General measures for separation in systems

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

Volumn 63, Issue 1 I, 2001, Pages 0111071-01110711

  Robinson, J W C ^a   Rung, J ^a   Bulsara, A R ^b   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: 33744890148     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.63.011107     Document Type: Article

References (53)

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2. To be more precise, for a Gaussian process the set of finitedimensional probability distribution functions (the law) is uniquely determined by the mean and autocovariance function, and the power spectrum is an equivalent representation of the autocovariance function.
3. Here we think of a SI simply as some real-valued functional of data; a figure of merit.
4. For good overviews see K. Wiesenfeld and F. Moss, Nature (London) 373, 33 (1995);
5. A. Bulsara and L. Gammaitoni, Phys. Today 49(3), 39 (1996);
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8. J.J. Collins, C.C. Chow, and T.T. Imhoff, Phys. Rev. E 52, 3321 (1995);
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13. I. Goychuk and P. Hänggi, ibid. 61, 4272 (2000).
14. J.W.C. Robinson, D.E. Asraf, A.R. Bulsara, and M.E. Inchiosa, Phys. Rev. Lett. 81, 2850 (1998).
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22. See, e.g., A. Bharucha-Reid, Elements of the Theory of Markov Processes and iheir Applications (McGraw-Hill, New York, 1960); N. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1992).
23. R.S. Liptser and A.N. Shiryayev, Statistics of Random Processes I: General Theory (Springer Verlag, New York, 1977).
24. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus (Springer-Verlag, New York, 1988).
25. Cf., e.g., Theorem 4.8 in [15].
26. 0).
27. Conditions guaranteeing a cyclostationary Markov solution to (1) can be found in Sec. III.5 of R.Z. Hasminskil, Stochastic Stability of Differential Equations (Sijthoff and Noordhoff, Al-phen an der Rijn, 1980).
28. Here and in the following, a number of measure-theoretic details are omitted, in particular the various er algebras (and filtrations) involved, since they are net 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 Bord 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].
29. 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 a algebras are omitted).
30. A thorough treatment of the associated theory of such densities can be found in Chapter 7 of Ref. [15].
31. Also for/linear, X, will be Gaussian.
32. Compare, e.g., Ref. [30], Sec. 2.4.
33. 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 informationtheoretic sense.
34. 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.
35. T.M. Cover and J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991).
36. D. Siegrnund, Sequential Analysis (Springer-Verlag, New York, 1985).
37. 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.
38. S. Kullback, Information Theory and Statistics (Dover, New York, 1997).
39. See. e.g., [36] for relations between these divergences.
40. This terminology is borrowed from information theory, where a related inequality with ihe same name holds for the mutual information; see [27].
41. For the injectivity to hold, it is sufficient that/satisfy a global
42. Lipschitz condition, but this is generally assumed in order to guarantee a strong solution to the SDE (1).
43. J. Rung and J.W.C. Robinson, in STOCHAOS: Stochastic and Chaotic Dynamics in the Lakes, edited by D.S. Broornhead, E.A. Luchinskaya, P.V.E. McClintock, and T. Mullin (American Institute of Physics, Melville. NY, 2000).
44. 1 is, or vice versa.
45. M. Basseville, Signal Proc. 18, 349 (1989).
46. 0-1.
47. 1.
48. 1 on C([0,T]) eventually separate completely so that perfect (zero error) detection becomes possible, yielding so-called
49. (asymptotically) singular detection [1].
50. 0) (and its modification) is no longer a sufficient statistic for the LR (8) (for any values of φ,T).
51. M.E. inchiosa and A.R. Bulsara, Phys. Rev. E 52, 327 (1995).
52. 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 tne same preservation of both divergences and SNR will occur then also.
53. 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.