Residual analysis for spatial point processes (with discussion)

Summary.  We define residuals for point process models fitted to spatial point pattern data, and we propose diagnostic plots based on them. The residuals apply to any point process model that has a conditional intensity; the model may exhibit spatial heterogeneity, interpoint interaction and dependence on spatial covariates. Some existing ad hoc methods for model checking (quadrat counts, scan statistic, kernel smoothed intensity and Berman's diagnostic) are recovered as special cases. Diagnostic tools are developed systematically, by using an analogy between our spatial residuals and the usual residuals for (non‐spatial) generalized linear models. The conditional intensity λ plays the role of the mean response. This makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context, giving recommendations for diagnostic plots. A plot of smoothed residuals against spatial location, or against a spatial covariate, is effective in diagnosing spatial trend or co‐variate effects. Q–Q‐plots of the residuals are effective in diagnosing interpoint interaction.

[1]  D. Horvitz,et al.  A Generalization of Sampling Without Replacement from a Finite Universe , 1952 .

[2]  M. S. Bartlett,et al.  The spectral analysis of two-dimensional point processes , 1964 .

[3]  M. S. Bartlett,et al.  207. Note: A Note on Spatial Pattern , 1964 .

[4]  David R. Cox,et al.  The statistical analysis of series of events , 1966 .

[5]  D. Vere-Jones.  Stochastic Models for Earthquake Occurrence , 1970 .

[6]  R. Gnanadesikan,et al.  A Probability Plotting Procedure for General Analysis of Variance , 1970 .

[7]  P. Lewis.  Recent results in the statistical analysis of univariate point processes , 1971 .

[8]  F. Papangelou,et al.  The conditional intensity of general point processes and an application to line processes , 1974 .

[9]  J. Besag.  Statistical Analysis of Non-Lattice Data , 1975 .

[10]  F. Kelly,et al.  A note on Strauss's model for clustering , 1976 .

[11]  B. Ripley.  The Second-Order Analysis of Stationary Point Processes , 1976 .

[12]  Hans-Otto Georgii,et al.  Canonical and grand canonical Gibbs states for continuum systems , 1976 .

[13]  Peter J. Diggle,et al.  Simple Monte Carlo Tests for Spatial Pattern , 1977 .

[14]  B. Ripley,et al.  Markov Point Processes , 1977 .

[15]  Gibbsian Description of Point Random Fields , 1977 .

[16]  B. Ripley.  Modelling Spatial Patterns , 1977 .

[17]  O. Kallenberg.  On conditional intensities of point processes , 1978 .

[18]  Peter J. Diggle,et al.  On Parameter Estimation for Spatial Point Processes , 1978 .

[19]  D. Brillinger.  Comparative Aspects of the Study of Ordinary Time Series and of Point Processes , 1978 .

[20]  Peter J. Diggle,et al.  On parameter estimation and goodness-of-fit testing for spatial point patterns , 1979 .

[21]  Hans Zessin,et al.  Integral and Differential Characterizations of the GIBBS Process , 1979 .

[22]  Lokale Energien und Potentiale für Punktprozesse , 1980 .

[23]  David Cox,et al.  Applied Statistics - Principles and Examples , 1981 .

[24]  D. Pregibon.  Logistic Regression Diagnostics , 1981 .

[25]  Y. Ogata,et al.  Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure , 1981 .

[26]  J. Kalbfleisch,et al.  The Statistical Analysis of Failure Time Data , 1980 .

[27]  J. Besag,et al.  Point process limits of lattice processes , 1982, Journal of Applied Probability.

[28]  P. V. D. Hoeven.  Une projection de processus ponctuels , 1982 .

[29]  O. Kallenberg.  Random Measures , 1983 .

[30]  Peter J. Diggle,et al.  Statistical analysis of spatial point patterns , 1983 .

[31]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[32]  D. Pregibon,et al.  Graphical Methods for Assessing Logistic Regression Models , 1984 .

[33]  A. Baddeley,et al.  A cautionary example on the use of second-order methods for analyzing point patterns , 1984 .

[34]  Thomas Fiksel,et al.  Estimation of Parametrized Pair Potentials of Marked and Non-marked Gibbsian Point Processes , 1984, J. Inf. Process. Cybern..

[35]  P. Diggle,et al.  Monte Carlo Methods of Inference for Implicit Statistical Models , 1984 .

[36]  Olav Kallenberg,et al.  An Informal Guide to the Theory of Conditioning in Point Processes , 1984 .

[37]  P. Diggle.  A Kernel Method for Smoothing Point Process Data , 1985 .

[38]  S. Weisberg.  Plots, transformations, and regression , 1985 .

[39]  M. Berman.  Testing for spatial association between a point process and another stochastic process , 1986 .

[40]  R. Takacs,et al.  Interaction Pair-potentials for a System of Ant's Nests , 1986 .

[41]  David Nualart,et al.  A Characterization of the Spatial Poisson Process and Changing Time , 1986 .

[42]  R. Takacs.  Estimator for the pair–potential of a gibbsian point process , 1986 .

[43]  J. Franklin,et al.  Second-Order Neighborhood Analysis of Mapped Point Patterns , 1987 .

[44]  E. Fowlkes.  Some diagnostics for binary logistic regression via smoothing , 1987 .

[45]  N. Cressie,et al.  Random set theory and problems of modeling , 1987 .

[46]  Yosihiko Ogata,et al.  Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes , 1988 .

[47]  Peter Breeze,et al.  Point Processes and Their Statistical Inference , 1991 .

[48]  B. Ripley.  Statistical inference for spatial processes , 1990 .

[49]  P. Diggle,et al.  Estimating weighted integrals of the second-order intensity of a spatial point process , 1989 .

[50]  S. Doguwa.  On Second Order Neighbourhood Analysis of Mapped Point Patterns , 1989 .

[51]  J. Besag,et al.  Generalized Monte Carlo significance tests , 1989 .

[52]  A. Baddeley,et al.  Nearest-Neighbour Markov Point Processes and Random Sets , 1989 .

[53]  Wilfrid S. Kendall.  A spatial Markov property for nearest-neighbour Markov point processes , 1990 .

[54]  P. Diggle.  A point process modeling approach to raised incidence of a rare phenomenon in the vicinity of a prespecified point , 1990 .

[55]  M. Gopalan Nair,et al.  Random Space Change for Multiparameter Point Processes , 1990 .

[56]  R. V. Ambartzumian,et al.  Stochastic point processes , 1990 .

[57]  N. Cressie,et al.  Statistics for Spatial Data. , 1992 .

[58]  J. L. Jensen,et al.  Pseudolikelihood for Exponential Family Models of Spatial Point Processes , 1991 .

[59]  D. Collett.  Modelling Binary Data , 1991 .

[60]  J. Besag,et al.  Sequential Monte Carlo p-values , 1991 .

[61]  Alan F. Karr,et al.  Point Processes and Their Statistical Inference , 1991 .

[62]  Merlise A. Clyde,et al.  Logistic regression for spatial pair-potential models , 1991 .

[63]  D. Harrington,et al.  Counting Processes and Survival Analysis , 1991 .

[64]  Dietrich Stoyan,et al.  Second-order Characteristics for Stochastic Structures Connected with Gibbs Point Processes† , 1991 .

[65]  J. Lindsey,et al.  Fitting and comparing probability distributions with log linear models , 1992 .

[66]  Martin Crowder,et al.  Statistical Theory and Modelling: In Honour of Sir David Cox, FRS. , 1992 .

[67]  M. N. M. van Lieshout,et al.  Object recognition using Markov spatial processes , 1992, Proceedings., 11th IAPR International Conference on Pattern Recognition. Vol.II. Conference B: Pattern Recognition Methodology and Systems.

[68]  Mark Berman,et al.  Approximating Point Process Likelihoods with Glim , 1992 .

[69]  A. B. Lawson,et al.  On Fitting Non-Stationary Markov Point Process Models on GLIM , 1992 .

[70]  James K. Lindsey,et al.  The Analysis of Stochastic Processes using GLIM , 1992 .

[71]  Coldplay,et al.  X/Y , 2020, The A–Z of Intermarriage.

[72]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[73]  Andrew B. Lawson,et al.  A Deviance Residual for Heterogeneous Spatial Poisson Processes , 1993 .

[74]  [The special case]. , 1993, Sportverletzung Sportschaden : Organ der Gesellschaft fur Orthopadisch-Traumatologische Sportmedizin.

[75]  Noel A. C. Cressie,et al.  Statistics for Spatial Data: Cressie/Statistics , 1993 .

[76]  Peter J. Diggle,et al.  A Conditional Approach to Point Process Modelling of Elevated Risk , 1994 .

[77]  B. Hambly.  Fractals, random shapes, and point fields , 1994 .

[78]  C. Geyer,et al.  Simulation Procedures and Likelihood Inference for Spatial Point Processes , 1994 .

[79]  David R. Brillinger,et al.  Time series, point processes, and hybrids* , 1994 .

[80]  Noel A Cressie,et al.  Statistics for Spatial Data, Revised Edition. , 1994 .

[81]  Andrew B. Lawson,et al.  Armadale: A Case‐Study in Environmental Epidemiology , 1994 .

[82]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

[83]  James Lindsey,et al.  Fitting Parametric Counting Processes by using Log-linear Models , 1995 .

[84]  R. Häggkvist,et al.  Second-order analysis of space-time clustering , 1995, Statistical methods in medical research.

[85]  Aila Särkkä,et al.  Parameter Estimation for Marked Gibbs Point Processes Through the Maximum Pseudo-likelihood Method , 1996 .

[86]  T. Mattfeldt.  Stochastic Geometry and Its Applications , 1996 .

[87]  Adrian Baddeley,et al.  Markov properties of cluster processes , 1996, Advances in Applied Probability.

[88]  William N. Venables,et al.  Modern Applied Statistics with S-Plus. , 1996 .

[89]  Natalie W. Harrington,et al.  The analysis of putative environmental pollution gradients in spatially correlated epidemiological data , 1996 .

[90]  D. Brillinger,et al.  Some wavelet analyses of point process data , 1997, Conference Record of the Thirty-First Asilomar Conference on Signals, Systems and Computers (Cat. No.97CB36136).

[91]  Brian D. Ripley,et al.  Modern Applied Statistics with S-Plus Second edition , 1997 .

[92]  J. Møller,et al.  Log Gaussian Cox Processes , 1998 .

[93]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[94]  Sven Erick Alm.  Approximation and Simulation of the Distributions of Scan Statistics for Poisson Processes in Higher Dimensions , 1998 .

[95]  Adrian Baddeley,et al.  Practical maximum pseudolikelihood for spatial point patterns , 1998, Advances in Applied Probability.

[96]  Wilfrid S. Kendall,et al.  Perfect Simulation for the Area-Interaction Point Process , 1998 .

[97]  M. Kulldorff.  Spatial Scan Statistics: Models, Calculations, and Applications , 1999 .

[98]  Frederic Paik Schoenberg,et al.  Transforming Spatial Point Processes into Poisson Processes , 1999 .

[99]  A. Baddeley,et al.  Non‐ and semi‐parametric estimation of interaction in inhomogeneous point patterns , 2000 .

[100]  Frank Mücklich,et al.  Statistical Analysis of Microstructures in Materials Science , 2000 .

[101]  A. Baddeley,et al.  Practical Maximum Pseudolikelihood for Spatial Point Patterns , 1998, Advances in Applied Probability.

[102]  van Marie-Colette Lieshout,et al.  Markov Point Processes and Their Applications , 2000 .

[103]  W. Kendall,et al.  Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes , 2000, Advances in Applied Probability.

[104]  J. Wakefield,et al.  Spatial epidemiology: methods and applications. , 2000 .

[105]  A. Baddeley.  Time-invariance estimating equations , 2000 .

[106]  Daniel Hug,et al.  On support measures in Minkowski spaces and contact distributions in stochastic geometry , 2000 .

[107]  Noel A Cressie,et al.  Analysis of spatial point patterns using bundles of product density LISA functions , 2001 .

[108]  P. Elliott,et al.  Disease clusters: should they be investigated, and, if so, when and how? , 2001 .

[109]  S. Mase,et al.  Packing Densities and Simulated Tempering for Hard Core Gibbs Point Processes , 2001 .

[110]  J. Yukich,et al.  Central limit theorems for some graphs in computational geometry , 2001 .

[111]  E. Renshaw,et al.  Gibbs point processes for studying the development of spatial-temporal stochastic processes , 2001 .

[112]  Noel A Cressie,et al.  Patterns in spatial point locations: Local indicators of spatial association in a minefield with clutter , 2001 .

[113]  Aila Särkkä,et al.  Interacting neighbour point processes: Some models for clustering , 2001 .

[114]  Andrew B. Lawson,et al.  Statistical Methods in Spatial Epidemiology , 2001 .

[115]  Adrian Baddeley,et al.  Nonparametric measures of association between a spatial point process and a random set, with geological applications , 2002 .

[116]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[117]  Eric Renshaw,et al.  Two-dimensional spectral analysis for marked point processes , 2002 .

[118]  Andrew B. Lawson,et al.  Spatial cluster modelling , 2002 .

[119]  Sung Nok Chiu,et al.  Goodness‐of‐fit test for complete spatial randomness against mixtures of regular and clustered spatial point processes , 2002 .

[120]  R. Peng,et al.  Multi-dimensional Point Process Models for Evaluating a Wildfire Hazard Index , 2003 .

[121]  J. Møller,et al.  Shot noise Cox processes , 2003, Advances in Applied Probability.

[122]  Jesper Møller,et al.  An Introduction to Simulation-Based Inference for Spatial Point Processes , 2003 .

[123]  J. Møller,et al.  Statistical Inference and Simulation for Spatial Point Processes , 2003 .

[124]  Laurence L. George,et al.  The Statistical Analysis of Failure Time Data , 2003, Technometrics.

[125]  Y. Ogata,et al.  Modelling heterogeneous space–time occurrences of earthquakes and its residual analysis , 2003 .

[126]  S. Scobie.  Spatial epidemiology: methods and applications , 2003 .

[127]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[128]  Frederic Paik Schoenberg,et al.  Multidimensional Residual Analysis of Point Process Models for Earthquake Occurrences , 2003 .

[129]  Frederic Paik Schoenberg,et al.  Rescaling Marked Point Processes , 2004 .

[130]  D. Vere-Jones,et al.  Analyzing earthquake clustering features by using stochastic reconstruction , 2004 .

[131]  Matthew A. Bognar.  Spatial Cluster Modeling , 2004 .

[132]  David R. Brillinger,et al.  Empirical examination of the threshold model of neuron firing , 1979, Biological Cybernetics.

[133]  Trevor Bailey,et al.  Statistical Analysis of Spatial Point Patterns. Second Edition. By PETER J. DIGGLE (London: Edward Arnold). [Pp. viii+159]. ISBN 0-340-74070-1. Price £40.00. Hardback , 2004, Int. J. Geogr. Inf. Sci..

[134]  Yosihiko Ogata,et al.  Space‐time model for regional seismicity and detection of crustal stress changes , 2004 .

[135]  D. Brillinger.  Maximum likelihood analysis of spike trains of interacting nerve cells , 2004, Biological Cybernetics.

[136]  Residual analysis for spatial point processes: Discussion , 2005 .

[137]  Carlos. Comas Rodriguez.  Modelling forest dynamics through the development of spatial and temporal marked point processes , 2005 .

[138]  Peter J. Diggle,et al.  Nonparametric estimation of spatial segregation in a multivariate point process: bovine tuberculosis in Cornwall, UK , 2005 .

[139]  Jiancang Zhuang,et al.  Multi-dimensional second-order residual analysis of space-time point processes and its applications in modelling earthquake data , 2005 .

[140]  Brian D. Ripley,et al.  Spatial Statistics: Ripley/Spatial Statistics , 2005 .

[141]  Adrian Baddeley,et al.  spatstat: An R Package for Analyzing Spatial Point Patterns , 2005 .

[142]  J. Symanzik.  Statistical Analysis of Spatial Point Patterns (2nd ed.) , 2005 .

[143]  Adrian Baddeley,et al.  Modelling Spatial Point Patterns in R , 2006 .

[144]  Eric Renshaw,et al.  The analysis of marked point patterns evolving through space and time , 2006, Comput. Stat. Data Anal..

[145]  Jiancang Zhuang,et al.  Diagnostic Analysis of Space-Time Branching Processes for Earthquakes , 2006 .

[146]  R. Waagepetersen.  An Estimating Function Approach to Inference for Inhomogeneous Neyman–Scott Processes , 2007, Biometrics.

[147]  Eric Renshaw,et al.  Disentangling mark/point interaction in marked-point processes , 2007, Comput. Stat. Data Anal..