Surrogate Hypothesis Tests - Calibrated Mutual Information
Are nonlinear signal processing methods ever really justified? Given the elegance and ubiquity of linear signal processing techniques, this is often a good question. When faced with a signal analysis problem, is it better to use a classical, linear technique that is simple and well-understood, or use something more complex? Surrogate data techniques are a formalized statistical approach to addressing this question. In the classical linear signal processing approach, if it is formally assumed that the signal can be represented as a stationary, linear Gaussian random process, this makes it possible to find the linear prediction (autoregression) coefficients by simple matrix inversion [1].
Given a real signal, does it actually support these assumptions? Surrogate data techniques can be formulated
to test the null hypothesis that the data has been generated by a (short-time) stationary, ergodic, Gaussian
linear random process. The desire is then to obtain sufficient significance
that the null hypothesis can be rejected, achieved by generating an appropriate number of surrogates using,
for example, the Fourier transform method (AAFT) [2].
Using this method, the surrogates should have the same power spectrum as the original signal,
yet have only linear, Gaussian statistical dependencies at different time lags.
There are however many pitfalls to trap the unwary with these simple surrogate techniques [3]. We have produced a new and very sensitive surrogate data method for testing the null hypothesis of a linear Gaussian random process, that circumvents many of the problems of previous methods that make them unreliable [3] by use of calibrated time-delayed mutual information and analytical entropy calculations [4] (see diagram above). With this test it is possible to rigorously check that the surrogates are purely linear as well as iron out the misleading effects of arbitrary parameters. See the software pages for code that implements these ideas.
[1] J.G. Proakis, D.G. Manolakis (1996), Digital signal processing : principles, algorithms, and applications. 3rd ed, Upper Saddle River, N.J.: Prentice Hall. 1 v. (various pagings).
[2] T. Schreiber, A. Schmitz (2000), Surrogate time series. Physica D-Nonlinear Phenomena, 142(3-4): pp. 346-382.
[3] D. Kugiumtzis (2001), On the reliability of the surrogate data test for nonlinearity in the analysis of noisy time series. International Journal of Bifurcation and Chaos, 11(7): pp. 1881-1896.
[4] M.A. Little, P.E. McSharry, I.M. Moroz, S.J. Roberts (2006), Testing the assumptions of linear prediction analysis in normal vowels. Journal of the Acoustical Society of America, 119(1): pp. 549-558.