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Significance tests for covariate-dependent trends in inhomogeneous spatio-temporal point processes

Stochastic Environmental Research and Risk Assessment volume 28, pages 593–609 (2014)Cite this article

Abstract

Modeling and inference for spatial and spatio-temporal point processes is an issue that has been broadly investigated in the last years. Application fields such as forestry, epidemiology and ecology have been the main engine driving such raised interest. The inclusion of spatially varying covariates in the models for the intensity function is becoming of particular interest, but little attention has been paid to testing the significance of such covariates. Testing the significance of covariates is important if one seeks to explain which covariates have an effect in the spatial or spatio-temporal distribution of the point pattern observed. We thus provide practical procedures to build statistical tests of significance for covariates that have an effect on the intensity function of a point pattern. Our approximation focuses on the conditional intensity function, by considering nonparametric kernel-based estimators. We calculate thinning probabilities under the conditions of absence and presence of a covariate and compare them through divergence measures. Based on Monte Carlo experiments, we approximate the statistical properties of our tests under a variety of practical scenarios. An application on testing the significance of a covariate in a spatio-temporal data set on wildfires is also developed.

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References

  • Assunçao R, Maia A (2007) A note on testing separability in spatial-temporal-marked point processes. Biometrics 63: 290–294.

    Article  Google Scholar 

  • Baddeley A, Turner R (2005). Spatstat: an R package for analyzing spatial point patterns. J Stat Softw 12: 1–42

    Google Scholar 

  • Baddeley A, van Lieshout MNM (1995). Area-interaction point processes. Ann Inst Stat Math 47: 601–619

    Article  Google Scholar 

  • Berman M., Diggle P.J. (1989). Estimating weighted integrals of the second-order intensity of a spatial point process. J R Stat Soc Ser B 51: 81–92

    Google Scholar 

  • Besag JE, Diggle PJ (1977) Simple Monte Carlo tests for spatial pattern. J R Stat Soc Ser C 26(3):327–333

    Google Scholar 

  • Chang C-H, Schoenberg FP (2011). Testing separability in marked multidimensional point processes with covariates. Ann Inst Stat Math 63:1103–1122

    Article  Google Scholar 

  • Comas C, Mateu J (2011) Statistical inference for Gibbs point processes based on field observations. Stoch Environ Res Risk Assess 25:287–300

    Article  Google Scholar 

  • Cressie NAC (1993) Statistics for spatial data (revised edn.). Wiley, New York

  • Daley D, Vere-Jones D (2003) An introduction to the theory of point processes, 2nd ed, vol I. Springer, New York

  • Dasgupta A, Raftery AE (1998) Detecting features in spatial point processes with clutter via model-based clustering. J Am Stat Assoc 93: 294–302

    Article  Google Scholar 

  • DÍaz-Avalos C, Juan P, Mateu J (2012) Similarity measures of conditional intensity functions to test separability in multidimensional point processes. Stoch Environ Res Risk Assess 27:1193–1205

    Google Scholar 

  • Diggle PJ (2003). Statistical analysis of spatial point patterns, 2nd edn. Arnold, London

  • Diggle PJ, Zheng P, Durr P (2005) Nonparametric estimation of spatial segregation in a multivariate point process: bovine tuberculosis in Cornwall, UK. Appl Stat 54(3): 645–658

    Google Scholar 

  • Gelfand AE, Diggle PJ, Guttorp P, Fuentes M (2010) Handbook of spatial statistics. CRC Press, Boca Raton.

  • Guan Y (2008a) A goodness-of-fit test for inhomogeneous spatial Poisson processes. Biometrika 95: 831–845

    Article  Google Scholar 

  • Guan Y. (2008b) On consistent nonparametric intensity estimation for inhomogeneous spatial point processes. J Am Stat Assoc 103: 1238–1247

    Article  CAS  Google Scholar 

  • Juan P, Mateu J, Saez M (2012) Pinpointing spatio-temporal interactions in wildfire patterns. Stoch Environ Res Risk Assess 26(8):1131–1150

    Article  Google Scholar 

  • Kelshall JE, Diggle PJ (1998) Spatial variation in risk of disease: a non parametric binary regression approach. Appl Stat 47: 539–573

    Google Scholar 

  • Møller J, Díaz-Avalos C (2010) Structured spatio-temporal shot-noise Cox point process models, with a view to modelling forest fires. Scand J Stat 37: 2–15

    Article  Google Scholar 

  • Møller J, Rasmussen JG (2005) Perfect simulation of Hawkes processes. Adv Appl Probab 37: 629–646.

    Google Scholar 

  • Møller J, Waagepetersen RP (2003) An introduction to simulation-based inference for spatial point processes. In Møller J (ed) Spatial statistics and computational methods, lecture notes in statistics, vol 173. Springer, New York, pp 143–198

  • Møller J, Waagepetersen RP (2004) Statistical inference and simulation for spatial point processes. Chapman and Hall/CRC, Boca Raton

  • Ripley B.D. (1976) The second-order analysis of stationary point processes. J Appl Probab 13:256–266

    Article  Google Scholar 

  • Schoenberg F.P. (2004) Testing separability in multi-dimensional point processes. Biometrics 60:471–481

    Article  Google Scholar 

  • Schoenberg FP (2006) A note on the separability of multidimensional point processes with covariates. UCLA Preprint Series, No. 496

  • Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, London

  • Turner R (2009) Point patterns of forest fire locations. Environ Ecol Stat 16: 197–223

    Article  Google Scholar 

  • Waagepetersen R (2007) An estimating function approach to inference for inhomogeneous Neyman–Scott processes. Biometrics 63:252–258

    Article  Google Scholar 

  • Waagepetersen R, Guan Y (2009) Two-step estimation for inhomogeneous spatial point processes. J R Stat Soc Ser B 71:685–702

    Article  Google Scholar 

Download references

Acknowledgments

Work partially funded by Grant MTM2010-14961 from the Spanish Ministry of Science and Education and by the program PASPA of the Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México. We are grateful to the reports of the reviewers and AE which improved the reading and quality of the manuscript.

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Authors and Affiliations

  1. Research Institute in Applied Mathematics and Systems, National University of Mexico, Mexico, DF, Mexico

    Carlos Díaz-Avalos

  2. Department of Mathematics, Universitat Jaume I, Campus Riu Sec, 12071, Castellón, Spain

    P. Juan & J. Mateu

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  1. Carlos Díaz-Avalos
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  2. P. Juan
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  3. J. Mateu
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Correspondence to Carlos Díaz-Avalos.

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Díaz-Avalos, C., Juan, P. & Mateu, J. Significance tests for covariate-dependent trends in inhomogeneous spatio-temporal point processes. Stoch Environ Res Risk Assess 28, 593–609 (2014). https://doi.org/10.1007/s00477-013-0775-1

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Keywords

  • Conditional intensity function
  • Covariates
  • Multidimensional spatial point processes
  • Nonparametric estimation
  • Significance
  • Wildfires