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

Year 2023, Volume: 72 Issue: 3, 778 - 802, 30.09.2023

Özge Tezel , Buğra Kaan Tiryaki , Eda Özkul , Orhan Kesemen

https://doi.org/10.31801/cfsuasmas.1159269

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

• Jammalamadaka, S. R., SenGupta, A., Topics in Circular Statistics, World Scientific Publishing Co. Pte. Ltd., 2001.
• 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
• Mardia, K. V., Statistics of Directional Data, Academic Press, 1972.
• Fisher, N., Statistical Analysis of Circular Data, Cambridge University Press, 1993.
• Mardia, K. V., Jupp, P. E., Directional Statistics, John Wiley & Sons Inc., 2000.
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Batschelet, E., Circular Statistics in Biology, Academic Press, 1981.
• Zar, J. H., Biostatistical Analysis 4th edition, Prentice Hill, 1999.
• Easton Jr, R. L., Topics in Circular Statistics, John Wiley & Sons, 2010.
• Rhoad, R., Milauskas G., Whipple, R., Geometry for Enjoyment and Challenge, McDougal Littell & Co., 1991.
• Ackermann, H., A note on circular nonparametrical classification, Biom. J., 39(5) (1997), 577-587. https://doi.org/10.1002/bimj.4710390506
• Lund, U., Cluster analysis for directional data, Commun. Stat.–Simul. Comput., 28(4) (1999), 1001-1009. https://doi.org/10.1080/03610919908813589
• 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
• 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
• Chapman, M., Underwood, A., Experimental designs for analyses of movements by molluscs, Proceedings of the third international symposium on littorinid biology, (1992), 169-180.
• 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
• 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
• 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
• Rossel, S., Wehner, R., Polarization vision in bees, Nature, 323(6084) (1986), 128-131. https://doi.org/10.1038/323128a0
• 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
• Brines, M. L., Gould, J. L., Bees have rules, Science, 206(4418) (1979), 571-573. https://doi.org/10.1126/science.206.4418.571