### Abstract

The scaling exponent of the root mean square (rms) displacement quantifies the roughness of fractal or multifractal time series; it is equivalent to other second-order measures of scaling, such as the power-law exponents of the spectral density and autocorrelation function. For self-similar time series, the rms scaling exponent equals the Hurst parameter, which is related to the fractal dimension. A scaling exponent of 0.5 implies that the process is normal diffusion, which is equivalent to an uncorrelated random walk; otherwise, the process can be modeled as anomalous diffusion. Higher exponents indicate that the increments of the signal have positive correlations, while exponents below 0.5 imply that they have negative correlations. Scaling exponent estimates of successive segments of the increments of a signal are used to test the null hypothesis that the signal is normal diffusion, with the alternate hypothesis that the diffusion is anomalous. Dispersional analysis, a simple technique which does not require long signals, is used to estimate the scaling exponent from the slope of the linear regression of the logarithm of the standard deviation of binned data points on the logarithm of the number of points per bin. Computing the standard error of the scaling exponent using successive segments of the signal is superior to previous methods of obtaining the standard error, such as that based on the sum of squared errors used in the regression; the regression error is more of a measure of the deviation from power-law scaling than of the uncertainty of the scaling exponent estimate. Applying this test to preterm neonate heart rate data, it is found that time intervals between heart beats can be modeled as anomalous diffusion with negatively correlated increments. This corresponds to power spectra between 1/f^{2} and 1/f, whereas healthy adults are usually reported to have 1/f spectra, suggesting that the immaturity of the neonatal nervous system affects the scaling properties of the heart rate.

Original language | English (US) |
---|---|

Pages (from-to) | 6440-6448 |

Number of pages | 9 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 58 |

Issue number | 5 B |

State | Published - Nov 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics

### Cite this

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*58*(5 B), 6440-6448.

**Detection of anomalous diffusion using confidence intervals of the scaling exponent with application to preterm neonatal heart rate variability.** / Bickel, David R.; Verklan, M. Terese; Moon, Jon.

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 58, no. 5 B, pp. 6440-6448.

}

TY - JOUR

T1 - Detection of anomalous diffusion using confidence intervals of the scaling exponent with application to preterm neonatal heart rate variability

AU - Bickel, David R.

AU - Verklan, M. Terese

AU - Moon, Jon

PY - 1998/11

Y1 - 1998/11

N2 - The scaling exponent of the root mean square (rms) displacement quantifies the roughness of fractal or multifractal time series; it is equivalent to other second-order measures of scaling, such as the power-law exponents of the spectral density and autocorrelation function. For self-similar time series, the rms scaling exponent equals the Hurst parameter, which is related to the fractal dimension. A scaling exponent of 0.5 implies that the process is normal diffusion, which is equivalent to an uncorrelated random walk; otherwise, the process can be modeled as anomalous diffusion. Higher exponents indicate that the increments of the signal have positive correlations, while exponents below 0.5 imply that they have negative correlations. Scaling exponent estimates of successive segments of the increments of a signal are used to test the null hypothesis that the signal is normal diffusion, with the alternate hypothesis that the diffusion is anomalous. Dispersional analysis, a simple technique which does not require long signals, is used to estimate the scaling exponent from the slope of the linear regression of the logarithm of the standard deviation of binned data points on the logarithm of the number of points per bin. Computing the standard error of the scaling exponent using successive segments of the signal is superior to previous methods of obtaining the standard error, such as that based on the sum of squared errors used in the regression; the regression error is more of a measure of the deviation from power-law scaling than of the uncertainty of the scaling exponent estimate. Applying this test to preterm neonate heart rate data, it is found that time intervals between heart beats can be modeled as anomalous diffusion with negatively correlated increments. This corresponds to power spectra between 1/f2 and 1/f, whereas healthy adults are usually reported to have 1/f spectra, suggesting that the immaturity of the neonatal nervous system affects the scaling properties of the heart rate.

AB - The scaling exponent of the root mean square (rms) displacement quantifies the roughness of fractal or multifractal time series; it is equivalent to other second-order measures of scaling, such as the power-law exponents of the spectral density and autocorrelation function. For self-similar time series, the rms scaling exponent equals the Hurst parameter, which is related to the fractal dimension. A scaling exponent of 0.5 implies that the process is normal diffusion, which is equivalent to an uncorrelated random walk; otherwise, the process can be modeled as anomalous diffusion. Higher exponents indicate that the increments of the signal have positive correlations, while exponents below 0.5 imply that they have negative correlations. Scaling exponent estimates of successive segments of the increments of a signal are used to test the null hypothesis that the signal is normal diffusion, with the alternate hypothesis that the diffusion is anomalous. Dispersional analysis, a simple technique which does not require long signals, is used to estimate the scaling exponent from the slope of the linear regression of the logarithm of the standard deviation of binned data points on the logarithm of the number of points per bin. Computing the standard error of the scaling exponent using successive segments of the signal is superior to previous methods of obtaining the standard error, such as that based on the sum of squared errors used in the regression; the regression error is more of a measure of the deviation from power-law scaling than of the uncertainty of the scaling exponent estimate. Applying this test to preterm neonate heart rate data, it is found that time intervals between heart beats can be modeled as anomalous diffusion with negatively correlated increments. This corresponds to power spectra between 1/f2 and 1/f, whereas healthy adults are usually reported to have 1/f spectra, suggesting that the immaturity of the neonatal nervous system affects the scaling properties of the heart rate.

UR - http://www.scopus.com/inward/record.url?scp=0000917173&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000917173&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0000917173

VL - 58

SP - 6440

EP - 6448

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 5 B

ER -