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

David R. Bickel, M. Terese Verklan, Jon Moon

Research output: Contribution to journalArticle

11 Citations (Scopus)

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/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.

Original languageEnglish (US)
Pages (from-to)6440-6448
Number of pages9
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume58
Issue number5 B
StatePublished - Nov 1998
Externally publishedYes

Fingerprint

Heart Rate Variability
heart rate
Anomalous Diffusion
Scaling Exponent
Confidence interval
confidence
exponents
intervals
scaling
Increment
Exponent
Heart Rate
Scaling
Standard error
Logarithm
Mean Square
regression analysis
Alternate hypothesis
Power Law
Time series

ASJC Scopus subject areas

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

Cite this

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title = "Detection of anomalous diffusion using confidence intervals of the scaling exponent with application to preterm neonatal heart rate variability",
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/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.",
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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.

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