Physiological time series

The physiological time series processed in the previous section clearly show that the complex phenomena supporting life, although they appear to be random, do in fact scale in time. This scaling indicates that the fluctuations that occur on multiple time scales are tied together, and the way we understand such interdependency in the physical sciences is through underlying mechanisms... 

Finally we show that physiologic time series are not mono-fractal, but have a fractal dimension that changes over time. The time series are multifractal, and as such they have a spectrum of dimensions. We review the procedure for constructing the multifractal spectrum and apply the technique to the SRV time series data obtained in our walking experiment [36] as a typical example of physiologic variability. 

The science of complexity, in so far as it can be said to be a science, has relinquished the signal plus noise paradigm for a different perspective. Physiological time series invariably contain fluctuations, so that when sampled N times the data set X7, j — 1,..., N, appears to be a sequence of random points. Examples of such data are the interbeat intervals of the human heart, interstride intervals of human gait, brain wave data from EEGs and interbreath intervals, to name a few. The processing of time series in each of these cases has made use of random walk concepts in both the processing of the data and in the interpretation of the results. So let us review some of what is known about random walks. 

Successive increments of mathematical fractal random processes are independent of the time step. Here D = 1.5 corresponds to a completely uncorrelated random process r = 0, such as Brownian motion, and D = 1.0 corresponds to a completely correlated process r= 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 

Joshua S. Richman, J. Randall Morman, Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol Heart Cite Physiol 278, 2000,pp. H2039-H2049 Pincus S.M., Approximate entropy in cardiology, Herzschr Elektro-phys,2000, pp. 11 139-150... 

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