Fisher Analysis of D st Time Series

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.9(2010) No.4,pp Fisher Analysis of D st Time Series L. Telesca 1, M. Lovallo 2 1,2 Istituto di Metodologie per l Analisi Ambientale, CNR, C.da S.Loja Tito(PZ), Italy (Received 9 May 2009, accepted 12 August 2009) Abstract: The daily means of D st time series has been investigated by using the Fisher Information Measure, powerful tool to analyze dynamical complexity in nonlinear time series. Our findings point out to the significant detection of a precursor of most intense magnetic storm. Keywords: Magnetic storm; Fisher Information Measure; Shannon Entropy 1 Introduction The complex interaction between the Earth s magnetic field and embedded hot plasma and particular magneto-plasma structures originating at the Sun and propagating to the near-earth space environment causes the onset of magnetic storms (M S). M Ss are responsible of several physical effects observed in the space, like acceleration of charged particles or increase of the electric current intensity, and on the ground, like aurora phenomena or global magnetic disturbance on the Earth s surface [1]. The monitoring of magnetic activity is performed by using the hourly D st index, which is calculated from an average over 4 mid-latitude magnetic observatories and freely available on the website In this paper the complexity of the daily means of D st from 1957 to 2003 is analyzed. During the observation period several MSs occurred. The most intense (D st 240nT ) magnetic storm (hereafter indicated as MS 0 )occurred on May 27, Fig. 1 shows the daily means of D st index and the down triangle indicates the time occurrences of the MS 0. We analyzed the data using the Fisher Information Measure (FIM), which is a powerful tool to investigate complex and nonstationary signals. The FIM was introduced by Fisher in 1925 in the context of statistical estimation [2]. In a seminal paper Frieden has shown FIM to be a versatile tool to describe the evolution laws of physical systems [3]. FIM permits to accurately describe the behavior of dynamic systems, and to characterize the complex signals generated by these systems [4]. This approach has been used by Martin et al. to characterize the dynamics of EEG signals [5]. Martin et al. have shown the informative content of FIM in detecting significant changes in the behavior of nonlinear dynamical systems [6], characterizing thus FIM as an important quantity involved in many aspects of the theoretical and observational description of natural phenomena. The FIM method was also applied in magnetotellurics [7]. 2 The methods Characterizing the complexity of signals issued from nonlinear dynamic systems implies accurate description of their behavior and suited methodologies have to be used in order to investigate their fluctuations. Let us introduce the relevant Fisher-associated quantities. Let f q 2 be a probability density in R N (N 1). Fisher s quantity of information associated to f (or to the probability amplitude q) is defined as the (possibly infinite) non-negative number I or in terms of the amplitudes Corresponding author. address: luciano.telesca@imaa.cnr.it I(f) = dx f 2, (1) R f N I(q) = dx( q q). R N (2) Copyright c World Academic Press, World Academic Union IJNS /366

2 L. Telesca, M. Lovallo: Fisher Analysis of D st Time Series 403 This formula defines a convex, isotropic functional I, which was first used by Fisher [2] for statistical purposes, and plays a fundamental role in information theory. It is clear that from Eq. 2 the integrand, being the scalar product of two vectors, is independent of the reference frame [6]. Let us focus the attention on one-dimensional case. Let us consider a random variable X whose probability density function is denoted as f X (x). Its FIM is defined as I(X) = ( x f x(x)) 2 dx f x (x). (3) Eq. 3 involves the calculation of the probability density function (pdf) f X (x).a rough approximation of the unknown probability density f is given by the histogram. Let us consider the statistical sample by {s i } N i=1, where N is the length of the sample. We consider a finite interval [a, b] such that a min i {s i } and b max i {s i }. Next we divide the interval [a, b] into n nonintersecting subintervals of equal length h = (b a)/n. A histogram is a function f N,n (x), constant on each of the subintervals [x k, x k+1 ), k = 1, 2,...n, defined as follows: f N,n (x) = #{s i [x k, x k+1 ) : x [x k, x k+1 )} (4) nh where # counts the number of data values falling into the specified intervals of the signal. The best convergence to the searched density function is obtained if the number of subintervals n is proportional to the cube root of the number N of observations [8]. A better approximation of the pdf f X (x) may be obtained by means of the kernel density estimator technique introduced in [9, 10]. For any real x the kernel density estimator provides an approximate value of the density in the form ˆf N (x) = 1 N N i=1 1 b N K( x s i b N ). (5) K(u) is the Epanechnikov kernel function [11], which is a continuous non-negative and symmetric function satisfying + K(u)du = 1, (6) whereas the window {b N } N=1,2,3,... is a sequence of positive real numbers such that lim N b N = 0 and lim N Nb N =. In our estimation procedure we used the Bertlett kernel K(u) = { 3 4 (1 u2 ) if u [ 1, 1] 0 if u [ 1, 1] (7) 3 Data analysis and discussion The FIM measures the degree of disorder of a system due to the spread of its f X (x): it is large for ordered states and small for disordered ones [3]. A higher degree of order implies a higher information content in the system, mainly linked with its higher regularity. In this paper, the daily means of D st series was analyzed using the FIM method. In order to identify significant changes in the dynamics of D st index, a time-varying FIM was performed. By using the concept of sliding window [12, 13] the temporal evolution of the local FIM was calculated. In our case we considered a sliding window of 250 days, and we calculated the local FIM in each window by means of the procedure given by Eqs The shift between two successive windows was set to 1 day, in order to smooth the results and evaluate sudden dynamical changes in the variation of FIM with a sufficiently good time resolution. This procedure is able to enhance the existence of transient phenomena. Fig. 2 shows the time variation of the local FIM. The curve is characterized by a quasi-spike-like behavior, which indicates that the system almost sharply changes its status. In order to detect possible anomalous values of FIM, we evaluated a threshold value, above which the FIM can be considered anomalous. To evaluate this threshold, we calculated the distribution of the obtained FIM values (Fig. 3), and identified F IM threshold = , which selects about the 1% of the total number of FIM values; therefore the region F IM indicates the anomalous FIM range. The horizontal dotted line in Fig. 2 indicates the F IM threshold value, above which the FIM values can be considered anomalous. Using IJNS homepage:

3 404 International Journal of NonlinearScience,Vol.9(2010),No.4,pp Figure 1: daily means of D st. The arrow indicates the occurrence of MS 0 occurred on May 27, 1967 Figure 2: Time variation of FIM. The vertical arrow indicates the time occurrences of MS 0 ; the horizontal dotted line indicate the F IM threshold Figure 3: Histogram of the FIM values. F IM characterizes the part of the histogram whose area is the 1% of the total area IJNS for contribution: editor@nonlinearscience.org.uk

4 L. Telesca, M. Lovallo: Fisher Analysis of D st Time Series 405 this criterion we can identify one anomalous FIM pattern just before the onset of MS 0. The magnetic storm forecasting is a very challenging issue due to the severity of technological impacts of space weather. In fact, spacecraft systems and subsystems are susceptible to effects of the space environment including communication, navigation, reconnaissance satellite operational anomalies. Furthermore, HF and VHF wireless communication links through the ionosphere, used for military and civil purposes, are significantly affected by changes in ionospheric properties induced by space weather [1]. Therefore, the complex physics of space storms and their impacts in technology and society give rise to the understanding of space weather dynamics and to developing forecasting methodologies. The low-dimensional behavior of magnetosphere [14, 15] and its description using a relatively small number of variables has stimulated the development of models and forecasting tools. Predictions of D st using nonlinear dynamical models based on the interactions solar wind-magnetosphere were given with good accuracy [16]. And very recently, Balasis et al. [17] applied the non-extensive Tsallis entropy to the 2001 hourly Dst time series, where in particular two intense MSs occurred on 31 March 2001 and 6 November They found that an effective dissimilarity of complexity can be revealed between pre-storm activity and intense MS. In this paper, the FIM is used on a long time series. The performed nonlinear analysis suggests that the D st index is characterized by a significant dynamical change induced by the MSs. Before the onset of a very strong MS, the FIM tends to increase, indicating a tendency of the whole system to behave more orderly, and the incoming of organized dynamics. Similar conclusions have been obtained by Balasis et al. [17], who found two different behaviors: one associated with the intense MSs, featured by a high degree of organization and persistent behavior, and the other associated with normal periods, characterized by a lower degree of organization and anti-persistence behavior. 4 Conclusions We applied a nonlinear approach, based on the FIM analysis, to investigate the complexity in the daily means of D st from 1957 to The FIM acts as a detector of changes in the dynamical behavior of the system generating magnetic data. We applied the Fisher methodology in order to identify the largest anomalous value preceding the strongest magnetic storm. Further studies will be performed in order to identify precursory patterns in FIM and other statistical measures before the occurrence of comparably large stroms. Our results could be of some significance concerning the automatic detection of precursory signatures of very intense MSs. References [1] I. A. Daglis, J. U. Kozyra, Y. Kamide, D. Vassiliadis, A. S. Sharma, M. W. Liemohn, W. D. Gonzales, B. T. Tsurutani, G. Lu: Intense space storms: Critical issues and open disputes. J. Geophys. Res. 108, 1208, doi: /2002ja (2003) [2] R. A. Fisher: Theory of statistical estimation. Proc. Cambridge Philos. Soc. 22(1925): [3] B. R. Frieden: Fisher information, disorder, and the equilibrium distributions of physics. Phys. Rev. A. 41(1990): [4] C. Vignat, J.-F. Bercher: Analysis of signals in the Fisher-Shannon information plane. Phys. Lett. A. 312(2003): [5] M. T. Martin, F. Pennini, A. Plastino: Fisher s information and the analysis of complex signals. Phys. Lett. A. 256(1999): [6] M. T. Martin, J. Perez, A. Plastino: Fisher information and nonlinear dynamics. Physica A. 291(2001): [7] M. Balasco, V. Lapenna, M. Lovallo, G. Romano, A. Siniscalchi and L. Telesca: Fisher Information Measure analysis of Earth s apparent resistivity. Int. J. Nonlinear Sci. 5(2008): [8] S. Mercik, K. Weron, Z. Siwy: Statistical analysis of ionic current fluctuations in membrane channels. Phys. Rev. E. 60(1999): [9] L. Devroye: A Course on Density Estimation. Birkhuser, Boston [10] A. Janicki, A. Weron: Simulation and Chaotic Behavior of -Stable Stochastic Processes. Marcel Dekker, New York [11] V. A. Epanechnikov: Non-parametric estimation of a multivariate probability density Theor. Prob. Appl. 14(1969): [12] L. Gamero, A. Plastino, M. E. Torres: Wavelet analysis and nonlinear dynamics in a nonextensive setting. Physica A. 246(1997): IJNS homepage:

5 406 International Journal of NonlinearScience,Vol.9(2010),No.4,pp [13] M. T. Martin, A. R. Plastino, A. Plastino: Tsallis-like information measures and the analysis of complex signals. Physica A. 275(2000): [14] D. Vassiliadis, A. S. Sharma, T. E. Eastman, K. Papadopoulos: Low-dimensional chaos in magnetospheric activity from AE time series. Geophys. Res. Lett. 17(1990): [15] A. S. Sharma: Assessing the magnetosphere s nonlinear behaviour:its dimension is low, its predictability is high. Rev. Geophys. 35(1995): [16] J. A. Valdivia, A. S. Sharma and K. Papadopoulos: Prediction of magnetic storms by nonlinear models. Geophys. Res. Lett. 23(1996): [17] G. Balasis, I. A. Daglis, C. Papadimitriou, M. Kalimeri, A. Anastasiadis, K. Eftaxias: Dynamical complexity in D st time series using non-extensive Tsallis entropy. Geophys. Res. Lett. 35, L14102, doi: /2008gl (2008) IJNS for contribution: editor@nonlinearscience.org.uk