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 et 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.

[1]  Eckmann,et al.  Liapunov exponents from time series. , 1986, Physical review. A, General physics.

[2]  Chi-Sang Poon,et al.  Decrease of cardiac chaos in congestive heart failure , 1997, Nature.

[3]  L. A. Aguirre,et al.  Validating Identified Nonlinear Models with Chaotic Dynamics , 1994 .

[4]  J E Skinner,et al.  Correlation dimension of heartbeat intervals is reduced in conscious pigs by myocardial ischemia. , 1991, Circulation research.

[5]  D. J. Marsh,et al.  Evidence of low dimensional chaos in renal blood flow control in genetic and experimental hypertension , 1995 .

[6]  Stephen A. Billings,et al.  Identification of models for chaotic systems from noisy data: implications for performance and nonlinear filtering , 1995 .

[7]  D J Marsh,et al.  Renal blood flow regulation and arterial pressure fluctuations: a case study in nonlinear dynamics. , 1994, Physiological reviews.

[8]  Giona,et al.  Functional reconstruction and local prediction of chaotic time series. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[9]  Ulrich Parlitz,et al.  Comparison of Different Methods for Computing Lyapunov Exponents , 1990 .

[10]  Clemens,et al.  Influence of roughness distributions and correlations on x-ray diffraction from superlattices. , 1993, Physical review. B, Condensed matter.

[11]  A. Provenzale,et al.  Finite correlation dimension for stochastic systems with power-law spectra , 1989 .

[12]  Alan A. Desrochers On an improved model reduction technique for nonlinear systems , 1981, Autom..

[13]  Leonard A. Smith,et al.  Distinguishing between low-dimensional dynamics and randomness in measured time series , 1992 .

[14]  N H Holstein-Rathlou,et al.  Chaos and non-linear phenomena in renal vascular control. , 1996, Cardiovascular research.

[15]  H. Akaike A new look at the statistical model identification , 1974 .

[16]  Serre,et al.  A choatic pulsating star: The case of R Scuti. , 1995, Physical review letters.

[17]  R. Hughson,et al.  Coarse-graining spectral analysis: new method for studying heart rate variability. , 1991, Journal of applied physiology.

[18]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[19]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[20]  Stephen P. Ellner,et al.  Chaos in a Noisy World: New Methods and Evidence from Time-Series Analysis , 1995, The American Naturalist.

[21]  R J Cohen,et al.  Detection of chaotic determinism in time series from randomly forced maps. , 1997, Physica D. Nonlinear phenomena.

[22]  S. Billings,et al.  A prediction-error and stepwise-regression estimation algorithm for non-linear systems , 1986 .

[23]  J E Skinner,et al.  A reduction in the correlation dimension of heartbeat intervals precedes imminent ventricular fibrillation in human subjects. , 1993, American heart journal.