The Annals of Applied Statistics

Nonstationary modelling of tail dependence of two subjects’ concentration

Abstract

We analyse eye-tracking data to understand how people collaborate. Our dataset consists of time series of measurements for eye movements, such as spatial entropy, calculated for each subject during an experiment when several pairs of participants collaborate to accomplish a task. We observe that pairs with high collaboration quality obtain their highest values of concentration (or equivalently lowest values of spatial entropy) occurring simultaneously. In this paper, we propose a flexible model that describes the tail dependence structure between two subjects’ entropy when the pair is collaborating. More generally, we develop a generalized additive model (GAM) framework for tail dependence coefficients in the presence of covariates. As for any GAM-type model, the methodology can be used to predict collaboration quality or to explore how joint concentration depends on other cognitive operations and varies over time.

Article information

Source
Ann. Appl. Stat., Volume 12, Number 2 (2018), 1293-1311.

Dates
Revised: October 2017
First available in Project Euclid: 28 July 2018

https://projecteuclid.org/euclid.aoas/1532743495

Digital Object Identifier
doi:10.1214/17-AOAS1111

Mathematical Reviews number (MathSciNet)
MR3834304

Citation

Sharma, Kshitij; Chavez-Demoulin, Valérie; Dillenbourg, Pierre. Nonstationary modelling of tail dependence of two subjects’ concentration. Ann. Appl. Stat. 12 (2018), no. 2, 1293--1311. doi:10.1214/17-AOAS1111. https://projecteuclid.org/euclid.aoas/1532743495

References

• Aas, K., Czado, C., Frigessi, A. and Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance Math. Econom. 44 182–198.
• Allopenna, P. D., Magnuson, J. S. and Tanenhaus, M. K. (1998). Tracking the time course of spoken word recognition using eye movements: Evidence for continuous mapping models. J. Mem. Lang. 38 419–439.
• Ballard, D. H., Hayhoe, M. M., Li, F., Whitehead, S. D., Frisby, J. P., Taylor, J. G. and Fisher, R. B. (1992). Hand-eye coordination during sequential tasks [and discussion]. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 337 331–339.
• Chase, W. G. and Simon, H. A. (1973). Perception in chess. Cogn. Psychol. 4 55–81.
• Coles, S. G. and Tawn, J. A. (1996). Modelling extremes of the areal rainfall process. J. Roy. Statist. Soc. Ser. B 58 329–347.
• Embrechts, P., Lindskog, F. and McNeil, A. (2003). Modelling dependence with copulas and applications to risk management. In Handbook of Heavy Tailed Distributions in Finance (S. Rachev, ed.) 329–384. Elsevier, Amsterdam.
• Ferreira, M. (2013). Nonparametric estimation of the tail-dependence coefficient. REVSTAT 11 1–16.
• Gardes, L. and Girard, S. (2015). Nonparametric estimation of the conditional tail copula. J. Multivariate Anal. 137 1–16.
• Grant, E. R. and Spivey, M. J. (2003). Eye movements and problem solving guiding attention guides thought. Psychol. Sci. 14 462–466.
• Green, P. J. (1987). Penalized likelihood for general semi-parametric regression models. Int. Stat. Rev. 55 245–259.
• Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Monographs on Statistics and Applied Probability 58. Chapman & Hall, London.
• Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Monographs on Statistics and Applied Probability 43. Chapman & Hall, London.
• Hmelo-Silver, C. E. (2006). Analyzing collaborative learning: Multiple approaches to understanding processes and outcomes. In Proceedings of the 7th International Conference on Learning Sciences 1059–1065. International Society of the Learning Sciences.
• Jacob, R. J. and Karn, K. S. (2003). Eye tracking in human-computer interaction and usability research: Ready to deliver the promises. Mind 2 4.
• Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. Chapman & Hall, London.
• Jones, G. (2003). Testing two cognitive theories of insight. J. Exper. Psychol., Learn., Mem., Cogn. 29 1017.
• Just, M. A. and Carpenter, P. A. (1976). Eye fixations and cognitive processes. Cogn. Psychol. 8 441–480.
• Knoblich, G., Ohlsson, S. and Raney, G. E. (2001). An eye movement study of insight problem solving. Mem. Cogn. 29 1000–1009.
• Li, F. (2016). Modeling covariate-contingent correlation and tail-dependence with copulas. Available at arXiv:1401.0100.
• McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton Univ. Press, Princeton, NJ.
• Nelsen, R. B. (1999). An Introduction to Copulas. Lecture Notes in Statistics 139. Springer, New York.
• Nüssli, M.-A. (2011). Dual eye-tracking methods for the study of remote collaborative problem solving.
• Pietinen, S., Bednarik, R. and Tukiainen, M. (2010). Shared visual attention in collaborative programming: A descriptive analysis. In Proceedings of the 2010 ICSE Workshop on Cooperative and Human Aspects of Software Engineering 21–24. ACM, New York.
• Richardson, D. C., Dale, R. and Kirkham, N. Z. (2007). The art of conversation is coordination common ground and the coupling of eye movements during dialogue. Psychol. Sci. 18 407–413.
• Sangin, M., Molinari, G., Nüssli, M.-A. and Dillenbourg, P. (2008). How learners use awareness cues about their peer’s knowledge?: Insights from synchronized eye-tracking data. In Proceedings of the 8th International Conference on International Conference for the Learning Sciences 2 287–294. International Society of the Learning Sciences.
• Schmidt, R. and Stadtmüller, U. (2006). Non-parametric estimation of tail dependence. Scand. J. Stat. 33 307–335.
• Schneider, B., Sharma, K., Cuendet, S., Zufferey, G., Dillenbourg, P. and Pea, R. D. (2015). 3D tangibles facilitate joint visual attention in dyads. In Proceedings of 11th International Conference of Computer Supported Collaborative Learning 1 156–165. EPFL-CONF-223609.
• Sharma, K., Chavez-Demoulin, V. and Dillenbourg, P. (2017). An application of extreme value theory to learning analytics: Predicting collaboration quality from eye-tracking data. J. Learn. Anal. 4 (3) 140–164.
• Sharma, K., Jermann, P., Nüssli, M.-A. and Dillenbourg, P. (2012). Gaze evidence for different activities in program understanding. In 24th Annual Conference of Psychology of Programming Interest Group. EPFL-CONF-184006.
• Sharma, K., Jermann, P., Nüssli, M.-A. and Dillenbourg, P. (2013). Understanding collaborative program comprehension: Interlacing gaze and dialogues. In Computer Supported Collaborative Learning (CSCL 2013).
• Sibuya, M. (1960). Bivariate extreme statistics. I. Ann. Inst. Statist. Math. Tokyo 11 195–210.
• Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8 229–231.
• Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript.
• Vatter, T. and Chavez-Demoulin, V. (2015). Generalized additive models for conditional dependence structures. J. Multivariate Anal. 141 147–167.
• Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC, Boca Raton, FL.
• Zelinsky, G. J. and Murphy, G. L. (2000). Synchronizing visual and language processing: An effect of object name length on eye movements. Psychol. Sci. 11 125–131.