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		2.2.4— Random Is High Dimensional; Chaos Is Low Dimensional

		The dimension of the phase space set reveals whether a random-looking set of data was generated by a random or by a deterministic mechanism. The dimension determines the number of equations and independent variables needed to generate the data. When the dimension is infinite, the data were generated by a random mechanism. When the dimension is small, the data were generated by a deterministic mechanism.

		(There is no rigorous limit on what value of the dimension corresponds to "small." In practice, it is often difficult to tell whether the dimension of the phase space set is finite or infinite. Depending on the amount of data, the largest finite values that can be detected are typically between 3 and 6. This suggests that "small" means a value less than about 6.)

		1— Random: Infinite Dimensional Phase Space Set

		When the dimension of the phase space set is very large, then the sequence of values measured in time was generated by a random mechanism based on chance. That is, the number of equations and variables is so large that there are no rules that can be used to predict the future values from the past values.

		What we mean by random is that so many different things are happening at once that it is not possible to understand how these mechanisms work, and thus we say that the future values are determined by chance.

		2— Deterministic Chaos: Low Dimensional Phase Space Set

		When the dimension of the phase space set is small, then the sequence of values measured in time was generated by a deterministic mechanism based on a small number of independent variables.

		What we mean by determinism is that a small number of equations and their variables describe how the values in the past can be used to compute the values in the future.

 

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