×

zbMATH — the first resource for mathematics

Residual analysis for spatial point processes (with discussion). (English) Zbl 1112.62302
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 \(\lambda\) 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.

MSC:
62-XX Statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andersen P., Statistical Models based on Counting Processes (1993) · Zbl 0769.62061 · doi:10.1007/978-1-4612-4348-9
[2] Anselin L., Geogr. Anal. 27 pp 93– (1995) · doi:10.1111/j.1538-4632.1995.tb00338.x
[3] Atkinson A., Plots, Transformations and Regression (1985) · Zbl 0582.62065
[4] A. Baddeley, J. Moller, M. Hazelton, and R. Turner (2005 ) Residuals for spatial point processes using the Papangelou conditional intensity . To be published.
[5] DOI: 10.1111/1467-9574.00144 · Zbl 1018.62027 · doi:10.1111/1467-9574.00144
[6] A. Baddeley, J. Moller, and R. Waagepetersen (2006 ) Residual versions of summary functions for spatial point patterns . To be published.
[7] DOI: 10.1111/1467-842X.00128 · Zbl 0981.62078 · doi:10.1111/1467-842X.00128
[8] Baddeley A., Lect. Notes Statist. 185 (2005)
[9] Baddeley A., J. Statist. Softwr. 12 (6) pp 1– (2005)
[10] Baddeley A., Research Report 2004/08 (2004)
[11] Berman M., Appl. Statist. 35 pp 54– (1986)
[12] Berman M., J. R. Statist. Soc. 51 pp 81– (1989)
[13] Berman M., Appl. Statist. 41 pp 31– (1992)
[14] Besag J., Statistician 24 pp 179– (1975)
[15] Besag J., Bull. Int. Statist. Inst. 44 pp 77– (1978)
[16] Besag J., Appl. Statist. 26 pp 327– (1977)
[17] Brillinger D., Developments in Statistics pp 33– (1978)
[18] DOI: 10.1007/BF00318010 · Zbl 0646.92007 · doi:10.1007/BF00318010
[19] Brillinger D., Can. J. Statist. 22 pp 177– (1994)
[20] Brillinger D., Proc. 13th Int. Biometric Conf., Seattle (1986)
[21] Clyde M., Spatial Statistics and Imaging pp 14– (1991) · doi:10.1214/lnms/1215460490
[22] Collett D., Modelling Binary Data (1991) · Zbl 1041.62058 · doi:10.1007/978-1-4899-4475-7
[23] Cox D. R., Point Processes (1980)
[24] Cox D., The Statistical Analysis of Series of Events (1966) · Zbl 0148.14005 · doi:10.1007/978-94-011-7801-3
[25] Cressie N., Statistics for Spatial Data (1991) · Zbl 0799.62002
[26] DOI: 10.1198/108571101300325292 · doi:10.1198/108571101300325292
[27] DOI: 10.1002/nav.1022 · Zbl 1005.90555 · doi:10.1002/nav.1022
[28] Daley D., An Introduction to the Theory of Point Processes (1988) · Zbl 0657.60069
[29] Davison A., Statistical Theory and Modelling (in Honour of Sir David Cox FRS) pp 83– (1991)
[30] Diggle P., J. R. Statist. Soc. 40 pp 178– (1978)
[31] Diggle P., Appl. Statist. 34 pp 138– (1985)
[32] Diggle P. J., J. R. Statist. Soc. 153 pp 349– (1990)
[33] Diggle P., Statistical Analysis of Spatial Point Patterns (2003) · Zbl 1021.62076
[34] Diggle P., Statist. Meth. Med. Res. 4 pp 124– (1995)
[35] Diggle P. J., J. R. Statist. Soc. 157 pp 433– (1994)
[36] Fowlkes E., Biometrika 74 pp 503– (1987)
[37] DOI: 10.1111/1467-9876.00261 · Zbl 1111.62324 · doi:10.1111/1467-9876.00261
[38] DOI: 10.1007/BF01609410 · doi:10.1007/BF01609410
[39] Getis A., Ecology 68 pp 473– (1987)
[40] Geyer C., Stochastic Geometry: Likelihood and Computation pp 79– (1999)
[41] Geyer C., Scand. J. Statist. 21 pp 359– (1994)
[42] Gnanadesikan R. B., J. R. Statist. Soc. 32 pp 88– (1970)
[43] Horvitz D., J. Am. Statist. Ass. 47 pp 663– (1952)
[44] Jensen J., Ann. Appl. Probab. 1 pp 445– (1991)
[45] Karr A., Point Processes and Their Statistical Inference (1985)
[46] Kulldorff M., Recent Advances on Scan Statistics pp 303– (1999)
[47] Landwehr J., J. Am. Statist. Ass. 79 pp 61– (1984)
[48] Lawson A., Computational Statistics pp 35– (1992) · doi:10.1007/978-3-662-26811-7_5
[49] Lawson A., Biometrics 49 pp 889– (1993)
[50] Lewis P., Stochastic Point Processes pp 1– (1972)
[51] Lieshout M., Markov Point Processes and Their Applications (2000) · Zbl 0968.60005 · doi:10.1142/9781860949760
[52] Lindsey J., The Analysis of Stochastic Processes using GLIM (1992) · Zbl 0767.62099 · doi:10.1007/978-1-4612-2888-2
[53] Lindsey J. K., Appl. Statist. 44 pp 201– (1995)
[54] DOI: 10.1023/A:1014662415827 · Zbl 1086.60512 · doi:10.1023/A:1014662415827
[55] Merzbach E., Ann. Probab. 14 pp 1380– (1986)
[56] DOI: 10.1239/aap/1059486821 · Zbl 1045.60007 · doi:10.1239/aap/1059486821
[57] DOI: 10.1111/1467-9469.00115 · Zbl 0931.60038 · doi:10.1111/1467-9469.00115
[58] Moller J., Statistical Inference and Simulation for Spatial Point Processes (2003)
[59] Moller J., Lect. Notes Statist. 173 pp 143– (2003) · doi:10.1007/978-0-387-21811-3_4
[60] Nair M., Ann. Probab. 18 pp 1222– (1990)
[61] Nguyen X., Math. Nacht. 88 pp 105– (1979)
[62] Numata M., Bull. Choshi Mar. Lab. (6) pp 27– (1964)
[63] Ogata Y., J. Am. Statist. Ass. 83 pp 9– (1988)
[64] Ogata Y., Ann. Inst. Statist. Math. 33 pp 315– (1981)
[65] Ogata Y., Proc. Pacific Statist. Congr. pp 150– (1986)
[66] DOI: 10.1007/BF00533242 · Zbl 0265.60047 · doi:10.1007/BF00533242
[67] Pregibon D., Ann. Statist. 9 pp 705– (1981)
[68] R Development Core Team, R: a Language and Environment for Statistical Computing (2004)
[69] Ripley B., J. R. Statist. Soc. 39 pp 172– (1977)
[70] Ripley B., Spatial Statistics (1981) · Zbl 0583.62087 · doi:10.1002/0471725218
[71] Ripley B., Statistical Inference for Spatial Processes (1988) · Zbl 0716.62100 · doi:10.1017/CBO9780511624131
[72] Ripley B., J. Lond. Math. Soc. 15 pp 188– (1977)
[73] Sarkka A., Pseudo-likelihood Approach for Pair Potential Estimation of Gibbs Processes (1993)
[74] DOI: 10.1016/S0304-4149(98)00098-2 · Zbl 0962.60029 · doi:10.1016/S0304-4149(98)00098-2
[75] Stoyan D., Math. Nacht. 151 pp 95– (1991)
[76] Stoyan D., Stochastic Geometry and Its Applications (1995) · Zbl 0838.60002
[77] Stoyan D., Fractals, Random Shapes and Point Fields (1995) · Zbl 0828.62085
[78] Venables W., Modern Applied Statistics with S-Plus (1997) · Zbl 0876.62001 · doi:10.1007/978-1-4757-2719-7
[79] Vere-Jones D., J. R. Statist. Soc. 32 pp 1– (1970)
[80] DOI: 10.1111/j.1467-842X.2004.00319.x · Zbl 1078.60039 · doi:10.1111/j.1467-842X.2004.00319.x
[81] Wand M., Kernel Smoothing (1995) · doi:10.1007/978-1-4899-4493-1
[82] Wartenberg D., Spatial Statistics: Past, Present and Future pp 133– (1990)
[83] Zhuang J., Lect. Notes Statist. 185 (2005)
[84] Baddeley A. J., Bernoulli 6 pp 783– (2000)
[85] Baddeley A., Proc. 11th International Association for Pattern Recognition Int. Conf. Pattern Recognition, Los Alamitos pp B136– (1992)
[86] Baddeley A., Int. Statist. Rev. 57 pp 89– (1989)
[87] A. Baddeley, and J. Moller (2005 ) Residuals for Markov random fields . To be published.
[88] A. Baddeley, J. Moller, M. Hazelton, and R. Turner (2005 ) Residuals for spatial point processes using the Papangelou conditional intensity . To be published.
[89] DOI: 10.1111/1467-9574.00144 · Zbl 1018.62027 · doi:10.1111/1467-9574.00144
[90] A. Baddeley, J. Moller, and R. Waagepetersen (2006 ) Residual versions of summary functions for spatial point patterns . To be published.
[91] Baddeley A. J., Biometrics 40 pp 1089– (1984)
[92] Baddeley A., J. Statist. Softwr. 12 pp 1– (2005)
[93] Bartlett M., Biometrics 20 pp 891– (1964)
[94] Besag J. E., Biometrika 76 pp 633– (1989)
[95] Besag J. E., Biometrika 78 pp 301– (1991)
[96] Besag J. E., J. Appl. Probab. 19 pp 210– (1982)
[97] Brillinger D. R., Technical Report (1997)
[98] C. Comas (2005 ) Modelling forest dynamics through the development of spatial and temporal marked point processes.PhD Thesis. University of Strathclyde, Glasgow.
[99] Cox D., Applied Statistics: Principles and Examples (1981) · Zbl 0612.62002
[100] Cressie N. A. C., Statistics for Spatial Data (1993) · Zbl 1347.62005 · doi:10.1002/9781119115151
[101] Diggle P., Appl. Statist. 34 pp 138– (1985)
[102] Diggle P. J., J. R. Statist. Soc. 153 pp 349– (1990)
[103] Diggle P., Statistical Analysis of Spatial Point Patterns (2003) · Zbl 1021.62076
[104] Diggle P. J., J. R. Statist. Soc. 157 pp 433– (1994)
[105] Diggle P., Appl. Statist. 54 pp 645– (2005) · Zbl 05188703 · doi:10.1111/j.1467-9876.2005.05373.x
[106] Elliott P., Spatial Epidemiology; Methods and Applications (2000)
[107] Geyer C., Stochastic Geometry: Likelihood and Computation pp 79– (1999)
[108] Geyer C., Scand. J. Statist. 21 pp 359– (1994)
[109] Goulard M., Scand. J. Statist. 23 pp 365– (1996)
[110] DOI: 10.1093/biomet/89.2.411 · Zbl 1019.62091 · doi:10.1093/biomet/89.2.411
[111] Grabarnik P., J. Statist. Computn Simuln 68 pp 113– (2001) · Zbl 1029.60039 · doi:10.1080/00949650108812059
[112] Kendall W., J. Appl. Probab. 28 pp 767– (1990)
[113] Lawson A., Biometrics 49 pp 889– (1993)
[114] Lawson A. B., Statistical Methods in Spatial Epidemiology (2001) · Zbl 1016.62125
[115] Lawson A., Spatial Cluster Modeling (2002) · doi:10.1201/9781420035414
[116] DOI: 10.1080/02664769624260 · doi:10.1080/02664769624260
[117] Lawson A. B., J. R. Statist. Soc. 157 pp 285– (1994)
[118] Moller J., Statistical Inference and Simulation for Spatial Point Processes (2003) · doi:10.1201/9780203496930
[119] Ogata Y., J. Geophys. Res. 109 pp B3– (2004)
[120] Ogata Y., Appl. Statist. 52 pp 499– (2003) · Zbl 1111.86301 · doi:10.1111/1467-9876.00420
[121] Ohser J., Statistical Analysis of Microstructures in Materials Science (2000) · Zbl 0960.62129
[122] Papangelou F., Stochastic Geometry pp 114– (1974)
[123] Penrose M. D., Random Geometric Graphs (2003) · Zbl 1029.60007 · doi:10.1093/acprof:oso/9780198506263.001.0001
[124] Penrose M. D., Ann. Appl. Probab. 11 pp 1005– (2001)
[125] Penttinen A., Jyvask. Stud. Comput. Sci. Econ. Statist. 7 (1984)
[126] DOI: 10.1002/1521-4036(200209)44:6<718::AID-BIMJ718>3.0.CO;2-6 · doi:10.1002/1521-4036(200209)44:6<718::AID-BIMJ718>3.0.CO;2-6
[127] E. Renshaw, J. Mateu, and F. Saura (2005 ) Disentangling mark/point interaction in marked point processes . To be published. · Zbl 1161.62409
[128] DOI: 10.1016/S0167-9473(00)00028-1 · Zbl 1080.62066 · doi:10.1016/S0167-9473(00)00028-1
[129] Sarkka A., Pseudo-likelihood Approach for Pair Potential Estimation of Gibbs Processes (1993)
[130] A. Sarkka, and E. Renshaw (2005 ) The analysis of marked point patterns evolving through space and time . To be published.
[131] A. Sarkka, and D. Stoyan (2005 ) Some residual analysis for Gibbs point process models . To be published.
[132] F. Saura, J. Mateu, and E. Renshaw (2006 ) Structure detection for one-dimensional patterns through wavelets . To be published.
[133] DOI: 10.1198/016214503000000710 · doi:10.1198/016214503000000710
[134] Stoyan D., Math. Nacht. 151 pp 95– (1991)
[135] Stoyan D., Fractals, Random Shapes and Point Fields (1995) · Zbl 0828.62085
[136] Tukey J. W., Statistical Papers in Honor of George W. Snedecor pp 293– (1972)
[137] R. Waagepetersen (2005 ) An estimating equation approach to inference for inhomogeneous Neyman-Scott processes . To be published.
[138] Zhuang J., J. R. Statist. Soc. (2005)
[139] DOI: 10.1029/2003JB002879 · doi:10.1029/2003JB002879
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.