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A new metric between distributions of point processes

Published online by Cambridge University Press:  01 July 2016

Dominic Schuhmacher*
The University of Western Australia
Aihua Xia*
The University of Melbourne
Current address: Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, CH-3012 Bern, Switzerland. Email address:
∗∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email address:
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Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric 1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about 1 and its induced Wasserstein metric 2 for point process distributions are given, including examples of useful 1-Lipschitz continuous functions, 2 upper bounds for the Poisson process approximation, and 2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of 1 in applications.

Stochastic Geometry and Statistical Applications
Copyright © Applied Probability Trust 2008 


Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Baddeley, A. and Turner, R. (2005). Spatstat: an R package for analyzing spatial point patterns. J. Statist. Software 12, 142.CrossRefGoogle Scholar
Barbour, A. D. and Brown, T. C. (1992a). Stein's method and point process approximation. Stoch. Process. Appl. 43, 931.CrossRefGoogle Scholar
Barbour, A. D. and Brown, T. C. (1992b). The Stein–Chen method, point processes and compensators. Ann. Prob. 20, 15041527.CrossRefGoogle Scholar
Barbour, A. D. and Jensen, J. L. (1989). Local and tail approximations near the Poisson limit. Scand. J. Statist. 16, 7587.Google Scholar
Barbour, A. D. and Månsson, M. (2002). Compound Poisson process approximation. Ann. Prob. 30, 14921537.Google Scholar
Barbour, A. D., Brown, T. C. and Xia, A. (1998). Point processes in time and Stein's method. Stoch. Stoch. Reports 65, 127151.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation (Oxford Studies Prob. 2). Oxford University Press.Google Scholar
Brown, T. C. and Xia, A. (1995a). On metrics in point process approximation. Stoch. Stoch. Reports 52, 247263.Google Scholar
Brown, T. C. and Xia, A. (1995b). On Stein–Chen factors for Poisson approximation. Statist. Prob. Lett. 23, 327332.CrossRefGoogle Scholar
Brown, T. C. and Xia, A. (2001). Stein's method and birth–death processes. Ann. Prob. 29, 13731403.Google Scholar
Chen, L. H. Y. and Xia, A. (2004). Stein's method, Palm theory and Poisson process approximation. Ann. Prob. 32, 25452569.Google Scholar
Conway, J. H. and Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups (Fundamental Principles Math. Sci. 290), 3rd edn. Springer, New York.CrossRefGoogle Scholar
Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks/Cole, Pacific Grove, CA.Google Scholar
Kallenberg, O. (1986). Random Measures, 4th edn. Academic Press, London.Google Scholar
Lee, A. J. (1990). U-Statistics (Statistics: Textbooks Monogr. 110). Marcel Dekker, New York.Google Scholar
Papadimitriou, C. H. and Steiglitz, K. (1998). Combinatorial Optimization: Algorithms and Complexity. Dover, Mineola, NY.Google Scholar
R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna. Available at Scholar
Reiss, R.-D. (1993). A Course on Point Processes. Springer, New York.CrossRefGoogle Scholar
Schuhmacher, D. (2005a). Estimation of distances between point process distributions. , University of Zurich. Available at Scholar
Schuhmacher, D. (2005b). Upper bounds for spatial point process approximations. Ann. Appl. Prob. 15, 615651.CrossRefGoogle Scholar
Schuhmacher, D. (2007a). Distance estimates for dependent thinnings of point processes with densities. Preprint. Available at Scholar
Schuhmacher, D. (2007b). Stein's method and Poisson process approximation for a class of Wasserstein metrics. Preprint. Available at Scholar
Xia, A. (1997a). On the rate of Poisson process approximation to a Bernoulli process. J. Appl. Prob. 34, 898907.CrossRefGoogle Scholar
Xia, A. (1997b). On using the first difference in the Stein–Chen method. Ann. Appl. Prob. 7, 899916.Google Scholar
Xia, A. (2005). Stein's method and Poisson process approximation. In An Introduction to Stein's Method (Lecture Notes Ser. Inst. Math. Sci. Nat. Univ. Singapore 4), Singapore University Press, pp. 115181.CrossRefGoogle Scholar
Xia, A. and Zhang, F. (2008). A polynomial birth–death point process approximation to the Bernoulli process. To appear in Stoch. Process. Appl. Google Scholar