Stochastic Models That Separate Fractal Dimension and the Hurst Effect

Fractal behavior and long-range dependence have been observed in an astonishing number of physical, biological, geological, and socioeconomic systems. Time series, profiles, and surfaces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coefficient, a measure of long-memory dependence. Both phenomena have been modeled and explained by self-affine random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical self-affinity implies a linear relationship between fractal dimension and Hurst coefficient and thereby links the two phenomena. This article introduces stochastic models that allow for any combination of fractal dimension and Hurst coefficient. Associated software for the synthesis of images with arbitrary, prespecified fractal properties and power-law correlations is available. The new models suggest a test for self-affinity that assesses coupling and decoupling of local and global behavior.

  • [1]  Google Scholar

  • [2]  J.‐M. Bardet, Testing for the presence of self‐similarity of Gaussian time series having stationary increments, J. Time Ser. Anal., 21 (2000), pp. 497–515. jtz JTSADL 0143-9782 J. Time Ser. Anal. CrossrefISIGoogle Scholar

  • [3]  O. E. Barndorff‐Nielsen, Superposition of Ornstein‐Uhlenbeck type processes, Theory Probab. Appl., 45 (2000), pp. 175–194. tba TPRBAU 0040-585X Theor. Probab. Appl. LinkISIGoogle Scholar

  • [4]  Google Scholar

  • [5]  Google Scholar

  • [6]  M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond and , and J. B. Bassingthwaighte, Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series, Phys. A, 241 (1997), pp. 606–626. pha PHYADX 0378-4371 Physica A CrossrefISIGoogle Scholar

  • [7]  Google Scholar

  • [8]  A. G. Constantine and  and P. Hall, Characterizing surface smoothness via estimation of effective fractal dimension, J. Roy. Statist. Soc. Ser. B, 56 (1994), pp. 97–113. Google Scholar

  • [9]  S. Davies and  and P. Hall, Fractal analysis of surface roughness by using spatial data, J. Roy. Statist. Soc. Ser. B, 61 (1999), pp. 3–37. CrossrefGoogle Scholar

  • [10]  C. R. Dietrich, A simple and efficient space‐domain implementation of the turning bands method, Water Resources Research, 31 (1995), pp. 147–156. wre WRERAQ 0043-1397 Water Resour. Res. CrossrefISIGoogle Scholar

  • [11]  C. R. Dietrich and  and G. N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput., 18 (1997), pp. 1088–1107. 8sm SJOCE3 1064-8275 SIAM J. Sci. Comput. (USA) LinkISIGoogle Scholar

  • [12]  B. Dubuc, J. F. Quiniou, C. Roques‐Carmes, C. Tricot and , and S. W. Zucker, Evaluating the fractal dimension of profiles, Phys. Rev. A, 39 (1989), pp. 1500–1512. pra PLRAAN 1050-2947 Phys. Rev. A CrossrefISIGoogle Scholar

  • [13]  H. Fairfield Smith, An empirical law describing heterogeneity in the yields of agricultural crops, J. Agric. Sci., 28 (1938), pp. 1–23. jaq JASIAB 0021-8596 J. Agric. Sci. CrossrefGoogle Scholar

  • [14]  Google Scholar

  • [15]  A. Gefferth, D. Veitch, I. Maricza, S. Molnár and , and I. Ruzsa, The nature of discrete second‐order self‐similarity, Adv. Appl. Probab., 35 (2003), pp. 395–416. aaz AAPBBD 0001-8678 Adv. Appl. Probab. CrossrefISIGoogle Scholar

  • [16]  T. Gneiting, On the derivatives of radial positive definite functions, J. Math. Anal. Appl., 236 (1999), pp. 86–93. jma JMANAK 0022-247X J. Math. Anal. Appl. CrossrefISIGoogle Scholar

  • [17]  T. Gneiting, A Pólya type criterion for radial characteristic functions in 2, Expo. Math., 17 (1999), pp. 181–183. 9zf ZZZZZZ 1058-6458 Exp. Math. Google Scholar

  • [18]  T. Gneiting, Power‐law correlations, related models for long‐range dependence, and their simulation, J. Appl. Probab., 37 (2000), pp. 1104–1109. jza JPRBAM 0021-9002 J. Appl. Probab. CrossrefISIGoogle Scholar

  • [19]  P. Hall and  and A. Wood, On the performance of box‐counting estimators of fractal dimension, Biometrika, 80 (1993), pp. 246–252. bik BIOKAX 0006-3444 Biometrika CrossrefISIGoogle Scholar

  • [20]  Google Scholar

  • [21]  H. E. Hurst, Long‐term storage capacity of reservoirs, Trans. Amer. Soc. Civil Engineers, 116 (1951), pp. 770–808. tal TACEAT 0066-0604 Trans. Am. Soc. Civ. Eng. ISIGoogle Scholar

  • [22]  J. W. Kantelhardt, E. Koscielny‐Bunde, H. H. A. Rego, S. Havlin and , and A. Bunde, Detecting long‐range correlations with detrended fluctuation analysis, Phys. A, 295 (2001), pp. 441–454. pha PHYADX 0378-4371 Physica A CrossrefISIGoogle Scholar

  • [23]  L. M. Kaplan and  and C.‐C. J. Kuo, Extending self‐similarity for fractional Brownian motion, IEEE Trans. Signal Process., 42 (1994), pp. 3526–3530. itl ITPRED 1053-587X IEEE Trans. Signal Process. CrossrefISIGoogle Scholar

  • [24]  E. Koscielny‐Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich and , and H.‐J. Schellnhuber, Indication of a universal persistence law governing atmospheric variability, Phys. Rev. Lett., 81 (1998), pp. 729–732. prl PRLTAO 0031-9007 Phys. Rev. Lett. CrossrefISIGoogle Scholar

  • [25]  D. Koutsoyiannis, A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series, Water Resources Research, 36 (2000), pp. 1519–1533. wre WRERAQ 0043-1397 Water Resour. Res. CrossrefISIGoogle Scholar

  • [26]  H. A. Makse, S. Havlin, M. Schwartz and , and H. E. Stanley, Method for generating long‐range correlations for large systems, Phys. Rev. E, 53 (1996), pp. 5445–5449. pre PLEEE8 1063-651X Phys. Rev. E CrossrefISIGoogle Scholar

  • [27]  Google Scholar

  • [28]  Google Scholar

  • [29]  B. B. Mandelbrot and  and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), pp. 422–437. sir SIREAD 0036-1445 SIAM Rev. LinkISIGoogle Scholar

  • [30]  G. Matheron, The intrinsic random functions and their applications, Adv. in Appl. Probab., 5 (1973), pp. 439–468. aaz AAPBBD 0001-8678 Adv. Appl. Probab. CrossrefGoogle Scholar

  • [31]  S. Orey, Gaussian sample functions and the Hausdorff dimension of level crossings, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 15 (1970), pp. 249–256. zwv ZWVGAA 0044-3719 Z. Wahrscheinlichkeitstheor. Verwandte Geb. CrossrefISIGoogle Scholar

  • [32]  J. D. Pelletier, Natural variability of atmospheric temperatures and geomagnetic intensity over a wide range of time scales, Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 2546–2553. pna PNASA6 0027-8424 Proc. Natl. Acad. Sci. U.S.A. CrossrefISIGoogle Scholar

  • [33]  C.‐K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley and , and A. L. Goldberger, Mosaic organization of DNA nucleotides, Phys. Rev. E, 49 (1994), pp. 1685–1689. pre PLEEE8 1063-651X Phys. Rev. E CrossrefISIGoogle Scholar

  • [34]  Google Scholar

  • [35]  A. H. Romero and  and J.‐M. Sancho, Generation of short and long range temporal correlated noises, J. Comput. Phys., 156 (1999), pp. 1–11. jct JCTPAH 0021-9991 J. Comput. Phys. CrossrefISIGoogle Scholar

  • [36]  M. Schlather, Simulation and analysis of random fields, R News, 1 (2) (2001), pp. 18–20. Google Scholar

  • [37]  Google Scholar

  • [38]  Google Scholar

  • [39]  Google Scholar

  • [40]  Google Scholar

  • [41]  M. S. Taqqu, V. Teverovsky and , and W. Willinger, Estimators for long‐range dependence: An empirical study, Fractals, 3 (1995), pp. 785–798. fra FRACEG 0218-348X Fractals CrossrefISIGoogle Scholar

  • [42]  C. C. Taylor and  and S. J. Taylor, Estimating the dimension of a fractal, J. Roy. Statist. Soc. Ser. B, 53 (1991), pp. 353–364. Google Scholar

  • [43]  Google Scholar

  • [44]  Google Scholar

  • [45]  P. D. Welch, The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms, IEEE Trans. Audio Electroacoustics, 15 (1967), pp. 70–73. ite ITADAS 0018-9278 IEEE Trans. Audio Electroacoust. CrossrefGoogle Scholar

  • [46]  P. Whittle, On the variation of yield variance with plot size, Biometrika, 43 (1956), pp. 337–343. bik BIOKAX 0006-3444 Biometrika CrossrefISIGoogle Scholar

  • [47]  P. Whittle, Topographic correlation, power‐law covariance functions, and diffusion, Biometrika, 49 (1962), pp. 305–314. bik BIOKAX 0006-3444 Biometrika CrossrefISIGoogle Scholar

  • [48]  A. T. A. Wood and  and G. Chan, Simulation of stationary Gaussian processes in [0,1]d, J. Comput. Graph. Statist., 3 (1994), pp. 409–432. b7x ZZZZZZ 1061-8600 J. Comput. Graph. Stat. Google Scholar