45.

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		2.2.7— Example of Phase Space Sets Constructed from the Measurement of One Variable

		Takens' theorem can be used to construct the phase space set from the measurement of the series of values of one variable x(t) in time t. The points in the N-dimensional phase space set have coordinates x(t), x(t+Dt), x(t+2Dt), . . . , x(t+(N-1) Dt). A series of such N-dimensional phase spaces are constructed with increasing N. If the dimension of the phase space set increases with increasing N, then the series of values x(t) was generated by a random mechanism. If the dimension of the phase space set reaches a constant value with increasing N, then the series of values x(t) was generated by a deterministic mechanism.

		For example, this procedure was used to analyze the time series x(n) in Data Set #1 that was generated by the random mechanism of choosing the value of x(n) at random and Data Set #2 that was generated by the deterministic mechanism that x(n+l) = 3.95 x(n) [1-x(n)]. The lag Dt was set equal to the time between consecutive points. The 2-dimensional phase space set was constructed from points with coordinates X=x(n) and Y=x(n+l). The 3-dimensional phase space set was constructed from points with coordinates X=x(n), Y=x(n+ 1), Z=x(n+2). And so on.

		1— Data Set #1: Random

		The fractal dimension of the phase space set increases as the embedding dimension increases. That is, the fractal dimension is infinite.

		Thus this time series was generated by a random mechanism. That is, it was generated by a mechanism with an infinite set of independent variables. This is what we mean by random, that there is a very large number of different things happening at once.

		2— Data Set #2: Deterministic Chaos

		The fractal dimension of the phase space set reaches a limiting value slightly less than 1 as the embedding dimension increases.

		Thus, this time series was generated by a deterministic rule that can be described by 1 equation with 1 independent variable.

 

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