Abstract

Two-dimensional fractional Brownian motion (FBM) is a good model for stellar wave fronts distorted by Kolmogorov turbulence. However, the time series generated by the movement of such a wave front past an observation point does not explain the reported predictability of measured wave-front slopes. Rescaled-range analysis, a technique for detecting dependence among samples in a series, and a correlation-dimension algorithm that tests for the presence of deterministic chaos were applied to series of real, measured, stellar wavefront slopes. These tests suggest that the source of correlation and predictability is the low-pass spatial filtering of the FBM wave front by the lenslets of an adaptive-optics wave-front sensor and that the process is not chaotic. A simple modeling procedure is described for generating time series of measured wave-front slopes with the correct temporal characteristics.

© 1997 Optical Society of America

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References

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  1. J. W. Goodman, in Statistcial Optics (Wiley, New York, 1985), pp. 429–431.
  2. P. M. Harrington, B. M. Welsh, “Frequency-domain analysis of an adaptive optical system’s temporal response,” Opt. Eng. 33, 2336–2342 (1994).
    [CrossRef]
  3. C. Schwartz, G. Baum, E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
    [CrossRef]
  4. E. E. Peters, Fractal Market Analysis (Wiley, New York, 1994).
  5. B. B. Mandelbrot, J. V. Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
    [CrossRef]
  6. M. Jorgenson, G Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
    [CrossRef] [PubMed]
  7. G. Aitken, D. McGaughey, “Predictability of atmospherically-distorted stellar wavefronts” in Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1996), pp. 89–94.
  8. M. Jorgenson, G. Aitken, E. Hege, “Evidence of a chaotic attractor in star-wander data,” Opt. Lett. 16, 64–66 (1991).
    [CrossRef] [PubMed]
  9. A. Armitage, Y. Guillo, F. Meyer, “Progress with a neural network in astronomical control,” in Practical Applications of Neural Networks in Control (IMC Symposium, London, 1994).
  10. M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1996), pp. 95–101.
  11. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1983), pp. 248–255 and 386–387.
  12. P. Grassberger, I. Proccaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
    [CrossRef]
  13. J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
    [CrossRef]
  14. H.-O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer-Verlag, New York, 1988).
  15. H. E. Hurst, “Long-term storage capacity of reservoirs,” Trans. Am. Soc. Civil Engineers 116, 770–808 (1951).
  16. J. Theiler, “Some comments on the correlation dimension of 1/fα noise,” Phys. Lett. A 135, 480–493 (1991).
    [CrossRef]
  17. J. Theiler, “Estimating the fractal dimension of chaotic time series,” Lincoln Lab. J. 3, 63–85 (1990).
  18. F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, D. Rank, L. S. Young, eds. (Springer-Verlag, Berlin, 1981), pp. 366–381.
  19. J. Theiler, “Estimating fractal dimension,” J. Opt. Soc. Am. A 7, 1055–1073 (1990).
    [CrossRef]
  20. J. Theiler, B. Galdrikian, A. Longtin, S. Eubank, J. D. Farmer, “Using surrogate data to detect nonlinear relationships between time series,” in Non-Linear Modeling and Forecasting, Vol. XII of SFI Studies in the Sciences of Complexity (Addison-Wesley, Reading, Mass., 1992).
  21. G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
    [CrossRef]
  22. F. Roddier, “The effects of atmospheric turbulence on optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  23. M. R. Dennis Montera, B. Welsh, D. Ruck, “Processing wave-front slope measurements using artificial neural networks,” Appl. Opt. 35, 4238–4251 (1996).
    [CrossRef]
  24. A. G. R. G. Lane, J. C. Dainty, “Simulation of Kolmogorov phase screens,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  25. B. Friedlander, B. Porat, “The modified Yule–Walker method of ARMA spectral estimation,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 158–173 (1984).
    [CrossRef]
  26. D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front-compensation systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
    [CrossRef]

1996 (1)

1995 (1)

1994 (2)

P. M. Harrington, B. M. Welsh, “Frequency-domain analysis of an adaptive optical system’s temporal response,” Opt. Eng. 33, 2336–2342 (1994).
[CrossRef]

C. Schwartz, G. Baum, E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
[CrossRef]

1992 (2)

M. Jorgenson, G Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
[CrossRef] [PubMed]

A. G. R. G. Lane, J. C. Dainty, “Simulation of Kolmogorov phase screens,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1991 (2)

M. Jorgenson, G. Aitken, E. Hege, “Evidence of a chaotic attractor in star-wander data,” Opt. Lett. 16, 64–66 (1991).
[CrossRef] [PubMed]

J. Theiler, “Some comments on the correlation dimension of 1/fα noise,” Phys. Lett. A 135, 480–493 (1991).
[CrossRef]

1990 (2)

J. Theiler, “Estimating the fractal dimension of chaotic time series,” Lincoln Lab. J. 3, 63–85 (1990).

J. Theiler, “Estimating fractal dimension,” J. Opt. Soc. Am. A 7, 1055–1073 (1990).
[CrossRef]

1984 (1)

B. Friedlander, B. Porat, “The modified Yule–Walker method of ARMA spectral estimation,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 158–173 (1984).
[CrossRef]

1983 (1)

P. Grassberger, I. Proccaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[CrossRef]

1976 (1)

1968 (1)

B. B. Mandelbrot, J. V. Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
[CrossRef]

1951 (1)

H. E. Hurst, “Long-term storage capacity of reservoirs,” Trans. Am. Soc. Civil Engineers 116, 770–808 (1951).

Aitken, G

Aitken, G.

M. Jorgenson, G. Aitken, E. Hege, “Evidence of a chaotic attractor in star-wander data,” Opt. Lett. 16, 64–66 (1991).
[CrossRef] [PubMed]

G. Aitken, D. McGaughey, “Predictability of atmospherically-distorted stellar wavefronts” in Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1996), pp. 89–94.

Armitage, A.

A. Armitage, Y. Guillo, F. Meyer, “Progress with a neural network in astronomical control,” in Practical Applications of Neural Networks in Control (IMC Symposium, London, 1994).

Baum, G.

Boyer, C.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Conan, J.-M.

Dainty, J. C.

A. G. R. G. Lane, J. C. Dainty, “Simulation of Kolmogorov phase screens,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Dennis Montera, M. R.

Eubank, S.

J. Theiler, B. Galdrikian, A. Longtin, S. Eubank, J. D. Farmer, “Using surrogate data to detect nonlinear relationships between time series,” in Non-Linear Modeling and Forecasting, Vol. XII of SFI Studies in the Sciences of Complexity (Addison-Wesley, Reading, Mass., 1992).

Farmer, J. D.

J. Theiler, B. Galdrikian, A. Longtin, S. Eubank, J. D. Farmer, “Using surrogate data to detect nonlinear relationships between time series,” in Non-Linear Modeling and Forecasting, Vol. XII of SFI Studies in the Sciences of Complexity (Addison-Wesley, Reading, Mass., 1992).

Fontanella, J. C.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Fried, D. L.

Friedlander, B.

B. Friedlander, B. Porat, “The modified Yule–Walker method of ARMA spectral estimation,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 158–173 (1984).
[CrossRef]

Gaffard, J. P.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Galdrikian, B.

J. Theiler, B. Galdrikian, A. Longtin, S. Eubank, J. D. Farmer, “Using surrogate data to detect nonlinear relationships between time series,” in Non-Linear Modeling and Forecasting, Vol. XII of SFI Studies in the Sciences of Complexity (Addison-Wesley, Reading, Mass., 1992).

Gigan, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, in Statistcial Optics (Wiley, New York, 1985), pp. 429–431.

Grassberger, P.

P. Grassberger, I. Proccaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[CrossRef]

Greenwood, D. P.

Guillo, Y.

A. Armitage, Y. Guillo, F. Meyer, “Progress with a neural network in astronomical control,” in Practical Applications of Neural Networks in Control (IMC Symposium, London, 1994).

Harrington, P. M.

P. M. Harrington, B. M. Welsh, “Frequency-domain analysis of an adaptive optical system’s temporal response,” Opt. Eng. 33, 2336–2342 (1994).
[CrossRef]

Hege, E.

Hurst, H. E.

H. E. Hurst, “Long-term storage capacity of reservoirs,” Trans. Am. Soc. Civil Engineers 116, 770–808 (1951).

Jagourel, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Jorgenson, M.

Kern, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Lane, A. G. R. G.

A. G. R. G. Lane, J. C. Dainty, “Simulation of Kolmogorov phase screens,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Lena, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Lloyd-Hart, M.

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1996), pp. 95–101.

Longtin, A.

J. Theiler, B. Galdrikian, A. Longtin, S. Eubank, J. D. Farmer, “Using surrogate data to detect nonlinear relationships between time series,” in Non-Linear Modeling and Forecasting, Vol. XII of SFI Studies in the Sciences of Complexity (Addison-Wesley, Reading, Mass., 1992).

Madec, P.-Y.

Mandelbrot, B. B.

B. B. Mandelbrot, J. V. Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
[CrossRef]

B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1983), pp. 248–255 and 386–387.

McGaughey, D.

G. Aitken, D. McGaughey, “Predictability of atmospherically-distorted stellar wavefronts” in Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1996), pp. 89–94.

McGuire, P.

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1996), pp. 95–101.

Merkle, F.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Meyer, F.

A. Armitage, Y. Guillo, F. Meyer, “Progress with a neural network in astronomical control,” in Practical Applications of Neural Networks in Control (IMC Symposium, London, 1994).

Ness, J. V.

B. B. Mandelbrot, J. V. Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
[CrossRef]

Peters, E. E.

E. E. Peters, Fractal Market Analysis (Wiley, New York, 1994).

Porat, B.

B. Friedlander, B. Porat, “The modified Yule–Walker method of ARMA spectral estimation,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 158–173 (1984).
[CrossRef]

Proccaccia, I.

P. Grassberger, I. Proccaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[CrossRef]

Ribak, E. N.

Rigaut, F.

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence on optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

Rousset, G.

J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

Ruck, D.

Schwartz, C.

Takens, F.

F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, D. Rank, L. S. Young, eds. (Springer-Verlag, Berlin, 1981), pp. 366–381.

Theiler, J.

J. Theiler, “Some comments on the correlation dimension of 1/fα noise,” Phys. Lett. A 135, 480–493 (1991).
[CrossRef]

J. Theiler, “Estimating the fractal dimension of chaotic time series,” Lincoln Lab. J. 3, 63–85 (1990).

J. Theiler, “Estimating fractal dimension,” J. Opt. Soc. Am. A 7, 1055–1073 (1990).
[CrossRef]

J. Theiler, B. Galdrikian, A. Longtin, S. Eubank, J. D. Farmer, “Using surrogate data to detect nonlinear relationships between time series,” in Non-Linear Modeling and Forecasting, Vol. XII of SFI Studies in the Sciences of Complexity (Addison-Wesley, Reading, Mass., 1992).

Welsh, B.

Welsh, B. M.

P. M. Harrington, B. M. Welsh, “Frequency-domain analysis of an adaptive optical system’s temporal response,” Opt. Eng. 33, 2336–2342 (1994).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Aerosp. Electron. Syst. (1)

B. Friedlander, B. Porat, “The modified Yule–Walker method of ARMA spectral estimation,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 158–173 (1984).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Lincoln Lab. J. (1)

J. Theiler, “Estimating the fractal dimension of chaotic time series,” Lincoln Lab. J. 3, 63–85 (1990).

Opt. Eng. (1)

P. M. Harrington, B. M. Welsh, “Frequency-domain analysis of an adaptive optical system’s temporal response,” Opt. Eng. 33, 2336–2342 (1994).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

J. Theiler, “Some comments on the correlation dimension of 1/fα noise,” Phys. Lett. A 135, 480–493 (1991).
[CrossRef]

Phys. Rev. Lett. (1)

P. Grassberger, I. Proccaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[CrossRef]

SIAM Rev. (1)

B. B. Mandelbrot, J. V. Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
[CrossRef]

Trans. Am. Soc. Civil Engineers (1)

H. E. Hurst, “Long-term storage capacity of reservoirs,” Trans. Am. Soc. Civil Engineers 116, 770–808 (1951).

Waves Random Media (1)

A. G. R. G. Lane, J. C. Dainty, “Simulation of Kolmogorov phase screens,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other (11)

F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, D. Rank, L. S. Young, eds. (Springer-Verlag, Berlin, 1981), pp. 366–381.

J. Theiler, B. Galdrikian, A. Longtin, S. Eubank, J. D. Farmer, “Using surrogate data to detect nonlinear relationships between time series,” in Non-Linear Modeling and Forecasting, Vol. XII of SFI Studies in the Sciences of Complexity (Addison-Wesley, Reading, Mass., 1992).

G. Rousset, J. C. Fontanella, P. Kern, P. Lena, P. Gigan, F. Rigaut, J. P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed. Proc. SPIE1237, 336–344 (1990).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence on optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

H.-O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer-Verlag, New York, 1988).

G. Aitken, D. McGaughey, “Predictability of atmospherically-distorted stellar wavefronts” in Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1996), pp. 89–94.

A. Armitage, Y. Guillo, F. Meyer, “Progress with a neural network in astronomical control,” in Practical Applications of Neural Networks in Control (IMC Symposium, London, 1994).

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1996), pp. 95–101.

B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1983), pp. 248–255 and 386–387.

J. W. Goodman, in Statistcial Optics (Wiley, New York, 1985), pp. 429–431.

E. E. Peters, Fractal Market Analysis (Wiley, New York, 1994).

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Figures (7)

Fig. 1
Fig. 1

Adjusted range. This diagram shows the cumulative sum, E, of a time series divided into segments d samples long. The most negative and positive cumulative departures from the mean are given by min[X(u)] and max[X(u)], respectively. The adjusted range of the function is R(d) = max[X(u)] − min[X(u)].

Fig. 2
Fig. 2

Wave-front slope time series 0303/y lenslet 9 (arbitrary units).

Fig. 3
Fig. 3

Temporal PSD for (a) 0303/y showing the noise floor and (b) 0303/y and (c) 0351/x with the noise floor removed.

Fig. 4
Fig. 4

Average RS analysis for all 20 lenslets for (a) 0303/y and (b) 0351/x.

Fig. 5
Fig. 5

(a) Correlation integrals for (a) 0303/y lenslet 1 and (b) surrogate data of 0303/y lenslet 1. The curves for vector dimensions of 2, 4, 6, 8, 10, 12, 14, and 16 appear from left to right in both graphs.

Fig. 6
Fig. 6

Ideal frequency response to model the aperture filter.

Fig. 7
Fig. 7

Filtered FBM time series with fk = 3 Hz: (a) time series, (b) PSD, (c) RS analysis.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

g ( r , t ) = m g ( r ) * ϕ ( r , t ) .
W g ( v ) = M g ^ ( v ) 2 W ϕ ( v ) ,
w ϕ ( f ) C N 2 d h V ( f V ) - 8 / 3 ,
w ϕ ( f ) C ( f V ) - 2 / 3 ,
w SH ( ν ) { C ( f V ) - 2 / 3 if f κ V D C ( f V ) - 11 / 3 if f > κ V D ,
f k = κ V / D .
[ B ( T + t ) - B ( T ) ] 2 t 2 H
P ( f ) = c f - β ,
β = 2 H + 1.
β = 2 H - 1.
β = 2 H + E .
E = i = 1 d ( i )
X ( u ) = i = 1 u [ ( i ) - d ^ ] ,
d ^ = 1 d i = 1 d ( i ) ,
R ( d ) = max 1 u d { [ i = 1 u ( i ) - u d ^ ] } - min 1 u d { [ i = 1 u ( i ) - u d ^ ] } .
Q ( d ) = R ( d ) S ( d ) ,
Q ( d ) = c d 1 / 2 .
Q ( d ) = c d J .
log [ Q ( d ) ] = J log d + log c .
X 1 ^ = { x ( 1 ) , x ( 1 + τ ) , x ( 1 + 2 τ ) , , x [ 1 + τ ( n - 1 ) ] } , X 2 ^ = { x ( 2 ) , x ( 2 + τ ) , x ( 2 + 2 τ ) , x [ 2 + τ ( n - 1 ) ] } , X M - τ ( n - 1 ) ^ = { x [ M - τ ( n - 1 ) ] , x [ M - τ ( n - 1 ) + τ ] , , x ( M ) } ,
N = m - τ ( n - 1 )
C ( N , r ) = n ( r ) N ,
C ( N , r ) = 2 N ( N - 1 ) i = 1 N j = i + 1 N θ ( r - X i ^ - X j ^ )
C ( N , r ) = r D ,
x ˙ ( i ) = x ( i + 1 ) - x ( i ) .

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