| | | | | | 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. | | | | | | | | | 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. | | | | |