Numerical algorithms are discussed for estimating dimensional complexity of an observed chaotic time series. Factors that enter the procedure are discussed and applied to a local estimate of the pointwise dimension or crowding index. The concepts are illustrated with the help of experimental time series obtained from speech signals. The temporal evolution of the crowding index shows oscillations that can be correlated with properties of the time series. In a mechanical context, think of a nonlinear oscillator which is kicked periodically. Because of the nonlinearity of the oscillators many modes will be excited and therefore the dimensionality of the signal will be large. Since the specific shape of the nonlinearities will determine the rate at which different modes decay; a monotonic decrease will be observed in dimensionality until only the resonant modes will still be active at small amplitudes before the next kick. Instead of observing the specific dynamics of the system, the dimensional complexity of the system is considered as a new dynamical variable. The time evolution of the dimensional complexity parameter is compared with the original time series and also with recurrence plots of the embedded time series.
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