Electronic Journal of Statistics

Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images

Abstract

We propose a class of goodness-of-fit tests for complete spatial randomness (CSR). In contrast to standard tests, our procedure utilizes a transformation of the data to a binary image, which is then characterized by geometric functionals. Under a suitable limiting regime, we derive the asymptotic distribution of the test statistics under the null hypothesis and almost sure limits under certain alternatives. The new tests are computationally efficient, and simulations show that they are strong competitors to other tests of CSR. The tests are applied to a real data set in gamma-ray astronomy, and immediate extensions are presented to encourage further work.

Article information

Source
Electron. J. Statist., Volume 12, Number 2 (2018), 2873-2904.

Dates
First available in Project Euclid: 18 September 2018

https://projecteuclid.org/euclid.ejs/1537257628

Digital Object Identifier
doi:10.1214/18-EJS1467

Citation

Ebner, Bruno; Henze, Norbert; Klatt, Michael A.; Mecke, Klaus. Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images. Electron. J. Statist. 12 (2018), no. 2, 2873--2904. doi:10.1214/18-EJS1467. https://projecteuclid.org/euclid.ejs/1537257628

References

• Acero, F. et al. (2015). Fermi Large Area Telescope third source catalog., The Astrophysical Journal Supplement Series, 218(2):23.
• Ade, P. a. R. et al. (2016a). Planck 2015 results - XVI. Isotropy and statistics of the CMB., Astronomy & Astrophysics, 594:A16.
• Ade, P. a. R. et al. (2016b). Planck 2015 results - XVII. Constraints on primordial non-Gaussianity., Astronomy & Astrophysics, 594:A17.
• Adler, R. J., Agami, S., and Pranav, P. (2017). Modeling and replicating statistical topology and evidence for CMB nonhomogeneity., Proceedings of the National Academy of Sciences, 114(45):11878–11883.
• Adler, R. J., Samorodnitsky, G., and Taylor, J. E. (2010). Excursion sets of three classes of stable random fields., Advances in Applied Probability, 42(2):293–318.
• Adler, R. J. and Taylor, J. E. (2007)., Random Fields and Geometry. Springer.
• Atwood, W. et al. (2009). The Large Area Telescope on the Fermi Gamma-Ray Space Telescope Mission., The Astrophysical Journal, 697(2):1071.
• Baddeley, A., Rubak, E., and Turner, R. (2015)., Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press, London.
• Baddeley, A. and Silverman, B. (1984). A cautionary example on the use of second-order methods for analyzing point patterns., Biometrics, 40:1089–1094.
• Berrendero, J., Cuevas, A., and Pateiro-López, B. (2012). Testing uniformity for the case of a planar unknown support., The Canadian Journal of Statistics, 40(2):378–395.
• Berrendero, J., Cuevas, A., and Vázquez-Grande, F. (2006). Testing Multivariate Uniformity: The Distance-to-Boundary Method., The Canadian Journal of Statistics, 34(4):693–707.
• Bordoloi, R. et al. (2017). Mapping the Nuclear Outflow of the Milky Way: Studying the Kinematics and Spatial Extent of the Northern Fermi Bubble., The Astrophysical Journal, 834(2):191.
• Byth, K. and Ripley, B. (1980). On sampling spatial patterns by distance methods., Biometrics, 36:279–284.
• Cabella, P., Hansen, F., Marinucci, D., Pagano, D., and Vittorio, N. (2004). Search for non-Gaussianity in pixel, harmonic and wavelet space: compared and combined., Physical Review D, 69(6).
• Chingangbam, P., Ganesan, V., Yogendran, K. P., and Park, C. (2017). On Minkowski Functionals of CMB polarization., Physics Letters B, 771:67–73.
• Coeurjolly, J. (2017). Median-based estimation of the intensity of a spatial point process., Annals of the Institute of Statistical Mathematics, 69:303–331.
• Core Team, R. (2016)., R: A language and environment for statistical computing. Statistical Computing.
• Cressie, N. (1993)., Statistics for Spatial Data. Wiley.
• Dai, X., Wang, Z., Vadakkumthani, J., and Xing, Y. (2016). Identification of candidate millisecond pulsars from fermi lat observations., Research in Astronomy and Astrophysics, 16(69):97–109.
• Dazzo, F. B., Yanni, Y. G., Jones, A., and Elsadany, A. Y. (2015). CMEIAS bioimage informatics that define the landscape ecology of immature microbial biofilms developed on plant rhizoplane surfaces., AIMS Bioengineering, 2(5):469–486.
• Ebner, B., Henze, N., and Yukich, J. E. (2018). Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances., Journal of Multivariate Analysis, 165:231–242.
• Fantaye, Y., Marinucci, D., Hansen, F., and Maino, D. (2015). Applications of the Gaussian kinematic formula to CMB data analysis., Physical Review D, 91(6).
• Gaetan, C. and Guyon, X. (2010)., Spatial Statistics and Modeling. Springer.
• Göring, D. (2012)., Gamma-Ray Astronomy Data Analysis Framework based on the Quantification of Background Morphologies using Minkowski Tensors. PhD thesis, Universität Erlangen-Nürnberg.
• Göring, D., Klatt, M. A., Stegmann, C., and Mecke, K. (2013). Morphometric analysis in gamma-ray astronomy using Minkowski functionals., Astronomy & Astrophysics, 555(A38).
• Gray, S. (1971). Local properties of binary images in two dimensions., IEEE Transactions on Computers, C-20(5):551–561.
• Heinrich, L. (2015). Gaussian limits of empirical multiparameter $K$-functions of homogeneous Poisson processes and tests for complete spatial randomness., Lithuanian Mathematical Journal, 55(1):72–90.
• Hopkins, B. (1954). A new method of determining thy type of distribution of plant individuals., Annals of Botany, 18:213–227.
• Illian, J., Penttinen, A., Stoyan, H., and Stoyan, D. (2008)., Statistical Analysis and Modelling of Spatial Point Patterns. Wiley.
• Justel, A., Pena, D., and Zamar, R. (1997). A multivariate Kolmogorov-Smirnov test of goodness of fit., Statistics & Probability Letters, 35:251–259.
• Karlis, D. and Xekalaki, E. (2005). Mixed Poisson Distributions., International Statistical Review, 73(1):35–58.
• Klatt, M. (2016)., Morphometry of random spatial structures in physics. PhD thesis.
• Klette, R. and Rosenfeld, A. (2004)., Digital Geometry. Morgan Kaufmann, San Francisco.
• Kong, T. and Rosenfeld, A. (1989). Digital topology: Introduction and survey., Computer Vision, graphics, and Image Processing, 48:357–393.
• Kong, Y. and Rosenfeld, A., editors (1996)., Topological Algorithms for Digital Image Processing. North Holland, Amsterdam.
• Kratz, M. and Vadlamani, S. (2017). Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian random Fields., Journal of Theoretical Probability.
• Lake, D. E. and Keenan, D. M. (1995). Identifying minefields in clutter via collinearity and regularity detection., SPIE, 2496:519–530.
• Legland, D., Kiêu, K., and Devaux, M.-F. (2007). Computation of Minkowski Measures on 2D and 3D binary images., Image Analysis & Stereology, 26:83–92.
• Liang, J., Fang, K., F., H., and Li, R. (2001). Testing Multivariate Uniformity and Its Applications., Mathematics of Computation, 70(233):337–355.
• Lloyd, C. (2007)., Local Models for Spatial Analysis. CRC Press, Boca Raton.
• Loosmore, N. and Ford, E. (2006). Statistical Inference using the G or K point pattern spatial statistics., Ecology, 87:1925–1931.
• Mantz, H., Jacobs, K., and Mecke, K. (2008). Utilising minkowski functionals for image analysis: a marching square algorithm., Journal of Statistical Mechanics: Theory and Experiment, 2008:P12015.
• Marhuenda, Y., Morales, D., and Pardo, M. (2005). A comparison of uniformity tests., Statistics, 39(4):315–328.
• Marinucci, D. (2004). Testing for Non-Gaussianity on Cosmic Microwave Background Radiation: A Review., Statistical Science, 19(2):294–307.
• Mecke, K. and Stoyan, D. (Eds.) (2000)., Statistical Physics and Spatial Statistics - The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, volume 554 of Lecture Notes in Physics. Springer.
• Møller, J. and Waagepetersen, R. (2003)., Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall.
• Müller, D. (2017). A central limit theorem for Lipschitz–Killing curvatures of Gaussian excursions., Journal of Mathematical Analysis and Applications, 452(2):1040–1081.
• Novikov, D., Schmalzing, J., and Mukhanov, V. F. (2000). On non–Gaussianity in the cosmic microwave background., Astron. Astrophys., 364:17–25.
• Okabe, B., Boots, B., and Sugihara, K. (1992)., Spatial Tesselations. Wiley.
• Penrose, M. (2004)., Random geometric graphs. Oxford University Press.
• Räth, C. et al. (2011). Scale-dependent non-Gaussianities in the WMAP data as identified by using surrogates and scaling indices: Non-Gaussianities in the WMAP data., Monthly Notices of the Royal Astronomical Society, 415(3):2205–2214.
• Reddy, T. R., Vadlamani, S., and Yogeshwaran, D. (2018). Central limit theorem for exponentially quasi-local statistics of spin models on cayley graphs., Journal of Statistical Physics.
• Rinott, J. and Rotar, V. (1996). A Multivariate CLT for Local Dependence with $n^-1/2\log n$ Rate and Applications to Multivariate Graph Related Statistics., Journal of Multivariate Analysis, 56:333–350.
• Schneider, R. and Weil, W. (2008)., Stochastic and Integral Geometry. Springer.
• Schröder-Turk, G. E. et al. (2011). Minkowski tensor shape analysis of cellular, granular and porous structures., Advanced Materials, 23:2535–2553.
• Shaked, M. (1980). On mixtures from exponential families., Journal of the Royal Statistical Society Series B, 42:192–198.
• Skellam, J. (1954). Appendix to article by Hopkins (1954)., Annals of Botany, 18:226–227.
• Su, M., Slatyer, T. R., and Finkbeiner, D. P. (2010). Giant Gamma-ray Bubbles from Fermi-LAT: Active Galactic Nucleus Activity or Bipolar Galactic Wind?, The Astrophysical Journal, 724(2):1044.
• Taheriyoun, A. R., Shafie, K., and Jozani, M. J. (2009). A note on the higher moments of the euler characteristic of the excursion sets of random fields., Statistics & Probability Letters, 79(8):1074–1082.
• Tenreiro, C. (2007). On the Finite Sample Behavior of Fixed Bandwidth Bickel-Rosenblatt Test for Univariate and Multivariate Uniformity., Communications in Statistics - Simulation and Computation, 36:827–846.
• Viktorova, I. I. and Chistyakov, V. P. (1966). Some generalizations of the test of empty boxes., Theory of Probability and its applications, 11(2):270–276.
• Wheeler, D. C. (2007). A comparison of spatial clustering and cluster detection techniques for childhood leukemia incidence in ohio, 1996-2003., International Journal of Health Geographics, 6(13).