Fractals and Chaos Simplified for the Life Sciences - page 2

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[Cover] [Abbreviated Contents] [Contents] [Index]
[Cover] [Abbreviated Contents] [Contents] [Index]

Page 1
PART I—
FRACTALS
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Objects or processes whose small pieces
resemble the whole.
1
Introduction
3
2
Self-Similarity
11
3
Scaling
27
4
Dimension
45
5
Statistical Properties
73
6
Summary
107

 
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[Cover] [Abbreviated Contents] [Contents] [Index]

Page 100
1.5.14—
More Examples of the Statistical Properties of Fractals
We have already seen that the statistical properties of fractals are present in the timing of the action potentials recorded from nerves that carry information about sound that we use for hearing, in the spatial distribution of blood flow in the muscle of the heart, in the sequence of the volumes of breaths, in the number of mutated cells produced by Darwinian evolution, in the timing of the electrical activity of the heartbeat, and in the spatial distribution of radioactive tracer in the liver.
Additional examples of these statistical properties include the changing electrical voltage across the cell membrane of white blood cells (T-lymphocytes), the sequence of base pairs in DNA where each base pair is assigned a number and the running sum of these numbers is analyzed, and the duration in time of consecutive breaths.

 
[Cover] [Abbreviated Contents] [Contents] [Index]
[Cover] [Abbreviated Contents] [Contents] [Index]

Page 101
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Page 102
1.5.15—
Biological Implications of the Statistical Properties of Fractals
1—
Not Gaussian or Asymptotically Gaussian
The statistical knowledge of most scientists is limited to the statistical properties of Gaussian distributions. Fractals do not have the properties of Gaussian distributions. In order to understand the many fractal objects and processes in the natural world, it is required to learn about the properties of stable distributions. Stable distributions are more general than Gaussian distributions.
2—
The Average and Variance Do Not Exist
The moments of a fractal, such as the mean and variance, do not exist. As more data are included, the measurements of these moments do not approach finite, limiting values.
3—
Large Variations
The variance of a fractal increases as more data are analyzed. The average values measured for the properties of the data will have wide variation from one time to another and among repetitions of the same experiment.
4—
When are these large variations significant?
When the variance of a fractal increases as more data are analyzed, we do not know how to perform statistical tests to determine if the parameters of the mechanism that generated the data have changed from one time to another or between experiments run under different conditions.
The statistical tests taught in the usual statistics courses are based on the assumption that the variance is finite. These tests are not valid to analyze fractal data where the variance is infinite. It would be very worthwhile for mathematicians to formulate statistical tests for fractal distributions where the variance is infinite.

 
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