## Spatial Statistics

*09 Mar 2024 13:37*

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.

- See also:
- Markov Models
- Network Data Analysis
- Optimal Linear Prediction and Estimation
- Pattern Formation
- Point Processes
- Random Fields
- Statistics

- Recommended, general:
- 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

- Recommended, of more specialized interest:
- A. C. Davison, S. A. Padoan and M. Ribatet, "Statistical Modeling
of Spatial
Extremes", Statistical
Science
**27**(2012): 161--186 [with discussion and response] - 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] - Morgan Kelly, "The Standard Errors of Persistence" [PDF preprint, 2019]
- 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]

- Recommended, of historical interest:
- 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...]

- Modesty forbids me to recommend:
- CRS, lecture notes for 36-467, Data over Space and Time

- To read, teaching:
- 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

- To read, learning:
- 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 - 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 - 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 - Peter St. Jean, Pockets of Crime: Broken Windows, Collective Efficacy, and the Criminal Point of View
- 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

- "Asymptotically Efficient Prediction of a Random Field with a Misspecified Covariance Function", Annals of Statistics
- 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]

- To write:
- CRS, Data over Space and Time