Optimal causal inference: Estimating stored information and approximating causal architecture

Chaos

Volumn 20, Issue 3, 2010, Pages

  Still, Susanne ^a   Crutchfield, James P ^b,c   Ellison, Christopher J ^b  

^a University of Hawai'i at Manoa   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c SANTA FE INSTITUTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 78649429079     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3489885     Document Type: Article

References (47)

1. Berge P. Pomeau Y. Vidal C. Order within Chaos 1986, and (Wiley, New York
2. Guckenheimer J. Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 1983, and (Springer-Verlag, New York
3. Wiggins S. Global Bifurcations and Chaos: Analytical Methods 1988, (Springer-Verlag, New York
4. Devaney R.L. An Introduction to Chaotic Dynamical Systems 1989, (Addison-Wesley, Redwood City, CA
5. Lieberman A.J. Lichtenberg M.A. Regular and Chaotic Dynamics 1993, and 2nd ed. (Springer-Verlag, New York
6. Ott E. Chaos in Dynamical Systems 1993, (Cambridge University Press, New York
7. Strogatz S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering 1994, (Addison-Wesley, Reading, MA
8. Packard N.H. Crutchfield J.P. Farmer J.D. Shaw R.S. Phys. Rev. Lett. 1980, 45:712. PRLTAO, 0031-9007, 10.1103/PhysRevLett.45.712, and.
9. Takens F. Symposium on Dynamical Systems and Turbulence 1981, 898:366. D. A. Rand, L. S. Young, in edited by and (Springer-Verlag, Berlin, ), Vol. p.
10. Fraser A. Information Dynamics 1991, 256:125. H. Atmanspacher, H. Scheingraber, in NATO Advanced Studies Institute, Series B: Physics Vol. edited by and (Plenum, New York, ), p.
11. Crutchfield J.P. McNamara B.S. Complex Syst. 1987, 1:417. CPSYEN, 0891-2513, and.
12. Nonlinear Modeling 1992, M. Casdagli, S. Eubank, SFI Studies in the Sciences of Complexity, edited by and (Addison-Wesley, Reading, MA
13. Sprott J.C. Chaos and Time-Series Analysis 2003, 2nd ed. (Oxford University Press, Oxford
14. Kantz H. Schreiber T. Nonlinear Time Series Analysis 2006, and 2nd ed. (Cambridge University Press, Cambridge
15. Crutchfield J.P. Young K. Phys. Rev. Lett. 1989, 63:105. PRLTAO, 0031-9007, 10.1103/PhysRevLett.63.105, and.
16. Tishby N. Pereira F. Bialek W. Proceedings of the 37th Annual Allerton Conference 1999, 368-377. B. Hajek, R. S. Sreenivas, and in edited by and (University of Illinois Press, Urbana, ), pp.
17. Still S. Bialek W. Neural Comput. 2004, 16:2483. NEUCEB, 0899-7667, 10.1162/0899766042321751, and.
18. Shalizi C.R. Crutchfield J.P. Adv. Complex Syst. 2002, 5:91. 0219-5259, 10.1142/S0219525902000481, and.
19. A more general approach is taken in Ref. 46, where both predictive modeling and decision making are considered. The scenario discussed here is a special case.
20. Shannon C.E. Shannon C.E. Shannon C.E. Weaver W. The Mathematical Theory of Communication 1949, 27:623. BSTJAN, 0005-8580, BSTJAN, 0005-8580, reprinted in and (University of Illinois Press, Urbana
21. Still S. Crutchfield J.P. and e-print arXiv:org/0708.0654.
22. Crutchfield J.P. Physica D 1994, 75:11. PDNPDT, 0167-2789, 10.1016/0167-2789(94)90273-9.
23. Crutchfield J.P. Shalizi C.R. Phys. Rev. E 1999, 59:275. PLEEE8, 1063-651X, 10.1103/PhysRevE.59.275, and.
24. To save space and improve readability, we use a simplified notation that refers to infinite sequences of random variables. The implication, however, is that one works with finite-length sequences into the past and into the future, whose infinite-length limit is taken at appropriate points. See, for example, Ref. 23 or, for measure-theoretic foundations, Ref. 47.
25. Cover T.M. Thomas J.A. Elements of Information Theory 2006, and 2nd ed. (Wiley-Interscience, New York
26. Crutchfield J.P. Ellison C.J. Mahoney J.R. Phys. Rev. Lett. 2009, 103:094101. PRLTAO, 0031-9007, 10.1103/PhysRevLett.103.094101, and.
27. Crutchfield J.P. Feldman D.P. Chaos 2003, 13:25. CHAOEH, 1054-1500, 10.1063/1.1530990, and.
28. Bialek W. de Ruyter van Steveninck R.R. Tishby N. 2006, 659-663. and Proceedings of the International Symposium on Information Theory, pp.
29. del Junco A. Rahe M. Proc. Am. Math. Soc. 1979, 75:259. PAMYAR, 0002-9939, 10.2307/2042753, and.
30. Crutchfield J.P. Packard N.H. Physica D 1983, 7:201. PDNPDT, 0167-2789, 10.1016/0167-2789(83)90127-6, and.
31. Shaw R. The Dripping Faucet as a Model Chaotic System 1984, (Aerial, Santa Cruz, CA
32. Grassberger P. Int. J. Theor. Phys. 1986, 25:907. IJTPBM, 0020-7748, 10.1007/BF00668821.
33. Li W. Complex Syst. 1991, 5:381. CPSYEN, 0891-2513.
34. Bialek W. Tishby N. and e-print arXiv:cond-mat/9902341v1.
35. Crutchfield J.P. Ellison C.J. Mahoney J.R. James R.G. Chaos 2010, 20:037105. CHAOEH, 1054-1500, and.
36. Arimoto S. IEEE Trans. Inf. Theory 1972, 18:14. IETTAW, 0018-9448, 10.1109/TIT.1972.1054753.
37. Blahut R.E. IEEE Trans. Inf. Theory 1972, 18:460. IETTAW, 0018-9448, 10.1109/TIT.1972.1054855.
38. Rose K. Gurewitz E. Fox G.C. Phys. Rev. Lett. 1990, 65:945. PRLTAO, 0031-9007, 10.1103/PhysRevLett.65.945, and.
39. Rose K. Proc. IEEE 1998, 86:2210. IEEPAD, 0018-9219, 10.1109/5.726788.
40. The algorithm follows that used in the information bottleneck (Ref. 16). The convergence arguments there apply to the OCF algorithm.
41. Ellison C.J. Mahoney J.R. Crutchfield J.P. J. Stat. Phys. 2009, 136:1005. JSTPBS, 0022-4715, 10.1007/s10955-009-9808-z, and.
42. Strelioff C.C. Crutchfield J.P. Hübler A. Phys. Rev. E 2007, 76:011106. PLEEE8, 1063-651X, 10.1103/PhysRevE.76.011106, and.
43. All quantities denoted with a {dot operator}̂ are evaluated at the estimate P̂.
44. Rabiner L.R. Juang B.H. 1986, and IEEE ASSP Magazine, January.
45. Rissanen J. Stochastic Complexity in Statistical Inquiry 1989, (World Scientific, Singapore
46. Still S. Europhys. Lett. 2009, 85:28005. EULEEJ, 0295-5075, 10.1209/0295-5075/85/28005.
47. Ay N. Crutchfield J.P. J. Stat. Phys. 2005, 210:659. JSTPBS, 0022-4715, 10.1007/s10955-005-6797-4, and.