M. Palus, Testing for nonlinearity using redundancies: Quantitative and qualitative aspects, Physica D 80 (1995) 186-205.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of values for different realisations of the same process, even if at these parameters no valid embedding could be expected. Of course, prediction errors are not the only class of nonlinearity measures that has been optimised for robustness. Notable other examples are coarse grained redundancies [14 16], and, at an even higher level of coarse graining, symbolic methods [17] The very popular method of false nearest neighbours [18] can be easily modified to yield a scalar quantity suitable for nonlinearity testing. The same is true for the concept of unstable periodic orbits (UPOs) 19, 20] 3 ....

M. Palus, Testing for nonlinearity using redundancies: Quantitative and qualitative aspects, Physica D 80, 186 (1995).

....power spectral density, but not higher order ones such as the bispectrum, which vanish. If the surrogate data yield results comparable to the original data set, then the inferred nonlinear invariants are very likely to be meaningless. This test can be combined with information theoretic criteria [Palus, 1995]. It is unavoidable when it comes to interpreting with confidence quantities such as Lyapunov exponents or fractal dimensions. 6.4 Nonlinear prediction The growing interest in space weather prediction has renewed the interest in linear and nonlinear prediction issues. It is now generally agreed ....

M. Palus [1995], Testing for nonlinearity using redundancies: quantitative and qualitative aspects, Physica D, 80, pp. 186--205.

....drawback of this method is that only linear dependencies are examined, which makes its application to non linear systems questionable. Linearized quantities, like the linear redundancy [Prichard Theiler, 1995] are frequently used to find non linear behaviour [Palus et al. 1993; Palus, 1993; Palus, 1995]. Furthermore, they are often applied together 1 One of the conditions in Taken s theorem is that the phase space is a smooth manifold with integer dimension. with other methods, like the method of surrogate data [Prichard Theiler, 1994; Theiler et al. 1992] Shaw [1981] was to our ....

Palus, M. [1995] "Testing for Nonlinearity using redundancies: quantitativ and qualitativ aspects ," Physica D 80, 186-205.

....[51] which are not suitable for estimating the metric entropy, but provide a classification of different systems (system states) equivalent to the classification based on the metric entropy. ffl Qualitative characterization of data (systems) under study, namely applications in nonlinearity tests [49], 48] and in the assessment of the possible existence of low dimensional chaos by searching for a region of linear decrease of n ( see Refs. 47, 52] and the next section. 7. Qualitative Characterization of Experimental Time Series Analyzing experimental data, before their ....

Palus M.: Testing for nonlinearity using redundancies: Quantitative and qualitative aspects. Physica D 80, 1995, 186-205.

....and Glass [151] is particularly suited to quantify deterministic structure in densely sampled data which permit the estimation of local flow vectors. Also the technique of false nearest neighbours advocated by Kennel, Brown, and Abarbanel [126] can be regarded in this way. Pompe [152] and Palus [153, 154] advocate the use of coarse grained redundancies, generalisations of the time delayed mutual information. Prichard and Theiler [64] among others) have pointed out that it can be computationally advantageous to estimate information theoretic quantities like redundancy and mutual information by ....

M. Palus, Testing for nonlinearity using redundancies: Quantitative and qualitative aspects, Physica D 80 (1995) 186.

....from different methods a clear picture about dependence structures in a series can be hardly made. Moreover, a number of methods for evaluating nonlinear aspects of time series can be influenced by numerical artifacts and spurious effects caused by linear properties of data under study (see [20] and references within) To appear in: Neural Network World, a special issue devoted to the 5th International Workshop on Parallel Applications in Statistics and Economics PASE 95: Non linear Data Analysis, Trier Mainz, 8 29 9 2, 1995. y Also: Santa Fe Institute, 1399 Hyde Park ....

....the redundancies of the type (6) 1 , or derived directly using relevant expressions for (multidimensional) Gaussian distributions. The general redundancies R detect all dependences in data under study, while the redundancies L are sensitive only to linear structures. For detailed discussion see [20]. 3 Surrogate Data The surrogate data method, related to the technique of bootstrap [5, 6] has been methodologically introduced in nonlinear dynamics by Theiler et al. 25, 26] as a method for testing nonlinearity. The basic idea in the surrogate data based nonlinearity test is to compute a ....

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Palus M.: Testing for nonlinearity using redundancies: Quantitative and qualitative aspects. Physica D, 80, 1995, 186-205

....the series under study. The FT1 surrogates are obtained by computing the Fourier transform (FT) of the series, keeping unchanged the magnitudes of the Fourier coefficients (the spectrum) but the phases of the Fourier coefficients are randomized and the inverse FT into the time domain is performed [3, 4]. The FT1 surrogates realize the null hypothesis of two linear stochastic processes which asynchronously oscillate with the same frequencies (power spectra) as the original series under study. 4. FT2 surrogates are realizations of a bivariate linear stochastic process which mimics individual ....

....time series. The method employs the technique of uni and multivariate surrogate data and information theoretic functionals called redundancies. The test for nonlinearity based on the redundancy linear redundancy approach, combined with the surrogate data is described in detail in Ref. [4], its multivariate version in Ref. 8] The surrogate data, briefly described above, have been introduced in Ref. 3] and the multivariate version in Ref. 7] More details about the information theoretic functionals can be found, e.g. in Ref. 9] Dealing with the problem of detecting ....

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Palus M: Testing for nonlinearity using redundancies: Quantitative and qualitative aspects. Physica D 80:186-205, 1995.

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M. Palus, Testing for nonlinearity using redundancies: Quantitative and qualitative aspects, Physica D 80 (1995) 186-205.

....process (GPER thereafter) is reduced to estimation of its spectrum. In the case of dynamical systems, Fraser (1989) proposed to estimate KSE from the asymptotic behavior of marginal redundancy, computed from a time series generated by the dynamical system. The 2 marginal redundancy n ( see Palus 1995, 1996a,b,c and references therein) for n = 2 also known as mutual information, measures the amount of information about a system in time t (n Gamma 1) contained in measurements done consecutively in times t, t , t (n Gamma 2) The asymptotic behaviour of n ( however, is ....

.... (1) correlation dimension computed by using the Grassberger Proccacia (1983) algorithm modified according to Dvor ak and Klaschka (1990) and a nonlinearity index computed as the L 2 difference between mutual information of the EEG signal and its isospectral surrogate data (Theiler et al. 1992, Palus 1995). The latter were generated as realizations of a linear stochastic process with the same spectrum as the analyzed segment of the EEG data. For comparison, CER s were also computed from surrogate data. All the quantities were computed in a window of 2048 samples, which was moved over the whole ....

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Palus, M. (1995) "Testing for nonlinearity using redundancies: Quantitative and qualitative aspects." Physica D 80 186-205.

....is frequently interpreted as a detection of a deterministic nonlinear relation (2 or 4) in data under study. This is, however, only one of possible alternatives. Other alternatives will be discussed in Sec. 3, in the following section we will briefly review a test for nonlinearity proposed by in [Palus, 1995]. Then we will present an example of a spurious nonlinearity detection due to a variable variance in a case of air surface pressure data (Sec. 4) Nonlinearity in a series of sunspot numbers is tested in Sec. 5. Section 6 introduces the amplitudefrequency correlation in nonlinear oscillators. ....

....from continuous variables on a continuous probability space, then the redundancies depend on a partition Detecting a nonlinear oscillator 3 chosen to discretize the space. Various strategies have been proposed to define an optimal partition for estimating redundancies of continuous variables (see [Palus, 1995, Weigend and Gershenfeld, 1993] and references therein) Here we use the marginal equiquantization method described in detail in [Palus, 1993, Palus, 1995] Now, let the n variables X 1 , Xn have zero means, unit variances and correlation matrix C. Then, we define the linear redundancy ....

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Palus, M. (1995). Testing for nonlinearity using redundancies: Quantitative and qualitative aspects. Physica D, 80:186--205.

....of surrogate data , advocated by Theiler et al. 31] In testing for nonlinearity, the surrogate data are artificially generated realizations of a linear stochastic process, which mimic linear properties of the studied data, namely the spectrum and autocorrelation function. For details see [31, 26]. Nonlinear structures, if present in the scrutinized data, are eliminated in the surrogates. Thus we can statistically compare the redundancies obtained from the analyzed data and from a set of surrogates to find out, whether the data is significantly different from the null hypothesis of a ....

....in numerical practice linear properties of the surrogates can differ from those of the data under study. The changes in linear properties are reflected in nonlinear measures 2 as well, and thus, a change in linear properties can lead to spurious detection of nonlinearity in linear data [32, 26]. Therefore, we evaluate also the statistic based on the linear redundancy L n ( reflecting specificaly the changes in linear properties. Then, only those significant differences in the nonlinear statistic can reliably count for nonlinearity, which are not detected in the 1 With a limited ....

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M. Palus, Testing for nonlinearity using redundancies: Quantitative and qualitative aspects, Santa Fe Institute Working Paper 93-12-076, submitted to Physica D.

....marginal boxes are defined in such a way that there is approximately the same number of points in each marginal box. Thus a partition is defined by a number Q of the equiquantal marginal boxes. For the embedding dimension n the total number of the partition boxes is Q n . For more details see [15, 17, 19]. 2 The effective series length N is N = N 0 Gamma (n Gamma 1) where N0 is the total series length, n is the embedding dimension and is the time delay used in the estimation of n ( 3 Of course, any digitized experimental time series is discrete, however, a marginal partition ....

.... numerical examples Consider an autoregressive process (ARP) given as y t = c 10 X k=1 a k y t Gammak oee t ; 11) where a k=1; 10 = 0; 0; 0; 0; 0; 19; 2; 2; 2; 2, oe = 0:01 and e t are Gaussian deviates with zero mean and unit variance. For c = 1 this ARP has long coherence time [19], for c 1 the coherence time decreases and the entropy rate increases. In particular, we can generate a number of the ARP s with different c s and thus with different entropy rates. The entropy rates of such ARP s should monotonically decrease with increasing c. Figure 2 presents the ....

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M. Palus, Testing for nonlinearity using redundancies: Quantitative and qualitative aspects, Physica D 80 (1995) 186-205.

....X 1 ; Xn are obtained from continuous variables on a continuous probability space, then the redundancies depend on a partition chosen to discretize the space. Various strategies have been proposed to define an optimal partition for estimating redundancies of continuous variables (see [12, 22] and references therein) Here we use the marginal equiquantization method described in detail in [10, 12] Now, let the n variables X 1 , Xn have zero means, unit variances and correlation matrix C. Then, we define the linear redundancy L(X 1 ; Xn ) of X 1 ; X 2 ; Xn as ....

....depend on a partition chosen to discretize the space. Various strategies have been proposed to define an optimal partition for estimating redundancies of continuous variables (see [12, 22] and references therein) Here we use the marginal equiquantization method described in detail in [10, 12]. Now, let the n variables X 1 , Xn have zero means, unit variances and correlation matrix C. Then, we define the linear redundancy L(X 1 ; Xn ) of X 1 ; X 2 ; Xn as L(X 1 ; Xn ) Gamma 1 2 n X i=1 log(oe i ) 6) where oe i are the eigenvalues of the n Theta ....

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Palus M. (1995) Testing for nonlinearity using redundancies: Quantitative and qualitative aspects. Physica D 80 186-205.

.... as finite dimensions or positive Lyapunov exponents, and reliability of chaos based algorithms in general, have recently come under question, and alternative methods for identifying possible nonlinear determinism in experimental time series have been proposed (see [Weigend Gershenfeld, 1993] [Palus, 1995], and references therein) Employing some of these methods, Palus [1996a] has detected a nonlinear component in EEG recordings of normal healthy volunteers, however, signatures of low dimensional chaos were not found. Similar results have been independently reported by Pritchard et al. 1995] ....

....chaotic, dynamical system, its description in terms of a spectral density is not sufficient. Indeed, realizations of isospectral Gaussian processes are used in the surrogate data based tests in order to discern nonlinear (possibly chaotic) processes from colored noises [Theiler et al. 1992] [Palus, 1995]. On the other hand, there are results indicating that some characteristic properties of nonlinear dynamical systems may be projected into their linear properties , i.e. into spectra, or equivalently, into autocorrelation functions: Sigeti [1995] has demonstrated that there may be a relation ....

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Palus, M. [1995] "Testing for nonlinearity using redundancies: Quantitative and qualitative aspects." Physica D 80 186-205.

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