A New Family of Random Graphs for Testing Spatial Segregation", Canadian Journal of Statistics 35 (2007): 27--50, arxiv:0802.0615
Sihao Cheng, Brice Ménard, "How to quantify fields or textures? A guide to the scattering transform", arxiv:2112.01288
David B. Chua, Eric D. Kolaczyk, and Mark Crovella, "Network Kriging", math.ST/0510013
S. Dachian, "Nonparametric estimation for Gibbs random fields specified through one-point systems", Statistical Inference for Stochastic Processes 1 (1998): 245--264
Jorn Davidsen, Peter Grassberger and Maya Paczuski, "Networks of Recurrent Events, a Theory of Records, and an Application to Finding Causal Signatures in Seismicity", physics/0701190
Tilman M. Davies, Martin L. Hazelton, Jonathan. C Marshall, "sparr: Analyzing Spatial Relative Risk Using Fixed and Adaptive Kernel Density Estimation in R", Journal of Statistical Software 39:1 (2011)
Mohamed El Machkouri, "Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields", arxiv:1008.1342
Samuel Elogne and Dionisis Hristopulos, "On the Inference of Spartan Spatial Random Field Models for Geostatistical Applications", math.ST/0603430
Bryan K. Epperson, Geographical Genetics
Thibault Espinasse, Jean-Michel Loubes, "A Kriging procedure for processes indexed by graphs", arxiv:1406.6592
Jacob Feldman and Manish Singh, "Bayesian estimation of the shape skeleton", Proceedings of the National Academy of Sciences (USA) 103 (2006): 18014--18019 [From the abstract, it sounds like this could really have been "penalized maximum likelihood estimation of the shape skeleton", since they're just doing MAP rather than any kind of averaging, or otherwise working with the posterior distribution.]
C. Fonseca, H. Ferreira, L. Pereira A.P., Martins, "Stability and contagion measures for spatial extreme value analyses", arxiv:1206.1228
Florence Forbes and Nathalie Peyrard, "Hidden Markov Random Field Model Selection Criteria Based on Mean Field-Like Approximations", IEEE Transactions on Pattern Analysis and Machine Intelligence 25 (2003): 1089--1101 [PostScript preprint]
Michael Friendly, "A.-M. Guerry's Moral Statistics of France: Challenges for Multivariable Spatial Analysis", Statistical Science 22 (2007): 368--399, arxiv:0801.4263
Alan E. Gelfand, Peter J. Diggle, Montserrat Fuentes and Peter Guttorp (eds.), Handbook of Spatial Statistics
Anandamohan Ghosh, V. Ravi Kumar and B. D. Kulkarni, "Parameter estimation in spatially extended systems: The Karhunen-Loeve and Galerkin multiple shooting approach," nlin.CD/0112029 = Physical Review E 64 (2001): 056222
Tilmann Gneiting, Hana Ševčíková, and Donald B. Percival, "Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data", Statistical Science 27 (2012): 247--277
Priscilla E. Greenwood and Wolfgang Wefelmeyer, "Characterizing Efficient Empirical Estimators for Local Interaction Gibbs Fields", Statistical Inference for Stochastic Processes 2 (1999): 119--134
Aude Grelaud, Christian Robert, Jean-Michel Marin, Francois Rodolphe, Jean-Francois Taly, "ABC likelihood-freee methods for model choice in Gibbs random fields", arxiv:0807.2767
Zachary T. Harmany, Roummel F. Marcia, Rebecca M. Willett, "This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction ALgorithms --- Theory and Practice", IEEE Transactions on Image Processing 21 (2012): 1084--1096, arxiv:1005.4274
Matthew J. Heaton, Abhirup Datta, Andrew Finley, Reinhard Furrer, Rajarshi Guhaniyogi, Florian Gerber, Robert B. Gramacy, Dorit Hammerling, Matthias Katzfuss, Finn Lindgren, Douglas W. Nychka, Furong Sun, Andrew Zammit-Mangion, "A Case Study Competition Among Methods for Analyzing Large Spatial Data", arxiv:1710.05013
D. T. Hristopulos and S. N. Elogne, "Fast Spatial Prediction from Inhomogeneously Sampled Data Based on Generalized Random Fields with Gibbs Energy Functionals", physics/0609071
John Hughes, Murali Haran, "Dimension reduction and alleviation of confounding for spatial generalized linear mixed models", Journal of the Royal Statistical Society B 75 (2013): 139--159
Janine Illian, Antti Penttinen, Helga Stoyan, Dietrich Stoyan, Statistical Analysis and Modelling of Spatial Point Patterns
Jun-ichi Inoue and Kazuyuki Tanaka, "Dynamics of the Maximum Marginal Likelihood Hyper-parameter Estimation in Image Restoration: Gradient Descent vs. EM Algorithm," cond-mat/0107023
Mark S. Kaiser, Soumendra N. Lahiri, and Daniel J. Nordman, "Goodness of fit tests for a class of Markov random field models", Annals of Statistics 40 (2012): 104--130, arxiv:1205.6086
Wolfgang Karcher, Elena Shmileva, Evgeny Spodarev, "Extrapolation of stable random fields", arxiv:1107.1654
Nhu D. Le and James V. Zidek, Statistical Analysis of Environmental Space-Time Processes
U. K. Lee, H. Choi, B. U. Park and K. S. Yu, "Local likelihood density estimation on random fields", Statistics and Probability Letters 68 (2004): 347--357
Loic Le Gratiet and Claire Cannamela, "Kriging-based sequential design strategies using fast cross-validation techniques with extensions to multi-fidelity computer codes", arxiv:1210.6187
Pei-Sheng Lin and Murray K. Clayton, "Analysis of binary spatial data by quasi-likelihood estimating equations", math.ST/0505602 = Annals of Statistics 33 (2005): 542--555
Jialuo Liu, Tingjin Chu, Jun Zhu, Haonan Wang, "Large spatial data modeling and analysis: A Krylov subspace approach", Scandinavian Journal of Statistics 49 (2022): 1115--1143
Zudi Lu and Xing Chen, "Spatial kernel regression estimation: weak consistency", Statistics and Probability Letters 68 (2004): 125--136
Gregoire Mariethoz and Jef Caers, Multiple‐Point Geostatistics: Stochastic Modeling with Training Images
Jorge Mateu and Francisco Montes, "Pseudo-likelihood Inference for Gibbs Processes with Exponential Families through Generalized Linear Models", Statistical Inference for Stochastic Processes 4 (2001): 125--154
Klaus R. Mecke and D. Stoyan (eds.)
- Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation
- Morphology of Condensed Matter: Physics and Geometry of Spatially Complex Systems
Werner G. Muller, Collecting Spatial Data: Optimum Design of Experiments for Random Fields
Saikat Nandy, Scott H. Holan, Michael Schweinberger, "A Socio-Demographic Latent Space Approach to Spatial Data When Geography is Important but Not All-Important", arxiv:2304.03331
Hien D. Nguyen, Geoffrey J. McLachlan, Ian A. Wood, "Mixtures of Spatial Spline Regressions", arxiv:1306.3014
Enza Orlandi, Eva Loecherbach, "On the neighborhood radius estimation in Variable-neighborhood Markov Random Fields", arxiv:1002.4850
Christopher J. Paciorek, "Spatial models for point and areal data using Markov random fields on a fine grid", Electronic Journal of Statistics 7 (2013): 946--972
Marcelo Pereyra, Nicolas Dobigeon, Hadj Batatia, Jean-Yves Tourneret, "Computing the Cramer-Rao bound of Markov random field parameters: Application to the Ising and the Potts models", arxiv:1206.3985
Xin Qi and Hongyu Zhao, "Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations", Annals of Statistics 38 (2010): 435--481
Laura M. Sangalli, James O. Ramsay, Timothy O. Ramsay, "Spatial spline regression models", Journal of the Royal Statistical Society B 75 (2013): 681--703
Evgeny Spodarev, Elena Shmileva, Stefan Roth, "Extrapolation of Stationary Random Fields", arxiv:1306.6205
Michael L. Stein
- "Asymptotically Efficient Prediction of a Random Field with a Misspecified Covariance Function", Annals of Statistics 16 (1988): 55--63
- "Uniform Asymptotic Optimality of Linear Predictions of a Random Field Using an Incorrect Second-Order Structure", Annals of Statistics 18 (1990): 850--872
- "The screening effect in Kriging", Annals of Statistics 30 (2002): 298--323
- "When does the screening effect hold?", Annals of Statistics 39 (2011): 2795--2819
Ne-Zheng Sun, "Structure reduction and robust experimental design for distributed parameter identification", Inverse Problems 21 (2005): 739--758
Youngchul Sung, Lang Tong and H. Vincent Poor, "A Large Deviations Apoproach to Sensor Scheduling for Detection of Correlated Random Fields", cs.IT/0501056
Lionel Truquet, "On a nonparametric resampling scheme for Markov random fields", Electronic Journal of Statistics 5 (2011): 1503--1536
Nicolas Verzelen, "Adaptive estimation of stationary Gaussian fields", Annals of Statistics 38 (2010): 1363--1402
Melanie W. Wall, "A close look at the spatial structure implied by the CAR and SAR models", Journal of Statistical Planning and Inference 121 (2004): 311-324
Wenjia Wang, Rui Tuo & C. F. Jeff Wu, "On Prediction Properties of Kriging: Uniform Error Bounds and Robustness", Journal of the American Statistical Association forthcoming as of May 2019
Wen-Yun Yang, John Novembre, Eleazar Eskin and Eran Halperin, "A model-based approach for analysis of spatial structure in genetic data", Nature Genetics 44 (2012): 725--731
Xiaoxi Zhang, Timothy D. Johnson, Roderick J. A. Little, Yue Cao, "Quantitative magnetic resonance image analysis via the EM algorithm with stochastic variation", Annals of Applied Statistics 2 (2008): 736--755, arxiv:0807.4672 [More of interest to me for the getting at uncertainty in estimation of hidden Markov random fields]

CRS, Data over Space and Time