Skip to main content
SpringerLink
Log in

Time Series and Stochastic Processes

  • Chapter
  • First Online:
Computational Psychometrics: New Methodologies for a New Generation of Digital Learning and Assessment

Part of the book series: Methodology of Educational Measurement and Assessment ((MEMA))

Abstract

This chapter addresses some statistical modeling approaches for time series data and discusses their potential for psychometric applications. We adopt a broad conceptualization of time series, including under this rubric any type of data that involves serial statistical dependence. Such dependence may be represented in continuous time, discrete time, or in a purely sequential manner. This chapter begins by discussing the relationships among these three representations and offers some general advice on when each might prove useful. We then provide an overview of three modeling frameworks that exemplify the different representations of statistical dependence: Markov decision processes, state-space modeling, and temporal point processes. For each modeling framework, we discuss its specification, its psychometric interpretation, and provide a brief numeric example including R code.

The R code for this chapter can be found at the GitHub repository of this book: https://github.com/jgbrainstorm/computational_psychometrics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    For identification purposes, both Eqs. (12.8) and (12.9) require specification of a reference regime where all parameters in the regression equation are zero.

  2. 2.

    The data were simulated using the R package Sim.DiffProc (Guidoum & Boukhetala 2016) with true parameters: ρ 1 = 0.2, ρ 2 = 0.4, a 12 = −0.1, a 21 = −0.3, K = 5, \(\sigma ^2_1 = \sigma ^2_2 = \sigma ^2_{\epsilon ,1} = \sigma ^2_{\epsilon ,2} = 0.01\), μ 1,1 = μ 1,2 = 2, \(\sigma ^2_{1,1} = \sigma ^2_{1,2} = 0.1\), σ 1,12 = 0.

References

  • Baker, C., Saxe, R., & Tenenbaum, J. (2009). Action understanding as inverse planning. Cognition, 113(3), 329–349.

    Article  Google Scholar 

  • Baker, C., Saxe, R., & Tenenbaum, J. (2011). Bayesian theory of mind: Modeling joint beliefdesire attribution. In Proceedings of the Thirty-Third Annual Conference of the Cognitive Science Society (pp. 2469–2474).

    Google Scholar 

  • Barabási, A.-L. (2005, May). The origin of bursts and heavy tails in human dynamics. Nature, 435, 207–211. https://doi.org/10.1038/nature03526.1

    Article  Google Scholar 

  • Blundell, C., Heller, K. A., & Beck, J. M. (2012). Modelling reciprocating relationships with Hawkes processes. Advances in Neural Information Processing Systems, 25, 2600–2608.

    Google Scholar 

  • Boker, S. M., Neale, M. C., & Klump, K. L. (2014). A differential equations model for the ovarian hormone cycle. In P. C. M. Molenaar, R. M. Lerner, & K. M. Newell (Eds.), Handbook of developmental systems theory and methodology (pp. 369–391). New York, NY: Guilford Press.

    Google Scholar 

  • Brillinger, D. R., Guttorp, P. M., & Schoenberg, F. P. (2002). Point processes, temporal. In A. H. El-Shaarawi & W. W. Piegorsch (Eds.), Encyclopedia of environmetrics (Vol. 3, pp. 1577–1581). Chichester, England: Wiley.

    Google Scholar 

  • Brown, E. N. & Luithardt, H. (1999). Statistical model building and model criticism for human circadian data. Journal of Biological Rhythms, 14, 609–616. https://doi.org/10.1177/074873099129000975

    Article  Google Scholar 

  • Chen, F., & Tan, W. H. (2018). Marked self-exciting point process modelling of information diffusion on twitter. Preprint, pp. 1–18. arXiv: 1802.09304

    Google Scholar 

  • Cho, S.-J., Brown-Schmidt, S., Boeck, P. D., & Shen, J. (2020). Modeling intensive polytomous time-series eye-tracking data: A dynamic tree-based item response model. Psychometrika, 85(1), 154–184. https://doi.org/10.1007/s11336-020-09694-6

    Article  Google Scholar 

  • Chow, S.-M. & Nesselroade, J. R. (2004). General slowing or decreased inhibition? Mathematical models of age differences in cognitive functioning. Journals of Gerontology B, 59(3), 101–109. https://doi.org/10.1093/geronb/59.3.P101

    Article  Google Scholar 

  • Chow, S.-M., Grimm, K. J., Guillaume, F., Dolan, C. V., & McArdle, J. J. (2013). Regimeswitching bivariate dual change score model. Multivariate Behavioral Research, 48(4), 463–502. https://doi.org/10.1080/00273171.2013.787870

    Article  Google Scholar 

  • Cox, D. R., & Isham, V. (1980). Point processes. New York: Chapman Hall/CRC.

    Google Scholar 

  • Crane, R., & Sornette, D. (2008). Robust dynamic classes revealed by measuring the response function of a social system. Proceedings of the National Academy of Sciences, 105(41), 15649–15653. https://doi.org/10.1073/pnas.0803685105

    Article  Google Scholar 

  • Gilbert, P. (2006 or later). Brief user’s guide: Dynamic systems estimation. Retrieved from http://cran.r-project.org/web/packages/dse/vignettes/Guide.pdf

  • Daley, D. J., & Vera-Jones, D. (2003). An introduction to the theory of point processes: Elementary theory and methods (Second). New York: Springer.

    Google Scholar 

  • De Boeck, P., & Jeon, M. (2019). An overview of models for response times and processes in cognitive tests. Frontiers in Psychology, 10. https://doi.org/10.3389/fpsyg.2019.00102

  • Dolan, C. V. (2005). MKFM6: Multi-group, multi-subject stationary time series modeling based on the Kalman filter. Retrieved from http://users.fmg.uva.nl/cdolan/

    Google Scholar 

  • Dolan, C. V. (2009). Structural equation mixture modeling. In R. E. Millsap & A. Maydeu-Olivares (Eds.), The SAGE handbook of quantitative methods in psychology (pp. 568–592). Thousand Oaks, CA: Sage.

    Chapter  Google Scholar 

  • Dolan, C. V., Jansen, B. R., & Van der Maas, H. L. J. (2004). Constrained and unconstrained multivariate normal finite mixture modeling of Piagetian data. Multivariate Behavioral Research, 39(1), 69–98. https://doi.org/10.1207/s15327906mbr3901_3

    Article  Google Scholar 

  • Driver, C. C., Oud, J. H. L., & Voelkle, M. C. (2017). Continuous time structural equation modelling with R package ctsem. Journal of Statistical Software, 77(5), 1–35. https://doi.org/10.18637/jss.v077.i05

    Article  Google Scholar 

  • Durrett, R. (2010). Probability: Theory and examples (4th ed.). Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge, New York: Cambridge University Press.

    Google Scholar 

  • Ercikan, K., & Pellegrino, J. W. (2017). Validation of score meaning for the next generation of assessments: The use of response processes. New York: Routledge.

    Book  Google Scholar 

  • Fox, E. W. (2015). Estimation and Inference for Self-Exciting Point Processes with Applications to Social Networks and Earthquake Seismology. Dissertation Thesis, University of Los Angeles.

    Google Scholar 

  • Gottman, J. M. (2002). The mathematics of marriage: Dynamic nonlinear models. Cambridge, MA: The MIT Press.

    Book  Google Scholar 

  • Guidoum, A. C., & Boukhetala, K. (2016). SIM.DIFFPROC: Simulation of diffusion processes. R package version 3.6. Retrieved from https://cran.r-project.org/package=Sim.DiffProc

  • Halpin, P. (2016). Psychometric Analysis of the Child Functioning Module: Comparisons Between Serbia and Mexico. UNICEF, Division of Data, Research, and Policy.

    Google Scholar 

  • Halpin, P. F., & De Boeck, P. (2013). Modelling dyadic interaction with Hawkes processes. Psychometrika, 78(4), 793–814. https://doi.org/10.1007/s11336-013-9329-1

    Article  Google Scholar 

  • Halpin, P. F., von Davier, A. A., Hao, J., & Liu, L. (2017). Measuring student engagement during collaboration. Journal of Educational Measurement, 54(1), 70–84. https://doi.org/10.1111/jedm.12133

    Article  Google Scholar 

  • Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57, 357–384. https://doi.org/10.2307/1912559

    Article  Google Scholar 

  • Harte, D. (2010). PtProcess: An R package for modelling marked point processes indexed by time. Journal of Statistical Software, 35(8). https://doi.org/10.18637/jss.v035.i08

  • Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. Journal of the Royal Statistical Society, Series B, 33(3), 438–443. https://doi.org/10.1073/pnas.0703993104. arXiv:1011.1669v3

  • Helske, J. (2017). KFAS: Exponential family state space models in R. Journal of Statistical Software, 78(10), 1–39. https://doi.org/10.18637/jss.v078.i10

    Article  Google Scholar 

  • Hofbauer, J., & Sigmund, K. (1988). The theory of evolution and dynamical systems: Mathematical aspects of selection (London Mathematical Society Student Texts). Cambridge: Cambridge University Press. Retrieved from http://www.worldcat.org/isbn/0521358388

    Google Scholar 

  • Holmes, E., Ward, E., & Wills, K. (2013). MARSS: Multivariate autoregressive state-space modeling. R package version 3.9. Retrieved from http://cran.r-project.org/web/packages/MARSS/

  • Holmes, E. E., Ward, E. J., & Wills, K. (2012). MARSS: Multivariate autoregressive statespace models for analyzing time-series data. The R Journal, 4(1), 30.

    Article  Google Scholar 

  • Hosenfeld, B. (1997). Indicators of discontinuous change in the development of analogical reasoning. Journal of Experimental Child Psychology, 64, 367–395. https://doi.org/10.1006/jecp.1996.2351

    Article  Google Scholar 

  • Howard, R. A. (1960). Dynamic programming and Markov processes (1st ed.). Cambridge, MA: The MIT Press.

    Google Scholar 

  • Jeon, M., & De Boeck, P. (2016). A generalized item response tree model for psychological assessments. Behavior Research Methods, 48(3), 1070–1085. https://doi.org/10.3758/s13428-015-0631-y

    Article  Google Scholar 

  • Karsai, M., Kivelä, M., Pan, R. K., Kaski, K., Kertész, J., Barabási, a.-L., & Saramäki, J. (2011). Small but slow world: How network topology and burstiness slow down spreading. Physical Review E, 83(2), 025102. https://doi.org/10.1103/PhysRevE.83.025102

  • Kim, C.-J., & Nelson, C. R. (1999). State-space models with regime switching: Classical and Gibbs-sampling approaches with applications. Cambridge, MA: MIT Press.

    Google Scholar 

  • Kohlberg, L., & Kramer, R. (1969). Continuities and discontinuities in childhood and adult moral development. Human development, 12(2), 93–120. https://doi.org/10.1159/000270857

    Article  Google Scholar 

  • Koopman, S. J., Shephard, N., & Doornik, J. A. (1999). Statistical algorithms for models in state space using ssfpack 2.2. Econometrics Journal, 2(1), 113–166. https://doi.org/10.1111/1368-423X.00023

    Article  Google Scholar 

  • LaMar, M. (2017). Markov decision process measurement model. Psychometrika, 1–22.

    Google Scholar 

  • Liniger, T. (2009). Multivariate Hawkes Processes. Dissertation Thesis, Swiss Federal Institute of Technology. https://doi.org/10.3929/ethz-a-006037599

    Google Scholar 

  • Lord, F., & Novick, M. (1968). Statistical theories of mental test scores. Addison-Wesley series in behavioral science. Addison-Wesley Pub. Co. Retrieved from https://books.google.com/books?id=FW5EAAAAIAAJ

    Google Scholar 

  • Lotka, A. J. (1925). Elements of physical biology. Baltimore, MD: Williams & Wilkins.

    Google Scholar 

  • Luethi, D., Erb, P., & Otziger, S. (2014). FKF: Fast Kalman filter. R package version 0.1.3. Retrieved from https://CRAN.R-project.org/package=FKF

  • Mahadevan, S. (1996). Average reward reinforcement learning: Foundations, algorithms, and empirical results. Machine Learning, 22, 159–195.

    Article  Google Scholar 

  • Matsubara, Y., Sakurai, Y., Prakash, B. A., Li, L., & Faloutsos, C. (2012). Rise and fall patterns of information diffusion: Model and implications. In KDD ’12: Proceedings of the 18th ACM SIGKDD (pp. 6–14). New York, NY: ACM Press.

    Google Scholar 

  • Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., …Hassabis, D. (2015). Human-level control through deep reinforcement learning. Nature, 518(7540), 529–533.

    Google Scholar 

  • Muthén, B. O. & Asparouhov, T. (2011). LTA in Mplus : Transition probabilities influenced by covariates. Mplus Web Notes: No. 13. Retrieved from http://www.statmodel.com/examples/LTAwebnote.pdf

  • Muthén, L. K., & Muthén, B. O. (1998–2012). Mplus user’s guide(7th ed.). Los Angeles, CA: Muthén & Muthén.

    Google Scholar 

  • Neale, M. C., Hunter, M. D., Pritikin, J. N., Zahery, M., Brick, T. R., Kirkpatrick, R. M., …Boker, S. M. (2016). OPENMX2.0: Extended structural equation and statistical modeling. Psychometrika, 80(2), 535–549. https://doi.org/10.1007/s11336-014-9435-8

  • Ng, A. Y., & Russell, S. (2000). Algorithms for inverse reinforcement learning. In Proceedings of the Seventeenth International Conference on Machine Learning (2000) (pp. 663–670).

    Google Scholar 

  • Oliveira, J. G., & Vazquez, A. (2009). Impact of interactions on human dynamics. Physica A: Statistical Mechanics and Its Applications, 388(2–3), 187–192. https://doi.org/10.1016/j.physa.2008.08.022. arXiv: physics/0603064

  • Ou, L., Hunter, M. D., & Chow, S.-M. (2017). DYNR: Dynamic modeling in R. R package 0.1.11-5 version.

    Google Scholar 

  • Ozaki, T. (1979). Maximum likelihood estimation of Hawkes’ self-exciting point processes. Annals of the Institute of Statistical Mathematics, 31(1), 145–155. https://doi.org/10.1007/BF02480272

    Article  Google Scholar 

  • Peng, R. D. (2003). Multi-Dimensional Point Process Models in R. Journal of Statistical Software, 8(16). https://doi.org/10.18637/jss.v008.i16

  • Petris, G. (2010). An R package for dynamic linear models. Journal of Statistical Software, 36(12), 1–16. Retrieved from http://www.jstatsoft.org/v36/i12/

    Article  Google Scholar 

  • Petris, G., Petrone, S., & Campagnoli, P. (2009). Dynamic linear models with R . useR!. New York, NY: Springer.

    Google Scholar 

  • Piaget, J., & Inhelder, B. (1969). The psychology of the child. New York, NY: Basic Books.

    Google Scholar 

  • Puterman, M. L. (1994). Markov decision processes: Discrete stochastic dynamic programming. New York: Wiley.

    Book  Google Scholar 

  • Qiao, X., & Jiao, H. (2018). Data mining techniques in analyzing process data: A didactic. Frontiers in Psychology, 9. https://doi.org/10.3389/fpsyg.2018.02231

  • Rafferty, A. N., LaMar, M. M., & Griffiths, T. L. (2015). Inferring learners’ knowledge from their actions. Cognitive Science, 39(3), 584–618.

    Article  Google Scholar 

  • Russell, S., & Norvig, P. (2009). Artificial intelligence: A modern approach (3rd ed.). Upper Saddle River: Pearson.

    Google Scholar 

  • Rust, J. (1994). Structural estimation of Markov decision processes. In R. Engle & D. Mc-Fadden (Eds.), Handbook of econometrics (Vol. 4, pp. 3081–3143). Elsevier Science.

    Google Scholar 

  • Scott, S. L. (2017). BSTS: Bayesian structural time series. R package version 0.7.1. Retrieved from https://CRAN.R-project.org/package=bsts

  • Tang, X., Zhang, S., Wang, Z., Liu, J., & Ying, Z. (2020). ProcData: An R Package for Process Data Analysis. arXiv:2006.05061 [cs, stat].

    Google Scholar 

  • Thatcher, R. W. (1998). A predator-prey model of human cerebral development. In K. M. Newell & P. C. M. Molenaar (Eds.), Applications of nonlinear dynamics to developmental process modeling (pp. 87–128). Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Thomas, E. A., & Martin, J. A. (1976). Analyses of parent-infant interaction. Psychological Review, 83(2), 141–156. https://doi.org/10.1037/0033-295X.83.2.141

    Article  Google Scholar 

  • van der Maas, H. L. J. & Molenaar, P. C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99(3), 395–417. https://doi.org/10.1037//0033-295x.99.3.395

    Article  Google Scholar 

  • van der Maas, H. L. J., Kolstein, R., & van der Pligt, J. (2003). Sudden transitions in attitudes. Sociological Methods & Research, 32, 125–152. https://doi.org/10.1177/0049124103253773

    Article  Google Scholar 

  • van der Maas, H. L., Dolan, C. V., Grasman, R. P., Wicherts, J. M., Huizenga, H. M., & Raijmakers, M. E. (2006). A dynamical model of general intelligence: The positive manifold of intelligence by mutualism. Psychological Review, 113(4), 842.

    Article  Google Scholar 

  • van Dijk, M., & van Geert, P. (2007). Wobbles, humps and sudden jumps: A case study of continuity, discontinuity and variability in early language development. Infant and Child Development, 16(1), 7–33. https://doi.org/10.1002/icd.506

    Article  Google Scholar 

  • Volterra, V. (1926). Fluctuations in the abundance of a species considered mathematically. Nature, 118, 558–560. https://doi.org/10.1038/118558a0

    Article  Google Scholar 

  • von Davier, A. A., & Halpin, P. F. (2013). Collaborative problem-solving and the assessment of cognitive skills: Psychometric considerations. ETS Research Report.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Peter Halpin

  2. ACT, Iowa City, IA, USA

    Lu Ou

  3. Educational Testing Service, Princeton, NJ, USA

    Michelle LaMar

Authors
  1. Peter Halpin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lu Ou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michelle LaMar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Halpin .

Editor information

Editors and Affiliations

  1. Duolingo and EdAstra Tech, LLC, Newton, MA, USA

    Alina A. von Davier

  2. Educational Testing Service, Princeton, NJ, USA

    Robert J. Mislevy

  3. Educational Testing Service, Princeton, NJ, USA

    Jiangang Hao

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Halpin, P., Ou, L., LaMar, M. (2021). Time Series and Stochastic Processes. In: von Davier, A.A., Mislevy, R.J., Hao, J. (eds) Computational Psychometrics: New Methodologies for a New Generation of Digital Learning and Assessment. Methodology of Educational Measurement and Assessment. Springer, Cham. https://doi.org/10.1007/978-3-030-74394-9_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-74394-9_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-74393-2

  • Online ISBN: 978-3-030-74394-9

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation