Method and apparatus for detecting nonlinearity and chaos in a dynamical system

• US 5,938,594 A
• Filed: 05/13/1998
• Issued: 08/17/1999
• Est. Priority Date: 05/14/1996
• Status: Expired due to Term

• Current Assignee: Massachusetts Institute of Technology
• Original Assignee: Massachusetts Institute of Technology
• Inventors: Barahona, Mauricio, Poon, Chi-Sang
• Primary Examiner(s): Kamm, William E.
• Application Number:
  US09/078,122
• Time in Patent Office: 461 Days
• Field of Search: 600/300, 600/509, 600/518, 600/519
• US Class Current: 600/300
• CPC Class Codes: G06F 17/18 for evaluating statistical data , e.g. average values, frequency distributions, probability functions, regression analysis forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04
• CPC Class Codes: G06F 2218/00 Aspects of pattern recognition specially adapted for signal processing

Abstract

Methods and apparatus are provided for detecting the presence of a nonlinear characteristic of an autonomous (i.e., non-driven and time-invariant), dynamical system and for determining whether such nonlinear dynamical system is chaotic. First, a system is determined to be either nonlinear or linear. If the system is determined to be nonlinear, then noise of increasing intensity is incrementally added to a data set representing the analyzed system until the resulting test signal appears to be linear. If the noise limit of the resulting test signal is significantly greater than zero, then the system is determined to be chaotic and the amount of noise added to the data set provides an indication of the relative strength of the chaos. Alternatively, if the noise limit of the resulting test signal is approximately zero, then the system is determined to be nonlinear with periodic or quasi-periodic limit cycles. An optional power spectrum test is described with which it can be confirmed that the system is nonlinear with periodic or quasi-periodic limit cycles.

Claims

1. A method for analyzing a biophysical signal to determine the presence of a nonlinear pattern in the biophysical signal, the method comprising the steps of:

(a) converting a series of intervals in the biophysical signal into time segments, each time segment having a value, with a sequence of such values representing the biophysical signal;

(b) determining if a nonlinear model predicts a value of the biophysical signal;

(c) in response to the nonlinear model predicting a value of the biophysical signal, adding a noise signal to the biophysical signal to provide a first test biophysical signal;

(d) determining if the nonlinear model predicts a value of the first test biophysical signal;

(e) in response to the nonlinear model predicting a value of the first test biophysical signal, adding a next noise signal to the first test biophysical signal to provide a next test biophysical signal; and

(f) repeating steps (d) and (e) until the nonlinear model no longer predicts a value of the next test biophysical signal.

2. The method of claim 1 further comprising the steps of:

(g) determining if a noise limit of the next test biophysical signal is greater than zero; and

(h) in response to the noise limit of the next test biophysical signal being greater than zero, providing an output signal indicating that the biophysical signal is chaotic.

3. The method of claim 2 further comprising the step of:

(i) in response to the noise limit of the next test biophysical signal being greater than zero, providing an output signal indicating the intensity of the chaos of the biophysical signal in response to the noise level of the next noise signal.

4. The method of claim 1 wherein the step of determining if the nonlinear model predicts a value of the first test biophysical signal comprises the steps of:

(i) generating a linear model to represent a first sequence of values of the first test biophysical signal;

(ii) generating a nonlinear model to represent the first sequence of values of the first test biophysical signal;

(iii) computing a performance measure of the linear model;

(iv) computing a performance measure of the nonlinear model;

(v) comparing the performance measure of the linear model to the performance measure of the nonlinear model; and

(vi) in response to the performance measure of the nonlinear model being a preferred performance measure, indicating that the nonlinear model predicts a value of the first test biophysical signal.

5. The method of claim 1 wherein the step of determining if a nonlinear model predicts a value of the biophysical signal comprises the steps of:

(i) generating a linear model to represent a first sequence of values of the biophysical signal;

(ii) generating a nonlinear model to represent the first sequence of values of the biophysical signal;

(iii) computing a performance measure of the linear model;

(iv) computing a performance measure of the nonlinear model;

(v) comparing the performance measure of the linear model to the performance measure of the nonlinear model; and

(vi) in response to the performance measure of the nonlinear model being a preferred performance measure, indicating that the nonlinear model predicts a value of the biophysical signal.

6. A method for detecting a chaotic signal component in a nonlinear pattern of a biosignal, comprising the steps of:

(a) adding a noise signal to the biosignal to provide a test biosignal;

(b) determining if a nonlinear model predicts a value of the test biosignal;

(c) in response to the nonlinear model predicting the value of the test biosignal, adding a next noise signal to the test biosignal to provide a next test biosignal; and

(d) repeating steps (a)-(c) until the nonlinear model no longer predicts a value of the next test biosignal.

7. The method of claim 6 further comprising the steps of:

(e) determining if a noise limit of the next test biosignal is greater than zero; and

(f) in response to the noise limit of the next test biosignal being greater than zero, providing an output signal indicating that the biosignal is chaotic.

8. The method of claim 7 further comprising the step of:

(g) in response to the noise limit of the next test biosignal being greater than zero, providing an output signal indicating the intensity of the chaos exhibited by the biosignal in response to the noise level of the next noise signal.

9. The method of claim 6 wherein the step of determining if a nonlinear model predicts a value in the test biosignal comprises the steps of:

(i) generating a linear model to represent a first sequence of values corresponding to measured intervals of the test biosignal;

(ii) generating a nonlinear model to represent the first sequence of values corresponding to measured intervals of the test biosignal;

(iii) computing a performance measure of the linear model;

(iv) computing a performance measure of the nonlinear model; and

(v) comparing the performance measure of the linear model to the performance measure of the nonlinear model.

10. The method of claim 9 further comprising the steps of:

(vi) identifying which of the first and second performance measures is a preferred performance measure; and

(vii) providing an output signal indicating whether the nonlinear model predicts a value in the test biosignal in response to which of the first and second performance measures is the preferred performance measure.

11. A method for detecting a chaotic signal component in a nonlinear pattern of a biosignal, comprising the steps of:

(a) adding a noise signal to the biosignal to provide a test biosignal;

(b) determining if a nonlinear model predicts a value in the test biosignal;

(c) in response to the nonlinear model not predicting a value in the test biosignal, determining if a noise limit of the test biosignal is greater than zero; and

(d) in response to the noise limit of the test biosignal being greater than zero, providing an output signal indicating that the biosignal is chaotic.

12. The method of claim 11 further comprising the step of:

(e) in response to the noise limit of the test biosignal being greater than zero, providing an output signal indicating the intensity of the chaos in response to the noise level of the noise signal.

13. The method of claim 11 further comprising the steps of:

(e) in response to the nonlinear model predicting a value in the test biosignal, adding a next noise signal to the test biosignal to provide a next test biosignal; and

(f) repeating steps (b)-(e) until the nonlinear model no longer predicts a value of the next test biosignal.

14. The method of claim 11 further comprising the steps of:

(e) computing the power spectrum of the test biosignal;

(f) determining if there are any discernible frequency peaks in the power spectrum of the test biosignal; and

(g) in response to there being no discernible frequency peaks in the power spectrum of the test biosignal, indicating that the biosignal is chaotic.

15. Apparatus for detecting a nonlinear component in an autonomous, dynamical system and for determining whether the system determined to include a nonlinear component is chaotic, comprising:

(a) a modeling system comprising;

(i) a modeling processor for representing a signal from the dynamical system with a linear model and a nonlinear model; and

(ii) a performance processor for computing a first performance measure of the linear model and for computing a second performance measure of the nonlinear model; and

(b) a performance measurement system coupled to the modeling system, said performance measurement system comprising;

(i) a comparison processor for receiving and comparing the first and second performance measures;

(ii) a selection processor for identifying which of the first and second performance measures is a preferred performance measure and for providing an output signal indicating whether the signal from the dynamical system includes a nonlinear component based on whether the preferred performance measure is associated with the linear model or the nonlinear model;

(iii) a noise processor for iteratively adding noise of predetermined intensity to the signal from the dynamical system in response to the output signal indicating that the signal from the dynamical system includes a nonlinear component in order to generate a test signal until the test signal appears linear; and

(iv) a noise limit comparator for determining whether the noise limit of the test signal is greater than zero and for providing an output signal indicating that the dynamical system is chaotic in response to the noise limit being greater than zero.

16. The apparatus of claim 15 wherein the performance measurement system further comprises:

(v) a power spectrum processor for computing the power spectrum of the signal representing the dynamical system in response to the output signal indicating that the signal from the dynamical system includes a nonlinear component; and

(vi) a frequency peak detector for detecting whether the power spectrum includes discernible frequency peaks and for providing an output signal indicating that the dynamical system is not chaotic in response to detection of discernible frequency peaks.

17. A method for detecting a chaotic signal component in a nonlinear pattern of a signal generated by an autonomous, dynamical system, comprising the steps of:

(a) adding a noise signal to the signal to provide a test signal;

(b) determining if a nonlinear model predicts a value of the test signal;

(c) in response to the nonlinear model predicting the next value of the test signal, adding a next noise signal to the test signal to provide a next test signal; and

(d) repeating steps (a)-(c) until the nonlinear model no longer predicts a value of the next test signal.