Ghost hunting in the nonlinear dynamic machine


Autoři: Jonathan E. Butner aff001;  Ascher K. Munion aff001;  Brian R. W. Baucom aff001;  Alexander Wong aff002
Působiště autorů: Department of Psychology, University of Utah, Salt Lake City, Utah, United States of America aff001;  Department of Psychology, California State University at Chico, Chico, California, United States of America aff002
Vyšlo v časopise: PLoS ONE 14(12)
Kategorie: Research Article
doi: 10.1371/journal.pone.0226572

Souhrn

Integrating dynamic systems modeling and machine learning generates an exploratory nonlinear solution for analyzing dynamical systems-based data. Applying dynamical systems theory to the machine learning solution further provides a pathway to interpret the results. Using random forest models as an illustrative example, these models were able to recover the temporal dynamics of time series data simulated using a modified Cusp Catastrophe Monte Carlo. By extracting the points of no change (set points) and the predicted changes surrounding the set points, it is possible to characterize the topology of the system, both for systems governed by global equation forms and complex adaptive systems. RESULTS: The model for the simulation was able to recover the cusp catastrophe (i.e. the qualitative changes in the dynamics of the system) even when applied to data that have a significant amount of error variance. To further illustrate the approach, a real-world accelerometer example was examined, where the model differentiated between movement dynamics patterns by identifying set points related to cyclic motion during walking and attraction during stair climbing. These example findings suggest that integrating machine learning with dynamical systems modeling provides a viable means for classifying distinct temporal patterns, even when there is no governing equation for the nonlinear dynamics. Results of these integrated models yield solutions with both a prediction of where the system is going next and a decomposition of the topological features implied by the temporal dynamics.

Klíčová slova:

Accelerometers – Climbing – Dynamical systems – Machine learning – Nonlinear dynamics – Simulation and modeling – Support vector machines – Adaptive systems


Zdroje

1. Jordan MI, Mitchell TM. Machine learning: Trends, perspectives, and prospects. Science. 2015;349(6245): 255–260. doi: 10.1126/science.aaa8415 26185243

2. Marx V. Biology: The big challenges of big data. Nature. 2013;498(7453): 255–260 doi: 10.1038/498255a 23765498

3. Harrington P. Machine learning in action. Greenwich: Manning Publications Co.; 2012.

4. Tyralis H, Papacharalampous GA, Langousis A. A brief review of random forests for water scientists and practitioners and their recent history in water resources. Water (Basel). 2019;11(5): 910. doi: 10.3390/w11050910

5. Abraham FD, Abraham RH, Shaw CD. A visual introduction to dynamical systems theory for psychology. Santa Cruz: Aerial Press; 1990.

6. Pfeifer R, Lungarella M, Iida F. Self-organization, embodiment, and biologically inspired robotics. Science. 2007;318(5853): 1088–1093. doi: 10.1126/science.1145803 18006736

7. Peters J, Vijayakumar, S, Schaal S. Reinforcement learning for humanoid robotics. In Proceedings of the third IEEE-RAS international conference on humanoid robots 2003 Sep 29 (pp. 1–20).

8. Müller KR, Tangermann M, Dornhege G, Krauledat M, Curio G, Blankertz B. Machine learning for real-time single-trial EEG-analysis: from brain–computer interfacing to mental state monitoring. J Neurosci Methods. 2008;167(1): 82–90. doi: 10.1016/j.jneumeth.2007.09.022 18031824

9. Fragkiadaki K, Levine S, Felsen P, Malik J. Recurrent network models for human dynamics. In Proceedings of the IEEE International Conference on Computer Vision 2015 (pp. 4346–4354).

10. Cao L, Gu Q. Dynamic support vector machines for non-stationary time series forecasting. Intell Data Anal. 2002;6(1): 67–83.

11. Weinan E. A proposal on machine learning via dynamical systems. Commun Math Stat. 2017;5: 1–11. doi: 10.1007/s40304-017-0103-z

12. Breiman L. Random forests. Mach Learn. 2001;45(1): 5–32. doi: 10.1023/A:1010933404324

13. Huang W, Nakamori Y, Wang SY. Forecasting stock market movement direction with support vector machine. Comput Oper Res. 2005;32(10): 2513–2522.

14. Aihara K, Suzuki H. Theory of hybrid dynamical systems and its applications to biological and medical systems. Philos Trans A Math Phys Eng Sci. 2010;368: 4893–4914. doi: 10.1098/rsta.2010.0237 20921003

15. Jopp F, Breckling B, Reuter H, editors. Modelling complex ecological dynamics. 1st ed. London: Springer; 2010.

16. Bongard J, Lipson H. Automated reverse engineering of nonlinear dynamical systems. Proc Natl Acad Sci U S A. 2007;104(24): 9943–9948. doi: 10.1073/pnas.0609476104 17553966

17. Schmidt M, Lipson H. Distilling free-form natural laws from experimental data. Science. 2009;324(5923): 81–85. doi: 10.1126/science.1165893 19342586

18. Brunton SL, Proctor JL, Kutz JN. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc Natl Acad Sci U S A. 2016;113(15): 3932–3937. doi: 10.1073/pnas.1517384113 27035946

19. Dekker H. Quantization of the linearly damped harmonic oscillator. Physical Review A, 1977;16(5): 2126–2134.

20. Holland J. (2014). Complexity: A very short introduction. 1st ed. Oxford: Oxford University Press; 2014.

21. Kugler P, Turvey MT. Information, natural law, and the self-assembly of rhythmic movement. Hillsdale: Lawrence Erlbaum Associates; 1987.

22. Skyrms B, Pemantle R. A dynamic model of social network formation. In: Gross T, Sayama H, editors. Adaptive networks. Berlin: Springer; 2009. pp. 231–251.

23. Butner JE, Wiltshire TJ, Munion AK. (2017). Modeling multi-agent self-organization through the lens of higher order attractor dynamics. Front Psychol. 2017;8: 380. doi: 10.3389/fpsyg.2017.00380 28373853

24. Chen G, Moiola JL, Wang HO. Bifurcation control: theories, methods, and applications. Int J Bifurcat Chaos. 2000;10(03): 511–548.

25. Segal MR. Machine learning benchmarks and random forest regression. Netherlands: Kluwer Academic Publishers; 2004.

26. Tyralis H, Papacharalampous GA, Burnetas A, Langousis A. Hydrological post-processing using stacked generalization of quantile regression algorithms: Large-scale application over CONUS. J Hydrol (Amst). 2019;577: 123957. doi: 10.1016/j.jhydrol.2019.123957

27. Hsu CW, Chang CC, Lin CJ. A practical guide to support vector classification. 2003.

28. Hsu CW, Lin CJ. A comparison of methods for multiclass support vector machines. IEEE Trans Neural Netw. 2002;13(2): 415–425. doi: 10.1109/72.991427 18244442

29. Burges CJ. A tutorial on support vector machines for pattern recognition. Data Min Knowl Discov. 1998;2(2): 121–167.

30. Smola AJ, Schölkopf B. A tutorial on support vector regression. Stat Comput. 2004;14(3): 199–222.

31. Xu Q, Zhang J, Jiang C, Huang X, He Y. Weighted quantile regression via support vector machine. Expert Syst Appl. 2015;42(13): 5441–5451.

32. Meinshausen N. Quantile regression forests. J Mach Learn Res, 2006;7(Jun): 983–999.

33. Kelso JA. Phase transitions and critical behavior in human bimanual coordination. Am J Physiol Regul Integr Comp Physiol. 1984:246(6); R1000–R1004.

34. Harrell FE Jr, Dupont C. Hmisc: Harrell miscellaneous. R package version 4.2–0. 2008;3(2). https://CRAN.R-project.org/package=Hmisc

35. Ishwaran H, Kogalur UB, Kogalur MUB. Package ‘randomForestSRC’. 2019.

36. Grasman R. cusp: Cusp-catastrophe model fitting using maximum likelihood. R package version 1.1.1. 2015. https://CRAN.R-project.org/package=cusp

37. Adler D, Murdoch D, Nenadic O, Urbanek S, Chen M, Gebhardt A. rgl: 3D Visualization Using OpenGL. R package version 0.100.26. 2019. https://CRAN.R-project.org/package=rgl

38. Fox J, Weisberg S, Price B. car: Companion to applied regression. R package version 3.0–3. 2019. https://CRAN.R-project.org/package=car

39. Wickham H. ggplot2: Elegant graphics for data analysis. New York: Springer-Verlag; 2016.

40. Zeeman EC. Catastrophe theory. Sci Am. 1976;234(4): 65–83.

41. Bernardo M, Budd C, Champneys A, Kowalczyk P. Piecewise-smooth dynamical systems: Theory and applications. London: Springer-Verlag London Limited; 2008.

42. Arecchi FT, Badii R, Politi A. Generalized multistability and noise-induced jumps in a nonlinear dynamical system. Phys Rev A Gen Phys. 1985;32(1): 402. doi: 10.1103/physreva.32.402 9896062

43. Butner JE, Dyer HL, Malloy TS, Kranz LV. Uncertainty in cost performance as a function of the cusp catastrophe in the NASA program performance management system. Nonlinear Dynamics Psychol Life Sci. 2014;18: 397–417. 25196707

44. Stewart IN, Peregoy PL. Catastrophe theory modeling in psychology. Psychol Bull. 1983;94(2): 336.

45. Chen DGD, Lin F, Chen XJ, Tang W, Kitzman H. Cusp catastrophe model: a nonlinear model for health outcomes in nursing research. Nurs Res. 2014;63(3): 211. doi: 10.1097/NNR.0000000000000034 24785249

46. Yarkoni T, Westfall j. Choosing prediction over explanation in psychology: Lessons from machine learning. Psychol Sci. 2017; 12(6): 1100–1122.

47. Guastello SJ. Moderator regression and the cusp catastrophe: Application of two‐stage personnel selection, training, therapy, and policy evaluation. Behav Sci. 1982;27(3): 259–272.

48. Haken H. Synergetics. An introduction. Nonequilibrium phase transitions and self-organization in physics, chemistry, and biology. 3rd rev. enl. ed. New York: Springer-Verlag, 1983.

49. Alexander RA, Herbert GR, DeShon RP, Hanges PJ. An examination of least-squares regression modeling of catastrophe theory. Psychol Bull. 1992;111(2), 366.

50. Banos O, Garcia R, Holgado JA, Damas M, Pomares H, Rojas I, et al. mHealthDroid: a novel framework for agile development of mobile health applications. Proceedings of the 6th International Work-conference on Ambient Assisted Living an Active Ageing (IWAAL 2014); 2014 December 2–5; Belfast, Northern Ireland.

51. Nguyen LT, Zeng M, Tague P, Zhang J. Recognizing New Activities with Limited Training Data. In IEEE International Symposium on Wearable Computers (ISWC); 2015.

52. Stewart I. Does God play dice?: The new mathematics of chaos. Penguin UK; 1997.

53. Turvey MT. Coordination. Am Psychol. 1990;45(8): 938. doi: 10.1037//0003-066x.45.8.938 2221565

54. Van der Maas HL, Molenaar PC. Stagewise cognitive development: an application of catastrophe theory. Psychol Rev. 1992;99(3): 395. doi: 10.1037/0033-295x.99.3.395 1502272

55. Friedman A. Stochastic differential equations and applications. Mineola: Dover Publications; 2012.

56. Butner JE, Diets-Lebehn C, Crenshaw AO, Wiltshire TJ, Perry NS, Kent de Grey RG, et al. A multivariate dynamic systems model for psychotherapy with more than one client. J Couns Psychol. 2017;64(6): 616. doi: 10.1037/cou0000238 29154574

57. Sims CA. Discrete approximations to continuous time lag distributions in econometrics. Econometrica. 1971; 67(337): 545–564.


Článek vyšel v časopise

PLOS One


2019 Číslo 12