Abstract
In modeling marked point processes, it is convenient to assume a separable or multiplicative form for the conditional intensity, as this assumption typically allows one to estimate each component of the model individually. Tests have been proposed in the simple marked point process case, to investigate whether the mark distribution is separable from the spatial–temporal characteristics of the point process. Here, we extend these tests to the case of a marked point process with covariates, and where one is interested in testing the separability of each of the covariates, as well as the mark and the coordinates of the point process. The extension is not at all trivial, and covariates must be treated in a fundamentally different way than marks and coordinates of the process, especially when the covariates are not uniformly distributed. An application is given to point process models for forecasting wildfire hazard in Los Angeles County, California, and solutions are proposed to the problem of how to proceed when the separability hypothesis is rejected.
Similar content being viewed by others
References
Andrews, P. L., Bradshaw, L. S. (1997). FIRES: Fire Information Retrieval and Evaluation System: A program for fire danger rating analysis. Gen. Tech. Rep. INT-GTR-367. Ogden, UT: US Department of Agriculture, Forest Service, Intermountain Research Station. p. 64
Assunção R., Maia A. (2007) A note on testing separability in spatial–temporal-marked point processes. Biometrics, 63: 290–294
Bowman A.W., Azzalini A. (1997) Applied smoothing techniques for data analysis: The kernel approach with S-Plus illustrations. Clarendon Press, Oxford
Bradshaw, L. S., Deeming, J. E., Burgan, R. E., Cohen, J. D. (1983). The 1978 national fire-danger rating system: Technical documentation. United States Department of Agriculture Forest Service General Technical Report INT-169. Ogden, UT: Intermountain Forest and Range Experiment Station. p. 46.
Cressie, N. A. (1993). Statistics for spatial data (revised ed.). New York: Wiley.
Daley, D., Vere-Jones, D. (2003). An introduction to the theory of point processes (Vol. I, 2nd ed.). New York: Springer.
Kagan Y.Y., Jackson D.D. (1994) Long-term probabilistic forecasting of earthquakes. Journal of Geophysical Research 99: 13685–13700
Keeley J. (2002) Fire management of California shrubland landscapes. Environmental Management 29: 395–408
Keeley J.E., Fotheringham C.J. (2003) Impact of past, present, and future fire regimes on North American Mediterranean shrublands. In: Veblen T.T., Baker W.L., Montenegro G., Swetnam T.W. (eds) Fire and climatic change in temperate ecosystems of the Western Americas. Springer, New York, pp 218–262
Ogata Y. (1988) Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association 83(401): 9–27
Ogata Y. (1998) Space–time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics 50(2): 379–402
Peng R.D., Schoenberg F P., Woods J. (2005) A space–time conditional intensity model for evaluating a wildfire hazard index. Journal of the American Statistical Association 100: 26–35
Pyne S.J., Andrews P.L., Laven R.D. (1996) Introduction to wildland fire (2nd ed). Wiley, New York
Schoenberg F.P. (2004) Testing separability in multi-dimensional point processes. Biometrics 60: 471–481
Schoenberg, F. P. (2006). A note on the separability of multidimensional point processes with covariates. UCLA Preprint Series, No. 496.
Schoenberg, F. P., Brillinger, D. R., Guttorp, P. M. (2002). Point processes, spatial–temporal. In A. El-Schaarawi, W. Piegorsch (Eds.), Encyclopedia of environmetrics (Vol. 3, pp. 1573–1577). New York: Wiley.
Schoengerg F.P., Peng R., Woods J. (2003) On the distribution of wildfire sizes. Environmetrics 14: 583–592
Schoenberg F.P., Chang C., Keeley J.E., Pompa J., Woods J., Xu H. (2007) A critical assessment of the Burning Index in Los Angeles County, California. International Journal of Wildland Fire 16: 473–483
Schoenberg, F. P., Pompa, J. L., Chang, C. (2009). A note on non-parametric and semi-parametric modeling of wildfire hazard in Los Angeles County, California. Journal of Environmental and Ecological Statistics, 16, in press.
Schroeder, M. J., Glovinsky, M., Hendricks, V., Hood, F., Hull, M., Jacobson, H., Kirkpatrick, R., Krueger, D., Mallory, L., Oertel, A., Reese, R., Sergius, L., Syverson, C. (1964). Synoptic weather types associated with critical fire weather. Washington, DC: Institute for Applied Technology, National Bureau of Standards, US Department of Commerce.
Sheather S.J., Jones M.C. (1991) A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B 53: 683–690
Silverman B.W. (1986) Density estimation for statistics and data analysis. Chapman and Hall, London
Stone C.J. (1984) An asymptotically optimal window selection rule for kernel density estimates. The Annals of Statistics 12: 1285–1297
Venables W.N., Ripley B.D. (2002) Modern applied statistics with S (4th ed). Springer, New York
Warren, J. R., Vance, D. L. (1981). Remote automatic weather station for resource and fire management agencies. Technical Report INT-116. Ogden, UT: USDA Forest Service, Intermountain Forest and Range Experiment Station.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Chang, CH., Schoenberg, F.P. Testing separability in marked multidimensional point processes with covariates. Ann Inst Stat Math 63, 1103–1122 (2011). https://doi.org/10.1007/s10463-010-0284-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-010-0284-7