Modeling nonlinear determinism in short time series from noise driven discrete and continuous systems

Current methods for detecting deterministic chaos in a time series require long, stationary, and relatively noise-free data records. This limits the utility of these methods in most experimental and clinical settings. Recently we presented a new method for detecting determinism in a time series, and for assessing whether this determinism has chaotic attributes, i.e. sensitivity to initial conditions. The method is based on fitting a deterministic nonlinear autoregressive (NAR) model to the data [Chon ct al., 1997]. This approach assumes that the noise in the model can be represented as a series of independent, identically distributed random variables. If this is not the case the accuracy of the algorithm may be compromised. To explicitly deal with the possibility of more complex noise structures, we present a method based on a stochastic NAR model. The method iteratively estimates NAR models for both the deterministic and the stochastic parts of the signal. An additional feature of the algorithm is that it includes only the significant autoregressive terms among the pool of candidate terms searched. As a result the algorithm results in a model with significantly fewer terms than a model obtained by traditional model order search criterions. Subsequently, Lyapunov exponents are calculated for the estimated models to examine if chaotic determinism (i.e, sensitivity to initial conditions) is present in the time series. The major advantages of this algorithm are: (1) it provides accurate parameter estimation with a small number of data points, (2) it is accurate for signal-to-noise ratios as low as -9 dB for discrete and -6 dB for continuous chaotic systems, and (3) it allows characterization of the dynamics of the system, and thus prediction of future states of the system, over short time scales. The stochastic NAR model is applied to renal tubular pressure data from normotensive and hypertensive rats. One form of hypertension was genetic, and the other was induced on normotensive rats by placing a restricting clip on one of their renal arteries. In both types of hypertensive rats, positive Lyapunov exponents were present, indicating that the fluctuations observed in the proximal tubular pressure were due to the operation of a system with chaotic determinism. In contrast, only negative exponents were found in the time series from normotensive rats.