General measures for signal-noise separation in nonlinear dynamical systems

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 63, Issue 1 II, 2001, Pages 1-11

  Robinson, J W C ^a   Rung, J ^b   Bulsara, A R ^c   Inchiosa, M E ^c  

^a SWEDISH DEFENCE RESEARCH AGENCY   (Sweden)

^b UPPSALA UNIVERSITY   (Sweden)

^c Naval Command   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 18644382053     PISSN: 15393755     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.63.011107     Document Type: Article

References (54)

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4. Here we think of a SI simply as some real-valued functional of data; a figure of merit.
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6. For good overviews see K. Wiesenfeld and F. Moss, Nature (London) 373, 33 (1995); A. Bulsara and L. Gammaitoni, Phys. Today 49(3), 39 (1996); L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
7. For good overviews see K. Wiesenfeld and F. Moss, Nature (London) 373, 33 (1995); A. Bulsara and L. Gammaitoni, Phys. Today 49(3), 39 (1996); L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
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12. J.J. Collins, C.C. Chow, and T.T. Imhoff, Phys. Rev. E 52, 3321 (1995); A. Bulsara and A. Zador, ibid. 54, R2185 (1996); C. Heneghan, C.C. Chow, J.J. Collins, T.T. Imhoff, S.B. Lowen, and M.C. Teich, ibid. 54, R2228 (1996); M. Stemmler, Network 7, 687 (1996); F. Chapeau-Blondeau, Phys. Rev. E 55, 2016 (1997); I. Goychuk and P. Hänggi, ibid. 61, 4272 (2000).
13. J.J. Collins, C.C. Chow, and T.T. Imhoff, Phys. Rev. E 52, 3321 (1995); A. Bulsara and A. Zador, ibid. 54, R2185 (1996); C. Heneghan, C.C. Chow, J.J. Collins, T.T. Imhoff, S.B. Lowen, and M.C. Teich, ibid. 54, R2228 (1996); M. Stemmler, Network 7, 687 (1996); F. Chapeau-Blondeau, Phys. Rev. E 55, 2016 (1997); I. Goychuk and P. Hänggi, ibid. 61, 4272 (2000).
14. J.J. Collins, C.C. Chow, and T.T. Imhoff, Phys. Rev. E 52, 3321 (1995); A. Bulsara and A. Zador, ibid. 54, R2185 (1996); C. Heneghan, C.C. Chow, J.J. Collins, T.T. Imhoff, S.B. Lowen, and M.C. Teich, ibid. 54, R2228 (1996); M. Stemmler, Network 7, 687 (1996); F. Chapeau-Blondeau, Phys. Rev. E 55, 2016 (1997); I. Goychuk and P. Hänggi, ibid. 61, 4272 (2000).
15. J.J. Collins, C.C. Chow, and T.T. Imhoff, Phys. Rev. E 52, 3321 (1995); A. Bulsara and A. Zador, ibid. 54, R2185 (1996); C. Heneghan, C.C. Chow, J.J. Collins, T.T. Imhoff, S.B. Lowen, and M.C. Teich, ibid. 54, R2228 (1996); M. Stemmler, Network 7, 687 (1996); F. Chapeau-Blondeau, Phys. Rev. E 55, 2016 (1997); I. Goychuk and P. Hänggi, ibid. 61, 4272 (2000).
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24. See, e.g., A. Bharucha-Reid, Elements of the Theory of Markov Processes and their Applications (McGraw-Hill, New York, 1960); N. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1992).
25. See, e.g., A. Bharucha-Reid, Elements of the Theory of Markov Processes and their Applications (McGraw-Hill, New York, 1960); N. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1992).
26. R.S. Liptser and A.N. Shiryayev, Statistics of Random Processes I: General Theory (Springer Verlag, New York, 1977).
27. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus (Springer-Verlag, New York, 1988).
28. Cf., e.g., Theorem 4.8 in [15].
29. 0).
30. Conditions guaranteeing a cyclostationary Markov solution to (1) can be found in Sec. III.5 of R.Z. Hasminskii, Stochastic Stability of Differential Equations (Sijthoff and Noordhoff, Alphen an der Rijn, 1980).
31. Here and in the following, a number of measure-theoretic details are omitted, in particular the various σ algebras (and filtrations) involved, since they are not essential (and a reader with the appropriate background can easily fill them in). Suffice it to say that there must exist a basic σ algebra F on Ω with respect to which P is defined and the various random variables and processes are measurable, and as basic σ algebra on C([0,T]) we take the Borel σ algebra B[C([0,T])]. A good introduction to (abstract) probability theory is given in D. Williams, Probability with Martingales (Cambridge University Press, Cambridge, 1991) and the measure-theoretic details of the continuous-time stochastic processes encountered here are covered by, e.g., [16].
32. If λ is a measure on χ and η:χ→γ, the map η induces a measure ρ on γ by ρ(B)=λ({x∈χ:η(x)∈B}) for B⊆γ (again, details about σ algebras are omitted).
33. A thorough treatment of the associated theory of such densities can be found in Chapter 7 of Ref. [15].
34. t will be Gaussian.
35. Compare, e.g., Ref. [30], Sec. 2.4.
36. Indeed, it represents a tremendous coding, since the trajectory lives in an infinite-dimensional space, whereas the LR takes values on the real line. Note, however, that we use the word coding a little bit loosely here, and not in its strict information-theoretic sense.
37. See, e.g., Sec. 3.5.D of Ref. [16] and Sec. 6.2 of Ref. [15], where further conditions guaranteeing that (12) is fulfilled can also be found.
38. T.M. Cover and J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991).
39. D. Siegmund, Sequential Analysis (Springer-Verlag, New York, 1985).
40. Here and in the following the word "information" is to be interpreted informally and not in its most common information-theoretic sense (which applies to communication). It is noteworthy, however, that Kullback [30] who mostly considered inference, quantified the word "information" by the value of the information divergence.
41. S. Kullback, Information Theory and Statistics (Dover, New York, 1997).
42. See, e.g., [36] for relations between these divergences.
43. This terminology is borrowed from information theory, where a related inequality with the same name holds for the mutual information; see [27].
44. For the injectivity to hold, it is sufficient that f satisfy a global Lipschitz condition, but this is generally assumed in order to guarantee a strong solution to the SDE (1).
45. J. Rung and J.W.C. Robinson, in STOCHAOS: Stochastic and Chaotic Dynamics in the Lakes, edited by D.S. Broomhead, E.A. Luchinskaya, P.V.E. McClintock, and T. Mullin (American Institute of Physics, Melville, NY, 2000).
46. 1 is, or vice versa.
47. M. Basseville, Signal Proc. 18, 349 (1989).
48. 0-1.
49. 1.
50. 1 on C([0,T]) eventually separate completely so that perfect (zero error) detection becomes possible, yielding so-called (asymptotically) singular detection [1].
51. 0) (and its modification) is no longer a sufficient statistic for the LR (8) (for any values of φ, T).
52. M.E. Inchiosa and A.R. Bulsara, Phys. Rev. E 52, 327 (1995).
53. For the system (1) with a potential such as that corresponding to Eq. (2), which both locally near the two local minima of the potential and for large |x| is parabolic, there are, moreover, two asymptotes that are to be expected in the deflection curves, provided the signal is small and T is large: when the input noise strength σ is small the system acts essentially linearly, and hence will preserve not only divergences but also SNR, and it will also appear linear for very large σ, and the same preservation of both divergences and SNR will occur then also.
54. It can be shown that functions of the LR that have the data processing property (14) must (under some technical conditions, cf., e.g., [10]) be of the form (13), with a strictly convex φ.