Computational Complexity

2012 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Correlations in Complex Systems

  • Renat M. Yulmetyev
  • Peter Hänggi
Reference work entry

Article Outline


Definition of the Subject


Correlation and Memory in Discrete Non-Markov Stochastic Processes

Correlation and Memory in Discrete Non‐Markov Stochastic Processes Generated by Random Events

Information Measures of Memory in Complex Systems

Manifestation of Strong Memory in Complex Systems

Some Perspectives on the Studies of Memory in Complex Systems



Deep Brain Stimulation Memory Effect Parkinson Disease Time Correlation Function Strong Memory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Primary Literature

  1. 1.
    Markov AA (1906) Two‐dimensional Brownian motion and harmonic functions. Proc Phys Math Soc Kazan Imp Univ 15(4):135–178; in RussianGoogle Scholar
  2. 2.
    Chapman S, Couling TG (1958) The mathematical theory of nonuniform gases. Cambridge University Press, CambridgeGoogle Scholar
  3. 3.
    Albeverio S, Blanchard P, Steil L (1990) Stochastic processes and their applications in mathematics and physics. Kluwer, DordrechtzbMATHCrossRefGoogle Scholar
  4. 4.
    Rice SA, Gray P (1965) The statistical mechanics of simple liquids. Interscience, New YorkGoogle Scholar
  5. 5.
    Kubo R, Toda M, Hashitsume N, Saito N (2003) Statistical physics II: Nonequilibrium statistical mechanics. In: Fulde P (ed) Springer Series in Solid-State Sciences, vol 31. Springer, Berlin, p 279Google Scholar
  6. 6.
    Ginzburg VL, Andryushin E (2004) Superconductivity. World Scientific, SingaporezbMATHCrossRefGoogle Scholar
  7. 7.
    Sachs I, Sen S, Sexton J (2006) Elements of statistical mechanics. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  8. 8.
    Fetter AL, Walecka JD (1971) Quantum theory of many‐particle physics. Mc Graw-Hill, New YorkGoogle Scholar
  9. 9.
    Chandler D (1987) Introduction to modern statistical mechanics. Oxford University Press, OxfordGoogle Scholar
  10. 10.
    Zwanzig R (2001) Nonequilibrium statistical mechanics. Cambridge University Press, CambridgeGoogle Scholar
  11. 11.
    Zwanzig R (1961) Memory effects in irreversible thermodynamics. Phys Rev 124:983–992zbMATHCrossRefGoogle Scholar
  12. 12.
    Mori H (1965) Transport, collective motion and Brownian motion. Prog Theor Phys 33:423–455; Mori H (1965) A continued fraction representation of the time correlation functions. Prog Theor Phys 34:399–416CrossRefGoogle Scholar
  13. 13.
    Grabert H, Hänggi P, Talkner P (1980) Microdynamics and nonlinear stochastic processes of gross variables. J Stat Phys 22:537–552Google Scholar
  14. 14.
    Grabert H, Talkner P, Hänggi P (1977) Microdynamics and time‐evolution of macroscopic non‐Markovian systems. Z Physik B 26:389–395Google Scholar
  15. 15.
    Grabert H, Talkner P, Hänggi P, Thomas H (1978) Microdynamics and time‐evolution of macroscopic non‐Markovian systems II. Z Physik B 29:273–280Google Scholar
  16. 16.
    Hänggi P, Thomas H (1977) Time evolution, correlations and linear response of non‐Markov processes. Z Physik B 26:85–92Google Scholar
  17. 17.
    Hänggi P, Talkner P (1983) Memory index of first‐passage time: A simple measure of non‐Markovian character. Phys Rev Lett 51:2242–2245Google Scholar
  18. 18.
    Hänggi P, Thomas H (1982) Stochastic processes: Time‐evolution, symmetries and linear response. Phys Rep 88:207–319Google Scholar
  19. 19.
    Lee MH (1982) Orthogonalization process by recurrence relations. Phys Rev Lett 49:1072–1072; Lee MH (1983) Can the velocity autocorrelation function decay exponentially? Phys Rev Lett 51:1227–1230CrossRefGoogle Scholar
  20. 20.
    Balucani U, Lee MH, Tognetti V (2003) Dynamic correlations. Phys Rep 373:409–492MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hong J, Lee MH (1985) Exact dynamically convergent calculations of the frequency‐dependent density response function. Phys Rev Lett 55:2375–2378CrossRefGoogle Scholar
  22. 22.
    Lee MH (2000) Heisenberg, Langevin, and current equations via the recurrence relations approach. Phys Rev E 61:3571–3578; Lee MH (2000) Generalized Langevin equation and recurrence relations. Phys Rev E 62:1769–1772CrossRefGoogle Scholar
  23. 23.
    Lee MH (2001) Ergodic theory, infinite products, and long time behavior in Hermitian models. Phys Rev Lett 87(1–4):250601CrossRefGoogle Scholar
  24. 24.
    Kubo R (1966) Fluctuation‐dissipation theorem. Rep Progr Phys 29:255–284CrossRefGoogle Scholar
  25. 25.
    Kawasaki K (1970) Kinetic equations and time correlation functions of critical fluctuations. Ann Phys 61:1–56CrossRefGoogle Scholar
  26. 26.
    Michaels IA, Oppenheim I (1975) Long-time tails and Brownian motion. Physica A 81:221–240CrossRefGoogle Scholar
  27. 27.
    Frank TD, Daffertshofer A, Peper CE, Beek PJ, Haken H (2001) H‑theorem for a mean field model describing coupled oscillator systems under external forces. Physica D 150:219–236zbMATHCrossRefGoogle Scholar
  28. 28.
    Vogt M, Hernandez R (2005) An idealized model for nonequilibrium dynamics in molecular systems. J Chem Phys 123(1–8):144109CrossRefGoogle Scholar
  29. 29.
    Sen S (2006) Solving the Liouville equation for conservative systems: Continued fraction formalism and a simple application. Physica A 360:304–324MathSciNetCrossRefGoogle Scholar
  30. 30.
    Prokhorov YV (1999) Probability and mathematical statistics (encyclopedia). Scien Publ Bolshaya Rossiyskaya Encyclopedia, MoscowGoogle Scholar
  31. 31.
    Yulmetyev R et al (2000) Stochastic dynamics of time correlation in complex systems with discrete time. Phys Rev E 62:6178–6194CrossRefGoogle Scholar
  32. 32.
    Yulmetyev R et al (2002) Quantification of heart rate variability by discrete nonstationarity non‐Markov stochastic processes. Phys Rev E 65(1–15):046107CrossRefGoogle Scholar
  33. 33.
    Reed M, Samon B (1972) Methods of mathematical physics. Academic, New YorkzbMATHGoogle Scholar
  34. 34.
    Graber H (1982) Projection operator technique in nonequilibrium statistical mechanics. In: Höhler G (ed) Springer tracts in modern physics, vol 95. Springer, BerlinGoogle Scholar
  35. 35.
    Yulmetyev RM (2001) Possibility between earthquake and explosion seismogram differentiation by discrete stochastic non‐Markov processes and local Hurst exponent analysis. Phys Rev E 64(1–14):066132CrossRefGoogle Scholar
  36. 36.
    Abe S, Suzuki N (2004) Aging and scaling of earthquake aftershocks. Physica A 332:533–538CrossRefGoogle Scholar
  37. 37.
    Tirnakli U, Abe S (2004) Aging in coherent noise models and natural time. Phys Rev E 70(1–4):056120CrossRefGoogle Scholar
  38. 38.
    Abe S, Sarlis NV, Skordas ES, Tanaka HK, Varotsos PA (2005) Origin of the usefulness of the natural‐time representation of complex time series. Phys Rev Lett 94(1–4):170601CrossRefGoogle Scholar
  39. 39.
    Stanley HE, Meakin P (1988) Multifractal phenomena in physics and chemistry. Nature 335:405–409CrossRefGoogle Scholar
  40. 40.
    Ivanov P Ch, Amaral LAN, Goldberger AL, Havlin S, Rosenblum MG, Struzik Z, Stanley HE (1999) Multifractality in human heartbeat dynamics. Nature 399:461–465CrossRefGoogle Scholar
  41. 41.
    Mokshin AV, Yulmetyev R, Hänggi P (2005) Simple measure of memory for dynamical processes described by a generalized Langevin equation. Phys Rev Lett 95(1–4):200601Google Scholar
  42. 42.
    Allegrini P et al (2003) Compression and diffusion: A joint approach to detect complexity. Chaos Soliton Fractal 15:517–535MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Amaral LAN et al (2001) Application of statistical physics methods and concepts to the study of science and technology systems. Scientometrics 51:9–36CrossRefGoogle Scholar
  44. 44.
    Arneodo A et al (1996) Wavelet based fractal analysis of DNA sequences. Physica D 96:291–320CrossRefGoogle Scholar
  45. 45.
    Ashkenazy Y et al (2003) Magnitude and sign scaling in power-law correlated time series. Physica A Stat Mech Appl 323:19–41zbMATHCrossRefGoogle Scholar
  46. 46.
    Ashkenazy Y et al (2003) Nonlinearity and multifractality of climate change in the past 420,000 years. Geophys Res Lett 30:2146CrossRefGoogle Scholar
  47. 47.
    Azbel MY (1995) Universality in a DNA statistical structure. Phys Rev Lett 75:168–171CrossRefGoogle Scholar
  48. 48.
    Baldassarri A et al (2006) Brownian forces in sheared granular matter. Phys Rev Lett 96:118002CrossRefGoogle Scholar
  49. 49.
    Baleanu D et al (2006) Fractional Hamiltonian analysis of higher order derivatives systems. J Math Phys 47:103503MathSciNetCrossRefGoogle Scholar
  50. 50.
    Blesic S et al (2003) Detecting long-range correlations in time series of neuronal discharges. Physica A 330:391–399MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Cajueiro DO, Tabak BM (2004) The Hurst exponent over time: Testing the assertion that emerging markets are becoming more efficient. Physica A 336:521–537MathSciNetCrossRefGoogle Scholar
  52. 52.
    Brecht M et al (1998) Correlation analysis of corticotectal interactions in the cat visual system. J Neurophysiol 79:2394–2407Google Scholar
  53. 53.
    Brouersa F, Sotolongo‐Costab O (2006) Generalized fractal kinetics in complex systems (application to biophysics and biotechnology). Physica A 368(1):165–175Google Scholar
  54. 54.
    Coleman P, Pietronero L (1992) The fractal structure of the universe. Phys Rep 213:311–389CrossRefGoogle Scholar
  55. 55.
    Goldberger AL et al (2002) What is physiologic complexity and how does it change with aging and disease? Neurobiol Aging 23:23–26CrossRefGoogle Scholar
  56. 56.
    Grau‐Carles P (2000) Empirical evidence of long-range correlations in stock returns. Physica A 287:396–404Google Scholar
  57. 57.
    Grigolini P et al (2001) Asymmetric anomalous diffusion: An efficient way to detect memory in time series. Fractal‐Complex Geom Pattern Scaling Nat Soc 9:439–449Google Scholar
  58. 58.
    Ebeling W, Frommel C (1998) Entropy and predictability of information carriers. Biosystems 46:47–55CrossRefGoogle Scholar
  59. 59.
    Fukuda K et al (2004) Heuristic segmentation of a nonstationary time series. Phys Rev E 69:021108CrossRefGoogle Scholar
  60. 60.
    Hausdorff JM, Peng CK (1996) Multiscaled randomness: A possible source of 1/f noise in biology. Phys Rev E 54:2154–2157CrossRefGoogle Scholar
  61. 61.
    Herzel H et al (1998) Interpreting correlations in biosequences. Physica A 249:449–459MathSciNetCrossRefGoogle Scholar
  62. 62.
    Hoop B, Peng CK (2000) Fluctuations and fractal noise in biological membranes. J Membrane Biol 177:177–185CrossRefGoogle Scholar
  63. 63.
    Hoop B et al (1998) Temporal correlation in phrenic neural activity. In: Hughson RL, Cunningham DA, Duffin J (eds) Advances in modelling and control of ventilation. Plenum Press, New York, pp 111–118Google Scholar
  64. 64.
    Ivanova K, Ausloos M (1999) Application of the detrended fluctuation analysis (DFA) method for describing cloud breaking. Physica A 274:349–354CrossRefGoogle Scholar
  65. 65.
    Ignaccolo M et al (2004) Scaling in non‐stationary time series. Physica A 336:595–637CrossRefGoogle Scholar
  66. 66.
    Imponente G (2004) Complex dynamics of the biological rhythms: Gallbladder and heart cases. Physica A 338:277–281MathSciNetCrossRefGoogle Scholar
  67. 67.
    Jefferiesa P et al (2003) Anatomy of extreme events in a complex adaptive system. Physica A 318:592–600CrossRefGoogle Scholar
  68. 68.
    Karasik R et al (2002) Correlation differences in heartbeat fluctuations during rest and exercise. Phys Rev E 66:062902CrossRefGoogle Scholar
  69. 69.
    Kulessa B et al (2003) Long-time autocorrelation function of ECG signal for healthy versus diseased human heart. Acta Phys Pol B 34:3–15Google Scholar
  70. 70.
    Kutner R, Switala F (2003) Possible origin of the non‐linear long-term autocorrelations within the Gaussian regime. Physica A 330:177–188MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Koscielny‐Bunde E et al (1998) Indication of a universal persistence law governing atmospheric variability. Phys Rev Lett 81:729–732Google Scholar
  72. 72.
    Labini F (1998) Scale invariance of galaxy clustering. Phys Rep 293:61–226CrossRefGoogle Scholar
  73. 73.
    Linkenkaer‐Hansen K et al (2001) Long-range temporal correlations and scaling behavior in human brain oscillations. J Neurosci 21:1370–1377Google Scholar
  74. 74.
    Mercik S et al (2000) What can be learnt from the analysis of short time series of ion channel recordings. Physica A 276:376–390CrossRefGoogle Scholar
  75. 75.
    Montanari A et al (1999) Estimating long-range dependence in the presence of periodicity: An empirical study. Math Comp Model 29:217–228MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Mark N (2004) Time fractional Schrodinger equation. J Math Phys 45:3339–3352MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    Niemann M et al (2008) Usage of the Mori–Zwanzig method in time series analysis. Phys Rev E 77:011117MathSciNetCrossRefGoogle Scholar
  78. 78.
    Nigmatullin RR (2002) The quantified histograms: Detection of the hidden unsteadiness. Physica A 309:214–230zbMATHCrossRefGoogle Scholar
  79. 79.
    Nigmatullin RR (2006) Fractional kinetic equations and universal decoupling of a memory function in mesoscale region. Physica A 363:282–298CrossRefGoogle Scholar
  80. 80.
    Ogurtsov MG (2004) New evidence for long-term persistence in the sun’s activity. Solar Phys 220:93–105CrossRefGoogle Scholar
  81. 81.
    Pavlov AN, Dumsky DV (2003) Return times dynamics: Role of the Poincare section in numerical analysis. Chaos Soliton Fractal 18:795–801zbMATHCrossRefGoogle Scholar
  82. 82.
    Paulus MP (1997) Long-range interactions in sequences of human behavior. Phys Rev E 55:3249–3256CrossRefGoogle Scholar
  83. 83.
    Peng C-K et al (1994) Mosaic organization of DNA nucleotides. Phys Rev E 49:1685–1689CrossRefGoogle Scholar
  84. 84.
    Peng C-K et al (1995) Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 5:82–87CrossRefGoogle Scholar
  85. 85.
    Poon CS, Merrill CK (1997) Decrease of cardiac chaos in congestive heart failure. Nature 389:492–495CrossRefGoogle Scholar
  86. 86.
    Rangarajan G, Ding MZ (2000) Integrated approach to the assessment of long range correlation in time series data. Phys Rev E 61:4991–5001MathSciNetCrossRefGoogle Scholar
  87. 87.
    Robinson PA (2003) Interpretation of scaling properties of electroencephalographic fluctuations via spectral analysis and underlying physiology. Phys Rev E 67:032902CrossRefGoogle Scholar
  88. 88.
    Rizzo F et al (2005) Transport properties in correlated systems: An analytical model. Phys Rev B 72:155113CrossRefGoogle Scholar
  89. 89.
    Shen Y et al (2003) Dimensional complexity and spectral properties of the human sleep EEG. Clinic Neurophysiol 114:199–209CrossRefGoogle Scholar
  90. 90.
    Schmitt D et al (2006) Analyzing memory effects of complex systems from time series. Phys Rev E 73:056204CrossRefGoogle Scholar
  91. 91.
    Soen Y, Braun F (2000) Scale‐invariant fluctuations at different levels of organization in developing heart cell networks. Phys Rev E 61:R2216–R2219CrossRefGoogle Scholar
  92. 92.
    Stanley HE et al (1994) Statistical‐mechanics in biology – how ubiquitous are long-range correlations. Physica A 205:214–253CrossRefGoogle Scholar
  93. 93.
    Stanley HE (2000) Exotic statistical physics: Applications to biology, medicine, and economics. Physica A 285:1–17zbMATHCrossRefGoogle Scholar
  94. 94.
    Tarasov VE (2006) Fractional variations for dynamical systems: Hamilton and Lagrange approaches. J Phys A Math Gen 39:8409–8425MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Telesca L et al (2003) Investigating the time‐clustering properties in seismicity of Umbria‐Marche region (central Italy). Chaos Soliton Fractal 18:203–217zbMATHCrossRefGoogle Scholar
  96. 96.
    Turcott RG, Teich MC (1996) Fractal character of the electrocardiogram: Distinguishing heart‐failure and normal patients. Ann Biomed Engin 24:269–293CrossRefGoogle Scholar
  97. 97.
    Thurner S et al (1998) Receiver‐operating‐characteristic analysis reveals superiority of scale‐dependent wavelet and spectral measures for assessing cardiac dysfunction. Phys Rev Lett 81:5688–5691CrossRefGoogle Scholar
  98. 98.
    Vandewalle N et al (1999) The moving averages demystified. Physica A 269:170–176CrossRefGoogle Scholar
  99. 99.
    Varela M et al (2003) Complexity analysis of the temperature curve: New information from body temperature. Eur J Appl Physiol 89:230–237CrossRefGoogle Scholar
  100. 100.
    Varotsos PA et al (2002) Long-range correlations in the electric signals that precede rupture. Phys Rev E 66:011902CrossRefGoogle Scholar
  101. 101.
    Watters PA (2000) Time‐invariant long-range correlations in electroencephalogram dynamics. Int J Syst Sci 31:819–825zbMATHCrossRefGoogle Scholar
  102. 102.
    Wilson PS et al (2003) Long‐memory analysis of time series with missing values. Phys Rev E 68:017103CrossRefGoogle Scholar
  103. 103.
    Yulmetyev RM et al (2004) Dynamical Shannon entropy and information Tsallis entropy in complex systems. Physica A 341:649–676MathSciNetCrossRefGoogle Scholar
  104. 104.
    Yulmetyev R, Hänggi P, Gafarov F (2000) Stochastic dynamics of time correlation in complex systems with discrete time. Phys Rev E 62:6178Google Scholar
  105. 105.
    Yulmetyev R, Gafarov F, Hänggi P, Nigmatullin R, Kayumov S (2001) Possibility between earthquake and explosion seismogram processes and local Hurst exponent analysis. Phys Rev E 64:066132Google Scholar
  106. 106.
    Yulmetyev R, Hänggi P, Gafarov F (2002) Quantification of heart rate variability by discrete nonstationary non‐Markov stochastic processes. Phys Rev E 65:046107Google Scholar
  107. 107.
    Yulmetyev R, Demin SA, Panischev OY, Hänggi P, Timashev SF, Vstovsky GV (2006) Regular and stochastic behavior of Parkinsonian pathological tremor signals. Physica A 369:655Google Scholar

Books and Reviews

  1. 108.
    Badii R, Politi A (1999) Complexity: Hierarchical structures and scaling in physics. Oxford University Press, New YorkzbMATHGoogle Scholar
  2. 109.
    Elze H-T (ed) (2004) Decoherence and entropy in complex systems. In: Selected lectures from DICE 2002 series: Lecture notes in physics, vol 633. Springer, HeidelbergGoogle Scholar
  3. 110.
    Kantz H, Schreiber T (2004) Nonlinear time series analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  4. 111.
    Mallamace F, Stanley HE (2004) The physics of complex systems (new advances and perspectives). IOS Press, AmsterdamzbMATHGoogle Scholar
  5. 112.
    Parisi G, Pietronero L, Virasoro M (1992) Physics of complex systems: Fractals, spin glasses and neural networks. Physica A 185(1–4):1–482Google Scholar
  6. 113.
    Sprott JC (2003) Chaos and time‐series analysis. Oxford University Press, New YorkzbMATHGoogle Scholar
  7. 114.
    Zwanzig R (2001) Nonequilibrium statistical physics. Oxford University Press, New YorkGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Renat M. Yulmetyev
    • 1
    • 2
  • Peter Hänggi
    • 3
  1. 1.Department of PhysicsKazan State UniversityKazanRussia
  2. 2.Tatar State University of Pedagogical and Humanities SciencesKazanRussia
  3. 3.University of AugsburgAugsburgGermany