54.

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

Page 150

		2.2.11— The Fractal Dimension Is Not Equal to the Fractal Dimension

		Unfortunately the same words, "the fractal dimension of the time series" have been used to refer to two different concepts.

		Each of these "fractal dimensions" is a different thing and they are not equal to each other.

		1— Fractal Dimension of the Time Series Itself

		This fractal dimension characterizes the self-similarity of the time series itself.

		The value of this fractal dimension describes how the small variations in the values measured over brief time intervals are related to the large variations measured over long time intervals.

		For example, it can be computed by using the box counting algorithm to analyze the plot of the time series values x(t) plotted versus the time t.

		2— Fractal Dimension of the Phase Space Set Constructed from the Time Series

		This fractal dimension characterizes the properties of the phase space set, not the original time series.

		The value of this fractal dimension tells us the number of independent variables needed to generate the time series from which the phase space was constructed.

		For example, it can be computed by using the box counting algorithm to analyze the phase space set constructed from the points with the coordinates x(t), x(t+Dt), x(t+2Dt), . . . , x(t+(N-1) Dt), where x(t) are the time series values, Dt is the lag, and N is the dimension of the phase space.

 

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