A new measure of preferred direction for circular data using angular wrapping

Abstract

The statistical techniques which are developed for the analysis of data in the linear number system cannot be applied to directional data directly. Circular data may be discontinuous in some principal interval. These discontinuities cause failure results in the circular statistics. Because of that the proposed wrapping operator must be used for data, which are defined in the discontinuous range. However, in both continuity and discontinuity, the wrapping operator works correctly. The most common preferred directions for circular data are circular mean and variance summarizing and comparing them. Although circular data has a very important role in statistics, the literature is weak in terms of statistical analysis of circular data. It creates a gap in this field. This study examines the preferred direction of circular data to fill this gap and presents a new measure of preferred direction for circular data using angular wrapping. Four different artificial and three real datasets are employed to evaluate the performance of the proposed methods. The results demonstrate the superiority of the proposed methods in terms of the absolute error and absolute percentage error. Consequently, it has been seen that the proposed methods giv e more consistent and more accurate results than thevectorial methods.

Keywords

• Angular wrapping
• circular data
• angular circular mean
• angular circular variance
• preferred directions

References

1. Jammalamadaka, S. R., SenGupta, A., Topics in Circular Statistics, World Scientific Publishing Co. Pte. Ltd., 2001.
2. Bowers, J. A., Morto, I. D., Mould, G. I., Directional statistics of the wind and waves, Appl. Ocean. Res., 22(1) (2000), 13-30. https://doi.org/10.1016/S0141-1187(99)00025-5
3. Mardia, K. V., Statistics of Directional Data, Academic Press, 1972.
4. Fisher, N., Statistical Analysis of Circular Data, Cambridge University Press, 1993.
5. Mardia, K. V., Jupp, P. E., Directional Statistics, John Wiley & Sons Inc., 2000.
6. Lark, L. M., Clifford, D., Waters, C. N., Modelling complex geological circular data with the projected normal distribution and mixtures of von Mises distribution, Solid Earth, 5(2) (2014) 631-639. https://doi.org/10.5194/se-5-631-2014
7. Kempter, R., Leibold, C., Buzs´aki, G., Diba, K., Schmidt, R., Quantifying circular–linear associations: Hippocampal phase precession, J. Neurosci. Methods, 207(1) (2012), 113-124. https://doi.org/10.1016/j.jneumeth.2012.03.007
8. La Sorte, F. A., Mannan, R. W., Reynolds, R. T., Grubb, T. G., Habitat associations ofsympatric red-tailed hawks and northern goshawks on the Kaibab Plateau, J. Wildl. Manage., 68(2) (2004), 307-317. https://doi.org/10.2193/0022-541X(2004)068[0307:HAOSRH]2.0.CO;2

9. Jones, M. C., Pewsey, A., Inverse Batschelet distributions for circular data, Biometrics, 68(1) (2012), 183-193. https://doi.org/10.1111/j.1541-0420.2011.01651.x
10. Baayen, C., Klugkist, I., Mechsner, F., Test of order-constrained hypotheses for circular data with applications to human movement science, J. Mot. Behav., 44(5) (2012), 351-363. https://doi.org/10.1080/00222895.2012.709549
11. Traa, J., Smaragdis, P., Multichannel source separation and tracking with RANSAC and directional statistics, IEEE/ACM Trans. Audio Speech. Lang. Process., 22(12) (2014), 2233-2243. https://doi.org/10.1109/TASLP.2014.2365701
12. Ehler, M., Galanis, J., Frame theory in directional statistics, Stat. Probab. Lett., 81(2) (2011), 1046-1051. https://doi.org/10.1016/j.spl.2011.02.027
13. Hawkins, D. M., Lombard, F., Segmentation of circular data, J. Appl. Stat., 42(1) (2015), 88-97. https://doi.org/10.1080/02664763.2014.934665
14. Klugkist, I., Bullens, J., Postma, A., Evaluating order-constrained hypotheses for circulardata using permutation tests, Br. J. Math. Stat. Psychol., 65(2) (2012), 222-236. https://doi.org/10.1111/j.2044-8317.2011.02018.x
15. Tasdan, F., Cetin, M., A simulation study on the influence of ties on uniform scores test for circular data, J. Appl. Stat., 41(5) (2014), 1137-1146. https://doi.org/10.1080/02664763.2013.862224
16. Thompson, L. M., van Manen, F. T., King, T. L., Geostatistical analysis of allele presence patterns among American black bears in eastern North Carolina, Ursus, 16(1) (2005), 59-69. https://doi.org/10.2192/1537-6176(2005)016[0059:GAOAPP]2.0.CO;2
17. Kubiak, T., Jonas, C., Applying circular statistics to the analysis of monitoring data, Eur. J. Psychol. Assess., 23(4) (2007), 227-237. https://doi.org/10.1027/1015-5759.23.4.227
18. Brunsdon, C., Corcoran, J., Using circular statistics to analyse time patterns in crime incidence, Comput. Environ. Urban Syst., 30(3) (2006), 300-319. https://doi.org/10.1016/j.compenvurbsys.2005.11.001
19. Huang, L., Helmke, B. P., A Semi-automatic method for image analysis of edge dynamics in living cells, Cell. Mol. Bioeng., 4(2) (2011), 205-219. https://doi.org/10.1007/s12195-010-0141-z
20. Abraham, C., Molinari, N., Servien, R., Unsupervised clustering of multivariate circular data, Stat. Med., 32(8) (2013), 1376-1382. https://doi.org/10.1002/sim.5589
21. Rocchi, M. B., Perlini, C., Is the time of suicide a random choice? A new statistical perspective, Crisis, 23(4) (2002), 161. https://doi.org/10.1027/0227-5910.23.4.161
22. Le, C. T., Liu, P., Lindgren, B. R., Daly, K. A., Giebink, G. S., Some statistical methods for investigating the date of birth as a disease indicator, Stat. Med., 22(13) (2003), 2127-2135. https://doi.org/10.1002/sim.1343
23. Chen, L., Singh, V. P., Guo, S., Fang, B., Liu, P., A new method for identification of flood seasons using directional statistics, Hydrol. Sci. J., 58(1) (2013), 28-40. https://doi.org/10.1080/02626667.2012.743661
24. Wang F., Gelfand, A. E., Modeling space and space-time directional data using projected Gaussian processes, J. Atmos. Ocean. Technol., 8(11) (2014), 1466-1485. https://doi.org/10.1080/01621459.2014.934454
25. Yurovskaya, M. V., Dulov, V. A., Chapron, B., Kudryavtsev, V. N., Directional short wind wave spectra derived from the sea surface photography, J. Geophys. Res. Oceans., 118(9) (2013), 4380-4394. https://doi.org/10.1002/jgrc.20296
26. Costa, M., Koivunen, V., Poor, H. V., Estimating directional statistics using wavefield modeling and mixtures of von-mises distributions, IEEE Signal Process. Lett., 21(12) (2014), 1496-1500. https://doi.org/10.1109/LSP.2014.2341651
27. Minguez, R., Espejo, A., Tomas, A., Mendez, F. J., Losada, I. J., Directional calibration of wave reanalysis databases using instrumental data, J. Atmos. Ocean. Technol., 28(11) (2011), 1466-1485. https://doi.org/10.1175/JTECH-D-11-00008.1
28. Schwartz, R. S., Barbosa, R. R. R., Meratnia, N., Heijenk, G., Scholten, H., A directional data dissemination protocol for vehicular environments, Comput. Commun., 34(17), (2011), 2057-2071. https://doi.org/10.1016/j.comcom.2011.03.007
29. Guo, C., Wu, X., Feng, C., Zeng, Z., Spectrum sensing for cognitive radios based on directionalstatistics of polarization vectors, IEEE J. Sel. Areas Commun., 31(3) (2013), 379-393. https://doi.org/10.1109/JSAC.2013.130305
30. Batschelet, E., Circular Statistics in Biology, Academic Press, 1981.
31. Zar, J. H., Biostatistical Analysis 4th edition, Prentice Hill, 1999.
32. Easton Jr, R. L., Topics in Circular Statistics, John Wiley & Sons, 2010.
33. Rhoad, R., Milauskas G., Whipple, R., Geometry for Enjoyment and Challenge, McDougal Littell & Co., 1991.
34. Ackermann, H., A note on circular nonparametrical classification, Biom. J., 39(5) (1997), 577-587. https://doi.org/10.1002/bimj.4710390506
35. Lund, U., Cluster analysis for directional data, Commun. Stat.–Simul. Comput., 28(4) (1999), 1001-1009. https://doi.org/10.1080/03610919908813589
36. Jander, R., Die optische richtungsorientierung der roten waldameise (formica ruea l.), Z. Vgl. Physiol., 40(2) (1957), 162-238. https://doi.org/10.1007/BF00297947
37. Chapman, M., Assessment of some controls in experimental transplants of intertidal gastropods, Journal of J. Exp. Mar. Biol. Ecol., 103(1-3) (1986), 181-201. https://doi.org/10.1016/0022-0981(86)90140-1
38. Chapman, M., Underwood, A., Experimental designs for analyses of movements by molluscs, Proceedings of the third international symposium on littorinid biology, (1992), 169-180.
39. Wehner R., Strasser, S., The POL area of the honey bee’s eye: behavioural evidence, Physiol. Entomol., 10(3) (1985), 337-349. https://doi.org/10.1111/j.1365-3032.1985.tb00055.x
40. Ravindran, P., Ghosh, S. K., Bayesian analysis of circular data using wrapped distributions, J. Stat. Theory Pract., 5(4) (2011), 547-561. https://doi.org/10.1080/15598608.2011.10483731
41. Otieno, B. S., Anderson-Cook, C. M., Measures of preferred direction for environmental and ecological circular data, Environ. Ecol. Stat., 13(3)(2006), 311-324. https://doi.org/10.1007/s10651-004-0014-5
42. Rossel, S., Wehner, R., Polarization vision in bees, Nature, 323(6084) (1986), 128-131. https://doi.org/10.1038/323128a0
43. Rossel, S., Wehner, R., The bee’s map of the e-vector pattern in the sky, Proc. Natl. Acad. Sci. U.S.A., 79(14) (1982), 4451-4455. https://doi.org/10.1073/pnas.79.14.4451
44. Brines, M. L., Gould, J. L., Bees have rules, Science, 206(4418) (1979), 571-573. https://doi.org/10.1126/science.206.4418.571