7.

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		1.5.16— Biological Implications of the Statistical Properties of Fractals (Continued)

		5— "Nonstationary" Does Not Mean That Things Are Changing

		When the measurements of the moments keep changing as more data are included, then a process is said to be "nonstationary. This is a poor choice of a word, because it suggests that the process is changing in time. This is not necessarily true. A generating mechanism whose operation is fixed in time can produce a fractal output whose moments keep changing in time. When the moments of the data are "nonstationary," this does not necessarily mean that the mechanism that produced the data is changing in time.

		6— When the Variance Looks Big, Maybe It's Infinite

		Sometimes there is a large variability in a property measured in repetitions of an experiment. For example, such variability is present in the number of mutant cells in evolution and in the time durations between the steps of growth and division (cell cycle time) in cancer cells. Scientists analyzing this kind of data who are familiar only with Gaussian statistics usually assume that the data were produced by a process with a finite variance. Any time such data with large variability are found, it may be worthwhile to determine if the variance does, or does not, have a finite, limiting value. This can be done by measuring how the variance depends on the amount of data included. If the variance increases with the amount of data included, then the data have fractal properties and the variance does not exist.

		7— Importance of the Scaling Relationship

		When the moments, such as the mean and variance, do not exist, their values will depend on the resolution used to measure them. Thus the measurement of the moments at one resolution is not meaningful. What is meaningful is to determine how the moments depend on the resolution used to measure them. The form of this dependency is called the scaling relationship, and it is characterized by the parameter called the fractal dimension.

 

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