241.

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		1.5.9— Example Where the Variance Does Not Exist: Volume of Breaths

		Hoop et al. measured the volume of air in each breath of a rat. We used the resealed range analysis developed by Hurst to analyze the values of the volumes of consecutive breaths. The rescaled range R/S is a measure of the variance. We evaluated the rescaled range over time windows of different length T. The length T of the time windows is called the lag. The values of the lag set the temporal resolution. Thus we determined how the rescaled range depended on the temporal resolution at which it was measured.

		A fractal set of values has small variations in the values over brief times that look similar to larger variations in the values over longer times. All the past values affect the future values. The rescaled range R/S of a fractal time series will have the power law scaling relationship that R/S is proportional to TH, where H is called the Hurst exponent.

		When H = 1/2, the differences between consecutive values are uncorrelated. When 1/2 < H < 1, the differences between consecutive values are said to be persistent. This means that increases at one time are more likely to be followed by increases at all later times, and decreases at one time are more likely to be followed by decreases at all later times. When 0 < H < 1/2, the differences between consecutive values are said to be anti-persistent. This means that increases at one time are more likely to be followed by decreases at all later times, and decreases at one time are more likely to be followed by increases at all later times.

		We found that the data were a straight line on a plot of Log (R/S) versus Log (T). This means that the rescaled range R/S of the volumes of the breaths was proportional to TH. Thus this power law scaling indicates that the sequence of volumes of these breaths is fractal. The Hurst exponent H was greater than 1/2. This means that there are persistent correlations in the volumes of the sequence of breaths. The discovery of the fractal nature of these volumes and their positive correlations may provide information about the feedback mechanisms that control the amount of air in each breath.

 

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