Volume 20, Issue 2 (12-2021)                   JIRSS 2021, 20(2): 1-28 | Back to browse issues page


XML Print


Department of Statistics, Shahid Beheshti University, Tehran, Iran. , faridrohani@sbu.ac.ir
Abstract:   (254 Views)

In this paper, we present the Anderson-Darling (AD) and Kolmogorov-Smirnov (KS) goodness of fit statistics for stationary and non-stationary random fields. Namely, we adopt an easy-to-apply method based on a random projection of a Hilbert-valued random field onto the real line R, and then, applying the well-known AD and KS goodness of fit tests. We conclude this paper by studying the behavior of the proposed approach in the wide range of simulation studies and in a case study of autistic and healthy individuals.

Full-Text [PDF 1447 kb]   (347 Downloads)    
Type of Study: Original Paper | Subject: 62Mxx: Inference from stochastic processes
Received: 2020/08/13 | Accepted: 2021/07/18 | Published: 2022/04/12

References
1. Achlioptas, D. (2003), Database-friendly random projections, Johnson-Lindenstrauss with binary coins. Journal of Computer and System Sciences , 66 (4), 671-687. [DOI:10.1016/S0022-0000(03)00025-4]
2. Adler, R. J. (1981). The Geometry of Random Fields . Wiley & Sons, New York.
3. Adler, R. J., and Taylor, J. E. (2004), Random Fields and Geometry . Springer. [DOI:10.1002/0471667196.ess2164]
4. Anderson, T. W., and Darling, D. A. (1952), Asymptotic theory of certain "goodness-of-fit" criteria based on stochastic processes. The Annals of Mathematical Statistics, 23 (2), 193-212. [DOI:10.1214/aoms/1177729437]
5. Baumgartner, W., Weiss , P., and Schindler, H. (1998), A nonparametric test for the general two-sample problem. Biometrics, 54 (3), 1129-1135. [DOI:10.2307/2533862]
6. Benjamini, Y., and Hochberg, Y. (1995), Controlling the false discovery rate, A practical and powerful approach to multiple testing. Journal of the Royal Statistical Society. Series B, 57 (1), 289-300. [DOI:10.1111/j.2517-6161.1995.tb02031.x]
7. Benjamini, Y., and Yekutieli, D. (2001), The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics, 29 , 1165-1188. [DOI:10.1214/aos/1013699998]
8. Berizzi, F., Dalle Mese, E., and Martorella, M. (2004), A sea surface fractal model for ocean remote sensing.{it International Journal of Remote Sensing, 25 , 1265-1270. [DOI:10.1080/01431160310001592157]
9. Cressie, N. A. C., and Huang, H. C. (1999), Classes of nonseparable, spatio-temporal stationary covariance functions. Journal of the American Statistical Association, 94 (448), 1330-1340. [DOI:10.1080/01621459.1999.10473885]
10. Cuesta-Albertos, J. A., Fraiman, R., and Ransford, T. (2006), Random projections and goodness-of-fit tests in infinite dimensional spaces. Bulletin of the Brazilian Mathematical Society, 37 , 477-501. [DOI:10.1007/s00574-006-0023-0]
11. Cuesta-Albertos, J. A., Fraiman, R., and Ransford, T. (2007a), A sharp form of the Cramer-Wold theorem. Journal of Theoretical Probability, 20 , 201-209. [DOI:10.1007/s10959-007-0060-7]
12. Cuesta-Albertos, J. A., del Barrio, E., Fraiman, R., and Matran, C. (2007b), The random projection method in goodness-of-fit for functional data. Computational Statistics and Data Analysis, 51 , 4814-4831. [DOI:10.1016/j.csda.2006.09.007]
13. Cuesta-Albertos, J. A., Gamboa, F., and Nieto-Reyes, A. (2009), A random-projection based procedure to test if a stationary process is Gaussian.arXiv, 1 ,1-31.
14. Cuevas, A., and Fraiman, R. (2009), On depth measures and dual statistics. a methodology for dealing with general data. Journal of Multivariate Analysis, 100 , 753-766. [DOI:10.1016/j.jmva.2008.08.002]
15. DasGupta, A. (2008), Asymptotic theory of statistics and probability. Springer Series in Statistics.
16. Di Bernardino, E., Estrade, A., and Le'on, J. (2017), A test of Gaussianity based on the Euler characteristic on excursion sets. Electronic journal of statistics, Shaker Heights, OH, Institute of Mathematical Statistics, 11 (1), 843-890. [DOI:10.1214/17-EJS1248]
17. Dorocic, I.P., Furth, D., Xuan, Y., Johansson, Y., Pozzi, L., Silberberg, G., Carl'en, M., and Meletis, K. (2014), A whole-brain atlas of inputs to serotonergic neurons of the dorsal and median raphe nuclei. Neuron, 83 (3), 663-678. [DOI:10.1016/j.neuron.2014.07.002]
18. Gneiting, T., Genton, M. G., and Guttorp, P. (2006), Geostatistical space-time models, stationarity, separability and full symmetry. Monographs in Statistics and Applied Probability, 107. Chapman and Hall/CRC Press. [DOI:10.1201/9781420011050.ch4]
19. Hsu, C. Y. , Schneller, B., Ghaffari, M., Alaraj, A., and Linninger, A. (2015), Medical image processing for fully integrated subject specific whole brain mesh generation. Technologies, 3 , 126-141. [DOI:10.3390/technologies3020126]
20. Johnson, B. W., and Lindenstrauss, J. (1984), Extensions of Lipschitz mappings into a Hilbert space. Contemporary Mathematics, 26 , 189-206. [DOI:10.1090/conm/026/737400]
21. Kanner, L. (1943), Autistic disturbances of affective contact. Nervous Child, 2 , 217-250.
22. Lehmann, E. L. (1999), Elements of large-sample theory . Springer Texts in Statistics. [DOI:10.1007/b98855]
23. Lejsek, H., A smundsson, F. H., Jonsson, B., and Amsaleg, L. (2005), Efficient and effective image copyright enforcement. 21e journees Bases de donnees avancees.
24. Li, H., Parikh, N. A., and He, L. (2018), A novel transfer learning approach to enhance deep neural network classification of brain functional connectomes. Frontiers in Neuroscience, 12 , 491. [DOI:10.3389/fnins.2018.00491]
25. Lim, S. C., and Teo, L. P. (2009), Gaussian fields and gaussian sheets with generalized cauchy [DOI:10.1016/j.spa.2008.06.011]
26. covariance structure. Stochastic Processes and their Applications, 119 (4), 1325-1356.
27. Mateu, J., Porcu, E., Christakos, G., and Bevilacqua, M. (2007), Fitting negative spatial covariances to geothermal field temperatures in nea kessani (greece). Envoronmetrics, 18 , 759-773. [DOI:10.1002/env.871]
28. Morariu, V. , Craciun, C., Neamtu, S., Iarinca, L., and Mihali, C. (2006), A fractal and longe-range correlation analysis of plant nucleus ultrastructure. Romanian Journal of Biophysics, 16 , 243-252.
29. Patilea, V., S'anchez-Sellero, C., and Saumard, M. (2012), Projection-based nonparametric testing for functional covariate effect. arXiv: 1205.5578.
30. Song, X., Meng, L., Shi, Q., and Lu, H. (2015), Learning tensor-based features for whole-brain f{MRI classification. Medical Image Computing and Computer-Assisted Intervention (MICCAI) , 613-620. [DOI:10.1007/978-3-319-24553-9_75]
31. Stein, M. L. (2005), Space-time covariance function. Journal of the American Statistical Association, 100 , 310-321. [DOI:10.1198/016214504000000854]
32. Stein, M. L. (2007), Seasonal variations in the spatial-temporal dependence of total column ozone. Envirometrics, 18 , 71-86. [DOI:10.1002/env.802]
33. Stephens, M. A. (1974), Edf statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69 , 730-737. [DOI:10.1080/01621459.1974.10480196]
34. Tejwani, R. , Liska, A., You, H., Reinen, J., and Das, P. (2017), Autism classification using brain functional connectivity dynamics and machine learning. NIPS workshop BigNeuro.
35. Tscheschel, A., Lacayo, J., and Stoyan, D. (2005), Statistical characterization of TEM images of silica-filled rubber. Journal of Microscopy, 217 , 75-82. [DOI:10.1111/j.0022-2720.2005.01426.x]
36. Xia, M., Wang, J., and He, Y. (2013), Brain {N et {V iewer, a network visualization tool for human brain connectomics. PLoS ONE, 8 (7)(e68910). [DOI:10.1371/journal.pone.0068910]

Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.