Multiscale Entropy Analysis: A New Method to Detect Determinism in a Time. Series. A. Sarkar and P. Barat. Variable Energy Cyclotron Centre

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1 Multiscale Entropy Analysis: A New Method to Detect Deterinis in a Tie Series A. Sarkar and P. Barat Variable Energy Cyclotron Centre /AF Bidhan Nagar, Kolkata , India PACS nubers: Tp, k, Bj Abstract In this letter we show that the Multiscale Entropy (MSE) analysis can detect the deterinis in a tie series.

2 The output variables (tie series) fro physical systes often exhibit coplex fluctuations containing inforation about the underlying dynaics. An iportant proble in the study of a tie series is deterining whether the tie series arises fro a stochastic process or has a deterinistic coponent that is generated fro chaotic dynaics having finite nuber of degrees of freedo. Whether a tie series has a deterinistic coponent or not in turn dictates what approaches are appropriate for investigating the tie series and its generating process. In this sense detecting the deterinis in a tie series is very iportant. Several ethods of nonlinear dynaical analysis have previously been developed to detect deterinis in tie series [-3]. These ethods are all based on the assuption that a trajectory in the state space reconstructed fro a deterinistic tie series behaves siilarly to nearby trajectories as tie evolves. Hence, a large nuber of data points are required to have sufficient inforation of the nearby trajectories to copare their future behaviors. In addition, the application of these ethods can lead to spurious results for nonstationary tie series. Recently Costa et al. [4] introduced a new ethod, Multiscale Entropy (MSE) analysis for easuring the coplexity of finite length tie series. In this paper we show that the MSE ethod can be used to detect the deterinis in a tie series. The MSE ethod easures coplexity taking into account the ultiple tie scales. This coputational tool can be quite effectively used to quantify the coplexity of a natural tie series. The MSE ethod uses Saple Entropy (SapEn) [5] to quantify the regularity of finite length tie series. SapEn is largely independent of the tie series length when the total nuber of data points is larger than approxiately 750 [5]. 2

3 Recently MSE has been successfully applied to quantify the coplexity of any Physiologic and Biological signals [6, 7]. The MSE ethod is based on the evaluation of SapEn on the ultiple scales. The prescription of the MSE analysis is: given a one-diensional discrete tie series, {x,...,x i,...,x N }, construct the consecutive coarse-grained tie series, { y (τ ) } by the scale factor, τ, according to the equation: jτ τ y j = / τ x i () i= ( j ) τ +, deterined where τ represents the scale factor and j N / τ. The length of each coarse-grained tie series is N / τ. For scale one, the coarse-grained tie series is siply the original tie series. Next we calculate the SapEn for each scale using the following ethod. Let { X } { x x,..., x } u i =,..., i N be a tie series of length N. { x, x,..., x }, i N ( i) = i i+ i+ be vectors of length. Let n i (r) represent the nuber of vectors u (j) within distance r of u (i), where j ranges fro to (N-) and j i to exclude the self atches. Ci ( r) = ni ( r) /( N ) is the probability that any u (j) is within r of u (i).we then define U N i= ( r) = /( N ) ln C ( r) (2) i The paraeter Saple Entropy (SapEn) [5] is defined as SapEn(, r) U li ln N U ( r) ( r + = ) (3) For finite length N the SapEn is estiated by the statistics 3

4 + U ( r) SapEn(, r, N) = ln (4) U ( r) Advantage of SapEn is that it is less dependent on tie series length and is relatively consistent over broad range of possible r, and N values. We have calculated SapEn for all the studied data sets with the paraeters =2 and r= 0.5äSD (SD is the standard deviation of the original tie series). Costa et al. had tested the MSE ethod on siulated white and /f noises [4]. They have shown that for the scale one, the value of entropy is higher for the white noise tie series in coparison to the /f noise. This ay apparently lead to the conclusion that the inherent coplexity is ore in the white noise in coparison to the /f noise. However, the application of the MSE ethod shows that the value of the entropy for the /f noise reains alost invariant for all the scales while the value of entropy for the white noise tie series onotonically decreases and for scales greater than 5, it becoes saller than the corresponding values for the /f noise. This result explains the fact that the /f noise contains coplex structures across ultiple scales in contrast to the white noise. With a view to understand the coplexity of deterinistic chaotic data we have applied the MSE ethod to the following synthetic chaotic data sets.. Logistic Map: xn+ = axn ( xn ) a= Henon Map: x y n+ n+ = = βx αx n 2 n + y n α=.4, β=0.3 4

5 3. Ikeda Map: ib xn+ = + cxn exp( ia ) a=0.4, b=6.0, c=0.9 + x n 4. Quadratic Map: x + = p x p= n n 5. Rossler Equation: dx = y z dy = x + ay dz = b + z( x c) a=0.2, b=0.2, c= Lorentz Equation: dx = ax + ay dy = bx y xz dz = cz + xy a=0, b=28, c=8/3 The result of the MSE analysis on the chaotic data sets together with the white noise, fractional Brownian noise (with Hurst exponent 0.7) [8] and the /f noise is shown in Fig.. It is seen that the entropy easure for the deterinistic chaotic tie series increases on sall scales and then gradually decreases indicating the reduction of coplexity on the larger scales. This trend of the variation of the SapEn with scale is entirely different 5

6 fro the white noise, fractional Brownian noise and the /f noise [4]. Moreover, the variation of the SapEn for all chaotic data sets showed a siilar behavior. This establishes the fact that the MSE analysis can be used to detect the deterinis in a tie series. In conclusion we have showed that the Saple Entropy is an iportant statistic for detecting deterinis in a tie series. 6

7 References:. KAPLAN D. T. and GLASS L., Phys. Rev. Lett. 68 (992) WAYLAND R., BROONLEY D., PICKETT D. and PASSARNATE A., Phys. Rev. Lett., 70 (993) SALVINO L. W. and CAWLEY R., Phys. Rev. Lett., 73 (994) COSTA M., GOLDBERGER A. L. and PENG C. K., Phys. Rev. Lett., 89 (2002) RICHMAN J. S. and MOORMAN J. R., A. J. Physiol., 278 (2000) H COSTA M., PENG C. K., GOLDBERGER A. L., and HAUSDORFF J. M. Physica A, 330 (2003) COSTA M., GOLDBERGER A. L. and PENG C. K., Phys. Rev. E, 7 (2005) MANDELBROT B. B., in The Fractal Geoetry of Nature, (San Francisco, Ca: Freean)

8 Figure caption Fig. MSE analysis of the various siulated chaotic data and white noise, fractional Brownian noise, /f noise each with data points. 8

9 Saple Entropy Scale Factor Logistic Map Henon Map Ikeda Map Quadratic Map Rossler Equation Lorenz Equation White Noise Fractional Brownian Noise /f Noise. Fig. 9