Deviation test construction and power comparison for marked spatial point patterns

The deviation test belong to core tools in point process statistics, where hypotheses are typically tested considering differences between an empirical summary function and its expectation under the null hypothesis, which depend on a distance variable r. This test is a classical device to overcome the multiple comparison problem which appears since the functional differences have to be considered for a range of distances r simultaneously. The test has three basic ingredients: (i) choice of a suitable summary function, (ii) transformation of the summary function or scaling of the differences, and (iii) calculation of a global deviation measure. We consider in detail the construction of such tests both for stationary and finite point processes and show by two toy examples and a simulation study for the case of the random labelling hypothesis that the points (i) and (ii) have great influence on the power of the tests.

[1]  Peter J. Diggle,et al.  Detecting dependence between marks and locations of marked point processes , 2004 .

[2]  Maximum likelihood estimation for cooperative sequential adsorption , 2009, Advances in Applied Probability.

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

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

[5]  E. D. Ford,et al.  Statistical inference using the g or K point pattern spatial statistics. , 2006, Ecology.

[6]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

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

[8]  A. Baddeley,et al.  Estimating the J function without edge correction , 1999, math/9910011.

[9]  Mari Myllymäki,et al.  Statistical models and inference for spatial point patterns with intensity-dependent marks , 2009 .

[10]  Aila Särkkä,et al.  Modelling the spatial structure of forest stands by multivariate point processes with hierarchical interactions. , 2009 .

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

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

[13]  B. Ripley.  Tests of 'Randomness' for Spatial Point Patterns , 1979 .

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

[15]  D. Stoyan,et al.  Statistical Analysis and Modelling of Spatial Point Patterns , 2008 .

[16]  I. Roberts,et al.  Statistical analysis of the distribution of gold particles over antigen sites after immunogold labelling , 1997 .

[17]  T. Hothorn,et al.  Multiple Comparisons Using R , 2010 .

[18]  Sung Nok Chiu,et al.  Using Weight Functions in Spatial Point Pattern Analysis with Application to Plant Ecology Data , 2009, Commun. Stat. Simul. Comput..

[19]  Martin Schlather,et al.  On the second-order characteristics of marked point processes , 2001 .

[20]  E. Thönnes,et al.  A comparative study on the power of van Lieshout and Baddeley's J-function , 1998 .

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

[22]  Dietrich Stoyan,et al.  Statistical analysis for a class of line segment processes , 1989 .

[23]  Dietrich Stoyan,et al.  Correct testing of mark independence for marked point patterns , 2011 .

[24]  C. Duby,et al.  Comparing the Performances of Diggle's Tests of Spatial Randomness for Small Samples with and without Edge‐Effect Correction: Application to Ecological Data , 1999, Biometrics.

[25]  G. A. Barnard,et al.  Discussion of Professor Bartlett''s paper , 1963 .

[26]  Murray Aitkin,et al.  The Fitting of Exponential, Weibull and Extreme Value Distributions to Complex Censored Survival Data using GLIM , 1980 .

[27]  Katja Schladitz,et al.  A third order point process characteristic , 1998, Advances in Applied Probability.

[28]  Testing the complete spatial randomness by Diggle's test without an arbitrary upper limit , 2006 .

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

[30]  A. Baddeley,et al.  A non-parametric measure of spatial interaction in point patterns , 1996, Advances in Applied Probability.

[31]  Yongtao Guan,et al.  Tests for Independence between Marks and Points of a Marked Point Process , 2006, Biometrics.

[32]  Jesper Møller,et al.  Transforming Spatial Point Processes into Poisson Processes Using Random Superposition , 2012, Advances in Applied Probability.

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

[34]  A. Penttinen,et al.  Conditionally heteroscedastic intensity‐dependent marking of log Gaussian Cox processes , 2009 .

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

[36]  L. P. Ho,et al.  Modelling marked point patterns by intensity-marked cox processes , 2008 .